diff --git a/.obsidian/graph.json b/.obsidian/graph.json index 8eeb2986..6545e108 100644 --- a/.obsidian/graph.json +++ b/.obsidian/graph.json @@ -53,6 +53,6 @@ "repelStrength": 20, "linkStrength": 1, "linkDistance": 120, - "scale": 0.25883865621816216, + "scale": 0.5087618855792573, "close": false } \ No newline at end of file diff --git a/.obsidian/workspace.json b/.obsidian/workspace.json index 64d444b6..e0b76e04 100644 --- a/.obsidian/workspace.json +++ b/.obsidian/workspace.json @@ -4,21 +4,17 @@ "type": "split", "children": [ { - "id": "477f1cf965ca606e", + "id": "3069b19e4433d230", "type": "tabs", "children": [ { - "id": "a31de1b302cdc5cf", + "id": "31299cd80f02d44f", "type": "leaf", "state": { - "type": "markdown", - "state": { - "file": "_templates/question.md", - "mode": "source", - "source": false - }, + "type": "empty", + "state": {}, "icon": "lucide-file", - "title": "question" + "title": "새 탭" } } ] @@ -195,11 +191,12 @@ "obsidian-git:Open Git source control": false } }, - "active": "a31de1b302cdc5cf", + "active": "31299cd80f02d44f", "lastOpenFiles": [ + "wiki/meta/dashboard.md", + "_templates/question.md", "wiki/canvases/main.canvas", "wiki/entities/ABAQUS.md", - "wiki/meta/dashboard.md", "wiki/meta/dashboard.base", "wiki/references/transport-fallback.md", "wiki/references/methodology-modes.md", diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_001.md b/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_001.md new file mode 100644 index 00000000..b2e910d4 --- /dev/null +++ b/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_001.md @@ -0,0 +1,292 @@ + + +this document downloaded from + +# vulcanhammer.net + +Since 1997, your complete online resource for information geotecnical engineering and deep foundations: + +The Wave Equation Page for Piling + +Online books on all aspects of soil mechanics, foundations and marine construction + +Free general engineering and geotechnical software + +And much more... + +# Terms and Conditions of Use: + +All of the information, data and computer software (“information”) presented on this web site is for general information only. While every effort will be made to insure its accuracy, this information should not be used or relied on for any specific application without independent, competent professional examination and verification of its accuracy, suitability and applicability by a licensed professional. Anyone making use of this information does so at his or her own risk and assumes any and all liability resulting from such use. The entire risk as to quality or usability of the information contained within is with the reader. In no event will this web page or webmaster be held liable, nor does this web page or its webmaster provide insurance against liability, for any damages including lost profits, lost savings or any other incidental or consequential damages arising from the use or inability to use the information contained within. + +This site is not an official site of Prentice-Hall, Pile Buck, the University of Tennessee at Chattanooga, or Vulcan Foundation Equipment. All references to sources of software, equipment, parts, service or repairs do not constitute an endorsement. + +Visit our companion site http://www.vulcanhammer.org + + + +# FINITE ELEMENTS IN + +# PLASTICITY: + +# Theory and Practice + +D. R. J. OWEN + +E. HINTON + +Department of Civil Engineering + +University College of Swansea, U.K. + + + +First published 1980 by + +Pineridge Press Limited + +91 West Cross Lane, West Cross, Swansea U.K. + +ISBN 0-906674-05-2 + +Copyright © 1980 by + +Pineridge Press Limited. + +OWEN, D. R. J. + +Finite Elements in Plasticity + +1. Elasticity + +2. Plasticity + +3. Finite Element Method + +I. Title II. HINTON, E. + +620.1'123 TA418 + +ISBN 0-906674-05-2 + +All rights reserved. + +No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior written permission of the publishers. + + + +# Contents + +# PART I + +# 1 Introduction 3 + +1.1 Introductory remarks 3 +1.2 Aims and layout 3 +1.3 Program structure 8 +1.4 References 11 + +# 2 One-dimensional nonlinear problems 13 + +2.1 Introduction 13 +2.2 Basic numerical solution processes for nonlinear problems 13 +2.3 Systems governed by a quasi-harmonic equation 22 +2.4 Nonlinear elastic problems 25 +2.5 Elasto-plastic problems in one dimension 26 +2.6 Problems 29 +2.7 References 31 + +# 3 Structure of computer programs for the solution of nonlinear problems 33 + +3.1 Introduction 33 +3.2 Input data subroutine, DATA 35 +3.3 Subroutine NONAL 40 +3.4 Subroutines for equation assembly and solution 42 +3.5 Output of results 58 +3.6 Subroutine INITIAL 59 +3.7 Load increment subroutine, INCLOD 60 +3.8 The master or controlling segment 61 +3.9 Program for the solution of one-dimensional quasi-harmonic problems by direct iteration 63 +3.10 Program for the solution of one-dimensional quasi-harmonic problems by the Newton–Raphson method 68 + + + +# CONTENTS + +3.11 Program for the solution of nonlinear elastic problems 74 +3.12 Program for the solution of elasto-plastic problems 78 +3.13 Problems 90 +3.14 References 94 + +# 4 Viscoplastic problems in one dimension 95 + +4.1 Introduction 95 +4.2 Basic theory 95 +4.3 Numerical solution process 99 +4.4 Limiting time-step length 102 +4.5 Computational procedure 103 +4.6 Program structure 104 +4.7 Element stiffness subroutine STUNVP 106 +4.8 Subroutine INCVP for the evaluation of end of time-step quantities and equilibrium correction terms 107 +4.9 Convergence monitoring subroutine, CONVP 109 +4.10 Subroutine INCLOD 110 +4.11 The main, master or controlling segment 111 +4.12 Numerical examples 113 +4.13 Problems 117 +4.14 References 119 + +# 5 Elasto-plastic Timoshenko beam analysis 121 + +5.1 Introduction 121 +5.2 The basic assumptions of Timoshenko beam theory 122 +5.3 Finite element idealisation for linear elastic Timoshenko beams 125 +5.4 Elasto-plastic nonlayered Timoshenko beams 129 +5.5 Elasto-plastic layered Timoshenko beams 141 +5.6 Problems 148 +5.7 References 152 + +# PART II + +# 6 Preliminary theory and standard subroutines for two-dimensional elasto-plastic applications 157 + +6.1 Introduction 157 +6.2 Virtual work expression for various solid mechanics applications 162 +6.3 Isoparametric finite element representation 169 + + + +# FINITE ELEMENTS IN PLASTICITY + +6.4 Standard subroutines for linear elastic finite element analysis 174 +6.5 Standard subroutines for elasto-plastic finite element analysis 205 +6.6 Problems 214 +6.7 References 214 + +# 7 Elasto-plastic problems in two dimensions 215 + +7.1 Introduction 215 +7.2 The mathematical theory of plasticity 215 +7.3 Matrix formulation 227 +7.4 Alternative form of the yield criteria for numerical computation 229 +7.5 Basic expressions for two-dimensional problems 232 +7.6 Singular points on the yield surface 234 +7.7 Finite element expressions and program structure 235 +7.8 Additional program subroutines 237 +7.9 Numerical examples 262 +7.10 Problems 265 +7.11 References 268 + +# 8 Elasto-viscoplastic problems in two dimensions 271 + +8.1 Introduction 271 +8.2 Theory of elasto-viscoplastic solids 272 +8.3 Selection of the time-step length 276 +8.4 Computational procedure 278 +8.5 Evaluation of matrix H 279 +8.6 Program structure 281 +8.7 Formulation of the tangential stiffness matrix 283 +8.8 Subroutine STEPVP for the evaluation of end of time step quantities and equilibrium correction terms 289 +8.9 Subroutine FLOWVP 294 +8.10 Subroutine STRESS 295 +8.11 Subroutine ZERO 297 +8.12 Subroutine STEADY for monitoring steady state convergence 297 +8.13 The main, master or controlling segment 299 +8.14 General comparison of implicit and explicit time integration schemes 302 +8.15 The overlay method for improved material response 304 +8.16 Numerical examples 310 +8.17 Problems 315 +8.18 References 317 + + + +# 9 Elasto-plastic Mindlin plate bending analysis 319 + +9.1 Introduction 319 +9.2 Equilibrium equations 321 +9.3 Discretisation 324 +9.4 Solution of nonlinear equations 326 +9.5 Software for the nonlayered approach 331 +9.6 Software for the layered approach 355 +9.7 Examples 370 +9.8 Problems 372 +9.9 References 373 + +# PART III + +# 10 Explicit transient dynamic analysis 377 + +10.1 Introduction 377 +10.2 Dynamic equilibrium equations 378 +10.3 Modelling of nonlinearities 381 +10.4 Explicit time integration scheme 388 +10.5 Critical time step 391 +10.6 Program DYNPAK 392 +10.7 Examples 420 +10.8 Problems 428 +10.9 References 429 + +# 11 Implicit-explicit transient dynamic analysis 431 + +11.1 Introduction 431 +11.2 Implicit time integration 432 +11.3 Implicit-explicit algorithm 434 +11.4 Evaluation of the tangential stiffness matrix 439 +11.5 Program MIXDYN 440 +11.6 Examples 458 +11.7 Problems 462 +11.8 References 462 + +# 12 Alternative formulations and further applications 465 + +12.1 Introduction 465 +12.2 List of subroutines 466 +12.3 Alternative material models 476 +12.4 Further applications 480 +12.5 Equation solving techniques 490 +12.6 Other enhancements in elasto-plastic analysis 493 +12.7 Concluding remarks 495 +12.8. References 496 + + + +# FINITE ELEMENTS IN PLASTICITY + +ix + +Appendix I Instructions for preparing input data for one-dimensional problems 503 + +Appendix II Instructions for preparing input data for plane, axisymmetric and plate bending problems 511 + +Appendix III Instructions for preparing input data for dynamic transient problems 521 + +Appendix IV Sample input data and line printer output for one- and two-dimensional applications 529 + +Author Index 585 + +Subject Index 589 + + + +# Preface + +The purpose of this text is to present and demonstrate the use of finite element based methods for the solution of problems involving plasticity. As well as the conventional quasi-static incremental theory of plasticity, attention is given to the slow transient phenomenon of elasto-viscoplastic behaviour and also to dynamic transient problems. We make no pretence that the text provides a complete treatment of any of these topics but rather we see it as an attempt to present numerical solution techniques, which have been well tried and tested, for selected important areas of application. + +In our earlier books on finite elements we have concentrated on linear applications. Here we attempt the much more daunting task of introducing, in detail, the use of finite elements for solving problems in which plasticity effects are present. To our knowledge it is the first such book. Our main idea is to present the theory and detailed algorithms in the form of modular routines written in FORTRAN which can be linked together to form 13 finite element plasticity programs. + +Writing this book has been in itself, rather like solving a nonlinear finite element problem. We have gone through many iterations and we hope that we have now converged to a reasonable ‘solution’. As in many real engineering situations our convergence criterion has been influenced by a deadline. In our case the deadline was largely self-imposed as we have already been engaged on this project for more than three years. We do not believe our solution to be unique or in any sense optimal. We merely offer it to fill a gap in the existing literature. + +The text is arranged in three main parts. Part I is devoted to one-dimensional problems. These relatively simple applications are possibly the most important in the book; since all the essential features of nonlinear finite element analysis are immediately recognisable without the distractions and complications that are present in general continuum problems. Part II deals with the two-dimensional applications of plane stress/strain and axisymmetric continua and plate bending problems. Finally in Part III we present some dynamic transient applications and briefly describe some further developments. + +All of the programs presented in this text have been specially written by the authors. In the development of the subroutines for the solution algorithms described, a conflict inevitably arose between computational efficiency + + + +and clarity of coding. Whatever sacrifices have been made have been biased towards satisfying the latter condition. However, we believe that the codes presented are both reasonably efficient and flexible and have potential usage in commercial as well as teaching and research environments. A total of 132 subroutines are presented which amount to more than 8,000 statements. The 13 assembled programs comprise approximately 20,000 statements. To aid readers wishing to implement the programs a magnetic tape of the computer codes together with the test input data listed in Appendix IV is available from the publishers. Although every attempt has been made to verify the programs, no responsibility can be accepted for their performance in practice. + +A further feature of the book is that each chapter contains several exercises for further study. + +We are indebted to many people for their direct or indirect assistance in the preparation of this text. This preface would not be complete without an acknowledgment of this debt and a record of our gratitude to the following: To Professor O. C. Zienkiewicz for his pioneering work and stimulating influence. To Professor G. C. Nayak whose work on numerical analysis of plasticity problems has significantly influenced the present text. To Dr. I. C. Cormeau whose thesis on viscoplasticity has been an invaluable source of information. To Professor K. J. Bathe for permission to use the profile equation solver included in Chapter 11. To N. Bicanic, D. K. Paul, H. H. Abdel Rahman and M. M. Huq for their generous assistance in the preparation of several chapters. To our colleagues and former research workers in the Department of Civil Engineering, University College of Swansea for helpful discussions and suggestions. To E. S. Caldis for his care in preparing annotated computer listings and, finally, to Mrs. M. J. Davies for her skill and patience in typing the manuscript. + +D. R. J. OWEN +E. HINTON + +Swansea, May 1980 diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_002.md b/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_002.md new file mode 100644 index 00000000..eed0da91 --- /dev/null +++ b/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_002.md @@ -0,0 +1,285 @@ + + +# Part I + + + + + +# Chapter 1 Introduction + +# 1.1 Introductory remarks + +The finite element method is now firmly accepted as a most powerful general technique for the numerical solution of a variety of problems encountered in engineering. Applications range from the stress analysis of solids to the solution of acoustical, neutron physics and fluid dynamics problems. Indeed the finite element process is now established as a general numerical method for the solution of partial differential equation systems, subject to known boundary and/or initial conditions. + +For linear analysis, at least, the technique is widely employed as a design tool. Similar acceptance for nonlinear situations is dependent on two major factors. Firstly, in view of the increased numerical operations associated with nonlinear problems, considerable computing power is required. Developments in the last decade or so have ensured that high-speed digital computers which meet this need are now available and present indications are that reductions in unit computing costs will continue. Secondly, before the finite element method can be used in design, the accuracy of any proposed solution technique must be proven. The development of improved element characteristics and more efficient nonlinear solution algorithms and the experience gained in their application to engineering problems have ensured that nonlinear finite element analyses can now be performed with some confidence. Hence barriers to the common use of nonlinear finite element techniques are being rapidly removed and the process is already economically acceptable for selected industrial applications. + +# 1.2 Aims and layout + +The object of this book is to describe in detail the application of the finite element method to the solution of materially nonlinear engineering analysis problems. Unlike other texts on linear and nonlinear finite element analysis $^{(1-4)}$ which have dealt predominantly with theoretical aspects, this book is intended to be more practical and therefore focuses attention on the computer implementation of nonlinear finite element schemes. + +Nonlinearities arise in engineering situations from several sources. For example a nonlinear material response can result from elasto-plastic material behaviour or from hyperelastic effects of some form. Additionally nonlinear + + + +characteristics can be associated with temporal effects such as viscoplastic behaviour or dynamic transient phenomena. Each of these nonlinearities may occur in a variety of structural types such as two- or three-dimensional solids, frames, plates or shells. Therefore it becomes clear that a textbook dealing with nonlinear finite element programming must at least be restricted to selected topics. For this reason three classes of problems will be examined in depth in the three parts of this text. + +Part I: One-dimensional materially nonlinear problems. All the essential features of a nonlinear finite element solution can be described in relation to one-dimensional models. The applications considered are: + +● Nonlinear quasi-harmonic problems +● Nonlinear elastic situations +- Elasto-plastic behaviour of axial bar systems +● Time dependent elasto-viscoplastic analysis of bar systems +- Elasto-plastic beam bending + +Part II: Two-dimensional materially nonlinear problems. In this part the ideas developed in Part I are extended to continuum problems. The following applications are presented: + +- Elasto-plastic analysis of plane stress, plane strain and axisymmetric solids +● Time dependent elasto-viscoplastic analysis of plane stress, plane strain and axisymmetric solids +- Elasto-plastic plate bending problems + +Part III: Nonlinear transient dynamic problems. In this time-dependent class of problems inertia effects are included in the analysis. In this part, the following topics are considered: + +- Elasto-plastic and geometrically nonlinear material behaviour +● Explicit and implicit time integration schemes +● Combined explicit/implicit algorithms + +It should be pointed out that several different programming options are open for solution of the above problems and the methods presented in this text are the ones which are physically the most clear and which experience indicates give reliable results for a wide range of applications. An important feature of this text is the step-by-step development of thirteen finite element programs to deal with the above problems. + +For the one-dimensional applications considered in Part I, only a 2-node element with linear displacement variation between nodes is considered. This allows the basic steps of a nonlinear finite element analysis to be presented without unnecessary distractions. In Parts II and III of the text, where two-dimensional continuum and plate bending problems are considered, isoparametric elements are exclusively employed. In particular, a + + + +4-node linear element and 8- and 9-node quadratic versions are used. These elements are illustrated in Fig. 1.1 and are extremely versatile, good performers which have been well tried and tested in both linear and nonlinear situations. A typical elasto-plastic application using 8-node isoparametric elements is shown in Fig. 1.2 where the incremental loading of a notched beam is illustrated. The progressive development of plastic zones with increasing load levels are compared for a Tresca and Von Mises yield criterion. + +![](images/page-015_53ac111761c9ad18723000ebb385b96414593b7d253b0ece3041c9ee3af55422.jpg) + +
+text_image + +y +x +(a) +
+ +![](images/page-015_82648261cf9624240e89d1e01be65045c13538e5949d4589fbf6efe3f34e2588.jpg) + +
+text_image + +General node +(b) +
+ +![](images/page-015_bb83187e781424ff0af7c9ba80c7bf9f8c86b520b442bd48341615f78678fb78.jpg) + +
+text_image + +Central node +(c) +
+ +Fig. 1.1 The two-dimensional isoparametric elements employed in the text: (a) Linear 4-node; (b) Serendipity 8-node; (c) Lagrangian 9-node. + +The layout of the book will now be briefly described. The remainder of Chapter 1 discusses the basic notation and style adopted in program presentation. + +Chapter 2 discusses the general nonlinear problem and some solution techniques are outlined. For the one-dimensional applications to be considered, basic theoretical expressions are developed in a form suitable for numerical solution. + +In Chapter 3, the solution techniques presented in Chapter 2 are programmed in FORTRAN and numerical examples are solved for each separate application. + +Chapter 4 is devoted to one-dimensional elasto-viscoplastic problems. The basic theory for this time-dependent phenomenon is first presented. The process is then coded and the program used to solve some numerical examples. + +In Chapter 5 elasto-plastic beam bending is considered. This topic forms a bridge between uniaxial and continuum applications since now more than one degree of freedom exists at each nodal point. Some measure of continuum behaviour is also introduced since a layered approach is used to trace the development of plasticity through the cross-section of the beam. + + + +![](images/page-016_cc7aad6598747466ef78508754fdf8451309c57a22e71f5b67adbcd31478c8a3.jpg) + +
+other + +| Dimension | Value | +| ----------------- | --------- | +| Height (in mm) | 0.5 IN | +| Width (in mm) | 12.7 mm | +| Height (in mm) | 0.75 IN | +| Width (in mm) | 19.05 mm | +| Height (in mm) | 6.0PY | +| Width (in mm) | 5.5PY | +| Height (in mm) | 6.0PY | +| Width (in mm) | 0.010 IN | +| Height (in mm) | 0.1667 IN | +| Width (in mm) | 4.2 mm | +| Height (in mm) | 0.3333 IN | +P (Top) | 22.2° | +
+ +NOTCHED BEND SPECIMEN + +$\mathbf{P}_{\gamma}$ -Initial yield load for Von + +Mises material = + +536 lb (2.39 kN) + +No strain hardening + +Elastic modulus, $E = 3 \times 10^{7}$ lb/in $^{2}$ + +$(2\times 10^{5}\mathrm{N / mm^{2}})$ + +Poisson's ratio, $\nu = 0.28$ + +Yield stress, $\sigma_{Y} = 6 \times 10^{4} \mathrm{~lb/in}^{2} = \frac{\mathrm{E}}{500}$ + +ZONES OF + +PLASTIC YIELD + +AT VARIOUS + +LOAD VALUES + +![](images/page-016_f27db06b79380f55875db51acee957ac608142384a4dee7cdc5f285783a0b53c.jpg) + +
+contour + +| Label | Value | +|-------------|-------| +| 1.2Pγ | 3.0 | +| 5.5 | 4.5 | +| 6.0 | 5.5 | +| 6.0 | 6.0 | +
+ +Fig. 1.2 Elasto-plastic analysis of a notched beam under bending showing plastic zone distributions for both a Von Mises and a Tresca yield criterion. + + + +Chapter 6 forms an introduction to two-dimensional continuum problems. The basic theory for two-dimensional isoparametric elements is presented and some standard subroutines required for applications described in later chapters are listed. These include routines which perform some standard linear elastic operations, such as nodal load generation, equation solution, etc., as well as nonlinear subroutines common to more than one application. + +Two-dimensional elasto-plastic problems are considered in Chapter 7. Basic theoretical expressions for a general continuum are first reviewed, and manipulated into forms convenient for numerical analysis. Particular expressions for plane stress/strain and axisymmetric situations are then developed and coded. Four different yield criteria are employed. The Tresca and Von Mises laws which closely approximate metal plasticity behaviour are considered and the Mohr–Coulomb and Drucker–Prager criteria, which are applicable to concrete, rocks and soil are presented. + +Chapter 8 is concerned with the transient phenomenon of elastoviscoplasticity where again the situations of plane stress/strain and axial symmetry are considered. Both explicit and implicit time integration schemes are presented and the four yield criteria considered in Chapter 7 are employed. The FORTRAN program developed is illustrated by application to some numerical examples. + +Elasto-plastic plate bending problems are discussed in Chapter 9. The basic theoretical expressions are presented in a form suitable for numerical analysis with both a layered and nonlayered approach to plastification through the plate thickness being considered. Treatment in this chapter is limited to the Tresca and Von Mises yield conditions. + +Chapters 10 and 11 deal with the transient dynamic analysis of two-dimensional continua. In this application inertia effects are included in the computation and problems such as blast loading and seismic phenomena are considered. Nonlinear effects due to both elasto-plastic material behaviour and gross geometric deformations are included. Both explicit and implicit techniques are employed for the time integration of the equations of motion as well as a combined implicit/explicit algorithm. The computer codes developed are applied to the solution of some practical problems. + +Finally in Chapter 12 further aspects of nonlinear material behaviour are discussed. Alternative solution techniques and material models are referred to and some additional fields of application indicated. + +Three appendices are included which contain user instructions for the computer programs described throughout the text. Appendices I and II provide user instructions for one-dimensional and two-dimensional continuum problems respectively. A user's guide for transient dynamic problems is provided in Appendix III. Finally in Appendix IV sample input data and lineprinter output are provided for both one- and two-dimensional applications. + + + +# 1.3 Program structure + +# 1.3.1 Introduction + +This section describes the main features of the computer programs to be developed later in the book. A modular approach is adopted, in that separate subroutines are employed to perform the various operations required in a nonlinear finite element analysis. Generally each program consists of 9 modules, each with a distinct operational function. Each module in turn is composed of one or more subroutines relevant only to its own needs and, in some cases, of subroutines which are common to several modules. Control of the modules is held by the main or master segment. + +The modules, shown schematically in Fig. 1.3, are described in relation to their general functions as follows: + +1. Initialisation or zeroing module—this is the first module entered and its function is to initialise to zero various vectors and matrices at the beginning of the solution process. +2. Data input and checking module—this is the second module entered. It handles input data defining the geometry, boundary conditions and material properties. This data is checked using diagnostic routines and if errors occur they are flagged and the remainder of the input data is printed out before the program is terminated. For isoparametric elements, Gaussian integration constants and mid-side nodal coordinates for straight-sided elements are also evaluated in this section. Once used this module is not needed again. +3. Loading module—this module organises the calculation of nodal forces due to the various forms of loading for two-dimensional application. These include pressure, gravity and concentrated loadings. +4. Load incrementing module—Any materially nonlinear finite element solution must proceed on an incremental basis. Therefore the function of this section is to control the incrementing of the applied loads evaluated by the loading module. It also ensures that any specified displacement values are also incrementally applied. +5. Stiffness module—this is the next module entered and organises the evaluation of the stiffness matrix for each element. The stiffness matrices are stored on disc and ordered in the sequence required for equation assembly and reduction. +6. Solution module—the general purpose of this routine is to assemble, reduce and solve the governing set of simultaneous equations to give the nodal displacements and force reactions at restrained nodal points. +7. Residual force module—the function of this module is to calculate the residual or 'out of balance' nodal forces at each stage of the analysis. +8. Convergence module—in this module the convergence of the nonlinear solution is checked against criteria given in later chapters. + + + +9. Output module—this module organises the output of the requested quantities. + +![](images/page-019_baf612902b02d3f830d3de370553915a03c9f703d975019b9bce9870d5f0fc63.jpg) + +
+flowchart + +```mermaid +graph TD + A["Main or master segment"] --> B["Initialising or zeroing module"] + A --> C["Data input and checking module"] + A --> D["Loading module"] + A --> E["Load incrementing module"] + A --> F["Stiffness module"] + A --> G["Solution module"] + A --> H["Residual force module"] + A --> I["Convergence module"] + A --> J["Output module"] + style A fill:#f9f,stroke:#333 + style B fill:#ccf,stroke:#333 + style C fill:#ccf,stroke:#333 + style D fill:#ccf,stroke:#333 + style E fill:#ccf,stroke:#333 + style F fill:#ccf,stroke:#333 + style G fill:#ccf,stroke:#333 + style H fill:#ccf,stroke:#333 + style I fill:#ccf,stroke:#333 + style J fill:#ccf,stroke:#333 + style F fill:#ccf,stroke:#333 + style G fill:#ccf,stroke:#333 + style H fill:#ccf,stroke:#333 + style I fill:#ccf,stroke:#333 + style J fill:#ccf,stroke:#333 + style A fill:#fff,stroke:#333 + style B fill:#fff,stroke:#333 + style C fill:#fff,stroke:#333 + style D fill:#fff,stroke:#333 + style E fill:#fff,stroke:#333 + style F fill:#fff,stroke:#333 + style G fill:#fff,stroke:#333 + style H fill:#fff,stroke:#333 + style I fill:#fff,stroke:#333 + style J fill:#fff,stroke:#333 + style A fill:#fff,stroke:#333 + style B fill:#fff,stroke:#333 + style C fill:#fff,stroke:#333 + style D fill:#fff,stroke:#333 + style E fill:#fff,stroke:#333 + style F fill:#fff,stroke:#333 + style G fill:#fff,stroke:#333 + note1["Increment loop"] --> A + note2["Iteration loop"] --> A +``` +
+ +Fig. 1.3 Program modules for nonlinear solution codes. + + + +The main purpose of the main or master segment is to call the above modules and to control the load increments and iteration procedure according to the solution algorithm being employed and the convergence rate of the solution process. + +# 1.3.2 Programming notation + +In the programs presented in this text an attempt has been made to name variables in a logical manner. By choosing descriptive names, the use of many of the variables becomes self-apparent, thus assisting the reader in the task of program assimilation. All variable names are chosen to be 5 characters in length; this occasionally causes a little difficulty in abbreviation but has an advantage with regard to neatness of program presentation. For example, the following names will be employed. + +NMATS The Number of different MATerialS +PROPS ( ) The array of material PROPERTieS +NEVAB The Number of Element VAriaBles +NNODE The Number of NODEs per Element +NDOFN The Number of Degrees Of Freedom per Node + +Furthermore a ‘common root’ principle will be adopted; where a single basic variable name is employed with different prefixes depending on its usage in the program. In particular: + +i) Prefix I, J or L will be used to indicate a DO loop variable +ii) Prefix K will indicate a counter +iii) Prefix M will indicate a maximum value +iv) Prefix N will indicate a given number + +For example IPOIN, NPOIN, MPOIN will indicate respectively a particular nodal point, the number of nodal points in the problem and the maximum permissible number of nodal points in the program. + +Similarly, any DO loop will be of the general form + +KEVAB=0 +DO 1 INODE=1, NNODE +DO 1 IDOFN=1, NDOFN +1 KEVAB=KEVAB+1 + +which indicates that the outer and inner DO loop indices range respectively over the number of nodes per element and the number of degrees of freedom per node. The prefix K is employed in KEVAB to indicate a counter over the number of element variables, NEVAB. + +All programming is undertaken in standard FORTRAN IV. A listing is presented for all subroutines described in this text and detailed notes on each group of statements are provided. Comment cards have also been used to assist in the understanding of the programs. diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_003.md b/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_003.md new file mode 100644 index 00000000..a772e8d5 --- /dev/null +++ b/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_003.md @@ -0,0 +1,342 @@ + + +# 1.4 References + +1. ZIENKIEWICZ, O. C. The Finite Element Method, McGraw-Hill, 1977. +2. ODEN, J. T., Finite Elements of Nonlinear Continua, McGraw-Hill, 1972. +3. DESAI, C. S. and ABEL, J. F., An Introduction to the Finite Element Method, Van Nostrand Reinhold, New York, 1972. +4. GALLAGHER, R. H., Finite Element Analysis—Fundamentals, Prentice Hall, 1975. +5. HINTON, E. and OWEN, D. R. J., Finite Element Programming, Academic Press, 1977. +6. HINTON, E. and OWEN, D. R. J., An Introduction to Finite Element Computations, Pineridge Press, Swansea, U.K., 1979. + + + + + +# Chapter 2 One-dimensional nonlinear problems + +# 2.1 Introduction + +Several classes of nonlinear problems of interest in many branches of science and engineering can be reduced to the solution of a system of simultaneous equations in which the equation coefficients are dependent on some function of the prime variables. $^{(1)}$ In this chapter some basic techniques for the numerical solution of such problems are examined. In order to introduce the essential details of the solution processes as simply as possible, the applications will be restricted to one-dimensional situations. In particular, elasto-plasticity, nonlinear elasticity problems and systems governed by a nonlinear quasi-harmonic equation will be considered. In each case a computer program will be developed and its use illustrated by application to simple problems. The aim of this chapter is to prepare the reader for the more comprehensive two-dimensional treatment of these topics which will be undertaken in Chapters 6–9. Indeed, all the essential features of nonlinear finite element analysis detailed in these later chapters will be recognisable from the simple treatment considered here. It should be emphasised that the subroutines developed in this chapter will not be used in the main finite element programs discussed in Parts II and III. + +# 2.2 Basic numerical solution processes for nonlinear problems + +The use of finite element discretisation in a large class of nonlinear problems results in a system of simultaneous equations of the form + +$$ +\boldsymbol {H} \boldsymbol {\varphi} + \boldsymbol {f} = 0, \tag {2.1} +$$ + +in which $\varphi$ is the vector of the basic unknowns, f is the vector of applied ‘loads’ and H is the assembled ‘stiffness’ matrix. For structural applications, the terms ‘load’ and ‘stiffness’ are directly applicable, but for other situations the interpretation of these quantities varies according to the physical problem under consideration. + +If the coefficients of the matrix H depend on the unknowns $\varphi$ or their derivatives, the problem clearly becomes nonlinear. In this case, direct solution of equation system (2.1) is generally impossible and an iterative scheme must be adopted. Many options remain open for the iterative + + + +sequence to be employed. Some of the most generally applicable methods available will now be outlined. + +# 2.2.1 Method of direct iteration (or successive approximations) + +In this approach $^{(2)}$ successive solutions are performed, in each of which the previous solution for the unknowns $\varphi$ is used to predict the current values of the coefficient matrix $H(\varphi)$ . Rewriting (2.1) as + +$$ +\varphi = - [ H (\varphi) ] ^ {- 1} f, \tag {2.2} +$$ + +then the iterative process yields the $(r + 1)^{\mathrm{th}}$ approximation to be + +$$ +\varphi^ {r + 1} = - [ H (\varphi^ {r}) ] ^ {- 1} f. \tag {2.3} +$$ + +If the process is convergent then in the limit as $r$ tends to infinity $\varphi^r$ tends to the true solution. + +It is seen from (2.3) that it is necessary to recalculate the 'stiffness' matrix $H$ for each iteration. To commence the process, an initial guess for the unknown $\varphi$ is required in order to calculate $H$ . Generally a value of $\varphi^0$ based on the solution for an average material property throughout the region is found to be satisfactory. If the nonlinearity of the material properties is very marked at certain values of $\varphi$ , an approximate prescription of the field variable at all nodes may be necessary. + +For practical purposes, the iterative process is deemed to have converged when some measure (usually a norm of the nodal unknowns) of the change in the unknown $\varphi$ between successive iterations has become tolerably small. The process is illustrated diagrammatically for a single variable in Figs 2.1 and 2.2, in which case the matrix $H$ and vector $\varphi$ reduce to the scalar equivalents $H$ and $\phi$ . The assumed dependence of $H$ on $\phi$ is a basic problem function which must be prescribed before solution can commence. This material property is included in Figs 2.1 and 2.2 and, for convenience, the relationship between $H(\phi). \phi$ and $\phi$ is prescribed rather than the $H(\phi) - \phi$ dependence. Figure 2.1 shows the convergence paths for initial trial values, $\phi^0$ , which are below and above the true solution, $\phi_T$ , and for a convex $H - \phi$ relation. From the initial trial value, $\phi^0$ , the corresponding value of $H$ is immediately given from the prescribed $H(\phi). \phi - \phi$ relationship, to be $H^0$ . Equation (2.3) is then solved to give $\phi^1$ . The value of $H$ corresponding to $\phi^1$ is then determined from the $H(\phi). \phi - \phi$ relationship and (2.3) then resolved to obtain $\phi^2$ . This cycling process is continued until $\phi^{n-1}$ and $\phi^n$ are deemed to be sufficiently close, indicating that convergence has occurred. The quantity $H^r$ is represented by the slope of the secant to the $H - \phi$ curve and decreases with increasing values of $\phi$ . Both the high and low initial trial solutions produce monotonic convergence paths. Figure 2.2 shows the unsuitability of the method for problems with a concave $H - \phi$ relationship. Both low and high initial trial solutions produce convergence paths which oscillate around the true solution. Although the solution converges for the + + + +![](images/page-025_dbee49abd4c05426368f38041d6147a3e2b726f5e9d7992f072c1e89d03bbb0f.jpg) + +
+line + +| Basic variable, φ | H(φ)φ (Slope H⁰) | H(φ)φ (Slope H¹) | H(φ)φ (Slope H²) | +| ----------------- | ---------------- | ---------------- | ---------------- | +| φ⁰ | ~0.0 | ~0.0 | ~0.0 | +| φ¹ | ~0.5 | ~0.3 | ~0.1 | +| φ² | ~1.0 | ~0.8 | ~0.3 | +| φ³ | ~1.5 | ~1.2 | ~0.6 | +
+ +![](images/page-025_0312473a92fe53299c399ee0c805683afb7f56dd00ef8e07844d5dd004f0e83c.jpg) + +
+line + +| x | φ | +|---|---| +| 0 | 0 | +| 1 | 0.5 | +| 2 | 0.75 | +| 3 | 0.9 | +| 4 | 0.95 | +| 5 | 0.98 | +
+ +(a) Low initial solution + +![](images/page-025_d7468ef1bbad5943cd1d9027e0f7fb2ad8ad2a10dd140d407c312ad886f984b7.jpg) + +
+text_image + +H(φ)φ +f +H^1 +H^2 +Slope +H^0 +φ^3 φ^2 φ^1 φ^0 +
+ +![](images/page-025_54ea7ca4dbe9144ee8dd4251952687ad6c8bfb5f9d5f13829ca596b3d97066b5.jpg) + +
+line + +| x | φ | +|---|---| +| 0 | 0 | +| 1 | φ | +| 2 | φ | +| 3 | φ | +| 4 | φ | +| 5 | φ | +
+ +(b) High initial solution +Fig. 2.1 Direct iteration method for a single variable problem—convex $H-\phi$ relation. + +single variable case, in multi-degree of freedom problems the coupling of stiffness terms is likely to lead to instability of the iterative process. A disadvantage of the direct iteration method is that convergence of the solution scheme is not guaranteed and cannot be predicted at the initial solution stage. + +# 2.2.2 The Newton-Raphson method + +During any step of an iterative process of solution, (2.1) will not be satisfied unless convergence has occurred. A system of residual forces can be assumed + + + +![](images/page-026_24da6b2750dcc375e900d65efed0c23da0cb7df19e14731a9261a1b77aef6a24.jpg) + +
+line + +| φ | H(φ)φ (H¹) | H(φ)φ (H²) | Slope H⁰ | +|-------|------------|------------|----------| +| 0 | 0 | 0 | 0 | +| φ⁰ | ~0.5 | ~0.2 | ~0.1 | +| φ² | ~1.0 | ~0.5 | ~0.2 | +| φ³ | ~1.5 | ~0.8 | ~0.3 | +| φ¹ | ~2.0 | ~1.0 | ~0.4 | +
+ +![](images/page-026_7ead789637eef39e460fdff244549655e1b5843604fc2a4672e05e0b51b1b124.jpg) + +
+line + +| x | φ | +|---|-------| +| 0 | 0 | +| 1 | 1 | +| 2 | 0.5 | +| 3 | 0.75 | +| 4 | 0.5 | +| 5 | 0.75 | +
+ +(a) Low initial solution + +![](images/page-026_3088db828b5f8b6a5c7ff01743aaca7a35af04268bef0916d1e82e4b3be78222.jpg) + +
+line + +| φ | H(φ)φ (H²) | H(φ)φ (H¹) | +|-------|------------|------------| +| 0 | 0 | 0 | +| φ¹ | H² | H¹ | +| φ³ | H² | H¹ | +| φ² | H² | H¹ | +| φ⁰ | H² | H¹ | +
+ +![](images/page-026_56398f8079052d2a958a8b6f27354cd95a0af8a0feab07cf9b2d9bb4d90eba53.jpg) + +
+line + +| x | φ | +|---|-------| +| 0 | φ | +| 1 | φ_T | +| 2 | φ_T | +| 3 | φ_T | +| 4 | φ_T | +| 5 | φ_T | +
+ +(b) High initial solution +Fig. 2.2 Direct iteration method for a single variable problem—concave $H-\phi$ relation. + +to exist, so that + +$$ +\psi = H \varphi + f \neq 0. \tag {2.4} +$$ + +These residual forces $\psi$ can be interpreted as a measure of the departure of (2.1) from equilibrium. Since H is a function of $\varphi$ and possibly its derivatives, then at any stage of the process, $\psi = \psi(\varphi)$ . + + + +If the true solution to the problem exists at $\varphi^r + \Delta \varphi^r$ then the Newton-Raphson approximation $^{(2)}$ for the general term of the residual force vector, $\psi^r$ corresponding to solution at $\varphi^r$ is + +$$ +\psi_ {i} ^ {r} = - \sum_ {j = 1} ^ {N} \Delta \phi_ {j} ^ {r} \left(\frac {\partial \psi_ {i}}{\partial \phi_ {j}}\right) ^ {r}, \tag {2.5} +$$ + +in which N is the total number of variables in the system and the superscript r denotes the $r^{th}$ approximation to the true solution. Substituting for $\psi_{i}$ from (2.4), the complete expression for all the residual components can be written in matrix form as + +$$ +\psi (\varphi^ {r}) = - J (\varphi^ {r}) \Delta \varphi^ {r}. \tag {2.6} +$$ + +in which a typical term of the Jacobian matrix $J$ is + +$$ +J _ {i j} = \left(\frac {\partial \psi_ {i}}{\partial \phi_ {j}}\right) ^ {r} = h _ {i j} ^ {r} + \sum_ {k = 1} ^ {m} \left(\frac {\partial h _ {i k}}{\partial \phi_ {j}}\right) ^ {r} \phi_ {k} ^ {r}, \tag {2.7} +$$ + +where $h_{ij}$ is the general term of matrix H. The last term in (2.7) gives rise to nonsymmetric terms in the Jacobian matrix. If these nonsymmetric terms are neglected in order to maintain symmetry, then substitution of (2.7) in (2.6) results in + +$$ +\boldsymbol {H} \left(\varphi^ {r}\right). \Delta \varphi^ {r} = - \psi \left(\varphi^ {r}\right). \tag {2.8} +$$ + +Or since + +$$ +\Delta \varphi^ {r} = \varphi^ {r + 1} - \varphi^ {r}, \tag {2.9} +$$ + +equation (2.8) reduces, on use of (2.4), to + +$$ +\boldsymbol {H} \left(\boldsymbol {\varphi} ^ {r}\right). \boldsymbol {\varphi} ^ {r + 1} + \boldsymbol {f} = 0. \tag {2.10} +$$ + +This equation is identical to equation (2.3), Section 2.2.1, which governs the method of direct iteration. Therefore in order to achieve the better convergence rate associated with the Newton–Raphson process it is essential that the unsymmetric terms in J be retained. + +The explicit form of the nonlinear terms in (2.7) will clearly depend on the way in which the stiffness matrix coefficients, $h_{ij}$ , depend on the unknowns, $\varphi$ . The terms of the Jacobian matrix, given in (2.7), can be assembled to give the general expression + +$$ +\boldsymbol {J} (\varphi) = \boldsymbol {H} (\varphi) + \boldsymbol {H} ^ {\prime} (\varphi), \tag {2.11} +$$ + +where the last term contains the unsymmetric terms only. The Newton-Raphson process can be finally written, using (2.6) and (2.11), in the form + +$$ +\Delta \varphi^ {r} = - [ J (\varphi^ {r}) ] ^ {- 1}. \psi (\varphi^ {r}) = - [ H (\varphi^ {r}) + H ^ {\prime} (\varphi^ {r}) ] ^ {- 1} \psi (\varphi^ {r}). \tag {2.12} +$$ + + + +This allows the correction to the vector of unknowns $\varphi$ to be obtained from the residual force vector $\psi$ for any iteration. Again an iterative approach must be followed, with the vector of unknowns $\varphi$ being corrected at each stage according to (2.12) until convergence of the process is deemed to have occurred. The technique is illustrated schematically in Figs 2.3 and 2.4 for + +![](images/page-028_edaa428362dd011b9e3f3e46f3cb84c50115eeddb676d87217b8095d8122eff9.jpg) + +
+line + +| φ | H(φ)φ (Slope J(φ⁰)) | H(φ)φ (J(φ¹)) | H(φ)φ (ψ⁰) | H(φ)φ (Δφ⁰) | H(φ)φ (Δφ¹) | +|-------|----------------------|---------------|------------|-------------|-------------| +| φ⁰ | Low | Low | Low | Low | Low | +| φ¹ | High | High | High | High | High | +| φ² | High | High | High | High | High | +
+ +![](images/page-028_8ae5c1144c8a6ace36ddad4766052730de209baa600743c9bae389170786a1a6.jpg) + +
+line + +| φ_T | φ | +|-----|------| +| 0 | 0.0 | +| 1 | 0.2 | +| 2 | 0.3 | +| 3 | 0.4 | +| 4 | 0.5 | +| 5 | 0.5 | +
+ +(a) Low initial solution + +![](images/page-028_3baca9f79ae33c60c648d45da1351c914cf6c42c066af5e9d619c7da022fd2ff.jpg) + +
+line + +| f | H(φ)φ (Slope J(φ⁰)) | Δφ⁰ (Slope J(φ⁰)) | ψ¹ (Slope J(φ⁰)) | Δφ¹ (Slope J(φ⁰)) | +|---------|----------------------|-------------------|------------------|-------------------| +| 0 | 0 | 0 | 0 | 0 | +| φ¹ | ~0.5 | ~0.5 | 0.5 | 0.5 | +| φ² | ~1.0 | ~1.0 | 1.0 | 1.0 | +| φ⁰ | ~1.5 | ~1.5 | 1.5 | 1.5 | +
+ +![](images/page-028_444bd2f939ff8a254a19317905a100316aa7b2c55de3263b5d2d2a69c6a03235.jpg) + +
+line + +| x | φ | φ_T | +|---|-------|------| +| 0 | 1.0 | 0.0 | +| 1 | 0.2 | 0.0 | +| 2 | 0.3 | 0.0 | +| 3 | 0.4 | 0.0 | +| 4 | 0.5 | 0.0 | +| 5 | 0.6 | 0.0 | +
+ +(b) High initial solution +Fig. 2.3 The Newton-Raphson method for a single variable problem—convex $H-\phi$ relation. + + + +![](images/page-029_d203c40f894967121e8e6cf6b6cc2228a4e2f3c4b54a74a3114746add2bab68c.jpg) +Fig. 2.4 The Newton–Raphson method for a single variable problem—concave $H-\phi$ relation. + +a single variable situation. Solution to the nonlinear problem will be achieved when the residual force $\psi$ vanishes, since this term directly measures the lack of equilibrium of the governing equation as indicated in (2.4). A trial value $\varphi^{0}$ of the basic unknown is assumed and the material stiffness associated with this value calculated according to the prescribed $H-\varphi$ relationship. + + + +The residual force, $\psi^0$ is then calculated from (2.4) and the Jacobian evaluated according to (2.7). The correction $\Delta \varphi^0$ to the first approximation for the basic unknown, can finally be found from (2.12). Thus an improved approximation to the solution has been found, as $\varphi^1 = \varphi^0 + \Delta \varphi^0$ . This process can then be continually repeated until the residual force, $\psi^n$ , is sufficiently small; or equivalently that $\varphi^{r-1}$ and $\varphi^r$ are sufficiently close. The Newton-Raphson process generally gives a more rapid and stable convergence path than the direct iteration method. + +# 2.2.3 The tangential stiffness method + +For structural applications the matrix H can be interpreted physically as the stiffness matrix of the structure. For nonlinear situations, in which the stiffness depends on the degree of displacement in some manner, H is equal to the local gradient of the force/displacement relationship of the structure at any point and is termed the tangential stiffness. The analysis of such problems must proceed in an incremental manner since the solution at any stage may not only depend on the current displacements of the structure, but also on the previous loading history. Consequently the problem can be linearised over any increment of load and therefore the matrix, which contains the nonlinear terms, can be discarded from (2.11) and (2.12). With this modification, the solution process is identical to that described in the previous section and for this reason the method is sometimes termed a generalised Newton–Raphson method. + +The solution algorithm is illustrated in Fig. 2.5; again for a single variable situation. Solution is commenced from a trial value $\varphi^{0}$ of the unknown (for structural problems the starting position of solution is almost invariably $\varphi^{0} = 0$ ). The tangential stiffness, $H(\varphi^{0})$ , corresponding to this displacement state is then determined and the residual force $\psi^{0}$ calculated according to (2.4). The correction, $\Delta\varphi^{0}$ , to the trial value is computed according to the linearised form of (2.12), which is + +$$ +\Delta \varphi^ {r} = - [ H (\varphi^ {r}) ] ^ {- 1}. \psi (\varphi^ {r}) \tag {2.13} +$$ + +An improved approximation to the unknown is then obtained as $\varphi^{1} = \varphi^{0} + \Delta\varphi^{0}$ . This iterative process is then continued until the solution converges to the nonlinear solution which is indicated by the condition that $\psi^{r}$ practically vanishes. + +# 2.2.4 The initial stiffness method + +In the methods described in the three previous sections, the complete factorisation (or reduction) and solution of the full set of simultaneous equations describing the discretised structure is essential for each iteration. For the method of direct iteration the equation solution indicated by (2.3) is necessary, whilst the Newton–Raphson technique and tangential stiffness method demand the equation solutions indicated by (2.12) and (2.13) diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_004.md b/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_004.md new file mode 100644 index 00000000..89e75423 --- /dev/null +++ b/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_004.md @@ -0,0 +1,436 @@ + + +![](images/page-031_34963c308dc70333f95678e42d5a568d89cc3cb3618464dce89aeead03982f5f.jpg) + +
+line + +| Basic variable φ | Applied force, f | Slope H(φ⁰) | Slope H(φ¹) | +| ---------------- | ---------------- | ----------- | ----------- | +| φ⁰ | ψ⁰ | - | - | +| φ¹ | ψ¹ | - | - | +| φ² | ψ² | - | - | +
+ +Fig. 2.5 Tangential stiffness solution algorithm for a single variable situation. + +respectively. If in (2.13) the tangential stiffness matrix is replaced, at all steps of the computation, by the stiffness corresponding to the initial trial value of $\varphi$ a complete factorisation, or reduction, of the assembled equations can be avoided. $^{(3)}$ In this case a complete equation solution need only be performed for the first iteration and subsequent approximations to the nonlinear solution performed, via the expression + +$$ +\Delta \varphi^ {r} = - [ H (\varphi^ {0}) ] ^ {- 1} \psi (\varphi^ {r}). \tag {2.14} +$$ + +Since the same stiffness matrix $H(\varphi^{0})$ is employed at each stage, the reduced equations can be stored in their reduced or factored form and a second or subsequent solution merely necessitates the reduction of the right-hand side $(\psi(\varphi^{r}))$ terms, together with a backsubstitution. This has the immediate advantage of significantly reducing the computing cost per iteration but reduces the convergence rate as can be seen from Fig. 2.6 where the scheme is schematically illustrated. The iterative algorithm is identical to that described in the preceding section. This method can be shown to be unconditionally convergent $^{(4)}$ and can even be employed in situations where the material exhibits negative stiffness. The relative economies of the initial stiffness and tangential stiffness methods depend to a large extent on the degree of nonlinearity inherent in the problem under consideration. The optimum algorithm is generally provided by an amalgamation of both processes, in which the stiffnesses are changed at selected iterative intervals only. + + + +![](images/page-032_a0573cb4c5dacac6feb1c5f31f4fcd20d8d8e72d37b98955769cb5f9d3d164e9.jpg) + +
+line + +| Basic variable, φ | Applied force, f | +| ----------------- | ---------------- | +| φ⁰ | ψ⁰ | +| φ¹ | ψ¹ | +| φ² | ψ² | +| φ³ | ψ³ | +
+ +Fig. 2.6 Initial stiffness solution algorithm for a single variable situation. + +# 2.3 Systems governed by a quasi-harmonic equation + +Many physical situations in engineering science are governed by a quasi-harmonic equation containing coefficients which are dependent on the unknown variable or its derivatives according to some prescribed law. The most common problem of this type occurs in heat conduction under steady-state conditions when the material conductivity is itself a function of temperature. This phenomenon also arises in diffusion problems where the diffusivity of the medium often varies with the concentration of the diffusing matter. Further physical examples are provided in Ref. (5). + +For a one-dimensional situation the governing equation to be considered is + +$$ +\frac {d}{d x} \left(K \frac {d \phi}{d x}\right) + Q = 0, \tag {2.15} +$$ + +in which $\phi$ is the unknown function and the terms K and Q may be functions of the position coordinate, x. The problem becomes nonlinear if K and/or Q are also functions of the unknown $\phi$ or its derivatives, according to some prescribed function. + +Two types of boundary condition will be considered: + +(a) The value of the unknown specified on the boundary + +$$ +\phi = \phi_ {B}. \tag {2.16} +$$ + +(b) The gradient of the unknown at the boundary specified to be zero + + + +$$ +\frac {d \phi}{d n} = \frac {d \phi}{d x} = 0. \tag {2.17} +$$ + +(A more general form of this latter boundary condition is considered in Ref. 6.) + +Equation (2.15) can be transformed to finite element form by suitable discretisation and use of the Galerkin weighted residual process. $^{(5,6)}$ The scalar product of equation (2.15) with any arbitrary weighting function, W, must be zero if $\phi$ satisfies (2.15) throughout any region $\Gamma$ , so that + +$$ +\int_ {\Gamma} \left(\frac {d}{d x} \left(K \frac {d \phi}{d x}\right) + Q\right) W d x = 0. \tag {2.18} +$$ + +Integrating the first term by parts results in + +$$ +\left[ W K \frac {d \phi}{d x} \right] _ {x _ {1}} ^ {x _ {2}} - \int_ {\Gamma} \left(K \frac {d W}{d x} \frac {d \phi}{d x} - Q W\right) d x = 0, \tag {2.19} +$$ + +where the limits of integration in the first term are the end points of the region $\Gamma$ . The unknown function $\phi$ may be approximated as + +$$ +\phi = \sum_ {i = 1} ^ {n} N _ {i} \phi_ {i}, \tag {2.20} +$$ + +in which n is the total number of nodes in the finite element idealisation and $N_{i}$ are the global shape functions. In the Galerkin process the number of weighting functions must equal the total number of unknown nodal values. The weighting function $W_{i}$ corresponding to node i can then be conveniently chosen such that $W_{i} = N_{i}$ . It should be noted that at nodes where the values of $\phi$ are prescribed, there is no associated unknown and consequently the weighting function for such nodes is zero. Therefore the first term in (2.19) always vanishes since at the two end points of the interval either $\phi$ is prescribed according to (2.16), in which case the weighting function for that point is zero, or $d\phi/dx$ is specified as zero according to (2.17). Substituting for $\phi$ and W in (2.19) and assembling all element contributions in the usual manner results in + +$$ +\boldsymbol {H} \varphi + \boldsymbol {f} = \mathbf {0}, \tag {2.21} +$$ + +in which typical element components are + +$$ +h _ {i j} ^ {(e)} = \int_ {\Gamma^ {(e)}} K \frac {d N _ {i} ^ {(e)}}{d x} \frac {d N _ {j} ^ {(e)}}{d x} d x, \tag {2.22} +$$ + +$$ +f _ {i} ^ {(e)} = \int_ {\Gamma^ {(e)}} Q N _ {i} ^ {(e)} d x, \tag {2.23} +$$ + + + +where $N_{i}^{(e)}$ are the element shape functions specifying the distribution of the unknown, $\phi$ , over the element. For the specific case of a two-noded element with a linear variation in $\phi$ as shown in Fig. 2.7, the shape functions are simply + +$$ +N _ {1} ^ {(e)} = \frac {1}{2} - \frac {x}{L}, \quad N _ {2} ^ {(e)} = \frac {1}{2} + \frac {x}{L}, \tag {2.24} +$$ + +where $L$ is the length of the element. + +![](images/page-034_ef3dbe5297fb2218dfad18a9a85eb82ab6ef41ec6c34297491c43c886eb3dafb.jpg) + +
+text_image + +N₁(e) +1 +L/2 +x +N₂(e) +2 +L/2 +
+ +Fig. 2.7 One-dimensional two-noded element with linear variation of the unknown, $\phi$ , showing element shape functions. + +Substituting in (2.22) and (2.23), and assuming no variation of K with position in the element, gives + +$$ +\boldsymbol {H} ^ {(e)} = \frac {K}{L} \left[ \begin{array}{c c} 1 & - 1 \\ - 1 & 1 \end{array} \right], \tag {2.25} +$$ + +and + +$$ +f _ {1} ^ {(e)} = f _ {2} ^ {(e)} = \frac {Q L}{2}. \tag {2.26} +$$ + +Provided that the variation of K with $\phi$ or its derivatives is specified, the problem falls into the category discussed in the previous section and can be solved by either the method of direct iteration or the Newton–Raphson approach. + +In the numerical examples considered later in this chapter a specific form of nonlinearity will be considered, namely + +$$ +K = K _ {0} (a + b \phi), \tag {2.27} +$$ + +in which $K_{0}$ is a reference value and a and b are known constants. For solution by the Newton–Raphson process the Jacobian matrix can be considered to be the sum of symmetric and nonsymmetric components as indicated in (2.11). The symmetric part has already been calculated in (2.25) and the nonsymmetric contribution must now be calculated according to the last + + + +term in (2.7). From (2.7), (2.22) and (2.27) the general term is given as + +$$ +h _ {i j} ^ {\prime} = \sum_ {k = 1} ^ {2} \left(\frac {\partial h _ {i k}}{\partial \phi_ {j}}\right) \phi_ {k} = \sum_ {k = 1} ^ {2} \left\{\phi_ {k} K _ {0} \int_ {- L / 2} ^ {L / 2} \frac {\partial}{\partial \phi_ {j}} [ a + b \phi ] \frac {d N _ {i} ^ {(e)}}{d x} \frac {d N _ {k} ^ {(e)}}{d x} d x \right\}. \tag {2.28} +$$ + +Noting that $\phi$ is given by (2.20) and that the shape functions are given by (2.24), the evaluation of (2.28) results in + +$$ +\boldsymbol {H} ^ {\prime (e)} = \frac {K _ {0} b}{2 L} (\phi_ {1} - \phi_ {2}) \left[ \begin{array}{c c} 1 & 1 \\ - 1 & - 1 \end{array} \right]. \tag {2.29} +$$ + +As expected, it is seen that the derivative matrix $H'(e)$ is unsymmetric. + +# 2.4 Nonlinear elastic problems + +The simplest case of nonlinear behaviour in structural problems arises from nonlinear elastic material action. The stress/strain relationship of the material is nonlinear but the material behaviour is elastic with all deformations and displacements recoverable on unloading. For example, this type of behaviour arises in hyperelastic problems $^{(7)}$ where the stresses are functions of a strain dependent material modulus. + +The nonlinear constitutive relation may be specified, for a one-dimensional situation, as + +$$ +\sigma = \frac {d W}{d \epsilon} = E _ {0}. g (\epsilon) \tag {2.30} +$$ + +where $\sigma$ is the stress, $\epsilon$ the strain and $E_{0}$ some reference value of the material modulus. The material performance will be nonlinear according to the form of the specified strain energy function, $W(\epsilon)$ . + +![](images/page-035_cfb75a943225e29d20f04e04b63c63e7673144dd3d7311381908887742d98769.jpg) + +
+text_image + +L +Cross sectional area A +δ +1 +2 +F +F +
+ +Fig. 2.8 Forces and displacements for a two-node element. + +The simplest form of one-dimensional finite element is the constant stress element shown in Fig. 2.8 in which a linear displacement variation is assumed between nodes 1 and 2. The force in the element is given, from (2.30), by + +$$ +F = E _ {0} A g (\delta / L), \tag {2.31} +$$ + + + +where A is the element cross-sectional area and $\delta$ the element extension. The tangential stiffness for the material is then + +$$ +K _ {T} = \frac {d F}{d \delta} = \frac {E _ {0} A}{L} \frac {d g}{d \epsilon} = \frac {E _ {0} A}{L} g ^ {\prime} (\epsilon). \tag {2.32} +$$ + +Or, in particular, the element tangential stiffness matrix is given by + +$$ +\boldsymbol {K} _ {T} ^ {(e)} = \frac {E _ {0} A}{L} g ^ {\prime} (\epsilon) \left[ \begin{array}{c c} 1 & - 1 \\ - 1 & 1 \end{array} \right]. \tag {2.33} +$$ + +Provided that $g'(\epsilon)$ is positive for all strain values, the tangential stiffness method of solution described in Section 2.2.3 can be employed in solution with $K_{T}^{(e)}$ being directly equivalent to $H(\varphi^{r})$ . If the tangential stiffness matrix becomes zero, the assembled stiffness equations will become singular and the inversion process required by (2.13) cannot be undertaken. Solution for situations in which the material tangential stiffness becomes non-positive can be performed by use of the initial stiffness method described in Section 2.2.4. Since the initial material stiffness is employed throughout this latter process, the assembled stiffness matrix will remain positive definite throughout the computation. + +# 2.5 Elasto-plastic problems in one dimension + +In this section the essential features of elasto-plastic material behaviour are introduced, and the basic expressions are developed in a form suitable for numerical solution by some of the methods described in the previous sections. + +Elasto-plastic behaviour is characterised by an initial elastic material response on to which a plastic deformation is superimposed after a certain level of stress has been reached. $^{(8)}$ Plastic deformation is essentially irreversible on unloading and is incompressible in nature. The onset of plastic deformation (or yielding) is governed by a yield criterion and post-yield deformation generally occurs at a greatly reduced material stiffness. Basic theoretical expressions for a general continuum are provided in Chapter 7. + +For one-dimensional situations, the material parameters required to completely define elasto-plastic behaviour are most conveniently obtained from a uniaxial tension test. Figure 2.9 shows an idealised stress-strain curve for a material and identical behaviour is assumed in tension and compression. The material initially deforms according to the elastic modulus, E, until the stress level reaches a value $\sigma_{Y}$ designated the uniaxial yield stress. On increasing the load further, the material is assumed to exhibit linear strain-hardening, characterised by the tangential modulus, $E_{T}$ . + +At some stage after initial yielding, consider a further load application resulting in an incremental increase of stress, $d\sigma$ , accompanied by a change of strain, $d\epsilon$ . Assuming that the strain can be separated into elastic and plastic + + + +![](images/page-037_0c8349f6b4bfb8553ae288c7cbd67011d9652f79a58e15b6533c191f7a0d2041.jpg) + +
+line + +| Strain, ε | Stress, σ | Elastic behaviour Slope, E | +| --------- | --------- | -------------------------- | +| 0 | 0 | 0 | +| ε | ε | dε_e | +| ε | ε | dε_p | +| ε | ε | dε_p | +| ε | ε | dε_p | +| ε | ε | dε_p | +| ε | ε | dε_p | +| ε | ε | dε_p | +| ε | ε | dε_p | +| ε | ε | dε_p | +| ε | ε | dε_p | +| σ_Y | σ_Y | Elastic behaviour Slope, E | +| σ_σ | σ_σ | Elastic behaviour Slope, E | +| SloE E | SloE E | Elastic response Slope, E_T | +| SloE E_T | SloE E_T | Elastic response Slope, E_T | +
+ +Fig. 2.9 Elastic, linear strain-hardening stress–strain behaviour for the uniaxial case. + +components, so that + +$$ +d \epsilon = d \epsilon_ {e} + d \epsilon_ {p}, \tag {2.34} +$$ + +we define a strain-hardening parameter, $H'$ , as + +$$ +H ^ {\prime} = \frac {d \sigma}{d \epsilon_ {p}}. \tag {2.35} +$$ + +This can be interpreted as the slope of the strain-hardening portion of the stress–strain curve after removal of the elastic strain component. Thus + +$$ +H ^ {\prime} = \frac {d \sigma}{d \epsilon - d \epsilon_ {e}} = \frac {E _ {T}}{1 - E _ {T} / E}. \tag {2.36} +$$ + +With reference to Fig. 2.8, consider the behaviour of a linear displacement element, which has a cross-sectional area A, when it is subjected to a gradually increasing axial force, F, which results in an extension, $\delta$ . Provided that F/A is less than or equal to the uniaxial yield stress, $\sigma_{Y}$ , the material behaviour will be elastic, exhibiting a stiffness of + +$$ +K _ {e} = \frac {F}{\delta} = \frac {E A}{L}, \tag {2.37} +$$ + + + +then the element stiffness matrix is simply + +$$ +\boldsymbol {K} _ {e} ^ {(e)} = \frac {E A}{L} \left[ \begin{array}{c c} 1 & - 1 \\ - 1 & 1 \end{array} \right]. \tag {2.38} +$$ + +Suppose F is increased until the material has yielded. Consider a further incremental increase in load dF which causes an additional element extension, $d\delta$ . Then + +$$ +d \delta = (d \epsilon_ {e} + d \epsilon_ {p}) L, \tag {2.39} +$$ + +where $L$ is the element length. Also, on use of (2.35) + +$$ +d F = d \sigma A = A H ^ {\prime} d \epsilon_ {p}. \tag {2.40} +$$ + +The tangential stiffness for the material is then + +$$ +K _ {e p} = \frac {d F}{d \delta} = \frac {A H ^ {\prime} d \epsilon_ {p}}{L \left(d \sigma / E + d \epsilon_ {p}\right)}. \tag {2.41} +$$ + +Or, using (2.35) and rearranging + +$$ +K _ {e p} = \frac {E A}{L} \left(1 - \frac {E}{E + H ^ {\prime}}\right). \tag {2.42} +$$ + +Finally, the element stiffness for elasto-plastic material behaviour is given by\* + +$$ +\boldsymbol {K} _ {e p} ^ {(e)} = \frac {E A}{L} \left(1 - \frac {E}{E + H ^ {\prime}}\right) \left[ \begin{array}{c c} 1 & - 1 \\ - 1 & 1 \end{array} \right]. \tag {2.43} +$$ + +In (2.42) it can be seen that the first term represents the elastic stiffness, as given by (2.38). The second term accounts for the reduction in stiffness from the elastic value due to yielding. + +\* The element stiffness matrix can be written in the standard finite element form + +$$ +\boldsymbol {K} _ {e} ^ {(e)} = \int_ {V} \boldsymbol {B} ^ {T} \boldsymbol {D} \boldsymbol {B} d V = A \int_ {0} ^ {L} \boldsymbol {B} ^ {T} \boldsymbol {D} \boldsymbol {B} d x, +$$ + +where integration is made over the volume of the element. For this one-dimensional application, D = E and + +$$ +\boldsymbol {B} = \left[ \frac {d N _ {1} ^ {(e)}}{d x}, \quad \frac {d N _ {2} ^ {(e)}}{d x} \right] = \left[ - \frac {1}{L}, \quad \frac {1}{L} \right], +$$ + +where $N_{1}^{(e)}$ and $N_{2}^{(e)}$ are given by (2.24). The tangential stiffness matrix for elastoplastic material behaviour is obtained by replacing D by + +$$ +\boldsymbol {D} _ {e p} = E \left(1 - \frac {E}{E + H ^ {\prime}}\right). +$$ + + + +For a perfectly plastic material behaviour, after initial yielding equation (2.36) implies that $H' = 0$ and it is then evident from (2.43) that $\boldsymbol{K}_{ep}^{(e)} = 0$ . This implies that the tangential (elasto-plastic) stiffness matrix for such a material is singular and the tangential stiffness method cannot generally be employed in solution. If a significant number of elements in the structure has yielded, the assembled tangential stiffness matrix will be singular, and the inversion or reduction demanded by (2.13) cannot be performed. This difficulty can be avoided by use of the initial stiffness method in which the elastic element stiffnesses are employed at every stage of the computation, thereby ensuring a positive definite assembled stiffness matrix. + +# 2.6 Problems + +In this section some tasks are set for the reader which illustrate some further points in connection with the topics discussed in the chapter. + +2.1 Use the direct iteration method to solve the following one degree of freedom problem, $H\phi + f = 0$ where $f = 10$ and $H$ depends on $\phi$ according to $H = 10(1 + e^{3\phi})$ . + +2.2 Repeat Problem 2.1 using the Newton–Raphson method. Compare the solutions and the computational effort required in each. + +2.3 Solve the following one degree of freedom problem by both the tangential stiffness and initial stiffness method. Apply the total load f as two equal increments + +$$ +H \phi + f = 0, \quad f = 1 0, \quad H = 2 0 (1 - \phi). +$$ + +2.4 The more general form of the boundary condition (2.17) in Section 2.3 is $d\phi / dx + q + \alpha \cdot \phi = 0$ , where $q$ and $\alpha$ are constants and $\phi$ is the undetermined value of the unknown at the boundary point. Repeat the Galerkin process of Section 2.3 to include these additional terms. In particular, determine the additional nodal force contribution and the discrete 'external' nodal stiffness which arise. + +2.5 For the two-noded element with linear variation in $\phi$ with shape functions as given by (2.24), evaluate the element stiffness matrix when $K$ is a function of $x$ . Assume that the spatial variation of $K$ within the element is linear and obtained by interpolation of the specified nodal values by use of the element shape functions. + +2.6 Suppose that a heat loss also occurs by convection from the surface area of an element, which is given by $h.\phi$ where h is the convection coefficient. If C is the circumference of the element, determine the additional contribution to $H^{(e)}$ resulting from this. $^{(9)}$ + +2.7 Determine the nonlinear portion, $H^{(e)}$ , of the Jacobian matrix for a material dependence $K = K_0(1 + e^{b\phi})$ . Assume a two-noded linear element. + +2.8 Evaluate the stiffness matrix $H^{(e)}$ for a three-noded element for a heat conduction problem. Assume that the element has shape functions + + + +$$ +N _ {1} ^ {(e)} = - \frac {2 x}{L ^ {2}} \left(\frac {L}{2} - x\right), \quad N _ {2} ^ {(e)} = \frac {4}{L ^ {2}} \left(\frac {L}{2} - x\right) \left(\frac {L}{2} + x\right), +$$ + +$$ +N _ {3} ^ {(e)} = \frac {2 x}{L ^ {2}} \left(\frac {L}{2} + x\right), +$$ + +and also that $K = K_0(a + b\phi)$ where $K_0, a$ and $b$ are constants. + +2.9 Repeat.Problem 2.8 for the case where $K_0$ is additionally a function of $x$ . Assume that the nodal values of $K_0$ are given. +2.10 Solve the nonlinear elastic problem of Fig. 2.10 by hand calculation. Use the tangential stiffness method and assume the total load to be applied in two equal increments. + +![](images/page-040_fc3edfd28234b9c5e6ba703473bd68f8818534b6a8f8a6917a72444499b43798.jpg) + +
+text_image + +A = 1.0 +σ = 20(ε - ε²) +1 +2 +P = 4.8 +5 +
+ +Fig. 2.10 Nonlinear elastic example—Problem 2.10. + +2.11 Solve Problem 2.10 if the structure is loaded by incrementally increasing the prescribed value of displacement at node 2. Increase the applied displacement in two equal increments up to a maximum value of $\phi_{2} = 3.0$ . Since the element stiffnesses become negative at the higher increment, use the initial stiffness method. +2.12 A locking material is one in which the stiffness increases with increasing strains. For example, if $g(\epsilon) = \epsilon^{2}$ can both the tangential stiffness and the initial stiffness methods be used to solve such material problems? + +![](images/page-040_df27ad1cb5e3d56e1763fbe5e8a2e28e30e96a31cf420c4064ff88abc48d5e5a.jpg) + +
+text_image + +1 +2 +20 +E = 1000 +A = 1.0 +σY = 10 +H' = 100 +10 +
+ +Fig. 2.11 Elasto-plastic example—Problem 2.13. + +2.13 Determine the nodal displacement of node 2 of the structure shown in Fig. 2.11 as the applied load is increased to 10 units in two equal increments. Assume elasto-plastic material behaviour and use the tangential stiffness approach for solution. + +![](images/page-040_570a87f73a30efd6200eb13cdf06cbe8584d8ac9969a00692a3b8229dbadb111.jpg) + +
+text_image + +1 +I +2 +20 +II +3 +10 +10 +
+
Element IElement II
E10001000
A1.01.0
$\sigma_{Y}$ 5.05.0
H'200-100
+ +Fig. 2.12 Bimaterial elasto-plastic example—Problem 2.14. diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_005.md b/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_005.md new file mode 100644 index 00000000..c15b6bf0 --- /dev/null +++ b/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_005.md @@ -0,0 +1,288 @@ + + +2.14 Determine the displacement of node 2 of the elasto-plastic structure shown in Fig. 2.12. Assume the load to be applied in two equal increments. What happens if $H_{I}^{\prime}=200$ , $H_{II}^{\prime}=-200$ ? + +# 2.7 References + +1. ZIENKIEWICZ, O. C., The Finite Element Method, McGraw-Hill, London, 1977. +2. BOOTH, A. D., Numerical Methods, Butterworth, London, 1966. +3. ZIENKIEWICZ, O. C., VALLIAPPAN, S. and KING, I. P., Elasto-plastic solutions of engineering problems. Initial stress, finite element approach, Int. J. Num. Meth. Engng., 1, 75–100 (1969). +4. ARGYRIS, J. H. and SCHARPF, D. W., Methods of elasto-plastic analysis, ISD, ISSC Symp. on Finite Element Tech., Stuttgart (1969). +5. LYNESS, J. F., OWEN, D. R. J. and ZIENKIEWICZ, O. C., The finite element analysis of engineering systems governed by a nonlinear quasi-harmonic equation, Computers and Structures, 5, 65–79 (1975). +6. HINTON, E. and OWEN, D. R. J., An Introduction to Finite Element Computations, Pineridge Press, Swansea, U.K., 1979. +7. ODEN, J. T., Finite Elements of Nonlinear Continua, McGraw-Hill, New York, 1972. +8. HILL, R., The Mathematical Theory of Plasticity, Oxford University Press, 1950. +9. SEEGERLIND, L. J., Applied Finite Element Analysis, John Wiley, New York, 1976. + + + + + +# Chapter 3 Solution of nonlinear problems + +# 3.1 Introduction + +A modular approach is adopted for the programs presented in this text, with the various main finite element operations being performed by separate subroutines. Any nonlinear finite element program must essentially contain all the subroutines necessary for elastic analysis. Briefly these consist of a subroutine to accept the input data, a subroutine for element stiffness formulation, subroutines for equation assembly and solution and a subroutine for output of the final results. + +In order to implement the solution algorithms described in Section 2.2, additional subroutines are clearly necessary. In particular two primary DO LOOPS are necessary to iterate the solution until convergence of the solution occurs and to increment the applied loading, if appropriate. Subroutines must be included to evaluate the residual forces and also to monitor convergence of the solution. Figure 3.1 shows the organisation of the programs presented in this chapter, particularly the sequence in which the subroutines are accessed. Four separate programs are developed to solve the following specific situations. + +- Solution of nonlinear quasi-harmonic situations by direct iteration. +- Solution of nonlinear quasi-harmonic situations by the Newton-Raphson method. +- Solution of nonlinear elastic problems by either the tangential stiffness or the initial stiffness method or a combination of both. +- Solution of elasto-plastic problems by either the tangential stiffness or the initial stiffness method or a combination of both approaches. + +With reference to Fig. 3.1, most of the subroutines are common to all four programs presented; the only exceptions being the subroutines necessary for stiffness matrix generation, residual force calculation and solution convergence checking. The element stiffness formulation subroutines for quasi-harmonic direct interaction, quasi-harmonic Newton–Raphson, nonlinear elastic situations and elasto-plastic problems are respectively named STIFF1, ASTIF1, STIFF2 and STIFF3. The evaluation of residual forces is not required in the direct iteration method and the appropriate subroutines for the quasi-harmonic Newton–Raphson, nonlinear elastic and elasto-plastic + + + +![](images/page-044_30b762f35ad25fc73a24946ed3ec1b7f05b8cd72cd47c3ff60182ea0b18cf575.jpg) + +
+flowchart + +```mermaid +graph TD + A["START"] --> B["DATA\nInput data defining geometry, loading, boundary conditions, material properties, etc."] + B --> C["INITIAL\nInitialise various arrays to zero"] + C --> D["INCLOD\nIncrement the applied loads"] + D --> E["NONAL\nSet indicator to identify type of solution algorithm, e.g., direct iteration, tangential stiffness, etc."] + E --> F["STIFF1/ASTIF1/STIFF2/STIFF3\nCalculate the element stiffnesses depending on the type of problem, e.g., quasi-harmonic, elasto-plasticity, etc."] + F --> G["ASSEMB\nAssemble the element loads and stiffnesses to give the global stiffness matrix and load vector"] + G --> H["GREDUC, BAKSUB & RESOLV\nSolve the resulting systems of simultaneous equations for the unknowns, φ"] + H --> I["REFOR1/REFOR2/REFOR3\nCalculate the residual force vector ψ for the Newton-Raphson, Tangential Stiffness and Initial Stiffness methods only"] + I --> J["MONITR/CONUND\nCheck to see if the solution has converged"] + J --> K["RESULT\nOutput the results"] + K --> L["END"] + A --> M["LOAD INCREMENT LOOP"] + M --> N["ITERATION LOOP"] + N --> O["NO"] + O --> K + K -->|YES| K +``` +
+ +Fig. 3.1 Program organisation for one-dimensional nonlinear applications. + +situations are named respectively REFOR1, REFOR2 and REFOR3. Finally, since the basis of solution convergence differs for the direct iteration method from that of the other procedures, it requires a separate convergence + + + +checking subroutine, termed MONITR. The equivalent subroutine for all other applications is named CONUND. + +The programs presented in this chapter also form the basis of an elastoviscoplastic program for one-dimensional applications developed in Chapter 4 and an elasto-plastic beam bending program considered in Chapter 5. In order to allow several of the subroutines developed in this chapter to be used for beam bending applications it will be necessary to permit the number of degrees of freedom per nodal point to be variable and to dimension some arrays to accommodate additional quantities. + +Sections 3.2 to 3.8 are devoted to the development of the subroutines which are common to the four programs presented. + +# 3.2 Input data subroutine, DATA + +For any finite element analysis the input data can be subdivided into three main classifications. Firstly the data required to define the geometry of the structure and the support conditions must be supplied. Secondly the material properties of the constituent materials must be supplied and finally the applied loading must be furnished. + +To allow a subroutine to be employed in more than one application, several control parameters must be supplied as input data. For example, the number of properties required to define the behaviour of a material will differ between quasi-harmonic problems and elasto-plastic situations. The use of variables in place of specific numerical values also generally aids program clarity. + +A list of control parameters required as input is now presented: + +NPOIN Total number of nodal points in the structure. + +NELEM Total number of elements in the structure. + +NBOUN Total number of boundary points, i.e. nodal points at which the value of the unknown is prescribed. In this context an internal node can be a boundary node. + +NMATS Total number of different materials in the structure. + +NPROP The number of material parameters required to define the characteristics of a material completely: + +4—For elasto-plastic problems, + +2—For all other applications. + +NNODE Number of nodes per element. For linear displacement one-dimensional elements this equals 2. + +NINCS The number of increments in which the total loading is to be applied. + +NALGO Indicator used to identify the type of solution algorithm to be employed: + +1—Direct iteration. + + + +2—Newton-Raphson method for quasi-harmonic problems. Tangential stiffness method for structural problems (nonlinear elastic and elastoplastic situations). + +3—Initial stiffness method. + +4—Combination of the initial and tangential stiffness methods, where the stiffnesses are recalculated on the first iteration of a load increment only. + +5—Combination of the initial and tangential stiffness methods, where the stiffnesses are recalculated on the second iteration of a load increment only. This can aid the rate of convergence considerably, if on the application of an increment of load there is substantial further yielding. When calculating the element stiffnesses the total plastic strains evaluated during the previous iteration are used to indicate whether the element has yielded or not. If the element stiffnesses are recalculated on the first iteration, the elements which have now yielded may have been elastic at the end of the previous load increment and consequently the reformulated stiffness will be based on elastic behaviour. This can reduce the convergence rate of the process since generally $H' \simeq 0.1E$ . From (2.42) the elasto-plastic stiffness is proportional to $E(1 - E/(E + H')) \simeq E/11$ , whereas the elastic stiffness depends linearly on E. Hence the tangential stiffness calculated grossly overestimates the true material response. This problem can be alleviated by reformulating the element stiffnesses during the second iteration of a load increment rather than the first, since the plastic strain evaluated on the first iteration will indicate yielding to have initiated. + +NDOFN The number of degrees of freedom per nodal point: + +1—For uniaxial problems. + +2—For beam bending problems (considered in Chapter 5). + +The geometry of the structure is completely defined on prescription of the nodal point coordinates and the element nodal connections. The coordinate of each nodal point must be defined with reference to a global coordinate system. For the one-dimensional situation being currently considered, the position of each nodal point is completely defined by a single coordinate whose value will be stored in the array + +# COORD (IPOIN) + +where IPOIN corresponds to the number of the nodal point. + +The origin of the coordinate system can be arbitrarily chosen. The geometry of each individual element must be specified by listing in a systematic way the numbers of the nodal points which define its outline. For the two-noded linear displacement element the nodal numbers can obviously be read in any + + + +order. The element topology is read into the array + +# LNODS (NUMEL, INODE) + +where NUMEL corresponds to the number of the element under consideration and subscript INODE ranges from 1 to NNODE. Since each element may conceivably be assigned different material properties, a material property identification number is also allocated to each element and stored in the array + +# MATNO (NUMEL) + +This implies that element number NUMEL has material properties of type MATNO (NUMEL). + +The material properties required for solution will differ for the various applications considered, but the same array will be employed for storage of this information. Namely + +# PROPS (NUMAT, IPROP) + +where NUMAT denotes the material identification number and the subscript IPROP the individual property. Each element is associated with a particular material type through the previously mentioned identification array MATNO (NUMEL). The relevant material properties associated with the different problem types considered here are listed below. + +(a) Quasi-harmonic problems + +PROPS (NUMAT, 1)—The reference value $K_0$ of the coefficient $K$ in equation (2.27). + +PROPS (NUMAT, 2)—The constant $b$ in equation (2.27) for a linear 'stiffness' variation. + +(b) Nonlinear elastic problems + +PROPS (NUMAT, 1)—The reference value $E_{0}$ in (2.30). + +PROPS (NUMAT, 2)—The cross-sectional area A, of the element. Each element with a different cross-sectional area must be assigned a different material property number. + +(c) Elasto-plastic problems + +PROPS (NUMAT, 1)—The elastic modulus, E, of the material. + +PROPS (NUMAT, 2)—The cross-sectional area, A, of the element. + +PROPS (NUMAT, 3)—The uniaxial yield stress of the material. + +PROPS (NUMAT, 4)—The linear strain hardening parameter, $H'$ , for the material (equation (2.35)). + +It should be mentioned here that the specific form of dependence of material stiffness on the unknown function for cases (a) and (b) will be directly incorporated into the program by use of a FORTRAN FUNCTION statement. + + + +Any nodal points at which a degree of freedom has a prescribed value must be identified by the temporary variable NODFX. To determine which degrees of freedom are to be prescribed at this node, the entries in the array + +# ICODE (IDOFN) + +are set to either 0 or 1. (Variable IDOFN ranges over the number of degrees of freedom per node NDOFN. In the present case NDOFN=1, but later in Chapter 5, NDOFN has the value 2.) If ICODE (IDOFN) is equal to 1, then degree of freedom IDOFN at node NODFX has a prescribed value. If NCODE (IDOFN) is equal to 0 then degree of freedom IDOFN at node NODFX is a free variable. + +The value for a prescribed degree of freedom is given by + +# VALUE (IDOFN) + +It should be noted that if ICODE (IDOFN)=0, then VALUE (IDOFN) is ignored. + +In order to simplify the solution process, the information stored in arrays ICODE and VALUE is transferred to much larger arrays IFPRE (NPOSN) and PEFIX (NPOSN) respectively, where NPOSN ranges over all the degrees of freedom for the whole finite element mesh. Both IFPRE and PEFIX are initially set equal to zero and as data for each restrained boundary node is read, they are modified if necessary. Unit entries in IFPRE indicate that the associated variable is prescribed. The prescribed value is obtained from the corresponding position in PEFIX. + +Finally, the loads applied to the structure must be specified. For the frontal method of equation solution employed in later chapters it is convenient to associate the applied loads with the elements on which they act. Thus for each element the nodal loads acting on the two nodes associated with the element must be input and these are stored in the array + +# RLOAD (IELEM, IEVAB) + +where IELEM indicates the element number and IEVAB relates to the degrees of freedom of the element (IEVAB ranges from 1 to NEVAB, the number of element variables, which is equal to 2 in the present case but which equals 4 in the applications described in Chapter 5). It should be noted that a nodal load may be arbitrarily assigned to any one of the elements connected to that node, since before eventual solution all element contributions are assembled to form a global load vector. Before entering the solution routines the loads are transferred to an array ELOAD (IELEM, IEVAB) as described later in Section 3.7. + +Subroutine DATA is now presented and should be largely self-explanatory. Descriptive comments are provided immediately after the FORTRAN listing of the subroutine. + + + +```csv +SUBROUTINE DATA DATA 1 +C************************** DATA 2 +C DATA 3 +C *** INPUTS DATA DEFINING GEOMETRY,LOADING,BOUNDARY CONDITIONS...ETC. DATA 4 +C DATA 5 +C************************** DATA 6 +COMMON/UNIM1/NPOIN.NELEM,NBOUN,NLOAD,NPROP,NNODE,IINCS,IITER, DATA 7 +KRESL,NCHEK,TOLER,NALGO,NSVAB,NDOFN,NINCS,NEVAB, DATA 8 +NITER,NOUTP,FACTO,PVALU DATA 9 +COMMON/UNIM2/PROPS(5,4),COORD(26),LNODS(25,2),IFPRE(52), DATA 10 +FIXED(52),TLOAD(25,4),RLOAD(25,4),ELOAD(25,4), DATA 11 +MATNO(25),STRES(25,2),PLAST(25),XDISP(52), DATA 12 +TDISP(26,2),TREAC(26,2),ASTIF(52,52),ASLOD(52), DATA 13 +REACT(52),FRESV(1352),PEFIX(52),ESTIF(4,4) DATA 14 +DIMENSION ICODE(2),VALUE(2),TITLE(18) DATA 15 +READ (5,965)TITLE DATA 16 +WRITE(6,965)TITLE DATA 17 +965 FORMAT(18A4) DATA 18 +READ(5,900) NPOIN,NELEM,NBOUN,NMATS,NPROP,NNODE,NINCS,NALGO,NDOFN DATA 19 +900 FORMAT(9I5) DATA 20 +WRITE(6,905)NPOIN,NELEM,NBOUN,NMATS,NPROP,NNODE,NINCS,NALGO,NDOFN DATA 21 +905 FORMAT(//1X,'NPOIN=',I5,3X,'NELEM=',I5,3X,'NBOUN=',I5,3X, DATA 22 +'NMATS=',I5//1X,'NPROP=',I5,3X,'NNODE=',I5,3X, DATA 23 +'NINCS=',I5,3X,'NALGO=',I5//1X,'NDOFN=',I5) DATA 24 +NEVAB=NDOFN*NNODE DATA 25 +NSVAB=NDOFN*NPOIN DATA 26 +WRITE(6,910) DATA 27 +910 FORMAT(1HO,5X,'MATERIAL PROPERTIES') DATA 28 +DO 10 IMATS=1,NMATS DATA 29 +READ (5,915) JMATS,(PROPS(JMATS,IPROP),IPROP=1,NPROP) DATA 30 +10 WRITE(6,915) JMATS,(PROPS(JMATS,IPROP),IPROP=1,NPROP) DATA 31 +915 FORMAT(I10,4F15.5) DATA 32 +WRITE(6,920) DATA 33 +920 FORMAT(1HO,3X,'EL NODES MAT.') DATA 34 +DO 20 IELEM=1,NELEM DATA 35 +READ (5,925) JELEM,(LNODS(JELEM,INODE),INODE=1,NNODE),MATNO(JELEM)DATA 36 +20 WRITE(6,925) JELEM,(LNODS(JELEM,INODE),INODE=1,NNODE),MATNO(JELEM)DATA 37 +925 FORMAT(4I5) DATA 38 +WRITE(6,930) DATA 39 +930 FORMAT(1HO,5X,'NODE',5X,'COORD.') DATA 40 +DO 30 IPOIN=1,NPOIN DATA 41 +READ (5,935) JPOIN,COORD(JPOIN) DATA 42 +30 WRITE(6,935) JPOIN,COORD(JPOIN) DATA 43 +935 FORMAT(I10,F15.5) DATA 44 +DO 40 ISVAB=1,NSVAB DATA 45 +IFPRE(ISVAB)=0 DATA 46 +40 PEFIX(ISVAB)=0.0 DATA 47 +IF(NDOFN.EQ.1) WRITE(6,940) DATA 48 +940 FORMAT(1HO,1X,'RES.NODE',2X,'CODE',3X,'PRES.VALUES') DATA 49 +IF(NDOFN.EQ.2) WRITE(6,945) DATA 50 +945 FORMAT(1HO,1X,'RES.NODE',2X,'CODE',3X,'PRES.VALUES',2X, DATA 51 +'CODE',3X,'PRES.VALUES') DATA 52 +DO 50 IBOUN=1,NBOUN DATA 53 +READ (5,950) NODFX,(ICODE(IDOFN),VALUE(IDOFN),IDOFN=1,NDOFN) DATA 54 +WRITE(6,950) NODFX,(ICODE(IDOFN),VALUE(IDOFN),IDOFN=1,NDOFN) DATA 55 +950 FORMAT(I10,2(I5,F15.5)) DATA 56 +NPOSN=(NODFX-1)*NDOFN DATA 57 +DO 50 IDOFN=1,NDOFN DATA 58 +NPOSN=NPOSN+1 DATA 59 +IFPRE(NPOSN)=ICODE(IDOFN) DATA 60 +50 PEFIX(NPOSN)=VALUE(IDOFN) DATA 61 +WRITE(6,955) DATA 62 +955 FORMAT(1HO,2X,'ELEMENT',10X,'NODAL LOADS') DATA 63 +DO 60 IELEM=1,NELEM DATA 64 +``` + + + +
DO 60 IEVAB=1,NEVABDATA65
60RLOAD(IELEM,IEVAB)=0.0DATA66
70READ (5,960) JELEM,(RLOAD(JELEM,IEVAB),IEVAB=1,NEVAB)DATA67
IF(JELEM.NE.NELEM) GO TO 70DATA68
DO 80 IELEM=1,NELEMDATA69
80WRITE(6,960) IELEM,(RLOAD(IELEM,IEVAB),IEVAB=1,NEVAB)DATA70
960FORMAT(I10,5F15.5)DATA71
RETURNDATA72
ENDDATA73
+ +DATA 16–18 Read and write the problem title. + +DATA 19-24 Read and write the control parameters for the problem. + +DATA 27–32 Read and write the material properties for each individual material. + +DATA 33–38 Read and write the nodal connection numbers and material identification number of each element. + +DATA 39–47 Read and write the coordinate of each nodal point. Also initialise the arrays for locating and recording prescribed values of the unknown. + +DATA 48–61 Read and write the node number and prescribed value for each degree of freedom for each boundary node and store in the global arrays IFPRE and PEFIX. + +DATA 62-71 Read and write the nodal loads for each element. + +# 3.3 Subroutine NONAL + +The main function of this subroutine is to control the solution process according to the value of the solution algorithm parameter, NALGO, input in subroutine DATA. The subroutine sets the value of indicator KRESL to either 1 or 2 according to NALGO and the current value of the iteration number IITER and increment number IINCS. A value of KRESL=1 indicates that the stiffnesses are to be reformulated and consequently a full system of simultaneous equations must be subsequently solved. If KRESL=2 the stiffnesses are not to be redefined and therefore only equation resolution need be undertaken. In this the reduced equations from the previous solution are stored and only the terms associated with the new loading need be reduced in the solution process. This results in a considerable saving in computation time with equation resolution generally requiring only 20% of the time required for complete analysis. For the algorithm options contained in the four programs presented, the value of KRESL is preset as follows. + +(a) Direct iteration. For this case the stiffnesses must be reformulated, according to (2.3), for every iteration. Consequently KRESL=1 at all stages. +(b) Newton–Raphson method for quasi-harmonic problems and tangential stiffness method for structural problems. Again the stiffnesses must be reformulated for every iteration according to (2.12) for quasi-harmonic situations and (2.13) for structural applications. Therefore KRESL=1 at all stages. diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_006.md b/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_006.md new file mode 100644 index 00000000..d76f6e0f --- /dev/null +++ b/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_006.md @@ -0,0 +1,450 @@ + + +(c) Initial stiffness method. In this approach the stiffnesses are calculated once and for all at the beginning of the computation, according to (2.14) and this value is then used throughout. Consequently KRESL=1 for the first iteration of the first load increment and is set equal to 2 thereafter. +(d) Combination of initial and tangential stiffness methods. In this algorithm the stiffnesses are recalculated only for the first iteration of any load increment and kept constant thereafter until convergence of solution under that particular loading is achieved. Therefore KRESL=1 for the first iteration of any load increment and is set to 2 at all other times. (Alternatively the element stiffnesses may be recomputed at the beginning of the second iteration as described in Section 3.2.) + +The final role of subroutine NONAL is to set the vector of prescribed unknowns to the correct values. For the method of direct iteration the problem is completely reanalysed for every iteration and therefore the vector of prescribed unknowns must be introduced unchanged into the solution sub-routines at each stage. However, for the three other solution algorithms considered, the processes are essentially accumulative with the value of the unknowns being totalled from the incremental values obtained for each iteration. Therefore, in order to maintain the fixed unknowns at their prescribed values, it is necessary to input the prescribed values into the solution routines for the first iteration of a load increment and then prescribe zero values for all subsequent iterations. In this way the final displacements will equal the prescribed values on convergence of the solution. If the structure is to be loaded by prescribing values of the unknowns then an incremental procedure may be adopted with factored values of the prescribed unknowns being applied sequentially. The prescribed displacements are factored by use of the variable FACTO, whose role is explained in terms of applied loads in Section 3.7. The prescribed values of the unknowns have been permanently stored in array PEFIX in subroutine DATA. These prescribed values, or zero values, required as described above, are transferred to the equation solution subroutines via the array FIXED. + +Subroutine NONAL is now presented and explanatory notes provided. + +SUBROUTINE NONAL NONL 1 +C**************************NONL 2 +C NONL 3 +C *** SETS INDICATOR TO IDENTIFY TYPE OF SOLUTION ALGORITHM NONL 4 +C NONL 5 +C**************************NONL 6 +COMMON/UNIM1/NPOIN,NELEM,NBOUN,NLOAD,NPROP,NNODE,IINCS,IITER, NONL 7 +. . KRESL,NCHEK,TOLER,NALGO,NSVAB,NDOFN,NINCS,NEVAB, NONL 8 +. . NITER,NOUTP,FACTO,PVALU NONL 9 +COMMON/UNIM2/PROPS(5,4),COORD(26),LNODS(25,2),IFPRE(52), NONL 10 +. . FIXED(52),TLOAD(25,4),RLOAD(25,4),ELOAD(25,4), NONL 11 +. . MATNO(25),STRES(25,2),PLAST(25),XDISP(52), NONL 12 + + + +
TDISP(26,2),TREAC(26,2),ASTIF(52,52),ASLOD(52),NONL13
REACT(52),FRESV(1352),PEFIX(52),ESTIF(4,4)NONL14
KRESL=2NONL15
IF(NALGO.EQ.1) KRESL=1NONL16
IF(NALGO.EQ.2) KRESL=1NONL17
IF(NALGO.EQ.3.AND.IINCS.EQ.1.AND.IITER.EQ.1) KRESL=1NONL18
IF(NALGO.EQ.4.AND.IITER.EQ.1) KRESL=1NONL19
IF(NALGO.EQ.5.AND.IINCS.EQ.1.AND.IITER.EQ.1) KRESL=1NONL20
IF(NALGO.EQ.5.AND.IITER.EQ.2) KRESL=1NONL21
IF(IITER.EQ.1.OR.NALGO.EQ.1) GO TO 20NONL22
DO 10 ISVAB=1,NSVABNONL23
FIXED(ISVAB)=0.0NONL24
RETURNNONL25
DO 30 ISVAB=1,NSVABNONL26
FIXED(ISVAB)=PEFIX(ISVAB)*FACTONONL27
RETURNNONL28
ENDNONL29
+ +NONL 15 Preset KRESL to the condition of equation resolution. + +NONL 16 For the direct iteration method set KRESL=1 for recomputation of the stiffnesses at all stages. + +NONL 17 For the Newton–Raphson method for quasi-harmonic problems or the tangential stiffness method for structural problems, recompute the stiffnesses at all stages. + +NONL 18 For the initial stiffness method for structural problems, compute the stiffnesses only at the beginning of the computation procedure. + +NONL 19 For the combined initial and tangential stiffness approach and NALGO=4, recompute the stiffnesses at the first iteration of each load increment only. + +NONL 20–21 For the initial/tangential approach with the option NALGO =5 (Section 3.2), the stiffnesses are recalculated on the 2nd iteration of any load increment. However, at the start of the computation the stiffnesses must be evaluated. + +NONL 22 For all stages of the direct iteration method or the first iteration of the other techniques, go to 20 to set the unknowns equal to the prescribed values. + +NONL 23–25 Set the vector of prescribed unknowns to zero and return. + +NONL 26–27 Set the vector of prescribed unknowns equal to the input prescribed values multiplied by a specified factor. + +# 3.4 Subroutines for equation assembly and solution + +For finite element analysis by the displacement process, the stiffness and load contributions of each element must be assembled into the global stiffness matrix and load vector respectively. The resulting set of simultaneous equations must then be solved to give the unknown nodal values. These aspects have been dealt with in detail elsewhere $^{(1-3)}$ and only the essential steps of the process will be reproduced here. + + + +# 3.4.1 Numerical example of equation assembly and solution + +In order to introduce the global stiffness matrix assembly and equation solution process we consider the example of a simple axial load structure shown in Fig. 3.2. The structure is subdivided into four elements in each of which a linear displacement variation is assumed. At each node i of the element there is an axial displacement degree of freedom, $\phi_{i}$ . + +![](images/page-053_4fdbe6252d17494de0f06b7a7743b8816459a9691a365175f241db750259b7c3.jpg) + +
+text_image + +5 IV 4 III 3 I 1 P₁ +LⅣ LⅢ LⅡ φₚ +Ⅰ 2 +
+ +Fig. 3.2 Structural example for illustration of equation solution process. + +The stiffness matrix for this element has already been derived in Section 2.5 and is given, for elastic material behaviour, by equation (2.38). The element stiffness matrices can be written as + +$$ +\boldsymbol {K} _ {\mathrm{I}} = k _ {\mathrm{I}} \left[ \begin{array}{c c} 1 & - 1 \\ - 1 & 1 \end{array} \right], \quad \boldsymbol {K} _ {\mathrm{II}} = k _ {\mathrm{II}} \left[ \begin{array}{c c} 1 & - 1 \\ - 1 & 1 \end{array} \right], +$$ + +$$ +\boldsymbol {K} _ {\mathrm{III}} = k _ {\mathrm{III}} \left[ \begin{array}{c c} 1 & - 1 \\ - 1 & 1 \end{array} \right], \quad \boldsymbol {K} _ {\mathrm{IV}} = k _ {\mathrm{IV}} \left[ \begin{array}{c c} 1 & - 1 \\ - 1 & 1 \end{array} \right], \tag {3.1} +$$ + +where + +$$ +k _ {\mathrm{I}} = \frac {E ^ {(\mathrm{I})} A ^ {(\mathrm{I})}}{L ^ {(\mathrm{I})}}, \text { etc. }, \tag {3.2} +$$ + +in which $E^{(I)}$ , $A^{(I)}$ and $L^{(I)}$ are respectively the elastic modulus, cross-sectional area and length of element I. The vector of applied nodal forces for each element is + +$$ +\boldsymbol {f} _ {\mathrm{I}} = \left[ \begin{array}{l} P _ {1} \\ 0 \end{array} \right], \quad \boldsymbol {f} _ {\mathrm{II}} = \left[ \begin{array}{l} 0 \\ 0 \end{array} \right], \quad \boldsymbol {f} _ {\mathrm{III}} = \left[ \begin{array}{l} 0 \\ 0 \end{array} \right], \quad \boldsymbol {f} _ {\mathrm{IV}} = \left[ \begin{array}{l} 0 \\ 0 \end{array} \right]. \tag {3.3} +$$ + +The vectors of the unknown nodal displacements for the elements are + +$$ +\boldsymbol {\delta} _ {\mathrm{I}} = \left[ \begin{array}{l} \phi_ {1} \\ \phi_ {3} \end{array} \right], \quad \boldsymbol {\delta} _ {\mathrm{II}} = \left[ \begin{array}{l} \phi_ {2} \\ \phi_ {3} \end{array} \right], \quad \boldsymbol {\delta} _ {\mathrm{III}} = \left[ \begin{array}{l} \phi_ {3} \\ \phi_ {4} \end{array} \right], \quad \boldsymbol {\delta} _ {\mathrm{IV}} = \left[ \begin{array}{l} \phi_ {4} \\ \phi_ {5} \end{array} \right]. \tag {3.4} +$$ + +We also assume the following prescribed displacement values + +$$ +\phi_ {2} = \phi_ {p}, \quad \phi_ {5} = 0. \tag {3.5} +$$ + + + +The Theorem of Minimum Total Potential Energy will now be used to derive the stiffness equations for this problem. The total potential energy for each element may be calculated separately. For example, the total potential energy of element I can be expressed as + +$$ +\pi_ {\mathrm{I}} = \frac {1}{2} [ \delta_ {\mathrm{I}} ] ^ {T} K _ {\mathrm{I}} \delta_ {\mathrm{I}} - [ \delta_ {\mathrm{I}} ] ^ {T} f _ {\mathrm{I}} = \frac {k _ {\mathrm{I}}}{2} \left(\phi_ {1} - \phi_ {3}\right) ^ {2} - P _ {1} \phi_ {1}. \tag {3.6} +$$ + +The augmented total potential energy of the assemblage is given by the sum of the individual element potentials plus extra terms to account for the prescribed values + +$$ +\pi = \pi_ {\mathrm{I}} + \pi_ {\mathrm{II}} + \pi_ {\mathrm{III}} + \pi_ {\mathrm{IV}} - R _ {2} \left(\phi_ {2} - \phi_ {p}\right) - R _ {5} \left(\phi_ {5} - 0\right) \tag {3.7} +$$ + +Note that $R_{2}$ and $R_{5}$ are the associated nodal reactions. + +Using the principle of minimum potential energy, we obtain + +$$ +\frac {\partial \pi}{\partial \phi_ {1}} = k _ {\mathrm{I}} (\phi_ {1} - \phi_ {3}) - P _ {1} = 0, +$$ + +$$ +\frac {\partial \pi}{\partial \phi_ {2}} = k _ {\mathrm{II}} (\phi_ {2} - \phi_ {3}) = R _ {2}, +$$ + +$$ +\frac {\partial \pi}{\partial \phi_ {3}} = k _ {\mathrm{I}} (\phi_ {3} - \phi_ {1}) + k _ {\mathrm{II}} (\phi_ {3} - \phi_ {2}) + k _ {\mathrm{III}} (\phi_ {3} - \phi_ {4}) = 0, \tag {3.8} +$$ + +$$ +\frac {\partial \pi}{\partial \phi_ {4}} = k _ {\mathrm{III}} (\phi_ {4} - \phi_ {3}) + k _ {\mathrm{IV}} (\phi_ {4} - \phi_ {5}) = 0, +$$ + +$$ +\frac {\partial \pi}{\partial \phi_ {5}} = k _ {\mathrm{IV}} \left(\phi_ {5} - \phi_ {4}\right) = R _ {5}. +$$ + +These equilibrium equations for the assembled elements of the structure can be expressed in matrix form as + +$$ +\left[ \begin{array}{c c c c c} 1 & 2 & 3 & 4 & 5 \\ k _ {\mathrm{I}} & 0 & - k _ {\mathrm{I}} & 0 & 0 \\ 0 & k _ {\mathrm{II}} & - k _ {\mathrm{II}} & 0 & 0 \\ - k _ {\mathrm{I}} & - k _ {\mathrm{II}} & k _ {\mathrm{I}} + k _ {\mathrm{II}} + k _ {\mathrm{III}} & - k _ {\mathrm{III}} & 0 \\ 0 & 0 & - k _ {\mathrm{III}} & k _ {\mathrm{III}} + k _ {\mathrm{IV}} & - k _ {\mathrm{IV}} \\ 0 & 0 & 0 & - k _ {\mathrm{IV}} & k _ {\mathrm{IV}} \end{array} \right] \left[ \begin{array}{l} \phi_ {1} \\ \phi_ {2} \\ \phi_ {3} \\ \phi_ {4} \\ \phi_ {5} \end{array} \right] = \left[ \begin{array}{l} P _ {1} \\ R _ {2} \\ 0 \\ 0 \\ R _ {5} \end{array} \right] \tag {3.9} +$$ + +The assembly process can be clearly appreciated by comparing the individual stiffness matrices (3.1), and load vectors (3.3), with the final assemblage. Obviously, the individual element contributions can be added directly into the overall stiffness matrix of the structure in positions appropriate to the element nodal connection numbers. + + + +It is noted that the global stiffness matrix is both symmetric and banded. By banded we mean that all the non-zero stiffness coefficients lie within a band adjacent to the leading diagonal. Banding of the stiffness equations is a direct consequence of the order in which the nodal points are numbered. + +In the equation solution subroutines presented later in Sections 3.4.2–3.4.5 no advantage will be taken of the banded symmetric form of the stiffness equations. + +Some elementary concepts of equation solution are now introduced. In particular we describe the Gaussian direct elimination process which will be used in a more efficient form in the main solution routine described later in Chapter 6. + +# 3.4.1.1 Gaussian direct elimination method for the solution of simultaneous equation systems + +Formulation of the global stiffness matrix resulted in equation system (3.9) which is of the general form + +$$ +k _ {1 1} \phi_ {1} + k _ {1 2} \phi_ {2} + k _ {1 3} \phi_ {3} + \dots k _ {1 n} \phi_ {n} = f _ {1} +$$ + +$$ +k _ {2 1} \phi_ {1} + k _ {2 2} \phi_ {2} + k _ {2 3} \phi_ {3} + \dots k _ {2 n} \phi_ {n} = f _ {2} +$$ + +$$ +\dots +$$ + +$$ +k _ {n 1} \phi_ {1} + k _ {n 2} \phi_ {2} + k _ {n 3} \phi_ {3} + \dots k _ {n n} \phi_ {n} = f _ {n}. \tag {3.10} +$$ + +The Gaussian direct elimination method seeks to reduce equation system (3.10) to the following triangular form $^{(4)}$ + +$$ +k _ {1 1} ^ {\prime} \phi_ {1} + k _ {1 2} ^ {\prime} \phi_ {2} + k _ {1 3} ^ {\prime} \phi_ {3} + \dots k _ {1, n - 1} ^ {\prime} \phi_ {n - 1} + k _ {1 n} ^ {\prime} \phi_ {n} = f _ {1} ^ {\prime} +$$ + +$$ +0 + k _ {2 2} ^ {\prime} \phi_ {2} + k _ {2 3} ^ {\prime} \phi_ {3} + \dots k _ {2, n - 1} ^ {\prime} \phi_ {n - 1} + k _ {2 n} ^ {\prime} \phi_ {n} = f _ {2} ^ {\prime} +$$ + +$$ +0 + 0 + k _ {3 3} ^ {\prime} \phi_ {3} + \dots k _ {3, n - 1} ^ {\prime} \phi_ {n - 1} + k _ {3 n} ^ {\prime} \phi_ {n} = f _ {3} ^ {\prime} +$$ + +$$ +k _ {n - 1, n - 1} ^ {\prime} \phi_ {n - 1} + k _ {n - 1, n} ^ {\prime} \phi_ {n} = f _ {n - 1} ^ {\prime} +$$ + +$$ +k _ {n n} ^ {\prime} \phi_ {n} = f _ {n} ^ {\prime}. \tag {3.11} +$$ + +Then all the unknowns can be systematically determined by taking these reduced equations in reverse order, since each new equation, proceeding in an upward direction, only introduces one additional unknown value. The last equation is solved for $\phi_{n}$ , then $\phi_{n-1}$ can be recovered from the next equation and so on. This phase of the solution scheme is termed back-substitution. + +# 3.4.1.2 The equation reduction or elimination phase + +Reduction of system (3.10) to the form (3.11) can be accomplished by employing the $i^{th}$ equation to eliminate $\phi_{i}$ from all equations below, i.e. from equations $i+1$ to n. Formally this can be done by subtracting from the $r^{th}$ equation ( $i < r \leqslant n$ ), the $i^{th}$ equation factored by $k_{ri}^{(i)}/k_{ii}^{(i)}$ , where the + + + +superscript i indicates that these coefficients have been already modified $(i-1)$ times prior to the elimination of the $i^{th}$ degree of freedom. For example, the first equation is used to eliminate $\phi_{1}$ from equations 2 to n as follows: + +$$ +\begin{array}{l} k _ {1 1} \phi_ {1} + k _ {1 2} \phi_ {2} + k _ {1 3} \phi_ {3} + \dots \quad k _ {1 n} \phi_ {n} = f _ {1} \\ 0. \phi_ {1} + \left(k _ {2 2} - k _ {1 2} \frac {k _ {2 1}}{k _ {1 1}}\right) \phi_ {2} + \left(k _ {2 3} - k _ {1 3} \frac {k _ {2 1}}{k _ {1 1}}\right) \phi_ {3} + \dots \left(k _ {2 n} - k _ {1 n} \frac {k _ {2 1}}{k _ {1 1}}\right) \phi_ {n} = f _ {2} - f _ {1} \frac {k _ {2 1}}{k _ {1 1}} \\ 0. \phi_ {1} + \left(k _ {n 2} - k _ {1 2} \frac {k _ {n 1}}{k _ {1 1}}\right) \phi_ {2} + \left(k _ {n 3} - k _ {1 3} \frac {k _ {n 1}}{k _ {1 1}}\right) \phi_ {3} + \dots \left(k _ {n n} - k _ {1 n} \frac {k _ {n 1}}{k _ {1 1}}\right) \phi_ {n} = f _ {n} - f _ {1} \frac {k _ {n 1}}{k _ {1 1}}. \tag {3.12} \\ \end{array} +$$ + +Then the second equation is used to eliminate $\phi_{2}$ from equations 3 to n and so on. Note that the modified terms in the equation system are still symmetric. + +# 3.4.1.3 The case of a prescribed displacement + +If a displacement is prescribed its value is known. Therefore the nodal force necessary to maintain the specified displacement becomes the unknown value associated with the node. Suppose for example that $\phi_{2}$ is prescribed to be some given value $\phi_{p}$ , in which case $f_{2}$ is the reaction value. In this case the elimination of $\phi_{2}$ is trivial and all that need be done is to substitute $\phi_{2}=\phi_{p}$ in equations 3 to n and transfer the now known quantity + +$$ +k _ {r 2} ^ {\prime} \phi_ {p} \quad (3 \leqslant r \leqslant n) +$$ + +to the right-hand side of each equation. This is illustrated below + +$$ +\begin{array}{l} k _ {1 1} \phi_ {1} + k _ {1 2} \phi_ {2} + k _ {1 3} \phi_ {3} + \dots k _ {1 n} \phi_ {n} = f _ {1} \\ 0. \phi_ {1} + k _ {2 2} ^ {\prime} \phi_ {2} + k _ {2 3} ^ {\prime} \phi_ {3} + \dots k _ {2 n} ^ {\prime} \phi_ {n} = f _ {2} \\ 0. \phi_ {1} + 0. \phi_ {2} + k _ {3 3} ^ {\prime} \phi_ {3} + \dots k _ {3 n} ^ {\prime} \phi_ {n} = f _ {3} - k _ {3 2} ^ {\prime} \phi_ {p} \\ 0. \phi_ {1} + 0. \phi_ {2} + k _ {n 3} ^ {\prime} \phi_ {3} + \dots k _ {n n} ^ {\prime} \phi_ {n} = f _ {n} - k _ {n 2} ^ {\prime} \phi_ {p}. \tag {3.13} \\ \end{array} +$$ + +For the particular case of a zero prescribed displacement value due to a pinned support, an alternative approach is to delete the row and column corresponding to the zero displacement from the equation system. The column can be deleted since it always multiplies a zero quantity and the row is removed since it only relates to equilibrium at the supported node. However this means that if the support reaction is required, it must be computed separately from the element forces meeting at the pinned node. + +The complete solution process is best illustrated by application to a particular problem. We will now substitute explicit values for the terms contained in (3.9) in order to permit numerical solution. Assume that + +$$ +k _ {\mathrm{I}} = 1, \quad k _ {\mathrm{II}} = 2, \quad k _ {\mathrm{III}} = 3, \quad k _ {\mathrm{IV}} = 4, \quad P _ {1} = 1 0, \quad \phi_ {p} = 2, \tag {3.14} +$$ + + + +then equations (3.9) can be written as + +$$ +\phi_ {1} + 0. \phi_ {2} - \quad \phi_ {3} + 0. \phi_ {4} + 0. \phi_ {5} = 1 0 \tag {3.15a} +$$ + +$$ +0. \phi_ {1} + 2 \phi_ {2} - 2 \phi_ {3} + 0. \phi_ {4} + 0. \phi_ {5} = R _ {2}; \quad \phi_ {2} = 2 \tag {3.15b} +$$ + +$$ +- \phi_ {1} - 2 \phi_ {2} + 6 \phi_ {3} - 3 \phi_ {4} + 0. \phi_ {5} = 0 \tag {3.15c} +$$ + +$$ +0. \phi_ {1} + 0. \phi_ {2} - 3 \phi_ {3} + 7 \phi_ {4} - 4 \phi_ {5} = 0 \tag {3.15d} +$$ + +$$ +0. \phi_ {1} + 0. \phi_ {2} + 0. \phi_ {3} - 4 \phi_ {4} + 4 \phi_ {5} = R _ {5}; \quad \phi_ {5} = 0. \tag {3.15e} +$$ + +where $R_{2}$ and $R_{5}$ are the nodal reactions associated with the displacement values prescribed at nodes 2 and 5. For example, $R_{2}$ must balance the sum of the elastic forces provided by all the elements meeting at node 2. We also imply by the notation adopted that $\phi_{2} = 2$ . + +To solve these equations by the Gaussian reduction process we first eliminate $\phi_{1}$ from all equations, except (3.15a). Then we eliminate $\phi_{2}$ from all equations below (3.15b), then $\phi_{3}$ is eliminated from all equations below (3.15c) and so on. Therefore, we eliminate a particular variable only below the current or active equation. (If we are eliminating $\phi_{r}$ , the $r^{th}$ equation is active.) + +We commence the process by eliminating $\phi_{1}$ from equations (3.15b)-(3.15e) by using (3.15a). In fact, we need only operate on (3.15c) since $\phi_{1}$ does not appear in the other equations. Thus we eliminate $\phi_{1}$ from (3.15c) by adding (3.15a) to (3.15c). This gives the first reduced set of equations as + +$$ +\phi_ {1} + 0. \phi_ {2} - \quad \phi_ {3} + 0. \phi_ {4} + 0. \phi_ {5} = 1 0 \tag {3.16a} +$$ + +$$ +0. \phi_ {1} + 2 \phi_ {2} - 2 \phi_ {3} + 0. \phi_ {4} + 0. \phi_ {5} = R _ {2}; \quad \phi_ {2} = 2 \tag {3.16b} +$$ + +$$ +0. \phi_ {1} - 2 \phi_ {2} + 5 \phi_ {3} - 3 \phi_ {4} + 0. \phi_ {5} = 1 0 \tag {3.16c} +$$ + +$$ +0. \phi_ {1} + 0. \phi_ {2} - 3 \phi_ {3} + 7 \phi_ {4} - 4 \phi_ {5} = 0 \tag {3.16d} +$$ + +$$ +0. \phi_ {1} + 0. \phi_ {2} + 0. \phi_ {3} - 4 \phi_ {4} + 4 \phi_ {5} = R _ {5}; \quad \phi_ {5} = 0. \tag {3.16e} +$$ + +Next we eliminate $\phi_{2}$ from (3.16c)-(3.16e) by using (3.16b). In fact, since $\phi_{2}$ is prescribed to be 2, all we need do is substitute $\phi_{2} = 2$ directly into the remaining equations. We also do this for (3.16b) in this case. + +$$ +\phi_ {1} + 0. \phi_ {2} - \quad \phi_ {3} + 0. \phi_ {4} + 0. \phi_ {5} = 1 0 \tag {3.17a} +$$ + +$$ +0. \phi_ {1} + 0. \phi_ {2} - 2 \phi_ {3} + 0. \phi_ {4} + 0. \phi_ {5} = - 4 + R _ {2}; +$$ + +$$ +\phi_ {2} = 2 \tag {3.17b} +$$ + + + +$$ +0. \phi_ {1} + 0. \phi_ {2} + 5 \phi_ {3} - 3 \phi_ {4} + 0. \phi_ {5} = 1 4 \tag {3.17c} +$$ + +$$ +0. \phi_ {1} + 0. \phi_ {2} - 3 \phi_ {3} + 7 \phi_ {4} - 4 \phi_ {5} = 0 \tag {3.17d} +$$ + +$$ +0. \phi_ {1} + 0. \phi_ {2} + 0. \phi_ {3} - 4 \phi_ {4} + 4 \phi_ {5} = R _ {5}; \quad \phi_ {5} = 0. \tag {3.17e} +$$ + +We then use (3.17c) to eliminate $\phi_{3}$ from (3.17d) and (3.17e). We need only operate on (3.17d), since $\phi_{3}$ does not appear in (3.17e), and in particular we add (3.17d) to 3/5 of (3.17c). + +$$ +\phi_ {1} + 0. \phi_ {2} - \quad \phi_ {3} + 0. \phi_ {4} + 0. \phi_ {5} = 1 0 \tag {3.18a} +$$ + +$$ +0. \phi_ {1} + 0. \phi_ {2} - 2 \phi_ {3} + 0. \phi_ {4} + 0. \phi_ {5} = - 4 + R _ {2}; \quad \phi_ {2} = 2 \tag {3.18b} +$$ + +$$ +0. \phi_ {1} + 0. \phi_ {2} + 5 \phi_ {3} - 3 \phi_ {4} + \phi_ {5} = 1 4 \tag {3.18c} +$$ + +$$ +0. \phi_ {1} + 0. \phi_ {2} + 0. \phi_ {3} + \frac {2 6}{5} \phi_ {4} - 4 \phi_ {5} = \frac {4 2}{5} \tag {3.18d} +$$ + +$$ +0. \phi_ {1} + 0. \phi_ {2} + 0. \phi_ {3} - 4 \phi_ {4} + 4 \phi_ {5} = R _ {5}; \quad \phi_ {5} = 0. \tag {3.18e} +$$ + +To complete the elimination process, we eliminate $\phi_{4}$ from (3.18e) by adding (3.18e) to 20/26 of (3.18d). + +$$ +\phi_ {1} + 0. \phi_ {2} - \quad \phi_ {3} + 0. \phi_ {4} + 0. \phi_ {5} = 1 0 \tag {3.19a} +$$ + +$$ +0. \phi_ {1} + 0. \phi_ {2} - 2 \phi_ {3} + 0. \phi_ {4} + 0. \phi_ {5} = - 4 + R _ {2}; \quad \phi_ {2} = 2 \tag {3.19b} +$$ + +$$ +0. \phi_ {1} + 0. \phi_ {2} + 5 \phi_ {3} - 3 \phi_ {4} + \phi_ {5} = 1 4 \tag {3.19c} +$$ + +$$ +0. \phi_ {1} + 0. \phi_ {2} + 0. \phi_ {3} + \frac {2 6}{5} \phi_ {4} - 4 \phi_ {5} = \frac {4 2}{5} \tag {3.19d} +$$ + +$$ +0. \phi_ {1} + 0. \phi_ {2} + 0. \phi_ {3} + 0. \phi_ {4} + \frac {1 2}{1 3} \phi_ {5} = \frac {8 4}{1 3} + R _ {5}; \quad \phi_ {5} = 0. \tag {3.19e} +$$ + +We now have a set of equations which can be solved directly if we take them in reverse order. Starting with (3.19e) we have $R_{5} = -84/13$ , since $\phi_{5} = 0$ . Knowing $\phi_{5}$ then (3.19d) gives $\phi_{4} = 21/13$ . Having obtained $\phi_{4}$ and $\phi_{5}$ equation (3.19c) gives $\phi_{3} = 49/13$ . Then knowing $\phi_{3}$ , $\phi_{4}$ , $\phi_{5}$ and with $\phi_{2}$ prescribed, (3.19b) gives $R_{2} = -46/13$ immediately. Finally we complete the back substitution process by determining $\phi_{1}$ from (3.19a) since $\phi_{2}$ , $\phi_{3}$ , $\phi_{4}$ are known at this stage. This gives $\phi_{1} = 179/13$ . Since the above procedure is quite systematic it can be readily programmed. + +The global stiffness matrix must be assembled and the stiffness equations reduced only if the element stiffnesses have been changed for the current iteration. The full assembly and reduction process must be followed if + + + +KRESL = 1, but only the global load vector need be formed and reduced if KRESL = 2. In this way a considerable number of arithmetic operations are avoided if only equation resolution is to be undertaken. This facility is incorporated in the equation solution subroutines presented in the following sections. + +The principles discussed in this section can now be repeated as a FORTRAN operation. Four subroutines are presented which undertake the respective tasks of equation assembly, equation reduction by Gaussian direct elimination, the back substitution process and reduction of subsequent load vectors for equation resolution. + +# 3.4.2 Subroutine ASSEMB + +This subroutine assembles the element nodal loads to form the global load vector. Also, the contributions of individual elements are assembled to form the global stiffness matrix. The variables employed in the subroutine are listed below and descriptive notes are again provided immediately after the FORTRAN listing. + +Dictionary of variable names (with dimensions) + +
ASLOD (MSVAB)ASsembled LOaD vector
ASTIF (MSVAB, MSVAB)Assembled global STIFfness matrix
RLOAD (MEVAB)Element load vector
ESTIF (MEVAB, MEVAB)Element STIFfness matrix
IELEM, NELEM, MELEMIndex, Number, Maximum of ELEMENTs
IFILEInput FILE
IDOFN, JDOFN, NODFNIndex, Index, Number of Degrees Of Freedom per Node
INODE, JNODE, NNODE, MNODEIndex, Index, Number, Maximum of NODEs per Element
ISVAB, JSVAB, MSVAB, NSVABIndex, Index, Maximum, Number of global Structural VAriaBles
JFILEOutput file
KRESLEquation resolution index
LNODS (MELEM, MNODE)ELEMENT NODE numberS listed for each element
NODEINODE I
NODEJNODE J
NCOLSNumber of the COLumn in the global Structural stiffness matrix
NROWSNumber of the ROW in the global Structural stiffness matrix and load vector
+ + + +# NCOLE + +# NROWE + +# MEVAB + +Number of the COLumn in the + +Element stiffness matrix + +Number of the ROW in the Element + +stiffness matrix and load vector + +Maximum of Element VAriaBles + +```fortran +SUBROUTINE ASSEMB ASEM 1 +C******************************* +ASEM 2 +C ASEM 3 +C *** ELEMENT ASSEMBLY ROUTINE ASEM 4 +C ASEM 5 +C******************************* +ASEM 6 +COMMON/UNIM1/NPOIN, NELEM, NBOUN, NLOAD, NPROP, NNODE, IINCS, IITER, ASEM 7 +KRESL, NCHEK, TOLER, NALGO, NSVAB, NDOFN, NINCS, NEVAB, ASEM 8 +NITER, NOUTP, FACTO, PVALU ASEM 9 +COMMON/UNIM2/PROPS(5,4), COORD(26), LNODS(25,2), IFPRE(52), ASEM 10 +FIXED(52), TLOAD(25,4), RLOAD(25,4), ELOAD(25,4), ASEM 11 +MATNO(25), STRES(25,2), PLAST(25), XDISP(52), ASEM 12 +TDISP(26,2), TREAC(26,2), ASTIF(52,52), ASLOD(52), ASEM 13 +REACT(52), FRESV(1352), PEFIX(52), ESTIF(4,4) ASEM 14 +ASEM 15 +C ELEMENT ASSEMBLY ROUTINE ASEM 16 +C ASEM 17 +REWIND 1 ASEM 18 +DO 10 ISVAB=1, NSVAB ASEM 19 +10 ASLOD(ISVAB)=0.0 ASEM 20 +IF(KRESL.EQ.2) GO TO 30 ASEM 21 +DO 20 ISVAB=1, NSVAB ASEM 22 +DO 20 JSVAB=1, NSVAB ASEM 23 +20 ASTIF(ISVAB, JSVAB)=0.0 ASEM 24 +30 CONTINUE ASEM 25 +C ASEM 26 +C ASSEMBLE THE ELEMENT LOADS ASEM 27 +C ASEM 28 +DO 50 IELEM=1, NELEM ASEM 29 +READ(1) ESTIF ASEM 30 +DO 40 INODE=1, NNODE ASEM 31 +NODEI=LNODS(IELEM, INODE) ASEM 32 +DO 40 IDOFN=1, NDOFN ASEM 33 +NROWS=(NODEI-1)*NDOFN + IDOFN ASEM 34 +NROWE=(INODE-1)*NDOFN + IDOFN ASEM 35 +ASLOD(NROWS)=ASLOD(NROWS) + ELOAD(IELEM, NROWE) ASEM 36 +C ASEM 37 +C ASSEMBLE THE ELEMENT STIFFNESS MATRICES ASEM 38 +C ASEM 39 +IF(KRESL.EQ.2) GO TO 40 ASEM 40 +DO 40 JNODE = 1, NNODE ASEM 41 +NODEJ=LNODS(IELEM, JNODE) ASEM 42 +DO 40 JDOFN =1, NDOFN ASEM 43 +NCOLS=(NODEJ-1)*NDOFN + JDOFN ASEM 44 +NCOLE=(JNODE-1)*NDOFN + JDOFN ASEM 45 +ASTIF(NROWS, NCOLS)=ASTIF(NROWS, NCOLS) + ESTIF(NROWE, NCOLE) ASEM 46 +40 CONTINUE ASEM 47 +50 CONTINUE ASEM 48 +RETURN ASEM 49 +END ASEM 50 +``` + +ASEM 18 Rewind file ready for reading the individual element stiffness matrices. + +ASEM 19–20 Set the global load vector, ASLOD, to zero. diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_007.md b/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_007.md new file mode 100644 index 00000000..75b64eb9 --- /dev/null +++ b/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_007.md @@ -0,0 +1,442 @@ + + +
ASEM 21-25If only equation resolution is to be performed during this iteration, do not set the global stiffness coefficients to zero.
ASEM 29Loop for each element.
ASEM 30Read ESTIF for the current element.
ASEM 31Loop for each node ‘INODE’ of current element.
ASEM 32From LNODS array identify node number of current node ‘INODE’.
ASEM 33Loop for each degree of freedom of the current node ‘INODE’.
ASEM 34Establish the row position in the global stiffness matrix and load vector.
ASEM 35Establish the row position in the element stiffness matrix and load vector.
ASEM 36Add the contribution to the global load vector from the element load vector.
ASEM 40If equation resolution is to be performed, avoid assembling the global stiffness matrix.
ASEM 41Loop for each node ‘JNODE’ of the current element.
ASEM 42From LNODS array identify node number of current node ‘JNODE’.
ASEM 43Loop for each degree of freedom of the current node ‘JNODE’.
ASEM 44Establish the column position in the global stiffness matrix.
ASEM 45Establish the column position in the element stiffness matrix.
ASEM 46Add the contribution to the global stiffness matrix from the element stiffness matrix.
ASEM 48End element loop.
+ +For the problem described in Section 3.4.1, the main variables have the following values + +$$ +\mathrm{NNODE} = 2, \text { NELEM } = 4, \text { NDOFN } = 1, \text { NSVAB } = 5, +$$ + +$$ +\text { LNODS } = \left[ \begin{array}{l l} 1 & 3 \\ 2 & 3 \\ 3 & 4 \\ 4 & 5 \end{array} \right] \begin{array}{l} - \text { Element I } \\ - \text { Element II } \\ - \text { Element III } \\ - \text { Element IV }. \end{array} +$$ + +# 3.4.3 Subroutine GREDUC + +This subroutine undertakes the equation elimination process for equation solution by Gaussian reduction as outlined in Section 3.4.1. The additional variable names employed are defined below. + +# Dictionary of variable names + +ASLOD (MEQNS) + +ASTIF (MEQNS, MEQNS) + +ASembled LOaD vector. + +Assembled global STIFFness matrix. + + + +IEQNS, NEQNS, MEQNS + +IFPRE (MEQNS) + +FIXED (MEQNS) + +ICOLS + +IROWS + +FACTR + +FRESV ( ) + +PIVOT + +Index, Number, Maximum of EQuatioNS. + +Vector of parameters defining the fixity of a node. 0 - free; 1 - fixed. + +Vector of prescribed displacements (zero if not prescribed). + +Index COLumn of Structural stiffness matrix. + +Index ROW of Structural stiffness matrix. + +Gaussian reduction FACToR. + +Stored Gaussian reduction factors. + +Diagonal term of variable which is currently being eliminated. + +```csv +SUBROUTINE GREDUC GRED 1 +C**************************GRED 2 +C GRED 3 +C *** GAUSSIAN REDUCTION ROUTINE GRED 4 +C GRED 5 +C**************************GRED 6 +COMMON/UNIM1/NPOIN,NELEM,NBOUN,NLOAD,NPROP,NNODE,IINCS,IITER, GRED 7 +. KRESL,NCHEK,TOLER,NALGO,NSVAB,NDOFN,NINCS,NEVAB, GRED 8 +. NITER,NOUTP,FACTO,PVALU GRED 9 +COMMON/UNIM2/PROPS(5,4),COORD(26),LNODS(25,2),IFPRE(52), GRED 10 +. FIXED(52),TLOAD(25,4),RLOAD(25,4),ELOAD(25,4), GRED 11 +. MATNO(25),STRES(25,2),PLAST(25),XDISP(52), GRED 12 +. TDISP(26,2),TREAC(26,2),ASTIF(52,52),ASLOD(52), GRED 13 +. REACT(52),FRESV(1352),PEFIX(52),ESTIF(4,4) GRED 14 +C GRED 15 +C GAUSSIAN REDUCTION ROUTINE GRED 16 +C GRED 17 +KOUNT=0 GRED 18 +NEQNS=NSVAB GRED 19 +DO 70 IEQNS=1,NEQNS GRED 20 +IF(IFPRE(IEQNS).EQ.1) GO TO 40 GRED 21 +C GRED 22 +C REDUCE EQUATIONS GRED 23 +C GRED 24 +PIVOT=ASTIF(IEQNS,IEQNS) GRED 25 +IF(ABS(PIVOT).LT.1.0E-10) GO TO 60 GRED 26 +IF(IEQNS.EQ.NEQNS) GO TO 70 GRED 27 +IEQN1=IEQNS+1 GRED 28 +DO 30 IROWS=IEQN1,NEQNS GRED 29 +KOUNT=KOUNT+1 GRED 30 +FACTR=ASTIF(IROWS.IEQNS)/PIVOT GRED 31 +FRESV(KOUNT)=FACTR GRED 32 +IF(FACTR.EQ.0.0) GO TO 30 GRED 33 +DO 10 ICOLS=IEQNS,NEQNS GRED 34 +ASTIF(IROWS,ICOLS)=ASTIF(IROWS,ICOLS)-FACTR*ASTIF(IEQNS,ICOLS) GRED 35 +10 CONTINUE GRED 36 +ASLOD(IROWS)=ASLOD(IROWS)-FACTR*ASLOD(IEQNS) GRED 37 +30 CONTINUE GRED 38 +GO TO 70 GRED 39 +C GRED 40 +C ADJUST RHS(LOADS) FOR PRESCRIBED DISPLACEMENTS GRED 41 +``` + + + +
CGRED42
40DO 50 IROWS=IEQNS,NEQNSGRED43
ASLOD(IROWS)=ASLOD(IROWS)-ASTIF(IROWS,IEQNS)*FIXED(IEQNS)GRED44
50CONTINUEGRED45
GO TO 70GRED46
60WRITE(6,900)GRED47
900FORMAT(5X,15HINCORRECT PIVOT)GRED48
STOPGRED49
70CONTINUEGRED50
RETURNGRED51
ENDGRED52
+ +GRED 18 Set the counter over the Gaussian reduction factorisation terms to zero. +GRED 19 Set the number of equations to be solved equal to the total number of variables in the structure, NSVAB. +GRED 20 Loop for each equation—this equation is associated with the variable about to be eliminated. +GRED 21 If this variable is fixed, skip to 40. +GRED 25 Extract PIVOT—the leading diagonal term. +GRED 26 Check for zero PIVOT in which case write a message and stop the program. +GRED 27–38 Alter equations below equation ‘IEQNS’, not those above, according to (3.12). Note that the Gaussian factorisation terms are stored for use during equation resolution. +GRED 43–45 For prescribed variables adjust the R.H.S. (or load) terms according to (3.13). +GRED 47-49 For an invalid pivot value, write a message and terminate execution of the program. + +For the problem considered in Section 3.4.1 the main variables have the following values: + +$$ +\mathrm{NEQNS} = 5, \quad \mathrm{ASLOD} = \left[ \begin{array}{l} 1 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array} \right], \quad \text { modified ASLOD } = \left[ \begin{array}{l} 1 0 \\ - 4 \\ 1 4 \\ 4 2 / 5 \\ 8 4 / 1 3 \end{array} \right] +$$ + +$$ +\mathbf {A S T I F} = \left[ \begin{array}{r r r r r} 1 & 0 & - 1 & 0 & 0 \\ 0 & 2 & - 2 & 0 & 0 \\ - 1 & - 2 & 6 & - 3 & 0 \\ 0 & 0 & - 3 & 7 & - 4 \\ 0 & 0 & 0 & - 4 & 4 \end{array} \right], \quad \text {modified} \quad \mathbf {A S T I F} = \left[ \begin{array}{r r r r r} 1 & 0 & - 1 & 0 & 0 \\ 0 & 0 & - 2 & 0 & 0 \\ 0 & 0 & 5 & - 3 & 1 \\ 0 & 0 & 0 & 2 6 / 5 & - 4 \\ 0 & 0 & 0 & 0 & 1 2 / 1 3 \end{array} \right] +$$ + + + +$$ +\text { IFPRE } = \left[ \begin{array}{l} 0 \\ 1 \\ 0 \\ 0 \\ 1 \end{array} \right], \quad \text { FIXED } = \left[ \begin{array}{l} 0 \\ 2 \\ 0 \\ 0 \\ 0 \end{array} \right]. +$$ + +The computational effort in this reduction process is proportional to $n^{3}$ . This can be approximately halved if we take advantage of the symmetry of the stiffness matrices. + +# 3.4.4 Subroutine BAKSUB + +The object of this subroutine is to perform the back substitution process required after equation elimination by Gaussian reduction. This results in sequential solution for all the unknowns and reactions at nodal points at which values of the unknown have been prescribed. In the nonlinear solution processes described in Chapter 2, the values of the unknown determined during any iteration may or may not be the total values depending on the solution algorithm being employed. If the method of direct iteration is being used, then, according to equation (2.3), the value of $\varphi$ determined during any iteration is the total value. For all other solution techniques considered the total values of the unknown are accumulated according to the corrections determined during each iteration, as indicated for example by (2.12). + +Therefore, for the direct iteration process, it is simply necessary to transfer the calculated values of the unknowns and the reactions to the arrays TDISP (ISVAB, IDOFN) and TREAC (ISVAB, IDOFN) for output later. This transfer is only necessary to allow the same subroutine to be employed for output of results for all four programs. + +Subroutine BAKSUB will now be presented in a form suitable for nonlinear solution dy direct iteration. + +Dictionary of variable names +
ASLOD (MEQNS)Reduced load vector.
ASTIF (MEQNS, MEQNS)Reduced global stiffness matrix.
IEQNS, NEQNS, MEQNSIndex, Number, Maximum of EQatioNS.
IFPRE (MEQNS)Vector of parameters defining the fixing of a node. 0 – free; 1 – fixed.
FIXED (MEQNS)Vector of prescribed displacements (zero if not prescribed).
PIVOTDiagonal term of variable currently being evaluated.
REACT (MEQNS)REACTIONS at nodes with prescribed displacements.
XDISP (MEQNS)Displacement at nodes.
+ + + +```fortran +SUBROUTINE BAKSUB +BAKS 1 +C***** +BAKS 2 +C +BAKS 3 +C *** BACK-SUBSTITUTION ROUTINE +BAKS 4 +C +BAKS 5 +C***** +BAKS 6 +COMMON/UNIM1/NPOIN.NELEM,NBOUN,NLOAD,NPROP,NNODE,IINCS,IITER, +BAKS 7 +KRESL,NCHEK,TOLER,NALGO,NSVAB,NDOFN,NINCS,NEVAB, +BAKS 8 +NITER,NOUTP,FACTO,PVALU +BAKS 9 +COMMON/UNIM2/PROPS(5,4),COORD(26),LNODS(25,2),IFPRE(52), +BAKS 10 +FIXED(52),TLOAD(25,4),RLOAD(25,4),ELOAD(25,4), +BAKS 11 +MATNO(25),STRES(25,2),PLAST(25),XDISP(52), +BAKS 12 +TDISP(26,2),TREAC(26,2),ASTIF(52,52),ASLOD(52), +BAKS 13 +REACT(52),FRESV(1352),PEFIX(52),ESTIF(4,4) +BAKS 14 +BAKS 15 +C +BACK-SUBSTITUTION ROUTINE +BAKS 16 +C +BAKS 17 +NEQNS=NSVAB +BAKS 18 +DO 10 IEQNS=1,NEQNS +BAKS 19 +REACT(IEQNS)=0.0 +BAKS 20 +10 CONTINUE +BAKS 21 +NEQN1=NEQNS+1 +BAKS 22 +DO 40 IEQNS=1,NEQNS +BAKS 23 +NBACK=NEQN1-IEQNS +BAKS 24 +PIVOT=ASTIF(NBACK,NBACK) +BAKS 25 +RESID=ASLOD(NBACK) +BAKS 26 +IF(NBACK.EQ.NEQNS) GO TO 30 +BAKS 27 +NBAC1=NBACK+1 +BAKS 28 +DO 20 ICOLS=NBAC1,NEQNS +BAKS 29 +RESID=RESID-ASTIF(NBACK,ICOLS)*XDISP(ICOLS) +BAKS 30 +20 CONTINUE +BAKS 31 +30 IF(IFPRE(NBACK).EQ.0) XDISP(NBACK)=RESID/PIVOT +BAKS 32 +IF(IFPRE(NBACK).EQ.1) XDISP(NBACK)=FIXED(NBACK) +BAKS 33 +IF(IFPRE(NBACK).EQ.1) REACT(NBACK)=-RESID +BAKS 34 +40 CONTINUE +BAKS 35 +KOUNT=0 +BAKS 36 +DO 50 IPOIN=1,NPOIN +BAKS 37 +DO 50 IDOFN=1,NDOFN +BAKS 38 +KOUNT=KOUNT+1 +BAKS 39 +TDISP(IPOIN,IDOFN)= XDISP(KOUNT) +BAKS 40 +50 TREAC(IPOIN,IDOFN)= REACT(KOUNT) +BAKS 41 +RETURN +BAKS 42 +END +BAKS 43 +``` + +BAKS 19-21 Zero space for reactions. + +BAKS 22–24 Loop backwards over each equation. + +BAKS 25 Use the same PIVOT as in subroutine GREDUC. + +BAKS 27 For the last equation (the first to be solved) we do not have any other variables to substitute (i.e. bypass the loop). + +BAKS 28–31 Evaluate RESID from previously calculated variables. + +BAKS 32 If the variable is not prescribed evaluate the variable. + +BAKS 34 If the variable is prescribed evaluate the R.H.S. reaction. + +BAKS 36–41 Store the solved variables and reactions in new arrays for output. + +For the problem described in Section 3.4.1, the arrays employed in addition to those utilised in Subroutine GREDUC have the following values: + + + +$$ +\mathrm{TDISP} = \mathrm{XDISP} = \left[ \begin{array}{c} 1 7 9 / 1 3 \\ 2 \\ 4 9 / 1 3 \\ 2 1 / 1 3 \\ 0 \end{array} \right], \quad \mathrm{TREAC} = \mathrm{REACT} = \left[ \begin{array}{c} 0 \\ - 4 6 / 1 3 \\ 0 \\ 0 \\ - 8 4 / 1 3 \end{array} \right]. +$$ + +It should be noted that nonzero reactions are obtained only for nodal positions at which the value of the unknown has been prescribed. For the Newton–Raphson, Tangential Stiffness and Initial Stiffness methods, the calculated unknowns and reactions must be accumulated from the values obtained during each iteration. Therefore, for these applications, statements BAKS 36–41 in the above listing must be replaced by + +$$ +\begin{array}{l l l l} \text {KOUNT = 0} & \text {BAKS} & 3 6 \\ \text {DO 50 IPOIN = 1,NPOIN} & \text {BAKS} & 3 7 \\ \text {DO 50 IDOFN = 1,NDOFN} & \text {BAKS} & 3 8 \\ \text {KOUNT = KOUNT + 1} & \text {BAKS} & 3 9 \\ \text {TDISP(IPOIN, IDOFN) = TDISP(IPOIN, IDOFN) + XDISP(KOUNT)} & \text {BAKS} & 4 0 \\ 5 0 \text {TREAC(IPOIN, IDOFN) = TREAC(IPOIN, IDOFN) + REACT(KOUNT)} & \text {BAKS} & 4 1 \end{array} +$$ + +with the arrays TDISP and TREAC being initially set to zero at the beginning of the program. + +For these three solution algorithms a final further programming addition must be made. When determining the residual forces according to (2.4), the contribution to f of the reactions at nodal points at which the value of the unknown is prescribed must be accounted for, since any reactions can be interpreted as additional applied loads necessary to maintain the prescribed value of the unknown. Therefore, the evaluated reactions must be added into the vector of applied nodal loads at every iteration. This task can be accomplished by the following coding inserted immediately before the RETURN statement: + +$$ +\begin{array}{l l} \text {DO 90 IPOIN = 1,NPOIN} & \text {BAKS 42} \\ \text {DO 60 IELEM = 1,NELEM} & \text {BAKS 43} \\ \text {DO 60 INODE = 1,NNODE} & \text {BAKS 44} \\ \text {NLOCA = LNODS(IELEM,INODE)} & \text {BAKS 45} \\ \text {60 IF(IPOIN.EQ.NLOCA) GO TO 70} & \text {BAKS 46} \\ \text {70 DO 80 IDOFN = 1,NDOFN} & \text {BAKS 47} \\ \text {NPOSN = (IPOIN - 1)*NDOFN+IDOFN} & \text {BAKS 48} \\ \text {IEVAB = (INODE - 1)*NDOFN+IDOFN} & \text {BAKS 49} \\ \text {80 TLOAD(IELEM,IEVAB) = TLOAD(IELEM,IEVAB) + REACT(NPOSN)} & \text {BAKS 50} \\ \text {90 CONTINUE} & \text {BAKS 51} \end{array} +$$ + +BAKS 42 Loop over each nodal point. + +BAKS 43–46 Search through the element nodal connections until one is found corresponding to the nodal point currently under consideration. As soon as one is found, abandon the search. Note that it is immaterial in which element the node is found since all element contributions will be finally assembled. + + + +BAKS 47–50 Add the nodal reaction into the appropriate position in the array of applied element loads. + +# 3.4.5 Subroutine RESOLV + +As stated in Section 3.4.1, for equation resolution (indicated by KRESL = 2) only the global load vector need be formed and reduced. Subroutine RESOLV merely reduces the R.H.S. (or load) terms by standard Gaussian elimination using the same operations as employed in Subroutine GREDUC, Section 3.4.3. The Gaussian factorisation terms were evaluated and stored in GREDUC and are now utilised in this subroutine. The programming logic follows that of Subroutine GREDUC and can be readily understood by reference to Section 3.4.3. + +```csv +SUBROUTINE RESOLV RSLV 1 +C************************(RSLV 2 +C RSLV 3 +C *** RESOLVING GAUSSIAN REDUCTION ROUTINE RSLV 4 +C RSLV 5 +C************************(RSLV 6 +COMMON/UNIM1/NPOIN,NELEM,NBOUN,NLOAD,NPROP,NNODE,IINCS,IITER, RSLV 7 +KRESL,NCHEK,TOLER,NALGO,NSVAB,NDOFN,NINCS,NEVAB, RSLV 8 +NITER,NOUTP,FACTO,PVALU RSLV 9 +COMMON/UNIM2/PROPS(5,4),COORD(26),LNODS(25,2),IFPRE(52), RSLV 10 +FIXED(52),TLOAD(25,4),RLOAD(25,4),ELOAD(25,4), RSLV 11 +MATNO(25),STRES(25,2),PLAST(25),XDISP(52), RSLV 12 +TDISP(26,2),TREAC(26,2),ASTIF(52,52),ASLOD(52), RSLV 13 +REACT(52),FRESV(1352),PEFIX(52),ESTIF(4,4) RSLV 14 +KOUNT=0 RSLV 15 +NEQNS=NSVAB RSLV 16 +DO 40 IEQNS=1,NEQNS RSLV 17 +IF(IFPRE(IEQNS).EQ.1) GO TO 20 RSLV 18 +C RSLV 19 +C REDUCE RHS RSLV 20 +C RSLV 21 +IF(IEQNS.EQ.NEQNS) GO TO 40 RSLV 22 +IEQN1=IEQNS+1 RSLV 23 +DO 10 IROWS=IEQN1,NEQNS RSLV 24 +KOUNT=KOUNT+1 RSLV 25 +FACTR=FRESV(KOUNT) RSLV 26 +IF(FACTR.EQ.0) GO TO 10 RSLV 27 +ASLOD(IROWS)=ASLOD(IROWS)-FACTR*ASLOD(IEQNS) RSLV 28 +10 CONTINUE RSLV 29 +GO TO 40 RSLV 30 +C RSLV 31 +C ADJUST RHS TO PRESCRIBED DISPLACEMENTS RSLV 32 +C RSLV 33 +20 DO 30 IROWS=IEQNS,NEQNS RSLV 34 +ASLOD(IROWS)=ASLOD(IROWS)-ASTIF(IROWS,IEQNS)*FIXED(IEQNS) RSLV 35 +30 CONTINUE RSLV 36 +40 CONTINUE RSLV 37 +RETURN RSLV 38 +END RSLV 39 +``` + + + +# 3.4.6 Improved numerical algorithm for equation solution + +Substantial economies can be achieved in both core storage requirements and execution times if advantage is taken of the banded symmetric form of the global stiffness matrix. Since: + +- By recognising that the global stiffness matrix is symmetric, it is necessary only to store the upper (or lower) triangular part of the stiffness matrix. +- By noting that all the non-zero coefficients in the global stiffness matrix occur in a band adjacent to the leading diagonal, further reductions in the core storage requirements can be made, as well as a significant reduction in the number of arithmetic operations undertaken in the equation reduction and backsubstitution phases. + +In order to introduce these enhancements it is convenient to store the global stiffness matrix as a one-dimensional array. The necessary programming changes required to the subroutines presented in Sections 3.4.2–3.4.5 are fully documented in Ref. 5. + +# 3.5 Output of results + +The next subroutine common to all four programs presented is subroutine RESULT whose function is to output the results at a frequency governed by a parameter input in Subroutine INCLOD described in Section 3.7. In order to make the subroutine applicable to all four cases, quantities will be output for some situations which are physically meaningless. In particular for quasi-harmonic problems, output items termed stress and plastic or nonlinear strain are output as zero values for this reason. For nonlinear elastic problems the latter term is the total strain, $\epsilon$ , defined in Section 2.4 and for elasto-plastic situations it is the plastic strain component, $\epsilon_{p}$ , defined in Section 2.5. For both cases the stress quantity output is the axial stress existing in each constant stress element employed. + +Subroutine RESULT will now be listed. + +[Unreadable] + +```txt +SUBROUTINE RESULT RSLT 1 +C************************(RSLT 2 +C RSLT 3 +C *** OUTPUTS DISPLACEMENT, REACTIONS AND STRESSES RSLT 4 +C RSLT 5 +C************************(RSLT 6 +COMMON/UNIM1/NPOIN, NELEM, NBOUN, NLOAD, NPROP, NNODE, IINCS, IITER, RSLT 7 +. KRESL, NCHEK, TOLER, NALGO, NSVAB, NDOFN, NINCS, NEVAB, RSLT 8 +. NITER, NOUTP, FACTO, PVALU RSLT 9 +COMMON/UNIM2/PROPS(5,4), COORD(26), LNODS(25,2), IFPRE(52), RSLT 10 +. FIXED(52), TLOAD(25,4), RLOAD(25,4), ELOAD(25,4), RSLT 11 +. MATNO(25), STRES(25,2), PLAST(25), XDISP(52), RSLT 12 +. TDISP(26,2), TREAC(26,2), ASTIF(52,52), ASLOD(52), RSLT 13 +. REACT(52), FRESV(1352), PEFIX(52), ESTIF(4,4) RSLT 14 +IF(NDOFN.EQ.1) WRITE(6,900) RSLT 15 +``` + + + +900 FORMAT(1H0,5X,'NODE',4X,'DISPL.',12X,'REACTIONS') RSLT 16 +IF(NDOFN.EQ.2) WRITE(6.910) RSLT 17 +910 FORMAT(1H0,5X,'NODE',4X,'DISPL.',12X,'REACTION', RSLT 18 +. 7X,'DISPL.',12X,'REACTION') RSLT 19 +DO 10 IPOIN=1,NPOIN RSLT 20 +10 WRITE(6,920) IPOIN,(TDISP(IPOIN,IDOFN),TREAC(IPOIN,IDOFN), RSLT 21 +.IDOFN=1,NDOFN) RSLT 22 +920 FORMAT(I10,2(E14.6,5X,E14.6)) RSLT 23 +IF(NDOFN.EQ.2) WRITE(6.930) RSLT 24 +930 FORMAT(1H0,2X,'ELEMENT',12X,'STRESSES',12X,'PL.STRAIN') RSLT 25 +IF(NDOFN.EQ.1) WRITE(6.940) RSLT 26 +940 FORMAT(1H0,2X,'ELEMENT',5X,'STRESSES',5X,'PL.STRAIN') RSLT 27 +DO 20 IELEM=1,NELEM RSLT 28 +20 WRITE(6,950) IELEM,(STRES(IELEM,IDOFN),IDOFN=1,NDOFN), RSLT 29 +. PLAST(IELEM) RSLT 30 +950 FORMAT(I10,3E14.6) RSLT 31 +RETURN RSLT 32 +END RSLT 33 + +RSLT 15–23 Write titles and output the calculated unknown and reaction at each nodal point. Non-zero reactions are only obtained for nodal points at which the value of the unknown is prescribed. + +RSLT 24–31 Write titles and output the stress and plastic or nonlinear elastic strain for each element. + +Note that provision is made for output of results for the beam bending application of Chapter 5. + +# 3.6 Subroutine INITIAL + +The function of this subroutine is to initialise to zero some arrays used by other subroutines. + +SUBROUTINE INITIAL INTL 1 +C************************** INTL 2 +C INTL 3 +C *** INITIALIZES TO ZERO ALL ACCUMULATIVE ARRAYS INTL 4 +C INTL 5 +C************************** INTL 6 +COMMON/UNIM1/NPOIN, NELEM, NBOUN, NLOAD, NPROP, NNODE, IINCS, IITER, INTL 7 +. KRESL, NCHEK, TOLER, NALGO, NSVAB, NDOFN, NINCS, NEVAB, INTL 8 +. NITER, NOUTP, FACTO, PVALU INTL 9 +COMMON/UNIM2/PROPS(5,4), COORD(26), LNODS(25,2), IFPRE(52), INTL 10 +. FIXED(52), TLOAD(25,4), RLOAD(25,4), ELOAD(25,4), INTL 11 +. MATNO(25), STRES(25,2), PLAST(25), XDISP(52), INTL 12 +. TDISP(26,2), TREAC(26,2), ASTIF(52,52), ASLOD(52), INTL 13 +. REACT(52), FRESV(1352), PEFIX(52), ESTIF(4,4) INTL 14 +DO 20 IELEM=1, NELEM INTL 15 +PLAST(IELEM)=0.0 INTL 16 +DO 10 IDOFN=1, NDOFN INTL 17 + + + +
10STRES(IELEM, IDOFN)=0.0INTL18
DO 20 IEVAB=1,NEVABINTL19
ELOAD(IELEM, IEVAB)=0.0INTL20
20TLOAD(IELEM, IEVAB)=0.0INTL21
DO 30 IPOIN=1,NPOININTL22
DO 30 IDOFN=1,NDOFNINTL23
TDISP(IPOIN, IDOFN)=0.0INTL24
30TREAC(IPOIN, IDOFN)=0.0INTL25
RETURNINTL26
ENDINTL27
+ +INTL 15–18 Initialise to zero the plastic or nonlinear strain vector and the stress vector. + +INTL 20 Initialise the array, ELOAD, which will contain the out of balance loading to be applied in solution for any iteration. For techniques other than the direct iteration method, this vector will contain the residual nodal forces and thus differs from the vector of applied loads. + +INTL 21 Initialise the vector of applied loads. + +INTL 22–25 Initialise the vector of total unknowns and total reactions to zero. + +# 3.7 Load increment subroutine, INCLOD + +This subroutine controls the incrementing of the applied loads. For each increment of load, data is input to this segment to control the upper limit to the number of iterations, the output frequency, the size of load increment and the convergence tolerance limit. These quantities are specifically input as: + +NITER Maximum permissible number of iterations. This is a safety measure to cover situations where the solution process does not converge. After performing NITER iteration cycles the program will then stop. + +NOUTP This parameter controls the frequency of output of results. In order to examine the iterative procedure the user may wish to obtain results at stages other than the converged solution. +0 – Print the results on convergence to the nonlinear solution only, for each load increment. +1 – Print the results after the first iteration and after convergence for each load increment. +2 – Print the results after every iteration for each load increment. + +FACTO This quantity controls the magnitude of any load increment. The applied loading is input in subroutine DATA into the array RLOAD as described in Section 3.2. The size of any load increment is then defined to be FACTO\*RLOAD diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_008.md b/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_008.md new file mode 100644 index 00000000..39885057 --- /dev/null +++ b/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_008.md @@ -0,0 +1,449 @@ + + +(IELEM, INODE) with the increment size factor, FACTO, being input for each increment. This permits unequal load increments to be taken. It should be noted that the applied loading at any instant is accumulative. Therefore, if FACTO is input for the first three increments as respectively 0·5, 0·3 and 0·1, the total loading applied to the structure during the third increment is 0·9 times the loading input in subroutine DATA. The above also holds for loading by incremental prescribed displacements. + +TOLER This item of data controls the tolerance permitted on the convergence process. Its use will be described in detail in Sections 3.9.2 and 3.9.3. + +Subroutine INCLOD is now presented and described: + +SUBROUTINE INCLOD INCL 1 +C************************** INCL 2 +C INCL 3 +C *** INPUTS DATA FOR CURRENT INCREMENT AND UPDATES LOAD VECTOR INCL 4 +C INCL 5 +C************************** INCL 6 +COMMON/UNIM1/NPOIN,NELEM,NBOUN,NLOAD,NPROP,NNODE,IINCS,IITER, INCL 7 +KRESL,NCHEK,TOLER,NALGO,NSVAB,NDOFN,NINCS,NEVAB, INCL 8 +NITER,NOUTP,FACTO,PVALU INCL 9 +COMMON/UNIM2/PROPS(5,4),COORD(26),LNODS(25,2),IFPRE(52), INCL 10 +FIXED(52),TLOAD(25,4),RLOAD(25,4),ELOAD(25,4), INCL 11 +MATNO(25),STRES(25,2),PLAST(25),XDISP(52), INCL 12 +TDISP(26,2),TREAC(26,2),ASTIF(52,52),ASLOD(52), INCL 13 +REACT(52),FRESV(1352),PEFIX(52),ESTIF(4,4) INCL 14 +READ (5,900) NITER,NOUTP,FACTO,TOLER INCL 15 +900 FORMAT(2I5,2F15.5) INCL 16 +WRITE(6,905) IINCS,NITER,NOUTP,FACTO,TOLER INCL 17 +905 FORMAT(1H0,5X,'IINCS=',I5,3X,'NITER=',I5,3X,'NOUTP=',I5, INCL 18 +3X,'FACTO=',E14.6,3X,'TOLER=',E14.6) INCL 19 +DO 10 IELEM=1,NELEM INCL 20 +DO 10 IEVAB=1,NEVAB INCL 21 +ELOAD(IELEM,IEVAB)=ELOAD(IELEM,IEVAB)+RLOAD(IELEM,IEVAB)*FACTO INCL 22 +TLOAD(IELEM,IEVAB)=TLOAD(IELEM,IEVAB)+RLOAD(IELEM,IEVAB)*FACTO INCL 23 +10 CONTINUE INCL 24 +RETURN INCL 25 +END INCL 26 + +INCL 15–19 Read and write the input data required for each load increment as described previously in this section. + +INCL 20-24 Add the current increment of load into the out of balance load array ELOAD and the total applied load vector TLOAD. + +# 3.8 The master or controlling segment + +The final portion of the program which will be common to all four programs (subject to the minor differences indicated in Fig. 3.1) is the master segment which controls the calling, in order, of the other subroutines. This program segment also controls the iterative process and also the incrementing of the applied loads, where appropriate. + + + +The following channel numbers are employed by the programs: 5 (card reader), 6 (line printer), 1 (scratch file). + +The MASTER segment will now be presented in the form required in the next section for the solution of one-dimensional quasi-harmonic problems by direct iteration. For other applications it is only necessary to arrange for the calling of appropriate subroutines as indicated in Fig. 3.1. + +```csv +MASTER UNIDIM +C***** +C +C *** PROGRAM FOR THE 1-D SOLUTION OF NONLINEAR PROBLEMS +C +C***** +COMMON/UNIM1/NPOIN, NELEM, NBOUN, NLOAD, NPROP, NNODE, IINCS, IITER, +KRESL, NCHEK, TOLER, NALGO, NSVAB, NDOFN, NINCS, NEVAB, +NITER, NOUTP, FACTO, PVALU +COMMON/UNIM2/PROPS(5,4), COORD(26), LNODS(25,2), IFPRE(52), +FIXED(52), TLOAD(25,4), RLOAD(25,4), ELOAD(25,4), +MATNO(25), STRES(25,2), PLAST(25), XDISP(52), +TDISP(26,2), TREAC(26,2), ASTIF(52,52), ASLOD(52), +REACT(52), FRESV(1352), PEFIX(52), ESTIF(4,4) +CALL DATA +CALL INITIAL +DO 30 IINCS=1, NINCS +CALL INCLOD +DO 10 IITER=1, NITER +CALL NONAL +IF(KRESL.EQ.1) CALL STIFF1 +CALL ASSEMB +IF(KRESL.EQ.1) CALL GREDUC +IF(KRESL.EQ.2) CALL RESOLV +CALL BAKSUB +CALL MONITR(RINTL) +IF(NCHEK.EQ.0) GO TO 20 +IF(IITER.EQ.1. AND.NOUTP.EQ.1) CALL RESULT +IF(NOUTP.EQ.2) CALL RESULT +10 CONTINUE +WRITE(6,900) +900 FORMAT(1H0,5X,'SOLUTION NOT CONVERGED') +STOP +20 CALL RESULT +30 CONTINUE +STOP +END +QUIT 1 +QUIT 2 +QUIT 3 +QUIT 4 +QUIT 5 +QUIT 6 +QUIT 7 +QUIT 8 +QUIT 9 +QUIT 10 +QUIT 11 +QUIT 12 +QUIT 13 +QUIT 14 +QUIT 15 +QUIT 16 +QUIT 17 +QUIT 18 +QUIT 19 +QUIT 20 +QUIT 21 +QUIT 22 +QUIT 23 +QUIT 24 +QUIT 25 +QUIT 26 +QUIT 27 +QUIT 28 +QUIT 29 +QUIT 30 +QUIT 31 +QUIT 32 +QUIT 33 +QUIT 34 +QUIT 35 +QUIT 36 +QUIT 37 +``` + +QUIT 15 Call the subroutine which reads the input data as described in Section 3.2. + +QUIT 16 Call the subroutine which initialises various arrays to zero. + +QUIT 17 Enter the DO LOOP over the number of load increments. + +QUIT 18 Call the subroutine which increments the applied loads. + +QUIT 19 Enter the DO LOOP over the maximum permissible number of iterations. + +QUIT 20 Call the subroutine which controls the solution process as described in Section 3.3. + +QUIT 21 If the element stiffnesses are to be reformulated, call the appropriate subroutine. + + + +QUIT 22–25 Call the subroutines which assemble the element stiffnesses and solve for the unknowns and reactions. + +QUIT 26 Call the subroutine which monitors the convergence process. This subroutine differs for the direct iteration method from that for the three other cases. + +QUIT 27 If the solution has converged, abandon the iterative process. + +QUIT 28–29 Output the results according to the display code, NOUTP, supplied as input for this particular load increment. + +QUIT 31–33 If the solution procedure reaches the maximum number of iterations permitted without convergence occurring, write a message and stop the program. + +QUIT 34 Otherwise output the converged results. + +QUIT 35 Return to process the next increment of load. + +# 3.9 Program for the solution of one-dimensional quasi-harmonic problems by direct iteration + +We now assemble a computer program which permits the solution of one-dimensional problems governed by a nonlinear quasi-harmonic equation. The behaviour of several physical situations can be described by such a model and some numerical examples will be provided at the end of this section. + +Most of the subroutines required for this program have been already described in the preceding sections of this chapter and, in particular, the master segment which controls the entire numerical process was described in Section 3.8. The additional subroutines, pertinent only to this application which must be developed, are the element stiffness generation subroutine, STIFF1, and the solution convergence monitoring subroutine, MONITR. Detailed ‘user instructions’, listing the required input data, are included in Appendix I. + +# 3.9.1 Element stiffness subroutine, STIFF1 + +The purpose of this subroutine is to formulate the stiffness matrix for each element in turn and store this data on a disc file. For solution by the method of direct iteration, the stiffness matrix for a one-dimensional element with a linear variation of the unknown is given by equation (2.25). The term K is, however, a specified function of the unknown or its derivatives which must be accounted for when formulating the element stiffnesses for each iteration of the solution sequence. In particular, K is assumed to vary according to + +$$ +K = K _ {0} f \left(\phi , \frac {d \phi}{d x}\right), \tag {3.20} +$$ + +where $K_{0}$ is a reference value of K and is specified as material property PROPS (NUMAT, 1) in subroutine DATA. The function $f(\phi, d\phi/dx)$ is + + + +defined by means of a FORTRAN FUNCTION statement and must be appropriately specified for each application. + +Subroutine STIFFI is now presented and descriptive notes provided. + +```fortran +SUBROUTINE STIFF1 STF1 1 +C************************** STF1 2 +C STF1 3 +C *** CALCULATES ELEMENT STIFFNESS MATRICES STF1 4 +C STF1 5 +C************************** STF1 6 +COMMON/UNIM1/NPOIN, NELEM, NBOUN, NLOAD, NPROP, NNODE, IINCS, IITER, STF1 7 +KRESL, NCHEK, TOLER, NALGO, NSVAB, NDOFN, NINCS, NEVAB, STF1 8 +NITER, NOUTP, FACTO, PVALU STF1 9 +COMMON/UNIM2/PROPS(5,4), COORD(26), LNODS(25,2), IFPRE(52), STF1 10 +FIXED(52), TLOAD(25,4), RLOAD(25,4), ELOAD(25,4), STF1 11 +MATNO(25), STRES(25,2), PLAST(25), XDISP(52), STF1 12 +TDISP(26,2), TREAC(26,2), ASTIF(52,52), ASLOD(52), STF1 13 +REACT(52), FRESV(1352), PEFIX(52), ESTIF(4,4) STF1 14 +REWIND 1 STF1 15 +DO 10 Ielem=1, NELEM STF1 16 +LPROP=MATNO(IELEM) STF1 17 +STERM=PROPS(LPROP,1) STF1 18 +NODE1=LNODS(IELEM,1) STF1 19 +NODE2=LNODS(IELEM,2) STF1 20 +ELENG=ABS(COORD(NODE1)-COORD(NODE2)) STF1 21 +AVERG=(TDISP(NODE1,1)+TDISP(NODE2,1))/2.0 STF1 22 +FMULT=STERM*VARIA(AVERG)/ELENG STF1 23 +ESTIF(1,1)=FMULT STF1 24 +ESTIF(1,2)=-FMULT STF1 25 +ESTIF(2,1)=-FMULT STF1 26 +ESTIF(2,2)=FMULT STF1 27 +WRITE(1) ESTIF STF1 28 +10 CONTINUE STF1 29 +RETURN STF1 30 +END STF1 31 +``` + +STF1 15 Rewind the file on which the stiffness matrix for each element will be stored in sequence. + +STF1 16 Loop over each element. + +STF1 17 Identify the material property of each element. + +STF1 18 Set STERM equal to $K_{0}$ . + +STF1 19-20 Identify the node numbers of the element. + +STF1 21 Calculate the element length. + +STF1 22 Calculate the element temperature as the average of the nodal values. + +STF1 23 Calculate the temperature gradient. + +STF1 24–27 Compute the components of the element stiffness matrix according to (2.25) with the function $f(\phi, d\phi/dx)$ being VARIA (AVERG). + +STF1 28 Write the element stiffness matrix on to disc file. + +STF1 29 Termination of DO LOOP over each element. + +The function $f(\phi, d\phi/dx)$ must be defined for each application. Below we show, for example, the appropriate function for the variation $K = K_{0}(1 + 10\phi)$ . + + + +FUNCTION VARIA(AVERG) STF1 32 +C**** STF1 33 +C MULTIPLYING FUNCTION FOR QUASI-HARMONIC STIFFNESS VARIATION STF1 34 +C**** STF1 35 +VARIA=1.0+10.0*AVERG STF1 36 +RETURN STF1 37 +END STF1 38 + +# 3.9.2 Solution convergence monitoring subroutine, MONITR + +Convergence of the numerical process to the nonlinear solution must be monitored by comparing, in some way, the values of the unknowns $\varphi$ determined during each iteration. One possible method is to compare each individual nodal value with the corresponding value obtained on the previous iteration. Then, provided that this change is negligibly small for all nodal points, convergence can be deemed to have occurred. In this chapter we will employ a global convergence check rather than such a local one. We will assume that the numerical process has converged if + +$$ +\frac {\left| \sqrt {\left[ \sum_ {i = 1} ^ {N} \left(\phi_ {i} ^ {r}\right) ^ {2} \right]} - \sqrt {\left[ \sum_ {i = 1} ^ {N} \left(\phi_ {i} ^ {r - 1}\right) ^ {2} \right]} \right|}{\sqrt {\left[ \sum_ {i = 1} ^ {N} \left(\phi_ {i} ^ {1}\right) ^ {2} \right]}} \times 1 0 0 \leqslant \text { TOLER }, \tag {3.21} +$$ + +where N denotes the total number of nodal points in the problem and r-1 and r denote successive iterations. It is assumed that the positive root is always considered and $\mid\mid$ signifies the absolute value of the numerator. The multiplication factor of 100 on the left-hand side allows the specified tolerance factor TOLER to be considered as a percentage term. Equation (3.21) states that convergence is assumed to have occurred if the difference in the norm of the unknowns between two successive iterations is less than or equal to TOLER times the norm of the unknowns on the first iteration. In practical situations a value of TOLER = 1·0 (i.e., 1%) is found to be adequate for the majority of applications. Convergence of the solution is indicated by the parameter NCHEK. A value of NCHEK = 1 indicates that convergence has not yet occurred, whereas NCHEK = 0, denotes a converged solution. Subroutine MONITR is now presented and descriptive notes provided. + +SUBROUTINE MONITR (RINTL) MNTR 1 +C*******************************MNTR 2 +C MNTR 3 +C *** CHECKS FOR SOLUTION CONVERGENCE MNTR 4 +C MNTR 5 +C*******************************MNTR 6 +COMMON/UNIM1/NPOIN, NELEM, NBOUN, NLOAD, NPROP, NNODE, IINCS, IITER, MNTR 7 +KRESL, NCHEK, TOLER, NALGO, NSVAB, NDOFN, NINCS, NEVAB, MNTR 8 +NITER, NOUTP, FACTO, PVALU MNTR 9 +COMMON/UNIM2/PROPS(5,4), COORD(26), LNODS(25,2), IFPRE(52), MNTR 10 +FIXED(52), TLOAD(25,4), RLOAD(25,4), ELOAD(25,4), MNTR 11 + + + +MATNO(25),STRES(25,2),PLAST(25),XDISP(52), MNTR 12 +TDISP(26,2),TREAC(26,2),ASTIF(52,52),ASLOD(52), MNTR 13 +REACT(52),FRESV(1352),PEFIX(52),ESTIF(4,4) MNTR 14 +NCHEK=0 MNTR 15 +RCURR=0.0 MNTR 16 +DO 10 IPOIN=1,NPOIN MNTR 17 +10 RCURR=RCURR+TDISP(IPOIN,1)*TDISP(IPOIN,1) MNTR 18 +IF(IITER.EQ.1) RINTL=RCURR MNTR 19 +IF(IITER.EQ.1) NCHEK=1 MNTR 20 +IF(IITER.EQ.1) GO TO 20 MNTR 21 +RATIO=100.0*SQRT(ABS(RCURR-PVALU))/SQRT(RINTL) MNTR 22 +IF(RATIO.GT.TOLER) NCHEK=1 MNTR 23 +20 PVALU=RCURR MNTR 24 +WRITE(6,900) NCHEK,RATIO MNTR 25 +900 FORMAT(1H0,5X,18HCONVERGENCE CODE =,I4,3X,28HNORM OF RESIDUAL SUM MNTR 26 +.RATIO =,E14.6) MNTR 27 +RETURN MNTR 28 +END MNTR 29 + +MNTR 15 Set the indicator monitoring convergence to zero. If convergence has not yet occurred this will be set to 1 later in the subroutine. + +MNTR 16–18 Compute the norm of the unknowns + +$$ +\sum_ {i = 1} ^ {N} \phi_ {i} ^ {2}, +$$ + +for the current iteration. + +MNTR 19 For the first iteration only compute the denominator of (3.21). + +MNTR 20-21 Convergence cannot possibly have occurred on the first iteration, therefore set NCHEK = 1 and skip the remainder of the checking procedure by going to 20. + +MNTR 22 Compute the left-hand side of (3.21). + +MNTR 23 If (3.21) is not satisfied (i.e., convergence not taken place), set NCHEK = 1. + +MNTR 24 Store the current value of the norm of the unknowns for use as + +$$ +\sum_ {i = 1} ^ {N} (\phi_ {i} r ^ {- 1}) ^ {2} +$$ + +during the next iteration. + +MNTR 25-27 Output the value of NCHEK and the left-hand side of (3.21). + +# 3.9.3 Numerical examples + +The first numerical example considered is illustrated in Fig. 3.3. The situation shown could physically represent the diffusion of a gas through a membrane in which case $\phi$ is the gas concentration and K is the diffusivity of the membrane. Alternatively, the problem also represents the conduction of heat through a one-dimensional solid in which case $\phi$ is the temperature and K the thermal conductivity. The boundary conditions assumed are + + + +![](images/page-077_f02046f242e0154e2d52fbd49794a8baa0a1de8a70a651bb48d7ac6fb386c5f0.jpg) + +
+line + +| Distance, x | Theoretical solution (φ) | Finite elements (φ) | +| ----------- | ------------------------ | ------------------- | +| 10 | 0.0 | 0.0 | +| 9 | 0.25 | 0.25 | +| 8 | 0.4 | 0.4 | +| 7 | 0.5 | 0.5 | +| 6 | 0.6 | 0.6 | +| 5 | 0.7 | 0.7 | +| 4 | 0.8 | 0.8 | +| 3 | 0.9 | 0.9 | +| 2 | 0.95 | 0.95 | +| 1 | 0.98 | 0.98 | +| 0 | 1.0 | 1.0 | +
+ +Fig. 3.3 Quasi-harmonic equation example—Problem of gas diffusion through a permeable membrane. + + + +specified values of the unknown at the two boundaries. The term K is assumed to vary with the unknown $\phi$ according to + +$$ +K = K _ {0} (1 + 1 0 \phi) = K _ {0} (1 + g (\phi)). \tag {3.22} +$$ + +An analytical solution $^{(6)}$ exists for this problem which enables $\phi$ to be determined from + +$$ +\frac {\phi_ {A} + F \left(\phi_ {A}\right) - \phi - F (\phi)}{\phi_ {A} + F \left(\phi_ {A}\right) - \phi_ {B} - F \left(\phi_ {B}\right)} = \frac {x}{L}, \tag {3.23} +$$ + +where + +$$ +F (\phi) = \int_ {0} ^ {\phi} g \left(\phi^ {\prime}\right) d \phi^ {\prime}. \tag {3.24} +$$ + +In the present case, $g(\phi) = 10\phi$ which gives on substitution in (3.24) and then in (3.23) + +$$ +\frac {6 - \phi - 5 \phi^ {2}}{6} = \frac {x}{1 0}, \tag {3.25} +$$ + +which allows $\phi$ to be determined for any value of $x$ and is shown as the full line in Fig. 3.3. The initial finite element solution (i.e., after the first iteration) is shown in Fig. 3.3 as the broken line and, as expected, is linear. The results upon convergence, after 10 iterations, of the process are then included as circles and it is seen that the numerical solution coincides with the theoretical values. For example, for $x = 6$ , the theoretical solution is $\phi = 0\cdot 6$ , whilst the finite element analysis yields $\phi = 0\cdot 599999$ (see Appendix IV). + +The second example considered includes the effect of the term Q in (2.15). For thermal problems this can be physically interpreted as a heat generation/unit length and must be specified as a loading, according to (2.26), in subroutine DATA. Figure 3.4 shows the problem to be considered. A bar with its surface insulated generates heat internally and the temperature at its ends is maintained at zero value. Due to symmetry only one half of the problem is analysed with the symmetry condition $d\phi/dx = 0$ at the centreline being invoked. The initial solution corresponding to $K = K_{0}$ is shown and is practically identical to the theoretical value. The process converged to the nonlinear solution after 12 iterations with the temperature being markedly reduced. The reduction is greater in regions of higher initial temperature due to the comparatively greater increase in material ‘stiffness’ in these areas. + +# 3.10 Program for the solution of one-dimensional quasi-harmonic problems by the Newton-Raphson method + +As seen in Section 2.3, use of this method results in the assembled stiffness equations being nonsymmetric. The equation assembly and solution routines developed in Section 3.4 made no use of the symmetry properties of the + + + +![](images/page-079_ea0ea723b0923e0f34a13e59da0fce4c9fbb2990b48109c91a470d9e1356bcde.jpg) + +
+line + +| Iteration | Initial linear solution | Direct iteration—12 iterations Newton-Raphson—7 iterations | Nonlinear solutions | +| --------- | ------------------------ | ------------------------------------------------------------- | ------------------- | +| 0 | 0.0 | 0.0 | 0.0 | +| 1 | 0.19 | 0.11 | 0.11 | +| 2 | 0.36 | 0.19 | 0.19 | +| 3 | 0.51 | 0.23 | 0.23 | +| 4 | 0.63 | 0.27 | 0.27 | +| 5 | 0.74 | 0.30 | 0.30 | +| 6 | 0.83 | 0.32 | 0.32 | +| 7 | 0.90 | 0.34 | 0.34 | +| 8 | 0.95 | 0.35 | 0.35 | +| 9 | 0.98 | 0.35 | 0.35 | +| 10 | 1.0 | 0.35 | 0.35 | +
+ +Fig. 3.4 Quasi-harmonic equation example—Heat generation in an axial bar. + + + +stiffness matrices. They are therefore applicable to this method of analysis without modification. + +Three additional subroutines need to be developed. These are the element stiffness subroutine ASTIF1 and, since solution convergence is now based on the elimination of the residual forces, subroutine REFOR1 must be formed to calculate these forces and subroutine CONVER to monitor their convergence to zero. The master segment controlling the solution process is again that developed in Section 3.8 and the remaining subroutines accessed by this segment have also been described previously. + +# 3.10.1 Element stiffness formulation subroutine, ASTIF1 + +For solution by the Newton–Raphson process, the ‘stiffness’ equations which require solution are summarised in (2.12) where it is seen that the total stiffness is the sum of symmetric, H, and nonsymmetric, $H'$ , contributions. The symmetric stiffness matrix is given by (2.25) and the nonsymmetric terms depend on the particular form of material nonlinearity. For a material nonlinearity of the form (2.27), the nonsymmetric portion of the stiffness matrix is given by (2.29). The subroutine which evaluates and sums these separate contributions is now presented below. + +```txt +SUBROUTINE ASTIF1 ASTF 1 +C************************** ASTF 2 +C ASTF 3 +C *** CALCULATES ELEMENT STIFFNESS MATRICES ASTF 4 +C ASTF 5 +C************************** ASTF 6 +COMMON/UNIM1/NPOIN.NELEM,NBOUN,NLOAD,NPROP,NNODE,IINCS,IITER, ASTF 7 +. KRESL,NCHEK,TOLER,NALGO.NSVAB,NDOFN,NINCS,NEVAB, ASTF 8 +. NITER.NOUTP,FACTO,PVALU ASTF 9 +COMMON/UNIM2/PROPS(5.4),COORD(26),LNODS(25,2),IFPRE(52), ASTF 10 +. FIXED(52),TLOAD(25,4),RLOAD(25,4),ELOAD(25,4), ASTF 11 +. MATNO(25),STRES(25,2),PLAST(25),XDISP(52), ASTF 12 +. TDISP(26.2),TREAC(26.2),ASTIF(52,52),ASLOD(52), ASTF 13 +. REACT(52),FRESV(1352),PEFIX(52),ESTIF(4.4) ASTF 14 +REWIND 1 ASTF 15 +DO 10 IELEM=1.NELEM ASTF 16 +LPROP=MATNO(IELEM) ASTF 17 +STERM=PROPS(LPROP,1) ASTF 18 +GRADU=PROPS(LPROP,2) ASTF 19 +NODE1=LNODS(IELEM,1) ASTF 20 +NODE2=LNODS(IELEM,2) ASTF 21 +ELENG=ABS(COORD(NODE1)-COORD(NODE2)) ASTF 22 +AVERG=(TDISP(NODE1.1)+TDISP(NODE2,1))/2.0 ASTF 23 +FMULT=STERM*VARIA(AVERG)/ELENG ASTF 24 +DIFFR=TDISP(NODE1.1)-TDISP(NODE2,1) ASTF 25 +COEFF=STERM*GRADU*DIFFR/(2.0*ELENG) ASTF 26 +ESTIF(1,1)=FMULT+COEFF ASTF 27 +ESTIF(1,2)=-FMULT+COEFF ASTF 28 +ESTIF(2,1)=-FMULT-COEFF ASTF 29 +ESTIF(2,2)=FMULT-COEFF ASTF 30 +WRITE(1) ESTIF ASTF 31 +10 CONTINUE ASTF 32 +RETURN ASTF 33 +END ASTF 34 +``` diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_009.md b/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_009.md new file mode 100644 index 00000000..eec3970a --- /dev/null +++ b/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_009.md @@ -0,0 +1,588 @@ + + +ASTF 15 Rewind the file on which the stiffness matrix of each element will be stored. + +ASTF 16 Loop over each element. + +ASTF 17 Identify the material property of each element. + +ASTF 18 Set STERM equal to $K_0$ in (2.27). + +ASTF 19 Set GRADU equal to $b$ in (2.27). + +ASTF 20-21 Identify the node numbers of the element. + +ASTF 22 Calculate the element length. + +ASTF 23 Calculate the element temperature as the average of the nodal values. + +ASTF 24 Calculate the multiplying term in (2.25) by use of FUNCTION statement VARIA. + +ASTF 25–26 Evaluate the multiplying term in (2.29). + +ASTF 27–30 Compute the components of the total stiffness matrix. + +ASTF 31 Write the clement stiffness matrix on to disc file. + +ASTF 32 Termination of DO LOOP over each element. + +# 3.10.2 Residual force calculation subroutine REFOR1 + +The residual forces after any step of the process are obtained from (2.4). The applied nodal forces, $f$ , are known and it only remains to evaluate the 'equivalent nodal forces', $H\varphi$ , which are the nodal forces consistent with the unknowns, $\varphi$ . It should be noted that $H$ is the linear symmetric matrix defined in (2.25). The equivalent nodal forces at the nodes 1 and 2 of the linear element can be explicitly written, using (2.25), as + +$$ +f _ {1} = \frac {K}{L} (\phi_ {1} - \phi_ {2}), +$$ + +$$ +f _ {2} = - \frac {K}{L} (\phi_ {1} - \phi_ {2}). \tag {3.26} +$$ + +The subroutine which evaluates these forces for each element is now presented. +```txt +SUBROUTINE REFOR1 RFR1 1 +C************************** RFR1 2 +C RFR1 3 +C *** CALCULATES INTERNAL EQUIVALENT NODAL FORCES RFR1 4 +C RFR1 5 +C************************** RFR1 6 +COMMON/UNIM1/NPOIN.NELEM,NBOUN,NLOAD,NPROP,NNODE.IINCS,IITER, RFR1 7 +. KRESL,NCHEK,TOLER,NALGO.NSVAB,NDOFN,NINCS.NEVAB, RFR1 8 +. NITER,NOUTP,FACTO,PVALU RFR1 9 +COMMON/UNIM2/PROPS(5.4),COORD(26),LNODS(25,2),IFPRE(52), RFR1 10 +. FIXED(52),TLOAD(25,4),RLOAD(25,4),ELOAD(25,4), RFR1 11 +. MATNO(25),STRES(25,2),PLAST(25),XDISP(52), RFR1 12 +. TDISP(26.2),TREAC(26.2),ASTIF(52,52),ASLOD(52), RFR1 13 +. REACT(52),FRESV(1352),PEFIX(52),ESTIF(4.4) RFR1 14 +DO 10 IELEM=1,NELEM RFR1 15 +DO 10 IEVAB=1,NEVAB RFR1 16 +``` + + + +
10ELOAD(IELEM,IEVAB)=0.0RFR117
DO 20 IELEM=1.NELEMRFR118
LPROP=MATNO(IELEM)RFR119
STERM=PROPS(LPROP,1)RFR120
NODE1=LNODS(IELEM,1)RFR121
NODE2=LNODS(IELEM,2)RFR122
ELENG=ABS(COORD(NODE1)-COORD(NODE2))RFR123
AVERG=(TDISP(NODE1.1)+TDISP(NODE2,1))/2.0RFR124
STIFF=STERM*VARIA(AVERG)/ELENGRFR125
ELOAD(IELEM,1)=STIFF*(TDISP(NODE1,1)-TDISP(NODE2,1))RFR126
20ELOAD(IELEM,2)=-STIFF*(TDISP(NODE1,1)-TDISP(NODE2,1))RFR127
RETURNRFR128
ENDRFR129
+ +RFR1 15–17 Initialise to zero the array in which the equivalent nodal forces for each element will be stored. + +RFR1 18 Loop over each element. + +RFR1 19 Identify the material property of each element. + +RFR1 20 Set STERM equal to $K_0$ in (2.27). + +RFR1 21–22 Identify the node numbers of the element. + +RFR1 23 Calculate the element length. + +RFR1 24 Calculate the element temperature as the average of the nodal values. + +RFR1 25 Calculate the multiplying term in (2.25). + +RFR1 26–27 Compute the equivalent nodal forces according to (3.26). + +# 3.10.3 Solution convergence monitoring subroutine, CONUND + +This subroutine must essentially differ from subroutine MONITR described in Section 3.9.2 since convergence is now based on the residual force values rather than values of the unknowns. The convergence criterion employed is similar to that described in (3.21) and is + +$$ +\frac {\sqrt {\left[ \sum_ {i = 1} ^ {N} \left(\psi_ {i} ^ {r}\right) ^ {2} \right]}}{\sqrt {\left[ \sum_ {i = 1} ^ {N} \left(f _ {i}\right) ^ {2} \right]}} \times 1 0 0 \leqslant \text { TOLER }, \tag {3.27} +$$ + +where N is the total number of nodal points in the problem and r denotes the iteration number. This criterion states that convergence occurs if the norm of the residual forces becomes less than TOLER times the norm of the total applied forces. Again the parameter NCHEK is used to indicate whether or not convergence has occurred. Three values of NCHEK are utilised: + +NCHEK = 0 Solution has converged. + += 1 Solution converging, with the norm of the residual forces being less for the $r^{th}$ iteration than the $(r-1)^{th}$ iteration. += 999 Solution diverging. The norm of the residual forces is greater for the $r^{\text{th}}$ iteration than the $(r - 1)^{\text{th}}$ iteration. + + + +Subroutine CONUND is now listed and descriptive notes provided. + +
SUBROUTINE CONUNDCOND1
C**********COND2
CCOND3
C ***CHECKS FOR SOLUTION CONVERGENCECOND4
CCOND5
C**********COND6
COMMON/UNIM1/NPOIN,NELEM,NBOUN,NLOAD,NPROP,NNODE.IINCS.IITER,COND7
KRESL,NCHEK,TOLER,NALGO,NSVAB,NDOFN,NINCS,NEVAB,COND8
NITER,NOUTP,FACTO,PVALUCOND9
COMMON/UNIM2/PROPS(5,4),COORD(26),LNODS(25,2),IFPRE(52),COND10
FIXED(52),TLOAD(25,4),RLOAD(25,4),ELOAD(25,4),COND11
MATNO(25),STRES(25,2),PLAST(25),XDISP(52),COND12
TDISP(26,2),TREAC(26,2),ASTIF(52,52),ASLOD(52),COND13
REACT(52),FRESV(1352),PEFIX(52),ESTIF(4,4)COND14
DIMENSION STFOR(52),TOFOR(52)COND15
NCHEK=0COND16
RESID=0.0COND17
RETOT=0.0COND18
DO 10 ISVAB=1,NSVABCOND19
STFOR(ISVAB)=0.0COND20
10TOFOR(ISVAB)=0.0COND21
DO 20 IELEM=1,NELEMCOND22
IEVAB=0COND23
DO 20 INODE=1,NNODECOND24
NODNO=LNODS(IELEM,INODE)COND25
DO 20 IDOFN=1,NDOFNCOND26
IEVAB=IEVAB+1COND27
NPOSN=(NCDNO-1)*NDOFN+IDOFNCOND28
STFOR(NPOSN)=STFOR(NPOSN)+ELOAD(IELEM,IEVAB)COND29
20TOFOR(NPOSN)=TOFOR(NPOSN)+TLOAD(IELEM,IEVAB)COND30
DO 30 ISVAB=1,NSVABCOND31
REFOR=TOFOR(ISVAB)-STFOR(ISVAB)COND32
RESID=RESID+REFOR*REFORCOND33
30RETOT=RETOT+TOFOR(ISVAB)*TOFOR(ISVAB)COND34
DO 40 IELEM=1,NELEMCOND35
DO 40 IEVAB=1,NEVABCOND36
40ELOAD(IELEM,IEVAB)=TLOAD(IELEM,IEVAB)-ELOAD(IELEM,IEVAB)COND37
RATIO=100.0*SQRT(RESID/RETOT)COND38
IF(RATIO.GT.TOLER) NCHEK=1COND39
IF(IITER.EQ.1) GO TO 50COND40
IF(RATIO.GT.PVALU) NCHEK=999COND41
50PVALU=RATIOCOND42
WRITE(6,900) IITER,NCHEK,RATIOCOND43
900FORMAT(1HO,5X,'ITERATION NUMBER=',I5/COND44
1HO,5X,'CONVERGENCE CODE=',I4,3X,COND45
'NORM OF RESIDUAL SUM RATIO=',E14.6)COND46
RETURNCOND47
ENDCOND48
+ +COND 16 Initialise the convergence indicator to zero. If convergence has not occurred during this iteration this value will be reset later in the subroutine. + +COND 17 Initialise to zero the norm of the residual forces. + +COND 18 Initialise to zero the norm of the total applied loads. + +COND 19–21 Initialise the arrays which will contain the equivalent nodal forces and the applied loads for each nodal point. + + + +
COND 22-30Assemble the equivalent nodal forces and applied load contributions of each element to give the total nodal values, as required for use in (3.27). This manipulation is necessary as we have decided to associate loads with an element rather than nodal points.
COND 32Calculate the nodal residual force according to (2.4).
COND 33Evaluate the norm of the residual forces.
COND 34Evaluate the norm of the total applied forces.
COND 35-37Calculate the residual nodal forces for each element, for application as forces for the next iteration according to (2.12).
COND 38Compute the left-hand side of (3.27)—the residual sum ratio.
COND 39If (3.27) is not satisfied reset NCHEK = 1 to indicate that convergence has not yet occurred.
COND 40-41For second and subsequent iterations check to see if the residual sum ratio has decreased from the previous iteration. If not, set NCHEK = 999.
COND 42Store the residual sum ratio, in order to perform the check indicated in COND 41 during the next iteration.
COND 43-46Write the convergence code and the residual sum ratio.
+ +# 3.10.4 Numerical examples + +The numerical example considered in Section 3.9.3 and illustrated in Fig. 3.3, was reanalysed using the Newton–Raphson approach. The process converged to the nonlinear solution in 5 iterations compared to the 10 cycles required for the direct iteration method. The reduction in the number of iterations must, however, be balanced against the increased computing effort required for the solution of nonsymmetric equations. This remark is applicable only when advantage of the symmetric property of the equations is taken in solution as is the case in the more sophisticated equation solver described later in Chapter 6. The numerical results are practically identical to those obtained by the method of direct iteration and consequently both solutions are represented by the full circles in Fig. 3.3. The problem of Fig. 3.4 was also reanalysed and a similar improvement in convergence behaviour was obtained with only 7 iterations being required in place of the 12 necessitated by direct iteration. + +# 3.11 Program for the solution of nonlinear elastic problems + +In this section a program is developed which permits the solution of nonlinear elastic problems by either the tangential stiffness or the initial stiffness approach or by a combination of both methods. The options open are controlled by the parameter NALGO, the possible values of which are described in Section 3.2. + + + +The structure of this program is identical to that described in Section 3.10 and it is only necessary to develop appropriate subroutines for element stiffness formulation, STIFF2, and residual force evaluation, REFOR2. + +# 3.11.1 Element stiffness subroutine, STIFF2 + +For any value of the total strain, $\epsilon$ , in an element, the tangential stiffness matrix is explicitly given by (2.33). It is seen from this expression that the first derivative of the strain function must be known. For the calculation of the residual forces, the strain function itself must be input. Since the computer cannot perform even the simplest differentiation it is necessary to supply both quantities in the form of FUNCTION statements. As an example, the strain function will be assumed to be of the form + +$$ +g (\epsilon) = \epsilon - 5 \epsilon^ {2}, \tag {3.28} +$$ + +in which case + +$$ +g ^ {\prime} (\epsilon) = 1 - 1 0 \epsilon . \tag {3.29} +$$ + +Subroutine STIFF2 is now listed below. +```fortran +SUBROUTINE STIFF2 STF2 1 +C************************** STF2 2 +C STF2 3 +C *** CALCULATES ELEMENT STIFFNESS MATRICES STF2 4 +C STF2 5 +C************************** STF2 6 +COMMON/UNIM1/NPOIN,NELEM,NBOUN,NLOAD,NPROP,NNODE,IINCS,IITER, STF2 7 +. KRESL,NCHEK,TOLER,NALGO,NSVAB,NDOFN,NINCS,NEVAB, STF2 8 +. NITER,NOUTP,FACTO,PVALU STF2 9 +COMMON/UNIM2/PROPS(5,4),COORD(26),LNODS(25,2),IFPRE(52), STF2 10 +. FIXED(52),TLOAD(25,4),RLOAD(25,4),ELOAD(25,4), STF2 11 +. MATNO(25),STRES(25,2),PLAST(25),XDISP(52), STF2 12 +. TDISP(26,2),TREAC(26,2),ASTIF(52,52),ASLOD(52), STF2 13 +. REACT(52),FRESV(1352),PEFIX(52),ESTIF(4,4) STF2 14 +REVIND 1 STF2 15 +DO 10 IELEM=1,NELEM STF2 16 +LPROP=MATNO(IELEM) STF2 17 +YOUNG=PROPS(LPROP,1) STF2 18 +XAREA=PROPS(LPROP,2) STF2 19 +NODE1=LNODS(IELEM,1) STF2 20 +NODE2=LNODS(IELEM,2) STF2 21 +ELENG=ABS(COORD(NODE1)-COORD(NODE2)) STF2 22 +PTRAN=PLAST(IELEM) STF2 23 +COEFF=YOUNG*XAREA/ELENG STF2 24 +FMULT=COEFF*STDIV(PTRAN) STF2 25 +ESTIF(1,1)=FMULT STF2 26 +ESTIF(1,2)=-FMULT STF2 27 +ESTIF(2,1)=-FMULT STF2 28 +ESTIF(2,2)=FMULT STF2 29 +WRITE(1) ESTIF STF2 30 +1C CONTINUE STF2 31 +RETURN STF2 32 +END STF2 33 +``` + + + +
STF2 15Rewind the file on which the stiffness matrix of each element will be stored.
STF2 16Loop over each element.
STF2 17Identify the material property of each element.
STF2 18Set YOUNG equal to the reference value of the material modulus, $E_0$ .
STF2 19Set XAREA equal to the cross-sectional area.
STF2 20–21Identify the node numbers of the element.
STF2 22Calculate the element length.
STF2 23Set PTRAN equal to the total strain, $\epsilon$ .
STF2 24–25Compute the multiplying term in (2.33) with $g'(\epsilon)$ given by STDIV (PTRAN).
STF2 26–29Compute the components of the stiffness matrix.
STF2 30Write the element stiffness matrix on to disc file.
STF2 31Termination of DO LOOP over each element.
+ +For a strain derivative function as defined by (3.29), the appropriate function statement is provided below. + +
FUNCTION STDIV(PTRAN)STF234
C****STF235
CSTRAIN DERIVATIVE FUNCTIONSTF236
C****STF237
STDIV=1.0-10.0*PTRANSTF238
RETURNSTF239
ENDSTF240
+ +# 3.11.2 Residual force calculation subroutine REFOR2 + +The residual forces existing at the end of any iteration must be calculated according to $(2.4)$ . The first step in this calculation entails the evaluation of the equivalent nodal forces, which are the forces required to produce the total displacements existing in the element. The element strain is simply + +$$ +\epsilon_ {E} = \left\{ \begin{array}{l l} \left(\phi_ {2} - \phi_ {1}\right) / L & \text { for } \quad x _ {2} > x _ {1} \\ \left(\phi_ {1} - \phi_ {2}\right) / L & \text { for } \quad x _ {2} < x _ {1}, \end{array} \right. \tag {3.30} +$$ + +where $x_{1}$ and $x_{2}$ denote the coordinates of the element nodes. This notation is required to ensure that tensile strains are positive and enables the nodal connections to be assigned in any order. + +Then from (2.30) the stress in the element is given by + +$$ +\sigma_ {E} = E _ {0} g (\epsilon_ {E}), \tag {3.31} +$$ + +and the equivalent nodal forces are + +$$ +f _ {1} = - f _ {2} = \left\{ \begin{array}{c c c} - \sigma_ {E} A & \text { for } & x _ {2} > x _ {1} \\ \sigma_ {E} A & \text { for } & x _ {2} < x _ {1}. \end{array} \right. \tag {3.32} +$$ + + + +Subroutine REFOR2 is now listed and described. +SUBROUTINE REFOR2 RFR2 1 +C**************************RFR2 2 +C RFR2 3 +C *** CALCULATES INTERNAL EQUIVALENT NODAL FORCES RFR2 4 +C RFR2 5 +C**************************RFR2 6 +COMMON/UNIM1/NPOIN,NELEM,NBOUN,NLOAD,NPROP,NNODE,IINCS,IITER, RFR2 7 +. KRESL,NCHEK,TOLER,NALGO,NSVAB,NDOFN,NINCS,NEVAB, RFR2 8 +. NITER,NOUTP,FACTO,PVALU RFR2 9 +COMMON/UNIM2/PROPS(5,4),COORD(26),LNODS(25,2),IFPRE(52), RFR2 10 +. FIXED(52),TLOAD(25,4),RLOAD(25,4),ELOAD(25,4), RFR2 11 +. MATNO(25),STRES(25,2),PLAST(25),XDISP(52), RFR2 12 +. TDISP(26,2),TREAC(26,2),ASTIF(52,52),ASLOD(52), RFR2 13 +. REACT(52),FRESV(1352),PEFIX(52),ESTIF(4,4) RFR2 14 +DO 10 IELEM=1,NELEM RFR2 15 +DO 10 IEVAB=1,NEVAB RFR2 16 +10 ELOAD(IELEM,IEVAB)=0.0 RFR2 17 +DO 30 IELEM=1,NELEM RFR2 18 +LPROP=MATNO(IELEM) RFR2 19 +YOUNG=PROPS(LPROP,1) RFR2 20 +XAREA=PROPS(LPROP,2) RFR2 21 +NODE1=LNODS(IELEM,1) RFR2 22 +NODE2=LNODS(IELEM,2) RFR2 23 +ELENG=ABS(COORD(NODE1)-COORD(NODE2)) RFR2 24 +IF(COORD(NODE2).GT.COORD(NODE1)) STRAN=(XDISP(NODE2)-XDISP(NODE1)) RFR2 25 +. /ELENG RFR2 26 +IF(COORD(NODE2).LT.COORD(NODE1)) STRAN=(XDISP(NODE1)-XDISP(NODE2)) RFR2 27 +. /ELENG RFR2 28 +PLAST(IELEM)=PLAST(IELEM)+STRAN RFR2 29 +PTRAN=PLAST(IELEM) RFR2 30 +STRES(IELEM,1)=YOUNG*STNFN(PTRAN) RFR2 31 +IF(COORD(NODE2).GT.COORD(NODE1)) GO TO 20 RFR2 32 +ELOAD(IELEM,1)=STRES(IELEM,1)*XAREA RFR2 33 +ELOAD(IELEM,2)=-STRES(IELEM,1)*XAREA RFR2 34 +GO TO 30 RFR2 35 +20 ELOAD(IELEM,1)=-STRES(IELEM,1)*XAREA RFR2 36 +ELOAD(IELEM,2)=STRES(IELEM,1)*XAREA RFR2 37 +30 CONTINUE RFR2 38 +RETURN RFR2 39 +END RFR2 40 + +RFR2 15–17 Initialise to zero the array in which the equivalent nodal forces for each element will be stored. + +RFR2 18 Loop over each element. + +RFR2 19 Identify the material property of each element. + +RFR2 20 Set YOUNG equal to the reference value of the material modulus, $E_{0}$ . + +RFR2 21 Set XAREA equal to the cross-sectional area. + +RFR2 22-23 Identify the node numbers of the element. + +RFR2 24 Calculate the element length. + +RFR2 25–28 Calculate the increase in element strain which occurred during the current iteration according to (3.30) (since XDISP measures the displacement change only). + +RFR2 29 Compute the total strain. + +RFR2 30–31 Compute the element stress according to (3.31). + +RFR2 32–37 Compute the equivalent nodal forces according to (3.32). + +RFR2 38 Termination of DO LOOP over the elements. + + + +For calculation of the element stress in steps RFR2 30–31 (equation (3.31)) the strain function $g(\epsilon)$ must be defined. The FUNCTION statement appropriate to the variation indicated in (3.28) is provided below. + +
FUNCTION STNFN(PTRAN)RFR241
C****RFR242
CSTRAIN FUNCTIONRFR243
C****RFR244
STNFN=PTRAN-5.0*PTRAN*PTRANRFR245
RETURNRFR246
ENDRFR247
+ +The equivalent nodal forces evaluated here are converted into residual forces $\psi$ in subroutine CONUND as described in Section 3.10.3. + +# 3.11.3 Numerical examples + +The first example considered is the uniaxial loading of a two-element system. The stress/strain relationship is assumed to be defined in terms of the nonlinear expression (3.28). The applied load is incrementally increased and the combined tangential/initial stiffness solution algorithm, NALGO = 4, is employed. Figure 3.5 shows the solution behaviour during iteration to the nonlinear solution. The element stiffnesses are initially assembled at the beginning of a load increment and then kept constant during iteration to the nonlinear solution. The convergence path is plotted and it is seen that the process converges within 7 iterations for the first load increment. For the second load increment the process requires 9 iterations before convergence takes place. The process diverged rapidly on further increase of load to a total value of 11; which is expected since no solution can exist for this load value. + +As an illustration of the application of the initial stiffness method to strain-softening problems, the above problem was reanalysed with the structure being loaded by prescribing an increasing value of displacement to node 3, rather than incrementing an applied load. For strain values at and beyond the peak load, the structural stiffness is either zero or negative and an initial stiffness approach must be employed. Figure 3.6 shows the results when the structure is strained beyond the peak load value. + +# 3.12 Program for the solution of elasto-plastic problems + +A computer program is now developed for the solution of one-dimensional elasto-plastic problems. Once again a tangential stiffness, initial stiffness or combined approach is permitted for solution. The program differs only from that described in the previous section in the explicit form of the element stiffness and residual force subroutines. + +# 3.12.1 Element stiffness subroutine, STIFF3 + +Before yielding, the stiffness matrix of an element with linear displacement variation is given by $(2.38)$ . After the onset of plastic deformation, as + + + +![](images/page-089_ec7b98e64af0773d3912bb1ac4ffbf46497bfdf2c34a18b772aaa9ade9f0a723.jpg) + +
+line + +| Axial extension Δ | Axial force P | Method | +| ----------------- | ------------- | -------------------------- | +| 0.5 | 8.0 | 7 iterations | +| 0.5 | 8.0 | 7 iterations | +| 0.5 | 8.0 | 7 iterations | +| 0.5 | 8.0 | 7 iterations | +| 0.5 | 8.0 | 7 iterations | +| 0.5 | 8.0 | 7 iterations | +| 0.5 | 10.0 | 7 iterations | +| 0.5 | 10.0 | 7 iterations | +| 0.5 | 10.0 | 7 iterations | +| 0.5 | 10.0 | 7 iterations | +| 0.5 | 10.0 | 7 iterations | +| 0.5 | 10.0 | 7 iterations, 9 iterations | +| 0.5 | 10.0 | 7 iterations, 9 iterations | +| 0.5 | 10.0 | 7 iterations, 9 iterations | +| 0.5 | 10.0 | 7 iterations, 9 iterations | +| 0.5 | 10.0 | 7 iterations, 9 iterations | +| 0.5 = 0.5 | 5.0 | 7 iterations, 9 iterations | +| 0.5 = 0.5 | 5.0 | 7 iterations, 9 iterations | +| 0.5 = 0.5 | 5.0 | 7 iterations, 9 iterations | +| 0.5 = 0.5 | 5.0 | 7 iterations, 9 iterations | +| 0.5 | 5.0 | 7 iterations, 9 iterations | +| 0.5 | 5.0 | 7 iterations, 9 iterations | +| 0.5 | 5.0 | 7 iterations, 9 iterations | +| 0.5 | 5.0 | 7 iterations, 9 iterations | +| 0.5 | 5.0 | 7 iterations, | +| 0.5 | 5.0 | 7 iterations, | +| 0.5 | 5.0 | 7 iterations, | +| 0.5 | 5.0 | 7 iterations, | +| 0.5 | 5.0 | 7 iterations, | +| 0.5 | 5.0 | 7 iterations, | +| 0, 0.5 | 5.0 | 7 iterations, | +| 0.5 | 5.0 | 7 iterations, | +| 0.5 | 5.0 | 7 iterations, | +| 0.5 | 5.0 | 7 iterations, | +| 0.5 | 5.0 | 7 iterations, | +| 0.5 | 5.5 | 7 iterations, | +| 0.5 | 5.5 | 7 iterations, | +| 0.5 | 5.5 | 7 iterations, | +| 0.5 | 5.5 | 7 iterations, | +| 0.5 | 5.5 | 7 iterations, | +| 0.5 | 5.5 | 7 iteration, | +| 0.5 | 5.5 | 7 iteration, | +| 0.5 | 5.5 | 7 iteration, | +| 0.5 | 5.5 | 7 iteration, | +| 0.5 | 5.5 | 7 iteration, | +| 0.5 | 5.5 | 7 iteration, | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ... (Δ) | +| ... | ... | ... (Δ) | +| ... | ... | ... (Δ) | +| ... | ... | ... (Δ) | +| ... | ... | ... (Δ) | +| ... | ... | ... (Δ) | +| ... | ... | ... (Δ) | +| ... | ... | ... (Δ) | +| ... | ... |... (Δ) | +| ... | ... | ... (Δ) | +| ... | ... | ... (Δ) | +| ... | ... | ... (Δ) | +| ... | ... | ... (Δ) | +| ... | ... | ... (Δ) | +| ... | ... | ... (Δ) | +| ... | ... | ... (Δ) | +| ... = 0.5 | 5.0 | 7 iterations, 9 iterations | +| ... | 5.0 | 7 iterations, 9 iterations | +| ... | 5.0 | 7 iterations, 9 iterations | +| ... | 5.0 | 7 iterations, 9 iterations | +| ... | 5.0 | 7 iterations, 9 iterations | +| ... | 5.0 | 7 iterations, 9 iterations | +| ... | 5.5 | 7 iterations, 9 iterations | +| ... | 5.5 | 7 iterations, 9 iterations | +| ... | 5.5 | 7 iterations, 9 iterations | +| ... | 5.5 | 7 iterations, 9 iterations | +| ... | 5.5 | 7 iterations, 9 iterations | +| ... | 5.5 | 7 (Δ) | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ...1 | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... |... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... |...1 | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... (Δ) | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... (Δ) | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ...3 | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... |... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... |... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | 5.0 | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | 10.0 | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | ... | ... | +| ... | <10.0 | ... | +| ... | <10.0 | ... | +| ... | <10.0 | ... | +| ... | <10.0 | ... | +| ... | <10.0 | ... | +| ... | <10.0 | ... | +| ... | <10.0 | ... | +| ... | <10.0 |... | +| ... | <10.0 | ... | +| ... | <10.0 | ... | +| ... | <10.0 | ... | +| ... | <10.0 | ... | +| ... | <10.0 | ... | +| ... | <10.0 | ... | +| ... | <10.0 | ... | +
+ +Fig. 3.5 Load/extension response of a nonlinear elastic bar under applied axial loading. + +governed by the uniaxial yield stress $\sigma_{Y}$ , the material stiffness is reduced and the elasto-plastic stiffness matrix is explicitly given by (2.43). Thus when forming the stiffness matrix for each element, it is first necessary to check whether the element behaviour is elastic or elasto-plastic. This can best be monitored by recording the plastic strain component, $\epsilon_{p}$ , for each element and noting that this will be zero for a completely elastic material response. + + + +![](images/page-090_99c47a2d240be0f0012b3d5c6077199dd388dcec73c1c6a97bb8bea7dce1c6eb.jpg) + +
+line + +| Axial extension, Δ | Axial force, P (Theoretical) | Axial force, P (Converged) | Axial force, P (Solution after 1st iteration) | +| ------------------ | ---------------------------- | -------------------------- | ------------------------------------------- | +| 0.0 | 0.0 | 0.0 | 0.0 | +| 0.6 | 8.5 | 8.5 | 8.5 | +| 0.8 | 10.0 | 10.0 | 10.0 | +| 1.0 | 10.0 | 10.0 | 10.0 | +| 1.2 | 10.0 | 10.0 | 10.0 | +| 1.4 | 8.5 | 8.5 | 8.5 | +| 1.6 | 6.0 | 6.0 | 6.0 | +| 1.8 | 3.5 | 3.5 | 3.5 | +| 2.0 | 0.0 | 0.0 | 0.0 | +| 2.2 | -4.0 | -4.0 | -4.0 | +| 2.6 | -10.0 | -10.0 | -10.0 | +
+ +Fig. 3.6 Solution for a nonlinear elastic bar by initial stiffness, incremented prescribed displacement approach. + +Subroutine STIFF3 can now be presented. + +
SUBROUTINE STIFF3STF31
C**********STF32
CSTF33
C ***CALCULATES ELEMENT STIFFNESS MATRICESSTF34
CSTF35
C**********STF36
diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_010.md b/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_010.md new file mode 100644 index 00000000..71015a53 --- /dev/null +++ b/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_010.md @@ -0,0 +1,379 @@ + + +```csv +COMMON/UNIM1/NPOIN, NELEM, NBOUN, NLOAD, NPROP, NNODE, IINCS, IITER, STF3 7 +. KRESL, NCHEK, TOLER, NALGO, NSVAB, NDOFN, NINCS, NEVAB, STF3 8 +. NITER, NOUTP, FACTO, PVALU STF3 9 +COMMON/UNIM2/PROPS(5,4), COORD(26), LNODS(25,2), IFPRE(52), STF3 10 +. FIXED(52), TLOAD(25,4), RLOAD(25,4), ELOAD(25,4), STF3 11 +. MATNO(25), STRES(25,2), PLAST(25), XDISP(52), STF3 12 +. TDISP(26,2), TREAC(26,2), ASTIF(52,52), ASLOD(52), STF3 13 +. REACT(52), FRESV(1352), PEFIX(52), ESTIF(4,4) STF3 14 +REWIND 1 STF3 15 +DO 10 IELEM=1, NELEM STF3 16 +LPROP=MATNO(IELEM) STF3 17 +YOUNG=PROPS(LPROP,1) STF3 18 +XAREA=PROPS(LPROP,2) STF3 19 +HARDS=PROPS(LPROP,4) STF3 20 +NODE1=LNODS(IELEM,1) STF3 21 +NODE2=LNODS(IELEM,2) STF3 22 +ELENG=ABS(COORD(NODE1)-COORD(NODE2)) STF3 23 +FMULT=YOUNG*XAREA/ELENG STF3 24 +IF(PLAST(IELEM).GT.0.0) FMULT=FMULT*(1.0-YOUNG/(YOUNG+HARDS)) STF3 25 +ESTIF(1,1)=FMULT STF3 26 +ESTIF(1,2)=-FMULT STF3 27 +ESTIF(2,1)=-FMULT STF3 28 +ESTIF(2,2)=FMULT STF3 29 +WRITE(1) ESTIF STF3 30 +10 CONTINUE STF3 31 +RETURN STF3 32 +END STF3 33 +``` + +
STF3 15Rewind the file on which the stiffness matrix of each element will be stored.
STF3 16Loop over each element.
STF3 17Identify the material property of each element.
STF3 18Set YOUNG equal to the material elastic modulus.
STF3 19Set XAREA equal to the cross-sectional area.
STF3 20Set HARDS equal to the strain hardening parameter, $H'$ .
STF3 21–22Identify the node numbers of the element.
STF3 23Calculate the element length.
STF3 24Compute the multiplying term in (2.38) as FMULT.
STF3 25Check if the element has yielded. If yes, compute FMULT as the multiplying term in (2.43).
STF3 26–29Compute the components of the stiffness matrix.
STF3 30Write the element stiffness matrix on to disc file.
STF3 31Termination of DO LOOP over each element.
+ +# 3.12.2 Residual force subroutine, REFOR3 + +The purpose of this subroutine is to calculate the equivalent nodal forces from which the residual nodal forces will be evaluated in subroutine CONUND. In view of the essentially incremental nature of the equations of plasticity, the subroutine is somewhat more intricate than the residual force + + + +subroutines developed to date. All stress and strain components must be accumulated from the values obtained during each iteration. The situation is further complicated by the fact that an element may yield when the residual forces are applied as loads for any iteration. The precise load at which yielding begins will generally lie somewhere between the total load corresponding to the previous iteration and the total load for the present cycle. Consequently the yield load must be determined and the plastic strain computed for only the post yield portion of the load. The general procedure adopted is to determine the stress in each element so that the yield criterion is satisfied. If the actual stress in any element is greater than this permissible value, then the additional part is removed but is included in the residual force vector to maintain equilibrium. + +Consider the situation existing for the $r^{th}$ iteration of any particular load increment. The solution algorithm employed is presented below. + +Step a The applied loads for the $r^{th}$ iteration are the residual forces $\psi^{r-1}$ calculated at the end of the $(r-1)^{th}$ iteration according to (2.4). These applied loads give rise to displacement increments, $\Delta\varphi^{r}$ , according to (2.12). Hence calculate the corresponding increment of strain $\Delta\epsilon^{r}$ . For the general element denote this value by $\Delta\epsilon^{r}$ and it is shown in Fig. 3.7. + +Step b Compute the incremental stress change assuming linear elastic behaviour. This will introduce errors if the element has yielded and the material is behaving elasto-plastically. However, we will correct any discrepancy when the residual forces are calculated. Therefore we calculate the stress change according to $\Delta\sigma_{e}^{r}=E\Delta\epsilon^{r}$ , where the subscript e is used to denote that this stress is based on elastic behaviour. + +Step c Accumulate the total stress for each element as $\sigma_{e}^{r} = \sigma^{r-1} + \Delta\sigma_{e}^{r}$ . The stress $\sigma^{r-1}$ will have been determined to satisfy the yield condition during the $(r-1)^{\text{th}}$ iteration. Consequently, the error in the stress $\sigma_{e}^{r}$ is limited to $\Delta\sigma_{e}^{r}$ . Again the subscript e denotes that $\sigma_{e}^{r}$ is based on an elastic behaviour. + +Step d The next step in the process depends on whether or not the element had previously yielded on the $(r-1)^{\text{th}}$ iteration. This can be checked from the known value of the yield stress for the $(r-1)^{\text{th}}$ iteration. The stress limit for this cycle is given from Fig. 2.9 as + +$$ +\sigma_ {Y} ^ {r - 1} = \sigma_ {Y} + H ^ {\prime} \epsilon_ {p} ^ {r - 1}. +$$ + +Since the plastic strain $\epsilon_{p}$ will differ from element to element, each element will generally have a different permissible stress level. + + + +![](images/page-093_cdebab83171b40011b56607ebb0f601021c2123a8a0d28b5ead13013ad6d89ff.jpg) + +
+text_image + +σe^r +Δσe^r +σ^r +σy +σr^-1 +RΔσe^r +A +E +B +Δσep^r +(1 - R)Δσe^r +D +C +Δεep^r +R = AB/AC = σe^r - σy/σe^r - σr^-1 +Δε^r +
+ +(a) +![](images/page-093_63dc8e93b23d7e7ed7e827cf2c194a21603e7e73778c17820a85ceaded0a161e.jpg) + +
+line + +| x-axis label | y-axis label | Description | +| ------------ | ------------ | ------------------- | +| Start | σₑʳ | | +| End | σʳ | Slope = E, Δσₑₚʳ, Δσₑₚʳ, r = 1 | +| End | σʳ | | +| End | σʳ⁻¹ | | +
+ +(b) +Fig. 3.7 Incremental stress and strain changes in a one-dimensional elasto-plastic material. (a) Initial yielding of material. (b) Material previously yielded. + + + +Therefore we check if $\sigma^{r-1} > \sigma_Y + H' \epsilon_p^{r-1}$ . If the answer is: + +# YES + +which implies that the element had already yielded during the previous iteration, then check to see if $\sigma_{e}^{r} > \sigma^{r-1}$ . If the answer is: + +NO + +YES + +The element is unloading which according to plasticity theory must take place elastically, and no further action need be taken. Go directly to Step g. + +The element had reached the threshold stress during the previous iteration and the stress is still increasing. Therefore all the excess stress $\sigma_{e}^{r}-\sigma^{r-i}$ must be reduced to the yield value as indicated in Fig. 3.7(b). Therefore the factor, R, which defines the portion of the stress which must be modified to satisfy the yield condition, is equal to 1 in this case as shown in Fig. 3.7(b). + +# NO + +which implies that the element had not previously yielded. We now check to see if $\sigma_{e}^{r} > \sigma_{Y}$ . If the answer is: + +NO + +YES + +The element is still elastic and no further action need be taken. Go directly to Step g. + +The element has yielded during the application of load corresponding to this iteration as illustrated in Fig. 3.7(a). Therefore the portion of the stress greater than the yield value must be reduced to the elastoplastic line. The removed portion will be included in the residual force vector. The reduction factor, R, is found, with reference to Fig. 3.7(a) to be + +$$ +\begin{array}{l} R = \frac {A B}{A C} \\ = \frac {\sigma_ {e} ^ {r} - \sigma_ {Y}}{\sigma_ {e} ^ {r} - \sigma^ {r - 1}}. \\ \end{array} +$$ + +Step e For yielded elements only, calculate the increment of stress $\Delta\sigma_{ep}^{k}$ , which is the portion after yielding, permitted by elasto-plastic theory. This stress value is shown in Fig. 3.7 for the two cases when (a) yielding has commenced during this iteration and (b) when the element had previously yielded. Using (2.4) we have + +$$ +\Delta \sigma_ {e p} ^ {r} = E \left(1 - \frac {E}{E + H ^ {\prime}}\right) \Delta \epsilon_ {e p} ^ {r}, \tag {3.33} +$$ + +where the subscript $ep$ denotes elasto-plastic behaviour. For the above to be generally true we must restrict ourselves to small increments of stress and strain. For the situation of Fig. 3.7(a), noting that triangles ADC and AEB are similar, we have + +$$ +\Delta \epsilon_ {e p} ^ {r} = R \Delta \epsilon^ {r}. \tag {3.34} +$$ + + + +Defining R = 1 for the situation of Fig. 3.7(b), then (3.34) is still correct. Therefore + +$$ +\Delta \sigma_ {e p} ^ {r} = E \left(1 - \frac {E}{E + H ^ {\prime}}\right) R \Delta \epsilon^ {r}. \tag {3.35} +$$ + +The total current stress is given by + +$$ +\sigma^ {r} = \sigma^ {r - 1} + (1 - R) \Delta \sigma_ {e} ^ {r} + \Delta \sigma_ {e p} ^ {r}, \tag {3.36} +$$ + +where the second term accounts for the elastic portion of the stress increment occurring before the onset of yielding. + +Step f For yielded elements only, evaluate the total plastic strain for the element as $\epsilon_{p}^{r} = \epsilon_{p}^{r-1} + \Delta\epsilon_{p}^{r}$ where the plastic strain increment for the iteration is calculated as follows. For the elastic component of strain, $\Delta\epsilon_{e}^{r}$ , we have + +$$ +\Delta \epsilon_ {e} ^ {r} = \frac {\Delta \sigma^ {r}}{E}. \tag {3.37} +$$ + +Substituting for $\Delta\sigma^{r}$ from the linearised form of (2.35) into (3.37) and then using (2.34) we obtain + +$$ +\Delta \epsilon_ {p} ^ {r} = \frac {\Delta \epsilon^ {r}}{1 + H ^ {\prime} / E}. \tag {3.38} +$$ + +Since the plastic strain component must be calculated for the part of the strain after the element yields, then, with reference to Fig. 3.7, $\Delta\epsilon^{r}$ must be replaced by $\Delta\epsilon_{ep}^{r}$ . Or, using (3.34), we have + +$$ +\Delta \epsilon_ {p} ^ {r} = \frac {R \Delta \epsilon^ {r}}{1 + H ^ {\prime} / E}. \tag {3.39} +$$ + +Then the total current plastic strain for the element is + +$$ +\epsilon_ {p} ^ {r} = \epsilon_ {p} ^ {r - 1} + \frac {R \Delta \epsilon^ {r}}{1 + H ^ {\prime} / E}. \tag {3.40} +$$ + +Step g For elastic elements only, store the correct current stress as + +$$ +\sigma^ {r} = \sigma^ {r - 1} + \Delta \sigma_ {e} ^ {r}. \tag {3.41} +$$ + +(This in fact repeats Step c.) + +Step h Finally, calculate the equivalent nodal forces from the element stress according to + +$$ +f _ {1} = - f _ {2} = \left\{ \begin{array}{c c c} - \sigma^ {r} A & \text {for} & x _ {2} > x _ {1} \\ \sigma^ {r} A & \text {for} & x _ {2} < x _ {1}. \end{array} \right. \tag {3.42} +$$ + + + +Subroutine REFOR3 is now presented below and explanatory notes provided. + +```fortran +SUBROUTINE REFOR3 RFR3 1 +C**************************RFR3 2 +C RFR3 3 +C *** CALCULATES INTERNAL EQUIVALENT NODAL FORCES RFR3 4 +C RFR3 5 +C**************************RFR3 6 +COMMON/UNIM1/NPOIN,NELEM,NBOUN,NLOAD,NPROP,NNODE,IINCS,IITER, RFR3 7 +. KRESL,NCHEK,TOLER,NALGO,NSVAB,NDOFN,NINCS,NEVAB, RFR3 8 +. NITER,NOUTP,FACTO,PVALU RFR3 9 +COMMON/UNIM2/PROPS(5,4),COORD(26),LNODS(25,2),IFPRE(52), RFR3 10 +. FIXED(52),TLOAD(25,4),RLOAD(25,4),ELOAD(25,4), RFR3 11 +. MATNO(25),STRES(25,2),PLAST(25),XDISP(52), RFR3 12 +. TDISP(26,2),TREAC(26,2),ASTIF(52,52),ASLOD(52), RFR3 13 +. REACT(52),FRESV(1352),PEFIX(52),ESTIF(4,4) RFR3 14 +DO 10 IELEM=1,NELEM RFR3 15 +DO 10 IEVAB=1,NEVAB RFR3 16 +10 ELOAD(IELEM,IEVAB)=0.0 RFR3 17 +DO 70 IELEM=1,NELEM RFR3 18 +LPROP=MATNO(IELEM) RFR3 19 +YOUNG=PROPS(LPROP,1) RFR3 20 +XAREA=PROPS(LPROP,2) RFR3 21 +YIELD=PROPS(LPROP,3) RFR3 22 +HARDS=PROPS(LPROP,4) RFR3 23 +NODE1=LNODS(IELEM,1) RFR3 24 +NODE2=LNODS(IELEM,2) RFR3 25 +ELENG=ABS(COORD(NODE1)-COORD(NODE2)) RFR3 26 +IF(COORD(NODE2).GT.COORD(NODE1)) STRAN=(XDISP(NODE2)-XDISP(NODE1)) RFR3 27 +./ELENG RFR3 28 +IF(COORD(NODE2).LT.COORD(NODE1)) STRAN=(XDISP(NODE1)-XDISP(NODE2)) RFR3 29 +./ELENG RFR3 30 +STLIN=YOUNG*STRAN RFR3 31 +STCUR=STRES(IELEM,1)+STLIN RFR3 32 +PREYS=YIELD+HARDS*ABS(PLAST(IELEM)) RFR3 33 +IF(ABS(STRES(IELEM,1)).GE.PREYS) GO TO 20 RFR3 34 +ESCUR=ABS(STCUR)-PREYS RFR3 35 +IF(ESCUR.LE.0.0) GO TO 40 RFR3 36 +RFACT=ESCUR/ABS(STLIN) RFR3 37 +GO TO 30 RFR3 38 +20 IF(STRES(IELEM,1).GT.0.0.AND.STLIN.LE.0.0) GO TO 40 RFR3 39 +IF(STRES(IELEM,1).LT.0.0.AND.STLIN.GT.0.0) GO TO 40 RFR3 40 +RFACT=1.0 RFR3 41 +30 REDUC=1.0-RFACT RFR3 42 +STRES(IELEM,1)=STRES(IELEM,1)+REDUC*STLIN+RFACT*YOUNG*(1.0- +. YOUNG/(YOUNG+HARDS))*STRAN RFR3 44 +PLAST(IELEM)=PLAST(IELEM)+RFACT*STRAN*YOUNG/(YOUNG+HARDS) RFR3 45 +GO TO 50 RFR3 46 +40 STRES(IELEM,1)=STRES(IELEM,1)+STLIN RFR3 47 +50 IF(COORD(NODE2).GT.COORD(NODE1)) GO TO 60 RFR3 48 +ELOAD(IELEM,1)=STRES(IELEM,1)*XAREA RFR3 49 +ELOAD(IELEM,2)=-STRES(IELEM,1)*XAREA RFR3 50 +GO TO 70 RFR3 51 +60 ELOAD(IELEM,1)=-STRES(IELEM,1)*XAREA RFR3 52 +ELOAD(IELEM,2)=STRES(IELEM,1)*XAREA RFR3 53 +70 CONTINUE RFR3 54 +RETURN RFR3 55 +END RFR3 56 +``` + + + +RFR3 15-17 Initialise to zero the array in which the equivalent nodal forces for each element will be stored. + +RFR3 18 Loop over each element. + +RFR3 19 Identify the material property of each element. + +RFR3 20 Set YOUNG equal to the elastic modulus, E, of the material. + +RFR3 21 Set XAREA equal to the cross-sectional area. + +RFR3 22 Set YIELD equal to the uniaxial yield stress, $\sigma_{Y}$ , of the material. + +RFR3 23 Set HARDS equal to the hardening parameter, $H'$ , of the material. + +RFR3 24-25 Identify the node numbers of the element. + +RFR3 26 Calculate the element length. + +RFR3 27–30 Calculate the element strain, so that a tensile strain is positive. + +RFR3 31 Calculate $\Delta \sigma_{e}^{r}$ according to Step b. + +RFR3 32 Calculate $\sigma_{e}^{r}$ according to Step c. + +RFR3 33–34 Check if the element had yielded on the previous iteration, i.e., if $\sigma^{r-1} > \sigma_{Y} + H' \epsilon_{p}^{r-1}$ which is the first operation of Step d. The absolute value of $\sigma^{r-1}$ is taken to account for yielding in compression. + +RFR3 35–36 If the element was previously elastic, check to see if it has yielded during this iteration. + +RFR3 37 For an element which yields during this iteration, calculate + +$$ +R = \frac {\sigma_ {e} ^ {r} - \sigma_ {Y}}{\sigma_ {e} ^ {r} - \sigma^ {r - 1}} +$$ + +(Fig. 3.7(a)). The absolute value sign is taken to account for compressive loading. + +RFR3 39–40 Check to see if an element which had previously yielded is unloading during this iteration. If yes, go to 40. + +RFR3 41 Otherwise, set R = 1. + +RFR3 42 Evaluate, $(1 - R)$ . + +RFR3 43–44 For plastic elements, calculate the correct current stress, $\sigma^{r}$ , according to (3.36). + +RFR3 45 Also calculate the plastic strain, $\epsilon_{p}^{r}$ , according to (3.40). + +RFR3 47 For elastic elements, calculate the current stress, $\sigma^{r}$ , according to Step g. + +RFR3 48–53 Evaluate the equivalent nodal forces, according to Step h. + +RFR3 54 Termination of DO LOOP over the elements. + +# 3.12.3 Numerical examples + +The first example considered is the yielding of a bar under self weight loading. The problem and finite element idealisation employed is illustrated in Fig. 3.8. Progressive yielding is induced in the system by increasing the gravitational field incrementally. The gravitational force due to self weight + + + +![](images/page-098_35b8167453b5fa4dd85d33baec6bf6c9892d77b967058a8eb9b87cd3a7e54a79.jpg) + +
+line + +| End displacement (node 6) | Reaction (node 1) | +| ------------------------- | ----------------- | +| 0.000 | 0.0 | +| 0.002 | 10.0 | +| 0.004 | 12.0 | +| 0.006 | 14.0 | +| 0.008 | 16.0 | +| 0.010 | 18.0 | +| 0.012 | 20.0 | +| 0.014 | 22.0 | +| 0.016 | 24.0 | +| 0.018 | 26.0 | +| 0.020 | 28.0 | +| 0.022 | 30.0 | +
+ +Fig. 3.8 Load/displacement response of a vertical bar loaded by a progressively increasing self-weight. + +acting on each element is equally distributed to its two nodes. The structure is capable of carrying load beyond first yield, due to the strain hardening characteristic of the material. + +The second example considered is the compound bar shown in Fig. 3.9. The two bars have a different yield stress and cross-sectional area in order to induce differential yielding. The structure is loaded by an end load, P, which is systematically incremented. The load/extension characteristics for the system are shown in Fig. 3.9. It is seen that there is an initial loss of stiffness corresponding to yielding of the first bar followed by a further reduction when the second bar becomes plastic. + +This simple example suggests a method by which more complex material responses can be generated. By connecting two bars with different properties in parallel we obtain a material behaviour made up of three linear portions. + + + +![](images/page-099_95b2fcfd25d0a63313102b8df5086fb4443c5b2b9c924020612ac9c2f750c2ee.jpg) + + + +By connecting n bars in parallel and choosing the yield stress and cross-sectional area of each appropriately we can approximate any arbitrary stress/strain response piecewise linearly by $(n+1)$ intervals. This is the basis of the ‘overlay method’ $^{(7)}$ which will be described later in Chapter 8. + +Also included in Fig. 3.9 are the results for the case when the load is cycled. First the load is incremented in tension up to a certain level, then removed and applied compressively, before final removal. It is immediately seen that a Bauschinger effect $^{(8)}$ is obtained with initial yield in compression taking place at a reduced value. This occurs even though we have assumed an equal yield stress in tension and compression. This behaviour is attributable to the differential straining of the two components and is a phenomenon evident in real materials. + +# 3.13 Problems + +3.1 Reanalyse the problem of Fig. 3.3, Section 3.9.3, for the case where the term K is assumed to vary with the unknown $\phi$ according to + +$$ +K = 1 0 (1 + e ^ {3 \phi}). +$$ + +Use the direct iteration solution code QUITER, user instructions for which are provided in Appendix I, Section A1.1, for solution. + +3.2 Resolve Problem 3.1 using the Newton-Raphson procedure which is coded in program QUNEWT. User instructions for this program are provided in Appendix I, Section A1.2. Compare the computation times required for the two different solution procedures. + +3.3 The quasi-harmonic equation described in Section 2.3 is also applicable to groundwater flow problems. $^{(5)}$ In this application $\phi$ is the pressure head potential, K is the material permeability and Q is the rate at which water is being injected per unit volume of material. The flow velocity at any point is then given by $v = -K(d\phi/dx)$ . Figure 3.10 illustrates the problem of water seeping through two permeable strata whose permeabilities depend on the seepage velocity as shown. By treating the problem as one-dimensional in the vertical direction obtain a numerical solution for the steady state potential and velocity distribution in the two strata. + +3.4 Following the approach of Section 2.3, develop the stiffness matrix $H^{(e)}$ and the load vector $f^{(e)}$ for the one-dimensional axisymmetric situation. In this application all quantities are symmetric with respect to a central axis and the radial coordinate r now replaces x. + +3.5 Implement the formulation of Problem 3.4 in program QUITER. + +3.6 Use the computer code developed in Problem 3.5 to solve the problem of water flow in the horizontal place of the confined aquifer shown in Fig. 3.11. In this case $\phi$ is the piezometric head, K is the material permeability and Q is the rate at which water is being injected per unit volume of material. diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_011.md b/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_011.md new file mode 100644 index 00000000..c2a51204 --- /dev/null +++ b/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_011.md @@ -0,0 +1,400 @@ + + +![](images/page-101_cf52779feb491906a1102171de4099983114596c28c816337ddee9e89db974e3.jpg) + +
+text_image + +10m +20m +K = 2(1 + 2v) +K = 1 + v +
+ +Fig. 3.10 Groundwater flow example—Problem 3.3. + +![](images/page-101_6da9c68ab204211c7135c0138c01cddcfa08f68d938afdda444e82067f71e73c.jpg) + +
+text_image + +φ = 100 +K = 1 - 10 dφ/dr +1000m +Well +point +
+ +Fig. 3.11 Water flow in a confined aquifer—Problem 3.6. + + + +The circular region shown in Fig. 3.11 has a central well point at which water is being extracted at a rate of $200 \, m^{3}/day$ . Determine the steady state potential distribution for this system assuming the material permeability to be nonlinear in the manner shown. + +3.7 The relationship between stress, $\sigma$ , and strain, $\epsilon$ , for a certain locking material is given by the relationship + +$$ +\sigma = \frac {E _ {0} \epsilon}{\epsilon_ {L} (\epsilon_ {L} - \epsilon)}, \tag {3.43} +$$ + +in which $E_{0}$ is the elastic modulus and $\epsilon_{L}$ is the limiting strain value of the material. Implement this relation in program NONLAS documented in Appendix I, Section A1.3, by modifying the strain derivative function in Section 3.11.1. Also allow the behaviour of certain elements to be linear elastic. Use this modified program to determine the force displacement/relationship of the central node in Fig. 3.12 for a total applied load of 100 units. + +![](images/page-102_0489c764bd10a78b7874855be312985168200715610dd7aa80be07f6148d0efa.jpg) + +
+text_image + +100 +Cross-sectional area, A = 1 +for both members +2 +2 +Nonlinear +elastic +E₀ = 1000 +εₗ = 0.1 +Linear elastic +material, E = 1000 +
+ +Fig. 3.12 Nonlinear elastic example—Problem 3.7. + +3.8 Use program ELPLAS, for which user instructions are provided in Appendix I, Section A1.4, to solve the one-dimensional elasto-plastic problem shown in Fig. 3.13. +3.9 Develop the elastic stiffness matrix, $K^{(e)}$ , for a two-node finite element in the form of a thin disc of thickness t which is to be subjected to axisymmetric in-plane loading. Assume a linear variation between nodes, as shown in Fig. 2.7, and note the following relationships + +$$ +\epsilon_ {r} = \frac {d u}{d r} = \frac {1}{E} (\sigma_ {r} - \nu \sigma_ {\theta}) +$$ + +$$ +\epsilon_ {\theta} = \frac {u}{r} = \frac {1}{E} (\sigma_ {\theta} - \nu \sigma_ {r}), \tag {3.44} +$$ + + + +![](images/page-103_f91950ee1fdef5b13f1cd340e16230c51156e0253a1e29e838e96fd9dcdf744d.jpg) + +
+other + +Prescribed axial +displacement. φ = 0.25 +| Section | Material | E | A | σγ | H' | +| :--- | :--- | :--- | :--- | :--- | :--- | +| Material 1 | 10 | 1000 | 1.0 | 10.0 | 0.0 | +| Material 2 | 10 | 1000 | 1.0 | 15.0 | -100.0 | +
+ +Fig. 3.13 Elasto-plastic example—Problem 3.8. + +in which u is the radial displacement and E and $\nu$ are respectively the elastic modulus and Poisson's ratio of the material. Also express the stresses $\sigma_{r}$ and $\sigma_{\theta}$ in terms of the nodal displacements $\phi_{1}$ and $\phi_{2}$ . + +3.10 Use the stiffness matrix evaluated in Problem 3.9 to modify program ELPLAS to allow solution of one-dimensional axisymmetric problems by the initial stiffness method. Assume a Tresca yield criterion (discussed in Chapter 7) where yielding is assumed to begin when the maximum shearing stress reaches a critical value. For the present application this implies commencement of yielding when either $\sigma_{r}$ or $\sigma_{\theta}$ reaches the uniaxial yield stress, $\sigma_{Y}$ . + +3.11 Employ the program developed in Problem 3.10 to determine the elasto-plastic stress distribution in a thin disc, of thickness 1 mm, subjected to internal pressure loading. Take the internal and external + +![](images/page-103_9be979aebddd3873b96201f1e4ebeaae5e0e82be36e6b7d261faa4ede290e3ed.jpg) + +
+text_image + +Axis of +symmetry +σθ +dr +σr +dr +σr + dσr +dr +dθ +r +f1 +φ1 +t +φ2 +f2 +r1 +r2 +
+ +Fig. 3.14 Axisymmetric membrane element—Problem 3.9 + + + +radii of the disc as 5 cm and 10 cm respectively, the elastic modulus $E=2\times10^{5}$ N/mm $^{2}$ , Poisson's ratio $\nu=0\cdot3$ and the uniaxial yield stress, $\sigma_{Y}=300$ N/mm $^{2}$ . Compare your solution with the theoretical expressions given in Ref. 8. + +# 3.14 References + +1. RALSTON, A., A First Course in Numerical Analysis, McGraw-Hill, 1965. +2. JENNINGS, A., Matrix Computation for Engineers and Scientists, John Wiley, 1977. +3. HINTON, E. and OWEN, D. R. J., Finite Element Programming, Academic Press, London, 1977. +4. SALVADORI, M. G. and BARON, M. L., Numerical Methods in Engineering, Prentice-Hall, New Jersey, 1964. +5. HINTON, E. and OWEN, D. R. J., An Introduction to Finite Element Computations, Pineridge Press, Swansea, U.K., 1979. +6. BARRER, R. M., Measurement of diffusion and thermal conductivity 'constants' in non-homogeneous media, and in media where these constants depend respectively on concentration or temperature, Proc. Phys. Soc. 58, 321–331 (1946). +7. OWEN, D. R. J., PRAKASH, A. and ZIENKIEWICZ, O. C., Finite element analysis of non-linear composite materials by use of overlay systems, Computers and Structures, 4, 1251–1267 (1974). +8. HILL, R., The Mathematical Theory of Plasticity, Oxford University Press, 1950. + + + +# Chapter 4 Viscoplastic problems in one dimension + +# 4.1 Introduction + +In this chapter the basic concepts of viscoplasticity are introduced by the consideration of one-dimensional situations. This topic is then studied further in Chapter 8 where the case of a general continuum is treated. + +Viscoplastic theory allows the modelling of time rate effects in the plastic deformation process. Thus after initial yielding of the material the plastic flow, and the resulting stresses and strains, are time dependent. Such effects are always present to some degree in all materials but they may or may not be significant depending on the physical situation being considered. + +The basic theory of viscoplasticity in one dimension is developed and a numerical solution process is then described. All the essential features of viscoplasticity can be demonstrated with reference to one-dimensional behaviour. Finally the solution process is coded in FORTRAN to form a working program and the basic characteristics of a viscoplastic material response are illustrated by the solution of numerical examples. + +# 4.2 Basic theory + +The concept of viscoplastic material behaviour is best introduced by means of the one-dimensional rheological model illustrated in Fig. 4.1. The friction slider component develops a stress $\sigma_{p}$ , becoming active only if $\sigma > Y$ , where $\sigma$ is the total applied stress and Y is some limiting yield value. The excess stress $\sigma_{d} = \sigma - \sigma_{p}$ is carried by the viscous dashpot. Instantaneous elastic response is, of course, provided by the linear spring. The presence of the dashpot allows the stress level to instantaneously exceed the value predicted by plasticity theory, the solution tending to this equilibrium level as steady state conditions are achieved in the system. + +The total strain in the model is given by the sum of the elastic and viscoplastic components as + +$$ +\epsilon = \epsilon_ {e} + \epsilon_ {v p}. \tag {4.1} +$$ + +The stress in the linear spring is equal to the total applied stress and is + + + +![](images/page-106_b5d17b5fdab269cd4a8f3dc2928428e0fd1c73a677d6a06b7b465f18c80bba4a.jpg) + +
+text_image + +σd = σ - σp. +σp +εvp +εe +σp +Inactive if +σp < Y +σp +
+ +Fig. 4.1 Basic one-dimensional elastic-viscoplastic model. + +related to the elastic strain by + +$$ +\sigma_ {e} = \sigma = E \epsilon_ {e}, \tag {4.2} +$$ + +where E is the elastic modulus of the linear spring. + +The stress level in the friction slider depends on whether or not the threshold or yield stress, Y, has been reached. The onset of viscoplastic deformation is governed by a uniaxial yield stress $\sigma_{Y}$ . The stress level for continuing viscoplastic flow depends on the strain-hardening characteristics of the material. Restricting discussion to a linear strain-hardening response as discussed in Section 2.5, the stress level for viscoplastic yielding at any stage is given by + +$$ +Y = \sigma_ {Y} + H ^ {\prime} \epsilon_ {v p}, \tag {4.3} +$$ + +in which $H'$ is the slope of the strain hardening portion of the stress-strain curve after removal of the elastic strain component and $\epsilon_{vp}$ is the current viscoplastic strain. Thus the stress in the friction slider is + +$$ +\begin{array}{r l} \sigma_ {p} = \sigma & \text { if } \left\{ \begin{array}{l} \sigma_ {p} < Y \\ \sigma_ {p} \geqslant Y. \end{array} \right. \\ = Y \end{array} \tag {4.4} +$$ + + + +The stress in the viscous dashpot, $\sigma_{d}$ , is related to the viscoplastic strain by + +$$ +\sigma_ {d} = \mu \frac {d \epsilon_ {v p}}{d t}, \tag {4.5} +$$ + +where $\mu$ is a viscosity coefficient and $t$ denotes the time. We note that + +$$ +\sigma = \sigma_ {d} + \sigma_ {p}. \tag {4.6} +$$ + +Before the onset of viscoplastic yielding $\epsilon_{vp}=0$ , giving $\sigma_{d}=0$ from (4.5) and consequently $\sigma_{p}=\sigma$ . It now remains to establish the constitutive relationship for the model under both elastic and elasto-viscoplastic conditions. + +Before viscoplastic yielding, $\epsilon_{vp} = 0$ and from (4.1) and (4.2) we have the elastic stress–strain relation to be + +$$ +\sigma = E \epsilon . \tag {4.7} +$$ + +Substituting from (4.4) and (4.5) in (4.6) gives + +$$ +\sigma_ {Y} + H ^ {\prime} \epsilon_ {v p} + \mu \frac {d \epsilon_ {v p}}{d t} = \sigma . \tag {4.8} +$$ + +Substituting for $\epsilon_{vp}$ from (4.1) and using (4.2) results in + +$$ +H ^ {\prime} E \epsilon + \mu E \frac {d \epsilon}{d t} = H ^ {\prime} \sigma + E (\sigma - \sigma_ {Y}) + \mu \frac {d \sigma}{d t}, \tag {4.9} +$$ + +which is a first order ordinary differential equation defining the time-dependent relationship between stress and strain under viscoplastic conditions. At this stage we introduce a fluidity parameter, $\gamma$ , such that + +$$ +\gamma = \frac {1}{\mu}. \tag {4.10} +$$ + +Substituting in (4.9) and rearranging + +$$ +\dot {\epsilon} = \frac {\dot {\sigma}}{E} + \gamma [ \sigma - (\sigma_ {Y} + H ^ {\prime} \epsilon_ {v p}) ], \tag {4.11} +$$ + +in which ( $\cdot$ ) denotes derivative with respect to time, t. Or + +$$ +\dot {\epsilon} = \dot {\epsilon} _ {e} + \dot {\epsilon} _ {v p}, \tag {4.12} +$$ + +where + +$$ +\dot {\epsilon} _ {e} = \frac {\dot {\sigma}}{E}, \tag {4.13} +$$ + +and + +$$ +\dot {\epsilon} _ {v p} = \gamma [ \sigma - (\sigma_ {Y} + H ^ {\prime} \epsilon_ {v p}) ]. \tag {4.14} +$$ + + + +Expression (4.14) defines the viscoplastic strain rate in terms of the portion of stress in excess of the steady state yield value. + +It is instructive to consider the closed form solution to (4.9). Consider the case when a constant applied stress $\sigma = \sigma_{A}$ is applied to the model. Then (4.9) reduces, (using (4.10)), to + +$$ +\gamma H ^ {\prime} \epsilon + \frac {d \epsilon}{d t} = \frac {\gamma H ^ {\prime}}{E} \sigma_ {A} + \gamma (\sigma_ {A} - \sigma_ {Y}). \tag {4.15} +$$ + +The solution to this first-order ordinary differential equation is elementary and is + +$$ +\epsilon = \frac {\sigma_ {A}}{E} + \frac {(\sigma_ {A} - \sigma_ {Y})}{H ^ {\prime}} [ 1 - \mathrm{e} ^ {- H ^ {\prime} \gamma t} ], \tag {4.16} +$$ + +![](images/page-108_c015ac0f80e97889a76d3bab7b42b9a0b44e40623c89a8a6689611f133947fc6.jpg) + +
+line +| Time, t | Strain, ε | +| ------- | --------- | +| 0 | 0 | +| t | σ_A/E | +| t | σ_A - σ_Y/H' | +
+ +(a) + +![](images/page-108_58153e5018a76b200a341cc975b30d3d84297fabfba132182da8b24b6fd777cd.jpg) + +
+line + +| Time, t | Strain, ε | +| ------- | --------- | +| 0 | 0 | +| 1 | 1 | +| 2 | 2 | +| 3 | 3 | +| 4 | 4 | +| 5 | 5 | +| 6 | 6 | +| 7 | 7 | +| 8 | 8 | +| 9 | 9 | +| 10 | 10 | +
+ +(b) +Fig. 4.2 Strain response with time for the model of Fig. 4.1 due to a constant applied load. (a) Linear strain hardening material. (b) Perfectly plastic material. + + + +provided that $H'$ is nonzero. The form of the response is shown in Fig. 4.2(a). Following an initial elastic response, the strain in the model attains the steady state value indicated in an exponential fashion. + +The case of a perfectly viscoplastic material in which $H' = 0$ , can be obtained by taking the limit as $H'$ tends to zero in (4.16) and applying L'Hopital's rule. This results in + +$$ +\epsilon = \frac {\sigma_ {A}}{E} + \left(\sigma_ {A} - \sigma_ {Y}\right) \gamma t. \tag {4.17} +$$ + +This response is shown in Fig. 4.2(b). In this case it is seen that a steady state condition is not achieved and that viscoplastic deformation continues indefinitely at a constant strain rate. The different behaviour shown in Figs. 4.2(a) and 4.2(b) arises from the fact that for a strain hardening material the viscoplastic yield stress increases according to (4.3) until it reaches the applied stress level $\sigma_{A}$ at which stage the viscoplastic strain rate becomes zero. On the other hand, for a perfectly viscoplastic material there is always a stress imbalance of $\sigma_{A}-\sigma_{Y}$ in the system which does not reduce and consequently steady state conditions cannot be achieved. + +We note that in (4.16) and (4.17) that the time $t$ only enters the expressions through the term $\gamma t$ . Therefore the solution for a material with a different fluidity parameter $\gamma$ can be obtained by a simple adjustment of the time scale. + +# 4.3 Numerical solution process + +Viscoplasticity is a transient phenomenon and therefore the essential objective of a numerical solution process is to determine the displacement, strains and stresses throughout the time interval of interest. Consequently some time stepping or time marching scheme must be introduced in order to allow the solution to be advanced from a time $t_{n}$ to time $t_{n+1} = t_{n} + \Delta t_{n}$ , where subscripts n and $n+1$ denote successive times and $\Delta t_{n}$ the interval between. The simplest method of incrementing quantities over a time interval is afforded by Euler's rule. In this the mean rate of change over the interval is taken as the value at the beginning of the interval and thus the predicted value of some quantity X at time $t_{n+1}$ is extrapolated from the value at time $t_{n}$ to be + +$$ +X ^ {n + 1} = X ^ {n} + (\dot {X}) ^ {n} \Delta t _ {n}. \tag {4.18} +$$ + +This scheme becomes unstable for time steps exceeding a critical value and estimation of the limiting step length is discussed in Section 4.4. The Euler method, however, remains attractive due to its simplicity. + +With the viscoplastic strain rate defined by (4.14) we can define the strain increment $\Delta\epsilon_{vp}^{n}$ occurring in a time interval $\Delta t_{n}=t_{n+1}-t_{n}$ , using (4.18), as + +$$ +\Delta \epsilon_ {v p} ^ {n} = \dot {\epsilon} _ {v p} ^ {n} \Delta t _ {n}. \tag {4.19} +$$ + + + +We note that the time step length can, in general, be different for each time interval. + +![](images/page-110_96db7ffbfe4d5a9f9f2433d08d37fc39b112173d0ae56e158c1f43435e3c2827.jpg) + +
+text_image + +Δφ₁ +f₁ +Δφ₂ +f₂ +1 +2 +L +
+ +Fig. 4.3 One-dimensional two-noded element with linear displacement variation. + +With reference to Fig. 4.3, consider the behaviour of a linear displacement element, which is of length L and has a cross-sectional area, A. The change of length in this element associated with strain increment (4.19) is + +$$ +\Delta \phi^ {n} = \Delta \epsilon_ {v p} ^ {n} L, \tag {4.20} +$$ + +or adding the displacement change due to a change in applied loading $\Delta f^{n}$ occurring between times $t_{n}$ and $t_{n+1}$ we obtain the total change in element length to be + +$$ +\Delta \phi^ {n} = \Delta \epsilon_ {v p} ^ {n} L + \frac {L}{A E} \Delta f ^ {n}. \tag {4.21} +$$ + +This can be rewritten in matrix form, in terms of the nodal displacements and forces as + +$$ +\Delta \varphi^ {n} = [ K ] ^ {- 1} \Delta V ^ {n}, \tag {4.22} +$$ + +where + +$$ +\Delta \varphi^ {n} = \left[ \begin{array}{l} \Delta \phi_ {1} ^ {n} \\ \Delta \phi_ {2} ^ {n} \end{array} \right], \tag {4.23} +$$ + +$$ +\Delta V ^ {n} = A E \dot {\epsilon} _ {v p} ^ {n} \Delta t _ {n} \left[ \begin{array}{c} 1 \\ - 1 \end{array} \right] + \Delta f ^ {n}, \tag {4.24} +$$ + +and + +$$ +\boldsymbol {K} ^ {(e)} = \frac {E A}{L} \left[ \begin{array}{c c} 1 & - 1 \\ - 1 & 1 \end{array} \right]. \tag {4.25} +$$ + +In the above, $\Delta V^{n}$ are termed the pseudo forces and $\Delta\varphi^{n}$ and $\Delta f^{n}$ are respectively the incremental changes in the nodal displacements and applied forces for the element. diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_012.md b/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_012.md new file mode 100644 index 00000000..ad8e8bc2 --- /dev/null +++ b/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_012.md @@ -0,0 +1,503 @@ + + +We note in passing that expressions (4.24) and (4.25) could be written in the standard finite element form + +$$ +\Delta V ^ {n} = \int_ {V} \boldsymbol {B} ^ {T} \boldsymbol {D} \epsilon d V + \Delta \boldsymbol {f} ^ {n} +$$ + +$$ +\boldsymbol {K} ^ {(e)} = \int_ {V} \boldsymbol {B} ^ {T} \boldsymbol {D} \boldsymbol {B} d V, \tag {4.26} +$$ + +since for the linear element considered + +$$ +\boldsymbol {B} = \left[ - \frac {1}{L}, \frac {1}{L} \right] +$$ + +$$ +\boldsymbol {D} = \boldsymbol {E} +$$ + +$$ +\int_ {V} d V = A L. \tag {4.27} +$$ + +The displacements at time $t_{n+1}$ are then obtained by simple accumulation as + +$$ +\varphi^ {n + 1} = \varphi^ {n} + \Delta \varphi^ {n}. \tag {4.28} +$$ + +The stress increment is given from (4.1) and (4.7) to be + +$$ +\Delta \sigma^ {n} = E \Delta \epsilon_ {e} ^ {n} = E \left(\Delta \epsilon^ {n} - \Delta \epsilon_ {v p} ^ {n}\right), \tag {4.29} +$$ + +or + +$$ +\Delta \sigma^ {n} = E \left(\frac {\Delta \phi_ {1} ^ {n} - \Delta \phi_ {2} ^ {n}}{L} - \dot {\epsilon} _ {v p} ^ {n} \Delta t _ {n}\right), \tag {4.30} +$$ + +where $\Delta\phi_{1}^{n}$ and $\Delta\phi_{2}^{n}$ are the displacement changes at the nodes of the element. + +The stress at time $t_{n+1}$ is then given by + +$$ +\sigma^ {n + 1} = \sigma^ {n} + \Delta \sigma^ {n}. \tag {4.31} +$$ + +The total viscoplastic strain at time $t_{n+1}$ is + +$$ +\epsilon_ {v p} ^ {n + 1} = \epsilon_ {v p} ^ {n} + \Delta \epsilon_ {v p} ^ {n}, \tag {4.32} +$$ + +and finally the viscoplastic strain rate at $t_{n+1}$ is given, from (4.14) as + +$$ +\dot {\epsilon} _ {v p} ^ {n + 1} = \gamma \left[ \sigma^ {n + 1} - \left(\sigma_ {Y} + H ^ {\prime} \epsilon_ {v p} ^ {n + 1}\right) \right]. \tag {4.33} +$$ + +In employing the Euler scheme for time-stepping, we are effectively linearising the variation of quantities over the increment. Therefore the total stresses $\sigma^{n+1}$ obtained by accumulating all such stress increments may not be in exact equilibrium with the applied forces. It is therefore necessary to introduce an equilibrium correction procedure into the numerical solution algorithm. The simplest approach is to evaluate the out-of-balance nodal forces at the end of each time step and consider these forces as additional forces to be applied at the beginning of the next time increment. + + + +The out-of-balance or residual forces, $\psi$ , for the general element are given as the algebraic sum of the applied nodal loads and the nodal forces equivalent to the element stress, so that + +$$ +\psi^ {n + 1} = A \sigma^ {n + 1} \left[ \begin{array}{c} 1 \\ - 1 \end{array} \right] + f ^ {n + 1}, \tag {4.34} +$$ + +in which $\sigma^{n+1}$ is the element stress and $f^{n+1}$ are the total applied forces at time $t_{n+1}$ . These residual forces are then added to the pseudo forces to give for the next time increment + +$$ +\Delta V ^ {n + 1} = A E \dot {\epsilon} _ {v p} ^ {n + 1} \Delta t _ {n + 1} \left[ \begin{array}{c} 1 \\ - 1 \end{array} \right] + \Delta f ^ {n + 1} + \psi^ {n + 1}. \tag {4.35} +$$ + +This sequence is repeated for each time step until solution is either obtained for the desired time duration or until steady state conditions are achieved. Steady state conditions are deemed to have been achieved when the viscoplastic strain rate, $\dot{\epsilon}_{vp}^{n}$ , becomes tolerably small. + +# 4.4 Limiting time-step length + +The critical time-step length for viscoplastic solution using the Euler time marching scheme has been established by Cormeau. $^{(1)}$ For the uniaxial case considered in this chapter the limiting value is + +$$ +\Delta t \leqslant \frac {\sigma_ {Y}}{\gamma E}. \tag {4.36} +$$ + +Alternatively the time-step length can be limited according to a semi-empirical relationship. Such an approach is essential for some general continuum problems where a theoretical value of the critical time-step length may not exist. The most obvious procedure is to limit the viscoplastic strain increment to be some specified factor, $\tau$ , of the total current strain, + +$$ +\dot {\epsilon} _ {v p} ^ {n} \Delta t _ {n} \leqslant \tau \epsilon^ {n}. \tag {4.37} +$$ + +Since each element generally has a different strain level, expression (4.37) will yield a different limiting step value when applied to each element in turn. Therefore the limiting value is restricted according to + +$$ +\Delta t _ {n} \leqslant \tau \left[ \frac {\epsilon^ {n}}{\dot {\epsilon} _ {v p} {} ^ {n}} \right] _ {\min}, \tag {4.38} +$$ + +where the minimum value of $\Delta t_{n}$ obtained after considering each element is taken. Stability of the solution process is also aided by restricting the length of successive time steps according to + +$$ +\Delta t _ {n + 1} \leqslant k \Delta t _ {n}, \tag {4.39} +$$ + +where k is a specified constant generally chosen in the range 1·5–2·0. + + + +# 4.5 Computational procedure + +Before proceeding with the development of a computer code for the solution of one-dimensional viscoplastic problems we will first summarise the essential steps of the computation. Solution to the problem must commence from the known initial conditions at time t = 0 which of course correspond to the initial elastic response. At this stage $\varphi^{0}, f^{0}, \epsilon^{0}, \sigma^{0}$ are known and $\epsilon_{vp}^{0} = 0$ . The general procedure for advancing the solution from a time $t_{n}$ to time $t_{n+1}$ is the following. + +Stage I At time $t = t_{n}$ the values of $\sigma^{n}$ , $\epsilon^{n}$ , $\epsilon_{vp}^{n}$ and $f^{n}$ are known for each element and the nodal displacements are also known. The viscoplastic strain rate for each element is then evaluated according to (4.14) as + +$$ +\dot {\epsilon} _ {v p} ^ {n} = \gamma \left[ \sigma^ {n} - \left(\sigma_ {Y} + H ^ {\prime} \epsilon_ {v p} ^ {n}\right) \right]. \tag {4.40} +$$ + +Stage 2 (a) Compute the displacement increments, $\Delta \varphi^n$ , according to (4.22)-(4.25), as + +$$ +\Delta \varphi^ {n} = [ K ] ^ {- 1} \Delta V ^ {n}, +$$ + +where + +$$ +\Delta V ^ {n} = A E \dot {\epsilon} _ {v p} ^ {n} \Delta t _ {n} \left[ \begin{array}{c} 1 \\ - 1 \end{array} \right] + \Delta f ^ {n}, +$$ + +and the stiffness matrix for an individual element is + +$$ +\boldsymbol {K} ^ {(e)} = \frac {E A}{L} \left[ \begin{array}{c c} 1 & - 1 \\ - 1 & 1 \end{array} \right]. +$$ + +(b) Calculate the stress increment $\Delta\sigma^{n}$ and the viscoplastic strain increment $\Delta\epsilon_{vp}^{n}$ for each element as + +$$ +\Delta \sigma^ {n} = E \left(\frac {\Delta \phi_ {1} ^ {n} - \Delta \phi_ {2} ^ {n}}{L} - \dot {\epsilon} _ {v p} ^ {n} \Delta t _ {n}\right), +$$ + +$$ +\Delta \epsilon_ {v p} ^ {n} = \dot {\epsilon} _ {v p} ^ {n} \Delta t _ {n}. +$$ + +Stage 3 Determine the total displacements, stresses and viscoplastic strain + +$$ +\varphi^ {n + 1} = \varphi^ {n} + \Delta \varphi^ {n}, +$$ + +$$ +\sigma^ {n + 1} = \sigma^ {n} + \Delta \sigma^ {n}, +$$ + +$$ +\epsilon_ {v p} ^ {n + 1} = \epsilon_ {v p} ^ {n} + \Delta \epsilon_ {v p} ^ {n}. +$$ + +Stage 4 Calculate the viscoplastic strain rate for each element + +$$ +\dot {\epsilon} _ {v p} ^ {n + 1} = \gamma [ \sigma^ {n + 1} - (\sigma_ {Y} + H ^ {\prime} \epsilon_ {v p} ^ {n + 1}) ]. +$$ + + + +Stage 5 Apply the equilibrium correction. Evaluate the residual forces, for each element, as + +$$ +\psi^ {n + 1} = A \sigma^ {n + 1} \left[ \begin{array}{c} 1 \\ - 1 \end{array} \right] + f ^ {n + 1}. +$$ + +Add these into the vector of incremental pseudo loads for use in the next time step + +$$ +\Delta V ^ {n + 1} = A E \dot {\epsilon} _ {v p} ^ {n + 1} \Delta t _ {n + 1} \left[ \begin{array}{c} 1 \\ - 1 \end{array} \right] + \Delta f ^ {n + 1} + \psi^ {n + 1}. +$$ + +Stage 6 Check to see if the viscoplastic strain rate $\dot{\epsilon}_{vp}^{n+1}$ in each element has become tolerably small. If so, steady state conditions have been reached and the solution is either terminated or the next load increment is applied. If $\dot{\epsilon}_{vp}^{n+1}$ is non-zero return to Stage 1 and repeat the entire procedure for the next time step. + +# 4.6 Program structure + +The organisation of the one-dimensional viscoplastic program is shown in Fig. 4.4 where, in particular, the order in which subroutines are accessed is indicated. The operations undertaken by the program are those described in Section 4.5. Many of the subroutines employed are common to the one-dimensional plasticity application described in Chapter 3 and, since they are used in the present program without modification, the reader will be referred to the appropriate section for details. Only the additional subroutines necessary to complete the computer package will be described in this chapter. + +With reference to Fig. 4.4 the following subroutines have been already described where indicated below: + +Subroutine ASSEMB —Section 3.4.2 + +Subroutine GREDUC—Section 3.4.3 + +Subroutine BAKSUB —Section 3.4.4 + +Subroutine RESOLV —Section 3.4.5 + +Subroutine RESULT —Section 3.5 + +Subroutine INITIAL —Section 3.6\* + +Also, Subroutine DATA described in Section 3.2 is used with some minor modifications. A viscoplastic material in one dimension requires five individual quantities to describe it completely. Thus NPROP becomes 5 and the following quantities must be specified as material properties. + +PROPS (NUMAT, 1)—The elastic modulus, E, of the material + +PROPS (NUMAT, 2)—The cross-sectional area, A, of the element + +PROPS (NUMAT, 3)—The uniaxial yield stress, $\sigma_{Y}$ , of the material + +PROPS (NUMAT, 4)—The linear strain hardening parameter, $H'$ , for the material + +PROPS (NUMAT, 5)—The fluidity parameter, $\gamma$ , controlling the viscoplastic strain rate. + +\* Subroutine NONAL, described in Section 3.3, is also employed but with IITER now replaced by the time step index, ISTEP. + + + +![](images/page-115_f42e61708dc6ef92588fcf0e5382f1942a78c61947d1e760ed792a6bb99722ed.jpg) + +
+flowchart + +```mermaid +graph TD + A["START"] --> B["DATA\nInput data defining geometry, loading, boundary conditions, material properties, etc."] + B --> C["STUNVP\nCalculate the stiffness matrix for each element"] + C --> D["ASSEMB\nAssemble the element loads and stiffnesses to give the global stiffness matrix and load vector"] + D --> E["GREDUC, BAKSUB & RESOLV\nSolve the resulting system of simultaneous equations for the displacements φ"] + E --> F["INCVP\na) Evaluate quantities at the end of the timestep\nb) Calculate the pseudo loads for application during the next time step"] + F --> G["CONVP\nCheck for convergence of the time stepping process to steady state conditions"] + G --> H["RESULT\nPrint the results for the current timestep"] + H --> I["END"] + I --> J["LOAD INCREMENT LOOP"] + J --> K["TIME STEPPING LOOP"] + K --> A +``` +
+ +Fig. 4.4 Operational sequence for the one-dimensional viscoplastic stress analysis program. + +Input data are also received by this segment which controls the time-stepping algorithm. The following information is input: + +TAUFT The parameter $\tau$ discussed in Section 4.4 + +DTINT The time-step length for the first time step + +FTIME The factor k defined in (4.39) which limits the relative length of successive time steps + +The additional subroutines which are required will now be described in turn. + + + +# 4.7 Element stiffness subroutine STUNVP + +In all stages of the viscoplastic solution the elastic element stiffness matrix is employed, as indicated in (4.25). Consequently the structure of subroutine STUNVP, which evaluates the stiffness matrix for each element in turn, is straightforward and can be presented without further comment. + +```fortran +SUBROUTINE STUNVP SNVP 1 +C**************************SNVP 2 +C SNVP 3 +C *** CALCULATES ELEMENT STIFFNESS MATRICES SNVP 4 +C SNVP 5 +C**************************SNVP 6 +COMMON/UNIM1/NPOIN,NELEM,NBOUN,NLOAD,NPROP,NNODE,IINCS,ISTEP, SNVP 7 +. KRESL,NCHEK,TOLER,NALGO,NSVAB,NDOFN,NINCS,NEVAB, SNVP 8 +. NSTEP,NOUTP,FACTO,TAUFT,DTINT,FTIME,FIRST,PVALU, SNVP 9 +. DTIME,TTIME SNVP 10 +COMMON/UNIM2/PROPS(5,5),COORD(26),LNODS(25,2),IFPRE(52), SNVP 11 +. FIXED(52),TLOAD(25,4),RLOAD(25,4),ELOAD(25,4), SNVP 12 +. MATNO(25),STRES(25,2),PLAST(25),XDISP(52), SNVP 13 +. TDISP(26,2),TREAC(26,2),ASTIF(52,52),ASLOD(52), SNVP 14 +. REACT(52),FRESV(1352),PEFIX(52),ESTIF(4,4),VIVEL(25) SNVP 15 +REWIND 1 SNVP 16 +DO 10 IELEM=1,NELEM SNVP 17 +LPROP=MATNO(IELEM) SNVP 18 +YOUNG=PROPS(LPROP,1) SNVP 19 +XAREA=PROPS(LPROP,2) SNVP 20 +NODE1=LNODS(IELEM,1) SNVP 21 +NODE2=LNODS(IELEM,2) SNVP 22 +ELENG=ABS(COORD(NODE1)-COORD(NODE2)) SNVP 23 +FMULT=YOUNG*XAREA/ELENG SNVP 24 +ESTIF(1,1)=FMULT SNVP 25 +ESTIF(1,2)=-FMULT SNVP 26 +ESTIF(2,1)=-FMULT SNVP 27 +ESTIF(2,2)=FMULT SNVP 28 +WRITE(1) ESTIF SNVP 29 +10 CONTINUE SNVP 30 +RETURN SNVP 31 +END SNVP 32 +``` + +SNVP 16 Rewind the file on which the stiffness matrix of each element will be stored. + +SNVP 17 Loop over each element. + +SNVP 18 Identify the material property of the current element. + +SNVP 19–20 Set YOUNG equal to the material elastic modulus and XAREA equal to the cross-sectional area. + +SNVP 21–22 Identify the node numbers of the element. + +SNVP 23 Calculate the element length. + +SNVP 24 Compute EA/L as FMULT. + +SNVP 25-28 Evaluate the components of the element stiffness matrix according to (4.25). + +SNVP 29 Write the element stiffness matrix on to disc file. + +SNVP 30 End of loop over each element. + + + +# 4.8 Subroutine INCVP for the evaluation of end of time-step quantities and equilibrium correction terms + +This subroutine evaluates quantities such as stresses and viscoplastic strains at the end of the current time step and also calculates the loading to be applied during the next time step. Essentially it undertakes Stages 3–5 described in Section 4.5. All quantities at the end of time step n are calculated as $()^{n+1}$ . + +The program presented is restricted to loading which is applied in discrete increments and is assumed to remain constant during the time-stepping process for any given increment. Thus in (4.35) $\Delta f^{n} = 0$ for all stages other than the first time step of a particular load increment. + +Subroutine INCVP is now presented and described. + +```fortran +SUBROUTINE INCVP INVP 1 +C********** INVP 2 +C INVP 3 +C *** CALCULATES INTERNAL EQUIVALENT NODAL FORCES INVP 4 +C INVP 5 +C********** INVP 6 +COMMON/UNIM1/NPOIN,NELEM,NBOUN,NLOAD,NPROP,NNODE,IINCS,ISTEP, INVP 7 +. KRESL,NCHEK,TOLER,NALGO,NSVAB,NDOFN,NINCS,NEVAB, INVP 8 +. NSTEP,NOUTP,FACTO,TAUFT,DTINT,FTIME,FIRST,PVALU, INVP 9 +. DTIME,TTIME INVP 10 +COMMON/UNIM2/PROPS(5,5),COORD(26),LNODS(25,2),IFPRE(52), INVP 11 +. FIXED(52),TLOAD(25,4),RLOAD(25,4),ELOAD(25,4), INVP 12 +. MATNO(25),STRES(25,2),PLAST(25),XDISP(52), INVP 13 +. TDISP(26,2),TREAC(26,2),ASTIF(52,52),ASLOD(52), INVP 14 +. REACT(52),FRESV(1352),PEFIX(52),ESTIF(4,4),VIVEL(25) INVP 15 +DO 10 IELEM=1,NELEM INVP 16 +DO 10 IEVAB=1,NEVAB INVP 17 +10 ELOAD(IELEM,IEVAB)=0.0 INVP 18 +DNEXT=FTIME*DTIME INVP 19 +DO 30 IELEM=1,NELEM INVP 20 +LPROP=MATNO(IELEM) INVP 21 +YOUNG=PROPS(LPROP,1) INVP 22 +XAREA=PROPS(LPROP,2) INVP 23 +YIELD=PROPS(LPROP,3) INVP 24 +HARDS=PROPS(LPROP,4) INVP 25 +GAMMA=PROPS(LPROP,5) INVP 26 +NODE1=LNODS(IELEM,1) INVP 27 +NODE2=LNODS(IELEM,2) INVP 28 +ELENG=ABS(COORD(NODE1)-COORD(NODE2)) INVP 29 +IF(COORD(NODE2).GT.COORD(NODE1)) STRAN=(XDISP(NODE2)-XDISP(NODE1)) INVP 30 +./ELENG INVP 31 +IF(COORD(NODE2).LT.COORD(NODE1)) STRAN=(XDISP(NODE1)-XDISP(NODE2)) INVP 32 +./ELENG INVP 33 +STRES(IELEM,1)=STRES(IELEM,1)+(STRAN-VIVEL(IELEM)*DTIME)*YOUNG INVP 34 +PLAST(IELEM)=PLAST(IELEM)+VIVEL(IELEM)*DTIME INVP 35 +IF(STRES(IELEM,1).LT.0.0) YIELD=-YIELD INVP 36 +PREYS=YIELD+HARDS*PLAST(IELEM) INVP 37 +IF(ABS(STRES(IELEM,1)).LE.ABS(PREYS)) GO TO 20 INVP 38 +VIVEL(IELEM)=GAMMA*(STRES(IELEM,1)-(YIELD+HARDS*PLAST(IELEM))) INVP 39 +SNTOT=(TDISP(NODE2,1)-TDISP(NODE1,1))/ELENG INVP 40 +DELTM=TAUFT*ABS(SNTOT/VIVEL(IELEM)) INVP 41 +IF(DELTMLT.DNEXT) DNEXT=DELTM INVP 42 +GO TO 30 INVP 43 +20 VIVEL(IELEM)=0.0 INVP 44 +``` + + + +
30 CONTINUEINVP45
DTIME=DNEXTINVP46
IF(ISTEP.EQ.1) DTIME=DTINTINVP47
DO 50 IELEM=1,NELEMINVP48
LPROP=MATNO(IELEM)INVP49
YOUNG=PROPS(LPROP,1)INVP50
XAREA=PROPS(LPROP,2)INVP51
FACTR=(YOUNG*VIVEL(IELEM)*DTIME-STRES(IELEM,1))*XAREAINVP52
IF(COORD(NODE2).GT.COORD(NODE1)) GO TO 40INVP53
ELOAD(IELEM,1)= FACTRINVP54
ELOAD(IELEM,2)=-FACTRINVP55
GO TO 50INVP56
40 ELOAD(IELEM,1)=-FACTRINVP57
ELOAD(IELEM,2)= FACTRINVP58
50 CONTINUEINVP59
DO 60 IELEM=1,NELEMINVP60
DO 60 IEVAB=1,NEVABINVP61
60 ELOAD(IELEM,IEVAB)=ELOAD(IELEM,IEVAB)+TLOAD(IELEM,IEVAB)INVP62
RETURNINVP63
ENDINVP64
+ +INVP 16–18 Zero the array in which the pseudo loads for the next time step will be stored. + +INVP 20 Loop over each element. + +INVP 21 Identify the element material property number. + +INVP 22–26 Store the elastic modulus as YOUNG, the cross-sectional area as XAREA, the uniaxial yield stress as YIELD, the uniaxial hardening parameter as HARDS and the fluidity parameter as GAMMA. + +INVP 27–28 Identify the element node numbers. + +INVP 29 Evaluate the length of the element. + +INVP 30–33 Calculate the element strain so that a tensile strain is positive. + +INVP 34 Evaluate the total current stress $\sigma^{n+1}$ according to (4.30) and (4.31). + +INVP 35 Evaluate the total viscoplastic strain $\epsilon_{vp}^{n+1}$ , according to (4.32). + +INVP 36 For a compressive stress take a negative value of the initial yield stress. + +INVP 37 Compute the current yield level $\sigma_{Y} + H^{\prime}\epsilon_{vp}^{n + 1}$ + +INVP 38 If the current stress is less than the current yield stress, avoid evaluation of the viscoplastic strain rate. + +INVP 39 Otherwise evaluate the viscoplastic strain rate according to (4.33). + +INVP 40–42 Evaluate the next time-step length according to (4.38). + +INVP 44 For elastic elements set the viscoplastic strain rate to zero. + +INVP 45 End of element loop. + +INVP 47 For the first timestep of a load increment choose the timestep as the initial value. + +INVP 48 Enter element loop to evaluate pseudo loads, $\Delta V^{n+1}$ , for the next time step. + +INVP 49 Identify the element material property number. + + + +INVP 50–51 Store the elastic modulus as YOUNG and the cross-sectional area as XAREA. + +INVP 52 Evaluate the factor $AE \dot{\epsilon}_{vp}^{n+1} \Delta t_{n+1} + A \sigma^{n+1}$ . + +INVP 53–62 Evaluate $\Delta V^{n+1}$ according to (4.34) and (4.35), taking the appropriate signs for tensile or compressive stresses and strains. Note that $f^{n+1} + \Delta f^{n+1}$ is the total load applied for time step $n+1$ which is stored as TLOAD. + +# 4.9 Convergence monitoring subroutine, CONVP + +Convergence of the numerical process to the steady state solution must be monitored by comparing, in some way, the values of the viscoplastic strain rate determined during each time step. This can be done in several ways and in this section we describe a procedure based on a global convergence check only. In particular we will assume that steady state conditions have been achieved if + +$$ +\frac {\sum_ {i = 1} ^ {M} \left| \left(\Delta \epsilon_ {v p} {} ^ {n}\right) _ {i} \right|}{\sum_ {i = 1} ^ {M} \left| \left(\Delta \epsilon_ {v p} {} ^ {1} {} _ {i}\right) \right|} \times 1 0 0 \leqslant \text { TOLER }, \tag {4.41} +$$ + +where M denotes the total number of elements in the problem and || denotes the absolute value. The multiplication factor of 100 on the left-hand side allows the specified tolerance factor TOLER to be considered as a percentage term. Equation (4.41) states that steady state conditions are deemed to have been achieved if the sum of the absolute values of the strain increment for any time step is less than or equal to TOLER times the corresponding value for the first time step. For practical purposes a value of TOLER $\leqslant 1\cdot0$ (i.e. 1%) is generally adequate. Parameter NCHEK indicates convergence of the solution to steady state, where; + +NCHEK = 1 indicates that the solution is converging to steady state, with the viscoplastic strain increment reducing between two successive time steps. + +NCHEK = 999 indicates a divergence, with the viscoplastic strain increment increasing between two successive time steps. + +NCHEK = 0 indicates that steady state conditions have been achieved. Subroutine CONVP is now presented and described. + +SUBROUTINE CONVP CNVP 1 +C*******************************CNVP 2 +C CNVP 3 +C *** CHECKS FOR SOLUTION CONVERGENCE CNVP 4 +C . CNVP 5 +C*******************************CNVP 6 +COMMON/UNIM1/NPOIN, NELEM, NBOUN, NLOAD, NPROP, NNODE, IINCS, ISTEP, CNVP 7 +KRESL, NCHEK, TOLER, NALGO, NSVAB, NDOFN, NINCS, NEVAB, CNVP 8 + + + +NSTEP, NOUTP, FACTO, TAUFT, DTINT, FTIME, FIRST, PVALU, CNVP 9 +DTIME, TTIME CNVP 10 +COMMON/UNIM2/PROPS(5,5), COORD(26), LNODS(25,2), IFPRE(52), CNVP 11 +FIXED(52), TLOAD(25,4), RLOAD(25,4), ELOAD(25,4), CNVP 12 +MATNO(25), STRES(25,2), PLAST(25), XDISP(52), CNVP 13 +TDISP(26,2), TREAC(26,2), ASTIF(52,52), ASLOD(52), CNVP 14 +REACT(52), FRESV(1352), PEFIX(52), ESTIF(4,4), VIVEL(25) CNVP 15 +NCHEK=1 CNVP 16 +TOTAL=0.0 CNVP 17 +DO 10 IELEM=1, NELEM CNVP 18 +TOTAL=TOTAL+ABS(VIVEL(IELEM))*DTIME CNVP 19 +IF(ISTEP.EQ.1) FIRST=TOTAL CNVP 20 +IF(FIRST.EQ.0.0) GO TO 20 CNVP 21 +RATIO=100.0*TOTAL/FIRST CNVP 22 +GO TO 30 CNVP 23 +20 RATIO=0.0 CNVP 24 +30 CONTINUE CNVP 25 +IF(RATIO.LE.TOLER) NCHEK=0 CNVP 26 +IF(RATIO.GT.PVALU) NCHEK=999 CNVP 27 +40 PVALU=RATIO CNVP 28 +WRITE(6,900) TTIME CNVP 29 +900 FORMAT(1H0,5X,12HTOTAL TIME =,E17.6) CNVP 30 +WRITE(6,910) NCHEK,RATIO CNVP 31 +910 FORMAT(1H0,5X,18HCONVERGENCE CODE =,I4,3X,28HNORM OF RESIDUAL SUM CNVP 32 +.RATIO =,E14.6) CNVP 33 +RETURN CNVP 34 +END CNVP 35 + +CNVP 16 Set the indicator monitoring convergence to 1. This will be reset later in the subroutine if necessary. + +CNVP 17-19 Compute + +$$ +\sum_ {i = 1} ^ {M} \left| \left(\Delta \epsilon_ {v p} ^ {n}\right) _ {i} \right| +$$ + +for the current time step as required in (4.41). + +CNVP 20 For the first time step evaluate the denominator in (4.41). + +CNVP 21–25 Evaluate the left-hand side in (4.41). If the denominator is zero there is no viscoplastic flow for the particular load increment, therefore set RATIO = 0 indicating a steady state condition. + +CNVP 26 If (4.41) is satisfied, set NCHEK = 0 indicating a steady state condition. + +CNVP 27 If the viscoplastic increment has increased from the value obtained on the previous time step set NCHEK = 999. + +CNVP 28 Store the current value of the left-hand side of (4.41) for use in Statement CNVP 27 during the next time step. + +CNVP 29-30 Output the current time. + +CNVP 31–33 Output the value of NCHEK and the left-hand side of (4.41). + +# 4.10 Subroutine INCLOD + +Subroutine INCLOD described in Section 3.7 is employed for this application with one minor change: The iteration limit NITER is now replaced by the time-step limit NSTEP. diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_013.md b/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_013.md new file mode 100644 index 00000000..80836c7a --- /dev/null +++ b/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_013.md @@ -0,0 +1,236 @@ + + +For each increment of load, data is accepted by INCLOD to control the upper limit to the number of time steps, the output frequency, the size of load increment and the convergence tolerance limit. These quantities are specifically input as: + +
NSTEPMaximum permissible number of time steps. This is a safety measure to cover situations where steady state conditions are not achieved. After performing NSTEP time steps the program will then stop.
NOUTPThis parameter controls the frequency of output of results:0—Print the results on convergence to steady state conditions only, for each load increment.1—Print the results after the first time step and at steady state, for each load increment.2—Print the results for each time step for each load increment.
FACTOThis quantity controls the magnitude of any load increment. The applied loading is accepted by subroutine DATA and stored in array RLOAD. The size of any load increment is then RLOAD factored by FACTO. Therefore if FACTO is input for the first three increments as respectively 0·3, 0·3 and 0·1, the total loading applied to the structure during the third increment is 0·7 times the loading input in subroutine DATA.
TOLERThis item of data controls the tolerance permitted on the steady state convergence process, and has been described in Section 4.9.
+ +Subject to the replacement of NITER by NSTEP, the form of this subroutine for the present application is identical to that provided in Section 3.7. + +# 4.11 The main, master or controlling segment + +This master segment controls the calling, in order, of the other sub-routines. This program segment also controls the time-stepping process and also the incrementing of the applied loads, where appropriate. + +The following channel numbers are employed by the program: 5 (card reader), 6 (line printer), 1 (scratch file). + +
MASTER UNVISCUVIS1
C**********UVIS2
CUVIS3
C *** PROGRAM FOR THE 1-D SOLUTION OF NONLINEAR PROBLEMSUVIS4
CUVIS5
C**********UVIS6
COMMON/UNIM1/NPOIN,NELEM,NBOUN,NLOAD,NPROP,NNODE,IINCS,ISTEP,UVIS7
KRESL,NCHEK,TOLER,NALGO,NSVAB,NDOFN,NINCS,NEVAB,UVIS8
NSTEP,NOUTP,FACTO,TAUFT,DTINT,FTIME,FIRST,PVALU,UVIS9
DTIME,TTIMEUVIS10
COMMON/UNIM2/PROPS(5,5),COORD(26),LNODS(25,2),IFPRE(52),UVIS11
FIXED(52),TLOAD(25,4),RLOAD(25,4),ELOAD(25,4),UVIS12
MATNO(25),STRES(25,2),PLAST(25),XDISP(52),UVIS13
TDISP(26,2),TREAC(26,2),ASTIF(52,52),ASLOD(52),UVIS14
REACT(52),FRESV(1352),PEFIX(52),ESTIF(4,4),VIVEL(25)UVIS15
+ + + +
TTIME=0.0UVIS 16
CALL DATAUVIS 17
CALL INITIALUVIS 18
CALL STUNVPUVIS 19
DO 30 IINCS=1,NINCSUVIS 20
CALL INCLODUVIS 21
DTIME=0.0UVIS 22
DO 10 ISTEP=1,NSTEPUVIS 23
TTIME=TTIME+DTIMEUVIS 24
CALL NONALUVIS 25
CALL ASSEMBUVIS 26
IF(KRESL.EQ.1) CALL GREDUCUVIS 27
IF(KRESL.EQ.2) CALL RESOLVUVIS 28
CALL BAKSUBUVIS 29
CALL INCVPUVIS 30
CALL CONVPUVIS 31
IF(NCHEK.EQ.0) GO TO 20UVIS 32
IF(ISTEP.EQ.1.AND.NOUTP.EQ.1) CALL RESULTUVIS 33
IF(NOUTP.EQ.2) CALL RESULTUVIS 34
10 CONTINUEUVIS 35
WRITE(6,900)UVIS 36
900 FORMAT(1H0,5X,'STEADY STATE NOT ACHIEVED')UVIS 37
STOPUVIS 38
20 CALL RESULTUVIS 39
30 CONTINUEUVIS 40
STOPUVIS 41
ENDUVIS 42
+ +UVIS 16 Initialise the total time to zero. + +UVIS 17 Call the subroutine which reads the input data as described in Section 3.2. + +UVIS 18 Call Subroutine INITIAL which: + +(i) Initialises to zero the viscoplastic strain vector and the stress vector. +(ii) Initialises the array, ELOAD, which will contain the pseudo loads to be applied during each time step. +(iii) Initialises the vector of applied loads. +(iv) Initialises the vector of total displacements and total reactions. + +UVIS 19 Call the subroutine which evaluates the stiffness matrix for each element. + +UVIS 20 Enter the DO LOOP over the number of load increments. + +UVIS 21 Call Subroutine INCLOD which: + +(i) Reads and writes the input data required for each load increment as described previously in Section 4.10. +(ii) Adds the current increment of load into the pseudo load vector, ELOAD, and into the total applied load vector, TLOAD. + +UVIS 23 Begin the time-stepping process. + +UVIS 24 Calculate the total time elapsed (note that the first time step corresponds to the elastic solution). + +UVIS 25 Call the subroutine which sets the parameter KRESL controlling equation resolution facility. + + + +UVIS 26-29 Call the subroutines which assemble the element stiffnesses and solve for the unknown displacements and reactions. +UVIS 30 Call the subroutine which evaluates quantities at the end of the time step and evaluates the loads for the next time step. +UVIS 31 Check whether or not steady state conditions have been achieved. +UVIS 32 If so, terminate the time-stepping process for the current load increment. +UVIS 33–34 Output the results at a frequency controlled by parameter, NOUTP. +UVIS 35 End of time-stepping loop. +UVIS 36–38 If steady state conditions have not been achieved when the upper time-step limit has been reached, write a message and terminate the execution. +UVIS 40 End of load increment loop. + +# 4.12 Numerical examples + +The first example considered is the viscoplastic deformation of a single element under constant applied loading. The element is of length 10 units and the applied load is 15 units. The material properties assumed are included in Fig. 4.5, where it is noted that the strain hardening parameter is taken to be zero. The finite element prediction is seen to be in excellent agreement with the theoretical result (4.17) for this problem. + +The problem was then reanalysed for a strain-hardening material with $H' = 5000$ . The finite element results are compared with the theoretical expression (4.16) in Fig. 4.6 for three different values of the time-stepping parameter, $\tau$ , defined in Section 4.4. For a value of $\tau = 0.01$ excellent agreement is obtained, but as the time-step length is increased ( $\tau = 0.05$ and $\tau = 0.1$ ) comparison with the theoretical solution deteriorates. In particular, an increase in the time-step length progressively overestimates the viscoplastic strain increment, which is a characteristic of the Euler method of time stepping. It is noted that the time-step length is not so critical in the perfectly viscoplastic case of Fig. 4.5 since the exact viscoplastic strain increment is in fact linear for this case. + +For the material properties assumed, the theoretical value of the limiting time step is given from (4.36) to be 1·0. It is seen from Figs. 4.5 and 4.6 that the time-step lengths employed in solution are well within this critical value. However, Fig. 4.6 shows that to achieve an accurate result even smaller time-step lengths must be taken. Thus although the theoretical value of the limiting time-step length guarantees numerical stability of the solution process it may not always lead to an accurate solution. + +The second example considered illustrates the redistribution of stress with time which generally takes place in viscoplastic problems. Figure 4.7 shows two members in parallel which are subjected to an end load P which + + + +![](images/page-124_d6bc359a7e3be14348a8d72b414d761503f050dcba1e21b236bc06fdf4802a27.jpg) + +
+line + +| Time | End displacement, φ₂ | +|------|---------------------| +| 0.0 | 0.015 | +| 0.1 | 0.020 | +| 0.2 | 0.025 | +| 0.3 | 0.030 | +| 0.4 | 0.035 | +| 0.5 | 0.040 | +| 0.6 | 0.045 | +| 0.7 | 0.050 | +| 0.8 | 0.055 | +| 0.9 | 0.060 | +| 1.0 | 0.065 | +
+ +Fig. 4.5 End displacement with time for a single viscoplastic element under constant applied load—No strain hardening. + + + +![](images/page-125_0e467fe8e943511b3d97600db3c0ed69cd530fa800c840989c462dce1abf2a16.jpg) + +
+line + +| Time | End displacement (Theoretical) | Finite element τ=0.1 Toler=0.1% | Finite element τ=0.05 | Finite element τ=0.01 | +|------|----------------------------------|----------------------------------|------------------------|------------------------| +| 0.0 | 0.016 | 0.015 | 0.016 | 0.016 | +| 0.1 | 0.018 | 0.018 | 0.019 | 0.019 | +| 0.2 | 0.020 | 0.020 | 0.021 | 0.021 | +| 0.3 | 0.022 | 0.022 | 0.023 | 0.023 | +| 0.4 | 0.023 | 0.023 | 0.024 | 0.024 | +| 0.5 | 0.024 | 0.024 | 0.024 | 0.024 | +| 0.6 | 0.024 | 0.024 | 0.024 | 0.024 | +| 0.7 | 0.024 | 0.024 | 0.024 | 0.024 | +| 0.8 | 0.024 | 0.024 | 0.024 | 0.024 | +| 0.9 | 0.024 | 0.024 | 0.024 | 0.024 | +| 1.0 | 0.024 | 0.024 | 0.024 | 0.024 | +
+ +Fig. 4.6 End displacement with time for a single viscoplastic element under constant applied load showing finite element results for different time-step lengths—Linear strain hardening. + + + +![](images/page-126_f9dd395efc3ebc04a9504aa429fc51d65121f845b083c5192461b7ab548c4e7e.jpg) + +
+line + +| Time | End displacement, φ₃ | +|-------|----------------------| +| 0.00 | 0.0011 | +| 0.05 | 0.0012 | +| 0.10 | 0.0012 | +| 0.15 | 0.0014 | +| 0.20 | 0.0015 | +| 0.25 | 0.0015 | +| 0.30 | 0.0016 | +| 0.35 | 0.0017 | +| 0.40 | 0.0017 | +| 0.45 | 0.0018 | +| 0.50 | 0.0019 | +| 0.55 | 0.0020 | +
+ +Fig. 4.7 End displacement with time for an elasto-viscoplastic parallel bar model subjected to an incrementally applied end load showing the attainment of steady state conditions. + + + +is incrementally applied. The material properties for each element are included in Fig. 4.7 with the only difference between the two members being the initial yield stress of the materials. The load is applied in four increments and steady state conditions are allowed to develop for each increment before application of further load. The end displacement with time is shown in Fig. 4.7. Steady state conditions are achieved for the first three load increments but not for the fourth since both elements, which behave perfectly plastically, have become yielded at this stage. + +# 4.13 Problems + +4.1 Develop the relationship between the applied stress, $\sigma$ , and the total strain, $\epsilon$ , for the rheological model shown in Fig. 4.8. Plot the strain response with time when the model is subjected to a constant applied stress, $\sigma_{A}$ . +4.2 Repeat Problem 4.1 for the rheological model shown in Fig. 4.9. In this case the friction slider becomes active for $\sigma \geqslant Y$ where, for a linear strain hardening material, $Y = \sigma_{Y} + H' \epsilon_{vp}$ . + +![](images/page-127_b655f4d2763b6d1ade5d91d5be38cfc240ee5f286f33fd615bae7b4f1a4c4b5b.jpg) + +
+text_image + +E₁ +E₂ +γ +σ +σ +ε +
+ +Fig. 4.8 Problem 4.1. + +![](images/page-127_cd1e38ef3c6c0dde715e75814e6d54f6dc0aef446d5340753e27f5bf5572447b.jpg) + +
+text_image + +E1 +E2 +Y +γ1 +γ2 +σ +εa +εw +εvp +ε +
+ +Fig. 4.9 Problem 4.2. + +· 4.3 Use the unidimensional computer code developed in this chapter to determine the stress relaxation with time when the Maxwell model shown in Fig. 4.10 is subjected to a constant displacement condition. The critical time-step length for this model can be shown to be + + + +$\Delta t = 2 / \gamma E$ . Solve the problem for several time-step lengths up to the critical value, thereby showing that numerical divergence occurs as soon as the limiting value is reached. For computation let $E = 100$ , $\gamma = 0.01$ and $\phi_p = 0.1$ . + +![](images/page-128_4a52a1e6bcef40c98310b1b9347146781893e0c0759e7d1bb0df0c16908b1ef9.jpg) + +
+text_image + +σ +E +γ +φp +
+ +Fig. 4.10 Problem 4.3. + +4.4 Modify the computer code developed in this chapter to allow solution of the material model of Problem 4.1. +4.5 In Section 4.9, Subroutine CONVP, monitoring convergence to steady state conditions, was based on a global criterion. Modify this subroutine so that convergence is based upon the condition + +$$ +\frac {\left| \Delta \epsilon_ {v p} {} ^ {n} \right|}{\left| \Delta \epsilon_ {v p} {} ^ {1} \right|} \times 1 0 0 \leqslant \text { TOLER }, \tag {4.42} +$$ + +for each individual element. + +4.6 Develop the elastic stiffness matrix, $K^{(e)}$ , for a two-node finite element in the form of a sphere and which is to be subjected to spherically symmetrical radial loading only. Assume a linear variation between nodes and note the following relationships + +$$ +\epsilon_ {r} = \frac {\partial u}{\partial r} = \frac {1}{E} [ \sigma_ {r} - \nu (\sigma_ {\theta} + \sigma_ {\phi}) ]; \quad \sigma_ {\theta} = \sigma_ {\phi}; +$$ + +$$ +\epsilon_ {\theta} = \epsilon_ {\phi} = \frac {u}{r} = \frac {1}{E} [ (1 - \nu) \sigma_ {\theta} - \nu \sigma_ {r} ], \tag {4.43} +$$ + +in which u is the radial displacement and $\epsilon_{r}$ , $\epsilon_{\theta}$ , $\epsilon_{\phi}$ and $\sigma_{r}$ , $\sigma_{\theta}$ , $\sigma_{\phi}$ are respectively the strain and stress components. Also express the stress components in terms of the nodal displacements. + +4.7 Use the stiffness matrix evaluated in Problem 4.6 to modify the one-dimensional viscoplastic program UNVIS to allow solution of spherically symmetrical problems. Assume a Tresca yield criterion which implies commencement of yielding when $\sigma_{r}-\sigma_{\theta}=\sigma_{Y}$ . +4.8 Employ the program developed in Problem 4.7 to determine the variation of the elasto-viscoplastic stress distribution with time in a sphere which is instantaneously loaded by an internal pressure of 500 N/mm $^{2}$ . The internal and external radii of the sphere are 10 cm and 25 cm + + + +respectively, the elastic modulus $E = 2 \times 10^{5} \, N/mm^{2}$ , Poisson's ratio $\nu = 0 \cdot 3$ , the uniaxial yield stress $\sigma_{Y} = 300 \, N/mm^{2}$ , hardening parameter, $H' = 0$ and take the fluidity parameter $\gamma = 0 \cdot 001$ . Compare your steady state solution with the theoretical elasto-plastic results of Ref. 2. + +# 4.14 References + +1. CORMEAU, I., Numerical stability in quasistatic elasto-visco-plasticity, Int. J. Num. Meth. Engng., 9, 109–127 (1975). +2. HILL, R., The Mathematical Theory of Plasticity, Oxford University Press, 1950. + + diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_014.md b/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_014.md new file mode 100644 index 00000000..d6c38912 --- /dev/null +++ b/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_014.md @@ -0,0 +1,404 @@ + + +# Chapter 5 + +# Elasto-plastic Timoshenko beam analysis + +Written in collaboration with H. H. Abdel Rahman + +# 5.1 Introduction + +In this chapter we introduce some elastoplastic beam formulations which are useful in their own right but which also provide insight into the elastoplastic plate formulations presented later. + +There are two main beam theories on which we could base our studies: + +(i) Euler–Bernoulli beam theory. This theory, which is usually favoured by engineers because of its simplicity, takes no account of transverse shear deformation. The simplest Euler–Bernoulli beam element based on the displacement method is the well-known Hermitian element $^{(1)}$ with cubic displacements. Bending moments may vary linearly over this element. +(ii) Timoshenko beam theory. This theory allows for transverse shear deformation effects. The simplest Timoshenko beam element is the Hughes element $^{(2)}$ with linear displacements and normal rotations. Bending moments are constant over this element. + +Although the Euler–Bernoulli theory is frequently adopted we choose the Timoshenko beam theory as a basis for our study of the elasto-plastic analysis of beams since we may make use of a finite element which involves constant bending moments and is more in keeping with the presentations given in the previous chapters. Furthermore, Timoshenko beam theory can rightly be considered as the one-dimensional precursor of Mindlin plate theory which is used in Chapter 9. + +Firstly in this chapter the basic assumptions of Timoshenko beam theory are outlined. The Hughes element formulation is then presented for the elastic case. + +There are two approaches to the elasto-plastic analysis of Timoshenko beams: + +(i) Non-layered approach. In this method, when the bending moment reaches the yield moment, the whole cross-section of the beam is assumed to become plastic instantaneously. This is however a convenient fiction as in reality there is always a gradual plastification of the beam with the outer + + + +fibres becoming plastic initially. The zone of plasticification then spreads inwards until the whole section ultimately becomes plastic. + +(ii) Layered approach. In this method we attempt to capture the spread of plasticity over the depth of the beam. The beam is thus divided into a number of layers each of which may become plastic separately. As the number of layers is increased, this model provides a more realistic representation of the gradual spread of plasticity over the beam cross-section. + +Both non-layered and layered approaches are described in detail and program TIMOSH for the non-layered beams and program TIMLAY for the layered beams are presented and their use is illustrated with the aid of some examples. + +# 5.2 The basic assumptions of Timoshenko beam theory + +# 5.2.1 Introductory comments + +There are several basic assumptions adopted-in the derivation of the governing equations of Timoshenko beam theory. Here we reiterate these assumptions for elastic, small deflection analysis and then in later sections we present some extensions of the theory to allow for elasto-plastic analysis. + +# 5.2.2 Assumed displacement field + +In a typical Timoshenko beam, such as the one shown in Fig. 5.1, it is usual to assume that normals to the neutral axis before deformation remain straight but not necessarily normal to the neutral axis after deformation. This implies that the axial displacement $\bar{u}$ at any point $(x, z)$ may be expressed directly in terms of $\theta(x)$ the rotation of the normal so that + +$$ +\bar {u} (x, z) = - z \theta (x) \tag {5.1} +$$ + +Note that the normal rotation $\theta(x)$ is equal to the slope of the neutral axis dw/dx minus a rotation $\beta$ which is due to the transverse shear deformation. + +![](images/page-132_75e89e0aadf652eb61cfa3b1e1ad86eb4940ededb01677de06e0fb86b299fe2f.jpg) + +
+text_image + +θ +ū, x +w, z +σₓ +τₓₓ +σₓ +τₓₓ +σₓ +σₓ +
+ +Fig. 5.1 Timoshenko beam. + + + +Thus we have + +$$ +\theta (x) = \frac {d \bar {w}}{d x} - \beta . \tag {5.2} +$$ + +Notice also that the lateral displacement $\bar{w}$ at any point $(x, z)$ is given by the lateral displacement at the neutral axis so that + +$$ +\bar {w} (x, z) = w (x) \tag {5.3} +$$ + +# 5.2.3 Stress-strain relationships + +In Timoshenko beam theory, the elastic stress–strain relationships used for plane stress analysis are usually adopted in a slightly modified form. For convenience we assume that the beam is loaded in the xz plane and thus for an isotropic elastic material the relevant stress–strain relationships are + +$$ +\left[ \begin{array}{l} \sigma_ {x} \\ \sigma_ {z} \\ \tau_ {x z} \end{array} \right] = \frac {E}{(1 - \nu^ {2})} \left[ \begin{array}{c c c} 1 & \nu & 0 \\ \nu & 1 & 0 \\ 0 & 0 & \frac {(1 - \nu)}{2} \end{array} \right] \left[ \begin{array}{l} \epsilon_ {x} \\ \epsilon_ {z} \\ \gamma_ {x z} \end{array} \right] \tag {5.4} +$$ + +where $E$ is the Young's modulus and $\nu$ is the Poisson's ratio. + +If $\sigma_z$ is assumed to be equal to zero then + +$$ +\epsilon_ {z} = - \nu \epsilon_ {x} \tag {5.5} +$$ + +and by eliminating $\epsilon_{z}$ from (5.4) and (5.5), it is possible to write the following stress-strain relationship + +$$ +\sigma_ {x} = E \epsilon_ {x} \quad \text { and } \quad \tau_ {x z} = G \gamma_ {x z} \tag {5.6} +$$ + +where for an isotropic material $G = E / [2(1 + \nu)]$ is the shear modulus. + +# 5.2.4 Strain-displacement relationships + +Usually small deflection theory is adopted and the axial strain $\epsilon_{x}$ is given as + +$$ +\epsilon_ {x} = \frac {\dot {c} \bar {u}}{\dot {c} x}. \tag {5.7} +$$ + +If approximation (5.1) is adopted then this strain can be written as + +$$ +\epsilon_ {x} = - z \frac {d \theta}{d x}. \tag {5.8} +$$ + +Similarly the shear strain $\gamma_{xz}$ is given as + +$$ +\gamma_ {x z} = \frac {\dot {c} \bar {u}}{\dot {c} z} + \frac {\dot {c} \bar {w}}{\dot {c} x} \tag {5.9} +$$ + + + +and if approximation (5.2) is adopted we obtain + +$$ +\gamma_ {x z} = - \theta + \frac {d w}{d x} = \beta . \tag {5.10} +$$ + +# 5.2.5 Virtual work expression + +Consider a Timoshenko beam of depth t in which the breadth b varies with depth symmetrically about the neutral axis. The beam is subjected to a distributed loading of intensity q. If the beam undergoes a set of virtual lateral displacements $\delta w$ , virtual normal rotations $\delta\theta$ and associated virtual curvatures $-z[d(\delta\theta)/dx]$ and virtual shear strains $\delta\beta$ then the virtual work equation can be written as + +$$ +\int_ {0} ^ {l} \int_ {- t / 2} ^ {t / 2} \int_ {b (- t / 2)} ^ {b (t / 2)} \left\{- z \frac {d (\delta \theta)}{d x} \sigma_ {x} + \delta \beta \tau_ {x z} \right\} d y d z d x - \int_ {0} ^ {l} \delta w q d x = 0 \tag {5.11} +$$ + +or + +$$ +\int_ {0} ^ {l} \left(- \frac {d (\delta \theta)}{d x} M + \delta \beta Q\right) d x - \int_ {0} ^ {l} \delta w q d x = 0 +$$ + +where the bending moment + +$$ +M = \int_ {- t / 2} ^ {t / 2} \int_ {b (- t / 2)} ^ {b (t / 2)} z \sigma_ {x} d y d z \tag {5.12} +$$ + +and the shear force + +$$ +Q = \int_ {- t / 2} ^ {t / 2} \int_ {b (- t / 2)} ^ {b (t / 2)} \tau_ {x z} d y d z. \tag {5.13} +$$ + +Using (5.12) and (5.13), if we substitute for $\sigma_x$ and $\tau_{xz}$ in (5.6) respectively we obtain + +$$ +M = \left(\int_ {- t / 2} ^ {t / 2} \int_ {b (- t / 2)} ^ {b (t / 2)} z ^ {2} E d y d z\right) \left(- \frac {d \theta}{d x}\right) = E I \left(- \frac {d \theta}{d x}\right) \tag {5.14} +$$ + +and + +$$ +Q = \left(\int_ {- t / 2} ^ {t / 2} \int_ {b (- t / 2)} ^ {b (t / 2)} G d y d z\right) (\beta) = G A \beta \tag {5.15} +$$ + +where $EI$ is the flexural rigidity and $GA$ , the shear rigidity, is replaced by $GA$ where the area $A$ is replaced by $A/\alpha$ . The parameter $\alpha$ is a correction factor to allow for cross-sectional warping. For a rectangular section $\alpha$ is usually taken as 1·5.\* + +\* Many different definitions of $\alpha$ have been presented in the various papers on Timoshenko beams. Cowper $^{(3)}$ summarises some definitions for beams of various cross-sections. For example, he shows that $\alpha$ may be taken as $(12 + 11\nu)/(10 + 10\nu)$ for rectangular cross-sections and $(7 + 6\nu)/(6 + 6\nu)$ for circular cross-sections. Here we take $\alpha = 1 \cdot 5$ unless otherwise stated. + + + +If we substitute for $M$ and $Q$ from (5.14) and (5.15) we can rewrite the virtual work equation (5.11) as + +$$ +\int_ {0} ^ {l} \left(\frac {d (\delta \theta)}{d x} E I \frac {d \theta}{d x} + \delta \beta G \hat {A} \beta - \delta w q\right) d x = 0 \tag {5.16} +$$ + +# 5.2.6 A comparison of various beam approximations + +In order to compare the various beam approximations consider a simply supported beam of rectangular cross-section, flexural rigidity $EI$ , Poisson's ratio $\nu$ , depth $t$ and length $L$ which is subjected to a uniformly distributed loading $q$ . The lateral deflection in the elastic range is given as + +$$ +w = \frac {q L ^ {4}}{2 4 E I} \left\{\left[ \left(\frac {x}{L}\right) ^ {4} - \frac {3}{2} \left(\frac {x}{L}\right) ^ {2} + \frac {5}{1 6} \right] + \left(\frac {t}{L}\right) ^ {2} \left[ \frac {1 2}{5} + \frac {3 \nu}{2} \right] \left[ \frac {1}{4} - \left(\frac {x}{L}\right) ^ {2} \right] \right\} \tag {5.17a} +$$ + +when plane stress (PS) assumptions are adopted, + +$$ +w = \frac {q L ^ {4}}{2 4 E I} \left\{\left[ \left(\frac {x}{L}\right) ^ {4} - \frac {3}{2} \left(\frac {x}{L}\right) ^ {2} + \frac {5}{1 6} \right] + \left(\frac {t}{L}\right) ^ {2} [ 2 a (1 + \nu) ] \left[ \frac {1}{4} - \left(\frac {x}{L}\right) ^ {2} \right] \right\} \tag {5.17b} +$$ + +when Timoshenko beam (TB) assumptions are adopted and + +$$ +w = \frac {q L ^ {4}}{2 4 E I} \left\{\left[ \left(\frac {x}{L}\right) ^ {4} - \frac {3}{2} \left(\frac {x}{L}\right) ^ {2} + \frac {5}{1 6} \right] \right\} \tag {5.17c} +$$ + +when Euler–Bernoulli (EB) assumptions are adopted. + +Thus, for long slender beams in which $(t/L)$ is small, EB theory is adequate. If we take Cowper's value ${}^{(3)}$ of $\alpha = (12 + 11\nu)/(10 + 10\nu)$ then the ratio of the second-order additional lateral deflections due to shear deformation obtained under TB and PS assumptions is $(24 + 22\nu)/(24 + 15\nu)$ which varies from 1·00 to 1·11 as $\nu$ varies from 0·0 to 0·5. Thus TB theory is an accurate theory for beams of all dimensions. + +# 5.3 Finite element idealisation for linear elastic Timoshenko beams + +# 5.3.1 Introduction + +The theoretical and programming aspects of the finite element analysis of linear elastic Timoshenko beams have been dealt with in detail in previous books by the authors $^{(1, 5)}$ . Here we derive the stiffness matrix and consistent load vector for a linear element and set the scene for the analysis of elasto-plastic Timoshenko beams which will be discussed later. + +# 5.3.2 Displacement and strain representation + +In the Hughes element representation, the lateral displacement w is represented by the relationship + + + +$$ +w ^ {(e)} = N _ {1} ^ {(e)} w _ {1} ^ {(e)} + N _ {2} ^ {(e)} w _ {2} ^ {(e)} \tag {5.18} +$$ + +where $w_{1}^{(e)}$ and $w_{2}^{(e)}$ are the nodal lateral displacements at local nodes 1 and 2 of element e and the shape functions (shown in Fig. 5.2) are + +$$ +N _ {1} ^ {(e)} = \left(x _ {2} ^ {(e)} - x ^ {(e)}\right) / l ^ {(e)} +$$ + +and + +$$ +N _ {2} ^ {(e)} = \left(x ^ {(e)} - x _ {1} ^ {(e)}\right) / l ^ {(e)} +$$ + +in which $x_{1}^{(e)}$ and $x_{2}^{(e)}$ are the x-coordinates of local nodes 1 and 2, $x^{(e)}$ is the x-coordinate of a point within the element and $l^{(e)}$ is the length of the element. + +![](images/page-136_55fd256a5513861bbd19e660637e0663fcfc456e614279bb2c60371c897a3e1a.jpg) + +
+text_image + +x₁⁽ᵉ⁾ +1 +x₂⁽ᵉ⁾ +2 +N₁⁽ᵉ⁾ +1 +N₂⁽ᵉ⁾ +1 +
+ +Fig. 5.2 Beam element shape functions. + +Similarly the normal rotation $\theta^{(e)}$ within element e is represented as + +$$ +\theta^ {(e)} = N _ {1} ^ {(e)} \theta_ {1} ^ {(e)} + N _ {2} ^ {(e)} \theta_ {2} ^ {(e)} \tag {5.19} +$$ + +where $\theta_{1}^{(e)}$ and $\theta_{2}^{(e)}$ are the normal rotations at local nodes 1 and 2 of element $e$ . + +The curvature-displacement relationship can be expressed as + +$$ +- \left(\frac {d \theta}{d x}\right) ^ {(e)} = - \left(\frac {d N _ {1}}{d x}\right) ^ {(e)} \theta_ {1} ^ {(e)} - \left(\frac {d N _ {2}}{d x}\right) ^ {(e)} \theta_ {2} ^ {(e)} \tag {5.20} +$$ + +or + +$$ +\epsilon_ {f} ^ {(e)} = \left[ 0, \frac {1}{l ^ {(e)}}, 0, - \frac {1}{l ^ {(e)}} \right] \left[ \begin{array}{l} w _ {1} ^ {(e)} \\ \theta_ {1} ^ {(e)} \\ w _ {2} ^ {(e)} \\ \theta_ {2} ^ {(e)} \end{array} \right] = \boldsymbol {B} _ {f} ^ {(e)} \boldsymbol {\varphi} ^ {(e)} +$$ + +where $B_{f}^{(e)}$ is the curvature–displacement matrix. + + + +The shear strain-displacement relationship is given as + +$$ +\left(\frac {d w}{d x} - \theta\right) ^ {(e)} = \left(\frac {d N _ {1}}{d x}\right) ^ {(e)} w _ {1} ^ {(e)} - N _ {1} ^ {(e)} \theta_ {1} ^ {(e)} + \left(\frac {d N _ {2}}{d x}\right) ^ {(e)} w _ {2} ^ {(e)} - N _ {2} ^ {(e)} \theta_ {2} ^ {(e)} \tag {5.21} +$$ + +or + +$$ +\epsilon_ {s} ^ {(e)} = \left[ - \frac {1}{l ^ {(e)}}, - \frac {\left(x _ {2} ^ {(e)} - x ^ {(e)}\right)}{l ^ {(e)}}, \frac {1}{l ^ {(e)}}, - \frac {\left(x ^ {(e)} - x _ {1} ^ {(e)}\right)}{l ^ {(e)}} \right] \left[ \begin{array}{l} w _ {1} ^ {(e)} \\ \theta_ {1} ^ {(e)} \\ w _ {2} ^ {(e)} \\ \theta_ {2} ^ {(e)} \end{array} \right] = \boldsymbol {B} _ {s} ^ {(e)} \boldsymbol {\varphi} ^ {(e)} +$$ + +where $B_{s}^{(e)}$ is the shear strain–displacement matrix. + +# 5.3.3 Stiffness matrix evaluation + +Given the element strain-displacement relationships outlined in Section 5.3.2, Hughes has shown that using a virtual work approach the governing equations can be expressed as + +$$ +[ \boldsymbol {K} _ {f} + \boldsymbol {K} _ {s} ] \varphi - \boldsymbol {f} = 0 \tag {5.22} +$$ + +where the submatrices of $K_{f}$ and $K_{s}$ and subvectors of f for element e can be written as + +$$ +\boldsymbol {K} _ {f} ^ {(e)} = \int_ {X _ {1} ^ {(e)}} ^ {X _ {2} ^ {(e)}} [ \boldsymbol {B} _ {f} ^ {(e)} ] ^ {T} (E I) ^ {(e)} \boldsymbol {B} _ {f} ^ {(e)} d x +$$ + +$$ +\boldsymbol {K} _ {s} ^ {(e)} = \int_ {X _ {1} ^ {(e)}} ^ {X _ {2} ^ {(e)}} [ \boldsymbol {B} _ {s} ^ {(e)} ] ^ {T} (G \hat {A}) ^ {(e)} \boldsymbol {B} _ {s} ^ {(e)} d x +$$ + +$$ +\boldsymbol {f} ^ {(e)} = \int_ {X _ {1} ^ {(e)}} ^ {X _ {2} ^ {(e)}} [ N _ {1} ^ {(e)}, 0, N _ {2} ^ {(e)}, 0 ] ^ {T} q d x. \tag {5.23} +$$ + +The flexural element stiffness matrix can be evaluated using a 1-point Gauss–Legendre rule and takes the form + +$$ +\boldsymbol {K} _ {f} ^ {(e)} = \left(\frac {E I}{l}\right) ^ {(e)} \left[ \begin{array}{c c c c} 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & - 1 \\ 0 & 0 & 0 & 0 \\ 0 & - 1 & 0 & 1 \end{array} \right] \tag {5.24} +$$ + +If $K_{s}^{(e)}$ is evaluated exactly using a 2-point Gauss–Legendre rule we obtain + + + +$$ +\boldsymbol {K} _ {s} ^ {(e)} = \left(\frac {G \hat {A}}{l}\right) ^ {(e)} \left[ \begin{array}{c c c c} 1 & \frac {l}{2} & - 1 & \frac {l}{2} \\ \frac {l}{2} & \frac {l ^ {2}}{3} & - \frac {l}{2} & \frac {l ^ {2}}{6} \\ - 1 & - \frac {l}{2} & 1 & - \frac {l}{2} \\ \frac {l}{2} & \frac {l ^ {2}}{6} & - \frac {l}{2} & \frac {l ^ {2}}{3} \end{array} \right] ^ {(e)} \tag {5.25} +$$ + +Unfortunately it has been shown that with this formulation, overstiff solutions are obtained. This phenomenon, known as locking, may be 'cured' by integrating $K_{s}^{(e)}$ with a 1-point Gauss-Legendre rule. If such a selectively integrated element is adopted we find that + +$$ +\boldsymbol {K} _ {s} ^ {(e)} = \left(\frac {G \hat {A}}{l}\right) ^ {(e)} \left[ \begin{array}{c c c c} 1 & \frac {l}{2} & - 1 & \frac {l}{2} \\ \frac {l}{2} & \frac {l ^ {2}}{4} & - \frac {l}{2} & \frac {l ^ {2}}{4} \\ - 1 & - \frac {l}{2} & 1 & - \frac {l}{2} \\ \frac {l}{2} & \frac {l ^ {2}}{4} & - \frac {l}{2} & \frac {l ^ {2}}{4} \end{array} \right] ^ {(e)} \tag {5.26} +$$ + +and the results obtained are excellent. + +The consistent nodal force vector is given as + +$$ +f ^ {(e)} = \left[ \frac {(q l) ^ {(e)}}{2}, 0, \frac {(q l) ^ {(e)}}{2}, 0 \right] \tag {5.27} +$$ + +which, unlike the Euler–Bernōulli cubic Hermitian element, only has lateral nodal point forces. + +For the nonlayered elasto-plastic Timoshenko beam finite element analysis, when the beam bending moment reaches the yield moment $M_{0}$ , the whole element becomes plastic and acts as a plastic hinge. In such a situation the flexural rigidity EI is replaced by an elasto-plastic flexural rigidity $(EI)_{ep}$ whereas the shear rigidity $G\hat{A}$ is assumed to be unchanged. + +# 5.3.4 Element stress resultants + +We can obtain expressions which enable us to calculate the bending moments and shear forces within each element using $(5.14)$ and $(5.15)$ . The + + + +bending moment, which is constant in each element $e$ , is given as + +$$ +\begin{array}{l} M ^ {(e)} = (E I) ^ {(e)} B _ {f} ^ {(e)} \varphi^ {(e)} = (E I) ^ {(e)} \left[ 0, \frac {1}{l ^ {(e)}}, 0, - \frac {1}{l ^ {(e)}} \right] \left[ \begin{array}{l} w _ {1} ^ {(e)} \\ \theta_ {1} ^ {(e)} \\ w _ {2} ^ {(e)} \\ \theta_ {2} ^ {(e)} \end{array} \right] \\ = \left(\frac {E I}{l}\right) ^ {(e)} \left(\theta_ {1} ^ {(e)} - \theta_ {2} ^ {(e)}\right). \tag {5.28} \\ \end{array} +$$ + +The shear force varies linearly over each element but we evaluate it at + +$$ +x = \frac {x _ {1} ^ {(e)} + x _ {2} ^ {(e)}}{2} +$$ + +and assume it to be constant over the element. This is consistent with the practice of using selective integration in the evaluation of $K^{(e)}$ . The shear force is therefore given as + +$$ +\begin{array}{l} \begin{array}{l} Q ^ {(e)} = (G \hat {A}) ^ {(e)} B _ {s} ^ {(e)} \varphi^ {(e)} = (G \hat {A}) ^ {(e)} \left[ - \frac {1}{l ^ {(e)}}, - \frac {1}{2}, \frac {1}{l ^ {(e)}}, - \frac {1}{2} \right] \left[ \begin{array}{l} w _ {1} ^ {(e)} \\ \theta_ {1} ^ {(e)} \\ w _ {2} ^ {(e)} \\ \theta_ {2} ^ {(e)} \end{array} \right] \\ \left(\left. w _ {2} ^ {(e)} - w _ {1} ^ {(e)}\right) \left. \left. \left. \left. \theta_ {1} ^ {(e)} + \theta_ {2} ^ {(e)}\right) \right. \right. \right. \right. \end{array} \\ = (G \hat {A}) ^ {(e)} \left\{\left(\frac {w _ {2} ^ {(e)} - w _ {1} ^ {(e)}}{l ^ {(e)}}\right) - \left(\frac {\theta_ {1} ^ {(e)} + \theta_ {2} ^ {(e)}}{2}\right) \right\}. \tag {5.29} \\ \end{array} +$$ + +# 5.4 Elasto-plastic nonlayered Timoshenko beams + +# 5.4.1 The yield moment + +Consider a Timoshenko beam subjected to a bending moment. Timoshenko's assumptions imply that the axial stress and strain vary linearly across the depth of the section. As the bending moment is increased the yield stress is attained at the top and bottom fibres and with a further increase the yield will spread from these outer fibres inwards until the two zones of yield meet. The cross-section is then said to be fully plastic. It should be noted that the interaction of $\sigma_{x}$ and $\tau_{xz}$ has been ignored during yield. This is inexact, but experience shows that the effect is not of prime importance especially when thin beams are considered. + +The value of this ultimate moment in the fully plastic condition can be calculated in terms of the yield stress $\sigma_{0}$ .\* Thus + +$$ +M _ {0} = \int_ {b (- t / 2)} ^ {b (t / 2)} \int_ {- t / 2} ^ {t / 2} z \sigma_ {0} d z d y \tag {5.30} +$$ + +\* Note that for beam and plate problems the uniaxial yield stress is designated by $\sigma_0$ and not $\sigma_Y$ . + + + +and for a rectangular beam of breadth b, $M_{0} = \sigma_{0}(bt^{2}/4)$ . However, it should be noted that the assumption used in the finite element solution implies that the whole cross-section becomes plastic as soon as the bending moment reaches its yield value $M_{0}$ . This means that, for the beam case shown in Fig. 5.3, the whole cross-section is assumed to be plastic when the bending moment of situation (c) becomes equal to the bending moment of situation (d)—in which case the extreme fibre stress in situation (c) exceeds the actual yield stress of the material. + +![](images/page-140_b5d42403a2df02c3f3ce8c4f6489b9ee78c11c42227d5fad18f035b81b434f5d.jpg) +Fig. 5.3 Yielding of non-layered beam. + +# 5.4.2 Elasto-plastic bending + +As mentioned earlier, elasto-plastic behaviour is characterised by an initial elastic material response with an additional plastic deformation when the bending moment $|M|$ exceeds the yield moment $M_{0}$ . The plastic deformation is irreversible on unloading and its onset is governed by a very simple yield criterion. Post-yield deformation usually occurs with a considerably reduced material stiffness. + +The moment-curvature relationship for a Timoshenko beam of elastoplastic material is shown in Fig. 5.4. The beam initially deforms elastically with a flexural rigidity of EI until the ultimate bending moment is reached at which stage the whole beam cross-section becomes plastic. On increasing the load further, the material is assumed to exhibit linear strain-hardening characterised by the tangential flexural rigidity $(EI)_{T}$ . + +At some stage after initial yielding consider a further load application resulting in an incremental increase of bending moment accompanied by a change of curvature $d\epsilon_{f}$ . Assuming that the curvature can be separated into elastic and plastic components, so that + +$$ +d \epsilon_ {f} = (d \epsilon_ {f}) _ {e} + (d \epsilon_ {f}) _ {p}, \tag {5.31} +$$ + +we define as a strain hardening parameter diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_015.md b/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_015.md new file mode 100644 index 00000000..6b350a90 --- /dev/null +++ b/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_015.md @@ -0,0 +1,412 @@ + + +![](images/page-141_c042553ebb990b23668258faec308a872c2aa80b6406581530e4a7a60cf59d2e.jpg) + +
+line + +| Curvature | Bending Moment | Slope EI | +| --------- | -------------- | -------- | +| dεf | 0 | 0 | +| dεf | M0 | (dεf)e | +| dεf | (dεf)p | (dεf)p | +
+ +Fig. 5.4 Moment curvature relationship for a Timoshenko beam. + +$$ +H ^ {\prime} = \frac {d M}{(d \epsilon_ {f}) _ {p}}. +$$ + +This can be interpreted as the slope of the strain-hardening portion of the moment-curvature curve after the removal of the elastic curvature component. Thus + +$$ +H ^ {\prime} = \frac {d M}{d \epsilon_ {f} - (d \epsilon_ {f}) _ {e}} = \frac {(E I) _ {T}}{1 - [ (E I) _ {T} / E I ]}. \tag {5.32} +$$ + +It is therefore possible to rewrite (5.31) as + +$$ +d \epsilon_ {f} = \frac {d M}{E I} + \frac {d M}{H ^ {\prime}} = \frac {d M (H ^ {\prime} + E I)}{E I H ^ {\prime}} \tag {5.33} +$$ + +and then the incremental moment–curvature relationship can be written in the form + +$$ +d M = \frac {E I H ^ {\prime}}{(E I + H ^ {\prime})} d \epsilon_ {f}. \tag {5.34} +$$ + +Thus during yielding the incremental stress–strain resultant relationship is + +$$ +d M = E I \left(1 - \frac {E I}{E I \pm H ^ {\prime}}\right) d \epsilon_ {f} +$$ + +$$ +d Q = G \hat {A} d \epsilon_ {s}. \tag {5.35} +$$ + + + +The shear force/shear strain relationship is always elastic whereas the moment–curvature relationship is elasto-plastic. After yielding the flexural rigidity EI is replaced by + +$$ +E I \left(1 - \frac {E I}{E I + H ^ {\prime}}\right). +$$ + +If the hardening parameter $H'$ is equal to zero then the material behaviour is elasto-perfectly plastic and as mentioned in Section 3.5 for elasto-plastic axial bar elements this may lead to tangential stiffness matrices which are singular. This difficulty can also be avoided by use of the initial stiffness method in which the elastic element stiffnesses are employed at every stage of the computation thereby guaranteeing a positive definite assembled stiffness matrix. + +# 5.4.3 Solution of nonlinear equations + +Let us now generate the nonlinear equilibrium equations using the virtual expression (5.11). In order to do this we require the global rather than the element expressions for the lateral displacements, rotation, curvature and shear strain. At any point in the finite element mesh the lateral displacement and rotation can be obtained from the expression + +$$ +\left[ \begin{array}{l} w \\ \theta \end{array} \right] = N \varphi \tag {5.36} +$$ + +where the shape function matrix is + +$$ +N = \left[ \begin{array}{l l l l l l} N _ {1}, & 0, & N _ {2}, & 0, & \dots , & N _ {n}, & 0 \\ 0, & N _ {1}, & 0, & N _ {2}, & \dots , & 0, & N _ {n} \end{array} \right] \tag {5.37} +$$ + +and the vector of nodal displacements is + +$$ +\varphi = [ w _ {1}, \theta_ {1}, w _ {2}, \theta_ {2}, \dots , w _ {n}, \theta_ {n} ] ^ {T} \tag {5.38} +$$ + +where $w_{i}$ , $\theta_{i}$ and $N_{i}$ are the lateral displacement, rotation and global shape functions associated with node i. + +The curvature and shear strain at any point within the entire finite element mesh is given as + +$$ +- \frac {d \theta}{d x} = B _ {f} \varphi \quad \text { and } \quad \frac {d w}{d x} - \theta = B _ {s} \varphi \tag {5.39} +$$ + +where + +$$ +\boldsymbol {B} _ {f} = \left[ 0, - \frac {d N _ {1}}{d x}, 0, - \frac {d N _ {2}}{d x}, \dots , 0, - \frac {d N _ {n}}{d x} \right] \tag {5.40} +$$ + +and + +$$ +\boldsymbol {B} _ {s} = \left[ \frac {d N _ {1}}{d x}, - N _ {1}, \frac {d N _ {2}}{d x}, - N _ {2}, \dots , \frac {d N _ {n}}{d x}, - N _ {n} \right] \tag {5.41} +$$ + + + +Virtual curvatures and shear strains are given as + +$$ +- \frac {d (\delta \theta)}{d x} = \boldsymbol {B} _ {f} \delta \varphi \quad \text { and } \quad \frac {d (\delta w)}{d x} - \delta \theta = \boldsymbol {B} _ {s} \delta \varphi \tag {5.42} +$$ + +respectively, where the vector of virtual nodal displacements is written as + +$$ +\delta \varphi = [ \delta w _ {1}, \delta \theta_ {1}, \delta w _ {2}, \delta \theta_ {2}, \dots , \delta w _ {n}, \delta \theta_ {n} ] ^ {T}. \tag {5.43} +$$ + +Thus the virtual work expression (5.11) can now be written as + +$$ +\begin{array}{l} \int_ {0} ^ {l} [ \delta \varphi ] ^ {T} [ \boldsymbol {B} _ {f} ] ^ {T} M d x + \int_ {0} ^ {l} [ \delta \varphi ] ^ {T} [ \boldsymbol {B} _ {s} ] ^ {T} Q d x \\ - \int_ {0} ^ {l} [ \delta \varphi ] ^ {T} [ \bar {\mathbf {N}} ] ^ {T} q d x = 0 \tag {5.44} \\ \end{array} +$$ + +where $\bar{\mathbf{N}} = [N_1, 0, N_2, 0, \dots, N_n, 0]$ . (5.45) + +Since (5.44) must be true for any set of virtual displacements $\delta \varphi$ then we have + +$$ +\left\{\int_ {0} ^ {l} \left[ \boldsymbol {B} _ {f} \right] ^ {T} M d x + \int_ {0} ^ {l} \left[ \boldsymbol {B} _ {s} \right] ^ {T} Q d x \right\} - \int_ {0} ^ {l} \left[ \bar {\mathbf {N}} \right] ^ {T} q d x = 0 \tag {5.46} +$$ + +or + +$$ +p - f = 0. +$$ + +In fact this equation is identical to (5.22) when there is no plasticity. + +Unfortunately in elasto-plastic problems M is a nonlinear function and in general we can only predict the vector p approximately. Thus (5.46) is nonlinear and since p is only approximately known than p-f will equal a residual value $\psi(\varphi)$ which we attempt to reduce to zero in our solution procedure. + +We evaluate contributions to p element by element and assemble in the usual manner. The contribution from element e has the form + +$$ +\begin{array}{l} \boldsymbol {p} ^ {(e)} = \int_ {x _ {1} ^ {(e)}} ^ {x _ {2} ^ {(e)}} \left[ \begin{array}{c} 0 \\ \frac {1}{l ^ {(e)}} \\ 0 \\ - \frac {1}{l ^ {(e)}} \end{array} \right] M ^ {(e)} d x + \int_ {x _ {1} ^ {(e)}} ^ {x _ {2} ^ {(e)}} \left[ \begin{array}{c} - \frac {1}{l ^ {(e)}} \\ \frac {x ^ {(e)} - x _ {2} ^ {(e)}}{l ^ {(e)}} \\ \frac {1}{l ^ {(e)}} \\ \frac {x _ {1} ^ {(e)} - x ^ {(e)}}{l ^ {(e)}} \end{array} \right] Q ^ {(e)} d x \\ = \left[ - Q ^ {(e)}, M ^ {(e)} - \frac {(Q l) ^ {(e)}}{2}, Q ^ {(e)}, - M ^ {(e)} - \frac {(Q l) ^ {(e)}}{2} \right] ^ {T}. \tag {5.47} \\ \end{array} +$$ + +\*The second integral evaluation is equivalent to using a 1-point Gauss rule. + + + +![](images/page-144_286badf9fdf174f5c6d48de7f661af3aedd8e1501ccaa009f381fe58e33ee0b8.jpg) + +
+flowchart + +```mermaid +graph TD + A["DATA"] --> B["Input data defining geometry, loading and boundary conditions, material properties, etc."] + B --> C["INITAL"] + C --> D["Input data for current increment."] + D --> E["INCREM"] + E --> F["Initialize accumulative arrays to zero. Update load vector."] + F --> G["NONAL"] + G --> H["Set indicator to identify type of solution algorithm."] + H --> I{Is new element stiffness matrix required?} + I -->|No| J["No"] + I -->|Yes| K["STIFFB"] + K --> L["Calculate the element stiffness matrices and store on disc."] + L --> M["ASSEMB and GREDUC or RESOLV and BAKSUB"] + M --> N["Assemble global stiffness matrix (or take previous one) and global load vector and solve the resulting equations for unknowns."] + N --> O["REFORB"] + O --> P["Calculate the residual force vector."] + P --> Q["CONUND"] + Q --> R["Has solution converged?"] + R -->|No| S["No"] + R -->|Yes| T["RESULT"] + T --> U["Output the results."] + U --> V["END"] + V --> W["LOAD INCREMENT LOOP"] + W --> X["ITERATION LOOP"] +``` +
+ +Fig. 5.5 Overall structure of program TIMOSH. + + + +Note that the appropriate value of bending moment $M^{(e)}$ is inserted in (5.47). + +Table 5.1 shows the complete sequence of nonlinear equation solving which is very similar to the one adopted for the axially-loaded bars in Chapter 3. + +1. Begin load increment. +Set $f = f + \Delta f$ , iteration counter i = 0 and $\psi^{i} = \Delta f + \psi$ (that is, include equilibrium correction from previous increment). +2. Evaluate the new tangential stiffness matrix $K_{T}$ if necessary. +3. Solve $\psi^i = \mathbf{K}_T\Delta \varphi^i$ +4. Evaluate $\varphi = \varphi + \Delta \varphi^i$ . +5. For each element evaluate $M^{(e)}$ and $Q^{(e)}$ . Check $M^{(e)}$ and adjust its value accordingly to account for any plastic behaviour. Evaluate the element residual force vector $[\psi^{(e)}]^{i+1} = \mathfrak{p}^{(e)} - \mathbf{f}^{(e)}$ and assemble into the global residual force vector $\psi^{i+1}$ . +6. Check $\Delta\varphi^{i}$ for convergence. +7. If solution has converged set $\psi = \psi^{i+1}$ and go to step 1, otherwise set $i = i+1$ and go to step 2. + +Table 5.1 Solution procedure for elasto-plastic nonlayered Timoshenko beam analysis. + +# 5.4.4 Overall program structure of TIMOSH + +A modular approach is adopted for program TIMOSH. In fact the overall structure is identical to the structure in the programs of Chapter 3. Figure 5.5 shows the overall structure of TIMOSH. Routines DATA, INITIAL, INCREM, NONAL, ASSEMB, GREDUC, BAKSUB, CONUND, RESOLV and RESULT have already been described in Chapter 3. The only new routines are STIFFB, REFORB and, of course, the MASTER routine BEAM. + +5.4.5 New routines for nonlayered elasto-plastic Timoshenko beam analysis Master BEAM The master calling routine BEAM simply organises the calling of the main routines as described in Fig. 5.5. + +
MASTER BEAMEPBM1
C**********EPBM2
CEPBM3
C ***ELSTO-PLASTIC NONLAYERED TIMOSHENKO BEAM PROGRAMEPBM4
CEPBM5
C**********EPBM6
.COMMON/UNIM1/NPOIN,NELEM,NBOUN,NLOAD,NPROP,NNODE,IINCS,IITER,EPBM7
KRESL,NCHEK,TOLER,NALGO,NSVAB,NDOFN,NINCS,NEVAB,EPBM8
NITER,NOUTP,FACTOEPBM9
COMMON/UNIM2/PROPS(5,4),COORD(26),LNODS(25,2),IFPRE(52),EPBM10
FIXED(52),TLOAD(25,4),RLOAD(25,4),ELOAD(25,4),EPBM11
MATNO(25),STRES(25,2),PLAST(25),XDISP(52),EPBM12
+ + + +```csv +TDISP(26,2),TREAC(26,2),ASTIF(52,52),ASLOD(52), +REACT(52),FRESV(1352),PEFIX(52),ESTIF(4,4) +CALL DATA +CALL INITIAL +DO 30 IINCS=1,NINCS +CALL INCLOD +DO 10 IITER=1,NITER +CALL NONAL +IF(KRESL.EQ.1) CALL STIFFB +CALL ASSEMB +IF(KRESL.EQ.1) CALL GREDUC +IF(KRESL.EQ.2) CALL RESOLV +CALL BAKSUB +CALL REFORB +CALL CONUND +IF(NCHEK.EQ.0) GO TO 20 +IF(IITER.EQ.1.AND.NOUTP.EQ.1) CALL RESULT +IF(NOUTP.EQ.2) CALL RESULT +10 CONTINUE +WRITE(6,900) +900 FORMAT(1H0,5X,'SOLUTION NOT CONVERGED') +STOP +20 CALL RESULT +30 CONTINUE +STOP +END +EPBM 13 +EPBM 14 +EPBM 15 +EPBM 16 +EPBM 17 +EPBM 18 +EPBM 19 +EPBM 20 +EPBM 21 +EPBM 22 +EPBM 23 +EPBM 24 +EPBM 25 +EPBM 26 +EPBM 27 +EPBM 28 +EPBM 29 +EPBM 30 +EPBM 31 +EPBM 32 +EPBM 33 +EPBM 34 +EPBM 35 +EPBM 36 +EPBM 37 +EPBM 38 +. +``` + +Subroutine STIFFB The purpose of this routine is to evaluate the element stiffness matrices and store them on disc prior to their use in the assembly and equation solving routines. + +```csv +SUBROUTINE STIFFB STFB 1 +C**************************STFB 2 +C STFB 3 +C *** CALCULATES ELEMENT STIFFNESS MATRICES STFB 4 +C STFB 5 +C**************************STFB 6 +COMMON/UNIM1/NPOIN,NELEM,NBOUN,NLOAD,NPROP,NNODE,IINCS,IITER, STFB 7 +KRESL,NCHEK,TOLER,NALGO,NSVAB,NDOFN,NINCS,NEVAB, STFB 8 +NITER,NOUTP,FACTO STFB 9 +COMMON/UNIM2/PROPS(5,4),COORD(26),LNODS(25,2),IFPRE(52), STFB 10 +FIXED(52),TLOAD(25,4),RLOAD(25,4),ELOAD(25,4), STFB 11 +MATNO(25),STRES(25,2),PLAST(25),XDISP(52), STFB 12 +TDISP(26,2),TREAC(26,2),ASTIF(52,52),ASLOD(52), STFB 13 +REACT(52),FRESV(1352),PEFIX(52),ESTIF(4,4) STFB 14 +REWIND 1 STFB 15 +DO 20 IELEM=1,NELEM STFB 16 +LPROP=MATNO(IELEM) STFB 17 +EIVAL=PROPS(LPROP,1) STFB 18 +SVALU=PROPS(LPROP,2) STFB 19 +HARDS=PROPS(LPROP,4) STFB 20 +NODE1=LNODS(IELEM,1) STFB 21 +NODE2=LNODS(IELEM,2) STFB 22 +ELENG=ABS(COORD(NODE2)-COORD(NODE1)) STFB 23 +IF(PLAST(IELEM).NE.0.0) EIVAL=EIVAL*(1.0-EIVAL/(EIVAL+HARDS)) STFB 24 +VALU1=0.5*SVALU STFB 25 +VALU2=SVALU/ELENG STFB 26 +VALU3=EIVAL/ELENG STFB 27 +VALU4=0.25*SVALU*ELENG STFB 28 +ESTIF(1,1)= VALU2 STFB 29 +ESTIF(1,2)= VALU1 STFB 30 +``` + + + +
ESTIF(1,3)=-VALU2STFB31
ESTIF(1,4)=VALU1STFB32
ESTIF(2,2)=VALU3+VALU4STFB33
ESTIF(2,3)=-VALU1STFB34
ESTIF(2,4)=-VALU3+VALU4STFB35
ESTIF(3,3)=VALU2STFB36
ESTIF(3,4)=-VALU1STFB37
ESTIF(4,4)=VALU3+VALU4STFB38
DO 10 ISTIF=1,4STFB39
DO 10 JSTIF=ISTIF,4STFB40
10 ESTIF(JSTIF,ISTIF)=ESTIF(ISTIF,JSTIF)STFB41
WRITE(1) ESTIFSTFB42
20 CONTINUESTFB43
RETURNSTFB44
ENDSTFB45
+ +STFB 15 Rewind disc ready for writing element stiffnesses. + +STFB 16–38 For each element evaluate the upper triangular portion of the element stiffness matrix $K^{(e)}$ . Note that if plasticity has taken place the elastic EI is replaced by the elasto-plastic $(EI)_{T}$ . + +STFB 39-41 Obtain using symmetry the lower triangular portion of $K^{(e)}$ . + +STFB 42 Write all element stiffness matrices on to disc. + +Subroutine REFORB This routine evaluates the equivalent nodal forces. + +
SUBROUTINE REFORBRFRB1
C**********RFRB2
CRFRB3
C *** CALCULATES INTERNAL EQUIVALENT NODAL FORCESRFRB4
CRFRB5
C**********RFRB6
COMMON/UNIM1/NPOIN.NELEM,NBOUN,NLOAD,NPROP,NNODE,IINCS,IITER,RFRB7
KRESL,NCHEK,TOLER,NALGO,NSVAB,NDOFN,NINCS,NEVAB,RFRB8
NITER,NOUTP,FACTORFRB9
COMMON/UNIM2/PROPS(5,4),COORD(26),LNODS(25,2),IFPRE(52),RFRB10
FIXED(52),TLOAD(25,4),RLOAD(25,4),ELOAD(25,4),RFRB11
MATNO(25),STRES(25,2),PLAST(25),XDISP(52),RFRB12
TDISP(26,2),TREAC(26,2),ASTIF(52,52),ASLOD(52),RFRB13
REACT(52),FRESV(1352),PEFIX(52),ESTIF(4,4)RFRB14
DO 10 IELEM=1,NELEMRFRB15
DO 10 IEVAB=1,NEVABRFRB16
10 ELOAD(IELEM,IEVAB)=0.0RFRB17
DO 70 IELEM=1,NELEMRFRB18
LPROP=MATNO(IELEM)RFRB19
EIVAL=PROPS(LPROP,1)RFRB20
SVALU=PROPS(LPROP,2)RFRB21
YIELD=PROPS(LPROP,3)RFRB22
HARDS=PROPS(LPROP,4)RFRB23
NODE1=LNODS(IELEM,1)RFRB24
NODE2=LNODS(IELEM,2)RFRB25
ELENG=ABS(COORD(NODE2)-COORD(NODE1))RFRB26
WNOD1=XDISP(NODE1*NDOFN-1)RFRB27
WNOD2=XDISP(NODE2*NDOFN-1)RFRB28
THTA1=XDISP(NODE1*NDOFN)RFRB29
THTA2=XDISP(NODE2*NDOFN)RFRB30
STRAN=(THTA1-THTA2)/ELENGRFRB31
STLIN=STRAN*EIVALRFRB32
STCUR=STRES(IELEM,1)+STLINRFRB33
PREYS=YIELD+HARDS*ABS(PLAST(IELEM))RFRB34
IF(ABS(STRES(IELEM,1)).GE.PREYS) GO TO 20RFRB35
+ + + +```txt +ESCUR=ABS(STCUR)-PREYS RFRB 36 +IF(ESCUR.LE.0.0) GO TO 40 RFRB 37 +RFACT=ESCUR/ABS(STLIN) RFRB 38 +GO TO 30 RFRB 39 +20 IF(STRES(IELEM,1).GT.0.0.AND.STLIN.LE.0.0) GO TO 40 RFRB 40 +IF(STRES(IELEM,1).LT.0.0.AND.STLIN.GE.0.0) GO TO 40 RFRB 41 +RFACT=1.0 RFRB 42 +30 REDUC=1.0-RFACT RFRB 43 +STRES(IELEM,1)=STRES(IELEM,1)+REDUC*STLIN+ RFRB 44 +RFACT*EIVAL*(1.0-EIVAL/(EIVAL+HARDS))*STRAN RFRB 45 +PLAST(IELEM)=PLAST(IELEM)+RFACT*STRAN*EIVAL/(EIVAL+HARDS) RFRB 46 +GO TO 50 RFRB 47 +40 STRES(IELEM,1)=STRES(IELEM,1)+STLIN RFRB 48 +50 STRES(IELEM,2)=STRES(IELEM,2)+(SVALU/ELENG)*(WNOD2-WNOD1) RFRB 49 +-0.5*SVALU*(THTA1+THTA2) RFRB 50 +ELOAD(IELEM,1)=ELOAD(IELEM,1)-STRES(IELEM,2) RFRB 51 +ELOAD(IELEM,2)=ELOAD(IELEM,2)+STRES(IELEM,1) RFRB 52 +-0.5*ELENG*STRES(IELEM,2) RFRB 53 +ELOAD(IELEM,3)=ELOAD(IELEM,3)+STRES(IELEM,2) RFRB 54 +ELOAD(IELEM,4)=ELOAD(IELEM,4)-STRES(IELEM,1) RFRB 55 +-0.5*ELENG*STRES(IELEM,2) RFRB 56 +70 CONTINUE RFRB 57 +RETURN RFRB 58 +END RFRB 59 +``` + +RFRB 15-17 Zero space for storing $\pmb{p}$ . + +RFRB 18-57 For each element evaluate $p^{(e)}$ and assemble into $p$ . + +# 5.4.6 Examples of nonlayered elasto-plastic Timoshenko beam analysis + +Two numerical examples are considered. The first example, Example 5.1, involves the yielding of a rectangular simple beam under uniformly distributed load. The beam material has the following properties: + +$$ +E = 2 1 0 \cdot 0 \mathrm{kN} / \mathrm{mm} ^ {2} +$$ + +$$ +\nu = 0 \cdot 3 +$$ + +$$ +\sigma_ {0} = 0 \cdot 2 5 \mathrm{kN} / \mathrm{mm} ^ {2} +$$ + +$$ +H ^ {\prime} = 0 \cdot 0 +$$ + +and the beam proportions are: + +$$ +b = 1 5 0 \mathrm{mm} +$$ + +$$ +t = 3 0 0 \mathrm{mm} +$$ + +$$ +l = 3 0 0 0 \mathrm{mm} +$$ + +Typical input data is provided in Appendix IV. + +The problem, finite element idealisation employed and the results are illustrated in Fig. 5.6, which shows that the beam fails as soon as a plastic hinge forms at the centre of the beam. Note that the beam material is assumed to have no strain hardening. + +The second example considered, Example 5.2, is the clamped I beam shown in Fig. 5.7. The beam has the same material properties as those of Example 5.1. + +The dimensions and finite element discretisation of the beam are given in Fig. 5.7; the load-displacement relationship at the beam centre is also provided. It is seen that there is an initial loss of stiffness corresponding to the + + + +![](images/page-149_e74f84438d5463fc913231dfde01b388b0d270c33f82df45ee677c8a957577fb.jpg) + +
+line + +| Central deflection (mm) | Applied load (KN) | +| ----------------------- | ----------------- | +| 0 | 0 | +| 10 | 1260 | +| 10 @ 300 mm | 1260 | +| 3000 mm | 1260 | +| 300 mm | 1260 | +| 150 mm | 1260 | +| 300 mm | 1260 | +
+ + + +![](images/page-150_f09a675eb77b8f34b357f897f1326ccf573ece3513d714eca9a0076183984a4e.jpg) +Fig. 5.7 Nonlayered elasto-plastic clamped beam. diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_016.md b/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_016.md new file mode 100644 index 00000000..d56ee6c2 --- /dev/null +++ b/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_016.md @@ -0,0 +1,340 @@ + + +yielding of the end sections followed by a further reduction when the central section becomes plastic resulting in a beam failure mechanism. + +# 5.5 Elasto-plastic layered Timoshenko beams + +# 5.5.1 Yielding of layered beams + +In the ‘layered’ approach the beam or the plate is subdivided into a chosen number of layers, as shown in Fig. 5.8. + +![](images/page-151_9b90c92a45ed30fcdfb217f4a692963c1cd26cecb09caa31b7b8ded73d3c000f.jpg) + +
+text_image + +(a) Layered beam +Layer i +
+ +![](images/page-151_6bfe52495bd45102f795b9dbb822019a54d76af7c74213f351a498f800e122f3.jpg) + +
+text_image + +(b) Layered plate +Layer i +
+ +Fig. 5.8 Layered subdivision of beam and plate. + +In the finite element solution it is assumed that as soon as the stress in the middle of the outer layers reaches the yield value, then the outer layers become plastic, while the rest of the layers remain elastic, as shown in + + + +![](images/page-152_7ce2772d93df636f14903dfb0cca9a9109385df4f97cdffdd7f63e65d6c910fa.jpg) +Fig. 5.9 Yielding of layered beam. +Fig. 5.9. Then, as plastification propagates, more layers become plastic, until the whole cross-section eventually becomes plastic. + +# 5.5.2 Formation of the stiffness matrix in the layered approach + +In the layered approach, we work in terms of stresses and not in terms of stress resultants as in the nonlayered approach. The state of stress at the middle of a layer is taken as representative for the entire layer. + +Contributions to the stress resultants M and Q are found for each layer separately by integrating over the layer thickness only. The bending moments and shear forces are then found from the contributions of all the layers of the beam element. + +The displacement field, stress-strain relationship and strain-displacement relationship are given in (5.1)-(5.10). + +The virtual work expression is given by (5.11) and when we evaluate the bending moment M and shear force Q we use a mid-ordinate rule as follows: + +$$ +M = E I \left(- \frac {d \theta}{d x}\right) \quad \text { and } \quad Q = G \hat {A} \epsilon_ {s} \tag {5.48} +$$ + +where + +$$ +E I = \sum_ {l} E _ {l} b _ {l} z _ {l} ^ {2} t _ {l} \tag {5.49} +$$ + +and + +$$ +G \hat {A} = \sum_ {l} G _ {l} b _ {l} t _ {l} \tag {5.50} +$$ + +and where $b_{l}$ is the layer breadth + +$t_{l}$ is the layer thickness + +$z_{l}$ is the $z$ -coordinate at the middle of the layer + +$E_{l}$ is the Young's modulus of the layer material + +and $G_{l}$ is the Shear modulus of the layer material. + + + +However, if the stress at the middle surface of a layer reaches the uniaxial yield stress of the layer material, the whole layer is considered to be plastic and $E_{l}$ is replaced by + +$$ +E _ {l} \left(1 - \frac {E _ {l}}{E _ {l} + H ^ {\prime}}\right), +$$ + +where $H'$ is the uniaxial strain hardening parameter. As mentioned before, the shear force–shear strain relationship is always elastic. + +# 5.5.3 Solution of nonlinear equations + +Recall that the virtual work expression (5.11) has the form + +$$ +\int_ {0} ^ {l} \int_ {- t / 2} ^ {t / 2} \int_ {b (- t / 2)} ^ {b (t / 2)} \left\{- z \frac {d (\delta \theta)}{d x} \sigma_ {x} + \delta \beta \tau_ {x z} \right\} d y d z d x - \int_ {0} ^ {l} \delta w q d x = 0. \tag {5.51} +$$ + +The mid-ordinate rule is again used to evaluate the first two integrals in (5.51) so that we obtain + +$$ +[ \delta \varphi ] ^ {T} [ \boldsymbol {p} _ {f} + \boldsymbol {p} _ {s} ] - [ \delta \varphi ] ^ {T} \boldsymbol {f} = 0 \tag {5.52} +$$ + +where + +$$ +\boldsymbol {p} _ {f} = \int_ {0} ^ {l} [ \boldsymbol {B} _ {f} ] ^ {T} \bar {M} d x +$$ + +and + +$$ +\boldsymbol {p} _ {s} = \int_ {0} ^ {l} [ \boldsymbol {B} _ {s} ] ^ {T} \bar {Q} d x +$$ + +in which $B_{f}$ , $B_{s}$ and $\delta\varphi$ have been defined in (5.40), (5.41) and (5.43) respectively and in which + +$$ +\overline {{{M}}} = \sum_ {l} b _ {l} \sigma_ {x l} z _ {l} t _ {l} \tag {5.53} +$$ + +and + +$$ +\bar {Q} = \sum_ {l} b _ {l} \tau_ {x z l} t _ {l}. \tag {5.54} +$$ + +Note that $\sigma_{xl}$ and $\tau_{xzl}$ are the direct and shear stresses in the layer respectively. Since (5.52) is true for any arbitrary set of virtual displacements then + +$$ +\boldsymbol {p} _ {f} + \boldsymbol {p} _ {s} - \boldsymbol {f} = 0. \tag {5.55} +$$ + +Contributions to $p_{f}$ and $p_{s}$ may be evaluated separately from each element so that + +$$ +\begin{array}{l} \boldsymbol {p} _ {f} ^ {(e)} = \int_ {x _ {1} ^ {(e)}} ^ {x _ {2} ^ {(e)}} [ \boldsymbol {B} _ {f} ^ {(e)} ] ^ {T} \bar {M} ^ {(e)} d x = \int_ {x _ {1} ^ {(e)}} ^ {x _ {2} ^ {(e)}} \left[ 0, \left(\frac {\bar {M}}{l}\right) ^ {(e)}, 0, - \left(\frac {\bar {M}}{l}\right) ^ {(e)} \right] ^ {T} d x \\ = [ 0, \bar {M} ^ {(e)}, 0, - \bar {M} ^ {(e)} ] ^ {T} \tag {5.56} \\ \end{array} +$$ + + + +and + +$$ +\begin{array}{l} \boldsymbol {p} _ {s} ^ {(e)} = \int_ {x _ {1} ^ {(e)}} ^ {x _ {2} ^ {(e)}} \left[ \boldsymbol {B} _ {s} ^ {(e)} \right] ^ {T} \bar {Q} ^ {(e)} d x = \int_ {x _ {1} ^ {(e)}} ^ {x _ {2} ^ {(e)}} \left[ - \frac {1}{l ^ {(e)}}, - \frac {1}{2}, \frac {1}{l ^ {(e)}}, - \frac {1}{2} \right] ^ {T} \bar {Q} ^ {(e)} d x \\ = \left[ - \bar {Q} ^ {(e)}, - \frac {(\bar {Q} l) ^ {(e)}}{2}, \bar {Q} ^ {(e)}, - \frac {(\bar {Q} l) ^ {(e)}}{2} \right] ^ {T}. \tag {5.57} \\ \end{array} +$$ + +The complete sequence of nonlinear equation solving is very similar to the one adopted in Table 5.1 for the nonlayered beam. Step 5 is now written as: + +5. For each element evaluate for each layer $\sigma_{xl}^{(e)}$ and $\tau_{xzl}^{(e)}$ . Check $\sigma_{xl}^{(e)}$ and adjust its value accordingly to account for any plastic behaviour. Evaluate the stress resultants $\bar{M}^{(e)}$ and $\bar{Q}^{(e)}$ and hence evaluate the residual force vector $[\psi^{(e)}]^{i+1} = p^{(e)} - f^{(e)}$ . Assemble $[\psi^{(e)}]^{i+1}$ into the global residual force vector $\psi^{i+1}$ . + +# 5.5.4 Overall structure of layered beam program TIMLAY + +The overall structure of the layered beam program is exactly the same as that of the nonlayered beam program given in Fig. 5.5. Subroutine STIFBL replaces STIFFB and subroutine RFORBL replaces REFORB. Subroutine STIFBL calls a further new routine called LAYER. The master routine BEML has minor changes as shown in the next section. + +# 5.5.5 Modified and new routines + +Master BEML This routine is almost identical to routine BEAM described earlier. +```txt +MASTER BEML LYBM 1 +C**************************LYBM 2 +C LYBM 3 +C *** ELSTO-PLASTIC LAYERED TIMOSHENKO BEAM PROGRAM LYBM 4 +C LYBM 5 +C**************************LYBM 6 +COMMON/UNIM1/NPOIN,NELEM,NBOUN,NLAYR,NPROP,NNODE,IINCS,IITER, LYBM 7 +. KRESL,NCHEK,TOLER,NALGO,NSVAB,NDOFN,NINCS,NEVAB, LYBM 8 +. NITER,NOUTP,FACTO LYBM 9 +COMMON/UNIM2/PROPS(5,25),COORD(26),LNODS(25,2),IFPRE(52), LYBM 10 +. FIXED(52),TLOAD(25,4),RLOAD(25,4),ELOAD(25,4), LYBM 11 +. MATNO(25),STRES(25,2),PLAST(250),XDISP(52), LYBM 12 +. TDISP(26,2),TREAC(26,2),ASTIF(52,52),ASLOD(52), LYBM 13 +. REACT(52),FRESV(1352),PEFIX(52),ESTIF(4,4), LYBM 14 +. STRSL(250,2) LYBM 15 +CALL DATA LYBM 16 +CALL INITIAL LYBM 17 +DO 30 IINCS=1,NINCS LYBM 18 +CALL INCLOD LYBM 19 +DO 10 IITER=1,NITER LYBM 20 +CALL NONAL LYBM 21 +IF(KRESL.EQ.1) CALL STIFBL LYBM 22 +CALL ASSEMB LYBM 23 +IF(KRESL.EQ.1) CALL GREDUC LYBM 24 +``` + + + +
IF(KRESL.EQ.2) CALL RESOLVLYBM25
CALL BAKSUBLYBM26
CALL RFORBLLYBM27
CALL CONUNDLYBM28
IF(NCHEK.EQ.0) GO TO 20LYBM29
IF(IITER.EQ.1.AND.NOUTP.EQ.1) CALL RESULTLYBM30
IF(NOUTP.EQ.2) CALL RESULTLYBM31
10 CONTINUELYBM32
WRITE(6,900)LYBM33
900 FORMAT(1H0,5X,'SOLUTION NOT CONVERGED')LYBM34
STOPLYBM35
20 CALL RESULTLYBM36
30 CONTINUELYBM37
STOPLYBM38
ENDLYBM39
+ +Subroutine STIFBL This routine calculates the element stiffness matrices for the elasto-plastic layered Timoshenko beam element. + +
SUBROUTINE STIFBLSTBL1
C********************STBL2
CSTBL3
C *** CALCULATES ELEMENT STIFFNESS MATRICESSTBL4
CSTBL5
C********************STBL6
COMMON/UNIM1/NPOIN,NELEM,NBOUN,NLAYR,NPROP,NNODE,IINCS,IITER,STBL7
KRESL,NCHEK,TOLER,NALGO,NSVAB,NDOFN,NINCS,NEVAB,STBL8
NITER,NOUTP,FACTOSTBL9
COMMON/UNIM2/PROPS(5,25),COORD(26),LNODS(25,2),IFPRE(52),STBL10
FIXED(52),TLOAD(25,4),RLOAD(25,4),ELOAD(25,4),STBL11
MATNO(25),STRES(25,2),PLAST(250),XDISP(52),STBL12
TDISP(26,2),TREAC(26,2),ASTIF(52,52),ASLOD(52),STBL13
REACT(52),FRESV(1352),PEFIX(52),ESTIF(4,4),STBL14
STRSL(250,2)STBL15
REWIND 1STBL16
DO 20 IELEM=1,NELEMSTBL17
LPROP=MATNO(IELEM)STBL18
CALL LAYER(IELEM,EIVAL,SVALU)STBL19
HARDS=PROPS(LPROP,4)STBL20
NODE1=LNODS(IELEM,1)STBL21
NODE2=LNODS(IELEM,2)STBL22
ELENG=ABS(COORD(NODE2)-COORD(NODE1))STBL23
VALU1=0.5*SVALUSTBL24
VALU2=SVALU/ELENGSTBL25
VALU3=EIVAL/ELENGSTBL26
VALU4=0.25*SVALU*ELENGSTBL27
ESTIF(1,1)=VALU2STBL28
ESTIF(1,2)=VALU1STBL29
ESTIF(1,3)=-VALU2STBL30
ESTIF(1,4)=VALU1STBL31
ESTIF(2,2)=VALU3+VALU4STBL32
ESTIF(2,3)=-VALU1STBL33
ESTIF(2,4)=-VALU3+VALU4STBL34
ESTIF(3,3)=VALU2STBL35
ESTIF(3,4)=-VALU1STBL36
ESTIF(4,4)=VALU3+VALU4STBL37
DO 10 ISTIF=1,4STBL38
DO 10 JSTIF=ISTIF,4STBL39
10ESTIF(JSTIF,ISTIF)=ESTIF(ISTIF,JSTIF)STBL40
WRITE(1) ESTIFSTBL41
20CONTINUESTBL42
RETURNSTBL43
ENDSTBL44
+ + + +STBL 19 Call routine LAYER which evaluates approximate values of EI and exact values of $GA$ using a mid-ordinate rule. Note that certain layers may be plastic. +Subroutine RFORBL This routine evaluates p for the layered beam using the mid-ordinate rule. + +
SUBROUTINE RFORBLRFRL1
C**********RFRL2
CRFRL3
C *** CALCULATES INTERNAL EQUIVALENT NODAL FORCESRFRL4
CRFRL5
C**********RFRL6
COMMON/UNIM1/NPOIN,NELEM,NBOUN,NLAYR,NPROP,NNODE,IINCS,IITER,RFRL7
KRESL,NCHEK,TOLER,NALGO,NSVAB,NDOFN,NINCS,NEVAB,RFRL8
NITER,NOUTP,FACTORFRL9
COMMON/UNIM2/PROPS(5,25),COORD(26),LNODS(25,2),IFPRE(52),RFRL10
FIXED(52),TLOAD(25,4),RLOAD(25,4),ELOAD(25,4),RFRL11
MATNO(25),STRES(25,2),PLAST(250),XDISP(52),RFRL12
TDISP(26,2),TREAC(26,2),ASTIF(52,52),ASLOD(52),RFRL13
REACT(52),FRESV(1352),PEFIX(52),ESTIF(4,4),RFRL14
STRSL(250,2)RFRL15
DIMENSION STRAN(2)RFRL16
DO 15 IELEM=1,NELEMRFRL17
DO 10 IEVAB=1,NEVABRFRL18
10 ELOAD(IELEM,IEVAB)=0.0RFRL19
DO 15 IDOFN=1,NDOFNRFRL20
15 STRES(IELEM,IDOFN)=0.0RFRL21
KLAYR=0RFRL22
DO 70 IELEM=1,NELEMRFRL23
LPROP=MATNO(IELEM)RFRL24
YOUNG=PROPS(LPROP,1)RFRL25
SHEAR=PROPS(LPROP,2)RFRL26
YIELD=PROPS(LPROP,3)RFRL27
HARDS=PROPS(LPROP,4)RFRL28
THKTO=PROPS(LPROP,5)RFRL29
NODE1=LNODS(IELEM,1)RFRL30
NODE2=LNODS(IELEM,2)RFRL31
ELENG=ABS(COORD(NODE2)-COORD(NODE1))RFRL32
WNOD1=XDISP(NODE1*NDOFN-1)RFRL33
WNOD2=XDISP(NODE2*NDOFN-1)RFRL34
THTA1=XDISP(NODE1*NDOFN)RFRL35
THTA2=XDISP(NODE2*NDOFN)RFRL36
STRAN(1)=(THTA1-THTA2)/ELENGRFRL37
STRAN(2)=(WNOD2-WNOD1)/ELENGRFRL38
-0.5*(THTA1+THTA2)RFRL39
ZMIDL=-THKTO/2.0RFRL40
KOUNT=5RFRL41
DO 50 ILAYR=1,NLAYRRFRL42
KLAYR=KLAYR+1RFRL43
KOUNT=KOUNT+1RFRL44
BRDTH=PROPS(LPROP,KOUNT)RFRL45
KOUNT=KOUNT+1RFRL46
THICK=PROPS(LPROP,KOUNT)RFRL47
ZMIDL=ZMIDL+THICK/2.0RFRL48
STLIN=YOUNG*STRAN(1)*ZMIDLRFRL49
STCUR=STRSL(KLAYR,1)+STLINRFRL50
PREYS=YIELD+HARDS*ABS(PLAST(KLAYR))RFRL51
IF(ABS(STRSL(KLAYR,1)).GE.PREYS) GO TO 20RFRL52
ESCUR=ABS(STCUR)-PREYSRFRL53
IF(ESCUR.LE.0.0) GO TO 40RFRL54
+ + + +```csv +RFACT=ESCUR/ABS(STLIN) RFRL 55 +GO TO 30 RFRL 56 +20 IF(STRSL(KLAYR,1).GT.0.0.AND.STLIN.LE.0.0) GO TO 40 RFRL 57 +IF(STRSL(KLAYR,1).LT.0.0.AND.STLIN.GE.0.0) GO TO 40 RFRL 58 +RFACT=1.0 RFRL 59 +30 REDUC=1.0-RFACT RFRL 60 +STRSL(KLAYR,1)=STRSL(KLAYR,1)+REDUC*STLIN+ RFRL 61 +• RFACT*YOUNG*(1.0-YOUNG/(YOUNG+HARDS))*STRAN(1)*ZMIDL RFRL 62 +PLAST(KLAYR)=PLAST(KLAYR)+RFACT*STRAN(1)*YOUNG/(YOUNG+HARDS) RFRL 63 +.*ZMIDL RFRL 64 +GO TO 45 RFRL 65 +40 STRSL(KLAYR,1)=STRSL(KLAYR,1)+STLIN RFRL 66 +45 STRSL(KLAYR,2)=STRSL(KLAYR,2)+STRAN(2)*SHEAR RFRL 67 +STRES(IELEM,1)=STRES(IELEM,1)+STRSL(KLAYR,1)* RFRL 68 +• BRDTH*THICK*ZMIDL RFRL 69 +STRES(IELEM,2)=STRES(IELEM,2)+STRSL(KLAYR,2)* RFRL 70 +• BRDTH*THICK RFRL 71 +ZMIDL=ZMIDL+THICK/2.0 RFRL 72 +50 CONTINUE RFRL 73 +ELOAD(IELEM,1)=ELOAD(IELEM,1)-STRES(IELEM,2) RFRL 74 +ELOAD(IELEM,2)=ELOAD(IELEM,2)+STRES(IELEM,1) RFRL 75 +• -0.5*ELENG*STRES(IELEM,2) RFRL 76 +ELOAD(IELEM,3)=ELOAD(IELEM,3)+STRES(IELEM,2) RFRL 77 +ELOAD(IELEM,4)=ELOAD(IELEM,4)-STRES(IELEM,1) RFRL 78 +• -0.5*ELENG*STRES(IELEM,2) RFRL 79 +70 CONTINUE RFRL 80 +RETURN RFRL 81 +END RFRL 82 +``` + +Subroutine LAYER This routine evaluates EI and $GA\hat{A}$ using the mid-ordinate rule. Note that certain layers may be plastic and therefore have a modified E. + +```txt +SUBROUTINE LAYER(IELEM,EIVAL,SVALU) LAYR 1 +C****************************************************************************************** +C LAYR 2 +C LAYR 3 +C *** CALCULATES INTEGRATED VALUES FOR EI AND GA THROUGH DEPTH LAYR 4 +C LAYR 5 +C****************************************************************************************** +COMMON/UNIM1/NPOIN,NELEM,NBOUN,NLAYR,NPROP,NNODE,IINCS,IITER, LAYR 7 +. KRESL,NCHEK,TOLER,NALGO,NSVAB,NDOFN,NINCS,NEVAB, LAYR 8 +. NITER,NOUTP,FACTO LAYR 9 +COMMON/UNIM2/PROPS(5,25),COORD(26),LNODS(25,2),IFPRE(52), LAYR 10 +. FIXED(52),TLOAD(25,4),RLOAD(25,4),ELOAD(25,4), LAYR 11 +. MATNO(25),STRES(25,2),PLAST(250),XDISP(52), LAYR 12 +. TDISP(26,2),TREAC(26,2),ASTIF(52,52),ASLOD(52), LAYR 13 +. REACT(52),FRESV(1352),PEFIX(52),ESTIF(4,4), LAYR 14 +. STRSL(250,2) LAYR 15 +EIVAL=0.0 LAYR 16 +SVALU=0.0 LAYR 17 +LPROP=MATNO(IELEM) LAYR 18 +KLAYR=(IELEM-1)*NLAYR LAYR 19 +SHEAR=PROPS(LPROP,2) LAYR 20 +HARDS=PROPS(LPROP,4) LAYR 21 +THKTO=PROPS(LPROP,5) LAYR 22 +ZMIDL=-THKTO/2.0 LAYR 23 +KOUNT=5 LAYR 24 +DO 10 ILAYR=1,NLAYR LAYR 25 +KLAYR=KLAYR+1 LAYR 26 +YOUNG=PROPS(LPROP,1) LAYR 27 +IF(PLAST(KLAYR).NE.0.0) YOUNG=YOUNG*(1.0-YOUNG/(YOUNG+HARDS)) LAYR 28 +``` + + + +KOUNT=KOUNT+1 LAYR 29 +BRDTH=PROPS(LPROP,KOUNT) LAYR 30 +KOUNT=KOUNT+1 LAYR 31 +THICK=PROPS(LPROP,KOUNT) LAYR 32 +ZMIDL=ZMIDL+THICK/2.0 LAYR 33 +EIVAL=EIVAL+YOUNG*BRDTH*THICK*ZMIDL*ZMIDL LAYR 34 +SVALU=SVALU+SHEAR*BRDTH*THICK LAYR 35 +ZMIDL=ZMIDL+THICK/2.0 LAYR 36 +10 CONTINUE LAYR 37 +RETURN LAYR 38 +END LAYR 39 + +# 5.5.6 Examples of layered elasto-plastic Timoshenko beam analysis + +The third example considered in this chapter is the elasto-plastic analysis of the simple beam of Example 5.1. The layered solution is adopted in this case. A typical input data listing is provided in Appendix IV. + +The results for both nonlayered and layered solutions to this beam problem are reproduced in Fig. 5.10. + +The last example to be considered here is the layered solution of the clamped I-beam given in Example 5.1. + +Again, both nonlayered and layered solution results are illustrated in Fig. 5.11. + +From Figs. 5.10 and 5.11 it is obvious that the layered solution is more realistic. Yielding takes place gradually through the layers, resulting in smoother curves representing the load-displacement relationship. + +# 5.6 Problems + +5.1 Derive the main expressions for the elasto-plastic analysis of Timoshenko beams using elements with + +(i) Quadratic shape functions + +$$ +N _ {1} ^ {(e)} = \frac {(x ^ {(e)} - x _ {2} ^ {(e)}) (x ^ {(e)} - x _ {3} ^ {(e)})}{(x _ {1} ^ {(e)} - x _ {2} ^ {(e)}) (x _ {1} ^ {(e)} - x _ {3} ^ {(e)})} +$$ + +$$ +N _ {2} ^ {(e)} = \frac {\big (x ^ {(e)} - x _ {1} ^ {(e)} \big) \big (x ^ {(e)} - x _ {3} ^ {(e)} \big)}{\big (x _ {2} ^ {(e)} - x _ {1} ^ {(e)} \big) \big (x _ {2} ^ {(e)} - x _ {3} ^ {(e)} \big)} +$$ + +$$ +N _ {3} ^ {(e)} = \frac {\left(x ^ {(e)} - x _ {1} ^ {(e)}\right) \left(x ^ {(e)} - x _ {2} ^ {(e)}\right)}{\left(x _ {3} ^ {(e)} - x _ {1} ^ {(e)}\right) \left(x _ {3} ^ {(e)} - x _ {2} ^ {(e)}\right)} \tag {5.58} +$$ + + + +![](images/page-159_8174ae8d5da81596c17c54c9a2288f0fba5472e4689445d81687bb0907911a06.jpg) + +
+line + +| Central deflection (mm) | Nonlayered solution (KN) | Layered solution (KN) | +| ----------------------- | ------------------------ | --------------------- | +| 0 | 0 | 0 | +| 5 | 600 | 550 | +| 10 | 1250 | 1150 | +| 15 | 1275 | 1200 | +| 20 | 1280 | 1230 | +| 25 | 1285 | 1250 | +
+ +Fig. 5.10 Load-deflection diagrams for simply supported beam. + + + +![](images/page-160_16aceee34f32a2e7a46460ee62635c2cc4a1fc931c2a2ccff3156327f37911a6.jpg) + +
+line + +| Layer number | Cross-section (mm) | Applied load intensity (KN/mm) | +| ------------ | ------------------- | ------------------------------ | +| 2 | 200 | 0.45 | +| 3 | 200 | 0.44 | +| 4 | 200 | 0.43 | +| 5 | 200 | 0.42 | +| 6 | 200 | 0.41 | +
diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_017.md b/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_017.md new file mode 100644 index 00000000..bbcaccfc --- /dev/null +++ b/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_017.md @@ -0,0 +1,277 @@ + + +(ii) Cubic shape functions + +$$ +N _ {1} ^ {(e)} = \frac {\left(x ^ {(e)} - x _ {2} ^ {(e)}\right) \left(x ^ {(e)} - x _ {3} ^ {(e)}\right) \left(x ^ {(e)} - x _ {4} ^ {(e)}\right)}{\left(x _ {1} ^ {(e)} - x _ {2} ^ {(e)}\right) \left(x _ {1} ^ {(e)} - x _ {3} ^ {(e)}\right) \left(x _ {1} ^ {(e)} - x _ {4} ^ {(e)}\right)} +$$ + +$$ +N _ {2} ^ {(e)} = \frac {\left(x ^ {(e)} - x _ {1} ^ {(e)}\right) \left(x ^ {(e)} - x _ {3} ^ {(e)}\right) \left(x ^ {(e)} - x _ {4} ^ {(e)}\right)}{\left(x _ {2} ^ {(e)} - x _ {1} ^ {(e)}\right) \left(x _ {2} ^ {(e)} - x _ {3} ^ {(e)}\right) \left(x _ {2} ^ {(e)} - x _ {4} ^ {(e)}\right)} +$$ + +$$ +N _ {3} ^ {(e)} = \frac {\left(x ^ {(e)} - x _ {1} ^ {(e)}\right) \left(x ^ {(e)} - x _ {2} ^ {(e)}\right) \left(x ^ {(e)} - x _ {4} ^ {(e)}\right)}{\left(x _ {3} ^ {(e)} - x _ {1} ^ {(e)}\right) \left(x _ {3} ^ {(e)} - x _ {2} ^ {(e)}\right) \left(x _ {3} ^ {(e)} - x _ {4} ^ {(e)}\right)} +$$ + +$$ +N _ {4} ^ {(e)} = \frac {\left(x ^ {(e)} - x _ {1} ^ {(e)}\right) \left(x ^ {(e)} - x _ {2} ^ {(e)}\right) \left(x ^ {(e)} - x _ {3} ^ {(e)}\right)}{\left(x _ {4} ^ {(e)} - x _ {1} ^ {(e)}\right) \left(x _ {1} ^ {(e)} - x _ {2} ^ {(e)}\right) \left(x _ {4} ^ {(e)} - x _ {3} ^ {(e)}\right)} \tag {5.59} +$$ + +For the quadratic and cubic elements use 2-point and 3-point Gauss-Legendre integration rules respectively. + +5.2 Develop a layered finite element Timoshenko beam program which allows for combined in-plane and bending behaviour of axially loaded beams or beams with cross-sections which are nonsymmetric about the neutral axis. Choose a displacement representation of the form + +$$ +\bar {u} (x, z) = u _ {0} (x) - z \theta_ {x} (x) \tag {5.60} +$$ + +in which $u_{0}(x)$ is the axial displacement at the neutral axis. + +5.3 Use the concepts developed in Chapters 4 and 5 to develop the necessary relationships for layered and nonlayered elastoviscoplastic Timoshenko beam analysis. + +5.4 (i) Evaluate the additional stiffness terms required to represent the Winkler foundation by a 2-node linear Timoshenko beam element. For a foundation modulus k note that the additional virtual work term associated with the elastic foundation is + +$$ +\int_ {0} ^ {l} \delta w k w d x +$$ + +in which $\delta w$ is the virtual lateral displacement. + +(ii) Modify programs TIMOSH and TIMLAY to allow for beams on elastic foundations. + +(iii) Use the program to analyse a uniformly loaded, simply supported beam on a Winkler foundation. The elastic closed form solution for an Euler–Bernoulli beam predicts lateral displacements + +$$ +w = \sum_ {n = 1, 3, 5, \dots} ^ {\infty} \frac {4 q L ^ {4} / \left(n ^ {5} \pi^ {5} E I\right)}{1 + k L ^ {4} / \left(n ^ {4} \pi^ {4} E I\right)} \sin \frac {n \pi x}{L} \tag {5.61} +$$ + + + +and bending moments + +$$ +M = \sum_ {n = 1, 3, 5, \dots} ^ {\infty} \frac {4 q L ^ {2} / (n \pi) ^ {3}}{1 + k L ^ {4} / (n ^ {4} \pi^ {4} E I)} \sin \frac {n \pi x}{L}. \tag {5.62} +$$ + +Compare the elastic results from the modified programs with the above solution for various values of $kL^{4}/EI$ and t/L where EI is the flexural rigidity, t is the thickness and L is the length of the beam. + +(iv) For a given yield stress, $\sigma_{0}$ , evaluate the ultimate load for various values of $kL^{4}/EI$ and t/L. + +5.5 (i) Consider the problem of finding the elastic deflections of a simply supported beam of length L, flexural rigidity EI, shear rigidity GA which is subjected to a uniform load q. The beam is elastically supported at mid-span by a single linear spring of stiffness K. Modify programs TIMOSH and TIMLAY to solve this problem. Check your finite element solutions by noting that the elastic Euler–Bernoulli solution is given as + +$$ +\begin{array}{l} w = \frac {4 q L ^ {4}}{E I} \sum_ {n = 1, 3, 5, \dots} ^ {\infty} \frac {\sin (n \pi x / L)}{n ^ {5}} \\ - \frac {2 K S L ^ {3}}{\pi^ {4} E I} \sum_ {n = 1, 3, 5, \dots} ^ {\infty} \left(\frac {\sin (n \pi / 2) \sin (n \pi x / L)}{n ^ {4}}\right) \tag {5.63} \\ \end{array} +$$ + +in which + +$$ +S = \frac {5 q L ^ {4}}{3 8 4 E I} / \left(1 + \frac {K L ^ {3}}{4 8 E I}\right). \tag {5.64} +$$ + +(ii) When the load carried by the elastic support reaches a value F the supported beam becomes perfectly plastic. How can this be catered for in the modified version of TIMOSH and TIMLAY? + +5.6 Use program TIMLAY to examine the effects of choosing + +(i) different load incrementations +(ii) various convergence tolerances +(iii) various numbers of layers +on the example given in Section 5.4 and also Problems 5.4 and 5.5. + +# 5.7 References + +1. HINTON, E. and OWEN, D. R. J., An Introduction to Finite Element Computations, Pineridge Press, Swansea, U.K., 1979. + + + +2. HUGHES, T. J. R., TAYLOR, R. L. and KANOKNUKULCHAI, S., A simple and efficient finite element for bending, Int. J. Num. Meth. Engng., 11, 1529–1543 (1977). +3. COWPER, G. R., The shear coefficient in Timoshenko's Beam Theory, J. Appl. Mech., 33, 335 (1966). +4. DYM, C. L. and SHAMES, I. H., Solid Mechanics: A Variational Approach, McGraw-Hill, New York, 1973. +5. HINTON, E. and OWEN, D. R. J., Finite Element Programming, Academic Press, London, 1977. + + + + + +# Part II + + + + + +# Chapter 6 Preliminary theory and standard subroutines for two-dimensional elasto-plastic applications + +# 6.1 Introduction + +In Part II of this text we extend the concepts and techniques developed in Part I for one-dimensional situations to now permit the solution of two-dimensional problems. In particular the following applications are presented: + +● Chapter 7 discusses the solution of elasto-plastic problems conforming to either plane stress, plane strain or axially symmetric conditions. +- Chapter 8 deals with plane stress/strain and axisymmetric problems where the material exhibits a time-dependent elasto-viscoplastic behaviour. +● Chapter 9 covers elasto-plastic plate bending situations. + +The nonlinear algorithms developed in Chapter 2 will be employed in solution. These processes are general and the main modifications necessary are those appropriate to two-dimensional continuum theory or plate bending expressions which must now be used. For example the level of initial yielding will now be dependent on three or more independent stress components in place of the uniaxial case considered earlier. + +The development of an elasto-plastic stress analysis program requires all of the basic features of the corresponding elastic program. In particular the same basic element formulation is employed and a wide choice of element types is available. In this text we consider three different element types all based on an isoparametric formulation. The elements included are illustrated in Fig. 6.1 and are: + +- The 4-node isoparametric quadrilateral element with linear displacement variation, Fig. 6.1(a). +- The 8-node Serendipity quadrilateral element with curved sides and a quadratic variation of the displacement field within the element, Fig. 6.1(b). +- The 9-node Lagrangian quadrilateral element which additionally has a central node, Fig. 6.1(c). + +The basic theoretical expressions for these elements are provided in Section 6.3. The use of these higher order elements leads to particularly efficient + + + +![](images/page-168_43cdc07545b197fc7b960da42f56375a617e4e4dca7dae8c27cc6fb0a6efe5e3.jpg) + +
+text_image + +4 +3 +η +ξ +1 +2 +
+ +$$ +N _ {i} (\xi , \eta) = \frac {1}{4} (1 + \xi \xi_ {i}) (1 + \eta \eta_ {i}). +$$ + +![](images/page-168_5f5f3f6bd03aa82de913d469087d4ff926d13c9de2d9306faa15c0fe985543c1.jpg) + +
+natural_image + +3D grid surface diagram with an inset showing curved contour lines (no text or symbols) +
+
Local node number $\xi_i$ $\eta_i$
1-1-1
21-1
311
4-11
+ +Fig. 6.1(a) The 4-node isoparametric quadrilateral element and shape functions. + +elasto-plastic solution packages. In order to simplify matters as much as possible consideration is restricted to isotropic situations.\* + +For all the plasticity applications presented in this text the classical incremental theory is employed with the full elasto-plastic material response being reproduced. Thus we are not concerned with limit state behaviour as predicted by rigid-plastic theories, etc. + +Consideration is limited to small deformation situations where the strains can be assumed to be infinitesimal and Lagrangian and Eulerian geometric descriptions then coincide. + +\- Extension to orthotropic situations is feasible and has indeed been dealt with in Ref. 1. + + + +![](images/page-169_9362d8d13e4c07dba4352a45ef2ce329426d1a77e9685da4afda724225d2cd84.jpg) + +
+text_image + +7 6 5 +8 η ξ +1 2 3 +
+ +8-node Serendipity element + +![](images/page-169_dae32b249ab4548670d88c52a219cdbe5ec5ffa94fadb267c1f00875ab39f61e.jpg) + +
+text_image + +7 +6 +5 +η +8 +9 +ξ +4 +1 +2 +3 +
+ +9-node Lagrangian element + +\- for corner nodes + +$$ +N _ {i} ^ {(e)} = \frac {1}{4} (1 + \xi \xi_ {i}) (1 + \eta \eta_ {i}) (\xi \xi_ {i} + \eta \eta_ {i} - 1), \quad i = 1, 3, 5, 7, +$$ + +• for midside nodes + +$$ +N _ {i} ^ {(e)} = \frac {\xi_ {i} ^ {2}}{2} (1 + \xi \xi_ {i}) (1 - \eta^ {2}) + \frac {\eta_ {i} ^ {2}}{2} (1 + \eta \eta_ {i}) (1 - \xi^ {2}), \quad i = 2, 4, 6, 8. +$$
Local node number $\xi$ $\eta$
1-1-1
20-1
31-1
410
511
601
7-11
8-10
900
+ +![](images/page-169_b8a0a934233fb4829ad9be346a3ea18c7104ff2a83705430838af19b0f0c5563.jpg) + +
+natural_image + +Abstract pattern of concentric curved lines forming a wave-like shape (no text or symbols) +
+ +![](images/page-169_8b4b79e8569ff88d2a6b5c049adcffedce312901ae1698219c421e8646b0503a.jpg) + +
+natural_image + +3D wireframe surface plot with grid lines, no text or symbols present +
+ +![](images/page-169_7ab9e9dc37ba51601d2d263a8b3a4da56528d7ed685857f6cef25ea669dab60a.jpg) + +
+natural_image + +3D wireframe surface plot with grid lines and directional arrows, no text or symbols present +
+ +![](images/page-169_84b42657afc0dbb68b9388bda4d07a1a4db91b5da5b0ec5be5d91572708a004d.jpg) + +
+natural_image + +Abstract contour line pattern with hatched shading, no text or symbols present +
+ +Fig. 6.1(b) The 8-node Serendipity quadrilateral element. + + + +![](images/page-170_74b58659bc7485b8a2ac51147f1b505b130ca9a4a3cba3650caa7fc901a7157e.jpg) + +
+natural_image + +3D wireframe surface plot with grid lines and a small inset showing a grid with a vertical axis (no text or symbols) +
+ +\- for corner nodes + +$$ +N ^ {(n)} = \frac {1}{4} (\xi^ {2} + \xi \xi_ {i})) (\eta^ {2} + \eta \eta_ {i}), \quad i = 1, 3, \quad 5, \quad 7, +$$ + +\- for midside nodes + +$$ +N _ {i} ^ {(s)} = \frac {1}{2} \eta_ {i} ^ {2} (\eta^ {2} - \eta \eta_ {i}) (1 - \xi^ {2}) + \frac {1}{2} \xi_ {i} ^ {2} (\xi^ {2} - \xi \xi_ {i}) (1 - \eta^ {2}), \quad i = 2, 4, 6, 8, +$$ + +\- for central node + +$$ +N _ {i} ^ {(e)} = (1 - \xi^ {2}) (1 - \eta^ {2}). +$$ + +![](images/page-170_e2734ee87400b781b87ff884c6bcd7c21d49653e5600dd00a2607abe69dce0d2.jpg) + +
+natural_image + +3D wireframe surface plot with a magnified circular pattern on the right (no text or symbols) +
+ +Fig. 6.1(c) The 9-node Lagrangian quadrilateral element. diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_018.md b/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_018.md new file mode 100644 index 00000000..59d296bf --- /dev/null +++ b/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_018.md @@ -0,0 +1,490 @@ + + +![](images/page-171_ee53d9f53bb1b85c6e5c506af4eca19b334993509bee8bbb4c13fb70dfbedf13.jpg) + +
+natural_image + +3D surface plot with grid lines and a small inset grid, no text or symbols present +
+ +Fig. 6.1(c) The 9-node Lagrangian quadrilateral element (continued). + +For each application, a computer code is developed which allows the solution of practical problems. The computation times of elasto-plastic problems are relatively high with solution costs being typically ten times those of the corresponding linear elastic analysis. Of course a direct comparison would depend on the extent of plastic yielding and how close to the ultimate load carrying capacity a solution is sought. In view of these relatively high computer costs it is essential that the codes developed should be as efficient as possible and that any numerical techniques which reduce the computational requirements be employed. Since the main aim of this text is to fulfil a teaching role some compromise must however be inevitably made between program clarity and efficiency. The applicability of the programs presented is demonstrated by the solution of practical examples. Detailed user instructions for all of the computer programs presented in Part II of this text are provided in Appendix II. + +In Section 6.2 the basic expressions for the linear elastic finite element analysis of two-dimensional continua and plate bending problems are presented. Section 6.3 outlines the principles of isoparametric element formulation with particular attention being given to the role of numerical integration. Standard subroutines pertaining to linear elastic finite element analysis are reviewed in Section 6.4 and some subroutines common to the three nonlinear applications considered in Chapters 7, 8 and 9 are presented in Section 6.5. + + + +# 6.2 Virtual work expressions for various solid mechanics applications + +# 6.2.1 Introduction + +In this section we briefly describe various two-dimensional solid mechanics finite element applications in the elastic range only. Later in Chapters 7–9 we demonstrate how elasto-plastic or elasto-viscoplastic behaviour may be included in these applications using finite elements. + +In Part I we presented some very simple finite element representations. By contrast, in Part II we include numerically integrated isoparametric quadrilateral elements. + +# 6.2.2 Virtual work expression + +If a body is subjected to a set of body forces b then by the Virtual Work Principle we can write + +$$ +\int_ {\Omega} [ \delta \epsilon ] ^ {T} \sigma d \Omega - \int_ {\Omega} [ \delta \boldsymbol {u} ] ^ {T} \boldsymbol {b} d \Omega - \int_ {\Gamma_ {t}} [ \delta \boldsymbol {u} ] ^ {T} \boldsymbol {t} d \Gamma = 0, \tag {6.1} +$$ + +where $\sigma$ is the vector of stresses, t is the vector of boundary tractions, $\delta u$ is the vector of virtual displacements, $\delta \epsilon$ is the vector of associated virtual strains, $\Omega$ is the domain of interest, $\Gamma_{t}$ is that part of the boundary on which boundary tractions are prescribed and $\Gamma_{u}$ is that part of the boundary on which displacements are prescribed. + +# 6.2.3 Plane stress + +Consider some typical plane stress problems shown in Fig. 6.2. Typically a thin plate is subjected to loads applied in the xy plane, that is the plane of the structure. $^{(2)}$ The thickness of the plate is assumed to be small compared with the plan dimensions in the xy plane. Stresses are assumed to be constant through the thickness of the plate and $\sigma_{z}$ , $\tau_{zx}$ and $\tau_{zy}$ are ignored. Thus the displacements may now be expressed as + +$$ +\boldsymbol {u} = [ u, v ] ^ {T}, \tag {6.2} +$$ + +where $u$ and $v$ are the in-plane displacements in the $x$ and $y$ directions respectively. + +The strain components may be listed in the vector + +$$ +\epsilon = [ \epsilon_ {x}, \epsilon_ {y}, \gamma_ {x y} ] ^ {T}, \tag {6.3} +$$ + +where for small displacements the normal strains are given as + +$$ +\epsilon_ {x} = \frac {\partial u}{\partial x}, \quad \epsilon_ {y} = \frac {\partial v}{\partial y}, +$$ + + + +![](images/page-173_46f658799121f7b0bdcfea7b6ea35511d2e9021a43d557fd136619f7883ffdc9.jpg) +Fig. 6.2 Typical plane stress problems. + +and the shear strain is given as + +$$ +\gamma_ {x y} = \frac {\dot {c} u}{\dot {c} y} + \frac {\dot {c} v}{\dot {c} x}. +$$ + +Note that virtual displacements are listed in the vector + +$$ +\delta \boldsymbol {u} = [ \delta u, \delta v ] ^ {T}, \tag {6.4} +$$ + +and the associated virtual strains are + +$$ +\delta \epsilon = \left[ \frac {\dot {c} (\delta u)}{\dot {c} x}, \frac {\dot {c} (\delta v)}{\dot {c} y}, \frac {\dot {c} (\delta u)}{\dot {c} y} + \frac {\dot {c} (\delta v)}{\dot {c} x} \right] ^ {T}. \tag {6.5} +$$ + +The relevant stress-strain relationships may be written as + +$$ +\sigma = D \epsilon , \tag {6.6} +$$ + +where + +$$ +\sigma = \left[ \sigma_ {x}, \sigma_ {y}, \tau_ {x y} \right] ^ {T}, +$$ + +in which $\sigma_{x}$ and $\sigma_{y}$ are the normal stresses and $\tau_{xy}$ is the shear stress. + +For linear elastic situations the stress-strain or constitutive matrix is given as + +$$ +\boldsymbol {D} = \frac {E}{(1 - \nu^ {2})} \left[ \begin{array}{c c c} 1 & \nu & 0 \\ \nu & 1 & 0 \\ 0 & 0 & \frac {(1 - \nu)}{2} \end{array} \right], \tag {6.7} +$$ + + + +in which $E$ and $\nu$ are the elastic modulus and Poisson's ratio respectively. + +The body forces b are written as + +$$ +\boldsymbol {b} = [ b _ {x}, b _ {y} ] ^ {T}, \tag {6.8} +$$ + +in which $b_{x}$ and $b_{y}$ are the body forces per unit volume in the x and y directions respectively. + +Boundary tractions t may be expressed as + +$$ +\boldsymbol {t} = [ t _ {x}, t _ {y} ] ^ {T}, \tag {6.9} +$$ + +in which $t_{x}$ and $t_{y}$ are the boundary tractions per unit length. + +An element of volume $d\Omega$ is given as + +$$ +d \Omega = t d x d y, \tag {6.10} +$$ + +where t is the plate thickness. + +# 6.2.4 Plane strain + +For plane strain problems the thickness dimension normal to a certain plane (say the xy plane) is large compared with the typical dimensions in the xy plane and the body is subjected to loads in the xy plane only. For plane strain problems $^{(2)}$ it may be assumed that the displacements in the z direction are negligible and that the in-plane displacements u and v are independent of z. Figure 6.3 illustrates some typical plane strain problems. + +The displacements are then listed in the vector + +$$ +\boldsymbol {u} = [ u, v ] ^ {T}, \tag {6.11} +$$ + +![](images/page-174_b448d0488010ef21b5c83cb612314109c56f3232e150186eae70c22cef9378e6.jpg) + +
+natural_image + +Pure geometric diagram of a trapezoidal structure with horizontal and vertical lines, no text or symbols present +
+ +(a) + +![](images/page-174_d8327c1c71db6131d8926bc0255582796b6008e238b22278280a1d6b89216610.jpg) + +
+natural_image + +Concentric circles with radial arrows, no text or symbols present +
+ +(b) +Fig. 6.3 Typical plane strain problems. + + + +in which $u$ and $v$ are the in-plane displacements in the $x$ and $y$ directions respectively. + +The in-plane strain components may be expressed as + +$$ +\epsilon = [ \epsilon_ {x}, \epsilon_ {y}, \gamma_ {x y} ] ^ {T}, \tag {6.12} +$$ + +where $\epsilon_{x}$ , $\epsilon_{y}$ and $\gamma_{xy}$ have the same meaning as the strain components in plane stress applications. + +Again the virtual displacements and associated virtual strains are respectively given as + +$$ +\delta \boldsymbol {u} = [ \delta u, \delta v ] ^ {T}, \tag {6.13} +$$ + +and + +$$ +\delta \epsilon = \left[ \frac {\partial (\delta u)}{\partial x}, \frac {\partial (\delta v)}{\partial y}, \frac {\partial (\delta u)}{\partial y} + \frac {\partial (\delta v)}{\partial x} \right] ^ {T}. \tag {6.14} +$$ + +The stress–strain relationships may be written in the form + +$$ +\sigma = D \epsilon , \tag {6.15} +$$ + +where the stresses $\sigma = [\sigma_x, \sigma_y, \tau_{xy}]^T$ have the same meaning as the stresses in plane stress applications. + +For linear elastic materials the stress-strain or constitutive matrix D is given as + +$$ +\boldsymbol {D} = \frac {E}{(1 + \nu) (1 - 2 \nu)} \left[ \begin{array}{c c c} (1 - \nu) & \nu & 0 \\ \nu & (1 - \nu) & 0 \\ 0 & 0 & \frac {(1 - 2 \nu)}{2} \end{array} \right]. \tag {6.16} +$$ + +Note that the stress normal to the $xy$ plane is nonzero and may be evaluated as + +$$ +\sigma_ {z} = \nu (\sigma_ {x} + \sigma_ {y}). \tag {6.17} +$$ + +The body forces b and surface tractions t have the same meaning as those adopted for plane stress problems. + +A typical element of volume is given as + +$$ +d \Omega = d x d y. \tag {6.18} +$$ + +under the assumption that a unit slice of the problem is being analysed. + +# 6.2.5 Axisymmetric solids + +For a three-dimensional solid which is symmetrical about its centreline axis (which coincides with the z axis) and which is subjected to loads and boundary conditions that are symmetrical about this axis, then the behaviour $^{(2)}$ is independent of the circumferential coordinate $\theta$ . Figure 6.4 shows a typical axisymmetric solid. + + + +![](images/page-176_28b875a3ec70467febd24adcd1a0760d532d9024bacc364d46ced0b535bc5741.jpg) + +
+text_image + +Axisymmetric loading +Axisymmetric loading +r, u +z, w +
+ +Fig. 6.4 A typical axisymmetric solid. + +The displacements may here be expressed as + +$$ +\boldsymbol {u} = [ u, w ] ^ {T}, \tag {6.19} +$$ + +where u and w are the displacements in the r and z directions respectively. + +The nonzero strains are given as + +$$ +\epsilon = [ \epsilon_ {r}, \epsilon_ {\theta}, \epsilon_ {z}, \gamma_ {r z} ] ^ {T}, \tag {6.20} +$$ + +where for small displacements, the normal strains are given as + +$$ +\epsilon_ {r} = \frac {\partial u}{\partial r}, \quad \epsilon_ {\theta} = \frac {u}{r} \quad \text { and } \quad \epsilon_ {z} = \frac {\partial w}{\partial z}, +$$ + +and the shear strain is + +$$ +\gamma_ {r z} = \frac {\partial u}{\partial z} + \frac {\partial w}{\partial r}. +$$ + +Virtual displacements and associated virtual strains are respectively given as + +$$ +\delta \boldsymbol {u} = [ \delta \boldsymbol {u}, \delta \boldsymbol {w} ] ^ {T}, \tag {6.21} +$$ + +and + +$$ +\delta \epsilon = \left[ \frac {\dot {c} (\delta u)}{\partial r}, \frac {\delta u}{r}, \frac {\dot {c} (\delta w)}{\dot {c} z}, \frac {\dot {c} (\delta u)}{\dot {c} z} + \frac {\dot {c} (\delta w)}{\dot {c} r} \right] ^ {T}. \tag {6.22} +$$ + +The stress-strain relationships are given as + +$$ +\sigma = D \epsilon , \tag {6.23} +$$ + +where $\sigma = [\sigma_{r}, \sigma_{\theta}, \sigma_{z}, \tau_{rz}]^{T}$ , in which $\sigma_{r}, \sigma_{\theta}$ and $\sigma_{z}$ are the normal stresses in the r, $\theta$ and z directions respectively and $\tau_{rz}$ is the shear stress in the rz plane. + + + +For linear elastic materials, the stress–strain matrix is given as + +$$ +D = \frac {E}{(1 + \nu) (1 - 2 \nu)} \left[ \begin{array}{c c c c} (1 - \nu) & \nu & 0 & 0 \\ \nu & (1 - \nu) & \nu & 0 \\ 0 & \nu & (1 - \nu) & 0 \\ 0 & 0 & 0 & \frac {(1 - 2 \nu)}{2} \end{array} \right] \tag {6.24} +$$ + +The body forces are given as + +$$ +\boldsymbol {b} = [ b _ {r}, b _ {z} ] ^ {T}, \tag {6.25} +$$ + +where $b_{r}$ and $b_{z}$ are the body forces/unit volume in the r and z direction respectively. + +The boundary tractions may be expressed as + +$$ +\boldsymbol {t} = [ t _ {r}, t _ {z} ] ^ {T}, \tag {6.26} +$$ + +where $t_{r}$ and $t_{z}$ are the boundary tractions/unit surface in the r and z directions. + +An elemental volume is given as + +$$ +d \Omega = 2 \pi r d r d z. \tag {6.27} +$$ + +# 6.2.6 Mindlin plates + +In Mindlin plate theory it is possible to allow for transverse shear deformation. It thus offers an alternative to classical Kirchhoff thin plate theory. The main assumptions are that: + +(a) displacements are small compared with the plate thickness, +(b) the stress normal to the midsurface of the plate is negligible, +(c) normals to the midsurface before deformation remain straight but not necessarily normal to the midsurface after deformation. + +A typical Mindlin plate is shown in Fig. 6.5. Note that Mindlin plate theory is the two-dimensional equivalent of Timoshenko beam theory which was discussed in Chapter 5. + +The main displacement parameters can be expressed + +$$ +\boldsymbol {u} = [ w, \theta_ {x}, \theta_ {y} ] ^ {T}, \tag {6.28} +$$ + +in which w is the lateral plate displacement normal to the xy plane and variables $\theta_{x}$ and $\theta_{y}$ are the normal rotations in the xz and yz planes. Here it should be noted that + +$$ +\theta_ {x} = \frac {\dot {c} w}{\partial x} - \phi_ {x} \quad \text { and } \quad \theta_ {y} = \frac {\dot {c} w}{\dot {c} y} - \phi_ {y}, \tag {6.29} +$$ + +where $\theta_{x}$ and $\theta_{y}$ are the rotations of the normal in the $xz$ and $yz$ planes + + + +![](images/page-178_a5ae519125815c12cc4cb63c0ad5768d8dbf47a6a27be5f604273ab2722c10de.jpg) + +
+text_image + +z, w +Qx +Mxy +Qy +Myx +My +Myx +Myx +Myx +Mxy +Mxy +Qy +Qx +θx +x +θy +Fig. 6.5. A typical Mindlin plate +
+ +Fig. 6.5 A typical Mindlin plate. + +respectively and are integrated measures of the transverse shear strain. In thin plate theory it is assumed that shear rotations $\phi_{x}$ and $\phi_{y}$ , defined below, are equal to zero. + +The strains, or more exactly the strain resultants, may be expressed as + +$$ +\epsilon = [ r _ {x}, r _ {y}, r _ {x y}, \phi_ {x}, \phi_ {y} ] ^ {T}, \tag {6.30} +$$ + +where the curvatures are given as + +$$ +r _ {x} = - \frac {\partial \theta_ {x}}{\partial x} \quad \text { and } \quad r _ {y} = - \frac {\partial \theta_ {y}}{\partial y}, +$$ + +and the twisting curvature is + +$$ +r _ {x y} = - \left(\frac {\partial \theta_ {y}}{\partial x} + \frac {\partial \theta_ {x}}{\partial y}\right). +$$ + +The shear strains are expressed as + +$$ +\phi_ {x} = \left(\frac {\partial w}{\partial x} - \theta_ {x}\right) \quad \text { and } \quad \phi_ {y} = \left(\frac {\partial w}{\partial y} - \theta_ {y}\right). \tag {6.31} +$$ + +Virtual displacements and rotations and associated virtual curvatures and shear strains are respectively given as + +$$ +\delta \boldsymbol {u} = [ \delta w, \delta \theta_ {x}, \delta \theta_ {y} ] ^ {T}, \tag {6.32} +$$ + +and + +$$ +\delta \epsilon = \left[ - \frac {\partial (\delta \theta_ {x})}{\partial x}, - \frac {\partial (\delta \theta_ {y})}{\partial y}, - \frac {\partial (\delta \theta_ {x})}{\partial y} - \frac {\partial (\delta \theta_ {y})}{\partial x}, \right. +$$ + +$$ +\left. \frac {\partial (\delta w)}{\partial x} - \delta \theta_ {x}, \frac {\partial (\delta w)}{\partial y} - \delta \theta_ {y} \right] ^ {T}. \tag {6.33} +$$ + + + +The constitutive relationships are given in the form + +$$ +\sigma = D \epsilon , \tag {6.34} +$$ + +where + +$$ +\sigma = [ M _ {x}, M _ {y}, M _ {x y}, Q _ {x}, Q _ {y} ] ^ {T}, +$$ + +in which $M_{x}$ and $M_{y}$ are the direct bending moments and $M_{xy}$ is the twisting moment. The quantities $Q_{x}$ and $Q_{y}$ are the shear forces in the xz and yz planes. + +For an isotropic elastic material + +$$ +\boldsymbol {D} = \left[ \begin{array}{c c c c c} D & \nu D & 0 & 0 & 0 \\ \nu D & D & 0 & 0 & 0 \\ 0 & 0 & \frac {(1 - \nu)}{2} D & 0 & 0 \\ 0 & 0 & 0 & S & 0 \\ 0 & 0 & 0 & 0 & S \end{array} \right], \tag {6.35} +$$ + +in which for a plate of thickness t + +$$ +D = \frac {E t ^ {3}}{1 2 (1 - \nu^ {2})} \quad \text { and } \quad S = \frac {G t}{1 . 2}, +$$ + +where $G$ is the shear modulus and the factor 1.2 is a shear correction term. + +Here we will not consider surface tractions. For a more complete discussion of this and other aspects of Mindlin plate theory the reader is directed to the work of Hughes and his coworkers. $^{(3)}$ We will only consider body forces of the form + +$$ +\boldsymbol {b} = [ q, 0, 0 ] ^ {T}, \tag {6.36} +$$ + +where $q$ is the lateral distributed loading per unit area. + +An elemental plate area is given as + +$$ +d \Omega = d x d y. \tag {6.37} +$$ + +# 6.3 Isoparametric finite element representation + +# 6.3.1 Governing equations + +In this section we present the discretised governing equations for the solid mechanics applications described in Sections 6.2.3–6.2.6. In a finite element representation, the displacements and strains and their virtual counterparts may be expressed by the relationships + +$$ +\boldsymbol {u} = \sum_ {i = 1} ^ {n} N _ {i} \boldsymbol {d} _ {i}, \quad \delta \boldsymbol {u} = \sum_ {i = 1} ^ {n} N _ {i} \delta \boldsymbol {d} _ {i}, \tag {6.38} +$$ + + + +$$ +\epsilon = \sum_ {i = 1} ^ {n} B _ {i} d _ {i}, \quad \delta \epsilon = \sum_ {i = 1} ^ {n} B _ {i} \delta d _ {i}, \tag {6.39} +$$ + +where, for node $i$ , $d_{i}$ is the vector of nodal variables, $^{*}$ $\delta d_{i}$ is the vector of virtual nodal variables, $N_{i} = I N_{i}$ is the matrix of global shape functions $\dagger$ and $B_{i}$ is the global strain-displacement matrix. The total number of nodes in the whole mesh is $n$ . + +If (6.38) and 6.39) are substituted into the virtual work expression (6.1) then we obtain + +$$ +\sum_ {i = 1} ^ {n} \left[ \delta \boldsymbol {d} _ {i} \right] ^ {T} \left\{\int_ {\Omega} \left[ \boldsymbol {B} _ {i} \right] ^ {T} \sigma d \Omega - \int_ {\Omega} \left[ N _ {i} \right] ^ {T} \boldsymbol {b} d \Omega - \int_ {\Gamma_ {t}} \left[ N _ {i} \right] ^ {T} \boldsymbol {t} d \Gamma \right\} = 0, \tag {6.40} +$$ + +and since (6.40) must be true for an arbitrary set of virtual displacements $\delta d_{i}$ then we have for each node i an equation of the form + +$$ +\int_ {\Omega} \left[ \boldsymbol {B} _ {i} \right] ^ {T} \sigma d \Omega - \int_ {\Omega} \left[ N _ {i} \right] ^ {T} \boldsymbol {b} d \Omega - \int_ {\Gamma_ {t}} \left[ N _ {t} \right] ^ {T} \boldsymbol {t} d \Gamma = 0. \tag {6.41} +$$ + +If we use $C(0)$ isoparametric finite element representations we can evaluate contributions to (6.41) separately from each element. + +The displacements can be expressed in the usual way as + +$$ +\boldsymbol {u} ^ {(e)} = \sum_ {i = 1} ^ {r} N _ {i} ^ {(e)} \boldsymbol {d} _ {i} ^ {(e)}, \tag {6.42} +$$ + +where, for local node i of element e, $N^{(e)} = I N^{(e)}$ is the matrix of shape functions and the vector of variables is $d_{i}^{(e)}$ . There are r local nodes in each element e. + +Typical 4-, 8- and 9-node isoparametric element shape functions are shown and listed in Figs. 6.1(a), (b) and (c) respectively. + +Note that in an isoparametric representation we may use the following representation for the $x$ and $y$ coordinates within an element diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_019.md b/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_019.md new file mode 100644 index 00000000..914ae0cb --- /dev/null +++ b/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_019.md @@ -0,0 +1,431 @@ + + +$$ +\left[ \begin{array}{l} x ^ {(e)} \\ y ^ {(e)} \end{array} \right] = \sum_ {i = 1} ^ {r} \left[ \begin{array}{c c} N _ {i} ^ {(e)} & 0 \\ 0 & N _ {i} ^ {(e)} \end{array} \right] \left[ \begin{array}{l} x _ {i} ^ {(e)} \\ y _ {i} ^ {(e)} \end{array} \right], \tag {6.43} +$$ + +in which $N_{i}^{(e)}$ are the same shape functions used in the displacement representation. We may then evaluate the Jacobian matrix as + +$$ +\boldsymbol {J} ^ {(e)} = \left[ \begin{array}{l l} \frac {\partial x}{\partial \xi} & \frac {\partial y}{\partial \xi} \\ \frac {\partial x}{\partial \eta} & \frac {\partial y}{\partial \eta} \end{array} \right] = \left[ \begin{array}{l l} \sum_ {i = 1} ^ {r} \frac {\partial N _ {i} ^ {(e)}}{\partial \xi} x _ {i} ^ {(e)} & \sum_ {i = 1} ^ {r} \frac {\partial N _ {i} ^ {(e)}}{\partial \xi} y _ {i} ^ {(e)} \\ \sum_ {i = 1} ^ {r} \frac {\partial N _ {i} ^ {(e)}}{\partial \eta} x _ {i} ^ {(e)} & \sum_ {i = 1} ^ {r} \frac {\partial N _ {i} ^ {(e)}}{\partial \eta} y _ {i} ^ {(e)} \end{array} \right]. \tag {6.44} +$$ + +The inverse of $J^{(e)}$ is then evaluated using the expression + +$$ +[ \boldsymbol {J} ^ {(e)} ] ^ {- 1} = \left[ \begin{array}{l l} \frac {\partial \xi}{\partial x} & \frac {\partial \eta}{\partial x} \\ \frac {\partial \xi}{\partial y} & \frac {\partial \eta}{\partial y} \end{array} \right] = \frac {1}{\det \boldsymbol {J} ^ {(e)}} \left[ \begin{array}{c c} \frac {\partial y}{\partial \eta} & - \frac {\partial y}{\partial \xi} \\ - \frac {\partial x}{\partial \eta} & \frac {\partial x}{\partial \xi} \end{array} \right]. \tag {6.45} +$$ + +The strain displacement relationships are expressed as + +$$ +\epsilon^ {(e)} = \sum_ {i = 1} ^ {r} B _ {i} ^ {(e)} d _ {i} ^ {(e)}, \tag {6.46} +$$ + +in which $B_{t}^{(e)}$ is the strain matrix. + +The discretised elemental volume (or area in the case of Mindlin plates) is given as + +$$ +d \Omega^ {(e)} = h ^ {(e)} \det J ^ {(e)} d \xi d \eta , \tag {6.47} +$$ + +where $h^{(e)}$ has been defined in Table 6.1 in which we also summarise the expressions for $d_{i}^{(e)}$ , $B_{i}^{(e)}$ and $d\Omega^{(e)}$ for the four applications. + +The Cartesian shape function derivatives used in the strain-displacement matrices in Table 6.1 may be obtained using the chain rule of differentiation + +$$ +\frac {\partial N _ {i} ^ {(e)}}{\partial x} = \frac {\partial N _ {i} ^ {(e)}}{\partial \xi} \frac {\partial \xi}{\partial x} + \frac {\partial N _ {i} ^ {(e)}}{\partial \eta} \frac {\partial \eta}{\partial x}, \tag {6.48} +$$ + +\* For axisymmetric problems replace x and y by r and z respectively. + + + +
Application $d_i^{(e)}$ $B_i^{(e)}$ $d\Omega^{(e)}$
Plane stress $\begin{bmatrix} u_i^{(e)} \\ v_i^{(e)}\end{bmatrix}$ $\begin{bmatrix} \left(\frac{\partial N_i}{\partial x}\right)^{(e)} & 0 \\ 0 & \left(\frac{\partial N_i}{\partial y}\right)^{(e)} \\ \left(\frac{\partial N_i}{\partial y}\right)^{(e)} & \left(\frac{\partial N_i}{\partial x}\right)^{(e)}\end{bmatrix}$ $t^{(e)} \det J^{(e)}d\xi d\eta$
Plane strain $\begin{bmatrix} u_i^{(e)} \\ v_i^{(e)}\end{bmatrix}$ $\begin{bmatrix} \left(\frac{\partial N_i}{\partial x}\right)^{(e)} & 0 \\ 0 & \left(\frac{\partial N_i}{\partial y}\right)^{(e)} \\ \left(\frac{\partial N_i}{\partial y}\right)^{(e)} \quad \left(\frac{\partial N_i}{\partial x}\right)^{(e)}\end{bmatrix}$ $\det J^{(e)}d\xi d\eta$
Axial symmetry $\begin{bmatrix} u_i^{(e)} \\ w_i^{(e)}\end{bmatrix}$ $\begin{bmatrix} \left(\frac{\partial N_i}{\partial r}\right)^{(e)} & 0 \\ \left(\frac{N_i}{r}\right)^{(e)} & 0 \\ 0 & \left(\frac{\partial N_i}{\partial z}\right)^{(e)} \\ \left(\frac{\partial N_i}{\partial z}\right)^{(e)} & \left(\frac{\partial N_i}{\partial r}\right)^{(e)}\end{bmatrix}$ $2\pi r^{(e)} \det J^{(e)}d\xi d\eta$
Mindlin plate $\begin{bmatrix} w_i^{(e)} \\ \theta_{xi^{(e)}} \\ \theta_{yi^{(e)}}\end{bmatrix}$ $\begin{bmatrix} 0 & \left(-\frac{\partial N_i}{\partial x}\right)^{(e)} & 0 \\ 0 & 0 & \left(-\frac{\partial N_i}{\partial y}\right)^{(e)} \\ 0 & \left(-\frac{\partial N_i}{\partial y}\right)^{(e)} & \left(-\frac{\partial N_i}{\partial x}\right)^{(e)} \\ \left(\frac{\partial N_i}{\partial x}\right)^{(e)} & -N_i^{(e)} & 0 \\ \left(\frac{\partial N_i}{\partial y}\right)^{(e)} & 0 & -N_i^{(e)}\end{bmatrix}$ $\det J^{(e)}d\xi d\eta$
+ +Table 6.1 Nodal displacements, strain matrices and elemental volumes or areas for two-dimensional solid mechanics applications. + + + +and + +$$ +\frac {\partial N _ {i} ^ {(e)}}{\partial y} = \frac {\partial N _ {i} ^ {(e)}}{\partial \eta} \frac {\partial \eta}{\partial y} + \frac {\partial N _ {i} ^ {(e)}}{\partial \xi} \frac {\partial \xi}{\partial y}, +$$ + +in which the terms $\partial \xi / \partial x$ , $\partial \eta / \partial x$ , $\partial \eta / \partial y$ and $\partial \xi / \partial y$ may be obtained from the inverse of the Jacobian matrix given in (6.45). + +Since we have a linear stress-strain relationship within each element of the form + +$$ +\sigma^ {(e)} = D ^ {(e)} \epsilon^ {(e)} = D ^ {(e)} \left(\sum_ {j = 1} ^ {r} B _ {j} ^ {(e)} d _ {j} ^ {(e)}\right), \tag {6.49} +$$ + +then the contribution from element $e$ to the first term in (6.41) is given as + +$$ +\sum_ {j = 1} ^ {r} \boldsymbol {K} _ {i j} ^ {(e)} \boldsymbol {d} _ {j} ^ {(e)} \equiv \int_ {\Omega^ {(e)}} \left[ \boldsymbol {B} _ {i} ^ {(e)} \right] ^ {T} \boldsymbol {D} ^ {(e)} \left(\sum_ {j = 1} ^ {r} \boldsymbol {B} _ {j} ^ {(e)} \boldsymbol {d} _ {j} ^ {(e)}\right) d \Omega , \tag {6.50} +$$ + +where $K_{ij}^{(e)}$ is the submatrix of element stiffness matrix $K^{(e)}$ . + +The contribution from element e to the second term in (6.41) is given as + +$$ +f _ {B _ {i} ^ {(e)}} = \int_ {\Omega^ {(e)}} [ N _ {i} ^ {(e)} ] ^ {T} b ^ {(e)} d \Omega . \tag {6.51} +$$ + +For the third term, the contribution from element e is + +$$ +f _ {T _ {i}} ^ {(e)} = \int_ {I _ {t} ^ {(e)}} [ N _ {i} ^ {(e)} ] ^ {T} t ^ {(e)} d \Gamma , \tag {6.52} +$$ + +where $\Gamma_t^{(e)}$ is that part of $\Gamma_t$ which coincides with a boundary of element $e$ . Of course for many elements there will be no contribution to $f_{Tt}^{(e)}$ . + +# 6.3.2 Evaluation of the stiffness matrix and consistent load vector + +Let us now consider the evaluation of K. + +The integration is now performed in the natural coordinate system. Thus the submatrix of the stiffness matrix $K^{(e)}$ linking nodes i and j has the form + +$$ +\boldsymbol {K} _ {i j} ^ {(e)} = \int_ {- 1} ^ {+ 1} \int_ {- 1} ^ {+ 1} \left[ \boldsymbol {B} _ {i} ^ {(e)} \right] ^ {T} \boldsymbol {D} ^ {(e)} \boldsymbol {B} _ {j} ^ {(e)} h ^ {(e)} \det \boldsymbol {J} ^ {(e)} d \xi d \eta . \tag {6.53} +$$ + +The elements of $K_{ij}^{(e)}$ are evaluated numerically. If the integrand in (6.53) is denoted as + +$$ +[ \boldsymbol {B} _ {i} ^ {(e)} ] ^ {T} \boldsymbol {D} ^ {(e)} \boldsymbol {B} _ {j} ^ {(e)} h ^ {(e)} \det \boldsymbol {J} ^ {(e)} = \boldsymbol {T} _ {i j} ^ {(e)}, \tag {6.54} +$$ + +then + +$$ +\boldsymbol {K} _ {i j} ^ {(e)} = \int_ {- 1} ^ {+ 1} \int_ {- 1} ^ {+ 1} \boldsymbol {T} _ {i j} ^ {(e)} d \xi d \eta . \tag {6.55} +$$ + + + +The numerical integration for a quadrilateral element with $n \times n$ sampling points leads to + +$$ +\boldsymbol {K} _ {i j} ^ {(e)} = \sum_ {p - 1} ^ {n} \sum_ {q = 1} ^ {n} \boldsymbol {T} \left(\xi_ {p}, \bar {\eta} _ {q}\right) _ {i j} W _ {p} W _ {q}, \tag {6.56} +$$ + +where $W_{p}$ and $W_{q}$ are weighting factors and $(\bar{\xi}_{p}, \bar{\eta}_{q})$ is a sampling position. The consistent nodal forces at node i caused by body forces are + +$$ +\boldsymbol {f} _ {B _ {i}} ^ {(e)} = \int_ {- 1} ^ {+ 1} \int_ {- 1} ^ {+ 1} \left[ \boldsymbol {N} _ {i} ^ {(e)} \right] ^ {T} \boldsymbol {b} ^ {(e)} h ^ {(e)} \det \boldsymbol {J} ^ {(e)} d \xi d \eta . \tag {6.57} +$$ + +The components of $f_{Bt}^{(e)}$ are evaluated numerically. If the integrand in (6.57) is denoted as + +$$ +\boldsymbol {g} _ {i} ^ {(e)} = \left[ N _ {i} ^ {(e)} \right] ^ {T} \boldsymbol {b} ^ {(e)} h ^ {(e)} \det \boldsymbol {J} ^ {(e)}, \tag {6.58} +$$ + +then + +$$ +\boldsymbol {f} _ {B _ {i}} ^ {(e)} = \int_ {- 1} ^ {+ 1} \int_ {- 1} ^ {+ 1} \boldsymbol {g} _ {i} ^ {(e)} d \xi d \eta . \tag {6.59} +$$ + +The numerical integration for a quadrilateral with $n \times n$ sampling points leads to + +$$ +f _ {B _ {i}} ^ {(e)} = \sum_ {p = 1} ^ {n} \sum_ {q = 1} ^ {n} \mathbf {g} (\bar {\xi} _ {p}, \bar {\eta} _ {q}) _ {i} ^ {(e)} W _ {p} W _ {q}, \tag {6.60} +$$ + +where $W_{p}$ and $W_{q}$ are weighting factors and $(\bar{\xi}_p,\bar{\eta}_q)$ is a sampling position. + +The consistent nodal forces for boundary tractions have been dealt with in the authors' previous book $^{(4)}$ and will be summarised in Section 6.4.5. + +The computer implementation of numerically integrated isoparametric elements has been described in detail in the text of Finite Element Programming. $^{(4)}$ Here we simply summarise in Fig. 6.6 the main steps involved in evaluating the element stiffness matrix. + +# 6.4 Standard subroutines for linear elastic finite element analysis + +Many of the subroutines required for elasto-plastic finite element analysis are common to the corresponding linear elastic application. In this section we present all the standard linear elastic subroutines required for later use in Chapters 7, 8 and 9. The function of each subroutine is explained and a FORTRAN listing is provided. The subroutines presented are drawn from Ref. 4 where a detailed description is provided. + +In order to make all subroutines modular in form we have adopted a type of dynamic dimensioning. Thus no COMMON blocks are used in the programs in Part II. Dimensions are fixed in the main or master routine and all necessary information is transmitted between routines by the use of + + + +# SUBROUTINE STIF2D + +Dimensions and common blocks. + +→ Enter loop over all elements. + +Retrieve element geometry and material properties for the current element. + +Zero the stiffness array. + +Call a routine which sets up $D^{(e)}$ the constitutive matrix. + +Enter loops covering all integration points. + +Look up sampling position for the current integration point $(\tilde{\xi}_p, \bar{\eta}_q)$ . + +Call shape function routine SFR2—given $(\bar{\xi}_{p}, \bar{\eta}_{q})$ this will return the shape functions $N_{i}^{(e)}$ and their derivatives $\partial N_{i}^{(e)}/\partial\xi$ and $\partial N_{i}^{(e)}/\partial\eta$ at the point $(\bar{\xi}_{p}, \bar{\eta}_{q})$ . + +Call JACOB2—given $N_{i}^{(e)}$ , $\partial N_{i}^{(e)}/\partial\xi$ and $\partial N_{i}^{(e)}/\partial\eta$ at point $(\bar{\xi}_{p},\bar{\eta}_{q})$ ; this will return Cartesian shape function derivatives $\partial N_{i}^{(e)}/\partial x$ and $\partial N_{i}^{(e)}/\partial y$ , the Jacobian matrix $J^{(e)}$ , its inverse $[J^{(e)}]^{-1}$ and its determinant $\det J^{(e)}$ and the x and y (or r and z) coordinates all at the point $(\bar{\xi}_{p},\bar{\eta}_{q})$ . + +Call strain matrix routine—given $N_{i}^{(e)}$ , $\partial N_{i}^{(e)}/\partial x$ and $\partial N_{i}^{(e)}/\partial y$ at $(\bar{\xi}_{p}, \bar{\eta}_{q})$ this will return the strain matrix $\boldsymbol{B}_{i}^{(e)}$ . + +Call a routine to evaluate $D^{(e)} B^{(e)}$ . + +Evaluate $[B_{i}^{(e)}]D^{(e)}B_{j}^{(e)}\det J^{(e)}\times\text{integration weights and assemble them into the element stiffness array }K_{ij}^{(e)}$ . + +Assemble $D^{(e)}B^{(e)}$ into a stress array for later evaluation of stresses from the nodal displacements. + +End integration loops. + +Write stiffness matrix and stress matrix onto file for use in the solution routine. + +End element loop. + +RETURN +END + +Fig. 6.6 Evaluation of element stiffness matrices for numerically integrated isoparametric elements. + + + +arguments (and also peripherals in certain instances). Apart from the modularity, this approach has the advantage that maximum dimensions can be updated in a very simple and straightforward manner. Only the DIMENSION statement in the main segment and some statements in a subroutine which sets the maximum dimensions sizes need modification. + +As an example, the relevant statements in a dynamically dimensioned program are listed below. + +```txt +PROGRAM FRED ( ) +DIMENSION AMATX (200, 5), ...* +. +. +. +CALL DIMENS (MROWS, MCOLS) +. +. +. +CALL DUMMY (AMATX, MROWS, MCOLS) +. +. +. +STOP +END +SUBROUTINE DIMENS (MROWS, MCOLS) +MROWS =200* +MCOLS = 5* +RETURN +END +SUBROUTINE DUMMY (AMATX, MROWS, MCOLS) +DIMENSION AMATX (MROWS, MCOLS) +. +. +. +RETURN +END +``` + +Note that AMATX ( ) has fixed dimensions in the main routine FRED. Subroutine DIMENS assigns values of 200 and 5 to the dimensions MROWS and MCOLS respectively.† In subroutine DUMMY we transmit AMATX, + +† Alternatively a DATA statement can be used. + + + +MROWS and MCOLS via the argument and therefore the DIMENSION statement in DUMMY refers to AMATX (MCOLS, MROWS) and not AMATX (200, 5). To update FRED for arrays AMATX with different maximum dimensions, we simply modify those statements indicated by an asterisk. + +Note also that the use of such arguments is not very expensive since only the address of the first term of an array is passed through the argument and not of all the terms in the array. + +More sophisticated versions of this approach can be implemented as illustrated in the book by Irons and Ahmad. $^{(5)}$ Such approaches undoubtedly save core storage but they do require careful housekeeping and checking procedures. + +In Part III we have generally dispensed with the use of maximum dimension variables in the programs. Thus main segment FRED would then be written as + +PROGRAM FRED ( ) +DIMENSION AMATX (200, 5), ... +. +. +. +CALL DUMMY (AMATX) +. +. +. +STOP +END +SUBROUTINE DUMMY (AMATX) +DIMENSION AMATX (200, 1)† +. +. +. +RETURN +END + +Although this approach uses nonstandard FORTRAN IV it does work on most machines and it has been adopted elsewhere in the literature. $^{(6)}$ If more than one subroutine such as DUMMY uses AMATX then the relevant dimensions must be identical in all of these subroutines. + +The list of variables in the argument list will differ between linear and nonlinear applications. For each subroutine presented in this section the form of the argument list and the dimension statements will be those required for two-dimensional elasto-plastic applications. + + + +# 6.4.1 Subroutine NODEXY for generating coordinate values for midside nodes + +For the quadratic 8- and 9-node elements described in Section 6.3 subroutine NODEXY checks each midside node (a midside node being recognisable from the element topology cards). If both coordinates of a midside node are found to be zero, its coordinates are linearly interpolated between the two adjacent corner nodes. Subroutine NODEXY is common to plane stress/strain, axisymmetric and plate bending situations. + +```fortran +SUBROUTINE NODEXY(COORD, LNODS, MELEM, MPOIN, NELEM, NNODE) NODE 1 +C******************************* +C +C**** THIS SUBROUTINE INTERPOLATES THE MIDE SIDE NODES OF STRAIGHT NODE 2 +C SIDES OF ELEMENTS AND THE CENTRAL NODE OF 9 NODED ELEMENTS NODE 3 +C +C******************************* +DIMENSION COORD(MPOIN, 2), LNODS(MELEM, 9) NODE 5 +IF(NNODE.EQ.4) RETURN NODE 6 +C +C*** LOOP OVER EACH ELEMENT NODE 7 +C +DO 30 IELEM=1, NELEM NODE 8 +C +C*** LOOP OVER EACH ELEMENT EDGE NODE 9 +C +NNOD1=9 NODE 10 +IF(NNODE.EQ.8) NNOD1=7 NODE 11 +DO 20 INODE=1, NNOD1, 2 NODE 12 +IF(INODE.EQ.9) GO TO 50 NODE 13 +C +C*** COMPUTE THE NODE NUMBER OF THE FIRST NODE NODE 14 +C +NODST=LNODS(IELEM, INODE) NODE 15 +IGASH=INODE+2 NODE 16 +IF(IGASH.GT.8) IGASH=1 NODE 17 +C +C*** COMPUTE THE NODE NUMBER OF THE LAST NODE NODE 18 +C +NODFN=LNODS(IELEM, IGASH) NODE 19 +MIDPT=INODE+1 NODE 20 +C +C*** COMPUTE THE NODE NUMBER OF THE LAST NODE NODE 21 +C +NODFN=LNODS(IELEM, IGASH) NODE 22 +MIDPT=INODE+1 NODE 23 +C +C*** COMPUTE THE NODE NUMBER OF THE LAST NODE NODE 24 +C +NODFN=LNODS(IELEM, IGASH) NODE 25 +MIDPT=INODE+1 NODE 26 +C +C*** COMPUTE THE NODE NUMBER OF THE LAST NODE NODE 27 +C +NODFN=LNODS(IELEM, IGASH) NODE 28 +MIDPT=INODE+1 NODE 29 +C +C*** COMPUTE THE NODE NUMBER OF THE LAST NODE NODE 30 +C +NODFN=LNODS(IELEM, IGASH) NODE 31 +MIDPT=INODE+1 NODE 32 +C +C*** COMPUTE THE NODE NUMBER OF THE LAST NODE NODE 33 +C +NODFN=LNODS(IELEM, MIDPT) NODE 34 +TOTAL=ABS(COORD(NODMD, 1))+ABS(COORD(NODMD, 2)) NODE 35 +C +C*** IF THE COORDINATES OF THE INTERMEDIATE NODE ARE BOTH ZERO NODE 36 +C INTERPOLATE BY A STRAIGHT LINE NODE 37 +C +IF(TOTAL.GT.0.0) GO TO 20 NODE 38 +KOUNT=1 NODE 39 +10 COORD(NODMD, KOUNT)=(COORD(NODST, KOUNT)+COORD(NODFN, KOUNT))/2.0 NODE 40 +KOUNT=KOUNT+1 NODE 41 +IF(KOUNT.EQ.2) GO TO 42 NODE 42 +20 CONTINUE NODE 43 +GO TO 30 NODE 44 +50 LNODE=LNODS(IELEM, INODE) NODE 45 +TOTAL=ABS(COORD(LNODE, 1))+ABS(COORD(LNODE, 2)) NODE 46 +IF(TOTAL.GT.0.0) GO TO 30 NODE 47 +50 LNODE=LNODS(IELEM, INODE) NODE 48 +TOTAL=ABS(COORD(LNODE, 1))+ABS(COORD(LNODE, 2)) NODE 49 +IF(TOTAL.GT.0.0) GO TO 30 NODE 50 +``` + + + +
LNOD1=LNODS(IELEM,1)NODE51
LNOD3=LNODS(IELEM,3)NODE52
LNOD5=LNODS(IELEM,5)NODE53
LNOD7=LNODS(IELEM,7)NODE54
KOUNT=1NODE55
40 COORD(LNODE,KOUNT)=(COORD(LNOD1,KOUNT)+COORD(LNOD3,KOUNT)NODE56
+COORD(LNOD5,KOUNT)+COORD(LNOD7,KOUNT))/4.0NODE57
KOUNT=KOUNT+1NODE58
IF(KOUNT.EQ.2) GO TO 40NODE59
30 CONTINUENODE60
RETURNNODE61
ENDNODE62
+ +# 6.4.2 Subroutine GAUSSQ for generating Gaussian quadrature data + +The function of this subroutine is to set up the sampling point positions and weighting factors for numerical integration. The Gauss quadrature processes utilised in this text are restricted to either two or three point integration rules.\* The role of numerical integration in the isoparametric formulation was discussed in detail in Section 6.3. The order of integration rule to be employed is defined by NGAUS and the sampling point positions and weighting factors are stored respectively in arrays POSGP() and WEIGP(). + +
SUBROUTINE GAUSSQ(NGAUS,POSGP,WEIGP)GAUS1
C***************GAUS2
CGAUS3
C****THIS SUBROUTINE SETS UP THE GAUSS-LEGENDRE INTEGRATION CONSTANTSGAUS4
CGAUS5
C***************GAUS6
DIMENSION POSGP(4),WEIGP(4)GAUS7
IF(NGAUS.GT.2) GO TO 4GAUS8
2POSGP(1)=-0.577350269189626GAUS9
WEIGP(1)=1.0GAUS10
GO TO 6GAUS11
4POSGP(1)=-0.774596669241483GAUS12
5POSGP(2)=0.0GAUS13
WEIGP(1)=0.555555555555556GAUS14
WEIGP(2)=0.888888888888889GAUS15
6KGAUS=NGAUS/2GAUS16
DO 8 IGASH=1,KGAUSGAUS17
JGASH=NGAUS+1-IGASHGAUS18
POSGP(JGASH)=-POSGP(IGASH)GAUS19
WEIGP(JGASH)=WEIGP(IGASH)GAUS20
8CONTINUEGAUS21
RETURNGAUS22
ENDGAUS23
+ +# 6.4.3 Subroutine SFR2 for evaluating the element shape functions + +The role of this subroutine is to evaluate the shape functions $N_{i}^{(e)}(\xi, \eta)$ and their derivatives $\partial N_{i}^{(e)} / \partial \xi$ , $\partial N_{i}^{(e)} / \partial \eta$ at any sampling point $\xi_{P}$ , $\eta_{P}$ within the element for each of the 4-, 8- or 9-noded elements described in Section 6.1. The shape functions for these elements are listed in Figs. 6.1(a), (b) and (c). The sampling point coordinates $\xi_{P}, \eta_{P}$ are specified as EXISP and ETASP respectively. The evaluated shape functions for each node of an element are stored in array SHAPE (INODE) and their derivatives in + +\- Except for selectively integrated 4-node Mindlin plates in which we modify GAUSSQ so that if NGAUS = 1 then POSGP(1) = 0·0 and WEIGP(1) = 2·0. + + + +array DERIV (INODE, IDIME) where INODE ranges over the element nodes and IDIME over the coordinate dimensions. + +```csv +SUBROUTINE SFR2(DERIV,ETASP,EXISP,NNODE,SHAPE) SFR2 1 +C***** THIS SUBROUTINE EVALUATES SHAPE FUNCTIONS AND THEIR DERIVATIVES SFR2 2 +C FOR LINEAR,QUADRATIC LAGRANGIAN AND SERENDIPITY SFR2 3 +C ISOPARAMETRIC 2-D ELEMENTS SFR2 4 +C SFR2 5 +C***** DIMENSION DERIV(2,9),SHAPE(9) SFR2 6 +S=EXISP SFR2 7 +T=ETASP SFR2 8 +IF(NNODE.GT.4) GO TO 10 SFR2 9 +ST=S*T SFR2 10 +C SFR2 11 +C *** SHAPE FUNCTIONS FOR 4 NODED ELEMENT SFR2 12 +C SFR2 13 +SHAPE(1)=(1-T-S+ST)*0.25 SFR2 14 +SHAPE(2)=(1-T+S-ST)*0.25 SFR2 15 +SHAPE(3)=(1+T+S+ST)*0.25 SFR2 16 +SHAPE(4)=(1+T-S-ST)*0.25 SFR2 17 +C SFR2 18 +C *** SHAPE FUNCTION DERIVATIVES SFR2 19 +C SFR2 20 +DERIV(1,1)=(-1+T)*0.25 SFR2 21 +DERIV(1,2)=(+1-T)*0.25 SFR2 22 +DERIV(1,3)=(+1+T)*0.25 SFR2 23 +DERIV(1,4)=(-1-T)*0.25 SFR2 24 +DERIV(2,1)=(-1+S)*0.25 SFR2 25 +DERIV(2,2)=(-1-S)*0.25 SFR2 26 +DERIV(2,3)=(+1+S)*0.25 SFR2 27 +DERIV(2,4)=(+1-S)*0.25 SFR2 28 +RETURN SFR2 29 +10 IF(NNODE.GT.8)GO TO 30 SFR2 30 +S2=S*2.0 SFR2 31 +T2=T*2.0 SFR2 32 +SS=S*S SFR2 33 +TT=T*T SFR2 34 +ST=S*T SFR2 35 +SST=S*S*T SFR2 36 +STT=S*T*T SFR2 37 +ST2=S*T*2.0 SFR2 38 +C SFR2 39 +C *** SHAPE FUNCTIONS FOR 8 NODED ELEMENT SFR2 40 +C SFR2 41 +SHAPE(1)=(-1.0+ST+SS+TT-SST-STT)/4.0 SFR2 42 +SHAPE(2)=(1.0-T-SS+SST)/2.0 SFR2 43 +SHAPE(3)=(-1.0-ST+SS+TT-SST+STT)/4.0 SFR2 44 +SHAPE(4)=(1.0+S-TT-STT)/2.0 SFR2 45 +SHAPE(5)=(-1.0+ST+SS+TT+SST+STT)/4.0 SFR2 46 +SHAPE(6)=(1.0+T-SS-SST)/2.0 SFR2 47 +SHAPE(7)=(-1.0-ST+SS+TT+SST-STT)/4.0 SFR2 48 +SHAPE(8)=(1.0-S-TT+STT)/2.0 SFR2 49 +C *** SHAPE FUNCTION DERIVATIVES SFR2 50 +C SFR2 51 +DERIV(1,1)=(T+S2-ST2-TT)/4.0 SFR2 52 +DERIV(1,2)=-S+ST SFR2 53 +DERIV(1,3)=(-T+S2-ST2+TT)/4.0 SFR2 54 +DERIV(1,4)=(1.0-TT)/2.0 SFR2 55 +DERIV(1,5)=(T+S2+ST2+TT)/4.0 SFR2 56 +DERIV(1,6)=-S-ST SFR2 57 +DERIV(1,7)=(-T+S2-ST2+TT)/4.0 SFR2 58 +DERIV(1,8)=(-T+S2-ST2+TT)/4.0 SFR2 59 +DERIV(1,9)=(-T+S2-ST2+TT)/4.0 SFR2 60 +DERIV(2,0)=(-T+S2-ST2+TT)/4.0 SFR2 61 +``` diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_020.md b/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_020.md new file mode 100644 index 00000000..26928056 --- /dev/null +++ b/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_020.md @@ -0,0 +1,2180 @@ + + +```txt +DERIV(1,7)=(-T+S2+ST2-TT)/4.0 SFR2 62 +DERIV(1,8)=(-1.0+TT)/2.0 SFR2 63 +DERIV(2,1)=(S+T2-SS-ST2)/4.0 SFR2 64 +DERIV(2,2)=(-1.0+SS)/2.0 SFR2 65 +DERIV(2,3)=(-S+T2-SS+ST2)/4.0 SFR2 66 +DERIV(2,4)=-T-ST SFR2 67 +DERIV(2,5)=(S+T2+SS+ST2)/4.0 SFR2 68 +DERIV(2,6)=(1.0-SS)/2.0 SFR2 69 +DERIV(2,7)=(-S+T2+SS-ST2)/4.0 SFR2 70 +DERIV(2,8)=-T+ST SFR2 71 +RETURN SFR2 72 +30 CONTINUE SFR2 73 +SS=S*S SFR2 74 +ST=S*T SFR2 75 +TT=T*T SFR2 76 +S1=S+1.0 SFR2 77 +T1=T+1.0 SFR2 78 +S2=S*2.0 SFR2 79 +T2=T*2.0 SFR2 80 +S9=S-1.0 SFR2 81 +T9=T-1.0 SFR2 82 +C SFR2 83 +C*** SHAPE FUNCTIONS FOR 9 NODED ELEMENT SFR2 84 +C SFR2 85 +SHAPE(1)=0.25*S9*ST*T9 SFR2 86 +SHAPE(2)=0.5*(1.0-SS)*T*T9 SFR2 87 +SHAPE(3)=0.25*S1*ST*T9 SFR2 88 +SHAPE(4)=0.5*S*S1*(1.0-TT) SFR2 89 +SHAPE(5)=0.25*S1*ST*T1 SFR2 90 +SHAPE(6)=0.5*(1.0-SS)*T*T1 SFR2 91 +SHAPE(7)=0.25*S9*ST*T1 SFR2 92 +SHAPE(8)=0.5*S*S9*(1.0-TT) SFR2 93 +SHAPE(9)=(1.0-SS)*(1.0-TT) SFR2 94 +C SFR2 95 +C*** SHAPE FUNCTION DERIVATIVES SFR2 96 +C SFR2 97 +DERIV(1,1)=0.25*T*T9*(-1.0+S2) SFR2 98 +DERIV(1,2)=-ST*T9 SFR2 99 +DERIV(1,3)=0.25*(1.0+S2)*T*T9 SFR2 100 +DERIV(1,4)=0.5*(1.0+S2)*(1.0-TT) SFR2 101 +DERIV(1,5)=0.25*(1.0+S2)*T*T1 SFR2 102 +DERIV(1,6)=-ST*T1 SFR2 103 +DERIV(1,7)=0.25*(-1.0+S2)*T*T1 SFR2 104 +DERIV(1,8)=0.5*(-1.0+S2)*(1.0-TT) SFR2 105 +DERIV(1,9)=-S2*(1.0-TT) SFR2 106 +DERIV(2,1)=0.25*(-1.0+T2)*S*S9 SFR2 107 +DERIV(2,2)=0.5*(1.0-SS)*(-1.0+T2) SFR2 108 +DERIV(2,3)=0.25*S*S1*(-1.0+T2) SFR2 109 +DERIV(2,4)=-ST*S1 SFR2 110 +DERIV(2,5)=0.25*S*S1*(1.0+T2) SFR2 111 +DERIV(2,6)=0.5*(1.0-SS)*(1.0+T2) SFR2 112 +DERIV(2,7)=0.25*S*S9*(1.0+T2) SFR2 113 +DERIV(2,8)=-ST*S9 SFR2 114 +DERIV(2,9)=-T2*(1.0-SS) SFR2 115 +20 CONTINUE SFR2 116 +RETURN SFR2 117 +END SFR2 118 +``` + +# 6.4.4 Subroutine JACOB2 for evaluating the Jacobian matrix + +This subroutine calculates, for any sampling position, $\xi_{P}$ , $\eta_{P}$ (usually the Gauss point), the following quantities: + + + +- The Cartesian coordinates of the Gauss point which are stored in the array GPCOD ( ). +- The Jacobian matrix which is stored in XJACM ( ). For two-dimensional applications the Jacobian matrix is defined by (6.44). +● The determinant of the Jacobian matrix, DJACB. +- The inverse of the Jacobian matrix which is stored as XJACI ( ). +- The Cartesian derivatives $\partial N_{t}^{(e)} / \partial x$ , $\partial N_{t}^{(e)} / \partial y$ (or $\partial N_{t}^{(e)} / \partial r$ , $\partial N_{t}^{(e)} / \partial z$ ), of the element shape functions. These quantities are defined in (6.48). + +```csv +SUBROUTINE JACOB2(CARTD,DERIV,DJACB,ELCOD,GPCOD,IELEM,KGASP, NNODE,SHAPE) JACB 1 +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +CREATE JACOBIAN MATRIX XJACM JACB 21 +C +DO 4 IDIME=1,2 JACB 22 +DO 4 JDIME=1,2 JACB 23 +XJACM(IDIME,JDIME)=0.0 JACB 24 +DO 4 INODE=1,NNODE JACB 25 +XJACM(IDIME,JDIME)=XJACM(IDIME,JDIME)+DERIV(IDIME,INODE)* JACB 26 +ELCOD(JDIME,INODE) JACB 27 +4 CONTINUE JACB 28 +JACB 29 +JACB 30 +JACB 31 +JACB 32 +JACB 33 +JACB 34 +JACB 35 +JACB 36 +JACB 37 +JACB 38 +JACB 39 +JACB 40 +JACB 41 +JACB 42 +JACB 43 +JACB 44 +JACB 45 +JACB 46 +JACB 47 +JACB 48 +JACB 49 +JACB 50 +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +D +D +D +D +D +D +D +D +D +D +D +D +D +D +D +D +D +D +D +D +D +D +D +D +D +D +D +D +D +D +D +D +D +D +D +D +D +D +D +D +D +D +D +D +D +D +D +D +D +D +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +O +O +O +O +O +O +O +O +O +O +O +O +O +O +O +O +O +O +O +O +O +O +O +O +O +O +O +O +O +O +O +O +O +O +O +O +O +O +O +O +O +O +O +O +O +O +O +O +O +O +N +N +N +N +N +N +N +N +N +N +N +N +N +N +N +N +N +N +N +N +N +N +N +N +N +N +N +N +N +N +N +N +N +N +N +N +N +N +N +N +N +N +N +N +N +N +N +N +N +N +O +O +O +O +O +O +O +O +O +O +O +O +O +O +O +O +O +O +O +O +O +O +O +O +O +O +O +O +O +O +O +O +O +O +O +O +O +O +O +O +O +O +O +O +O +O +O +O +O +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +N +N +N +N +N +N +N +N +N +N +N +N +N +N +N +N +N +N +N +N +N +N +N +N +N +N +N +N +N +N +N +N +N +N +N +N +N +N +N +N +N +N +N +N +N +N +N +N +N +M +N +N +N +N +N +N +N +N +N +N +N +N +N +N +N +N +N +N +N +N +N +N +N +N +N +N +N +N +N +N +N +N +N +N +N +N +N +N +N +N +N +N +N +N +N +N +N +N +N +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +3 +3 +3 +3 +3 +3 +3 +3 +3 +3 +3 +3 +3 +3 +3 +3 +3 +3 +3 +3 +3 +3 +3 +3 +3 +3 +3 +3 +3 +3 +3 +3 +3 +3 +3 +3 +3 +3 +3 +3 +3 +3 +3 +3 +3 +3 +3 +3 +3 +3 +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +A +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +M +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +J +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +I +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +P +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +B +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +L +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +U +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +S +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +G +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +F +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +E +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +T +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +R +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +c +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +| +| +| +| +| +| +| +| +| +| +| +| +| +| +| +| +| +| +| +| +| +| +| +| +| +| +| +| +| +| +| +| +| +| +| +| +| +| +| +| +| +| +| +| +| +| +| +| +| +| +| + +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +``` + + + +
10 CONTINUEJACB51
600 FORMAT(//,36H PROGRAM HALTED IN SUBROUTINE JACOB2,/,11X, .22H ZERO OR NEGATIVE AREA,/,10X,16H ELEMENT NUMBER ,I5)JACB52
JACB53
RETURNJACB54
ENDJACB55
+ +# 6.4.5 Subroutine LOADPS for evaluating the element nodal forces for plane and axisymmetric situations + +The role of this subroutine is to evaluate the consistent nodal forces for each element due to discrete point loads, gravity loading and distributed edge loading/unit length of element. This subroutine is described in detail in Chapter 7, Ref. 4. The types of loading to be considered are controlled by input parameters IPLOD, IGRAV, IEDGE. Nonzero values of these respective items indicate that point loads, gravity loading or distributed edge loading is to be considered. + +The consistent nodal loads are evaluated for each element separately and stored in the array RLOAD (IELEM, IEVAB) where IELEM indicates the element and IEVAB ranges over the degrees of freedom of the element. For equation solution by the frontal process it is not necessary to evaluate the total applied load acting at each node, with instead each element contribution being assembled directly into the global load vector during equation assembly and solution. + +# Point loads + +If parameter IPLOD is nonzero the applied nodal loads are read as input. For each particular node the applied forces are associated with any one of the elements attached to it; since each element contribution will be assembled before equation solution. Thus a search is performed over all elements until the node number is found in an element and the nodal loads are then associated with the appropriate degrees of freedom of that element. + +# Gravity loading + +For plane stress or plane strain problems the direction in which gravity acts need not coincide with either of the coordinate axes. Therefore the direction in which gravity acts must be defined as shown in Fig. 6.7 by specifying the angle $\theta$ which the gravity axis makes with the positive y axis. The intensity of the loading is defined by specifying the gravitational acceleration, g, which acts. For axisymmetric problems, of course, the gravity axis must coincide with the z axis. + +The consistent nodal forces for node i of an element are then given by + +$$ +\left[ \begin{array}{l} P _ {x i} \\ P _ {y i} \end{array} \right] ^ {(e)} = \int_ {\underline {{\Omega}} ^ {(e)}} N _ {i} ^ {(e)} \rho g \left[ \begin{array}{c} \sin \theta \\ - \cos \theta \end{array} \right] d \Omega , \tag {6.61} +$$ + +in which $\rho$ is the material mass density. Integrated numerically this becomes + + + +$$ +\left[ \begin{array}{l} P _ {x t} \\ P _ {y t} \end{array} \right] ^ {(e)} = \sum_ {n = 1} ^ {N G A U S} \sum_ {m = 1} ^ {N G A U S} \rho g t \left[ \begin{array}{c} \sin \theta \\ - \cos \theta \end{array} \right] N _ {t} (\xi_ {n}, \eta_ {m}) W _ {n} W _ {m} \det J, \tag {6.62} +$$ + +where t is the element thickness for plane problems. For axisymmetric applications t is replaced by $2\pi r_{P}$ , where $r_{P}$ is the radial distance to the Gauss point under consideration. + +![](images/page-194_73fddd4e53f983503cb0c0745f599206e6b691f07bc07863d5f06ddcd032a31d.jpg) + +
+text_image + +Consideration. +η = +1 +5 +-ξ = +1 +η +ξ = -1 +8 +y, z +2 +η = -1 +3 +θ = 0 for axisymmetric problems +x, r +Direction in which gravity acts +
+ +Fig. 6.7 Specification of the gravity axis for two-dimensional problems. + +# Distributed edge loading + +Any element edge can have a distributed loading per unit length in a normal and tangential direction prescribed to it as shown in Fig. 6.8. These distributed forces can vary (independently) along the edges. For the quadratic elements considered in this text, a quadratic loading distribution can, at best, be accommodated. The variation is defined by prescribing the normal and tangential values at the three nodal points forming the element edge to which the loads are applied. For linear quadrilateral elements, only a linear distributed load variation can be accommodated. In order to be consistent with the order of listing of nodal connection numbers in the element topology definition, the three (or two) nodes forming the loaded edge must also be listed in an anticlockwise sequence with respect to the loaded element. The positive directions of normal and tangential loading are indicated in Fig. 6.8. + +The consistent nodal forces for node i can be shown to be $^{(4)}$ + +$$ +P _ {x i} ^ {(e)} = \int_ {I ^ {(e)}} N _ {i} ^ {(e)} \left(p _ {t} \frac {\dot {c} x}{\dot {c} \xi} - p _ {n} \frac {\dot {c} y}{\dot {c} \xi}\right) d \xi +$$ + +$$ +P _ {y i} ^ {(e)} = \int_ {\Gamma^ {(e)}} N _ {i} ^ {(e)} \left(p _ {n} \frac {\dot {c} x}{\dot {c} \xi} + p _ {t} \frac {\dot {c} y}{\dot {c} \xi}\right) d \xi , \tag {6.63} +$$ + +where $p_{n}$ and $p_{t}$ are the normal and tangential distributed loads respectively. Integration is taken along the loaded element edge $\Gamma^{(e)}$ , which is arbitrarily chosen to be defined by $\eta = -1$ , as shown in Fig. 6.8. + + + +![](images/page-195_f990afdf4a36ce9eff3a1c174c46fa8ecd8ab62c509d30ea5316d3315eae399d.jpg) + +
+text_image + +numbering +sequence +y, z +x, r +Pn +P1 +η +ξ +dy +dx +dT +Pt +Pn +
+ +Fig. 6.8 Normal and tangential distributed loading on an element edge. + +For axisymmetric problems the edge loading is in fact a distributed loading/unit area, since integration is additionally made over the circumferential direction. + +If more than one type of loading acts on an element, the total nodal forces are accumulated and stored in array RLOAD. This total loading is then applied incrementally during elasto-plastic solution. + +```asm +SUBROUTINE LOADPS(COORD, LNODS, MATNO, MELEM, MMATS, MPOIN, NELEM, LDPS 1 +. NEVAB, NGAUS, NNODE, NPOIN, NSTRE, NTYPE, POSGP, LDPS 2 +. PROPS, RLOAD, WEIGP, NDOFN) LDPS 3 +. LDPS 4 +C*************** LDPS 5 +C LDPS 6 +C**** THIS SUBROUTINE EVALUATES THE CONSISTENT NODAL FORCES FOR EACH LDPS 7 +C ELEMENT LDPS 8 +C LDPS 9 +C*************** LDPS 10 +DIMENSION CARTD(2,9), COORD(MPOIN,2), DERIV(2,9), DGASH(2), LDPS 11 +. DMATX(4,4), ELCOD(2,9), LNODS(MELEM,9), MATNO(MELEM), LDPS 12 +. NOPRS(4), PGASH(2), POINT(2), POSGP(4), PRESS(4,2), LDPS 13 +. PROPS(MMATS,7), RLOAD(MELEM,18), SHAPE(9), STRAN(4), LDPS 14 +. STRES(4), TITLE(12), LDPS 15 +. WEIGP(4), GPCOD(2,9) LDPS 16 +TWOPI=6.283185308 LDPS 17 +DO 10 IELEM=1, NELEM LDPS 18 +DO 10 IEVAB=1, NEVAB LDPS 19 +10 RLOAD(IELEM,IEVAB)=0.0 LDPS 20 +READ(5,901) TITLE LDPS 21 +901 FORMAT(12A6) LDPS 22 +WRITE(6,903) TITLE LDPS 23 +903 FORMAT(1H0,12A6) LDPS 24 +``` + + + +```csv +C +C*** READ DATA CONTROLLING LOADING TYPES TO BE INPUTTED +C +READ(5,919) IPLOD,IGRAV,IEDGE +WRITE(6,919) IPLOD,IGRAV,IEDGE +919 FORMAT(3I5) +C +C*** READ NODAL POINT LOADS +C +IF(IPLOD.EQ.0) GO TO 500 +20 READ(5,919) LODPT,(POINT(IDOFN),IDOFN=1,2) +WRITE(6,931) LODPT,(POINT(IDOFN),IDOFN=1,2) +931 FORMAT(I5,2F10.3) +C +C*** ASSOCIATE THE NODAL POINT LOADS WITH AN ELEMENT +C +DO 30 IELEM=1,NELEM +DO 30 INODE=1,NNODE +NLOCA=IABS(LNODS(IELEM,INODE)) +30 IF(LODPT.EQ.NLOCA) GO TO 40 +40 DO 50 IDOFN=1,2 +NGASH=(INODE-1)*2+IDOFN +50 RLOAD(IELEM,NGASH)=POINT(IDOFN) +IF(LODPT.LT.NPOIN) GO TO 20 +500 CONTINUE +IF(IGRAV.EQ.0) GO TO 600 +C +C*** GRAVITY LOADING SECTION +C +C +C*** READ GRAVITY ANGLE AND GRAVITATIONAL CONSTANT +C +READ(5,906) THETA,GRAVY +906 FORMAT(2F10.3) +WRITE(6,911) THETA,GRAVY +911 FORMAT(1H0,16H GRAVITY ANGLE =,F10.3,19H GRAVITY CONSTANT =,F10.3) +THETA=THETA/57.295779514 +C +C*** LOOP OVER EACH ELEMENT +C +DO 90 IELEM=1,NELEM +C +C*** SET UP PRELIMINARY CONSTANTS +C +LPROP=MATNO(IELEM) +THICK=PROPS(LPROP,3) +DENSE=PROPS(LPROP,4) +IF(DENSE.EQ.0.0) GO TO 90 +GXCOM=DENSE*GRAVY*SIN(THETA) +GYCOM=-DENSE*GRAVY*COS(THETA) +C +C*** COMPUTE COORDINATES OF THE ELEMENT NODAL POINTS +C +DO 60 INODE=1,NNODE +LNODE=IABS(LNODS(IELEM,INODE)) +DO 60 IDIME=1,2 +60 ELCOD(IDIME,INODE)=COORD(LNODE,IDIME) +C +C*** ENTER LOOPS FOR AREA NUMERICAL INTEGRATION +C +KGASP=0 +DO 80 IGAUS=1,NGAUS +DO 80 JGAUS=1,NGAUS +EXISP=POSGP(IGAUS) +ETASP=POSGP(JGAUS) +LDPS 25 +LDPS 26 +LDPS 27 +LDPS 28 +LDPS 29 +LDPS 30 +LDPS 31 +LDPS 32 +LDPS 33 +LDPS 34 +LDPS 35 +LDPS 36 +LDPS 37 +LDPS 38 +LDPS 39 +LDPS 40 +LDPS 41 +LDPS 42 +LDPS 43 +LDPS 44 +LDPS 45 +LDPS 46 +LDPS 47 +LDPS 48 +LDPS 49 +LDPS 50 +LDPS 51 +LDPS 52 +LDPS 53 +LDPS 54 +LDPS 55 +LDPS 56 +LDPS 57 +LDPS 58 +LDPS 59 +LDPS 60 +LDPS 61 +LDPS 62 +LDPS 63 +LDPS 64 +LDPS 65 +LDPS 66 +LDPS 67 +LDPS 68 +LDPS 69 +LDPS 70 +LDPS 71 +LDPS 72 +LDPS 73 +LDPS 74 +LDPS 75 +LDPS 76 +LDPS 77 +LDPS 78 +LDPS 79 +LDPS 80 +LDPS 81 +LDPS 82 +LDPS 83 +LDPS 84 +LDPS 85 +LDPS 86 +LDPS 87 +LDPS 88 +LDPS 89 +``` + + + +```csv +C +C*** COMPUTE THE SHAPE FUNCTIONS AT THE SAMPLING POINTS AND ELEMENTAL +C VOLUME +C +CALL SFR2(DERIV,ETASP,EXISP,NNODE,SHAPE) +KGASP=KGASP+1 +CALL JACOB2(CARTD,DERIV,DJACB,ELCOD,GPCOD,IELEM,KGASP, +NNODE,SHAPE) +DVOLU=DJACB*WEIGP(IGAUS)*WEIGP(JGAUS) +IF(THICK.NE.0.0) DVOLU=DVOLU*THICK +IF(NTYPE.EQ.3) DVOLU=DVOLU*TWOPI*GPCOD(1,KGASP) +C +C*** CALCULATE LOADS AND ASSOCIATE WITH ELEMENT NODAL POINTS +C +DO 70 INODE=1,NNODE +NGASH=(INODE-1)*2+1 +MGASH=(INODE-1)*2+2 +RLOAD(IELEM,NGASH)=RLOAD(IELEM,NGASH)+GXCOM*SHAPE(INODE)*DVOLU +70 RLOAD(IELEM,MGASH)=RLOAD(IELEM,MGASH)+GYCOM*SHAPE(INODE)*DVOLU +80 CONTINUE +90 CONTINUE +600 CONTINUE +IF(IEDGE.EQ.0) GO TO 700 +C +C*** DISTRIBUTED EDGE LOADS SECTION +C +READ(5,932) NEDGE +-932 FORMAT(I5) +WRITE(6,912) NEDGE +912 FORMAT(1H0,5X,21HNO. OF LOADED EDGES =,I5) +WRITE(6,915) +915 FORMAT(1H0,5X,38HLIST OF LOADED EDGES AND APPLIED LOADS) +NODEG=3 +NCODE=NNODE +IF(NNODE.EQ.4) NODEG=2 +IF(NNODE.EQ.9) NCODE=8 +C +C*** LOOP OVER EACH LOADED EDGE +C +DO 160 IEDGE=1,NEDGE +C +C*** READ DATA LOCATING THE LOADED EDGE AND APPLIED LOAD +C +READ(5,902) NEASS,(NOPRS(IODEG),IODEG=1,NODEG) +902 FORMAT(4I5) +WRITE(6,913) NEASS,(NOPRS(IODEG),IODEG=1,NODEG) +913 FORMAT(I10,5X,3I5) +READ(5,914) ((PRESS(IODEG,IDOFN),IDOFN=1,2),IODEG=1,NODEG) +WRITE(6,914) ((PRESS(IODEG,IDOFN),IDOFN=1,2),IODEG=1,NODEG) +914 FORMAT(6F10.3) +ETASP=-1.0 +C +C*** CALCULATE THE COORDINATES OF THE NODES OF THE ELEMENT EDGE +C +DO 100 IODEG=1,NODEG +LNODE=NOPRS(IODEG) +DO 100 IDIME=1,2 +100 ELCOD(IDIME,IODEG)=COORD(LNODE,IDIME) +C +C*** ENTER LOOP FOR LINEAR NUMERICAL INTEGRATION +DO 150 IGAUS=1,NGAUS +EXISP=POSGP(IGAUS) +C +C*** EVALUATE THE SHAPE FUNCTIONS AT THE SAMPLING POINTS +C +``` + + + +```csv +CALL SFR2(DERIV, ETASP, EXISP, NNODE, SHAPE) LDPS 155 +C +C*** CALCULATE COMPONENTS OF THE EQUIVALENT NODAL LOADS LDPS 157 +C +DO 110 IDOFN=1,2 LDPS 156 +PGASH(IDOFN)=0.0 LDPS 160 +DGASH(IDOFN)=0.0 LDPS 161 +DO 110 IODEG=1, NODEG LDPS 162 +PGASH(IDOFN)=PGASH(IDOFN)+PRESS(IODEG, IDOFN)*SHAPE(IODEG) LDPS 163 +110 DGASH(IDOFN)=DGASH(IDOFN)+ELCOD(IDOFN, IODEG)*DERIV(1, IODEG) LDPS 164 +DVOLU=WEIGP(IGAUS) LDPS 165 +PXCOM=DGASH(1)*PGASH(2)-DGASH(2)*PGASH(1) LDPS 166 +PYCOM=DGASH(1)*PGASH(1)+DGASH(2)*PGASH(2) LDPS 167 +IF(NTYPE.NE.3) GO TO 115 LDPS 168 +RADUS=0.0 LDPS 169 +DO 125 IODEG=1, NODEG LDPS 170 +125 RADUS=RADIUS+SHAPE(IODEG)*ELCOD(1, IODEG) LDPS 171 +DVOLU=DVOLU*TWOPI*RADIUS LDPS 172 +115 CONTINUE LDPS 173 +C +C*** ASSOCIATE THE EQUIVALENT NODAL EDGE LOADS WITH AN ELEMENT LDPS 175 +C +DO 120 INODE=1, NNODE LDPS 176 +NLOCA=IABS(LNODS(NEASS, INODE)) LDPS 177 +120 IF(NLOCA.EQ.NOPRS(1)) GO TO 130 LDPS 178 +130 JNODE=INODE+NODEG-1 LDPS 179 +KOUNT=0 LDPS 180 +DO 140 KNODE=INODE, JNODE LDPS 181 +KOUNT=KOUNT+1 LDPS 182 +NGASH=(KNODE-1)*NDOFN+1 LDPS 183 +MGASH=(KNODE-1)*NDOFN+2 LDPS 184 +IF(KNODE.GT.NCODE) NGASH=1 LDPS 185 +IF(KNODE.GT.NCODE) MGASH=2 LDPS 186 +RLOAD(NEASS, NGASH)=RLOAD(NEASS, NGASH)+SHAPE(KOUNT)*PXCOM*DVOLU LDPS 187 +140 RLOAD(NEASS, MGASH)=RLOAD(NEASS, MGASH)+SHAPE(KOUNT)*PYCOM*DVOLU LDPS 188 +150 CONTINUE LDPS 189 +160 CONTINUE LDPS 190 +700 CONTINUE LDPS 191 +WRITE(6,907) LDPS 192 +907 FORMAT(1HO,5X,36H TOTAL NODAL FORCES FOR EACH ELEMENT) LDPS 193 +DO 290 IELEM=1, NELEM LDPS 194 +290 WRITE(6,905) IELEM,(RLOAD(IELEM, IEVAB), IEVAB=1, NEVAB) LDPS 195 +905 FORMAT(1X,14,5X,8E12.4/(10X,8E12.4)) LDPS 196 +RETURN LDPS 197 +END LDPS 198 +LDPS 199 +``` + +# 6.4.6 Subroutine LOADPB for evaluating the element nodal forces for plate bending applications + +For plate bending applications two forms of loading will be considered. Firstly load components corresponding to the permissible generalised forces may be prescribed at the nodal points. Thus with respect to Fig. 6.9, a load in the z direction and couples acting in both the xz and yz planes may be input at each nodal point. Secondly a uniformly distributed load acting normal to the plate (i.e. in the z direction) may be applied. As in Section 6.4.5 such a loading must be converted into equivalent nodal forces before equation solution takes place. The equivalent nodal forces for node i take the form $^{(4)}$ + + + +$$ +\left[ \begin{array}{l} P _ {i} \\ M _ {x i} \\ M _ {y i} \end{array} \right] ^ {(e)} = \int_ {A ^ {(e)}} N _ {i} ^ {(e)} \left[ \begin{array}{l} q \\ 0 \\ 0 \end{array} \right] d A, \tag {6.64} +$$ + +where q is the distributed load intensity and integration is taken over the element area. The structure of the subroutine is similar to that of subroutine LOADPS described in Section 6.4.5. + +![](images/page-199_12f7eebe561571e7d62ec3478907e328773403722387073b60a691fbee292c98.jpg) + +
+text_image + +q/unit area +P +My +Mx +z +y +x +
+ +Fig. 6.9 Applied nodal and distributed forces for plate applications +```fortran +SUBROUTINE LOADPB (COORD, LNODS, MATNO, MELEM, MMATS, MPOIN, LOAD 1 +NELEM, NEVAB, NGAUS, NNODE, NPOIN, PROPS, LOAD 2 +RLOAD) LOAD 3 +C********** LOAD 4 +C LOAD 5 +C*** COMPUTE NODAL FORCES AFTER READING RELEVANT DATA LOAD 6 +C*** FOR MINDLIN PLATE ELEMENTS LOAD 7 +C LOAD 8 +C********** LOAD 9 +DIMENSION CARTD(2,9), COORD(MPOIN,2), DERIV(2,9), ELCOD(2,9), LOAD 10 +GPCOD(2,9), LNODS(MELEM,9), MATNO(MELEM), LOAD 11 +POINT(3), POSGP(4), PROPS(MMATS,8), RLOAD(MELEM,27), LOAD 12 +SHAPE(9), TITLE(12), WEIGP(4) LOAD 13 +DO 10 IELEM=1, NELEM LOAD 14 +DO 10 IEVAB=1, NEVAB LOAD 15 +10 RLOAD(IELEM, IEVAB)=0.0 LOAD 16 +READ(5,901) TITLE LOAD 17 +901 FORMAT(12A6) LOAD 18 +WRITE(6,903) TITLE LOAD 19 +903 FORMAT(1H0,12A6) LOAD 20 +C LOAD 21 +C*** READ DATA CONTROLLING LOADING TYPES TO BE INPUTTED LOAD 22 +C LOAD 23 +READ(5,919) IPLOD LOAD 24 +WRITE(6,919) IPLOD LOAD 25 +919 FORMAT(4I5) LOAD 26 +C LOAD 27 +C*** READ NODAL POINT LOADS LOAD 28 +C LOAD 29 +IF(IPLOD.EQ.0) GO TO 500 LOAD 30 +20 READ(5,931) LODPT, (POINT(IDOFN), IDOFN=1,3) LOAD 31 +WRITE(6,931) LODPT, (POINT(IDOFN), IDOFN=1,3) LOAD 32 +931 FORMAT(I5,2F10.3) LOAD 33 +C LOAD 34 +``` + + + +```csv +C*** ASSOCIATE THE NODAL POINT LOADS WITH AN ELEMENT +C +DO 30 IELEM=1,NELEM +DO 30 INODE=1,NNODE +NLOCA=IABS(LNODS(IELEM,INODE)) +30 IF(LODPT.EQ.NLOCA) GO TO 40 +40 DO 50 IDOFN=1,3 +NGASH=(INODE-1)*3+IDOFN +50 RLOAD(IELEM,NGASH)=POINT(IDOFN) +IF(LODPT.LT.NPOIN) GO TO 20 +500 CONTINUE +C +C*** LOOP OVER EACH ELEMENT +C +DO 220 IELEM=1,NELEM +LPROP=MATNO(IELEM) +UDLOD=PROPS(LPROP,4) +IF(UDLOD.EQ.0.0)GO TO 220 +C +C*** EVALUATE THE COORDINATES OF THE ELEMENT NODAL POINTS +C +DO 140 INODE=1,NNODE +LNODE=LNODS(IELEM,INODE) +LNODE=IABS(LNODE) +DO 140 IDIME=1,2 +ELCOD(IDIME,INODE)=COORD(LNODE,IDIME) +140 CONTINUE +KGASP=0 +CALL GAUSSQ (NGAUS,POSGP,WEIGP) +C +C*** ENTER LOOPS FOR NUMERICAL INTEGRATION +C +DO 200 IGAUS=1,NGAUS +EXISP=POSGP(IGAUS) +DO 200 JGAUS=1,NGAUS +ETASP=POSGP(JGAUS) +KGASP=KGASP+1 +C +C*** EVALUATE THE SHAPE FUNCTIONS AT THE SAMPLING +C POINTS AND ELEMENTAL AREA +C +CALL SFR2 (DERIV,ETASP,EXISP,NNODE,SHAPE) +CALL JACOB2 (CARTD,DERIV,DJACB,ELCOD,GPCOD,IELEM, +KGASP,NNODE,SHAPE) +DAREA=DJACB*WEIGP(IGAUS)*WEIGP(JGAUS) +C +C*** CALCULATE LOADS AND ASSOCIATE WITH ELEMENT NODALPOINTS +C +DO 180 INODE=1,NNODE +NPOSN=(INODE-1)*3+1 +RLOAD(IELEM,NPOSN)=RLOAD(IELEM,NPOSN)+ +SHAPE(INODE)*UDLOD*DAREA +180 CONTINUE +200 CONTINUE +220 CONTINUE +WRITE(6,907) +907 FORMAT(1H0,5X,36H TOTAL NODAL FORCES FOR EACH ELEMENT) +DO 290 IELEM=1,NELEM +290 WRITE(6,905) IELEM,(RLOAD(IELEM,IEVAB),IEVAB=1,NEVAB) +905 FORMAT(1X,I4,5X,8E12.4/(10X,8E12.4)) +RETURN +END +LOAD 35 +LOAD 36 +LOAD 37 +LOAD 38 +LOAD 39 +LOAD 40 +LOAD 41 +LOAD 42 +LOAD 43 +LOAD 44 +LOAD 45 +LOAD 46 +LOAD 47 +LOAD 48 +LOAD 49 +LOAD 50 +LOAD 51 +LOAD 52 +LOAD 53 +LOAD 54 +LOAD 55 +LOAD 56 +LOAD 57 +LOAD 58 +LOAD 59 +LOAD 60 +LOAD 61 +LOAD 62 +LOAD 63 +LOAD 64 +LOAD 65 +LOAD 66 +LOAD 67 +LOAD 68 +LOAD 69 +LOAD 70 +LOAD 71 +LOAD 72 +LOAD 73 +LOAD 74 +LOAD 75 +LOAD 76 +LOAD 77 +LOAD 78 +LOAD 79 +LOAD 80 +LOAD 81 +LOAD 82 +LOAD 83 +LOAD 84 +LOAD 85 +LOAD 86 +LOAD 87 +LOAD 88 +LOAD 89 +LOAD 90 +LOAD 91 +LOAD 92 +LOAD 93 +LOAD 94 +LOAD 95 +LOAD 96 +``` diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_021.md b/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_021.md new file mode 100644 index 00000000..6a790390 --- /dev/null +++ b/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_021.md @@ -0,0 +1,759 @@ + + +# 6.4.7 Subroutine BMATPS for evaluating the strain matrix B for plane and axisymmetric situations + +The function of this subroutine is to evaluate the strain matrix B at any position within an element. The relevant expressions are given in Table 6.1. The B matrix is stored in array BMATX ( ). + +```txt +SUBROUTINE BMATPS(BMATX,CARTD,NNODE,SHAPE,GPCOD,NTYPE,KGASP) BMPS 1 +C*************** BMPS 2 +C BMPS 3 +C**** THIS SUBROUTINE EVALUATES THE STRAIN-DISPLACEMENT MATRIX BMPS 4 +C BMPS 5 +C*************** BMPS 6 +DIMENSION BMATX(4,18),CARTD(2,9),SHAPE(9),GPCOD(2,9) BMPS 7 +NGASH=0 BMPS 8 +DO 10 INODE=1,NNODE BMPS 9 +MGASH=NGASH+1 BMPS 10 +NGASH=MGASH+1 BMPS 11 +BMATX(1,MGASH)=CARTD(1,INODE) BMPS 12 +BMATX(1,NGASH)=0.0 BMPS 13 +BMATX(2,MGASH)=0.0 BMPS 14 +BMATX(2,NGASH)=CARTD(2,INODE) BMPS 15 +BMATX(3,MGASH)=CARTD(2,INODE) BMPS 16 +BMATX(3,NGASH)=CARTD(1,INODE) BMPS 17 +IF(NTYPE.NE.3) GO TO 10 BMPS 18 +BMATX(4,MGASH)=SHAPE(INODE)/GPCOD(1,KGASP) BMPS 19 +BMATX(4,NGASH)=0.0 BMPS 20 +10 CONTINUE BMPS 21 +RETURN BMPS 22 +END BMPS 23 +``` + +# 6.4.8 Subroutine BMATPB for evaluating the strain matrix B for plate bending problems + +This subroutine evaluates the strain matrix B within any point of an element for plate bending applications according to Table 6.1. The B matrix is partitioned into plane, BPLAN, flexural, BFLEX, and shear, BSHER, contributions. + +```csv +SUBROUTINE BMATPB (BFLEX,BPLAN,BSHER,CARTD,KNODE,SHAPE, BMAT 1 +IFPLA,IFFLE,IFSHE) BMAT 2 +C********** BMAT 3 +C BMAT 4 +C*** EVALUATES STRAIN-DISPLACEMENT MATRIX FOR BMAT 5 +C*** MINDLIN PLATE BMAT 6 +C BMAT 7 +C********** BMAT 8 +DIMENSION BFLEX(3,3),BPLAN(3,2),BSHER(2,3), BMAT 9 +CARTD(2,9),SHAPE(9) BMAT 10 +DNKDX=CARTD(1,KNODE) BMAT 11 +DNKDY=CARTD(2,KNODE) BMAT 12 +C*** FORM BPLAN BMAT 13 +IF(IFPLA.EQ.0) GO TO 10 BMAT 14 +DO 1 IROWS=1,3 BMAT 15 +DO 1 JCOLS=1,2 BMAT 16 +1 BPLAN(IROWS,JCOLS)=0.0 BMAT 17 +BPLAN(1,1)=DNKDX BMAT 18 +BPLAN(2,2)=DNKDY BMAT 19 +BPLAN(3,1)=DNKDY BMAT 20 +BPLAN(3,2)=DNKDX BMAT 21 +``` + + + +
C*** FORM BFLEXBMAT22
10 IF(IFFLE.EQ.0) GO TO 20BMAT23
DO 2 IROWS=1,3BMAT24
DO 2 JCOLS=1,3BMAT25
2 BFLEX(IROWS,JCOLS)=0.0BMAT26
BFLEX(1,2)=-DNKDXBMAT27
BFLEX(2,3)=-DNKDYBMAT28
BFLEX(3,2)=-DNKDYBMAT29
BFLEX(3,3)=-DNKDXBMAT30
C*** FORM BSHERBMAT31
20 IF(IFSHE.EQ.0) RETURNBMAT32
DO 3 IROWS=1,2BMAT33
DO 3 JCOLS=1,3BMAT34
3 BSHER(IROWS,JCOLS)=0.0BMAT35
BSHER(1,1)=DNKDXBMAT36
BSHER(1,2)=-SHAPE(KNODE)BMAT37
BSHER(2,1)=DNKDYBMAT38
BSHER(2,3)=-SHAPE(KNODE)BMAT39
RETURNBMAT40
ENDBMAT41
+ +# 6.4.9 Subroutine MODPS for evaluating the $D$ matrix for plane and axisymmetric situations + +This subroutine simply evaluates the elasticity matrix D for either plane stress, plane strain or axisymmetric situations according to (6.7), (6.16) or (6.24) respectively. The D matrix is stored in the array DMATX(). + +```csv +SUBROUTINE MODPS(DMATX,LPROP,MMATS,NTYPE,PROPS) MDPS 1 +C***** +C +C**** +C +C +C***** +DIMENSION DMATX(4,4),PROPS(MMATS,7) MDPS 7 +YOUNG=PROPS(LPROP,1) MDPS 8 +POISS=PROPS(LPROP,2) MDPS 9 +DO 10 ISTR1=1,4 MDPS 10 +DO 10 JSTR1=1,4 MDPS 11 +10 DMATX(ISTR1,JSTR1)=0.0 MDPS 12 +IF(NTYPE.NE.1) GO TO 4 MDPS 13 +C +C**** +C +C +CONST=YOUNG/(1.0-POISS*POISS) MDPS 17 +DMATX(1,1)=CONST MDPS 18 +DMATX(2,2)=CONST MDPS 19 +DMATX(1,2)=CONST*POISS MDPS 20 +DMATX(2,1)=CONST*POISS MDPS 21 +DMATX(3,3)=(1.0-POISS)*CONST/2.0 MDPS 22 +RETURN MDPS 23 +4 IF(NTYPE.NE.2) GO TO 6 MDPS 24 +C +C**** +C +C +CONST=YOUNG*(1.0-POISS)/((1.0+POISS)*(1.0-2.0*POISS)) MDPS 28 +DMATX(1,1)=CONST MDPS 29 +DMATX(2,2)=CONST MDPS 30 +DMATX(1,2)=CONST*POISS/(1.0-POISS) MDPS 31 +DMATX(2,1)=CONST*POISS/(1.0-POISS) MDPS 32 +``` + + + +```asm +DMATX(3,3)=(1.0-2.0*POISS)*CONST/(2.0*(1.0-POISS)) +MDPS 33 +RETURN +MDPS 34 +6 IF(NTYPE.NE.3) GO TO 8 +MDPS 35 +C +MDPS 36 +C*** D MATRIX FOR AXISYMMETRIC CASE +MDPS 37 +C +CONST=YOUNG*(1.0-POISS)/((1.0+POISS)*(1.0-2.0*POISS)) +MDPS 38 +CONSS=POISS/(1.0-POISS) +MDPS 39 +DMATX(1,1)=CONST +MDPS 40 +DMATX(2,2)=CONST +MDPS 41 +DMATX(3,3)=CONST*(1.0-2.0*POISS)/(2.0*(1.0-POISS)) +MDPS 42 +DMATX(1,2)=CONST*CONSS +MDPS 43 +DMATX(1,4)=CONST*CONSS +MDPS 44 +DMATX(2,1)=CONST*CONSS +MDPS 45 +DMATX(2,4)=CONST*CONSS +MDPS 46 +DMATX(4,1)=CONST*CONSS +MDPS 47 +DMATX(4,2)=CONST*CONSS +MDPS 48 +DMATX(4,4)=CONST +MDPS 49 +8 CONTINUE +MDPS 50 +RETURN +MDPS 51 +END +MDPS 52 +MDPS 53 +``` + +# 6.4.10 Subroutine MODPB for evaluating the D matrix for plate bending applications + +This subroutine evaluates the elasticity matrix D for plate bending situations according to (6.35). Again the result is partitioned into plane, DPLAN, flexural, DFLEX, and shear, DSHER, contributions. + +```txt +SUBROUTINE MODPB (DFLEX, DPLAN, DSHER, LPROP, MMATS, PROPS, IFPLA, IFFLE, IFSHE) MODP 1 +C*************** MODP 2 +C +C*** CALCULATES MATRIX OF ELASTIC RIGIDITIES MODP 3 +C*** FOR MINDLIN PLATE MODP 4 +C +C*************** MODP 5 +DIMENSION DFLEX(3,3), DPLAN(3,3), DSHER(2,2), MODP 6 +PROPS(MMATS,8) MODP 7 +YOUNG=PROPS(LPROP,1) MODP 8 +POISS=PROPS(LPROP,2) MODP 9 +THICK=PROPS(LPROP,3) MODP 10 +C*** FORM DPLAN MODP 11 +IF(IFPLA.EQ.0) GO TO 10 MODP 12 +DO 1 IROWS=1,3 MODP 13 +DO 1 JCOLS=1,3 MODP 14 +1 DPLAN(IROWS, JCOLS)=0.0 MODP 15 +CONST=(YOUNG*THICK)/(1.0-POISS*POISS) MODP 16 +DPLAN(1,1)=CONST MODP 17 +DPLAN(2,2)=CONST MODP 18 +DPLAN(1,2)=CONST*POISS MODP 19 +DPLAN(2,1)=CONST*POISS MODP 20 +DPLAN(3,3)=CONST*(1.0-POISS)/2.0 MODP 21 +C*** FORM DFLEX MODP 22 +10 IF(IFFLE.EQ.0) GOTO 20 MODP 23 +DO 2 IROWS=1,3 MODP 24 +DO 2 JCOLS=1,3 MODP 25 +2 DFLEX(IROWS, JCOLS)=0.0 MODP 26 +CONST=(YOUNG*THICK**3)/(12.*(1.-POISS*POISS)) MODP 27 +DFLEX(1,1)=CONST MODP 28 +DFLEX(2,2)=CONST MODP 29 +DFLEX(1,2)=CONST*POISS MODP 30 +DFLEX(1,2)=CONST*POISS MODP 31 +DFLEX(1,2)=CONST*POISS MODP 32 +DFLEX(1,2)=CONST*POISS MODP 33 +``` + + + +```csv +DFLEX(2,1)=CONST*POISS +DFLEX(3,3)=CONST*(1.-POISS)/2. +C*** FORM DSHER +20 IF(IFSHE.EQ.0) RETURN +DO 3 IROWS=1,2 +DO 3 JCOLS=1,2 +3 DSHER(IROWS,JCOLS)=0.0 +DSHER(1,1)=(YOUNG*THICK)/(2.4+2.4*POISS) +DSHER(2,2)=(YOUNG*THICK)/(2.4+2.4*POISS) +RETURN +END +MODP 34 +MODP 35 +MODP 36 +MODP 37 +MODP 38 +MODP 39 +MODP 40 +MODP 41 +MODP 42 +MODP 43 +MODP 44 +``` + +# 6.4.11 Subroutine DBE for formulating the matrix product DB + +This subroutine simply multiplies the elasticity matrix D by the strain matrix B. + +```fortran +SUBROUTINE DBE(BMATX,DBMAT,DMATX,MEVAB,NEVAB,NSTRE,NSTR1) DBYB 1 +C******************************* +C +C**** THIS SUBROUTINE MULTIPLIES THE D-MATRIX BY THE B-MATRIX DBYB 4 +C +C******************************* +DIMENSION BMATX(NSTR1,MEVAB),DBMAT(NSTR1,MEVAB), DBYB 7 +. DMATX(NSTR1,NSTR1) DBYB 8 +DO 2 ISTRE=1,NSTRE DBYB 9 +DO 2 IEVAB=1,NEVAB DBYB 10 +DBMAT(ISTRE,IEVAB)=0.0 DBYB 11 +DO 2 JSTRE=1,NSTRE DBYB 12 +DBMAT(ISTRE,IEVAB)=DBMAT(ISTRE,IEVAB)+ +.DMATX(ISTRE,JSTRE)*BMATX(JSTRE,IEVAB) DBYB 13 +2 CONTINUE DBYB 14 +RETURN DBYB 15 +END DBYB 16 +``` + +# 6.4.12 Subroutine FRONT for equation solution by the frontal method + +The function of this subroutine is to assemble the contributions from each element to form the global stiffness matrix and global load vector and to solve the resulting set of simultaneous equations by Gaussian direct elimination. The main feature of the frontal solution technique is that it assembles the equations and eliminates the variables at the same time. Complete details of the frontal process can be found in Chapter 8, Ref. 4. The subroutine presented in Ref. 4 differs from the one listed in this section in three important ways: + +\- As described in Sections 3.3 and 3.4 for one-dimensional problems, a full equation solution need only be undertaken for iterations during which the element stiffnesses are being modified. Such a situation is recognised by the resolution counter KRESL = 1. On the other hand if the element stiffnesses have not been changed during the iteration, signified by KRESL = 2, only the R.H.S. or load terms need be reduced during the elimination phase. This situation is identical to the case of solution for second and subsequent loading cases in elastic problems. + + + +\- The reduced equations corresponding to eliminated variables are stored in core in a temporary array termed a buffer area. As soon as this array is full, the information is then transferred to disc. The number of reduced equations that can be accommodated in the buffer area is governed by the specified parameter, MBUFA. Thus on elimination of a variable a counter over the number of eliminated variables is incremented by one and the reduced equations stored in core. The counter is checked against the permissible buffer length, MBUFA. If this has been reached, the buffer array is transferred to disc file and the counter reset to zero. On back-substitution the contents of a complete buffer length are read from discfile by backspacing. + +\- The displacement and reaction values evaluated by subroutine FRONT during each iteration are incremental values and must be accumulated to give the total displacements, TDISP ( ) and total reactions, TREAC ( ). Also the incremental reactions must be added into the vector of total applied loads, TLOAD ( ), in order to check for convergence of the iteration process; since equilibrium is satisfied when the applied loads and reactions at restrained nodes balance with the nodal forces equivalent to the internal stress field. + +The displacements and reactions evaluated in Subroutine FRONT are stored for output by Subroutine OUTPUT described in Section 7.8.8. + +```csv +SUBROUTINE FRONT(ASDIS,ELOAD,EQRHS,EQUAT,ESTIF,FIXED,IFFIX,IINCS, FRNT 1 +. IITER,GLOAD,GSTIF,LOCEL,LNODS,KRESL,MBUFA,MELEM, FRNT 2 +. MEVAB,MFRON,MSTIF,MTOTV,MVFIX,NACVA,NAMEV,NDEST, FRNT 3 +. NDOFN,NELEM,NEVAB,NNODE,NOFIX,NPIVO,NPOIN, FRNT 4 +. NTOTV,TDISP,TLOAD,TREAC,VECRV) FRNT 5 +C************************** FRNT 6 +C +C**** THIS SUBROUTINE UNDERTAKES EQUATION SOLUTION BY THE FRONTAL FRNT 7 +C METHOD FRNT 8 +C +C************************** FRNT 9 +C +DIMENSION ASDIS(MTOTV),ELOAD(MELEM,MEVAB),EQRHS(MBUFA), FRNT 10 +. EQUAT(MFRON,MBUFA),ESTIF(MEVAB,MEVAB),FIXED(MTOTV), FRNT 11 +. IFFIX(MTOTV),NPIVO(MBUFA),VECRV(MFRON),GLOAD(MFRON), FRNT 12 +. GSTIF(MSTIF),LNODS(MELEM,9),LOCEL(MEVAB),NACVA(MFRON), FRNT 13 +. NAMEV(MBUFA),NDEST(MEVAB),NOFIX(MVFIX),NOUTP(2), FRNT 14 +. TDISP(MTOTV),TLOAD(MELEM,MEVAB),TREAC(MVFIX,NDOFN) FRNT 15 +NFUNC(I,J)=(J*J-J)/2+I FRNT 16 +C +C*** CHANGE THE SIGN OF THE LAST APPEARANCE OF EACH NODE FRNT 17 +C +IF(IINCS.GT.1.OR.IITER.GT.1) GO TO 455 FRNT 18 +DO 140 IPOIN=1,NPOIN FRNT 19 +KLAST=0 FRNT 20 +DO 130 IELEM=1,NELEM FRNT 21 +DO 120 INODE=1,NNODE FRNT 22 +IF(LNODS(IELEM,INODE).NE.IPOIN) GO TO 23 FRNT 23 +KLAST=IELEM FRNT 24 +NLAST=INODE FRNT 25 +120 CONTINUE FRNT 26 +FRNT 27 +FRNT 28 +FRNT 29 +FRNT 30 +``` + + + +```txt +130 CONTINUE +IF(KLAST.NE.0) LNODS(KLAST,NLAST)=-IPOIN +140 CONTINUE +455 CONTINUE +C +C*** START BY INITIALIZING EVERYTHING THAT MATTERS TO ZERO +C +DO 450 IBUFA=1,MBUFA +450 EQRHS(IBUFA)=0.0 +DO 150 ISTIF=1,MSTIF +150 GSTIF(ISTIF)=0.0 +DO 160 IFRON=1,MFRON +GLOAD(IFRON)=0.0 +VECRV(IFRON)=0.0 +NACVA(IFRON)=0 +DO 160 IBUFA=1,MBUFA +160 EQUAT(IFRON,IBUFA)=0.0 +C +C*** AND PREPARE FOR DISC READING AND WRITING OPERATIONS +C +NBUFA=0 +IF(KRESL.GT.1) NBUFA=MBUFA +REWIND 1 +REWIND 2 +REWIND 3 +REWIND 4 +REWIND 8 +C +C*** ENTER MAIN ELEMENT ASSEMBLY-REDUCTION LOOP +C +NFRON=0 +KELVA=0 +DO 320 IELEM=1,NELEM +IF(KRESL.GT.1) GO TO 400 +KEVAB=0 +READ(1) ESTIF +DO 170 INODE=1,NNODE +DO 170 IDOFN=1,NDOFN +NPOSI=(INODE-1)*NDOFN+IDOFN +LOCNO=LNODS(IELEM,INODE) +IF(LOCNO.GT.0) LOCEL(NPOSI)=(LOCNO-1)*NDOFN+IDOFN +IF(LOCNO.LT.0) LOCEL(NPOSI)=(LOCNO+1)*NDOFN-IDOFN +170 CONTINUE +C +C*** START BY LOOKING FOR EXISTING DESTINATIONS +C +DO 210 IEVAB=1,NEVAB +NIKNO=IABS(LOCEL(IEVAB)) +KEXIS=0 +DO 180 IFRON=1,NFRON +IF(NIKNO.NE.NACVA(IFRON)) GO TO 180 +KEVAB=KEVAB+1 +KEXIS=1 +NDEST(KEVAB)=IFRON +180 CONTINUE +IF(KEXIS.NE.0) GO TO 210 +C +C*** WE NOW SEEK NEW EMPTY PLACES FOR DESTINATION VECTOR +C +DO 190 IFRON=1,MFRON +IF(NACVA(IFRON).NE.0) GO TO 190 +NACVA(IFRON)=NIKNO +KEVAB=KEVAB+1 +NDEST(KEVAB)=IFRON +GO TO 200 +FRNT 31 +FRNT 32 +FRNT 33 +FRNT 34 +FRNT 35 +FRNT 36 +FRNT 37 +FRNT 38 +FRNT 39 +FRNT 40 +FRNT 41 +FRNT 42 +FRNT 43 +FRNT 44 +FRNT 45 +FRNT 46 +FRNT 47 +FRNT 48 +FRNT 49 +FRNT 50 +FRNT 51 +FRNT 52 +FRNT 53 +FRNT 54 +FRNT 55 +FRNT 56 +FRNT 57 +FRNT 58 +FRNT 59 +FRNT 60 +FRNT 61 +FRNT 62 +FRNT 63 +FRNT 64 +FRNT 65 +FRNT 66 +FRNT 67 +FRNT 68 +FRNT 69 +FRNT 70 +FRNT 71 +FRNT 72 +FRNT 73 +FRNT 74 +FRNT 75 +FRNT 76 +FRNT 77 +FRNT 78 +FRNT 79 +FRNT 80 +FRNT 81 +FRNT 82 +FRNT 83 +FRNT 84 +FRNT 85 +FRNT 86 +FRNT 87 +FRNT 88 +FRNT 89 +FRNT 90 +FRNT 91 +FRNT 92 +FRNT 93 +FRNT 94 +FRNT 95 +``` + + + +```txt +190 CONTINUE FRNT 96 +C FRNT 97 +C*** THE NEW PLACES MAY DEMAND AN INCREASE IN CURRENT FRONTWIDTH FRNT 98 +C FRNT 99 +200 IF(NDEST(KEVAB).GT.NFRON) NFRON=NDEST(KEVAB) FRNT 100 +210 CONTINUE FRNT 101 +WRITE(8) LOCEL,NDEST,NACVA,NFRON FRNT 102 +400 IF(KRESL.GT.1) READ(8) LOCEL,NDEST,NACVA,NFRON FRNT 103 +C FRNT 104 +C*** ASSEMBLE ELEMENT LOADS FRNT 105 +C FRNT 106 +DO 220 IEVAB=1,NEVAB FRNT 107 +IDEST=NDEST(IEVAB) FRNT 108 +GLOAD(IDEST)=GLOAD(IDEST)+ELOAD(IELEM,IEVAB) FRNT 109 +C FRNT 110 +C*** ASSEMBLE THE ELEMENT STIFFNESSES-BUT NOT IN RESOLUTION FRNT 111 +C FRNT 112 +IF(KRESL.GT.1) GO TO 402 FRNT 113 +DO 222 JEVAB=1,IEVAB FRNT 114 +JDEST=NDEST(JEVAB) FRNT 115 +NGASH=NFUNC(IDEST,JDEST) FRNT 116 +NGISH=NFUNC(JDEST,IDEST) FRNT 117 +IF(JDEST.GE.IDEST) GSTIF(NGASH)=GSTIF(NGASH)+ESTIF(IEVAB,JEVAB) FRNT 118 +IF(JDEST.LT.IDEST) GSTIF(NGISH)=GSTIF(NGISH)+ESTIF(IEVAB,JEVAB) FRNT 119 +222 CONTINUE FRNT 120 +402 CONTINUE FRNT 121 +220 CONTINUE FRNT 122 +C FRNT 123 +C*** RE-EXAMINE EACH ELEMENT NODE, TO ENQUIRE WHICH CAN BE ELIMINATED FRNT 124 +C FRNT 125 +DO 310 IEVAB=1,NEVAB FRNT 126 +NIKNO=-LOCEL(IEVAB) FRNT 127 +IF(NIKNO.LE.0) GO TO 310 FRNT 128 +C FRNT 129 +C*** FIND POSITIONS OF VARIABLES READY FOR ELIMINATION FRNT 130 +C FRNT 131 +DO 300 IFRON=1,NFRON FRNT 132 +IF(NACVA(IFRON).NE.NIKNO) GO TO 300 FRNT 133 +NBUFA=NBUFA+1 FRNT 134 +C FRNT 135 +C*** WRITE EQUATIONS TO DISC OR TO TAPE FRNT 136 +C FRNT 137 +IF(NBUFA.LE.MBUFA) GO TO 406 FRNT 138 +NBUFA=1 FRNT 139 +IF(KRESL.GT.1) GO TO 408 FRNT 140 +WRITE(2) EQUAT,EQRHS,NPIVO,NAMEV FRNT 141 +GO TO 406 FRNT 142 +408 WRITE(4) EQRHS FRNT 143 +READ(2) EQUAT,EQRHS,NPIVO,NAMEV FRNT 144 +406 CONTINUE FRNT 145 +C FRNT 146 +C*** EXTRACT THE COEFFICIENTS OF THE NEW EQUATION FOR ELIMINATION FRNT 147 +C FRNT 148 +IF(KRESL.GT.1) GO TO 404 FRNT 149 +DO 230 JFRON=1,MFRON FRNT 150 +IF(IFRON.LT.JFRON) NLOCA=NFUNC(IFRON,JFRON) FRNT 151 +IF(IFRON.GE.JFRON) NLOCA=NFUNC(JFRON,IFRON) FRNT 152 +EQUAT(JFRON,NBUFA)=GSTIF(NLOCA) FRNT 153 +230 GSTIF(NLOCA)=0.0 FRNT 154 +404 CONTINUE FRNT 155 +C FRNT 156 +C*** AND EXTRACT THE CORRESPONDING RIGHT HAND SIDES FRNT 157 +C FRNT 158 +EQRHS(NBUFA)=GLOAD(IFRON) FRNT 159 +GLOAD(IFRON)=0.0 FRNT 160 +``` + + + +```csv +KELVA=KELVA+1 FRNT 161 +NAMEV(NBUFA)=NIKNO FRNT 162 +NPIVO(NBUFA)=IFRON FRNT 163 +C FRNT 164 +C*** DEAL WITH PIVOT FRNT 165 +C FRNT 166 +PIVOT=EQUAT(IFRON,NBUFA) FRNT 167 +IF(PIVOT.GT.0.0) GO TO 235 FRNT 168 +WRITE(6,900) NIKNO,PIVOT FRNT 169 +900 FORMAT(1H0,3X,52HNEGATIVE OR ZERO PIVOT ENCOUNTERED FOR VARIABLE NFRNT 170 +.O.,14,10H OF VALUE,E17.6) FRNT 171 +STOP FRNT 172 +235 CONTINUE FRNT 173 +EQUAT(IFRON,NBUFA)=0.0 FRNT 174 +C FRNT 175 +C*** ENQUIRE WHETHER PRESENT VARIABLE IS FREE OR PRESCRIBED FRNT 176 +C FRNT 177 +IF(IFFIX(NIKNO).EQ.0) GO TO 250 FRNT 178 +C FRNT 179 +C*** DEAL WITH A PRESCRIBED DEFLECTION FRNT 180 +C FRNT 181 +DO 240 JFRON=1,NFRON FRNT 182 +240 GLOAD(JFRON)=GLOAD(JFRON)-FIXED(NIKNO)*EQUAT(JFRON,NBUFA) FRNT 183 +GO TO 280 FRNT 184 +C FRNT 185 +C*** ELIMINATE A FREE VARIABLE - DEAL WITH THE RIGHT HAND SIDE FIRST FRNT 186 +C FRNT 187 +250 DO 270 JFRON=1,NFRON FRNT 188 +GLOAD(JFRON)=GLOAD(JFRON)-EQUAT(JFRON,NBUFA)*EQRHS(NBUFA)/PIVOT FRNT 189 +C FRNT 190 +C*** NOW DEAL WITH THE COEFFICIENTS IN CORE FRNT 191 +C FRNT 192 +IF(KRESL.GT.1) GO TO 418 FRNT 193 +IF(EQUAT(JFRON,NBUFA).EQ.0.0) GO TO 270 FRNT 194 +NLOCA=NFUNC(0,JFRON) FRNT 195 +CUREQ=EQUAT(JFRON,NBUFA) FRNT 196 +DO 260 LFRON=1,JFRON FRNT 197 +NGASH=LFRON+NLOCA FRNT 198 +260 GSTIF(NGASH)=GSTIF(NGASH)-CUREQ*EQUAT(LFRON,NBUFA) FRNT 199 +./PIVOT FRNT 200 +418 CONTINUE FRNT 201 +270 CONTINUE FRNT 202 +280 EQUAT(IFRON,NBUFA)=PIVOT FRNT 203 +C FRNT 204 +C*** RECORD THE NEW VACANT SPACE, AND REDUCE FRONTWIDTH IF POSSIBLE FRNT 205 +C FRNT 206 +NACVA(IFRON)=0 FRNT 207 +GO TO 290 FRNT 208 +C FRNT 209 +C*** COMPLETE THE ELEMENT LOOP IN THE FORWARD ELIMINATION FRNT 210 +C FRNT 211 +300 CONTINUE FRNT 212 +290 IF(NACVA(NFRON).NE.0) GO TO 310 FRNT 213 +NFRON=NFRON-1 FRNT 214 +IF(NFRON.GT.0) GO TO 290 FRNT 215 +310 CONTINUE FRNT 216 +320 CONTINUE FRNT 217 +IF(KRESL.EQ.1) WRITE(2) EQUAT,EQRHS,NPIVO,NAMEV FRNT 218 +BACKSPACE 2 FRNT 219 +C FRNT 220 +C*** ENTER BACK-SUBSTITUTION PHASE. LOOP BACKWARDS THROUGH VARIABLES FRNT 221 +C FRNT 222 +DO 340 IELVA=1,KELVA FRNT 223 +C FRNT 224 +C***READ A NEW BLOCK OF EQUATIONS - IF NEEDED FRNT 225 +``` + + + +```csv +C +IF(NBUFA.NE.0) GO TO 412 +BACKSPACE 2 +READ(2) EQUAT,EQRHS,NPIVO,NAMEV +BACKSPACE 2 +NBUFA=MBUFA +IF(KRESL.EQ.1) GO TO 412 +BACKSPACE 4 +READ(4) EQRHS +BACKSPACE 4 +412 CONTINUE +C +C*** PREPARE TO BACK-SUBSTITUTE FROM THE CURRENT EQUATION +C +IFRON=NPIVO(NBUFA) +NIKNO=NAMEV(NBUFA) +PIVOT=EQUAT(IFRON,NBUFA) +IF(IFFIX(NIKNO).NE.0) VECRV(IFRON)=FIXED(NIKNO) +IF(IFFIX(NIKNO).EQ.0) EQUAT(IFRON,NBUFA)=0.0 +C +C*** BACK-SUBSTITUTE IN THE CURRENT EQUATION +C +DO 330 JFRON=1,MFRON +330 EQRHS(NBUFA)=EQRHS(NBUFA)-VECRV(JFRON)*EQUAT(JFRON,NBUFA) +C +C*** PUT THE FINAL VALUES WHERE THEY BELONG +C +IF(IFFIX(NIKNO).EQ.0) VECRV(IFRON)=EQRHS(NBUFA)/PIVOT +IF(IFFIX(NIKNO).NE.0) FIXED(NIKNO)=-EQRHS(NBUFA) +NBUFA=NBUFA-1 +ASDIS(NIKNO)=VECRV(IFRON) +340 CONTINUE +C +C*** ADD DISPLACEMENTS TO PREVIOUS TOTAL VALUES +C +DO 345 ITOTV=1,NTOTV +345 TDISP(ITOTV)=TDISP(ITOTV)+ASDIS(ITOTV) +C +C*** STORE REACTIONS FOR PRINTING LATER +C +KBOUN=1 +DO 370 IPOIN=1,NPOIN +NLOCA=(IPOIN-1)*NDOFN +DO 350 IDOFN=1,NDOFN +NGUSH=NLOCA+IDOFN +IF(IFFIX(NGUSH).GT.0) GO TO 360 +350 CONTINUE +GO TO 370 +360 DO 510 IDOFN=1,NDOFN +NGASH=NLOCA+IDOFN +510 TREAC(KBOUN, IDOFN)=TREAC(KBOUN, IDOFN)+FIXED(NGASH) +KBOUN=KBOUN+1 +370 CONTINUE +C +C*** ADD REACTIONS INTO THE TOTAL LOAD ARRAY +C +DO 700 IPOIN=1,NPOIN +DO 710 IELEM=1,NELEM +DO 710 INODE=1,NNODE +NLOCA=IABS(LNODS(IELEM, INODE)) +710 IF(IPOIN.EQ.NLOCA) GO TO 720 +720 DO 730 IDOFN=1,NDOFN +NGASH=(INODE-1)*NDOFN+IDOFN +MGASH=(IPOIN-1)*NDOFN+IDOFN +730 TLOAD(IELEM, NGASH)=TLOAD(IELEM, NGASH)+FIXED(MGASH) +``` + + + +700 CONTINUE +RETURN +END + +FRNT 291 +FRNT 292 +FRNT 293 + +# 6.4.13 Data error diagnostic subroutine CHECK1 + +The function of this subroutine is to scrutinise the problem control parameters, which are accepted by the data input subroutine, INPUT, which will be described in Section 6.5.1. Since subroutine INPUT is common to plane stress/strain, axisymmetric and plate bending applications, subroutine CHECK1 will only check that the control parameters are within the bounds defined by the correct values for the four cases. + +A counter, KEROR, is employed to indicate whether or not any errors have been detected. If errors have been found (indicated by KEROR = 1), subroutine ECHO, described in the next section, is called to list the remainder of the input data. + +Any errors detected are signalled by means of printed error numbers. The interpretation of each error message is given in Table 6.2. + +```csv +SUBROUTINE CHECK1(NDOFN,NELEM,NGAUS,NMATS,NNODE,NPOIN,NESTRE,NTYPE,NVFIX,NCRIT,NALGO,NINCS) CEK1 1 +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +EITHER RETURN,OR ELSE PRINT THE ERRORS DIAGNOSED +KEROR=0 +DO 20 IEROR=1,12 +IF(NEROR(IEROR).EQ.0) GO TO 20 +KEROR=1 +WRITE(6,900) IEROR +900 FORMAT(//31H *** DIAGNOSIS BY CHECK1, ERROR,I3) +20 CONTINUE +IF(KEROR.EQ.0) RETURN +CEK1 1 +CEK1 2 +CEK1 3 +CEK1 4 +CEK1 5 +CEK1 6 +CEK1 7 +CEK1 8 +CEK1 9 +CEK1 10 +CEK1 11 +CEK1 12 +CEK1 13 +CEK1 14 +CEK1 15 +CEK1 16 +CEK1 17 +CEK1 18 +CEK1 19 +CEK1 20 +CEK1 21 +CEK1 22 +CEK1 23 +CEK1 24 +CEK1 25 +CEK1 26 +CEK1 27 +CEK1 28 +CEK1 29 +CEK1 30 +CEK1 31 +CEK1 32 +CEK1 33 +CEK1 34 +CEK1 35 +CEK1 36 +``` diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_022.md b/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_022.md new file mode 100644 index 00000000..cab2c5df --- /dev/null +++ b/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_022.md @@ -0,0 +1,411 @@ + + +
CCEK137
C*** OTHERWISE ECHO ALL THE REMAINING DATA WITHOUT FURTHER COMMENTCEK138
CCEK139
CALL ECHOCEK140
ENDCEK141
+ +Table 6.2 Errors diagnosed by Subroutine CHECK1. + +
Error LabelInterpretation
1The specified total number of node points, NPOIN, in the structure is less than or equal to zero.
2The possible maximum total number of node points in the structure is less than the specified total, NPOIN.
3The number of restrained nodal points is less than 2 or greater than NPOIN (for plane problems at least 2 points must be restrained to eliminate rigid body motions).
4The total number of load increments is less than 1.
5The problem type parameter, NTYPE, is not specified as either 1, 2 or 3.
6The number of nodes/element is less than 4 (linear quadrilateral) or greater than 9 (quadratic Lagrangian elements).
7The number of degrees of freedom per node is not equal to 2 (plane) or 3 (plate problems).
8The total number of different materials is less than or equal to zero or greater than the total number of elements in the structure.
9The parameter specifying the yield criterion to be employed is outside the permissible range.
10The number of Gaussian integration points in each direction is not equal to either 2 or 3.
11The parameter specifying the nonlinear solution algorithm to be employed is outside the permissible range.
12The size of the stress matrix is less than 3 (plane) or greater than 5 (plate problems).
+ +# 6.4.14 Data echo subroutine, ECHO + +The function of this subroutine is to list all the remaining data cards after at least one error has been detected by either of the diagnostic subroutines CHECK1 or CHECK2. This is accomplished by means of a simple read and write operation in alphanumeric format. + +
SUBROUTINE ECHOECHO1
C**********ECHO2
CECHO3
C***IF DATA ERRORS HAVE BEEN DETECTED BY SUBROUTINES CHECK1 ORECHO4
CCHECK2,THIS SUBROUTINE READS AND WRITES THE REMAINING DATA CARDSECHO5
CECHO6
C**********ECHO7
DIMENSION NTITL(80)ECHO8
WRITE(6,900)ECHO9
+ + + +
900FORMAT(//50H NOW FOLLOWS A LISTING OF POST-DISASTER DATA CARDS/)ECHO10
10READ(5,905) NTITLECHO11
905FORMAT(80A1)ECHO12
WRITE(6,910) NTITLECHO13
910FORMAT(20X,80A1)ECHO14
GO TO 10ECHO15
ENDECHO16
+ +# 6.4.15 Data error diagnostic subroutine, CHECK2 + +If the problem control parameters have passed the scrutiny of subroutine CHECK1, the geometric data, boundary conditions and material properties are then assimilated by subroutine INPUT. This data is then scrutinised for possible errors in subroutine CHECK2 where error types 13 to 24, listed in Table 6.3, are checked for. + +Probably the most useful check in this subroutine is the one which ensures that the maximum frontwidth does not exceed the dimensions specified in subroutine FRONT. Subroutine CHECK2 is described in detail in Chapter 9, Ref. 4. + +```fortran +SUBROUTINE CHECK2(COORD, IFFIX, LNODS, MATNO, MELEM, MFRON, MPOIN, MTOTV, CEK2 1 +. MVFIX, NDFRO, NDOFN, NELEM, NMATS, NNODE, NOFIX, NPOIN, CEK2 2 +. NVFIX) CEK2 3 +C*************** CEK2 4 +C CEK2 5 +C**** THIS SUBROUTINE CHECKS THE REMAINDER OF THE INPUT DATA CEK2 6 +C CEK2 7 +C*************** CEK2 8 +DIMENSION COORD(MPOIN, 2), IFFIX(MTOTV), LNODS(MELEM, 9), CEK2 9 +MATNO(MELEM), NDFRO(MELEM), NEROR(24), NOFIX(MVFIX) CEK2 10 +C CEK2 11 +C*** CHECK AGAINST TWO IDENTICAL NONZERO NODAL COORDINATES CEK2 12 +C CEK2 13 +DO 5 IEROR=13,24 CEK2 14 +5 NEROR(IEROR)=0 CEK2 15 +DO 10 IELEM=1, NELEM CEK2 16 +10 NDFRO(IELEM)=0 CEK2 17 +DO 50 IPOIN=2, NPOIN CEK2 18 +KPOIN=IPOIN-1 CEK2 19 +DO 30 JPOIN=1, KPOIN CEK2 20 +DO 20 IDIME=1,2 CEK2 21 +IF(COORD(IPOIN, IDIME).NE.COORD(JPOIN, IDIME)) GO TO 30 CEK2 22 +20 CONTINUE CEK2 23 +NEROR(13)=NEROR(13)+1 CEK2 24 +30 CONTINUE CEK2 25 +40 CONTINUE CEK2 26 +C CEK2 27 +C*** CHECK THE LIST OF ELEMENT PROPERTY NUMBERS CEK2 28 +C CEK2 29 +DO 50 IELEM=1, NELEM CEK2 30 +50 IF(MATNO(IELEM).LE.0.OR.MATNO(IELEM).GT.NMATS) NEROR(14)=NEROR(14) CEK2 31 +. +1 CEK2 32 +C CEK2 33 +C*** CHECK FOR IMPOSSIBLE NODE NUMBERS CEK2 34 +C CEK2 35 +DO 70 IELEM=1, NELEM CEK2 36 +DO 60 INODE=1, NNODE CEK2 37 +IF(LNODS(IELEM, INODE).EQ.0) NEROR(15)=NEROR(15)+1 CEK2 38 +``` + + + +```csv +60 IF(LNODS(IELEM,INODE).LT.0.OR.LNODS(IELEM,INODE).GT.NPOIN) NEROR( CEK2 39 +. 16)=NEROR(16)+1 +70 CONTINUE +CEK2 40 +CEK2 41 +CEK2 42 +CEK2 43 +CEK2 44 +DO 140 IPOIN=1,NPOIN +CEK2 45 +KSTAR=0 +CEK2 46 +DO 100 IELEM=1,NELEM +CEK2 47 +KZERO=0 +CEK2 48 +DO 90 INODE=1,NNODE +CEK2 49 +IF(LNODS(IELEM,INODE).NE.IPOIN) GO TO 90 +CEK2 50 +KZERO=KZERO+1 +CEK2 51 +IF(KZERO.GT.1) NEROR(17)=NEROR(17)+1 +CEK2 52 +CEK2 53 +C*** SEEK FIRST,LAST AND INTERMEDIATE APPEARANCES OF NODE IPOIN +CEK2 54 +C +IF(KSTAR.NE.0) GO TO 80 +CEK2 55 +KSTAR=IELEM +CEK2 56 +CEK2 57 +CEK2 58 +C*** CALCULATE INCREASE OR DECREASE IN FRONTWIDTH AT EACH ELEMENT STAGE +CEK2 59 +C +NDFRO(IELEM)=NDFRO(IELEM)+NDOFN +CEK2 60 +80 CONTINUE +CEK2 61 +CEK2 62 +C +C*** AND CHANGE THE SIGN OF THE LAST APPEARANCE OF EACH NODE +CEK2 63 +CEK2 64 +CEK2 65 +KLAST=IELEM +CEK2 66 +NLAST=INODE +CEK2 67 +90 CONTINUE +CEK2 68 +100 CONTINUE +CEK2 69 +IF(KSTAR.EQ.0) GO TO 110 +CEK2 70 +IF(KLAST.LT.NELEM) NDFRO(KLAST+1)=NDFRO(KLAST+1)-NDOFN +CEK2 71 +LNODS(KLAST,NLAST)=-IPOIN +CEK2 72 +GO TO 140 +CEK2 73 +CEK2 74 +C*** CHECK THAT COORDINATES FOR AN UNUSED NODE HAVE NOT BEEN SPECIFIED +CEK2 75 +CEK2 76 +110 WRITE(6,900) IPOIN +CEK2 77 +900 FORMAT(/15H CHECK WHY NODE,I4,14H NEVER APPEARS) +CEK2 78 +NEROR(18)=NEROR(18)+1 +CEK2 79 +SIGMA=0.0 +CEK2 80 +DO 120 IDIME=1,2 +CEK2 81 +120 SIGMA=SIGMA+ABS(COORD(IPOIN,IDIME)) +CEK2 82 +IF(SIGMA.NE.0.0) NEROR(19)=NEROR(19)+1 +CEK2 83 +CEK2 84 +C*** CHECK THAT AN UNUSED NODE NUMBER IS NOT A RESTRAINED NODE +CEK2 85 +CEK2 86 +DO 130 IVFIX=1,NVFIX +CEK2 87 +130 IF(NOFIX(IVFIX).EQ.IPOIN) NEROR(20)=NEROR(20)+1 +CEK2 88 +140 CONTINUE +CEK2 89 +CEK2 90 +C*** CALCULATE THE LARGEST FRONTWIDTH +CEK2 91 +CEK2 92 +NFRON=0 +CEK2 93 +KFRON=0 +CEK2 94 +DO 150 IELEM=1,NELEM +CEK2 95 +NFRON=NFRON+NDFRO(IELEM) +CEK2 96 +150 IF(NFRON.GT.KFRON) KFRON=NFRON +CEK2 97 +WRITE(6,905) KFRON +CEK2 98 +905 FORMAT(/33H MAXIMUM FRONTWIDTH ENCOUNTERED =,I5) +CEK2 99 +IF(KFRON.GT.MFRON) NEROR(21)=1 +CEK2 100 +CEK2 101 +C*** CONTINUE CHECKING THE DATA FOR THE FIXED VALUES +CEK2 102 +CEK2 103 +``` + + + +
DO 170 IVFIX=1,NVFIXCEK2 104
IF(NOFIX(IVFIX).LE.O.OR.NOFIX(IVFIX).GT.NPOIN) NEROR(22)=NEROR(22)CEK2 105
. +1CEK2 106
KOUNT=0CEK2 107
NLOCA=NOFIX(IVFIX)-1)*NDOFNCEK2 108
DO 160 IDOFN=1,NDOFNCEK2 109
NLOCA=NLOCA+1CEK2 110
160 IF(IFFIX(NLOCA).GT.0) KOUNT=1CEK2 111
IF(KOUNT.EQ.0) NEROR(23)=NEROR(23)+1CEK2 112
KVFIX=IVFIX-1CEK2 113
DO 170 JVFIX=1,KVFIXCEK2 114
170 IF(IVFIX.NE.1.AND.NOFIX(IVFIX).EQ.NOFIX(JVFIX)) NEROR(24)=NEROR(24CEK2 115
. )+1CEK2 116
KEROR=0CEK2 117
DO 180 IEROR=13,24CEK2 118
IF(NEROR(IEROR).EQ.0) GO TO 180CEK2 119
KEROR=1CEK2 120
WRITE(6,910) IEROR,NEROR(IEROR)CEK2 121
910 FORMAT(//31H *** DIAGNOSIS BY CHECK2, ERROR,I3,6X,18H ASSOCIATED NCEK2122
.UMBER,I5)CEK2 123
180 CONTINUECEK2 124
IF(KEROR.NE.0) GO TO 200CEK2 125
CCEK2 126
C*** RETURN ALL NODAL CONNECTION NUMBERS TO POSITIVE VALUESCEK2 127
CCEK2 128
DO 190 IELEM=1,NELEMCEK2 129
DO 190 INODE=1,NNODECEK2 130
190 LNODS(IELEM,INODE)=IABS(LNODS(IELEM,INODE))CEK2 131
RETURNCEK2 132
200 CALL ECHOCEK2 133
ENDCEK2 134
+ +Table 6.3 Errors diagnosed by Subroutine CHECK2 + +
Error LabelInterpretation
13A total of x identical nodal coordinates have been detected, i.e. x nodal points have coordinates which are identical to those of one or more of the remaining nodes.
14A total of x element material identification numbers are less than or equal to zero or greater than the total number of elements in the structure.
15A total of x nodal connection numbers have a zero value.
16A total of x nodal connection numbers are negative or greater than the specified maximum value, NPOIN.
17A total of x repetitions of node numbers within individual elements have been detected.
18A total of x nodes exist in the list of nodal points which do not appear anywhere in the list of element nodal connection numbers.
19Non-zero coordinates have been specified for a total of x nodes which do not appear in the list of element nodal connection numbers.
20A total of x node numbers which do not appear in the element nodal connections list have been specified as restrained nodal points.
21The largest frontwidth encountered in the problem has exceeded the maximum value specified in subroutine FRONT of the program.
+ + + +22 A total of x restrained nodal points have numbers less than or equal to zero or greater than the specified maximum value, NPOIN. + +23 A total of $x$ restrained nodal points at which the fixity code is less than or equal to zero have been detected. + +24 A total of $x$ repetitions in the list of restrained nodal points have been detected. + +# 6.5 Standard subroutines for elasto-plastic finite element analysis + +In this section we describe four additional subroutines which are common to all the elasto-plastic and elasto-viscoplastic applications presented in Chapters 7, 8 and 9. For each subroutine presented, the form of the argument list and common block structure will be that required for two-dimensional elasto-plastic applications. + +# 6.5.1 Data input subroutine, INPUT + +The role of this subroutine is to accept most of the input data required for analysis of elasto-plastic problems. The structure of this subroutine follows closely that of subroutine DATA described in Section 3.2. Subroutine INPUT also closely resembles the data input subroutine presented in Chapter 3, Ref. 4 for linear elastic problems. + +The control parameters necessary for two-dimensional applications extend beyond those required for one-dimensional analysis and are presented below. + +NPOIN Total number of nodal points in the structure. + +NELEM Total number of elements in the structure. + +NVFIX Total number of boundary points, i.e. nodal points at which one or more degrees of freedom are restrained. + +NTYPE Problem type parameter: + +1—Plane stress, + +2—Plain strain, + +3—Axial symmetry. + +NNODE Number of nodes per element: + +4—Linear isoparametric quadrilateral element, + +8—Quadratic isoparametric Serendipity element, + +9—Quadratic isoparametric Langrangian element. + +NMATS Total number of different materials in the structure. + +NGAUS The order of Gaussian quadrature rule to be employed for numerical integration of the element stiffness matrices, etc., as described in Section 6.3.2. If NGAUS is prescribed as 2 a two-point Gauss rule is to be employed; if NGAUS is input as 3 a three-point rule will be used. + + + +NALGO Parameter controlling nonlinear solution algorithm: + +1—Initial stiffness method. The element stiffnesses are computed at the beginning of the analysis and remain unchanged thereafter. +2—Tangential stiffness method. The element stiffnesses are recomputed during each iteration of each load increment. +3—Combined algorithm. The element stiffnesses are recomputed for the first iteration of each load increment only. +4—Combined algorithm. The element stiffnesses are recomputed for the second iteration of each load increment only. (Of course for the first load increment, the element stiffnesses must be calculated for the first iteration also.) + +NCRIT The yield criterion to be employed: + +1—Tresca, +2—Von Mises, +3—Mohr-Coulomb, +4—Drucker-Prager. + +NINCS The total number of increments in which the final loading is to be applied. + +NSTRE The number of independent stress components for the application: + +3—Plane stress/strain, +4—Axial symmetry. + +For the present two-dimensional applications two coordinate components are required to locate each nodal point. With reference to Figs. 6.2–6.4 the x, y components must be specified for plane stress or plane strain problems and the r, z components for axisymmetric situations. This information is stored in the array + +# COORD (IPOIN, IDIME) + +where IPOIN corresponds to the number of the nodal point and IDIME refers to the coordinate component. As mentioned in Section 6.4.1 nodal coordinates need not be supplied for mid-side nodes of 8- and 9-noded elements if they lie on a straight line between corner nodes. The coordinates of such intermediate nodes are evaluated by subroutine NODEXY by linear interpolation. + +For each nodal point at which the displacement value corresponding to one or more degrees of freedom are prescribed, input data must be supplied specifying these fixity conditions. The nodes at which one or more degrees of freedom are restrained are stored in array + +# NOFIX (IVFIX) + +which signifies that the IVFIX $^{th}$ boundary node has a nodal point number NOFIX (IVFIX). Input parameter IFPRE controls which degrees of freedom of a particular node are to have a specified displacement value. For + + + +example, for plane or axisymmetric problems, integer code IFPRE may have the following values: + +10 Displacement in the $x(r)$ direction specified, +01 Displacement in the $y(z)$ direction specified, +11 Displacements in both $x(r)$ and $y(z)$ directions specified. + +This information is then transferred, for permanent storage, into array IFFIX (ITOTV) + +where ITOTV ranges over the total number of degrees of freedom of the structure. The prescribed displacement value associated with a restrained degree of freedom is stored in array + +PRESC (IVFIX, IDOFN) + +where IVFIX indicates that the prescribed displacements pertain to the IVFIX $^{th}$ boundary node and IDOFN ranges over the degrees of freedom of that node. + +The list of material properties for two-dimensional applications differs from the corresponding one-dimensional case considered in Section 3.2. In particular for plane and axisymmetric elasto-plastic problems the following material parameters must be input. + +PROPS (NUMAT, 1) Elastic modulus, $E$ . +PROPS (NUMAT, 2) Poisson's ratio, $\nu$ . +PROPS (NUMAT, 3) Material thickness, t (applicable to plane problems only). +PROPS (NUMAT, 4) Material mass density, $\rho$ . +PROPS (NUMAT, 5) Uniaxial yield stress, $\sigma_{Y}$ (Tresca and Von Mises solids); Cohesion c (Mohr–Coulomb and Drucker–Prager materials). +PROPS (NUMAT, 6) Hardening parameter $H'$ for linear strain hardening. +PROPS (NUMAT, 7) Angle of internal friction for Mohr-Coulomb and Drucker-Prager materials only. + +Consequently NPROP = 7 for two-dimensional elasto-plastic applications. The corresponding material data for plate bending problems is listed in Chapter 9. + +Subroutine INPUT also calls subroutine GAUSSQ, described in Section 6.4.2, whose function is to generate the sampling point position and weighting factors for numerical integration of the element stiffness matrices, etc., by Gaussian quadrature. The order of integration rule to be employed has been specified, through NGAUS, in the control data. + +Subroutine INPUT is now presented and is self-explanatory. + + + +```fortran +SUBROUTINE INPUT(COORD, IFFIX, LNODS, MATNO, MELEM, MEVAB, MFRON, MMATS, INPT 1 +. MPOIN, MTOTV, MVFIX, NALGO, INPT 2 +. NCRIT, NDFRO, NDOFN, NELEM, INPT 3 +. NEVAB, NGAUS, NGAU2, INPT 4 +. NINCS, NMATS, NNODE, NOFIX, NPOIN, NPROP, NSTRE, NSTR1, INPT 5 +. NTOTG, NTOTV, NTYPE, NVFIX, POSGP, PRESC, PROPS, WEIGP) INPT 6 +C***** +C +C*** THIS SUBROUTINE ACCEPTS MOST OF THE INPUT DATA INPT 7 +C +C***** +DIMENSION COORD(MPOIN, 2), IFFIX(MTOTV), LNODS(MELEM, 9), INPT 10 +. MATNO(MELEM), NDFRO(MELEM), 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 155, 156, 157, 158, 159, 160, 161, 162, 163, 164, 165, 166, 167, 168, 169, 170, 171, 172, 173, 174, 175, 176, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 211, 212, 213, 214, 215, 216, 217, 218, 219, 220, 221, 222, 223, 224, 225, 226, 227, 228, 229, 230, 231, 232, 233, 234, 235, 236, 237, 238, 239, 240, 241, 242, 243, 244, 245, 246, 247, 248, 249, 250, 251, 252, 253, 254, 255, 256, 257, 258, 259, 260, 261, 262, 263, 264, 265, 266, 267, 268, 269, 270, 271, 272, 273, 274, 275, 276, 277, 278, 279, 280, 281, 282, 283, 284, 285, 286, 287, 288, 289, 290, 291, 292, 293, 294, 295, 296, 297, 298, 299, 300, 301, 302, 303, 304, 305, 306, 307, 308, 309, 310, 311, 312, 313, 314, 315, 316, 317, 318, 319, 320, 321, 322, 323, 324, 325, 326, 327, 328, 329, 330, 331, 332, 333, 334, 335, 336, 337, 338, 339, 340, 341, 342, 343, 344, 345, 346, 347, 348, 349, 350, 351, 352, 353, 354, 355, 356, 357, 358, 359, 360, 361, 362, 363, 364, 365, 366, 367, 368, 369, 370, 371, 372, 373, 374, 375, 376, 377, 378, 379, 380, 381, 382, 383, 384, 385, 386, 387, 388, 389, 390, 391, 392, 393, 394, 395, 396, 397, 398, 399, 400, 401, 402, 403, 404, 405, 406, 407, 408, 409, 410, 411, 412, 413, 414, 415, 416, 417, 418, 419, 420, 421, 422, 423, 424, 425, 426, 427, 428, 429, 430, 431, 432, 433, 434, 435, 436, 437, 438, 439, 440, 441, 442, 443, 444, 445, 446, 447, 448, 449, 450, 451, 452, 453, 454, 455, 456, 457, 458, 459, 460, 461, 462, 463, 464, 465, 466, 467, 468, 469, 470, 471, 472, 473, 474, 475, 476, 477, 478, 479, 480, 481, 482, 483, 484, 485, 486, 487, 488, 489, 490, 491, 492, 493, 494, 495, 496, 497, 498, 499, 500, 501, 502, 503, 504, 505, 506, 507, 508, 509, 510, 511, 512, 513, 514, 515, 516, 517, 518, 519, 520, 521, 522, 523, 524, 525, 526, 527, 528, 529, 530, 531, 532, 533, 534, 535, 536, 537, 538, 539, 540, 541, 542, 543, 544, 545, 546, 547, 548, 549, 550, 551, 552, 553, 554, 555, 556, 557, 558, 559, 560, 561, 562, 563, 564, 565, 566, 567, 568, 569, 570, 571, 572, 573, 574, 575, 576, 577, 578, 579, 580, 581, 582, 583, 584, 585, 586, 587, 588, 589, 590, 591, 592, 593, 594, 595, 596, 597, 598, 599, 600, 601, 602, 603, 604, 605, 606, 607, 608, 609, 610, 611, 612, 613, 614, 615, 616, 617, 618, 619, 620, 621, 622, 623, 624, 625, 626, 627, 628, 629, 630, 631, 632, 633, 634, 635, 636, 637, 638, 639, 640, 641, 642, 643, 644, 645, 646, 647, 648, 649, 650, 651, 652, 653, 654, 655, 656, 657, 658, 659, 660, 661, 662, 663, 664, 665, 666, 667, 668, 669, 670, 671, 672, 673, 674, 675, 676, 677, 678, 679, 680, 681, 682, 683, 684, 685, 686, 687, 688, 689, 690, 691, 692, 693, 694, 695, 696, 697, 698, 699, 700, 701, 702, 703, 704, 705, 706, 707, 708, 709, 710, 711, 712, 713, 714, 715, 716, 717, 718, 719, 720, 721, 722, 723, 724, 725, 726, 727, 728, 729, 730, 731, 732, 733, 734, 735, 736, 737, 738, 739, 740, 741, 742, 743, 744, 745, 746, 747, 748, 749, 750, 751, 752, 753, 754, 755, 756, 757, 758, 759, 760, 761, 762, 763, 764, 765, 766, 767, 768, 769, 770, 771, 772, 773, 774, 775, 776, 777, 778, 779, 780, 781, 782, 783, 784, 785, 786, 787, 788, 789, 790, 791, 792, 793, 794, 795, 796, 797, 798, 799, 800, 801, 802, 803, 804, 805, 806, 807, 808, 809, 810, 811, 812, 813, 814, 815, 816, 817, 818, 819, 820, 821, 822, 823, 824, 825, 826, 827, 828, 829, 830, 831, 832, 833, 834, 835, 836, 837, 838, 839, 840, 841, 842, 843, 844, 845, 846, 847, 848, 849, 850, 851, 852, 853, 854, 855, 856, 857, 858, 859, 860, 861, 862, 863, 864, 865, 866, 867, 868, 869, 870, 871, 872, 873, 874, 875, 876, 877, 878, 879, 880, 881, 882, 883, 884, 885, 886, 887, 888, 889, 890, 891, 892, 893, 894, 895, 896, 897, 898, 899, 900, 901, 902, 903, 904, 905, 906, 907, 908, 909, 910, 911, 912, 913, 914, 915, 916, 917, 918, 919, 920, 921, 922, 923, 924, 925, 926, 927, 928, 929, 930, 931, 932, 933, 934, 935, 936, 937, 938, 939, 940, 941, 942, 943, 944, 945, 946, 947, 948, 949, 950, 951, 952, 953, 954, 955, 956, 957, 958, 959, 960, 961, 962, 963, 964, 965, 966, 967, 968, 969, 970, 971, 972, 973, 974, 975, 976, 977, 978, 979, 980, 981, 982, 983, 984, 985, 986, 987, 988, 989, 990, 991, 992, 993, 994, 995, 996, 997, 998, 999, 1000, 1001, 1002, 1003, 1004, 1005, 1006, 1007, 1008, 1009, 1010, 1011, 1012, 1013, 1014, 1015, 1016, 1017, 1018, 1019, 1020, 1021, 1022, 1023, 1024, 1025, 1026, 1027, 1028, 1029, 1030, 1031, 1032, 1033, 1034, 1035, 1036, 1037, 1038, 1039, 1040, 1041, 1042, 1043, 1044, 1045, 1046, 1047, 1048, 1049, 1050, 1051, 1052, 1053, 1054, 1055, 1056, 1057, 1058, 1059, 1060, 1061, 1062, 1063, 1064, 1065, 1066, 1067, 1068, 1069, 1070, 1071, 1072, 1073, 1074, 1075, 1076, 1077, 1078, 1079, 1080, 1081, 1082, 1083, 1084, 1085, 1086, 1087, 1088, 1089, 1090, 1091, 1092, 1093, 1094, 1095, 1096, 1097, 1098, 1099, 1100, 1101, 1102, 1103, 1104, 1105, 1106, 1107, 1108, 1109, 1110, 1111, 1112, 1113, 1114, 1115, 1116, 1117, 1118, 1119, 1120, 1121, 1122, 1123, 1124, 1125, 1126, 1127, 1128, 1129, 1130, 1131, 1132, 1133, 1134, 1135, 1136, 1137, 1138, 1139, 1140, 1141, 1142, 1143, 1144, 1145, 1146, 1147, 1148, 1149, 1150, 1151, 1152, 1153, 1154, 1155, 1156, 1157, 1158, 1159, 1160, 1161, 1162, 1163, 1164, 1165, 1166, 1167, 1168, 1169, 1170, 1171, 1172, 1173, 1 +``` + + + +```csv +6 READ(5,905) IPOIN,(COORD(IPOIN,IDIME),IDIME=1,2) INPT 65 +905 FORMAT(I5,6F10.5) INPT 66 +IF(IPOIN.NE.NPOIN) GO TO 6 INPT 67 +C INPT 68 +C*** INTERPOLATE COORDINATES OF MID-SIDE NODES INPT 69 +C CALL NODEXY(COORD,LNODS,MELEM,MPOIN,NELEM,NNODE) INPT 71 +DO 10 IPOIN=1,NPOIN INPT 72 +10 WRITE(6,906) IPOIN,(COORD(IPOIN,IDIME),IDIME=1,2) INPT 73 +906 FORMAT(1X,I5,3F10.3) INPT 74 +C INPT 75 +C*** READ THE FIXED VALUES. INPT 76 +C INPT 77 +WRITE(6,907) INPT 78 +907 FORMAT(//5H NODE,6X,4HCODE,6X,12HFIXED VALUES) INPT 79 +DO 8 IVFIX=1,NVFIX INPT 80 +READ(5,908) NOFIX(IVFIX),IFPRE,(PRESC(IVFIX,IDOFN),IDOFN=1,NDOFN) INPT 81 +WRITE(6,908) NOFIX(IVFIX),IFPRE,(PRESC(IVFIX,IDOFN),IDOFN=1,NDOFN) INPT 82 +NLOCA=(NOFIX(IVFIX)-1)*NDOFN INPT 83 +IFDOF=10**(NDOFN-1) INPT 84 +DO 8 IDOFN=1,NDOFN INPT 85 +NGASH=NLOCA+IDOFN INPT 86 +IF(IFPRE.LT.IFDOF) GO TO 8 INPT 87 +IFFIX(NGASH)=1 INPT 88 +IFPRE=IFPRE-IFDOF INPT 89 +8 IFDOF=IFDOF/10 INPT 90 +908 FORMAT(1X,I4,5X,I5,5X,5F10.6) INPT 91 +C INPT 92 +C*** READ THE AVAILABLE SELECTION OF ELEMENT PROPERTIES. INPT 93 +C INPT 94 +16 WRITE(6,910) INPT 95 +910 FORMAT(//7H NUMBER,6X,18HELEMENT PROPERTIES) INPT 96 +DO 18 IMATS=1,NMATS INPT 97 +READ(5,900) NUMAT INPT 98 +READ(5,930) (PROPS(NUMAT,IPROP),IPROP=1,NPROP) INPT 99 +930 FORMAT(8F10.5) INPT 100 +18 WRITE(6,911) NUMAT,(PROPS(NUMAT,IPROP),IPROP=1,NPROP) INPT 101 +911 FORMAT(1X,I4,3X,8E14.6) INPT 102 +C INPT 103 +C*** SET UP GAUSSIAN INTEGRATION CONSTANTS INPT 104 +C INPT 105 +CALL GAUSSQ(NGAUS,POSGP,WEIGP) INPT 106 +CALL CHECK2(COORD,IFFIX,LNODS,MATNO,MELEM,MFRON,MPOIN,MTOTV,INPT 107 +. MVFIX,NDFRO,NDOFN,NELEM,NMATS,NNODE,NOFIX,NPOIN,INPT 108 +. NVFIX) INPT 109 +RETURN INPT 110 +END INPT 111 +``` + +# 6.5.2 Subroutine ALGOR + +The function of this subroutine is to control the solution process according to the value of the solution algorithm parameter, NALGO, input in subroutine INPUT. This subroutine is similar in form to subroutine NONAL presented in Section 3.3 for one-dimensional applications. The subroutine sets the value of indicator KRESL to either 1 or 2 according to NALGO and the current values of the iteration number IITER and increment number IINCS. A value of KRESL = 1 indicates reformulation of the element stiffnesses accompanied by a full equation solution and KRESL = 2 indicates that the element stiffnesses are not to be modified and consequently only equation resolution takes place. + + + +With the definitions of the permissible values of NALGO given in Section 6.5.1, subroutine ALGOR is self-explanatory and is listed below.\* + +SUBROUTINE ALGOR(FIXED,IINCS,IITER,KRESL, MTOTV,NALGO,NTOTV) ALGR 1 +C***** THIS SUBROUTINE SETS EQUATION RESOLUTION INDEX,KRESL ALGR 3 +C ALGR 4 +C***** THIS SUBROUTINE SETS EQUATION RESOLUTION INDEX,KRESL ALGR 5 +C ALGR 6 +C***** DIMENSION FIXED(MTOTV) ALGR 7 +KRESL=2 ALGR 8 +IF(NALGO.EQ.1.AND.IINCS.EQ.1.AND.IITER.EQ.1) KRESL=1 ALGR 10 +IF(NALGO.EQ.2) KRESL=1 ALGR 11 +IF(NALGO.EQ.3.AND.IITER.EQ.1) KRESL=1 ALGR 12 +IF(NALGO.EQ.4.AND.IINCS.EQ.1.AND.IITER.EQ.1) KRESL=1 ALGR 13 +IF(NALGO.EQ.4.AND.IITER.EQ.2) KRESL=1 ALGR 14 +IF(IITER.EQ.1) RETURN ALGR 15 +DO 100 ITOTV = 1,NTOTV ALGR 16 +FIXED(ITOTV)=0.0 ALGR 17 +100 CONTINUE ALGR 18 +RETURN ALGR 19 +END ALGR 20 + +# 6.5.3 Subroutine INCREM + +The role of subroutine INCREM is to increment the applied loading or any prescribed displacements according to the load factors specified as input. This subroutine is accessed on the first iteration of each load increment. For each increment of load the following items of information are input as data and are similar to those described in Section 3.7. + +FACTO This controls the magnitude of the load increment. The applied loading for each element is evaluated in Subroutine LOADPS for plane and axisymmetric situations, or Subroutine LOADPB for plate problems, and is stored in the array RLOAD (IELEM, IEVAB) as described in Section 6.4.5. The additional element load applied during the increment is RLOAD (IELEM, IEVAB)\*FACTO. The applied loading is accumulative so that if FACTO is input as 0·8, 0·2 and 0·1 for the first three increments, the total load acting on the structure during the third load increment is 1·1 times the loads calculated in Subroutine LOADPS. This method of load factoring permits unequal load increments to be taken. If loading is by prescribed displacements the same factoring process holds. + +TOLER This controls the tolerance permitted on the convergence process and its use has been described in Section 3.9.3. + +MITER Maximum permissible number of iterations. This is a safety measure to cover situations where the solution process does + +\* For elasto-viscoplastic applications described in Chapter 8, iteration number IITER is replaced by timestep number, ISTEP. diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_023.md b/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_023.md new file mode 100644 index 00000000..bc8a69ef --- /dev/null +++ b/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_023.md @@ -0,0 +1,391 @@ + + +not converge. After performing MITER iteration cycles the program will then stop. + +NOUTP (1) This parameter controls the output of the unconverged results after the first iteration. In order to examine the convergence process the user can vary the frequency of output for each load increment: + +1—Print the displacements only after the first iteration. +2—Print the displacements and nodal reactions after the first iteration. +3—Print the displacements, reactions and stresses after the first iteration. + +NOUTP (2) This parameter controls the output of the converged results: + +1—Print the final displacements only. +2—Print the final displacements and nodal reactions. +3—Print the final displacements, reactions and stresses. + +The loading to which the structure is subjected is monitored by the arrays ELOAD (IELEM, IEVAB) and TLOAD (IELEM, IEVAB). The total loading applied to the structure at any stage of the analysis is accumulated in the TLOAD array. On the other hand ELOAD contains the loading to be applied to the structure for each iteration of the solution process. Initially (the first iteration of the first load increment) ELOAD contains the first increment of applied load. For the second and subsequent iterations ELOAD contains the residual nodal forces which must be redistributed as described in Section 3.7. After convergence has occurred, the next increment of load is assimilated into ELOAD, so that at this stage ELOAD contains the new applied load increment together with any residual forces still remaining after convergence of the solution for the previous load increment. These residual forces should be negligibly small if the convergence tolerance factor, TOLER, is correctly chosen. However, since any residual forces are retained in ELOAD and applied as nodal forces during the next load increment, it is noted that equilibrium is maintained at every stage of the computation process. + +The final role of this subroutine is to insert appropriate values in the fixity array to control any prescribed displacements. As described in Section 3.3, in order to arrive at the correct value of a displacement whose value is prescribed for a load increment, it is necessary to prescribe the given value for equation solution during the first iteration and then prescribe a zero value for all subsequent iterations. Since the displacements occurring during each iteration accumulate to give the total displacement then clearly the prescribed value will be obtained by this process. + +Subroutine INCREM will now be presented and explanatory notes provided. + + + +SUBROUTINE INCREM(ELOAD, FIXED, IINCS, MELEM, MEVAB, MITER, INCR 1 +. MTOTV, MVFIX, NDOFN, NELEM, NEVAB, NOUTP, INCR 2 +. NOFIX, NTOTV, NVFIX, PRESC, RLOAD, TFACT, INCR 3 +. TLOAD, TOLER) INCR 4 +C********** INCR 5 +C INCR 6 +C*** THIS SUBROUTINE INCREMENTS THE APPLIED LOADING INCR 7 +C INCR 8 +C********** INCR 9 +DIMENSION ELOAD(MELEM, MEVAB), FIXED(MTOTV), INCR 10 +. IFFIX(MTOTV), INCR 11 +. NOUTP(2), NOFIX(MVFIX), INCR 12 +. PRESC(MVFIX, NDOFN), RLOAD(MELEM, MEVAB), TLOAD(MELEM, MEVAB) INCR 13 +WRITE(6, 900) IINCS INCR 14 +900 FORMAT(1H0, 5X, 17HINCREMENT NUMBER, I5) INCR 15 +READ(5, 950) FACTO, TOLER, MITER, NOUTP(1), NOUTP(2) INCR 16 +950 FORMAT(2F10.5, 3I5) INCR 17 +TFACT=TFACT+FACTO INCR 18 +WRITE(6, 960) TFACT, TOLER, MITER, NOUTP(1), NOUTP(2) INCR 19 +960 FORMAT(1H0, 5X, 13HLOAD FACTOR =, F10.5, 5X, INCR 20 +.24H CONVERGENCE TOLERANCE =, F10.5, 5X, 24HMAX. NO. OF ITERATIONS =, INCR 21 +. I5, //27H INITIAL OUTPUT PARAMETER =, I5, 5X, 24HFINAL OUTPUT PARAMETINCR 22 +.ER =, I5) INCR 23 +DO 80 IELEM=1, NELEM INCR 24 +DO 80 IEVAB=1, NEVAB INCR 25 +ELOAD(IELEM, IEVAB)=ELOAD(IELEM, IEVAB)+RLOAD(IELEM, IEVAB)*FACTO INCR 26 +80 TLOAD(IELEM, IEVAB)=TLOAD(IELEM, IEVAB)+RLOAD(IELEM, IEVAB)*FACTO INCR 27 +C INCR 28 +C*** INTERPRET FIXITY DATA IN VECTOR FORM INCR 29 +C INCR 30 +DO 100 ITOTV=1, NTOTV INCR 31 +100 FIXED(ITOTV)=0.0 INCR 32 +DO 110 IVFIX=1, NVFIX INCR 33 +NLOCA=(NOFIX(IVFIX)-1)*NDOFN INCR 34 +DO 110 IDOFN=1, NDOFN INCR 35 +NGASH=NLOCA+IDOFN INCR 36 +FIXED(NGASH)=PRESC(IVFIX, IDOFN)*FACTO INCR 37 +110 CONTINUE INCR 38 +RETURN INCR 39 +END INCR 40 + +INCR 14–15 Write the number of the load increment which is being currently solved. + +INCR 16-23 Read and write the load increment control parameters. Note that the incremental load factor, FACTO, is input whereas the total load factor, TFACT, is output. + +INCR 24–27 Accumulate the incremental loading into array ELOAD for equation solution and also into TLOAD to record the total load applied to the structure. + +INCR 31–32 Zero the global vector of prescribed displacements. + +INCR 33–38 Insert any prescribed displacement values, factored by the load increment factor, into the appropriate position in the global vector. + +# 6.5.4 Solution convergence monitoring subroutine CONVER + +This subroutine monitors convergence of the nonlinear solution iteration process. It is almost identical to subroutine CONUND for one-dimensional + + + +applications described in Section 3.10.3. Since for two-dimensional and plate bending problems we have more than one degree of freedom per nodal point, summation in (3.27) must now be made over the total number of degrees of freedom in the structure. As an additional check on the nonlinear solution process we also arrange to evaluate the maximum individual residual force $\psi_{i}^{r}$ existing in the structure. + +Subroutine CONVER is now presented and can be understood with the aid of Section 3.10.3. + +```csv +SUBROUTINE CONVER(ELOAD,IITER,LNODS,MELEM,MEVAB,MTOTV,NCHEK,CONV 1 +NDOFN,NELEM,NEVAB,NNODE,NTOTV,PVALU,STFOR,CONV 2 +TLOAD,TOFOR,TOLER)CONV 3 +C***** +THIS SUBROUTINE CHECKS FOR CONVERGENCE OF THE ITERATION PROCESS +CONV 4 +C +CONV 5 +C***** +DIMENSION ELOAD(MELEM,MEVAB),LNODS(MELEM,9),STFOR(MTOTV),CONV 6 +CONV 7 +C***** +CONV 8 +DIMENSION ELOAD(MELEM,MEVAB),LNODS(MELEM,9),STFOR(MTOTV),CONV 9 +TOFOR(MTOTV),TLOAD(MELEM,MEVAB)CONV 10 +NCHEK=0CONV 11 +RESID=0.0CONV 12 +RETOT=0.0CONV 13 +REMAX=0.0CONV 14 +DO 5 ITOTV=1,NTOTVCONV 15 +STFOR(ITOTV)=0.0CONV 16 +TOFOR(ITOTV)=0.0CONV 17 +5 CONTINUECONV 18 +DO 40 IELEM=1,NELEMCONV 19 +KEVAB=0CONV 20 +DO 40 INODE=1,NNODECONV 21 +LOCNO=IABS(LNODS(IELEM,INODE))CONV 22 +DO 40 IDOFN=1,NDOFNCONV 23 +KEVAB=KEVAB+1CONV 24 +NPOSI=(LOCNO-1)*NDOFN+IDOFNCONV 25 +STFOR(NPOSI)=STFOR(NPOSI)+ELOAD(IELEM,KEVAB)CONV 26 +40 TOFOR(NPOSI)=TOFOR(NPOSI)+TLOAD(IELEM,KEVAB)CONV 27 +DO 50 ITOTV=1,NTOTVCONV 28 +REFOR=TOFOR(ITOTV)-STFOR(ITOTV)CONV 29 +RESID=RESID+REFOR*REFORCONV 30 +RETOT=RETOT+TOFOR(ITOTV)*TOFOR(ITOTV)CONV 31 +AGASH=ABS(REFOR)CONV 32 +50 IF(AGASH.GT.REMAX) REMAX=AGASHCONV 33 +DO 10 IELEM=1,NELEMCONV 34 +DO 10 IEVAB=1,NEVABCONV 35 +10 ELOAD(IELEM,IEVAB)=TLOAD(IELEM,IEVAB)-ELOAD(IELEM,IEVAB)CONV 36 +RESID=SQRT(RESID)CONV 37 +RETOT=SQRT(RETOT)CONV 38 +RATIO=100.0*RESID/RETOTCONV 39 +IF(RATIO.GT.TOLER) NCHEK=1CONV 40 +IF(IITER.EQ.1) GO TO 20CONV 41 +IF(RATIO.GT.PVALU) NCHEK=999CONV 42 +20 PVALU=RATIOCONV 43 +WRITE(6,30) NCHEK,RATIO,REMAXCONV 44 +30 FORMAT(1HO,3X,18HCONVERGENCE CODE =,I4,3X,28HNORM OF RESIDUAL SUM CONV 45 +.RATIO =,E14.6,3X,18HMAXIMUM RESIDUAL =,E14.6)CONV 46 +RETURNCONV 47 +ENDCONV 48 +``` + + + +# 6.6 Problems + +6.1 Using the subroutines described in this chapter devise programs to evaluate the stiffness matrices and load vectors for 4-, 8- and 9-node quadrilateral isoparametric elements for plane stress, plane strain, axisymmetric and Mindlin plate applications. +6.2 Use the shape functions $L_{t}^{(e)}(\xi, \eta)$ from the 9-node Lagrangian quadrilateral isoparametric element to devise a new family of 8-node Serendipity quadrilateral element shape functions $N_{t}^{(e)}(\xi, \eta)$ of the form + +$$ +N _ {i} ^ {(e)} = L _ {i} ^ {(e)} + a L _ {9} ^ {(e)} \quad i = 1, 3, 5 \text { and } 7 \text { (corner nodes) }, +$$ + +$$ +N _ {i} ^ {(e)} = L _ {i} ^ {(e)} + b L _ {9} ^ {(e)} \quad i = 2, 4, 6 \text { and } 8 \text { (midside nodes) }, +$$ + +where $L_{9}^{(e)}$ is the shape function of the central node of the Lagrangian element. What limits are there on $a$ and $b$ ? + +6.3 Determine some further diagnostic checks on the input, other than those described in Sections 6.4.13 and 6.4.15. Apart from the check on the Jacobian determinant given in Subroutine JACOB2 in Section 6.4.4, are there any other checks which could be incorporated into the program after the input has been successfully read and checked? +6.4 Determine the consistent nodal forces for the case when a point load with components $P_x, P_y$ acts at an arbitrary point along an element edge defined by Cartesian coordinates $(x_P, y_P)$ , which correspond to local coordinates $(\xi, \eta) = (\xi_P, -1)$ . + +# 6.7 References + +1. HILL, R., The Mathematical Theory of Plasticity, Oxford University Press, 1950. +2. TIMOSHENKO, S. P. and GOODIER, J. N., Theory of Elasticity, McGraw-Hill, New York, 1951. +3. HUGHES, T. J. R., COHEN, M. and HAROUN, M., Reduced and selective integration techniques in the finite element analysis of plates, Nucl. Eng. Design, 46, 203–222 (1978). +4. HINTON, E. and OWEN, D. R. J., Finite Element Programming, Academic Press, London, 1977. +5. IRONS, B. M. and AHMAD, S., Techniques of Finite Elements, Ellis Horwood, Chichester, 1980. +6. BATHE, K. J. and WILSON, E. L., Numerical Methods in Finite Element Analysis, Prentice-Hall, Englewood Cliffs, New Jersey, 1977. + + + +# Chapter 7 Elasto-plastic problems in two dimensions + +# 7.1 Introduction + +In this chapter we consider the elasto-plastic stress analysis of solids which conform to plane stress, plane strain or axisymmetric conditions. Most of the problems encountered in engineering can be approximated to satisfy one of these classifications. + +The basic laws governing elasto-plastic material behaviour in a two-dimensional solid must be presented before the numerical aspects of the problem can be considered and to this end new concepts, such as the plastic potential and the normality condition will be introduced. Only the essential expressions will be provided in this text and the reader will be directed to other sources for a more complete theoretical treatment. + +The situation is complicated by the fact that different classes of materials exhibit different elasto-plastic characteristics. In this chapter four different yield criteria are employed. The Tresca and Von Mises laws, which closely approximate metal plasticity behaviour, are considered and the Mohr-Coulomb and Drucker-Prager criteria, which are applicable to concrete, rocks and soils, are presented. + +In the latter sections of this chapter a computer code is developed to allow the solution of practical problems. Many of the subroutines required for elasto-plastic solution have been reviewed in Chapter 6. In this chapter the additional subroutines are developed and assembled to provide a working program. + +# 7.2 The mathematical theory of plasticity + +The object of the mathematical theory of plasticity is to provide a theoretical description of the relationship between stress and strain for a material which exhibits an elasto-plastic response. In essence, plastic behaviour is characterised by an irreversible straining which is not time dependent and which can only be sustained once a certain level of stress has been reached. In this section we outline the basic assumptions and associated theoretical expressions for a general continuum. For a more complete treatment the reader is directed to Refs. 1–3. In order to formulate a theory which models elasto-plastic material deformation three requirements have to be met: + + + +\- An explicit relationship between stress and strain must be formulated to describe material behaviour under elastic conditions, i.e. before the onset of plastic deformation. + +● A yield criterion indicating the stress level at which plastic flow commences must be postulated. + +\- A relationship between stress and strain must be developed for post-yield behaviour, i.e. when the deformation is made up of both elastic and plastic components. + +Before the onset of plastic yielding the relationship between stress and strain is given by the standard linear elastic expression.\* + +$$ +\sigma_ {i j} = C _ {i j k l} \epsilon_ {k l}, \tag {7.1} +$$ + +where $\sigma_{ij}$ and $\epsilon_{kl}$ are the stress and strain components respectively and $C_{ijkl}$ is the tensor of elastic constants which for an isotropic material has the explicit form + +$$ +C _ {i j k l} = \lambda \delta_ {i j} \delta_ {k l} + \mu \delta_ {i k} \delta_ {j l} + \mu \delta_ {i l} \delta_ {j k}, \tag {7.2} +$$ + +where $\lambda$ and $\mu$ are the Lamé constants and $\delta_{ij}$ is the Kronecker delta defined by + +$$ +\delta_ {i j} = \left\{ \begin{array}{l l} 1 & \text { if } \quad i = j \\ 0 & i \neq j. \end{array} \right. \tag {7.3} +$$ + +# 7.2.1 The yield criterion + +The yield criterion determines the stress level at which plastic deformation begins and can be written in the general form + +$$ +f (\sigma_ {i j}) = k (\kappa), \tag {7.4} +$$ + +where f is some function and k a material parameter to be determined experimentally. The term k may be a function of a hardening parameter $\kappa$ discussed later in Section 7.2.2. On physical grounds, any yield criterion should be independent of the orientation of the coordinate system employed and therefore it should be a function of the three stress invariants only + +$$ +J _ {1} = \sigma_ {i i} +$$ + +$$ +J _ {2} = \frac {1}{2} \sigma_ {i j} \sigma_ {i j} +$$ + +$$ +J _ {3} = \frac {1}{3} \sigma_ {i j} \sigma_ {j k} \sigma_ {k i}. \tag {7.5} +$$ + +Experimental observations, notably by Bridgeman, $^{(4)}$ indicate that plastic deformation of metals is essentially independent of hydrostatic pressure. Consequently the yield function can only be of the form + +$$ +f (J _ {2} ^ {\prime}, J _ {3} ^ {\prime}) = k (\kappa), \tag {7.6} +$$ + +\- In the indicial notation employed, Einstein's summation convention is invoked, whereby it is implicitly assumed that a summation from 1 to 3 is performed over any index which is repeated in any term of an expression. Also indices 1, 2, 3 refer to Cartesian components $x, y, z$ respectively. Note that $\sigma_{11} = \sigma_{xx} = \sigma_x, \sigma_{12} = \sigma_{xy}$ , etc. + + + +where $J_{2}'$ and $J_{3}'$ are the second and third invariants of the deviatoric stresses, + +$$ +\sigma_ {i j} ^ {\prime} = \sigma_ {i j} - \frac {1}{3} \delta_ {i j} \sigma_ {k k}. \tag {7.7} +$$ + +Most of the various yield criteria that have been suggested for metals are now only of historic interest, since they conflict with experimental predictions. The two simplest which do not have this fault are the Tresca criterion and the Von Mises criterion. + +# The Tresca yield criterion (1864) + +This states that yielding begins when the maximum shear stress reaches a certain value. If the principal stresses are $\sigma_{1}$ , $\sigma_{2}$ , $\sigma_{3}$ where $\sigma_{1} \geqslant \sigma_{2} \geqslant \sigma_{3}$ then yielding begins when + +$$ +\sigma_ {1} - \sigma_ {3} = Y (\kappa), \tag {7.8} +$$ + +where Y is a material parameter to be experimentally determined and which may be a function of the hardening parameter $\kappa$ . By considering all other possible maximum shearing stress values (e.g. $\sigma_{2}-\sigma_{1}$ if $\sigma_{2}\geqslant\sigma_{3}\geqslant\sigma_{1}$ ) it can be shown that this yield criterion may be represented in the $\sigma_{1}\sigma_{2}\sigma_{3}$ stress space by the surface of an infinitely long regular hexagonal cylinder as shown in Fig. 7.1. The axis of the cylinder coincides with the space diagonal, defined by points $\sigma_{1}=\sigma_{2}=\sigma_{3}$ , and since each normal section of the cylinder is identical, (a consequence of the assumption that a hydrostatic stress does not influence yielding), it is convenient to represent the yield surface geometrically by projecting it onto the so-called $\pi$ plane, $\sigma_{1}+\sigma_{2}+\sigma_{3}=0$ as shown in Fig. 7.2(a). When the yield function f depends on $J_{2}'$ and $J_{3}'$ alone it can be + +![](images/page-227_84ed560e82e9c0f35106ef29bec36764ce231a20828abdf8bc2d28a8d4c54073.jpg) + +
+text_image + +π plane +σ₁ + σ₂ + σ₃ = 0 +Space diagonal +σ₁ = σ₂ = σ₃ +Von Mises +Tresca +
+ +Fig. 7.1 Geometrical representation of the Tresca and Von Mises yield surfaces in principal stress space. + + + +![](images/page-228_07e84cb70d60d2b854966febc7b8be4293f2dbc4044c947bc7c43f32c4cdcfa4.jpg) +Fig. 7.2 Two-dimensional representations of the Tresca and Von Mises yield criteria. (a) $\pi$ plane representation. (b) Conventional engineering representation. + +written in the form $f(\sigma_{1}-\sigma_{3},\sigma_{2}-\sigma_{3})$ and a two-dimensional plot of the surface f=k is then possible as shown in Fig. 7.2(b). It can be shown generally $(1,2)$ that yield surfaces must be convex (except for local flat areas, possibly) and that they must contain the stress origin. + +# The Von Mises yield criterion (1913) + +Von Mises suggested that yielding occurs when $J_{2}^{\prime}$ reaches a critical value, or + +$$ +(J _ {2} ^ {\prime}) ^ {\ddagger} = k (\kappa), \tag {7.9} +$$ + +in which k is a material parameter to be determined. The second deviatoric stress invariant, $J_{2}'$ , can be explicitly written as + +$$ +\begin{array}{l} J _ {2} ^ {\prime} = \frac {1}{2} \sigma_ {i j} ^ {\prime} \sigma_ {i j} ^ {\prime} = \frac {1}{6} [ (\sigma_ {1} - \sigma_ {2}) ^ {2} + (\sigma_ {2} - \sigma_ {3}) ^ {2} + (\sigma_ {3} - \sigma_ {1}) ^ {2} ] \\ = \frac {1}{2} \left[ \sigma_ {x} ^ {\prime 2} + \sigma_ {y} ^ {\prime 2} + \sigma_ {z} ^ {\prime 2} \right] + \tau_ {x y} ^ {2} + \tau_ {y z} ^ {2} + \tau_ {x z} ^ {2}. \tag {7.10} \\ \end{array} +$$ + +Yield criterion (7.9) may be further written as + +$$ +\bar {\sigma} = \sqrt {3} (J _ {2} ^ {\prime}) ^ {\frac {1}{2}} = \sqrt {3} k, \tag {7.11} +$$ + +where + +$$ +\tilde {\sigma} = \sqrt {(3 / 2) \left\{\sigma_ {i j} ^ {\prime} \sigma_ {i j} ^ {\prime} \right\} ^ {\frac {1}{2}}}, \tag {7.12} +$$ + +and $\bar{\sigma}$ is termed the effective stress, generalised stress or equivalent stress. Some physical insight into the definition of $\bar{\sigma}$ will be apparent later from Section 7.2.4 where the case of uniaxial yielding is considered. There are two physical interpretations of the Von Mises yield condition. Nadai (1937) introduced the so-called octahedral shear stress $\tau_{oct}$ , which is the shear stress on the planes of a regular octahedron, the apices of which coincide with the + + + +principal axes of stress. The value of $\tau_{oct}$ is related to $J_{2}'$ by + +$$ +\tau_ {\mathrm{oct}} = \sqrt {(2 J _ {2} ^ {\prime} / 3)}. \tag {7.13} +$$ + +Thus yielding can be interpreted to begin when $\tau_{oct}$ reaches a critical value. Hencky (1924) pointed out that the Von Mises law implies that yielding begins when the (recoverable) elastic energy of distortion reaches a critical value. + +Fig. 7.1 shows the geometrical interpretation of the Von Mises yield surface to be a circular cylinder whose projection onto the $\pi$ plane is a circle of radius $\sqrt{(2)}k$ as shown in Fig. 7.2(a). The two dimensional plot of the Von Mises yield surface is the ellipse shown in Fig. 7.2(b). A physical meaning of the constant k can be obtained by considering the yielding of materials under simple stress states. The case of pure shear ( $\sigma_{1} = -\sigma_{2}, \sigma_{3} = 0$ ) requires on use of (7.9) and (7.10) that k must equal the yield shear stress. Alternatively the case of uniaxial tension ( $\sigma_{2} = \sigma_{3} = 0$ ) requires that $\sqrt{(3)}k$ is the uniaxial yield stress. + +The Tresca yield locus is a hexagon with distances of $\sqrt{(2/3)}Y$ from origin to apex on the $\pi$ plane whereas the Von Mises yield surface is a circle of radius $\sqrt{(2)}k$ . By suitably choosing the constant Y, the criteria can be made to agree with each other, and with experiment, for a single state of stress. This may be selected arbitrarily; it is conventional to make the circle pass through the apices of the hexagon by taking the constant $Y = \sqrt{(3)}k$ , the yield stress in simple tension. The criteria then differ most for a state of pure shear, where the Von Mises criterion gives a yield stress $2/\sqrt{(3)} (\approx 1 \cdot 15)$ times that given by the Tresca criterion. For most metals Von Mises' law fits the experimental data more closely than Tresca's, but it frequently happens that the Tresca criterion is simpler to use in theoretical applications. + +# The Mohr-Coulomb yield criterion + +This is a generalisation of the Coulomb (1773) friction failure law defined by + +$$ +\tau = c - \sigma_ {n} \tan \phi , \tag {7.14} +$$ + +where $\tau$ is the magnitude of the shearing stress, $\sigma_{n}$ is the normal stress (tensile stress is positive), $c$ is the cohesion and $\phi$ the angle of internal friction. Graphically (7.14) represents a straight line tangent to the largest principal stress circle as shown in Fig. 7.3 and was first demonstrated by Mohr (1882). From Fig. 7.3, and for $\sigma_{1} \geqslant \sigma_{2} \geqslant \sigma_{3}$ (7.14) can be rewritten as + +$$ +- \frac {1}{2} \left(\sigma_ {1} - \sigma_ {3}\right) \cos \phi = c - \left(\frac {\sigma_ {1} + \sigma_ {3}}{2} - \frac {\left(\sigma_ {1} - \sigma_ {3}\right)}{2} \sin \phi\right) \tan \phi , \tag {7.15} +$$ + +or rearranging + +$$ +\left(\sigma_ {1} - \sigma_ {3}\right) = 2 c \cos \phi - \left(\sigma_ {1} + \sigma_ {3}\right) \sin \phi . \tag {7.16} +$$ + + + +![](images/page-230_93e316717cb7dbfafeddf4a26ebbeeeb8c68e355f199140220b817cd3a181272.jpg) + +
+text_image + +P +τ +F E +φ +c +φ +C D B A -σ₁ 0 c Cot φ Q σₙ +-σ₂ +-σₙ +-(σ₁ + σ₂)/2 +-σ₃ +
+ +Fig. 7.3 Mohr circle representation of the Mohr–Coulomb yield criterion. + +Again, as for the Tresca criterion, the complete yield surface is obtained by considering all other stress combinations which can cause yielding (e.g. $\sigma_{3} \geqslant \sigma_{1} \geqslant \sigma_{2}$ ). In principal stress space this gives a conical yield surface whose normal section at any point is an irregular hexagon as shown in Fig. 7.4. The conical, rather than cylindrical, nature of the yield surface is a consequence of the fact that a hydrostatic stress does influence yielding which is evident from the last term in (7.14). When $\sigma_{1} = \sigma_{2} = \sigma_{3}$ we have from (7.16) that the mean hydrostatic stress, $\sigma_{m} = c \cot \phi$ and therefore the apex of the hexagonal pyramid, 0, in Fig. 7.4, lies along the space diagonal at the point $\sigma_{1} = \sigma_{2} = \sigma_{3} = c \cot \phi$ . This criterion is applicable to concrete, rock and soil problems. + +# The Drucker-Prager yield criterion + +An approximation to the Mohr–Coulomb law was presented by Drucker and Prager (1952) as a modification of the Von Mises yield criterion. The influence of a hydrostatic stress component on yielding was introduced by inclusion of an additional term in the Von Mises expression to give + +$$ +a J _ {1} + (J _ {2} ^ {\prime}) ^ {\frac {1}{2}} = k ^ {\prime}. \tag {7.17} +$$ + +This yield surface has the form of a circular cone. In order to make the Drucker–Prager circle coincide with the outer apices of the Mohr–Coulomb hexagon at any section, it can be shown that + +$$ +a = \frac {2 \sin \phi}\sqrt {(3) (3 - \sin \phi)}, \quad k ^ {\prime} = \frac {6 c \cos \phi}{\sqrt {(3) (3 - \sin \phi)}}. \tag {7.18} +$$ + +Coincidence with the inner apices of the Mohr–Coulomb hexagon is provided by + +$$ +\alpha = \frac {2 \sin \phi}{\sqrt {(3) (3 + \sin \phi)}}, \quad k ^ {\prime} = \frac {6 c \cos \phi}{\sqrt {(3) (3 + \sin \phi)}}. \tag {7.19} +$$ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_024.md b/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_024.md new file mode 100644 index 00000000..569a6a03 --- /dev/null +++ b/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_024.md @@ -0,0 +1,467 @@ + + +![](images/page-231_a0648648fd91b4dbb8ca8eb789dc62d520365b0f1cf2bb70c365c167fb31957e.jpg) + +
+text_image + +Drucker-Prager +Mohr-Coulomb +σ₃ +c Cot φ +O +σ₂ +σ₁ +
+ +Fig. 7.4 (a) Geometrical representation of the Mohr-Coulomb and Drucker-Prager yield surfaces in principal stress space. + +![](images/page-231_fb37979afda202c2a6af931de83d1e75b4445d641f3c7d41490d0357191a1eb5.jpg) + +
+text_image + +σ₃ +Mohr-Coulomb +C +B +D +0 +A +θ +E +σ₁ +F +σ₂ +Drucker-Prager +Line of pure +shear (θ = 0) +
+ +Fig. 7.4 (b) Two-dimensional, $\pi$ plane, representation of the Mohr-Coulomb and Drucker-Prager yield criteria. + +However, the approximation given by either the inner or outer cone to the true failure surface can be poor for certain stress combinations. $^{(5)}$ + + + +# 7.2.2 Work or strain hardening + +After initial yielding, the stress level at which further plastic deformation occurs may be dependent on the current degree of plastic straining. Such a phenomenon is termed work hardening or strain hardening. Thus the yield surface will vary at each stage of the plastic deformation, with the subsequent yield surfaces being dependent on the plastic strains in some way. Some alternative models which describe strain hardening in a material are illustrated in Fig. 7.5. A perfectly plastic material is shown in Fig. 7.5(a) where the yield stress level does not depend in any way on the degree of plastification. If the subsequent yield surfaces are a uniform expansion of the original yield curve, without translation, as shown in Fig. 7.5(b) the strain-hardening model is said to be isotropic. On the other hand if the subsequent yield surfaces preserve their shape and orientation but translate in the stress space as a rigid body as shown in Fig. 7.5(c), kinematic hardening is said to take place. Such a hardening model gives rise to the experimentally observed Bauschinger effect on cyclic loading. + +![](images/page-232_5d318734b4c9b6725086ed9f8d3d8d35ad18f695a5f28cc7dc4660db83692739.jpg) +Fig. 7.5 Mathematical models for representation of strain hardening behaviour. + + + +For some materials, notably soils, the yield surface may not strain harden but strain soften instead, so that the yield stress level at a point decreases with increasing plastic deformation. Therefore, for an isotropic model, the original yield curve contracts progressively without translation. Consequently yielding implies local failure and the yield surface becomes a failure criterion. + +The progressive development of the yield surface can be defined by relating the yield stress k to the plastic deformation by means of the hardening parameter $\kappa$ . This can be done in two ways. Firstly the degree of work hardening can be postulated to be a function of the total plastic work, $W_{p}$ , only. Then, + +$$ +\kappa = W _ {p}, \tag {7.20} +$$ + +where + +$$ +W _ {p} = \int \sigma_ {i j} (d \epsilon_ {i j}) _ {p}, \tag {7.21} +$$ + +in which $(d\epsilon_{ij})_{p}$ are the plastic components of strain occurring during a strain increment. Alternatively $\kappa$ can be related to a measure of the total plastic deformation termed the effective, generalised or equivalent plastic strain which is defined incrementally as + +$$ +d \bar {\epsilon} _ {p} = \sqrt {\left(\frac {2}{3}\right)} \left\{\left(d \epsilon_ {i j}\right) _ {p} \left(d \epsilon_ {i j}\right) _ {p} \right\} ^ {\frac {1}{2}}. \tag {7.22} +$$ + +A physical insight of this definition is provided in Section 7.2.4 where uniaxial yielding is considered. For situations where the assumption that yielding is independent of any hydrostatic stress is valid, $(d\epsilon_{ii})_{p}=0$ and hence $(d\epsilon_{ij}^{\prime})_{p}=(d\epsilon_{ij})_{p}$ . Consequently (7.22) can be rewritten as + +$$ +d \bar {\epsilon} _ {p} = \sqrt {\left(\frac {2}{3}\right)} \left\{\left(d \epsilon_ {i j} ^ {\prime}\right) _ {p} \left(d \epsilon_ {i j} ^ {\prime}\right) _ {p} \right\} ^ {\frac {1}{2}}. \tag {7.23} +$$ + +Then the hardening parameter, $\kappa$ , is assumed to be defined as + +$$ +\kappa = \bar {\epsilon} _ {p}, \tag {7.24} +$$ + +where $\bar{\epsilon}_{p}$ is the result of integrating $d\bar{\epsilon}_{p}$ over the strain path. This behaviour is termed strain hardening. Only an isotropic hardening model will be considered in this text. + +Stress states for which f = k represent plastic states, while elastic behaviour is characterised by f < k. At a plastic state, f = k, the incremental change in the yield function due to an incremental stress change is + +$$ +d f = \frac {\partial f}{\partial \sigma_ {i j}} d \sigma_ {i j}. \tag {7.25} +$$ + +Then if:- + +df<0 elastic unloading occurs (elastic behaviour) and the stress point returns inside the yield surface + +df=0 neutral loading (plastic behaviour for a perfectly plastic material) and the stress point remains on the yield surface + + + +df>0 plastic loading (plastic behaviour for a strain hardening material) and the stress point remains on the expanding yield surface. + +It can also be shown $^{(1-3)}$ that, for a stable material that the initial and all subsequent yield surfaces must be convex. + +# 7.2.3 Elasto-plastic stress/strain relation + +After initial yielding the material behaviour will be partly elastic and partly plastic. During any increment of stress, the changes of strain are assumed to be divisible into elastic and plastic components, so that + +$$ +d \epsilon_ {i j} = (d \epsilon_ {i j}) _ {e} + (d \epsilon_ {i j}) _ {p}. \tag {7.26} +$$ + +The elastic strain increment is related to the stress increment by (7.1). Or, decomposing the stress terms into their deviatoric and hydrostatic components + +$$ +(d \epsilon_ {i j}) _ {e} = \frac {d \sigma_ {i j} ^ {\prime}}{2 \mu} + \frac {(1 - 2 \nu)}{E} \delta_ {i j} d \sigma_ {k k}, \tag {7.27} +$$ + +where $E$ and $\nu$ are respectively the elastic modulus and Poisson's ratio of the material. + +In order to derive the relationship between the plastic strain component and the stress increment a further assumption on the material behaviour must be made. In particular it will be assumed that, the plastic strain increment is proportional to the stress gradient of a quantity termed the plastic potential Q, so that + +$$ +(d \epsilon_ {i j}) _ {p} = d \lambda \frac {\partial Q}{\partial \sigma_ {i j}}, \tag {7.28} +$$ + +where $d\lambda$ is a proportionality constant termed the plastic multiplier. A theoretical basis for this assumption is developed in Ref. 1. Equation (7.28) is termed the flow rule since it governs the plastic flow after yielding. The potential $Q$ must be a function of $J_2'$ and $J_3'$ but as yet it cannot be determined in its most general form. However the relation $f \equiv Q$ has a special significance in the mathematical theory of plasticity, since for this case certain variational principles and uniqueness theorems can be formulated. The identity $f \equiv Q$ is a valid one since it has been postulated that both are functions of $J_2'$ and $J_3'$ and such an assumption gives rise to an associated theory of plasticity. In this case (7.28) becomes + +$$ +(d \epsilon_ {i j}) _ {p} = d \lambda \frac {\partial f}{\partial \sigma_ {i j}}, \tag {7.29} +$$ + +and is termed the normality condition since $\partial f / \partial \sigma_{ij}$ is a vector directed normal to the yield surface at the stress point under consideration as shown in Fig. 7.6. It is seen that the components of the plastic strain increment are required to combine vectorially in $n$ -dimensional space to give a vector + + + +![](images/page-235_757ccffcf662a7c0c70bc579c4b917eabcf6ccf409819201f370d0981544535b.jpg) + +
+text_image + +Surface, f = k +σ₂ +∂f/∂σᵢⱼ +∂f/∂σ₂ +∂f/∂σ₁ +σ₁ +
+ +Fig. 7.6 Geometrical representation of the normality rule of associated plasticity. + +which is normal to the yield surface. For the particular case of $f = J_{2}'$ we have + +$$ +\frac {\partial f}{\partial \sigma_ {i j}} = \frac {\partial J _ {2} ^ {\prime}}{\partial \sigma_ {i j}} = \sigma_ {i j} ^ {\prime}. \tag {7.30} +$$ + +Then (7.29) becomes + +$$ +(d \epsilon_ {i j}) _ {p} = d \lambda \sigma_ {i j} ^ {\prime}, \tag {7.31} +$$ + +which are known as the Prandtl–Reuss equations $^{(1)}$ and have been extensively employed in theoretical work. Experimental observations indicate that the normality condition is an acceptable assumption for metals, but the question of normality in rocks and soils is still open to debate $^{(6)}$ and is discussed further in Chapter 12. Thus on use of (7.26), (7.27) and (7.29) the complete incremental relationship between stress and strain for elasto-plastic deformation is found to be + +$$ +d \epsilon_ {i j} = \frac {d \sigma_ {i j} ^ {\prime}}{2 \mu} + \frac {(1 - 2 \nu)}{E} \delta_ {i j} d \sigma_ {k k} + d \lambda \frac {\partial f}{\partial \sigma_ {i j}}. \tag {7.32} +$$ + +# 7.2.4 Uniaxial yield test on a strain-hardening material + +Consider the uniaxial testing of an elasto-plastic material which produces the stress-strain curve shown in Fig. 7.7. The behaviour is initially elastic characterised by an elastic modulus E until yielding commences at the uniaxial yield stress $\sigma_{Y}$ . Thereafter the material response is elasto-plastic with the local tangent to the curve continually varying and is termed the elasto-plastic tangent modulus, $E_{T}$ . The hardening law $k = k(\kappa)$ could just as easily be expressed in terms of the effective stress, $\bar{\sigma}$ (since it is proportional to $J_{2}'$ ) to give, for the strain hardening hypothesis (7.24) + +$$ +\bar {\sigma} = H (\bar {\epsilon} _ {p}), \tag {7.33} +$$ + + + +![](images/page-236_96e6f34e07830c785e9ad0085d34499f323fe671d83b39bc64ed36dc4cda5e6f.jpg) + +
+line + +| Strain, ε | Stress, σ (Eτ-Elasto-plastic tangent modulus) | Stress, σ (E-Elastic modulus) | +| --------- | --------------------------------------------- | ------------------------------ | +| 0 | 0 | 0 | +| ε | dσ | σγ | +| εe | dεe | σγ | +| ε | dεp | σγ | +| εe | dεp | σγ | +
+ +Fig. 7.7 Elasto-plastic strain hardening behaviour for the uniaxial case. + +or differentiating, + +$$ +\frac {d \bar {\sigma}}{d \bar {\epsilon} _ {p}} = H ^ {\prime} (\bar {\epsilon} _ {p}). \tag {7.34} +$$ + +For the uniaxial case under consideration $\sigma_{1} = \sigma$ , $\sigma_{2} = \sigma_{3} = 0$ and thus from (7.12) + +$$ +\bar {\sigma} = \sqrt {\left(\frac {3}{2}\right)} \{\sigma_ {i j} ^ {\prime} \sigma_ {i j} ^ {\prime} \} ^ {1 / 2} = \sigma . \tag {7.35} +$$ + +If the plastic strain increment in the direction of loading is $d\epsilon_{p}$ , then $(d\epsilon_{1})_{p} = d\epsilon_{p}$ and since plastic straining is assumed to be incompressible, Poisson's ratio is effectively 0.5 and $(d\epsilon_{2})_{p} = -\frac{1}{2} d\epsilon_{p}$ and $(d\epsilon_{3})_{p} = -\frac{1}{2} d\epsilon_{p}$ . Then from (7.23) the effective plastic strain becomes + +$$ +d \bar {\epsilon} _ {p} = \sqrt {\left(\frac {2}{3}\right)} \left\{\left(\epsilon_ {i j} ^ {\prime}\right) _ {p} \left(\epsilon_ {i j} ^ {\prime}\right) _ {p} \right\} ^ {1 / 2} = d \epsilon_ {p}. \tag {7.36} +$$ + +Expressions (7.35) and (7.36) explain the apparent arbitrary constants employed in the definition of $\bar{\sigma}$ and $\bar{\epsilon}_{p}$ , since these terms are required to become the actual stress and strain for uniaxial yielding. Using (7.35) and (7.36) then (7.34) becomes + +$$ +H ^ {\prime} (\bar {\epsilon} _ {p}) = \frac {d \sigma}{d \epsilon_ {p}} = \frac {d \sigma}{d \epsilon - d \epsilon_ {e}} = \frac {1}{d \epsilon / d \sigma - d \epsilon_ {e} / d \sigma}, +$$ + +or + +$$ +H ^ {\prime} = \frac {E _ {T}}{1 - E _ {T} / E}. \tag {7.37} +$$ + + + +Thus the hardening function $H'$ can be determined experimentally from a simple uniaxial yield test. (For numerical computation it will be shown in the next section that it is $H'$ and not H that is required). + +# 7.3 Matrix formulation + +The theoretical expressions developed in Section 7.2 will now be converted to matrix form. $^{(7,8)}$ The yield function, first defined in (7.4), can be rewritten as + +$$ +f (\sigma) = k (\kappa), \tag {7.38} +$$ + +where $\sigma$ is the stress vector and $\kappa$ is the hardening parameter which governs the expansion of the yield surface. In particular, from (7.20) and (7.21), $d\kappa = \sigma^{T}d\epsilon_{p}$ for the work hardening hypothesis and from (7.24) $d\kappa = d\epsilon_{p}$ for the strain hardening hypothesis. Rearranging (7.38) we get + +$$ +F (\sigma , \kappa) = f (\sigma) - k (\kappa) = 0. \tag {7.39} +$$ + +By differentiating (7.39) we have + +$$ +d F = \frac {\partial F}{\partial \sigma} d \sigma + \frac {\partial F}{\partial \kappa} d \kappa = 0, \tag {7.40} +$$ + +or + +$$ +\boldsymbol {a} ^ {T} d \sigma - A d \lambda = 0, \tag {7.41} +$$ + +with the definitions + +$$ +\boldsymbol {a} ^ {T} = \frac {\partial F}{\partial \sigma} = \left[ \frac {\partial F}{\partial \sigma_ {x}}, \frac {\partial F}{\partial \sigma_ {y}}, \frac {\partial F}{\partial \sigma_ {z}}, \frac {\partial F}{\partial \tau_ {y z}}, \frac {\partial F}{\partial \tau_ {z x}}, \frac {\partial F}{\partial \tau_ {x y}} \right], \tag {7.42} +$$ + +and + +$$ +A = - \frac {1}{d \lambda} \frac {\partial F}{\partial \kappa} d \kappa . \tag {7.43} +$$ + +The vector a is termed the flow vector. Expression (7.32) can be immediately rewritten as + +$$ +d \epsilon = [ D ] ^ {- 1} d \sigma + d \lambda \frac {\partial F}{\partial \sigma}, \tag {7.44} +$$ + +where D is the usual matrix of elastic constants. Premultiplying both sides of (7.44) by $d_{D}^{T} = a^{T}D$ and eliminating $a^{T}d\sigma$ by use of (7.41) we obtain the plastic multiplier $d\lambda$ to be + +$$ +d \lambda = \frac {1}{[ A + a ^ {T} D a ]} a ^ {T} d _ {D} d \epsilon . \tag {7.45} +$$ + +Or substituting (7.45) into (7.44) we obtain the complete elasto-plastic incremental stress-strain relation to be + +$$ +d \sigma = D _ {e p} d \epsilon , \tag {7.46} +$$ + + + +with + +$$ +\boldsymbol {D} _ {e p} = \boldsymbol {D} - \frac {\boldsymbol {d} _ {D} \boldsymbol {d} _ {D} ^ {T}}{\boldsymbol {A} + \boldsymbol {d} _ {0} ^ {T} \boldsymbol {a}}; \quad \boldsymbol {d} _ {D} = \boldsymbol {D} \boldsymbol {a}. \tag {7.47} +$$ + +This expression for $D_{ep}$ is similar in form to that for one dimensional application given in Page 28, Chapter 2. It now remains to determine the explicit form of the scalar term, A. The work hardening hypothesis is more general from a thermodynamic viewpoint $^{(9)}$ than the strain hardening hypothesis and will be employed for numerical work in this text. Therefore + +$$ +d \kappa = \sigma^ {T} d \epsilon_ {p}. \tag {7.48} +$$ + +Equation (7.39) can be rewritten in the form + +$$ +F (\sigma , \kappa) = f (\sigma) - \sigma_ {Y} (\kappa) = 0, \tag {7.49} +$$ + +since the uniaxial yield stress, $\sigma_{Y} = \sqrt{(3)} k$ . Thus from (7.43) + +$$ +A = - \frac {1}{d \lambda} \frac {\partial F}{\partial \kappa} d \kappa = \frac {1}{d \lambda} \frac {d \sigma_ {Y}}{d \kappa} d \kappa . \tag {7.50} +$$ + +Note that the full differential may be employed in the last term since $\sigma_{Y}$ is a function of $\kappa$ only. Employing the normality condition in (7.48) to express $d\epsilon_{p}$ we have + +$$ +d \kappa = \sigma^ {T} d \epsilon_ {p} = \sigma^ {T} d \lambda a = d \lambda a ^ {T} \sigma . \tag {7.51} +$$ + +Or, for the uniaxial case $\sigma = \bar{\sigma} = \sigma_{Y}$ and $d\epsilon_{p} = d\bar{\epsilon}_{p}$ where $\bar{\sigma}$ and $\bar{\epsilon}_{p}$ are respectively the effective stress and strain. Thus (7.51) becomes + +$$ +d \kappa = \sigma_ {Y} d \bar {\epsilon} _ {p} = d \lambda a ^ {T} \sigma . \tag {7.52} +$$ + +Also, from (7.34) we have + +$$ +\frac {d \bar {\sigma}}{d \bar {\epsilon} _ {p}} = \frac {d \sigma_ {Y}}{d \bar {\epsilon} _ {p}} = H ^ {\prime}. \tag {7.53} +$$ + +Using Euler's theorem† applicable to all homogeneous functions of order one, we can write from (7.49) + +$$ +\frac {\partial f}{\partial \sigma} \sigma = \sigma_ {Y}. \tag {7.54} +$$ + +Or from (7.42) + +$$ +\boldsymbol {a} ^ {T} \boldsymbol {\sigma} = \sigma_ {Y}. \tag {7.55} +$$ + +Substituting (7.53) and (7.55) into (7.52) and (7.50) we obtain + +$$ +d \lambda = d \bar {\epsilon} _ {p} +$$ + +$$ +A = H ^ {\prime}. \tag {7.56} +$$ + +† Euler's theorem on homogeneous functions states that if $f(\mathbf{x})$ is homogeneous and of degree $n$ then $(\partial f / \partial \mathbf{x})$ . $\mathbf{x} = nf$ . + + + +Thus A is obtained to be the local slope of the uniaxial stress/plastic strain curve and can be determined experimentally from (7.37). + +# 7.4 Alternative form of the yield criteria for numerical computation + +For numerical computations it is convenient to rewrite the yield function in terms of alternative stress invariants. This formulation is due to Nayak $^{(10)}$ and its main advantage is that it permits the computer coding of the yield function and the flow rule in a general form and necessitates only the specification of three constants for any individual criterion. + +The principal deviatoric stresses $\sigma_{1}^{\prime}$ , $\sigma_{2}^{\prime}$ , $\sigma_{3}^{\prime}$ are given as the roots of the cubic equation $^{(11)}$ + +$$ +t ^ {3} - J _ {2} ^ {\prime} t - J _ {3} ^ {\prime} = 0. \tag {7.57} +$$ + +Noting the trigonometric identity + +$$ +\sin^ {3} \theta - \frac {3}{4} \sin \theta + \frac {1}{4} \sin 3 \theta = 0, \tag {7.58} +$$ + +and substituting $t = r \sin \theta$ into (7.57) we have + +$$ +\sin^ {3} \theta - \frac {J _ {2} ^ {\prime}}{r ^ {2}} \sin \theta - \frac {J _ {3} ^ {\prime}}{r ^ {3}} = 0. \tag {7.59} +$$ + +Comparing (7.58) and (7.59) gives + +$$ +r = \frac {2}{\sqrt {3}} (J _ {2} ^ {\prime}) ^ {1 / 2}, \tag {7.60} +$$ + +$$ +\sin 3 \theta = - \frac {4 J _ {3} ^ {\prime}}{r ^ {3}} = - \frac {3 \sqrt {3}}{2} \frac {J _ {3} ^ {\prime}}{\left(J _ {2} ^ {\prime}\right) ^ {3 / 2}}. \tag {7.61} +$$ + +The first root of (7.61) with $\theta$ determined for $3\theta$ in the range $\pm\pi/2$ is a convenient alternative to the third invariant, $J_{3}^{\prime}$ . By noting the cyclic nature of $\sin(3\theta+2n\pi)$ we have immediately the three (and only three) possible values of $\sin\theta$ which define the three principal stresses. The deviatoric principal stresses are given by $t=r\sin\theta$ on substitution of the three values of $\sin\theta$ in turn. Substituting for $r$ from (7.60) and adding the mean hydrostatic stress component gives the total principal stresses to be + +$$ +\left\{ \begin{array}{l} \sigma_ {1} \\ \sigma_ {2} \\ \sigma_ {3} \end{array} \right\} = \frac {2 \left(J _ {2} ^ {\prime}\right) ^ {\frac {1}{2}}}{\sqrt {3}} \left\{ \begin{array}{l} \sin \left(\theta + \frac {2 \pi}{3}\right) \\ \sin \theta \\ \sin \left(\theta + \frac {4 \pi}{3}\right) \end{array} \right\} + \frac {J _ {1}}{3} \left\{ \begin{array}{l} 1 \\ 1 \\ 1 \end{array} \right\}, \tag {7.62} +$$ + +with $\sigma_{1} > \sigma_{2} > \sigma_{3}$ and $-\pi / 6 \leqslant \theta \leqslant \pi / 6$ . The term $\theta$ is essentially similar to the Lode parameter(1) $\Gamma$ defined by $\Gamma = -\sqrt{(3)} \tan \theta$ . The four yield criteria + + + +considered in Section 7.2.1 can now be rewritten in terms of $J_{1}, J_{2}'$ and $\theta$ as follows. + +The Tresca yield criterion + +Substitute for $\sigma_{1}$ and $\sigma_{3}$ from (7.62) into (7.8) gives + +$$ +\frac {2}{\sqrt {3}} (J _ {2} ^ {\prime}) ^ {\frac {1}{2}} \left[ \sin \left(\theta + \frac {2 \pi}{3}\right) - \sin \left(\theta + \frac {4 \pi}{3}\right) \right] = Y (\kappa), +$$ + +or expanding we have + +$$ +2 (J _ {2} ^ {\prime}) ^ {\frac {1}{2}} \cos \theta = Y (\kappa) = \sqrt {(3)} k (\kappa) = \sigma_ {Y} (\kappa). \tag {7.63} +$$ + +The physical interpretation of $\theta$ is evident from Fig. 7.2. + +The Von Mises yield criterion + +There is no change in this case since this yield function depends on $J_2'$ only. From (7.9) + +$$ +(J _ {2} ^ {\prime}) ^ {\frac {1}{2}} = k (\kappa), +$$ + +or $\sqrt{3}(J_2')^{\frac{1}{2}} = \sigma_Y(\kappa).$ (7.64) + +The Mohr-Coulomb yield criterion + +Substituting from (7.62) for $\sigma_{1}$ and $\sigma_{3}$ into (7.16) results in + +$$ +\frac {1}{3} J _ {1} \sin \phi + (J _ {2} ^ {\prime}) ^ {1 / 2} \left(\cos \theta - \frac {1}{\sqrt {3}} \sin \theta \sin \phi\right) = c \cos \phi . \tag {7.65} +$$ + +The Drucker-Prager yield criterion + +There is no change for this criterion and we can write directly from (7.17) that + +$$ +a J _ {1} + (J _ {2} ^ {\prime}) ^ {\frac {1}{2}} = k ^ {\prime}, \tag {7.66} +$$ + +where $a$ and $k'$ are defined in (7.18) or (7.19). + +In order to calculate the $D_{ep}$ matrix in (7.47) we require to express the flow vector a in a form suitable for numerical computation. We can always write + +$$ +\boldsymbol {a} ^ {T} = \frac {\partial F}{\partial \sigma} = \frac {\partial F}{\partial J _ {1}} \frac {\partial J _ {1}}{\partial \sigma} + \frac {\partial F}{\partial (J _ {2} ^ {\prime}) ^ {1 / 2}} \frac {\partial (J _ {2} ^ {\prime}) ^ {1 / 2}}{\partial \sigma} + \frac {\partial F}{\partial \theta} \frac {\partial \theta}{\partial \sigma}, \tag {7.67} +$$ + +where + +$$ +\sigma^ {T} = \left\{\sigma_ {x}, \sigma_ {y}, \sigma_ {z}, \tau_ {y z}, \tau_ {z x}, \tau_ {x y} \right\}. +$$ + +Differentiating (7.61) we obtain + +$$ +\frac {\partial \theta}{\partial \sigma} = \frac {- \sqrt {3}}{2 \cos 3 \theta} \left[ \frac {1}{(J _ {2} ^ {\prime}) ^ {3 / 2}} \frac {\partial J _ {3}}{\partial \sigma} - \frac {3 J _ {3}}{(J _ {2} ^ {\prime}) ^ {2}} \frac {\partial (J _ {2} ^ {\prime}) ^ {1 / 2}}{\partial \sigma} \right]. \tag {7.68} +$$ + +Substituting this in (7.67) and using (7.61), we can then write diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_025.md b/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_025.md new file mode 100644 index 00000000..f0af929d --- /dev/null +++ b/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_025.md @@ -0,0 +1,574 @@ + + +$$ +\boldsymbol {a} = C _ {1} \boldsymbol {a} _ {1} + C _ {2} \boldsymbol {a} _ {2} + C _ {3} \boldsymbol {a} _ {3}, \tag {7.69} +$$ + +where + +$$ +\boldsymbol {a} _ {1} ^ {T} = \frac {\partial J _ {1}}{\partial \sigma} = \{1, 1, 1, 0, 0, 0 \} +$$ + +$$ +\boldsymbol {a} _ {2} ^ {T} = \frac {\partial (J _ {2} ^ {\prime}) ^ {1 / 2}}{\partial \sigma} = \frac {1}{2 (J _ {2} ^ {\prime}) ^ {1 / 2}} \left\{\sigma_ {x} ^ {\prime}, \sigma_ {y} ^ {\prime}, \sigma_ {z} ^ {\prime}, 2 \tau_ {y z}, 2 \tau_ {z x}, 2 \tau_ {x y} \right\} +$$ + +$$ +\boldsymbol {a} _ {3} ^ {T} = \frac {\partial J _ {3}}{\partial \sigma} = \left\{\left(\sigma_ {y} ^ {\prime} \sigma_ {z} ^ {\prime} - \tau_ {y z} ^ {2} + \frac {J _ {2} ^ {\prime}}{3}\right), \quad \left(\sigma_ {x} ^ {\prime} \sigma_ {z} ^ {\prime} - \tau_ {x z} ^ {2} + \frac {J _ {2} ^ {\prime}}{3}\right), \right. +$$ + +$$ +\left(\sigma_ {x} ^ {\prime} \sigma_ {y} ^ {\prime} - \tau_ {x y} ^ {2} + \frac {J _ {2} ^ {\prime}}{3}\right), \quad 2 \left(\tau_ {x z} \tau_ {x y} - \sigma_ {x} ^ {\prime} \tau_ {y z}\right), +$$ + +$$ +\left. 2 \left(\tau_ {x y} \tau_ {y z} - \sigma_ {y} ^ {\prime} \tau_ {x z}\right), \quad 2 \left(\tau_ {y z} \tau_ {x z} - \sigma_ {z} ^ {\prime} \tau_ {x y}\right) \right\}, \tag {7.70} +$$ + +and + +$$ +C _ {1} = \frac {\partial F}{\partial J _ {1}}, \quad C _ {2} = \left(\frac {\partial F}{\partial \left(J _ {2} ^ {\prime}\right) ^ {1 / 2}} - \frac {\tan 3 \theta}{\left(J _ {2} ^ {\prime}\right) ^ {1 / 2}} \frac {\partial F}{\partial \theta}\right), +$$ + +$$ +C _ {3} = \frac {- \sqrt {3}}{2 \cos 3 \theta} \frac {1}{(J _ {2} ^ {\prime}) ^ {3 / 2}} \frac {\partial F}{\partial \theta}. \tag {7.71} +$$ + +Only the constants $C_{1}$ , $C_{2}$ and $C_{3}$ are then necessary to define the yield surface. Thus we can achieve a simplicity of programming as only these three constants have to be varied between one yield surface and another. The constants $C_{i}$ are given in Table 7.1 for the four yield criteria considered in Section 7.2.1 and other yield functions can be expressed in the same form with equal ease. + +Table 7.1 Constants defining the yield surface in a form suitable for numerical analysis. +
Yield Criterion $C_1$ $C_2$ $C_3$
Tresca0 $2\cos\theta(1+\tan\theta\tan3\theta)$ $\frac{\sqrt{3}}{J_2'}\frac{\sin\theta}{\cos3\theta}$
Von Mises0 $\sqrt{3}$ 0
Mohr–Coulomb $\frac{1}{3}\sin\phi$ $\cos\theta[(1+\tan\theta\tan3\theta)+\sin\phi(\tan3\theta-\tan\theta)/\sqrt{3}]$ $\frac{(\sqrt{3}\sin\theta+\cos\theta\sin\phi)}{(2J_2'\cos3\theta)}$
Drucker–Prager $\alpha$ 1.00
+ + + +# 7.5 Basic expressions for two dimensional problems + +For two dimensional problems, the general expressions derived so far in this chapter have to be modified. Primarily the main alteration required is the deletion of the stress (and strain) components which vanish under the conditions of plane stress, plane strain or axial symmetry. We have only four non-zero stress or strain components, namely + +$$ +\sigma^ {T} = \{\sigma_ {x}, \sigma_ {y}, \tau_ {x y}, \sigma_ {z} \}, \quad \sigma_ {z} = 0 \quad \text { for Plane Stress } +$$ + +$$ +\{\sigma_ {x}, \sigma_ {y}, \tau_ {x y}, \sigma_ {z} \}, \quad \epsilon_ {z} = 0 \quad \text { Plane Strain } +$$ + +$$ +\left\{\sigma_ {r}, \sigma_ {z}, \tau_ {r z}, \sigma_ {\theta} \right\} \quad \text { Axial Symmetry. } \tag {7.72} +$$ + +From Fig. 7.8 it is seen that the z direction is taken as the coordinate independent direction for plane stress and plane strain. It is also found convenient to order the stress components as indicated in (7.72) with the stress in the coordinate independent direction being last. + +![](images/page-242_f042f7a046babbdad5f8e3244aa8b82b53b57852471ad71fadd568e6f85dd679.jpg) +Fig. 7.8 Two-dimensional applications showing coordinate systems employed. + +The explicit form of the elasticity matrix D can be written + +$$ +\boldsymbol {D} = \frac {E (1 - \nu)}{(1 + \nu) (1 - 2 \nu)} \left[ \begin{array}{c c c c} 1 & \frac {\nu}{1 - \nu} & 0 & \frac {\nu}{1 - \nu} \\ \frac {\nu}{1 - \nu} & 1 & 0 & \frac {\nu}{1 - \nu} \\ 0 & 0 & \frac {1 - 2 \nu}{2 (1 - \nu)} & 0 \\ \frac {\nu}{1 - \nu} & \frac {\nu}{1 - \nu} & 0 & 1 \end{array} \right] \text { for plane strain and axial symmetry, } +$$ + + + +$$ +\boldsymbol {D} = \frac {E}{1 - \nu^ {2}} \left[ \begin{array}{c c c c c} 1 & \nu & 0 & 0 \\ \nu & 1 & 0 & 0 \\ 0 & 0 & \frac {1 - \nu}{2} & 0 \\ - \frac {2}{0} & 0 & 0 & 1 \end{array} \right] \quad \text { for plane stress. } \tag {7.73} +$$ + +Note that the components corresponding to the coordinate independent direction have been included for the plane stress and strain cases. These terms will be excluded for element stiffness formulation and only the first $3 \times 3$ portion indicated will be employed. By eliminating the appropriate stress terms the expressions developed to date can be readily modified. The flow vector a becomes + +$$ +\boldsymbol {a} ^ {T} = \left\{\frac {\partial F}{\partial \sigma_ {x}}, \frac {\partial F}{\partial \sigma_ {y}}, \frac {\partial F}{\partial \tau_ {x y}}, \frac {\partial F}{\partial \sigma_ {z}} \right\}, \tag {7.74} +$$ + +with x, y and z being replaced by r, z and $\theta$ respectively for the case of axial symmetry. The specific form of the vector, a is still given by (7.69) but in this case we have from (7.70) + +$$ +\boldsymbol {a} _ {1} ^ {T} = \{1, 1, 0, 1 \} +$$ + +$$ +\boldsymbol {a} _ {2} ^ {T} = \frac {1}{2 (J _ {2} ^ {\prime}) ^ {1 / 2}} \left\{\sigma_ {x} ^ {\prime}, \sigma_ {y} ^ {\prime}, 2 \tau_ {x y}, \sigma_ {z} ^ {\prime} \right\} +$$ + +$$ +\boldsymbol {a} _ {3} ^ {T} = \left\{\left(\sigma_ {y} ^ {\prime} \sigma_ {z} ^ {\prime} + \frac {J _ {2} ^ {\prime}}{3}\right), \left(\sigma_ {x} ^ {\prime} \sigma_ {z} ^ {\prime} + \frac {J _ {2} ^ {\prime}}{3}\right), \right. +$$ + +$$ +\left. - 2 \sigma_ {z} ^ {\prime} \tau_ {x y}, \left(\sigma_ {x} ^ {\prime} \sigma_ {y} ^ {\prime} - \tau_ {x y} ^ {2} + \frac {J _ {2} ^ {\prime}}{3}\right) \right\}, \tag {7.75} +$$ + +and the deviatoric stress invariants become, from (7.5) + +$$ +J _ {2} ^ {\prime} = \frac {1}{2} \left(\sigma_ {x} ^ {\prime 2} + \sigma_ {y} ^ {\prime 2} + \sigma_ {z} ^ {\prime 2}\right) + \tau_ {x y} ^ {2} +$$ + +$$ +J _ {3} ^ {\prime} = \sigma_ {z} ^ {\prime} (\sigma_ {z} ^ {\prime 2} - J _ {2} ^ {\prime}). \tag {7.76} +$$ + +To complete the prescription of the elasto-plastic matrix $D_{ep}$ given in (7.47) we require $d_{D}$ . Employing the relevant form of D from (7.73) in (7.47) results in, for plane strain and axial symmetry + + + +$$ +\boldsymbol {d} _ {D} = \left\{ \begin{array}{l} d _ {1} \\ d _ {1} \\ d _ {3} \\ d _ {4} \end{array} \right\} = \left\{ \begin{array}{c} \frac {E}{1 + \nu} a _ {1} + M _ {1} \\ \frac {E}{1 + \nu} a _ {2} + M _ {1} \\ G a _ {3} \\ \frac {E}{1 + \nu} a _ {4} + M _ {1} \end{array} \right\}, \quad M _ {1} = \frac {E \nu \left(a _ {1} + a _ {2} + a _ {4}\right)}{(1 + \nu) (1 - 2 \nu)}, \tag {7.77} +$$ + +where $G = E / 2(1 + \nu)$ is the shear modulus and $a_1 \ldots a_4$ are the components of $\pmb{a}$ . For plane stress we have + +$$ +\boldsymbol {d} _ {D} = \left\{ \begin{array}{c} \frac {E}{1 + \nu} a _ {1} + M _ {2} \\ \frac {E}{1 + \nu} a _ {2} + M _ {2} \\ G a _ {3} \\ \frac {E}{1 + \nu} a _ {4} + M _ {2} \end{array} \right\}, \quad M _ {2} = \frac {E \nu \left(a _ {1} + a _ {2}\right)}{1 - \nu^ {2}}. \tag {7.78} +$$ + +# 7.6 Singular points on the yield surface + +For many yield surfaces the flow vector a is not uniquely defined for certain stress combinations. For example this arises at the corners of the Tresca and Mohr–Coulomb criteria located by $\theta = \pm30^{\circ}$ and the direction of plastic straining there is indeterminate. Koiter $^{(12)}$ has provided limits within which the incremental plastic strain vector must lie. Numerical difficulties will be encountered as $\theta$ approaches $\pm30^{\circ}$ for the Tresca and Mohr–Coulomb laws since it is seen from Table 7.1 that for these values of $\theta$ both $C_{2}$ and $C_{3}$ become indeterminate. This difficulty can be overcome by returning to the original expressions (7.63) for the Tresca law and (7.65) for the Mohr–Coulomb criterion and rewriting these for the explicit values $\theta = \pm30^{\circ}$ . Thus we have for the Tresca law + +$$ +\sqrt {(3)} \left(J _ {2} ^ {\prime}\right) ^ {\frac {1}{2}} = Y (\kappa) = \sqrt {(3)} k (\kappa), \tag {7.79} +$$ + +and thus from (7.71) we have + +$$ +C _ {1} = 0, \quad C _ {2} = \sqrt {(3)}, \quad C _ {3} = 0 \quad \text {for} \quad \theta = \pm 3 0 ^ {\circ}. \tag {7.80} +$$ + +Physically, since (7.79) is the Von Mises criterion, this is equivalent to stating that the direction of plastic straining at the corners of the Tresca criterion is that given by the Von Mises circle which also passes through the corner (see Fig. 7.2). Similarly for the Mohr-Coulomb criterion we have + + + +from (7.65), + +$$ +\frac {1}{3} J _ {1} \sin \phi + \left(J _ {2} ^ {\prime}\right) ^ {1 / 2} \frac {1}{2} \left(\sqrt {3} - \frac {\sin \phi}{\sqrt {3}}\right) - c \cos \phi = 0 \quad \text { for } \quad \theta = + 3 0 ^ {0} +$$ + +$$ +\frac {1}{3} J _ {1} \sin \phi + (J _ {2} ^ {\prime}) ^ {1 / 2} \frac {1}{2} \left(\sqrt {3 + \frac {\sin \phi}{\sqrt {3}}}\right) - c \cos \phi = 0 \quad \theta = - 3 0 ^ {0}, \tag {7.81} +$$ + +or from (7.71) we have + +$$ +C _ {1} = \frac {1}{3} \sin \phi , C _ {2} = \frac {1}{2} \left(\sqrt {3} - \frac {\sin \phi}{\sqrt {3}}\right), C _ {3} = 0 \quad \text { for } \quad \theta = + 3 0 ^ {0} +$$ + +$$ +C _ {1} = \frac {1}{3} \sin \phi , C _ {2} = \frac {1}{2} \left(\sqrt {3} + \frac {\sin \phi}{\sqrt {3}}\right), C _ {3} = 0 \quad \theta = - 3 0 ^ {0}. \tag {7.82} +$$ + +The practical approach adopted in this text is to use the general expressions for $C_1, C_2, C_3$ given in Table 7.1 for all values of $|\theta| \leqslant 29^\circ$ and to then employ either (7.80) for Tresca or (7.82) for Mohr-Coulomb in the vicinity of the corners. This makes the direction of straining unique, and also satisfies the Koiter requirements. Physically this artifice corresponds to a 'rounding off' of the yield surface corners. + +# 7.7 Finite element expressions and program structure + +The basic expressions required for solution can be again obtained by use of the principle of virtual work. Consider the solid, in which the internal stresses $\sigma$ , the distributed loads/unit volume $b$ and external applied forces $f$ form an equilibrating field, to undergo an arbitrary virtual displacement pattern $\delta d^{*}$ which result in compatible strains $\delta \epsilon^{*}$ and internal displacements $\delta u^{*}$ . Then the principle of virtual work requires that + +$$ +\int_ {\Omega} \left(\delta \epsilon^ {* T} \sigma - \delta u ^ {* T} b\right) d \Omega - \delta d ^ {* T} f = 0. \tag {7.83} +$$ + +Then the normal finite element discretising procedure leads to the following expressions for the displacements and strains within any element + +$$ +\delta \boldsymbol {u} ^ {*} = N \delta \boldsymbol {d} ^ {*}, \quad \delta \boldsymbol {\epsilon} ^ {*} = \boldsymbol {B} \delta \boldsymbol {d} ^ {*}, \tag {7.84} +$$ + +where N and B are respectively the usual matrix of shape functions and the elastic strain matrix. Then the element assembly process gives + +$$ +\int_ {\Omega} \delta \boldsymbol {d} ^ {* T} (\boldsymbol {B} ^ {T} \boldsymbol {\sigma} - \boldsymbol {N} ^ {T} \boldsymbol {b}) d \Omega - \delta \boldsymbol {d} ^ {* T} \boldsymbol {f} = 0, \tag {7.85} +$$ + +where the volume integration over the solid is the sum of the individual element contributions. Since this expression must hold true for any arbitrary $\delta d^{*}$ value + + + +$$ +\int_ {\Omega} \boldsymbol {B} ^ {T} \sigma d \Omega - \boldsymbol {f} - \int_ {\Omega} \boldsymbol {N} ^ {T} \boldsymbol {b} d \Omega = 0. \tag {7.86} +$$ + +For the solution of nonlinear problems as described in Chapter 2, (7.86) will not generally be satisfied at any stage of the computation, and + +$$ +\psi = \int_ {\Omega} \boldsymbol {B} ^ {T} \sigma d \Omega - \left(f + \int_ {\Omega} N ^ {T} \boldsymbol {b} d \Omega\right) \neq 0, \tag {7.87} +$$ + +where $\psi$ is the residual force vector. For an elasto-plastic situation the material stiffness is continually varying, and instantaneously the incremental stress/strain relationship is given by (7.46). For the purpose of evaluating the material tangential stiffness matrix $K_{T}$ at any stage, the incremental form of (7.87) must be employed. Thus within an increment of load we have + +$$ +\Delta \psi = \int_ {\Omega} \boldsymbol {B} ^ {T} \Delta \sigma d \Omega - \left(\Delta f + \int_ {\Omega} \boldsymbol {N} ^ {T} \Delta \boldsymbol {b} d \Omega\right). \tag {7.88} +$$ + +Substituting for $\Delta\sigma$ from (7.46) results in + +$$ +\Delta \psi = K _ {T} d - \left(\Delta f + \int_ {\Omega} N ^ {T} \Delta b d \Omega\right), \tag {7.89} +$$ + +where + +$$ +\boldsymbol {K} _ {T} = \int_ {\Omega} \boldsymbol {B} ^ {T} \boldsymbol {D} _ {e p} \boldsymbol {B} d \Omega . \tag {7.90} +$$ + +Expression (7.89) is essentially identical to (2.4) and therefore the solution procedures developed in Chapter 2 can be again employed. + +The programming philosophy adopted for this application follows that employed in Chapter 3 for one-dimensional elasto-plastic problems. It is suggested that the reader reviews the appropriate sections of Chapter 3 before proceeding to the remainder of this chapter. The solution techniques discussed in Chapters 2 and 3 are utilised and in particular an initial stiffness algorithm, a tangential stiffness algorithm and two options of the combined initial/tangential stiffness approach are included. An outline of the program is provided in Fig. 7.9. Many of the subroutines required are common to the corresponding linear elastic solution program and their function and structure have already been described. In particular, subroutines BMATS, CHECK1, CHECK2, DBE, ECHO, FRONT, GAUSSQ, JACOB2, LOADPS, MODPS, NODEXY and SFR2 have been described in Section 6.4. Also the standard nonlinear subroutines ALGOR, CONVER, INCREM and INPUT have been presented in Section 6.5. We will now formulate the additional subroutines required and assemble them to form a working program. + + + +![](images/page-247_70f1f32e35fa7973c290612732c7bf46a5ee8ef0377b73c3fd1aa0e0c2902e2e.jpg) + +
+flowchart + +```mermaid +graph TD + A["START"] --> B["DIMEN
Presents the variables associated with the dynamic dimensioning process."] + B --> C["INPUT
Inputs data defining geometry, boundary conditions and material properties."] + C --> D["LOADPS
Evaluates the equivalent nodal forces for pressure loading, gravity loading, etc."] + D --> E["ZERO
Sets to zero arrays required for accumulation of data."] + E --> F["INCREM
Increments the applied loads according to specified load factors."] + F --> G["ALGOR
Sets indicator to identify the type of solution algorithm e.g. initial stiffness, tangential stiffness, etc."] + G --> H["STIFFP
Calculates the element stiffnesses for elastic and elasto-plastic material behaviour."] + H --> I["FRONT
Solves the simultaneous equation system by the frontal method."] + I --> J["RESIDU
Calculates the residual force vector, ψ."] + J --> K["CONVER
Checks to see if the solution process has converged."] + K -->|NO| L["END"] + K -->|YES| M["OUTPUT
Prints the results for this load increment."] + L --> N["ITERATION LOOP"] + M --> N +``` +
+ +Fig. 7.9 Program organisation for two-dimensional elasto-plastic applications. + +# 7.8 Additional program subroutines + +A total of eight additional subroutines are required some of which will be common to other nonlinear applications considered in later chapters of this text. + + + +# 7.8.1 Subroutine DIMEN + +The function of this subroutine is to preset the values of variables employed in the program. In particular the variables associated with the dynamic dimensioning process described in Chapter 6 are defined. Thus if it is required to upgrade the magnitude of the maximum problem size which can be solved it is only necessary to modify the dimension statements in the main or master subroutine together with the variables set in subroutine DIMEN. All the variables preset in this subroutine have been previously defined and their specified values are indicated in the following listing. + +```txt +SUBROUTINE DIMEN(MBUFA,MELEM,MEVAB,MFRON,MMATS,MPOIN,MSTIF,MTOTG,MTOTV,MVFIX,NDOFN,NPROP,NSTRE) DIMN 1 +C*************** DIMN 2 +C +C**** THIS SUBROUTINE PRESETS VARIABLES ASSOCIATED WITH DYNAMIC DIMN 3 +C DIMENSIONING DIMN 4 +C +C*************** DIMN 5 +MBUFA = 10 DIMN 6 +MELEM=40 DIMN 7 +MFRON=80 DIMN 8 +MMATS = 5 DIMN 9 +MPOIN=150 DIMN 10 +MSTIF=(MFRON*MFRON-MFRON)/2.0+MFRON DIMN 11 +MTOTG = MELEM*9 DIMN 12 +NDOFN = 2 DIMN 13 +MTOTV = MPOIN*NDOFN DIMN 14 +MVFIX=30 DIMN 15 +NPROP=7 DIMN 16 +MEVAB = NDOFN*9 DIMN 17 +RETURN DIMN 18 +END DIMN 19 +DIMN 20 +DIMN 21 +DIMN 22 +``` + +# 7.8.2 Subroutine ZERO + +This subroutine merely sets to zero the contents of several arrays employed in the program. These arrays will be employed to accumulate data as the incremental and iterative process continues and they therefore require to be initialised to zero. This subroutine is self-explanatory and is presented without further comment. + +```txt +SUBROUTINE ZERO(ELOAD,MELEM,MEVAB,MPOIN,MTOTG,MTOTV,NDOFN,NELEM, ZRO1 1 +. NEVAB,NGAUS,NSTR1,NTOTG,EPSTN,EFFST, ZRO1 2 +. NTOTV,NVFIX,STRSG,TDISP,TFACT, ZRO1 3 +. TLOAD,TREAC,MVFIX) ZRO1 4 +C******************************* +C******************************* +C******************************* +C**** THIS SUBROUTINE INITIALISES VARIOUS ARRAYS TO ZERO ZRO1 7 +C ZRO1 8 +C******************************* +DIMENSION ELOAD(MELEM,MEVAB),STRSG(4,MTOTG),TDISP(MTOTV), ZRO1 10 +. TLOAD(MELEM,MEVAB),TREAC(MVFIX,2),EPSTN(MTOTG), ZRO1 11 +. EFFST(MTOTG) ZRO1 12 +TFACT=0.0 ZRO1 13 +DO 30 IELEM=1,NELEM ZRO1 14 +DO 30 IEVAB=1,NEVAB ZRO1 15 +ELOAD(IELEM,IEVAB)=0.0 ZRO1 16 +``` + + + +
30TLOAD(IELEM,IEVAB)=0.0ZRO117
DO 40 ITOTV=1,NTOTVZRO118
40TDISP(ITOTV)=0.0ZRO119
DO 50 IVFIX=1,NVFIXZRO120
DO 50 IDOFN=1,NDOFNZRO121
50TREAC(IVFIX,IDOFN)=0.0ZRO122
DO 60 ITOTG=1,NTOTGZRO123
EPSTN(ITOTG)=0.0ZRO124
EFFST(ITOTG)=0.0ZRO125
DO 60 ISTR1=1,NSTR1ZRO126
60STRSG(ISTR1,ITOTG)=0.0ZRO127
RETURNZRO128
ENDZRO129
+ +# 7.8.3 Subroutine INVAR + +The role of this subroutine is to evaluate the various functions of stress used to indicate either initiation of or continuing plastic deformation for the four yield criteria considered in this text. More explicitly we need to calculate the items listed in Table 7.2. + +Table 7.2 Effective stress and uniaxial yield stress levels for the yield criteria included in the elasto-plastic computer code. + +
Equation No.Yield criterionStress level (effective stress)Uniaxial (or equivalent yield stress)
(7.63)Tresca $2(J_2')^{1/2} \cos \theta$ $\sigma_Y$
(7.64)Von Mises $\sqrt{3} (J_2')^{1/2}$ $\sigma_Y$
(7.65)Mohr–Coulomb $\frac{1}{3} J_1 \sin \phi + (J_2')^{1/2} \times (\cos \theta - \sin \theta \sin \phi/\sqrt{3})$ $c \cos \phi$
(7.66)Drucker–Prager $a J_1 + (J_2')^{1/2}$ $k'$
+ +Whether or not plastic deformation takes place at any point is governed by its stress level as monitored by the functions in the third column of Table 7.2. For plastic flow to occur this stress level must achieve the values given in the final column of Table 7.2. For the Tresca and Von Mises criteria this value is precisely the uniaxial yield stress but for the Mohr–Coulomb and Drucker–Prager criteria it is an equivalent value defined by the stress-independent terms in (7.65) and (7.66) respectively. Note that all the values given in the final column of Table 7.2 can be functions of the hardening parameter, $\kappa$ . + +Subroutine INVAR merely computes the effective or deviatoric stress components and then evaluates the appropriate function in the third column of Table 7.2 depending on the yield criterion being employed. The choice of yield criterion is governed by the parameter NCRIT, input in subroutine INPUT, and the available options are provided below + + + +```txt +NCRIT = 1 Tresca yield criterion +2 Von Mises +3 Mohr-Coulomb +4 Drucker-Prager +``` + +Subroutine INVAR is now presented and descriptive notes provided. + +```csv +SUBROUTINE INVAR(DEVIA,LPROP,MMATS,NCRIT,PROPS,SINT3,STEFF,STEMP,INVR 1 +THETA,VARJ2,YIELD) INVR 2 +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +DIMENSION DEVIA(4),PROPS(MMATS,7),STEMP(4) INVR 9 +ROOT3=1.73205080757 INVR 10 +SMEAN=(STEMP(1)+STEMP(2)+STEMP(4))/3.0 INVR 11 +DEVIA(1)=STEMP(1)-SMEAN INVR 12 +DEVIA(2)=STEMP(2)-SMEAN INVR 13 +DEVIA(3)=STEMP(3) INVR 14 +DEVIA(4)=STEMP(4)-SMEAN INVR 15 +VARJ2=DEVIA(3)*DEVIA(3)+0.5*(DEVIA(1)*DEVIA(1)+DEVIA(2)*DEVIA(2) INVR 16 +.+DEVIA(4)*DEVIA(4)) INVR 17 +VARJ3=DEVIA(4)*(DEVIA(4)*DEVIA(4)-VARJ2) INVR 18 +STEFF=SQRT(VARJ2) INVR 19 +IF(STEFF.EQ.0.0) GO TO 10 INVR 20 +-SINT3=-3.0*ROOT3*VARJ3/(2.0*VARJ2*STEFF) INVR 21 +IF(SINT3.GT.1.0) SINT3=1.0 INVR 22 +GO TO 20 INVR 23 +10 SINT3=0.0 INVR 24 +20 CONTINUE INVR 25 +IF(SINT3.LT.-1.0) SINT3=-1.0 INVR 26 +IF(SINT3.GT.1.0) SINT3=1.0 INVR 27 +THETA=ASIN(SINT3)/3.0 INVR 28 +GO TO (1,2,3,4) NCRIT INVR 29 +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +3 +PHIRA=PROPS(LPROP,7)*0.017453292 INVR 37 +SNPHI=SIN(PHIRA) INVR 38 +YIELD=SMEAN*SNPHI+STEFF*(COS(THETA)-SIN(THETA)*SNPHI/ROOT3) INVR 39 +RETURN INVR 40 +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +1 +YIELD=2.0*COS(THETA)*STEFF INVR 31 +RETURN INVR 32 +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +``` + +INVR 11-15 Compute the deviatoric stresses according to (7.7) with the order of the components being as indicated in (7.72). + +INVR 16–17 Calculate the second deviatoric stress invariant, $J_{2}^{\prime}$ . + +INVR 18 Calculate the third deviatoric stress invariant, $J_{3}'$ . diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_026.md b/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_026.md new file mode 100644 index 00000000..b9f22625 --- /dev/null +++ b/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_026.md @@ -0,0 +1,594 @@ + + +INVR 19 Compute, $(J_{2}^{\prime})^{\dagger}$ . + +INVR 20-26 Evaluate sin 3θ according to (7.61). + +INVR 27 Then compute, $\theta$ . Note that the principal value is obtained as required in Section 7.4. + +INVR 28 Branch according to the yield criterion being employed. + +INVR 30 Evaluate the yield function in Column 3, Table 7.2 for the Tresca criterion. + +INVR 33 Evaluate the yield function in Column 3, Table 7.2 for the Von Mises criterion. + +INVR 36–38 Evaluate the yield function in Column 3, Table 7.2 for the Mohr–Coulomb criterion. + +INVR 41-43 Evaluate the yield function in Column 3, Table 7.2 for the Drucker-Prager criterion. + +# 7.8.4.1 Subroutine YIELDF + +The function of this subroutine is to determine the flow vector a defined in (7.74). Vector a is given by (7.69) where $C_{1}$ , $C_{2}$ and $C_{3}$ are given in Table 7.1 for the various yield criteria considered and the vectors $a_{1}$ , $a_{2}$ and $a_{3}$ are given by (7.75) for two dimensional applications. For the Tresca and Mohr–Coulomb yield surfaces which have singular points at $\theta = \pm30^{\circ}$ the alternative values of $C_{1}$ , $C_{2}$ and $C_{3}$ given respectively in (7.80) and (7.82) must be employed. + +Subroutine YIELDF is now presented and described. + +```csv +SUBROUTINE YIELDF(AVECT,DEVIA,LPROP,MMATS,NCRIT,NSTR1, + PROPS,SINT3,STEFF,THETA,VARJ2) +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +``` + + + +```csv +C +C*** CALCULATE VECTOR A3 +C +VECA3(1)=DEVIA(2)*DEVIA(4)+VARJ2/3.0 +VECA3(2)=DEVIA(1)*DEVIA(4)+VARJ2/3.0 +VECA3(3)=-2.0*DEVIA(3)*DEVIA(4) +VECA3(4)=DEVIA(1)*DEVIA(2)-DEVIA(3)*DEVIA(3)+VARJ2/3.0 +GO TO (1,2,3,4) NCRIT +C +C*** TRESCA +C +1 CONS1=0.0 +ABTHE=ABS(THETA*57.29577951308) +IF(ABTHE.LT.29.0) GO TO 20 +CONS2=ROOT3 +CONS3=0.0 +GO TO 40 +20 CONS2=2.0*(COSTH+SINTH*TANT3) +CONS3=ROOT3*SINTH/(VARJ2*COST3) +GO TO 40 +C +C*** VON MISES +C +2 CONS1=0.0 +CONS2=ROOT3 +CONS3=0.0 +GO TO 40 +C +C*** MOHR-COULOMB +C +3 CONS1=SIN(FRICT*0.017453292)/3.0 +ABTHE=ABS(THETA*57.29577951308) +IF(ABTHE.LT.29.0) GO TO 30 +CONS3=0.0 +PLUMI=1.0 +IF(THETA.GT.0.0) PLUMI=-1.0 +CONS2=0.5*(ROOT3+PLUMI*CONS1*ROOT3) +GO TO 40 +30 CONS2=COSTH*((1.0+TANTH*TANT3)+CONS1*(TANT3-TANTH)*ROOT3) +CONS3=(ROOT3*SINTH+3.0*CONS1*COSTH)/(2.0*VARJ2*COST3) +GO TO 40 +C +C*** DRUCKER-PRAGER +C +4 SNPHI=SIN(FRICT*0.017453292) +CONS1=2.0*SNPHI/(ROOT3*(3.0-SNPHI)) +CONS2=1.0 +CONS3=0.0 +40 CONTINUE +DO 50 ISTR1=1,NSTR1 +50 AVECT(ISTR1)=CONS1*VECA1(ISTR1)+CONS2*VECA2(ISTR1)+CONS3* +. VECA3(ISTR1) +RETURN +END +YLDF 31 +YLDF 32 +YLDF 33 +YLDF 34 +YLDF 35 +YLDF 36 +YLDF 37 +YLDF 38 +YLDF 39 +YLDF 40 +YLDF 41 +YLDF 42 +YLDF 43 +YLDF 44 +YLDF 45 +YLDF 46 +YLDF 47 +YLDF 48 +YLDF 49 +YLDF 50 +YLDF 51 +YLDF 52 +YLDF 53 +YLDF 54 +YLDF 55 +YLDF 56 +YLDF 57 +YLDF 58 +YLDF 59 +YLDF 60 +YLDF 61 +YLDF 62 +YLDF 63 +YLDF 64 +YLDF 65 +YLDF 66 +YLDF 67 +YLDF 68 +YLDF 69 +YLDF 70 +YLDF 71 +YLDF 72 +YLDF 73 +YLDF 74 +YLDF 75 +YLDF 76 +YLDF 77 +YLDF 78 +YLDF 79 +YLDF 80 +YLDF 81 +YLDF 82 +YLDF 83 +YLDF 84 +``` + +YLDF 10 For the (unlikely) case of a Gauss point with zero stress (identified by $J_{2}^{\prime}=J_{3}^{\prime}=0$ ) avoid evaluation of the flow vector. + +YLDF 11 Identify FRICT as the friction angle $\phi$ for Mohr-Coulomb and Drucker-Prager materials. + + + +YLDF 12-13 Evaluate $\tan \theta$ and $\tan 3\theta$ . + +YLDF 14-16 Evaluate $\sin \theta$ , $\cos \theta$ and $\cos 3\theta$ . + +YLDF 17 Compute $\sqrt{(3)}$ . + +YLDF 21-24 Evaluate $a_1$ according to (7.75). + +YLDF 28–30 Evaluate $a_{2}$ according to (7.75). Note that STEFF and DEVIA are transferred via the argument list from subroutine INVAR. + +YLDF 34-37 Evaluate $a_{3}$ according to (7.75). + +YLDF 38 Branch according to the yield criterion being employed. + +YLDF 41–49 Compute the constants $C_{1}$ , $C_{2}$ and $C_{3}$ for a Tresca material according to Table 7.1. In the vicinity of a singular point, identified by $|\theta|>29.0^{\circ}$ evaluate $C_{1}$ , $C_{2}$ and $C_{3}$ according to (7.80). + +YLDF 53–55 Compute $C_{1}$ , $C_{2}$ and $C_{3}$ for a Von Mises material according to Table 7.1. + +YLDF 61–67 Compute $C_{1}$ , $C_{2}$ and $C_{3}$ for the Mohr–Coulomb criterion. In the vicinity of a singular point defined by $|\theta|>29.0^{\circ}$ evaluate $C_{1}$ , $C_{2}$ and $C_{3}$ according to (7.82). + +YLDF 75–78 Calculate $C_{1}$ , $C_{2}$ and $C_{3}$ for the Drucker–Prager yield criterion. + +YLDF 80–82 Evaluate a according to (7.69). + +# 7.8.4.2 Subroutine FLOWPL + +The main purpose of this subroutine is to determine the vector $d_{D}$ according to either (7.77) or (7.78) depending on the type of analysis being undertaken. In the program presented in this chapter only a linear form of strain hardening is explicitly considered, with the coding of alternative models being left as an exercise for the reader. In this case the term $H'$ in (7.37) becomes a constant and is specified as a material property. + +Subroutine FLOWPL is now listed and described. + +
SUBROUTINE FLOWPL(AVECT,ABETA,DVECT,NTYPE,PROPS,LPROP,NSTR1,MMATS)FLPL1
C***************FLPL 2
CFLPL 3
C**** THIS SUBROUTINE EVALUATES THE PLASTIC D VECTORFLPL 4
CFLPL 5
C***************FLPL 6
DIMENSION AVECT(4),DVECT(4),PROPS(MMATS,7)FLPL 7
YOUNG=PROPS(LPROP,1)FLPL 8
POISS=PROPS(LPROP,2)FLPL 9
HARDS=PROPS(LPROP,6)FLPL 10
FMUL1=YOUNG/(1.0+POISS)FLPL 11
IF(NTYPE.EQ.1) GO TO 60FLPL 12
FMUL2=YOUNG*POISS*(AVECT(1)+AVECT(2)+AVECT(4))/(1.0+POISS)*FLPL 13
.(1.0-2.0*POISS))FLPL 14
DVECT(1)=FMUL1*AVECT(1)+FMUL2FLPL 15
DVECT(2)=FMUL1*AVECT(2)+FMUL2FLPL 16
DVECT(3)=0.5*AVECT(3)*YOUNG/(1.0+POISS)FLPL 17
DVECT(4)=FMUL1*AVECT(4)+FMUL2FLPL 18
GO TO 70FLPL 19
+ + + +
60 FMUL3=YOUNG*POISS*(AVECT(1)+AVECT(2))/(1.0-POISS*POISS)FLPL20
DVECT(1)=FMUL1*AVECT(1)+FMUL3FLPL21
DVECT(2)=FMUL1*AVECT(2)+FMUL3FLPL22
DVECT(3)=0.5*AVECT(3)*YOUNG/(1.0+POISS)FLPL23
DVECT(4)=FMUL1*AVECT(4)+FMUL3FLPL24
70 DENOM=HARDSFLPL25
DO 80 ISTR1=1,NSTR1FLPL26
80 DENOM=DENOM+AVECT(ISTR1)*DVECT(ISTR1)FLPL27
ABETA=1.0/DENOMFLPL28
RETURNFLPL29
ENDFLPL30
+ +FLPL 8 Identify YOUNG as the elastic modulus, E. + +FLPL 9 Identify POISS as the Poisson's ratio, $\nu$ . + +FLPL 10 Identify HARDS as $H'$ for linear strain hardening. + +FLPL 13-18 Evaluate $d_D$ according to (7.77) for plane strain and axisymmetric situations. + +FLPL 20-24 Evaluate $d_D$ according to (7.78) for plane stress problems. + +FLPL 26-28 Compute $1 / (H' + d_D^T a)$ for later evaluation of the elastoplastic matrix $D_{ep}$ in (7.47). + +# 7.8.5 Subroutine STIFFP + +This subroutine evaluates the stiffness matrix for each element in turn and differs from the linear elastic version, described in Section 6.3.2, only in that the elasticity matrix D is replaced (for the tangential stiffness approach at least) by the elasto-plastic matrix $D_{ep}$ defined in (7.47). This subroutine is called only when the element stiffnesses are to be reformulated as controlled by variable KRESL defined in subroutine ALGOR. Obviously the element stiffnesses must be calculated for the first iteration of the first load increment and elastic behaviour must be assumed. Every other time this subroutine is accessed the stiffnesses are to be recalculated to account for any plastic deformation of the material and consequently the $D_{ep}$ matrix must be employed. Apart from this change the element stiffness formulation process is identical to that for elastic materials as described in Section 6.3.2. + +Subroutine STIFFP will now be described and explanatory notes provided. + +
SUBROUTINE STIFFP(COORD,EPSTN,IINCS,LNODS,MATNO,MEVAB,MMATS,STFP1
. MPOIN,MTOTV,NELEM,NEVAB,NGAUS,NNODE,NSTRE,STFP2
. NSTR1,POSGP,PROPS,WEIGP,MELEM,MTOTG,STFP3
. STRSG,NTYPE,NCRIT)STFP4
C**********STFP5
CSTFP6
C**** THIS SUBROUTINE EVALUATES THE STIFFNESS MATRIX FOR EACH ELEMENTSTFP7
C IN TURNSTFP8
CSTFP9
C**********STFP10
DIMENSION BMATX(4,18),CARTD(2,9),COORD(MPOIN,2),DBMAT(4,18),STFP11
. DERIV(2,9),DEVIA(4),DMATX(4,4),STFP12
. ELCOD(2,9),EPSTN(MTOTG),ESTIF(18,18),LNODS(MELEM,9),STFP13
. MATNO(MELEM),POSGP(4),PROPS(MMATS,7),SHAPE(9),STFP14
. WEIGP(4),STRES(4),STRSG(4,MTOTG),STFP15
. DVECT(4),AVECT(4),GPCOD(2,9)STFP16
TWOPI=6.283185308STFP17
REWIND 1STFP18
+ + + +```csv +KGAUS=0 +STFP 19 +C +C*** LOOP OVER EACH ELEMENT +STFP 20 +C +DO 70 IELEM=1,NELEM +STFP 21 +LPROP=MATNO(IELEM) +STFP 22 +STFP 23 +STFP 24 +C +C*** EVALUATE THE COORDINATES OF THE ELEMENT NODAL POINTS +STFP 25 +C +DO 10 INODE=1,NNODE +STFP 26 +LNODE=IABS(LNODS(IELEM,INODE)) +STFP 27 +IPOSN=(LNODE-1)*2 +STFP 28 +DO 10 IDIME=1,2 +STFP 29 +IPOSN=IPOSN+1 +STFP 30 +STFP 31 +STFP 32 +10 ELCOD(IDIME,INODE)=COORD(LNODE,IDIME) +STFP 33 +THICK=PROPS(LPROP,3) +STFP 34 +C +C*** INITIALIZE THE ELEMENT STIFFNESS MATRIX +STFP 35 +C +DO 20 IEVAB=1,NEVAB +STFP 36 +DO 20 JEVAB=1,NEVAB +STFP 37 +20 ESTIF(IEVAB,JEVAB)=0.0 +STFP 38 +KGASP=0 +STFP 39 +KGASP=0 +STFP 40 +STFP 41 +C +C*** ENTER LOOPS FOR AREA NUMERICAL INTEGRATION +STFP 42 +C +DO 50 IGAUS=1,NGAUS +STFP 43 +EXISP=POSGP(IGAUS) +STFP 44 +DO 50 JGAUS=1,NGAUS +STFP 45 +ETASP=POSGP(JGAUS) +STFP 46 +KGASP=KGASP+1 +STFP 47 +KGASP=KGASP+1 +STFP 48 +STFP 49 +STFP 50 +C +C*** EVALUATE THE D-MATRIX +STFP 51 +C +CALL MODPS(DMATX,LPROP,MMATS,NTYPE,PROPS) +STFP 52 +C +C*** EVALUATE THE SHAPE FUNCTIONS,ELEMENTAL VOLUME,ETC. +STFP 53 +C +CALL SFR2(DERIV,ETASP,EXISP,NNODE,SHAPE) +STFP 54 +CALL JACOB2(CARTD,DERIV,DJACB,ELCOD,GPCOD,IELEM,KGASP, +NNODE,SHAPE) +STFP 55 +DVOLU=DJACB*WEIGP(IGAUS)*WEIGP(JGAUS) +STFP 56 +IF(NTYPE.EQ.3) DVOLU=DVOLU*TWOPI*GPCOD(1,KGASP) +STFP 57 +IF(THICK.NE.0.0) DVOLU=DVOLU*THICK +STFP 58 +C +C*** EVALUATE THE B AND DB MATRICES +STFP 59 +STFP 60 +STFP 61 +STFP 62 +STFP 63 +C +C*** EVALUATE THE B AND DB MATRICES +STFP 64 +C +CALL BMATPS(BMATX,CARTD,NNODE,SHAPE,GPCOD,NTYPE,KGASP) +STFP 65 +IF(IINCS.EQ.1) GO TO 80 +STFP 66 +IF(EPSTN(KGAUS),EQ.0.0) GO TO 80 +STFP 67 +DO 90 ISTR1=1,NSTR1 +STFP 68 +STFP 69 +STFP 70 +90 STRES(ISTR1)=STRSG(ISTR1,KGAUS) +STFP 71 +CALL INVAR(DEVIA,LPROP,MMATS,NCRIT,PROPS,SINT3,STEFF,STRES, +THETA,VARJ2,YIELD) +STFP 72 +CALL YIELDF(AVECT,DEVIA,LPROP,MMATS,NCRIT,NSTR1, +PROPS,SINT3,STEFF,THETA,VARJ2) +STFP 73 +CALL FLOWPL(AVECT,ABETA,DVECT,NTYPE,PROPS,LPROP,NSTR1,MMATS) +STFP 74 +DO 100 ISTRE=1,NSTRE +STFP 75 +DO 100 JSTRE=1,NSTRE +STFP 76 +STFP 77 +100 DMATX(ISTRE,JSTRE)=DMATX(ISTRE,JSTRE)-ABETA*DVECT(ISTRE) +STFP 78 +DVECT(JSTRE) +STFP 79 +STFP 80 +80 CONTINUE +STFP 81 +CALL DBE(BMATX,DBMAT,DMATX,MEVAB,NEVAB,NSTRE,NSTR1) +STFP 82 +``` + + + +
CSTFP83
C*** CALCULATE THE ELEMENT STIFFNESSESSTFP84
CSTFP85
DO 30 IEVAB=1,NEVABSTFP86
DO 30 JEVAB=IEVAB,NEVABSTFP87
DO 30 ISTRE=1,NSTRESTFP88
30 ESTIF(IEVAB,JEVAB)=ESTIF(IEVAB,JEVAB)+BMATX(ISTRE,IEVAB)*STFP89
. DBMAT(ISTRE,JEVAB)*DVOLUSTFP90
50 CONTINUESTFP91
CSTFP92
C*** CONSTRUCT THE LOWER TRIANGLE OF THE STIFFNESS MATRIXSTFP93
CSTFP94
DO 60 IEVAB=1,NEVABSTFP95
DO 60 JEVAB=1,NEVABSTFP96
60 ESTIF(JEVAB,IEVAB)=ESTIF(IEVAB,JEVAB)STFP97
CSTFP98
C*** STORE THE STIFFNESS MATRIX,STRESS MATRIX AND SAMPLING POINTSTFP99
C COORDINATES FOR EACH ELEMENT ON DISC FILESTFP100
CSTFP101
WRITE(1) ESTIFSTFP102
70 CONTINUESTFP103
RETURNSTFP104
ENDSTFP105
+ +STFP 17 Compute the value of $2\pi$ . + +STFP 18 Rewind the disc file on which the element stiffness matrices will be stored in turn. + +STFP 19 Set to zero the counter which indicates the overall Gauss point location. So KGAUS ranges from 1 to NGAUS\*NGAUS\*NELEM. + +STFP 23 Enter the loop over each element in the structure. + +STFP 24 Identify the material property type of the current element. + +STFP 28-33 Store the element nodal coordinates in the local array ELCOD for convenient use later. + +STFP 34 Identify the element thickness. + +STFP 38–40 Zero the element stiffness array. + +STFP 41 Set to zero the element Gauss point counter. So KGASP ranges from 1 to NGAUS\*NGAUS. + +STFP 45–48 Enter the numerical integration loops and locate the position $(\xi, \eta)$ of the current point. + +STFP 49–50 Increment the local and global Gauss point counters. + +STFP 54 Call subroutine MODPS to evaluate the elasticity matrix, D. + +STFP 58 Evaluate the shape functions $N_{i}$ and the derivatives $\partial N_{i}/\partial\xi$ , $\partial N_{i}/\partial\eta$ for the current Gauss point. + +STFP 59–60 Evaluate the Gauss point coordinates, GPCOD(IDIME, KGASP), the determinant of the Jacobian matrix, $|J|$ and the Cartesian derivatives of the shape functions $\partial N_{i}/\partial x$ , $\partial N_{i}/\partial y$ (or $\partial N_{i}/\partial r$ , $\partial N_{i}/\partial z$ for axisymmetric problems). + +STFP 61–63 Calculate the elemental volume for numerical integration as $|J|W_{\xi}W_{\eta}$ taking care to multiply by the appropriate thickness or by $2\pi r$ for axisymmetric problems. Note that if a zero thickness is specified it is automatically taken to be unity. + + + +STFP 67 Evaluate the B matrix. + +STFP 68 For the first time avoid the replacement of D by $D_{ep}$ , as defined in (7.47). + +STFP 69 Also for Gauss points at which the behaviour is elastic avoid the replacement of D by $D_{ep}$ . + +STFP 70-71 Store the total current stresses in the array STRES. + +STFP 72-76 Call subroutines INVAR, YIELDF and FLOWPL to evaluate the vectors $a$ , (AVECT) and $d_{D}$ , (DVECT) and ABETA = $1/(H' + d_{D}^{T}a)$ . + +STFP 77-80 Evaluate $D_{ep}$ according to (7.47). + +STFP 82 Evaluate $D_{ep}B$ . + +STFP 86-90 Compute the upper triangle of the element stiffness matrix as + +$$ +\int_ {\Omega} \boldsymbol {B} ^ {T} \boldsymbol {D} _ {e p} \boldsymbol {B} d \Omega +$$ + +STFP 91 End of loop for numerical integration. + +STFP 95–97 Complete the lower triangle of the element stiffness matrix by symmetry. + +STFP 102 Store the element stiffness matrix on disc file 1. + +STFP 103 Return to process the next element. + +# 7.8.6 Subroutine LINEAR + +The purpose of this subroutine is merely to determine the stresses from given displacements assuming linear elastic behaviour. This subroutine is employed in the residual force calculation to be described in the next section. The element displacement components, ELDIS(IDOFN, INODE) are entered into the subroutine, the strain components at the Gauss point under consideration, STRAN(ISTR1) calculated and finally the stress components are evaluated and stored in STRES(ISTR1). + +The subroutine is now listed and described. +```csv +SUBROUTINE LINEAR(CARTD,DMATX,ELDIS,LPROP,MMATS,NDOFN,NNODE,NSTRE,LINR 1 +NTYPE,PROPS,STRAN,STRES,KGASP,GPCOD,SHAPE) LINR 2 +C********** LINR 3 +C LINR 4 +C**** THIS SUBROUTINE EVALUATES STRESSES AND STRAINS ASSUMING LINEAR LINR 5 +C ELASTIC BEHAVIOUR LINR 6 +C LINR 7 +C********** LINR 8 +DIMENSION AGASH(2,2),CARTD(2,9),DMATX(4,4),ELDIS(2,9), LINR 9 +PROPS(MMATS,7),STRAN(4),STRES(4), LINR 10 +GPCOD(2,9),SHAPE(9) LINR 11 +POISS=PROPS(LPROP,2) LINR 12 +DO 20 IDOFN=1,NDOFN LINR 13 +DO 20 JDOFN=1,NDOFN LINR 14 +BGASH=0.0 LINR 15 +DO 10 INODE=1,NNODE LINR 16 +``` + + + +10 BGASH=BGASH+CARTD(JDOFN,INODE)*ELDIS(IDOFN,INODE) LINR 17 +20 AGASH(IDOFN,JDOFN)=BGASH LINR 18 +C LINR 19 +C*** CALCULATE THE STRAINS LINR 20 +C LINR 21 +STRAN(1)=AGASH(1,1) LINR 22 +STRAN(2)=AGASH(2,2) LINR 23 +STRAN(3)=AGASH(1,2)+AGASH(2,1) LINR 24 +STRAN(4)=0.0 LINR 25 +DO 30 INODE=1,NNODE LINR 26 +30 STRAN(4)=STRAN(4)+ELDIS(1,INODE)*SHAPE(INODE)/GPCOD(1,KGASP) LINR 27 +C LINR 28 +C*** AND THE CORRESPONDING STRESSES LINR 29 +C LINR 30 +DO 40 ISTRE=1,NSTRE LINR 31 +STRES(ISTRE)=0.0 LINR 32 +DO 40 JSTRE=1,NSTRE LINR 33 +40 STRES(ISTRE)=STRES(ISTRE)+DMATX(ISTRE,JSTRE)*STRAN(JSTRE) LINR 34 +IF(NTYPE.EQ.1) STRES(4)=0.0 LINR 35 +IF(NTYPE.EQ.2) STRES(4)=POISS*(STRES(1)+STRES(2)) LINR 36 +RETURN LINR 37 +END LINR 38 + +LINR 12 Identify POISS as the Poisson's ratio of the element material. + +LINR 13-18 Calculate the Cartesian derivatives of the Gauss point displacement components $\partial u / \partial x$ , $\partial u / \partial y$ , $\partial v / \partial x$ , $\partial v / \partial y$ . + +LINR 22-27 Evaluate the strain components at the Gauss point according to + +$$ +\epsilon = \left\{ \begin{array}{l} \epsilon_ {x} \\ \epsilon_ {y} \\ \gamma_ {x y} \\ \epsilon_ {z}. \end{array} \right\} = \left\{ \begin{array}{c} \frac {\partial u}{\partial x} \\ \frac {\partial v}{\partial y} \\ \frac {\partial u}{\partial y} + \frac {\partial v}{\partial x} \\ 0 \end{array} \right\} \text { for plane problems }, +$$ + +$$ +\epsilon = \left\{ \begin{array}{l} \epsilon_ {r} \\ \epsilon_ {z} \\ \gamma_ {r z} \\ \epsilon_ {\theta} \end{array} \right\} = \left\{ \begin{array}{c} \frac {\partial u}{\partial r} \\ \frac {\partial w}{\partial z} \\ \frac {\partial u}{\partial z} + \frac {\partial w}{\partial r} \\ \frac {u}{r} \end{array} \right\} \text { for axisymmetric problems. } +$$ + +LINR 31-34 Calculate the stress components, assuming elastic behaviour, according to $\sigma = D\epsilon$ . + + + +LINR 35-36 For a plane stress problem set $\sigma_z = 0$ and set $\sigma_z = \nu(\sigma_x + \sigma_y)$ for plane strain situations. + +# 7.8.7 Subroutine RESIDU + +The function of this subroutine is to evaluate the nodal forces which are statically equivalent to the stress field satisfying elasto-plastic conditions. Comparison of these equivalent nodal forces with the applied loads gives the residual forces, according to (2.4), and this operation is carried out in subroutine CONVER. Therefore RESIDU performs the same task for two-dimensional continua as subroutine REFOR3 undertook for uniaxial situations, and the reader is urged to review Section 3.12.2 before proceeding further. The logic applied in this subroutine is almost identical to that applied in Section 3.12.2. Below we reproduce the essential steps in an abbreviated form and expand only the steps which pertain to the case of two dimensional solids. + +During the application of an increment of load an element, or part of an element, may yield. All stress and strain quantities are monitored at each Gaussian integration point and therefore we can determine whether or not plastic deformation has occurred at such points. Consequently an element can behave partly elastically and partly elasto-plastically if some, but not all, Gauss points indicate plastic yielding. For any load increment it is necessary to determine what proportion is elastic and which part produces plastic deformation and then adjust the stress and strain terms until the yield criterion and the constitutive laws are satisfied. The procedure adopted is as follows. + +Step a. The applied loads for the $r^{\text{th}}$ iteration are the residual forces $\psi^{r-1}$ , given by (2.4) which give rise to displacement increments $dd^{r}$ , according to (2.12), and strain increments $d\epsilon^{r}$ . + +Step b. Compute the incremental stress changes, $d\sigma_{e}^{r}$ as $d\sigma_{e}^{r} = Dd\varepsilon^{r}$ where the subscript e denotes that we are assuming elastic behaviour. + +Step c. Accumulate the total stress for each element Gauss point as $\sigma_{e}^{r} = \sigma^{r-1} + d\sigma_{e}^{r}$ where $\sigma^{r-1}$ are the converged stresses for iteration r-1. + +Step d. The next step depends on whether or not yielding took place at the Gauss point during the $(r-1)^{\text{th}}$ iteration. Therefore we check if $\bar{\sigma}^{r-1} > \sigma_{Y} = \sigma_{Y}^{\circ} + H' \bar{\epsilon}_{p}^{r-1}$ , where $\bar{\sigma}^{r-1}$ is the effective stress given by Column 3, Table 7.2, $\sigma_{Y}$ is the uniaxial yield stress, (Column 4, Table 7.2), $H'$ is the linear strain hardening parameter and $\bar{\epsilon}_{p}^{r-1}$ is the effective plastic strain existing at the end of the $(r-1)^{\text{th}}$ iteration. This expression is identical to the uniaxial case, Section 3.12.2, with all quantities replaced by the effective or equivalent values. If the answer is: + + + +# YES + +The Gauss point had previously yielded. Now check to see if $\bar{\sigma}_{e}^{r} > \bar{\sigma}^{r-1}$ where $\bar{\sigma}_{e}^{r}$ is the effective stress, Col. 3, Table 7.2 based on stresses $\sigma_{e}^{r}$ . If the answer is: + +NO + +YES + +The Gauss point is unloading elastically and therefore go directly to Step g. + +The Gauss point had yielded previously and the stress is still increasing. Therefore all the excess stress $\sigma_{e}^{r}-\sigma^{r-1}$ must be reduced to the yield surface as indicated in + +Fig. 7.10(a). Therefore the factor R which defines the portion of stress which must be modified to satisfy the yield criterion is equal to 1. + +# NO + +Which implies that the Gauss point had not previously yielded. Now check to see if $\bar{\sigma}_{e}^{r} > \sigma_{Y}^{0}$ . If the answer is: + +NO + +YES + +The Gauss point is still elastic and therefore go directly to Step g. + +The Gauss point has yielded during application of load corresponding to this iteration as shown in + +Fig. 7.10(b). The portion of the stress greater than the yield value must be reduced to the yield surface. The reduction factor R is given from + +Fig. 7.10(b) to be + +$$ +R = \frac {A B}{A C} = \frac {\bar {\sigma} _ {e} ^ {r} - \sigma_ {Y}}{\bar {\sigma} _ {e} ^ {r} - \bar {\sigma} ^ {r - 1}}. +$$ + +![](images/page-260_3115adacbb056a4c2f3c30be94f5c9db6bebf2cbbec0f7ed62db8d1087a57621.jpg) + +
+text_image + +F=0 +r dσP^r = dσe^r = Ddε^r +Ddλa = dλd_D +σ^r-1 +σ_2 +σ_1 +σ_3 +σ^r = σ^r \left( \frac{\sigma_1^0 + H' \bar{\epsilon}_p^r}{\bar{\sigma}^r} \right) +σ_3 +σ^r +σ^r +Ddλa = Ddε_p^r +
+ +Fig. 7.10(a) Incremental stress changes in an already yielded point in an elastoplastic continuum. diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_027.md b/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_027.md new file mode 100644 index 00000000..affba4b5 --- /dev/null +++ b/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_027.md @@ -0,0 +1,573 @@ + + +![](images/page-261_6df8063e48ad99ead0dbb23c781c567da2d9b64534aa06296e01aac3cbbc0f19.jpg) + +
+text_image + +F=0 +dσe^r=Ddε^r +(1-r)dσe^r +rdσe^r +σ2 +σr-1 +C +B +Ddλa= +dλd_D +σε^r +Ddλa=Ddε_p^r +σr +σ3 +σ1 +σr=σr(σγ^0+H'ε_p^r)/σ̃^r +D' +σr +
+ +Fig. 7.10(b) Incremental stress changes at a point in an elasto-plastic continuum at initial yield. + +Step e. For yielded Gauss points only compute the portion of the total stress which satisfies the yield criterion as $\sigma^{r-1} + (1 - R)d\sigma_{e}^{r}$ . + +Step f. The remaining portion of stress, $Rd\sigma_{e}^{r}$ must be effectively eliminated in some way. The point A must be brought onto the yield surface by allowing plastic deformation to occur. Physically this can be described as follows. On loading from point C, the stress point moves elastically until the yield surface is met at B. Elastic behaviour beyond this point would result in a final stress state defined by point A. However in order to satisfy the yield criterion, the stress point cannot move outside the yield surface and consequently the stress point can only traverse the surface until both equilibrium conditions and the constitutive relation are satisfied. From (7.45), (7.46) and (7.47) we have + +$$ +d \sigma^ {r} = D d \epsilon^ {r} - d \lambda d _ {D}, \tag {7.91} +$$ + +or + +$$ +\sigma^ {r} = \sigma^ {r - 1} + d \sigma_ {e} ^ {r} - d \lambda d _ {D}, \tag {7.92} +$$ + +which gives the total stresses $\sigma^{r}$ satisfying elasto-plastic conditions when the stresses are incremented from $\sigma^{r-1}$ . Expression (7.92) is illustrated vectorially in Fig. 7.10 and the reader should note the similarity to Fig. 3.7(a). It is seen that if a finite sized stress increment is taken, the final stress point D, corresponding to $\sigma^{r}$ , may depart from the yield surface. This discrepancy can be practically eliminated + + + +by ensuring that the load increments considered in solution are sufficiently small. However the point D can be reduced to the yield surface by simply scaling the vector $\sigma^{r}$ . Denoting the effective stress, given by Col. 3, Table 7.2, due to stress $\sigma^{r}$ as $\bar{\sigma}^{r}$ and noting that this value should coincide with $\sigma_{Y} = \sigma_{Y}^{\circ} + H' \bar{\epsilon}_{p}^{r}$ if the point D lies on the yield surface, the appropriate scaling factor is readily seen to be + +$$ +\sigma^ {r} = \sigma^ {r} \left(\frac {\sigma_ {Y} ^ {0} + H ^ {\prime} \bar {\epsilon} _ {p} ^ {r}}{\bar {\sigma} ^ {r}}\right). \tag {7.93} +$$ + +This represents a scaling of the vector $\sigma^{r}$ which implies that the individual stress components are proportionally reduced. The normality condition for the plastic strain increment is evident from Fig. 7.10 since $Dd\lambda a = Dd\varepsilon_{p}$ . + +![](images/page-262_3527177768779bc7234199ebd27b665d568c9c1203da8f687ee43c28e87163a3.jpg) + +
+text_image + +σ₂ +σᵣ₋₁ +C +σᵣ = σᵣ \left( \frac{\sigma_{\gamma^{0}} + H' \epsilon_{p}}{\bar{\sigma}^r} \right) +σ₁ +σ₃ +A +B +E' +D' +D +σ₃ +
+ +Fig. 7.11 Refined process for reducing a stress point to the yield surface. + +If relatively large load increment sizes are to be permitted the process described above can lead to an inaccurate prediction of the final point D on the yield surface if the stress point is in the vicinity of a region of large curvature of the yield surface. This is illustrated in Fig. 7.11 where the process of reducing the elastic stress to the yield surface is shown to end in the stress point D which is then scaled down to the yield surface to give point $D'$ . Greater accuracy can be achieved by relaxing the excess stress to the yield surface in several stages.\* Fig. 7.11 shows the case where the excess stress is divided into three equal parts and each increment reduced to the yield surface in turn. After the three reduction cycles to the stress point E the drift away from the yield surface can be corrected by simple scaling to give the final stress point $E'$ . It is seen that the final + +• Alternative procedures for this operation are presented in Refs. 18 and 19 whilst a completely different approach to stress projection is followed in Ref. 20. + + + +points $D'$ and $E'$ can be significantly different. An additional refinement which can be introduced is to scale the stress point to the yield surface after the reduction process for each cycle and not only after the final cycle as shown in Fig. 7.11. Obviously the greater the number of steps into which the excess stress AB is divided, the greater the accuracy. However the computation for each step is relatively expensive since the vectors a and $d_{D}$ have to be calculated at each stage. Clearly a balance must be sought and in this text the following criterion is adopted. The excess stress $Rd\sigma_{e}^{r}$ is divided into m parts where m is given by the nearest integer which is less than + +$$ +\left(\frac {\bar {\sigma} _ {e} ^ {r} - \sigma_ {Y}}{\sigma_ {Y} \cdot^ {0}}\right) 8 \div 1, \tag {7.94} +$$ + +where $\bar{\sigma}_{e}^{r}-\sigma_{Y}$ gives a measure of the excess stress AB and $\sigma_{Y}^{\circ}$ is the initial uniaxial yield stress in Col. 4, Table 7.2 before the onset of work hardening. This criterion can be readily amended by the user. + +Step g. For elastic Gauss points only calculate $\sigma^r$ as $\sigma^r = \sigma^{r-1} + d\sigma_e^r$ . + +Step h. Finally, calculate the equivalent nodal forces from the element stresses according to + +$$ +(f ^ {(e)}) ^ {r} = \int_ {\Omega^ {(e)}} \boldsymbol {B} ^ {T} \boldsymbol {\sigma} ^ {r} d \Omega . \tag {7.95} +$$ + +Subroutine RESIDU is now listed and described. + +SUBROUTINE RESIDU(ASDIS,COORD,EFFST,ELOAD,FACTO,IITER,LNODS, RSDU 1 +. LPROP,MATNO,MELEM,MMATS,MPOIN,MTOTG,MTOTV,NDOFN, RSDU 2 +. NELEM,NEVAB,NGAUS,NNODE,NSTR1,NTYPE,POSGP,PROPS, RSDU 3 +. NSTRE,NCrit,STRSG,WEIGP,TDISP,EPSTN) RSDU 4 +C********** RSDU 5 +C RSDU 6 +C*** THIS SUBROUTINE REDUCES THE STRESSES TO THE YIELD SURFACE AND RSDU 7 +C EVALUATES THE EQUIVALENT NODAL FORCES RSDU 8 +C RSDU 9 +C********** RSDU 10 +DIMENSION ASDIS(MTOTV),AVECT(4),CARTD(2,9),COORD(MPOIN,2), RSDU 11 +. DEVIA(4),DVECT(4),EFFST(MTOTG),ELCOD(2,9),ELDIS(2,9), RSDU 12 +. ELOAD(MELEM,18),LNODS(MELEM,9),POSGP(4),PROPS(MMATS,7), RSDU 13 +. STRAN(4),STRES(4),STRSG(4,MTOTG), RSDU 14 +. WEIGP(4),DLCOD(2,9),DESIG(4),SIGMA(4),SGTOT(4), RSDU 15 +. DMATX(4,4),DERIV(2,9),SHAPE(9),GPCOD(2,9), RSDU 16 +. EPSTN(MTOTG),TDISP(MTOTV),MATNO(MELEM),BMATX(4,18) RSDU 17 +ROOT3=1.73205080757 RSDU 18 +TWOPI=6.283185308 RSDU 19 +DO 10 IELEM=1,NELEM RSDU 20 +DO 10 IEVAB=1,NEVAB RSDU 21 +10 ELOAD(IELEM,IEVAB)=0.0 RSDU 22 +KGAUS=0 RSDU 23 +DO 20 IELEM=1,NELEM RSDU 24 +LPROP=MATNO(IELEM) RSDU 25 +UNIAX=PROPS(LPROP,5) RSDU 26 +HARDS=PROPS(LPROP,6) RSDU 27 + + + +```csv +FRICT=PROPS(LPROP,7) RSDU 28 +IF(NCRIT.EQ.3)UNIAX=PROPS(LPROP,5)*COS(FRICT*0.017453292) RSDU 29 +IF(NCRIT.EQ.4)UNIAX=6.0*PROPS(LPROP,5)*COS(FRICT*0.017453292)/ RSDU 30 +.(ROOT3*(3.0-SIN(FRICT*0.017453292))) RSDU 31 +C RSDU 32 +C*** COMPUTE COORDINATE. AND INCREMENTAL DISPLACEMENTS OF THE RSDU 33 +C ELEMENT NODAL POINTS RSDU 34 +C RSDU 35 +DO 30 INODE =1,NNODE RSDU 36 +LNODE=IABS(LNODS(IELEM,INODE)) RSDU 37 +NPOSN=(LNODE-1)*NDOFN RSDU 38 +DO 30 IDOFN=1,NDOFN RSDU 39 +NPOSN=NPOSN+1 RSDU 40 +ELCOD(IDOFN,INODE)=COORD(LNODE,IDOFN) RSDU 41 +30 ELDIS(IDOFN,INODE)=ASDIS(NPOSN) RSDU 42 +CALL MODPS(DMATX,LPROP,MMATS,NTYPE,PROPS) RSDU 43 +THICK=PROPS(LPROP,3) RSDU 44 +KGASP=0 RSDU 45 +DO 40 IGAUS=1,NGAUS RSDU 46 +DO 40 JGAUS=1,NGAUS RSDU 47 +EXISP=POSGP(IGAUS) RSDU 48 +ETASP=POSGP(JGAUS) RSDU 49 +KGAUS=KGAUS+1 RSDU 50 +KGASP=KGASP+1 RSDU 51 +CALL SFR2(DERIV,ETASP,EXISP,NNODE,SHAPE) RSDU 52 +CALL JACOB2(CARTD,DERIV,DJACB,ELCOD,GPCOD,IELEM,KGASP, RSDU 53 +NNODE,SHAPE) RSDU 54 +DVOLU=DJACB*WEIGP(IGAUS)*WEIGP(JGAUS) RSDU 55 +IF(NTYPE.EQ.3)DVOLU=DVOLU*TWOPI*GPCOD(1,KGASP) RSDU 56 +IF(THICK.NE.0.0)DVOLU=DVOLU*THICK RSDU 57 +CALL BMATPS(BMATX,CARTD,NNODE,SHAPE,GPCOD,NTYPE,KGASP) RSDU 58 +CALL LINEAR(CARTD,DMATX,ELDIS,LPROP,MMATS,NDOFN,NNODE,NSTRE, RSDU 59 +NTYPE,PROPS,STRAN,STRES,KGASP,GPCOD,SHAPE) RSDU 60 +PREYS=UNIAX+EPSTN(KGAUS)*HARDS RSDU 61 +DO 150 ISTR1=1,NSTR1 RSDU 62 +DESIG(ISTR1)=STRES(ISTR1) RSDU 63 +150 SIGMA(ISTR1)=STRSG(ISTR1,KGAUS)+STRES(ISTR1) RSDU 64 +CALL INVAR(DEVIA,LPROP,MMATS,NCRIT,PROPS,SINT3,STEFF,SIGMA, RSDU 65 +THETA,VARJ2,YIELD) RSDU 66 +ESPRE=EFFST(KGAUS)-PREYS RSDU 67 +IF(ESPRE.GE.0.0)GO TO 50 RSDU 68 +ESCUR=YIELD-PREYS RSDU 69 +IF(ESCUR.LE.0.0)GO TO 60 RSDU 70 +RFACT=ESCUR/(YIELD-EFFST(KGAUS)) RSDU 71 +GO TO 70 RSDU 72 +50 ESCUR=YIELD-EFFST(KGAUS) RSDU 73 +IF(ESCUR.LE.0.0)GO TO 60 RSDU 74 +RFACT=1.0 RSDU 75 +70 MSTEP=ESCUR*8.0/UNIAX+1.0 RSDU 76 +ASTEP=MSTEP RSDU 77 +REDUC=1.0-RFACT RSDU 78 +DO 80 ISTR1=1,NSTR1 RSDU 79 +SGTOT(ISTR1)=STRSG(ISTR1,KGAUS)+REDUC*STRES(ISTR1) RSDU 80 +80 STRES(ISTR1)=RFACT*STRES(ISTR1)/ASTEP RSDU 81 +DO 90 ISTEP=1,MSTEP RSDU 82 +CALL INVAR(DEVIA,LPROP,MMATS,NCRIT,PROPS,SINT3,STEFF,SGTOT, RSDU 83 +THETA,VARJ2,YIELD) RSDU 84 +CALL YIELDF(AVECT,DEVIA,LPROP,MMATS,NCRIT,NSTR1, RSDU 85 +PROPS,SINT3,STEFF,THETA,VARJ2) RSDU 86 +CALL FLOWPL(AVECT,ABETA,DVECT,NTYPE,PROPS,LPROP,NSTR1,MMATS) RSDU 87 +AGASH=0.0 RSDU 88 +DO 100 ISTR1=1,NSTR1 RSDU 89 +100 AGASH=AGASH+AVECT(ISTR1)*STRES(ISTR1) RSDU 90 +DLAMD=AGASH*ABETA RSDU 91 +``` + + + +
IF(DLAMD.LT.0.0) DLAMD=0.0RSDU92
BGASH=0.0RSDU93
DO 110 ISTR1=1,NSTR1RSDU94
BGASH=BGASH+AVECT(ISTR1)*SGTOT(ISTR1)RSDU95
110 SGTOT(ISTR1)=SGTOT(ISTR1)+STRES(ISTR1)-DLAMD*DVECT(ISTR1)RSDU96
EPSTN(KGAUS)=EPSTN(KGAUS)+DLAMD*BGASH/YIELDRSDU97
90 CONTINUERSDU98
CALL INVAR(DEVIA,LPROP,MMATS,NCRIT,PROPS,SINT3,STEFF,SGTOT,THETA,VARJ2,YIELD)RSDU99
CURYS=UNIAX+EPSTN(KGAUS)*HARDSRSDU100
BRING=1.0RSDU101
IF(YIELD.GT.CURYS) BRING=CURYS/YIELDRSDU102
DO 130 ISTR1=1,NSTR1RSDU103
130 STRSG(ISTR1,KGAUS)=BRING*SGTOT(ISTR1)RSDU104
EFFST(KGAUS)=BRING*YIELDRSDU105
C*** ALTERNATIVE LOCATION OF STRESS REDUCTION LOOP TERMINATION CARDRSDU106
C 90 CONTINUERSDU107
C***RSDU108
GO TO 190RSDU109
60 DO 180 ISTR1=1,NSTR1RSDU110
180 STRSG(ISTR1,KGAUS)=STRSG(ISTR1,KGAUS)+DESIG(ISTR1)RSDU111
EFFST(KGAUS)=YIELDRSDU112
CRSDU113
C*** CALCULATE THE EQUIVALENT NODAL FORCES AND ASSOCIATE WITH THERSDU114
C ELEMENT NODESRSDU115
190 MGASH=0RSDU116
DO 140 INODE=1,NNODERSDU117
DO 140 IDOFN=1,NDOFNRSDU118
MGASH=MGASH+1RSDU119
DO 140 ISTRE=1,NSTRERSDU120
140 ELOAD(IELEM,MGASH)=ELOAD(IELEM,MGASH)+BMATX(ISTRE,MGASH)*RSDU121
.STRSG(ISTRE,KGAUS)*DVOLURSDU122
40 CONTINUERSDU123
20 CONTINUERSDU124
RETURNRSDU125
ENDRSDU126
RSDU127
+ +RSDU 18–19 + +Compute $\sqrt{(3)}$ and $2\pi$ . + +RSDU 20-22 + +Zero the array in which the equivalent nodal forces, calculated in Step h, will be stored. + +RSDU 23 + +Zero the Gauss point counter over all elements. + +RSDU 24 + +Loop over each element. + +RSDU 25 + +Identify the element material property number. + +RSDU 26-28 + +Identify the initial uniaxial yield stress, $\sigma_{Y}^{\circ}$ (or $c$ for Mohr-Coulomb or Drucker-Prager criteria), the linear strain hardening parameter $H'$ and the friction angle $\phi$ for Mohr-Coulomb and Drucker-Prager materials. + +RSDU 29 + +For a Mohr-Coulomb material evaluate the equivalent yield stress as $c \cos \phi$ . + +RSDU 30-31 + +For a Drucker–Prager material evaluate the equivalent yield stress as $k'$ according to (7.18). + +RSDU 36-42 + +Store the element nodal coordinates in array ELCOD and the nodal displacements due to the application of the residual forces in array ELDIS. + + + +RSDU 43 + +RSDU 44 + +RSDU 45 + +RSDU 46–49 + +RSDU 50-51 + +RSDU 52 + +RSDU 53-54 + +RSDU 55-57 + +RSDU 58 + +RSDU 59–60 + +RSDU 61 + +RSDU 62-64 + +RSDU 65-66 + +RSDU 67-68 + +RSDU 69-70 + +RSDU 71 + +Evaluate the elastic D matrix. + +Identify the element thickness. + +Zero the local Gauss point counter. + +Enter the loops for numerical integration and evaluate the local coordinates $(\xi, \eta)$ at the sampling point. + +Increment the local and global Gauss point counters. + +Evaluate the shape functions $N_{i}$ and their derivatives $\partial N_{i}/\partial\xi$ , $\partial N_{i}/\partial\eta$ . + +Evaluate the Gauss point coordinates GPCOD(IDIME, KGASP), the determinant of the Jacobian matrix $|J|$ and the Cartesian derivatives of the shape functions $\partial N_{i} / \partial_{x}$ , $\partial N_{i} / \partial y$ (or $\partial N_{i} / \partial r$ , $\partial N_{i} / \partial z$ for axisymmetric problems). + +Calculate the elemental volume for numerical integration as $|J|W_{\xi}W_{\eta}$ taking care to multiply by the appropriate thickness or by $2\pi r$ for axisymmetric problems. The default value of the thickness is 1.0. + +Compute the strain matrix $\pmb{B}$ for the Gauss point. + +Compute the stress increment STRES(ISTR1), assuming elastic behaviour as $d\sigma_{e}^{r} = Dd\epsilon^{r}$ . + +Compute the yield stress for the $(r - 1)^{\mathrm{th}}$ iteration as $\sigma_{Y}^{\circ} + H^{\prime}\bar{\epsilon}_{p}^{r - 1}$ . + +Store $d\sigma_{e}^{r}$ as DESIG(ISTR1) and $\sigma_{e}^{r}$ as SIGMA(ISTR1). + +Evaluate the effective stress in Col. 3, Table 7.2 and store as YIELD. + +Check if the Gauss point had yielded on the previous iteration, i.e. if $\bar{\sigma}^{r-1} > \sigma_Y^\circ + H' \bar{\epsilon}_p^{r-1}$ which is the first operation of Step $d$ . + +If the Gauss point was previously elastic, check to see if it has yielded during this iteration. + +For a Gauss point which yields during the iteration calculate + +$$ +R = \frac {\bar {\sigma} _ {e} ^ {r} - \sigma_ {Y}}{\bar {\sigma} _ {e} ^ {r} - \bar {\sigma} ^ {r - 1}}. +$$ + +RSDU 73-74 +RSDU 75 +RSDU 76–77 +RSDU 78 +RSDU 79-81 +Check to see if a Gauss point which had previously yielded is unloading during this iteration. If yes, go to 60. +Otherwise, set $R = 1$ . +Evaluate the number of steps into which the excess stress, $Rd\sigma_{e}^{r}$ is to be divided according to (7.94). +Compute $(1 - R)$ . + +Compute $\sigma^{r-1}+(1-R)d\sigma_{e}^{r}$ according to Step $e$ and store in SGTOT(ISTR1) and evaluate $Rd\sigma_{e}^{r}/m$ and store in STRES(ISTR1). + +RSDU 82 + +RSDU 83-87 + +Loop over each stress reduction step. + +Compute the vectors $a$ and $d_{D}$ . + + + +RSDU 88-92 Compute $d\lambda$ according to (7.45) and store as DLAMD. + +RSDU 93-96 Compute $\sigma^r = \sigma^{r-1} + (1 - R)d\sigma_e^r + Rd\sigma_e^r/m - d\lambda d_D/m$ . When the summation process from 1 to $m$ required in DO LOOP to index 90 is completed this will result in $\sigma^r = \sigma^{r-1} + d\sigma_e^r - d\lambda d_D$ to give the stress point $E$ in Fig. 7.11. + +RSDU 97 Compute the effective plastic strain as follows. From (7.51) we have + +$$ +d \kappa = d \lambda a ^ {T} \sigma = \sigma^ {T} d \epsilon_ {p}, +$$ + +or rewriting the right hand side in terms of the effective stress $\bar{\sigma}$ and effective plastic strain $\bar{\epsilon}_{p}$ we have + +$$ +d \lambda a ^ {T} \sigma = \bar {\sigma} d \bar {\epsilon} _ {p}, +$$ + +and therefore + +$$ +\bar {\epsilon} _ {p} ^ {r} = \bar {\epsilon} _ {p} ^ {r - 1} + \frac {d \lambda \boldsymbol {a} ^ {T} \boldsymbol {\sigma}}{\bar {\sigma}}. \tag {7.96} +$$ + +RSDU 98 Return to loop over the next stress reduction step. This statement is so placed that the final stresses $\sigma^{r}$ are scaled down to lie on the yield surface only after all the reduction steps have been completed. An additional refinement can be introduced where, with reference to Fig. 7.11, the stresses are scaled to the yield surface after each reduction step. Such a refinement is not normally required; however it can be introduced by moving statement RSDU 98 to the position indicated in RSDU 108. + +RSDU 99-100 Compute the effective stress $\bar{\sigma}^r$ . + +RSDU 101 Evaluate $\sigma_{Y}^{\circ} + H^{\prime}\bar{\epsilon}_{p}^{r}$ . + +RSDU 102–105 Factor the stresses $\sigma^{r}$ to ensure that they lie on the yield surface, according to $\sigma^{r} = \sigma^{r}(\sigma_{Y}^{\circ} + H' \bar{\epsilon}_{p}^{r}) / \bar{\sigma}^{r}$ as indicated in Fig. 7.11. + +RSDU 106 Store the effective stress $\bar{\sigma}^{r}$ in array EFFST. + +RSDU 108 Location of end of loop if the refinement indicated in RSDU 98 is to be included. + +RSDU 111-113 For elastic Gauss points compute $\sigma^r$ as $\sigma^{r-1} + d\sigma_e^r$ and store $\bar{\sigma}^r$ in EFFST. + +RSDU 117-123 Compute the equivalent nodal forces as + +$$ +(f ^ {(e)}) ^ {r} = \int_ {\Omega} \boldsymbol {B} ^ {T} \sigma^ {r} d \Omega . +$$ + +RSDU 124–125 Termination of loop for numerical integration and over each element respectively. + + + +# 7.8.8 Subroutine OUTPUT + +This subroutine outputs the results at a frequency determined by the output parameters NOUTP(1) and NOUTP(2) whose role is described in Section 6.5.3. The principal stresses and direction are also calculated in this subroutine and these are given by the following expressions + +$$ +\sigma_ {\max} = \frac {\sigma_ {x} + \sigma_ {y}}{2} + \sqrt {\left(\frac {(\sigma_ {x} - \sigma_ {y}) ^ {2}}{4} + \tau_ {x y} ^ {2}\right)}, +$$ + +$$ +\sigma_ {\min} = \frac {\sigma_ {x} + \sigma_ {y}}{2} - \sqrt {\left(\frac {(\sigma_ {x} - \sigma_ {y}) ^ {2}}{4} + \tau_ {x y} ^ {2}\right)} +$$ + +$$ +\theta = \tan^ {- 1} \left(\frac {2 \tau_ {x y}}{\sigma_ {x} - \sigma_ {y}}\right). \tag {7.97} +$$ + +with x and y being replaced by r and z for the axisymmetric case. The term $\theta$ defines the angle which the maximum principal stress makes with the y (or z) axis; a positive angle being measured anticlockwise. + +This subroutine is largely self-explanatory and is listed below. +```csv +SUBROUTINE OUTPUT(IITER,MTOTG,MTOTV,MVFIX,NELEM,NGAUS,NOFIX, OTPT 1 +. NOUTP,NPOIN,NVFIX,STRSG,TDISP,TREAC,EPSTN, OTPT 2 +. NTYPE,NCHEK) OTPT 3 +C******************************* +C +C**** THIS SUBROUTINE OUTPUTS DISPLACEMENTS.REACTIONS AND STRESSES OTPT 6 +C +C******************************* +DIMENSION NOFIX(MVFIX),NOUTP(2),STRSG(4,MTOTG),STRSP(3), OTPT 9 +. TDISP(MTOTV),TREAC(MVFIX,2),EPSTN(MTOTG) OTPT 10 +KOUTP=NOUTP(1) OTPT 11 +IF(IITER.GT.1) KOUTP=NOUTP(2) OTPT 12 +IF(IITER.EQ.1.AND.NCHEK.EQ.0) KOUTP=NOUTP(2) OTPT 13 +C +C*** OUTPUT DISPLACEMENTS OTPT 15 +C +IF(KOUTP.LT.1) GO TO 10 OTPT 16 +WRITE(6,900) OTPT 17 +900 FORMAT(1HO,5X,13HDISPLACEMENTS) OTPT 18 +IF(NTYPE.NE.3) WRITE(6,950) OTPT 19 +950 FORMAT(1HO,6X,4HNODE,6X,7HX-DISP.,7X,7HY-DISP.) OTPT 20 +IF(NTYPE.EQ.3) WRITE(6,955) OTPT 21 +955 FORMAT(1HO,6X,4HNODE,6X,7HR-DISP.,7X,7HZ-DISP.) OTPT 22 +DO 20 IPOIN=1,NPOIN OTPT 23 +NGASH=IPOIN*2 OTPT 24 +NGISH=NGASH-2+1 OTPT 25 +20 WRITE(6,910) IPOIN,(TDISP(IGASH),IGASH=NGISH,NGASH) OTPT 26 +910 FORMAT(I10,3E14.6) OTPT 27 +10 CONTINUE OTPT 28 +C +C*** OUTPUT REACTIONS OTPT 30 +C +IF(KOUTP.LT.2) GO TO 30 OTPT 31 +WRITE(6,920) OTPT 32 +920 FORMAT(1HO,5X,9HREACTIONS) OTPT 33 +IF(NTYPE.NE.3) WRITE(6,960) OTPT 34 +OTPT 35 +OTPT 36 +``` + + + +960 FORMAT(1HO,6X,4HNODE,6X,7HX-REAC.,7X,7HY-REAC.) OTPT 37 +IF(NTYPE.EQ.3) WRITE(6,965) OTPT 38 +965 FORMAT(1HO,6X,4HNODE,6X,7HR-REAC.,7X,7HZ-REAC.) OTPT 39 +DO 40 IVFIX=1,NVFIX OTPT 40 +40 WRITE(6,910) NOFIX(IVFIX),(TREAC(IVFIX,IDOFN),IDOFN=1,2) OTPT 41 +30 CONTINUE OTPT 42 +C OTPT 43 +C*** OUTPUT STRESSES OTPT 44 +C OTPT 45 +IF(KOUTP.LT.3) GO TO 50 OTPT 46 +IF(NTYPE.NE.3) WRITE(6,970) OTPT 47 +970 FORMAT(1HO,1X,4HG.P.,6X,9HXX-STRESS,5X,9HYY-STRESS,5X,9HXY-STRESS,OTPT 48 +.5X,9HZZ-STRESS,6X,8HMAX P.S.,6X,8HMIN P.S.,3X,5HANGLE,3X, OTPT 49 +.6HE.P.S.) OTPT 50 +IF(NTYPE.EQ.3) WRITE(6,975) OTPT 51 +975 FORMAT(1HO,1X,4HG.P.,6X,9HRR-STRESS,5X,9HZZ-STRESS,5X,9HRZ-STRESS,OTPT 52 +.5X,9HTT-STRESS,6X,8HMAX P.S.,6X,8HMIN P.S.,3X,5HANGLE,3X, OTPT 53 +.6HE.P.S.) OTPT 54 +KGAUS=0 OTPT 55 +DO 60 IELEM=1,NELEM OTPT 56 +KELGS=0 OTPT 57 +WRITE(6,930) IELEM OTPT 58 +930 FORMAT(1HO,5X,13HELEMENT NO. =,I5) OTPT 59 +DO 60 IGAUS=1,NGAUS OTPT 60 +DO 60 JGAUS=1,NGAUS OTPT 61 +KGAUS=KGAUS+1 OTPT 62 +KELGS=KELGS+1 OTPT 63 +XGASH=(STRSG(1,KGAUS)+STRSG(2,KGAUS))*0.5 OTPT 64 +XGISH=(STRSG(1,KGAUS)-STRSG(2,KGAUS))*0.5 OTPT 65 +XGESH=STRSG(3,KGAUS) OTPT 66 +XGOSH=SQRT(XGISH*XGISH+XGESH*XGESH) OTPT 67 +STRSP(1)=XGASH+XGOSH OTPT 68 +STRSP(2)=XGASH-XGOSH OTPT 69 +IF(XGISH.EQ.0.0) XGISH=0.1E-20 OTPT 70 +STRSP(3)=ATAN(XGESH/XGISH)*28.647889757 OTPT 71 +60 WRITE(6,940) KELGS,(STRSG(ISTR1,KGAUS),ISTR1=1,4), OTPT 72 +. (STRSP(ISTRE),ISTRE=1,3),EPSTN(KGAUS) OTPT 73 +940 FORMAT(I5,2X,6E14.6,F8.3,E14.6) OTPT 74 +50 CONTINUE OTPT 75 +RETURN OTPT 76 +END OTPT 77 + +OTPT 11–13 Set the output indicator, KOUTP, according to whether or not this is the first iteration of a load increment or not. If it is the first iteration the results will be output according to NOUTP(1) but for a converged solution the results are output according to NOUTP(2). + +OTPT 17-29 For an output code value of 1 or greater, output the nodal displacements after printing the appropriate headings. + +OTPT 33-42 For an output code of 2 or greater, output appropriate headings and the reactions at restrained nodal points. + +OTPT 46 For an output code of 3 output the Gauss point stresses. + +OTPT 47-54 Write appropriate headings. + +OTPT 56–59 Loop over each element and write the element number. + +OTPT 60–61 Loop over each element Gauss point. + +OTPT 62–71 Evaluate the principal stresses and direction for each Gauss point according to (7.97). + + + +OTPT 72–74 Output the Cartesian stress components, the principal stresses and direction and the total effective plastic strain for each Gauss point. This latter quantity gives an immediate indication whether the Gauss point has yielded or not, since it will be zero for all elastic points. + +# 7.8.9 The main, master or controlling segment + +This segment controls the calling, in order, of the other subroutines and is similar in structure to the segment described in Section 3.8 for one-dimensional situations. Its other function is to control the iterative process and also the incrementing of the applied loads. + +The following channel numbers are employed by the program: 5 (card reader), 6 (line printer), 1, 2, 3, 4, 8 (scratch files). + +This routine is self-explanatory and is presented below without further comment. + +```txt +MASTER PLAST +C********** PROGRAM FOR THE ELASTO-PLASTIC ANALYSIS OF PLANE STRESS, +C PLANE STRAIN AND AXISYMMETRIC SOLIDS +C********** DIMENSION ASDIS(300),COORD(150,2),ELOAD(40,18),ESTIF(18,18), +EQRHS(10),EQUAT(80,10),FIXED(300),GLOAD(80),GSTIF(3240), +IFFIX(300),LNODS(40,9),LOCEL(18),MATNO(40), +NACVA(80),NAMEV(10),NDEST(18),NDFRO(40),NOFIX(30), +NOUTP(2),NPIVO(10), +POSGP(4),PRESC(30,2),PROPS(5,7),RLOAD(40,18), +STFOR(300),TREAC(30,2),VECRV(80),WEIGP(4), +STRSG(4,360),TDISP(300),TLOAD(40,18), +TOFOR(300),EPSTN(360),EFFST(360) +PLAS 6 +PLAS 7 +PLAS 8 +PLAS 9 +PLAS 10 +PLAS 11 +PLAS 12 +PLAS 13 +PLAS 14 +PLAS 15 +C +C*** PRESET VARIABLES ASSOCIATED WITH DYNAMIC DIMENSIONING +C +CALL DIMEN(MBUFA,MELEM,MEVAB,MFRON,MMATS,MPOIN,MSTIF,MTOTG,MTOTV, +MVFIX,NDOFN,NPROP,NSTRE) +PLAS 16 +PLAS 17 +PLAS 18 +PLAS 19 +PLAS 20 +C +C*** CALL THE SUBROUTINE WHICH READS MOST OF THE PROBLEM DATA +C +CALL INPUT(COORD,IFFIX,LNODS,MATNO,MELEM,MEVAB,MFRON,MMATS, +MPOIN,MTOTV,MVFIX,NALGO, +NCRIT,NDFRO,NDOFN,NELEM,NEVAB,NGAUS,NGAU2, +NINCS,NMATS,NNODE,NOFIX,NPOIN,NPROP,NSTRE, +NSTR1,NTOTG,NTOTV, +NTYPE,NVFIX,POSGP,PRES C,PROPS,WEIGP) +PLAS 21 +PLAS 22 +PLAS 23 +PLAS 24 +PLAS 25 +PLAS 26 +PLAS 27 +PLAS 28 +PLAS 29 +C +C*** CALL THE SUBROUTINE WHICH COMPUTES THE CONSISTENT LOAD VECTORS +C FOR EACH ELEMENT AFTER READING THE RELEVANT INPUT DATA +PLAS 30 +PLAS 31 +PLAS 32 +PLAS 33 +PLAS 34 +PLAS 35 +PLAS 36 +C +C*** INITIALISE CERTAIN ARRAYS +PLAS 37 +PLAS 38 +``` diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_028.md b/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_028.md new file mode 100644 index 00000000..1a7deed1 --- /dev/null +++ b/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_028.md @@ -0,0 +1,594 @@ + + +```csv +CALL ZERO(ELOAD,MELEM,MEVAB,MPOIN,MTOTG,MTOTV,NDOFN,NELEM, PLAS 39 +. NEVAB,NGAUS,NSTR1,NTOTG,EPSTN,EFFST, PLAS 40 +. NTOTV,NVFIX,STRSG,TDISP,TFACT, PLAS 41 +. TLOAD,TREAC,MVFIX) PLAS 42 +C +C*** LOOP OVER EACH INCREMENT PLAS 43 +C +DO 100 IINCS = 1,NINCS PLAS 44 +C +C*** READ DATA FOR CURRENT INCREMENT PLAS 45 +C +CALL INCREM(ELOAD,FIXED,IINCS,MELEM,MEVAB,MITER,MTOTV, PLAS 46 +. MVFIX,NDOFN,NELEM,NEVAB,NOUTP,NOFIX,NTOTV, PLAS 47 +. NVFIX,PRESC,RLOAD,TFACT,TLOAD,TOLER) PLAS 48 +C +C*** LOOP OVER EACH ITERATION PLAS 50 +C +DO 50 IITER = 1,MITER PLAS 51 +C +C*** CALL ROUTINE WHICH SELECTS SOLUTION ALORITHM VARIABLE KRESL PLAS 52 +C +CALL ALGOR(FIXED,IINCS,IITER,KRESL,MTOTV,NALGO, PLAS 53 +. NTOTV) PLAS 54 +C +C*** CHECK WHETHER A NEW EVALUATION OF THE STIFFNESS MATRIX IS REQUIRED PLAS 55 +C +IF(KRESL.EQ.1) CALL STIFFP(COORD,EPSTN,IINCS,LNODS,MATNO, PLAS 56 +. MEVAB,MMATS,MPOIN,MTOTV,NELEM,NEVAB,NGAUS,NNODE, PLAS 57 +. NSTRE,NSTR1,POSGP,PROPS,WEIGP,MELEM,MTOTG, PLAS 58 +. STRSG,NTYPE,NCRIT) PLAS 59 +C +C +C*** SOLVE EQUATIONS PLAS 60 +C +C +CALL FRONT(ASDIS,ELOAD,EQRHS,EQUAT,ESTIF,FIXED,IFFIX,IINCS,IITER, PLAS 61 +. GLOAD,GSTIF,LOCEL,LNODS,KRESL,MBUFA,MELEM,MEVAB,MFRON, PLAS 62 +. MSTIF,MTOTV,MVFIX,NACVA,NAMEV,NDEST,NDOFN,NELEM,NEVAB, PLAS 63 +. NNODE,NOFIX,NPIVO,NPOIN,NTOTV,TDISP,TLOAD,TREAC, PLAS 64 +. VECRV) PLAS 65 +C +C*** CALCULATE RESIDUAL FORCES PLAS 66 +C +CALL RESIDU(ASDIS,COORD,EFFST,ELOAD,FACTO,IITER,LNODS, PLAS 67 +. LPROP,MATNO,MELEM,MMATS,MPOIN,MTOTG,MTOTV,NDOFN, PLAS 68 +. NELEM,NEVAB,NGAUS,NNODE,NSTR1,NTYPE,POSGP,PROPS, PLAS 69 +. NSTRE,NCRIT,STRSG,WEIGP,TDISP,EPSTN) PLAS 70 +C +C*** CHECK FOR CONVERGENCE PLAS 71 +C +CALL CONVER(ELOAD,IITER,LNODS,MELEM,MEVAB,MTOTV,NCHEK,NDOFN, PLAS 72 +. NELEM,NEVAB,NNODE,NTOTV,PVALU,STFOR,TLOAD,TOFOR,TOLER) PLAS 73 +C +C*** OUTPUT RESULTS IF REQUIRED PLAS 74 +IF(IITER.EQ.1.AND.NOUTP(1).GT.0) PLAS 75 +.CALL OUTPUT(IITER,MTOTG,MTOTV,MVFIX,NELEM,NGAUS,NOFIX,NOUTP, PLAS 76 +. NPOIN,NVFIX,STRSG,TDISP,TREAC,EPSTN,NTYPE,NCHEK) PLAS 77 +C +C*** IF SOLUTION HAS CONVERGED STOP ITERATING AND OUTPUT RESULTS PLAS 78 +C +IF(NCHEK.EQ.0) GO TO 75 PLAS 79 +50 CONTINUE PLAS 80 +C +C*** PLAS 81 +C +PLAS 82 +PLAS 83 +PLAS 84 +PLAS 85 +PLAS 86 +PLAS 87 +PLAS 88 +PLAS 89 +PLAS 90 +PLAS 91 +PLAS 92 +PLAS 93 +PLAS 94 +PLAS 95 +PLAS 96 +PLAS 97 +PLAS 98 +PLAS 99 +PLAS 100 +PLAS 101 +PLAS 102 +PLAS 103 +``` + + + +
IF(NALGO.EQ.2) GO TO 75PLAS 104
STOPPLAS 105
75 CALL OUTPUT(IITER,MTOTG,MTOTV,MVFIX,NELEM,NGAUS,NOFIX,NOUTP,NPOIN,NVFIX,STRSG,TDISP,TREAC,EPSTN,NTYPE,NCHEK)PLAS 106
100 CONTINUEPLAS 107
STOPPLAS 108
ENDPLAS 109
PLAS 110
+ +# 7.9 Numerical examples + +The first numerical example considered is illustrated in Fig. 7.12(a). The problem studied is that of a thick cylinder subjected to a gradually increasing internal pressure, with plane strain conditions being assumed in the axial direction. A Von Mises yield criterion is assumed and the numerical solutions obtained compared with the theoretical results of Reference 14. The pressure/radial displacement characteristics are shown in Fig. 7.12(b) and good + +![](images/page-272_d5e3706c62e3aa04f486ba4a7a967b3d5e5628b289d4d42a4b0811eca2c234c2.jpg) + +
+radar + +| Angle (°) | Value | +| --------- | ----- | +| 40 | 50 | +| 45 | 50 | +| 50 | 50 | +| 55 | 50 | +| 60 | 50 | +| 65 | 50 | +| 70 | 50 | +| 75 | 50 | +| 80 | 50 | +| 85 | 50 | +| 90 | 50 | +| 95 | 50 | +| 100 | 50 | +| 105 | 50 | +| 110 | 50 | +| 115 | 50 | +| 120 | 50 | +| 125 | 50 | +| 130 | 50 | +| 135 | 50 | +| 140 | 50 | +| 145 | 50 | +| 150 | 50 | +| 155 | 50 | +| 160 | 50 | +| 165 | 50 | +| 170 | 50 | +| 175 | 50 | +| 180 | 50 | +| 185 | 50 | +| 190 | 50 | +| 195 | 50 | +| 200 | 50 | +| 205 | 50 | +| 210 | 50 | +| 215 | 50 | +| 220 | 50 | +| 225 | 50 | +| 230 | 50 | +| 235 | 50 | +| 240 | 50 | +| 245 | 50 | +| 250 | 50 | +| 255 | 50 | +| 260 | 50 | +| 265 | 50 | +| 270 | 50 | +| 275 | 50 | +| 280 | 50 | +| 285 | 50 | +| 290 | 50 | +| 295 | 50 | +| 300 | 50 | +| 305 | 50 | +| 310 | 50 | +| 315 | 50 | +| 320 | 50 | +| 325 | 50 | +| 330 | 50 | +| 335 | 50 | +| 340 | 50 | +| 345 | 50 | +| 350 | 50 | +| 355 | 50 | +| 360 | 50 | +| 365 | 50 | +| 370 | 50 | +| 375 | 50 | +| 380 | 50 | +| 385 | 50 | +| 390 | 50 | +| 395 | 50 | +| 400 | 50 | +| 405 | 50 | +| 410 | 50 | +| 415 | 50 | +| 420 | 50 | +| 425 | 50 | +| 430 | 50 | +| 435 | 50 | +| 440 | 50 | +| 445 | 50 | +| 450 | 50 | +| 455 | 50 | +| 460 | 50 | +| 465 | 50 | +| 470 | 50 | +| 475 | 50 | +| 480 | 50 | +| 485 | 50 | +| 490 | 50 | +| 495 | 50 | +| 500 | 50 | +| 505 | 50 | +| 510 | 50 | +| 515 | 50 | +| 520 | 50 | +| 525 | 50 | +| 530 | 50 | +| 535 | 50 | +| 540 | 50 | +| 545 | 50 | +| 550 | 50 | +| 555 | 50 | +| 560 | 50 | +| 565 | 50 | +| 570 | 50 | +| 575 | 50 | +| 580 | 50 | +| 585 | 50 | +| 590 | 50 | +| 595 | 50 | +| 600 | 50 | +| 605 | 50 | +| 610 | 50 | +| 615 | 50 | +| 620 | 50 | +| 625 | 50 | +| 630 | 50 | +| 635 | 50 | +| 640 | 50 | +| 645 | 50 | +| 650 | 50 | +| 655 | 50 | +| 660 | 50 | +| 665 | 50 | +| 670 | 50 | +| 675 | 50 | +| 680 | 50 | +| 685 | 50 | +| 690 | 50 | +| 695 | 50 | +| 700 | 50 | +| 705 | 50 | +| 710 | 50 | +| 715 | 50 | +| 720 | 50 | +| 725 | 50 | +| 730 | 50 | +| 735 | 50 | +| 740 | 50 | +| 745 | 50 | +| 750 | 50 | +| 755 | 50 | +| 760 | 50 | +| 765 | 50 | +| 770 | 50 | +| 775 | 50 | +| 780 | 50 | +| 785 | 50 | +| 790 | 50 | +| 795 | 50 | +| 800 | 50 | +| 805 | 50 | +| 810 | 50 | +| 815 | 50 | +| 820 | 50 | +| 825 | 50 | +| 830 | 50 | +| 835 | 50 | +| 840 | 50 | +| 845 | 50 | +| 850 | 50 | +| 855 | 50 | +| 860 | 50 | +| 865 | 50 | +| 870 | 50 | +| 875 | 50 | +| 880 | 50 | +| 885 | 50 | +| 890 | 50 | +| 895 | 50 | +| 900 | 50 | +| 905 | 50 | +| 910 | 50 | +| 915 | 50 | +| 920 | 50 | +| 925 | 50 | +| 930 | 50 | +| 935 | 50 | +| 940 | 50 | +| 945 | 50 | +| 950 | 50 | +| 955 | 50 | +| 960 | 50 | +| 965 | 50 | +| 970 | 50 | +| 975 | 50 | +| 980 | 50 | +| 985 | 50 | +| 990 | 50 | +| 995 | 50 | +| 1000 | 50 | +
+ +Elastic modulus, $E = 2.1 \times 10^{4} \, dN/mm^{2}$ + +Poissons ratio, $\nu = 0.3$ + +Uniaxial yield stress, $\sigma_{r}=24.0\ dN/mm^{2}$ + +Strain hardening parameter, $\mathbf{H}' = 00$ + +Von mises yield criterion + +(a) +![](images/page-272_f209ae3ddae5cfe9803a92d90c08f695f6a1d13acb0fa457b0e0a4cda02079e4.jpg) + +
+line + +| displacement of inner face (× 10⁵ mm) | 3 point gauss rule | 2 point gauss rule | theory (Ref. 14) | +| ------------------------------------ | ------------------ | ------------------ | ---------------- | +| 0 | 0 | 0 | 0 | +| 8 | 8 | 8 | 8 | +| 12 | 12 | 12 | 12 | +| 16 | 16 | 16 | 16 | +| 20 | 18 | 18 | 18 | +| 24 | 19 | 19 | 19 | +| 28 | 19 | 19 | 19 | +
+ +Fig. 7.12 (a) Mesh and material properties employed in the elasto-plastic analysis of an internally pressurised thick cylinder under plane strain conditions. (b) Displacement of the inner surface with increasing pressure for the problem of Fig. 7.12(a). + + + +agreement between the numerical and analytical solutions is evident. In the numerical studies, collapse was deemed to have occurred if the iterative procedure diverged for an incremental load increase. + +![](images/page-273_949697ecb66898432a5ec8607c8ad817468f384080b1769dcf89c8974402d2e8.jpg) + +
+line + +| x | σθ | +| --- | --- | +| 100 | 13 | +| 110 | 12 | +| 120 | 11 | +| 130 | 10 | +| 140 | 9 | +| 150 | 8 | +| 160 | 7 | +| 170 | 6.5 | +| 180 | 6 | +| 190 | 5.5 | +| 200 | 5 | +
+ +![](images/page-273_3b2638e93ca693401baaa5436c9378c37c12a063be49af751c41b15ed9477243.jpg) + +
+line + +| x | σₙ | +| ---- | --- | +| 100 | 16 | +| 120 | 15 | +| 140 | 13 | +| 160 | 11 | +| 180 | 9 | +| 200 | 8 | +
+ +![](images/page-273_a143d68fb20704fb894dd76419381215a619f8943c650b54af42b09fdaf5d1e5.jpg) + +
+line + +| x | σ_e | +| ---- | ---- | +| 100 | 13 | +| 110 | 16 | +| 120 | 19 | +| 130 | 18 | +| 140 | 16 | +| 150 | 14 | +| 160 | 12 | +| 170 | 11 | +| 180 | 10 | +| 190 | 9 | +| 200 | 8 | +
+ +![](images/page-273_7b1aff624c5a8a846dc5831bd90b11f242091b8d6bd8418cb69e2644e2e0123d.jpg) + +
+line + +| x | σθ | +| ---- | ---- | +| 100 | 10 | +| 120 | 15 | +| 140 | 20 | +| 160 | 22 | +| 180 | 20 | +| 200 | 18 | +
+ +![](images/page-273_907240ec3c67cdaad19bb886e17751c07226b053ebaccf48056a32defe284128.jpg) + +
+text_image + +Two rows of circular symbols, likely representing data points or indicators, with dots inside each row. +
+ +![](images/page-273_b2e0c3bf872562ea76364bd2a9b16ddfdf5d9c9e4daddaf390a5c1eb36413810.jpg) + +
+text_image + +P 2 3 4 5 6 7 8 9 10 +
+ +Fig. 7.13 Hoop stress distributions at various pressure values for the problem of Fig. 7.12(a). + +Fig. 7.13 shows the circumferential (hoop) stress distributions for specified pressure values. Again a good agreement is evident. In solution both a two-point and three-point Gaussian integration rule was considered. Whilst the nodal displacements obtained by use of both rules are practically identical, it is seen from Fig. 7.13 that use of a $2 \times 2$ integrating rule gives superior stress values to a $3 \times 3$ rule. This is a general result for elasto-plastic problems and therefore use of a two-point rule is recommended. This phenomenon is an example of the benefit of a reduced integration order for parabolic isoparametric elements. $^{(15)}$ + + + +![](images/page-274_6b4b2340bbaf0114d7bee2defef827db84aa11d2f83cbce3a5ff61c801107c11.jpg) + +
+line + +| Central deflection, w | Intensity of UDL, P | +| --------------------- | ------------------- | +| 0.00 | 0 | +| 0.05 | 100 | +| 0.10 | 150 | +| 0.15 | 200 | +| 0.20 | 250 | +| 0.25 | 275 | +| 0.30 | 280 | +
+ +Fig. 7.14 Load/central deflection response for a uniformly loaded simply supported circular plate. + +The second example considered is the simply supported circular plate shown in Fig. 7.14. + +The plate is modelled by five axisymmetric elements and the loading takes the form of a progressively increasing uniformly distributed load. The growth in central deflection with increasing load is shown in Fig. 7.14. A converged solution was obtained for P = 270 but the numerical process diverged for P = 280 and consequently the collapse load is taken to be 270. This is in good agreement with the value of 260 quoted in Ref. 16, particularly in view of the coarse mesh employed in the present study. Fig. 7.15 shows the deflection profile with increasing applied load. + +![](images/page-274_39af2daf93374d3ac1b1550f4d07b82d159306b0400395a2d54c07847827635f.jpg) + +
+line + +| r/a | w/h (P = 100) | w/h (P = 200) | w/h (P = 260 lb/in²) | +| --- | --- | --- | --- | +| 0.0 | 0.05 | 0.15 | 0.20 | +| 0.2 | 0.05 | 0.15 | 0.20 | +| 0.4 | 0.05 | 0.15 | 0.20 | +| 0.6 | 0.05 | 0.15 | 0.20 | +| 0.8 | 0.05 | 0.15 | 0.20 | +| 1.0 | 0.05 | 0.15 | 0.20 | +
+ +Fig. 7.15 Deflection profiles for the problem of Fig. 7.14 at various applied load values. + + + +# 7.10 Problems + +7.1 In Section 7.2.1 it was stated that the Von Mises law implies that yielding begins when the (recoverable) elastic energy of distortion, $D$ , reaches a critical value. Prove this by showing that $J_2'$ is proportional to $D$ , since $D$ can be written as + +$$ +D = \frac {1}{2} \sigma_ {i j} \epsilon_ {i j} - \frac {(1 - 2 \nu)}{1 2 \mu (1 + \nu)} (\sigma_ {i i}) ^ {2}. \tag {7.98} +$$ + +![](images/page-275_d5b5d6eee6767c228ab22cb5f3542c10b1fcdc5f89469c3cb4d42d382e899c18.jpg) + +
+text_image + +σ₃ +S +√3P₁ +0 +σ₂ +σ₁ +√3P₀ +
+ +Fig. 7.16 Geometric representation of the Berg yield criterion—Problem 7.2. + +7.2 A yield criterion has been proposed by Berg $^{(17)}$ which attempts to account for the tensile failure of a material due to the formation of voids at a sufficiently high strain level. The yield surface is illustrated in Fig. 7.16 and can be seen to be made up of two distinct portions. For stress levels below a mean hydrostatic tension of $P_{I}$ the material yields according to the Von Mises cylinder of radius S. The yield surface in the tensile range is terminated by an elliptic cap whose extremity is defined by $P_{0}$ . The three constants S, $P_{I}$ and $P_{0}$ are material constants and must be experimentally determined. The two distinct portions of the yield surface can be expressed as + + + +$$ +\sqrt {2} (J _ {2} ^ {\prime}) ^ {\ddagger} = S \quad \text { for } \sigma_ {m} \leqslant P _ {I} +$$ + +$$ +[ 2 J _ {2} ^ {\prime} + H (\sigma_ {m} - P _ {I}) ^ {2} ] ^ {\frac {1}{2}} = S \quad P _ {I} \leqslant \sigma_ {m} \leqslant P _ {0}, \tag {7.99} +$$ + +where $H = S^2 / (P_I - P_0)^2$ and $\sigma_m$ is the mean hydrostatic pressure. + +Show that this yield criterion can be expressed in the form of three constants $C_{1}$ , $C_{2}$ and $C_{3}$ as indicated in Section 7.4 where + +$$ +C _ {1} = 0, \quad C _ {2} = \sqrt {2}, \quad C _ {3} = 0 \quad \text { for } \quad \sigma_ {m} \leqslant P _ {I} +$$ + +$$ +C _ {1} = H \left(\sigma_ {m} - P _ {I}\right) / S, \quad C _ {2} = 2 J _ {2} ^ {\prime} / S, \quad C _ {3} = 0 \quad P _ {I} \leqslant \sigma_ {m} \leqslant P _ {0}. +$$ + +7.3 A certain material yields when the maximum principal stress reaches a critical value, Y. Assuming identical behaviour in tension and compression, determine the geometrical form of the yield surface. The solution is given in Fig. 7.17. + +![](images/page-276_3c1cd65038d00e90078076b655fe98057341973c73b7258d782449fa767b93ba.jpg) + +
+text_image + +σ₃ +30° +√2/3 Y +σ₁ +σ₂ +
+ +Fig. 7.17 $\pi$ plane representation of a yield criterion based on maximum principal stress values—Problem 7.3. + +7.4 The assumption of a linear strain hardening material law may prove to be inadequate for certain situations. If the uniaxial stress/strain test curve for the material is known, then it is possible to represent the stress-plastic strain relationship in a piecewise linear fashion as shown in Fig. 7.18 and the instantaneous yield stress can be written in the form $\sigma_{Y} = \sigma_{Y^{0}} + S(\bar{\epsilon}_{p})$ where $S(\bar{\epsilon}_{p})$ is the piecewise linear function describing the increase (or decrease) in the initial yield stress $\sigma_{Y^{0}}$ with the increase of effective plastic strain $\bar{\epsilon}_{p}$ . The program modifications required to describe this behaviour will all be included in subroutine RESIDU, except for changes in material property specification which will need to be made in subroutine INPUT. Carry out all necessary modifications. + + + +![](images/page-277_65547797836e77bb9e4da8221a6464c7c6b87c2f69202b1640f838d8e1f1ea59.jpg) + +
+line + +| Effective plastic strain ε̅p | Stress, σ | +| --------------------------- | --------- | +| Δε̅p (1) | σ⁰_γ | +| Δε̅p (2) | σ²_γ | +| n-1 (3) | σ²_γ | +| n (n) | σ²_γ | +
+ +Fig. 7.18 Piecewise-linear representation of material strain hardening—Problem 7.4. + +7.5 By using the mesh of Fig. 7.12(a) and solving as an axisymmetric problem, use program PLANET (documented in Appendix II, Section A2.1) to determine the elasto-plastic stress and displacement distributions in a sphere when it is loaded by an incrementally applied internal pressure. The dimensions and material properties of the sphere are given by reference to Fig. 7.12. Assume a Tresca yield criterion for solution and compare your results with the solution given in Ref. 1. +7.6 Use program PLANET to solve the problem illustrated in Fig. 1.2, Chapter 1. Use both a Tresca and Von Mises yield criterion and compare the plastic zone distributions obtained with those of Fig. 1.2. +7.7 Subroutine CONVER, described in Section 6.5.4, bases convergence of the nonlinear solution process on the global norm of the residual force vector. Modify subroutine CONVER so that convergence is based on expression (3.27) in which the summation signs are absent; so that convergence is monitored locally at each of the nodes 1 to N in turn. +7.8 Modify subroutine CONVER, Section 6.5.4 so that convergence is monitored locally at each node according to the displacement changes that occur during a particular iteration, r, as follows. + +$$ +\frac {| \Delta d ^ {r} |}{| d ^ {1} |} \times 1 0 0 \leqslant \text { TOLER }, \tag {7.100} +$$ + +where $d^{1}$ is the elastic displacement occurring upon application of the load increment and $\Delta d^{r}$ is the change in nodal displacement during the $r^{th}$ iteration. + + + +7.9 Modify program PLANET to undertake the elasto-plastic solution of three-dimensional solids. To simplify the task consider only the Von Mises yield criterion and assume that the solid is loaded by nodal point loads only. +7.10 The yield criterion to be employed in program PLANET is specified by means of control parameter NCRIT in subroutine INPUT described in Section 6.5.1. In some applications, such as steel-concrete composites, it is necessary to employ a different yield surface for different parts of the structure. Modify program PLANET so that the yield criterion governing elasto-plastic behaviour is separately specified for each element in the solid. + +# 7.11 References + +1. HILL, R., The Mathematical Theory of Plasticity, Oxford University Press, 1950. +2. PRAGER, W., An Introduction to Plasticity, Addison-Wesley, Amsterdam and London, 1959. +3. HOFFMAN, O. and SACHS, G., Introduction to the Theory of Plasticity for Engineers, McGraw-Hill, 1953. +4. BRIDGMAN, P. W., Studies in Large Plastic Flow and Fracture, McGraw-Hill, New York, 1952. +5. BISHOP, A. W., The strength of soils as engineering materials, Geotechnique, 16, 89–130 (1966). +6. DAVIS, E. M., Theories of plasticity and the failure of soil masses, Ch. 6 Soil Mechanics, Ed. I. K. Lee, Butterworths, London, 1968. +7. ZIENKIEWICZ, O. C., VALLIAPPAN, S. and KING, I. P., Elasto-plastic solutions of engineering problems; Initial stress finite element approach, Int. J. Num. Meth. Engng. 1, 75–100 (1969). +8. YAMADA, Y., YOSHIMURA, N. and SAKURAI, T., Plastic stress-strain matrix and its application for the solution of elastic-plastic problems by Finite Element Method, Int. J. Mech. Sci. 10, 343–354 (1968). +9. BLAND, D. R., The associated flow rule of plasticity, J. Mech. Phy. of Solids, 6, 71–78 (1957). +10. NAYAK, G. C. and ZIENKIEWICZ, O. C., Convenient form of stress invariants for Plasticity, Journ. of the Struct. Div. Proc. of A.S.C.E., 949-953, April 1972. +11. FREDERICK, D. and CHANG, T. S., Continuum Mechanics, Allyn and Bacon, 1965. +12. KOITER, W. T., Stress-strain relations, uniqueness and variational theorems for elastic-plastic materials with singular yield surface, Quart. Appl. Math., 11, 350–354 (1953). +13. HINTON, E. and OWEN, D. R. J., Finite Element Programming, Academic Press, London, 1977. +14. HODGE, P. G. and WHITE, G. N., A quantitative comparison of flow and deformation theories of plasticity' J. Appl. Mech. 17, 180–184 (1950). +15. ZIENKIEWICZ, O. C. and HINTON, E., Reduced integration, function smoothing and non-conformity in finite element analysis (with special reference to thick plates), J. of the Franklin Institute, 302, Nos. 5 and 6, Nov./Dec. 1976. +16. ARMEN, H., Jr., PIFKO, A. and LEVINE, H. S., Finite element analysis of structures in the plastic range, N.A.S.A. Contractor Report, CR.1649 (1971). + + + +17. BERG, C. A., Plastic dilation and void interaction, Battelle Inst. Material Science Colloquia, Sept. 1969. Inelastic Behaviour of Solids, Ed. M. F. Kanninen et al., McGraw-Hill, 171–210, 1970. +18. KRIEG, R. D. and KRIEG, D. B., Accuracies of numerical solution methods for the elastic-perfectly plastic mode, Trans. ASME, J. Pressure Vessel Technology, 99, 4, 510–515 (1977). +19. SCHREYER, H. L., KULAK, R. F. and KRAMER, J. M., Accurate numerical solutions for elastic-plastic models, Trans. ASME, J. Pressure Vessel Technology, 101, 3, 226–234 (1979). +20. SAMUELSSON, A. and FRÖIER, M., Finite elements in plasticity—A variational inequality approach, in The Mathematics of Finite Elements and Applications III, Ed. J. R. Whiteman, Academic Press, 1979. + + diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_029.md b/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_029.md new file mode 100644 index 00000000..eebf2858 --- /dev/null +++ b/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_029.md @@ -0,0 +1,431 @@ + + +# Chapter 8 Elasto-viscoplastic problems in two dimensions + +# 8.1 Introduction + +In all inelastic deformations time rate effects are always present to some degree. Whether or not their exclusion has a significant influence on the prediction of the material behaviour depends upon several factors. In the study of structural components under static loading conditions at normal temperatures it is accepted that time rate effects are generally not important and the conventional theory of plasticity, as described in Chapter 7, then models the behaviour adequately. However metals, especially under high temperatures, exhibit simultaneously the phenomena of creep and visco-plasticity. The former is essentially a redistribution of stress and/or strains with time under elastic material response while the latter is a time dependent plastic deformation. Experimental observations cannot distinguish between the two phenomena and their separation has been largely an analytical convenience rather than a physical requirement. Numerical processes, as described in this chapter, allow the simultaneous description of both effects. + +A further situation in which time rate effects are important is in the dynamic transient loading of structures. For example, it can be experimentally demonstrated that the instantaneous yield stress of materials under high strain rates can be significantly greater than the corresponding quasi-static value. This class of problem is dealt with in Chapter 10. + +In this chapter we utilise the theory of viscoplasticity to provide a unified approach to problems of creep and plasticity. As well as providing solutions to time-dependent situations the viscoplastic algorithm can provide economic solution for classic elasto-plastic problems since it can be readily shown that the steady-state solution of the viscoplastic problem is identical to the corresponding conventional static elasto-plastic solution. Furthermore, by reducing the yield stress of the material to zero, elastic creep problems can be solved. + +The concept of ‘overlay models’ is also introduced in this chapter. In this, the solid is assumed, for mathematical convenience only, to be composed of several layers or overlays each of which undergo the same deformation. By assigning different properties to each overlay a composite behaviour can be + + + +obtained which exhibits all the essential characteristics of the visco-elastic-plastic response of many real materials. + +The basic one-dimensional rheological model developed in Chapter 4 is now extended to the case of a general continuum and the essential steps employed in the numerical solution algorithm are discussed. Since most of the matrix expressions involved in viscoplastic analysis are common to conventional elasto-plastic theory, the majority of the subroutines developed in Chapter 7 can be again used with little or no change. The additional subroutines required are then constructed and assembled to form a working program. Finally it is briefly demonstrated how the overlay principle can be used to simulate a complex material response. + +# 8.2 Theory of elasto-viscoplastic solids + +# 8.2.1 Basic expressions + +In the usual manner for nonlinear continua problems it is assumed that the total strain, $\epsilon$ , can be separated into elastic, $\epsilon_{e}$ , and viscoplastic, $\epsilon_{vp}$ , components, so that the total strain rate can be expressed as $^{(1-3)}$ + +$$ +\dot {\epsilon} = \dot {\epsilon} _ {e} + \dot {\epsilon} _ {v p}, \tag {8.1} +$$ + +where ( $\cdot$ ) represents differentiation with respect to time. The total stress rate depends on the elastic strain rate according to + +$$ +\dot {\sigma} = D \dot {\epsilon} _ {e}, \tag {8.2} +$$ + +where D is the elasticity matrix. The onset of viscoplastic behaviour is governed by a scalar yield condition of the form + +$$ +F (\sigma , \epsilon_ {v p}) - F _ {0} = 0, \tag {8.3} +$$ + +in which $F_{0}$ is the uniaxial yield stress which may itself be a function of a hardening parameter, $\kappa$ . For frictional materials $F_{0}$ is the equivalent yield stress as given by Column 4, Table 7.2. It is assumed that viscoplastic flow occurs for values of $F > F_{0}$ only. + +It is now necessary to choose a specific law defining the viscoplastic strains. The simplest option is one in which the viscoplastic strain rate depends only on the current stresses, so that + +$$ +\dot {\epsilon} _ {v p} = f (\sigma). \tag {8.4} +$$ + +This relationship can be generalised to include strain hardening and temperature dependence and the influence of state dependent variables, such as damage parameters for rupture theories, can also be considered. + + + +One explicit form of (8.4) which has wide applicability, is offered by the following viscoplastic flow rule. $^{(4)}$ + +$$ +\dot {\epsilon} _ {v p} = \gamma \langle \Phi (F) \rangle \frac {\partial Q}{\partial \sigma}, \tag {8.5} +$$ + +in which $Q = Q(\sigma, \epsilon_{vp}, \kappa)$ is a 'plastic' potential and $\gamma$ is a fluidity parameter controlling the plastic flow rate. The term $\Phi(x)$ is a positive monotonic increasing function for $x > 0$ and the notation $\langle \rangle$ implies + +$$ +\begin{array}{l} \langle \Phi (x) \rangle = \Phi (x) \text { for } x > 0 \\ \langle \Phi (x) \rangle = 0 \quad x \leqslant 0. \tag {8.6} \\ \end{array} +$$ + +Comparison of (8.5) with (7.28) shows an analogy between the flow rule of conventional non-associated plasticity and the present definition of viscoplastic flow rate. If, once again, we restrict ourselves to associated plasticity situations, in which case $F \equiv Q$ , expression (8.5) reduces to + +$$ +\dot {\epsilon} _ {v p} = \gamma \langle \Phi (F) \rangle \frac {\partial F}{\partial \sigma} = \gamma \langle \Phi \rangle a, \tag {8.7} +$$ + +where the same definition of the flow vector a is employed as in (7.42). Different choices have been recommended $^{(5)}$ for the function $\Phi$ . The two most common versions are + +$$ +\Phi (F) = e ^ {M \left(\frac {F - F _ {0}}{F _ {0}}\right)} - 1, \tag {8.8} +$$ + +and + +$$ +\Phi (F) = \left(\frac {F - F _ {0}}{F _ {0}}\right) ^ {N}, \tag {8.9} +$$ + +in which M and N are arbitrary prescribed constants. The latter option, when employed in (8.7) can be made to model the Norton power law of metallic creep by assigning the threshold uniaxial yield value, $F_{0}$ , to zero (or to an arbitrarily small value for numerical convenience). + +# 8.2.2 The viscoplastic strain increment + +With the strain rate law expressed by (8.7) we can define a strain increment $\Delta\epsilon_{vp}^{n}$ occurring in a time interval $\Delta t_{n}=t_{n+1}-t_{n}$ using an implicit time stepping scheme, as $^{(6)}$ + +$$ +\Delta \epsilon_ {v p} ^ {n} = \Delta t _ {n} \left[ (1 - \Theta) \dot {\epsilon} _ {v p} ^ {n} + \Theta \dot {\epsilon} _ {v p} ^ {n + 1} \right]. \tag {8.10} +$$ + +For $\Theta = 0$ we obtain the Euler time integration scheme which is also referred to as 'fully explicit' (or forward difference method) since the strain increment is completely determined from conditions existing at time, $t_n$ . On the other + + + +hand $\Theta = 1$ gives a 'fully implicit' (or backward difference) scheme with the strain increment being determined from the strain rate corresponding to the end of the time interval. The case $\Theta = \frac{1}{2}$ results in the so-called 'implicit trapezoidal' scheme which is also known generally as the Crank-Nicolson rule in the context of linear equations. + +To define $\dot{\epsilon}_{vp}^{n+1}$ in (8.10) we can use a limited Taylor series expansion and write + +$$ +\dot {\epsilon} _ {v p} ^ {n + 1} = \dot {\epsilon} _ {v p} ^ {n} + H ^ {n} \Delta \sigma^ {n}, \tag {8.11} +$$ + +where + +$$ +\boldsymbol {H} ^ {n} = \left(\frac {\partial \dot {\epsilon} _ {v p}}{\partial \sigma}\right) ^ {n} = \boldsymbol {H} ^ {n} (\sigma^ {n}), \tag {8.12} +$$ + +and $\Delta\sigma^{n}$ is the stress change occurring in the time interval $\Delta t_{n}=t_{n+1}-t_{n}$ . Thus (8.10) can be rewritten as + +$$ +\Delta \epsilon_ {v p} ^ {n} = \dot {\epsilon} _ {v p} ^ {n} \Delta t _ {n} + C ^ {n} \Delta \sigma^ {n}, \tag {8.13} +$$ + +where + +$$ +\boldsymbol {C} ^ {n} = \Theta \Delta t _ {n} \boldsymbol {H} ^ {n}. \tag {8.14} +$$ + +We draw the attention of the reader to the fact that the matrix H defined in (8.12) is the matrix whose eigenvalues determine the limiting time step length, $\Delta t_{n}$ which can be employed in the explicit integration schemes. The matrix H depends on the stress level and no difficulty arises in its evaluation and specific forms will be developed in Section 8.5. + +# 8.2.3 Stress increments + +Using the incremental form of (8.2) we obtain + +$$ +\Delta \sigma^ {n} = D \Delta \epsilon_ {e} ^ {n} = D (\Delta \epsilon^ {n} - \Delta \epsilon_ {v p} ^ {n}). \tag {8.15} +$$ + +Or expressing the total strain increment in terms of the displacement increment as + +$$ +\Delta \epsilon^ {n} = B ^ {n} \Delta d ^ {n}, \tag {8.16} +$$ + +and substituting for $\Delta \epsilon_{vp}^n$ from (8.13), then (8.15) becomes + +$$ +\Delta \sigma^ {n} = \hat {D} ^ {n} (B ^ {n} \Delta d ^ {n} - \dot {\epsilon} _ {v p} ^ {n} \Delta t _ {n}), \tag {8.17} +$$ + +where + +$$ +\hat {\boldsymbol {D}} ^ {n} = (\boldsymbol {I} + \boldsymbol {D} \boldsymbol {C} ^ {n}) ^ {- 1} \boldsymbol {D} = (\boldsymbol {D} ^ {- 1} + \boldsymbol {C} ^ {n}) ^ {- 1}. \tag {8.18} +$$ + +In (8.16) and (8.17) the notation $B^{n}$ is employed to denote the possibility that the strain matrix may not be constant throughout the solution. For example, if large deformations are to be considered, the strain matrix for a Lagrangian formulation is nonlinear and can be written + + + +$$ +\boldsymbol {B} ^ {n} = \boldsymbol {B} _ {0} + \boldsymbol {B} _ {N L} ^ {n}, \tag {8.19} +$$ + +where $B_{0}$ represents the standard linear terms which do not vary during solution and $B_{NL}^{n}$ contains the nonlinear quadratic terms. These latter expressions are dependent on the current displacements and therefore vary throughout the solution process. + +The matrix $D^{n}$ is a symmetric matrix when the visco-plastic law is associative. For the non-associated case, the matrix $C^{n}$ is unsymmetric, requiring unsymmetric equation solvers for analysis. + +For the solution of linear elastic problems by the explicit scheme ( $\Theta = 0$ ), equation (8.17) simplifies considerably to give + +$$ +\Delta \sigma^ {n} = D (B \Delta d ^ {n} - \dot {\epsilon} _ {v p} ^ {n} \Delta t _ {n}). \tag {8.20} +$$ + +# 8.2.4 Equations of equilibrium + +The equations of equilibrium to be satisfied at any instant of time, $t_n$ , are + +$$ +\int_ {\Omega} [ \boldsymbol {B} ^ {n} ] ^ {T} \boldsymbol {\sigma} ^ {n} d \Omega + \boldsymbol {f} ^ {n} = \mathbf {0}, \tag {8.21} +$$ + +where $f^{n}$ is the vector of equivalent nodal loads due to applied surface tractions, body forces, thermal loads, etc. During a time increment the equilibrium equations which must be satisfied are given by the incremental form of (8.21) to be + +$$ +\int_ {\Omega} [ \boldsymbol {B} ^ {n} ] ^ {T} \Delta \sigma^ {n} d \Omega + \Delta \boldsymbol {f} ^ {n} = \mathbf {0}, \tag {8.22} +$$ + +in which $\Delta f^{n}$ represents the change in loads during the time interval $\Delta t_{n}$ . In the majority of problems encountered in engineering the load increments are applied as discrete steps and thus $\Delta f^{n} = 0$ for all time steps other than the first within an increment. + +Using (8.13) and (8.20) the displacement increment occurring during time step $\Delta t_{n}$ can be calculated as + +$$ +\Delta d ^ {n} = \left[ K _ {T} ^ {n} \right] ^ {- 1} \Delta V ^ {n} +$$ + +$$ +\Delta V ^ {n} = \int_ {\Omega} [ B ^ {n} ] ^ {T} \hat {D} ^ {n} \dot {\epsilon} _ {v p} ^ {n} \Delta t _ {n} d \Omega + \Delta f ^ {n}, \tag {8.23} +$$ + +where $K_{T}^{n}$ is the tangential stiffness matrix with the following form + +$$ +\boldsymbol {K} _ {T} ^ {n} = \int_ {\Omega} [ \boldsymbol {B} ^ {n} ] ^ {T} \hat {\boldsymbol {D}} ^ {n} \boldsymbol {B} ^ {n} d \Omega , \tag {8.24} +$$ + +and $\Delta V^{n}$ are termed the incremental pseudo-loads. The displacement increments, $\Delta d^{n}$ , when substituted back into (8.20) give the stress increments + + + +$\Delta \sigma^n$ and thus + +$$ +\sigma^ {n + 1} = \sigma^ {n} + \Delta \sigma^ {n} +$$ + +$$ +\boldsymbol {d} ^ {n + 1} = \boldsymbol {d} ^ {n} + \Delta \boldsymbol {d} ^ {n}. \tag {8.25} +$$ + +Use of (8.15) and (8.16) gives + +$$ +\Delta \epsilon_ {v p} ^ {n} = B ^ {n} \Delta d ^ {n} - D ^ {- 1} \Delta \sigma^ {n}, \tag {8.26} +$$ + +and then + +$$ +\epsilon_ {v p} ^ {n + 1} = \epsilon_ {v p} ^ {n} + \Delta \epsilon_ {v p} ^ {n}. \tag {8.27} +$$ + +Arrival at stationary or steady state conditions can be monitored by examination of the strain rates. In particular $\dot{\epsilon}_{vp}$ , as given by (8.7), is calculated at each time interval and the time marching process halted as soon as this quantity becomes tolerably small. + +# 8.2.5 Equilibrium correction + +The stress increment calculation is based on a linearised form of the incremental equilibrium equations (8.22). Therefore the total stresses, $\sigma^{n+1}$ , obtained by accumulating all such stress increments are not strictly correct and will not exactly satisfy the equations of equilibrium, (8.21). There are several solution procedures available for applying the necessary correction and Reference 7 discusses the relative merits of various options. The simplest approach is to evaluate $\sigma^{n+1}$ according to (8.20) and (8.25) and then compute the residual, or out-of-balance, forces, $\psi$ , as + +$$ +\psi^ {n + 1} = \int_ {\Omega} [ B ^ {n + 1} ] ^ {T} \sigma^ {n + 1} d \Omega + f ^ {n + 1} \neq 0, \tag {8.28} +$$ + +noting, for geometrically nonlinear problems, that $B^{n+1}$ is evaluated for a displacement state $d^{n+1}$ . This residual force is then added to the applied force increment at the next time step. Such a technique avoids an iteration process and at the same time achieves a reduction in error. + +# 8.3 Selection of the time step length + +It can be shown $^{(14)}$ that the time integration scheme formally represented by (8.10) is unconditionally stable for values of $\Theta\geqslant\frac{1}{2}$ . This implies that the time marching scheme is numerically stable but does not guarantee the accuracy of the solution at any stage; so that in practice even for values of $\Theta\geqslant\frac{1}{2}$ limits must be placed on the time step length in order to achieve a valid solution. + +For $\Theta<\frac{1}{2}$ the integration process is only conditionally stable and numerical time integration can only proceed for values of $\Delta t_{n}$ less than some critical value. We now proceed to establish rules for choosing the time step length for computation. + + + +Schemes can be employed in which the time step length can be either constant or vary for each time interval. In the variable scheme the magnitude of the time step is controlled by a factor $\tau$ which limits the maximum effective viscoplastic strain increment, $\Delta \bar{\epsilon}_{vp^n}$ as a fraction of the total effective strain, $\bar{\epsilon}^n$ , so that + +$$ +\Delta \tilde {\epsilon} _ {v p} ^ {n} = (\sqrt {\frac {2}{3}}) \left\{\dot {\epsilon} _ {i j} ^ {n}\right) _ {v p} \left(\dot {\epsilon} _ {i j} ^ {n}\right) _ {v p} \} ^ {1 / 2} \Delta t _ {n} \leqslant \tau \bar {\epsilon} ^ {n}. \tag {8.29} +$$ + +For isoparametric elements, all strains are evaluated at the Gaussian integration points. Therefore $\Delta t_{n}$ must be computed to satisfy (8.29) at each such point and the least value taken for analysis. A variant on the above is to limit the time step length according to + +$$ +\left\{\dot {\epsilon} _ {i i} ^ {n} \right\} ^ {\frac {1}{2}} v p \Delta t _ {n} \leqslant \tau \left\{\epsilon_ {i i} ^ {n} \right\} ^ {\frac {1}{2}}, \tag {8.30} +$$ + +in which $\epsilon_{tt}^{n}$ is the first total strain invariant and $(\dot{\epsilon}_{ii}^{n})_{vp}$ is the first viscoplastic strain rate invariant. Thus $\Delta t_{n}$ can be formally written for this case as + +$$ +\Delta t _ {n} \leqslant \tau \left[ \epsilon_ {i i} ^ {n} / \left(\dot {\epsilon} _ {i i} ^ {n}\right) _ {v p} \right] ^ {\frac {1}{2}} \min. \tag {8.31} +$$ + +The minimum in (8.31) is that taken over all integrating points in the solid. The value of the time increment parameter $\tau$ must be specified by the user and for explicit time marching schemes accurate results have been obtained $^{(4,8)}$ in the range $0.01 < \tau < 0.15$ . For implicit schemes, values of $\tau$ up to 10 have been found to be stable though the accuracy deteriorates. + +Another useful limit can be imposed while using the variable time stepping scheme. The change in the time step length between any two intervals is limited according to + +$$ +\Delta t _ {n + 1} \leqslant k \Delta t _ {n}, \tag {8.32} +$$ + +where k is a specified constant. Experience suggests a value of $k = 1 \cdot 5$ to be suitable although there are no fixed criteria for its specification. + +The above time step limiting values are basically empirical. Theoretical restrictions on the time step length have been provided by Cormeau $^{(9)}$ for specific forms of the viscoplastic flow rule and for explicit time integration only. In particular, for associated viscoplasticity $Q \equiv F$ and a linear function $\Phi(F) = F$ we have the following limits on the time step length. + +$$ +\Delta t \leqslant \frac {(1 + \nu) F _ {0}}{\gamma E} \quad \text { for Tresca materials } +$$ + +$$ +\Delta t \leqslant \frac {4 (1 + \nu) F _ {0}}{3 \gamma E} \quad \text { Von Mises } +$$ + +$$ +\Delta t \leqslant \frac {4 (1 + \nu) (1 - 2 \nu) F _ {0}}{\gamma (1 - 2 \nu + \sin^ {2} \phi) E} \quad \text { Mohr - Coulomb }, \tag {8.33} +$$ + + + +where $\gamma$ is the fluidity parameter and $\phi$ is the angle of internal friction. The term $F_{0}$ is the uniaxial yield stress for Tresca and Von Mises solids and is the equivalent value ( $c \cos \phi$ ) for Mohr–Coulomb materials where c is the cohesion. No simple expression exists for the limiting time step length in Drucker–Prager solids. + +# 8.4 Computational procedure + +The essential steps in the solution process can be summarised as follows. Solution to the problem must begin from the known initial conditions at time t = 0, which are, of course, the solution of the static elastic situation. At this stage $d^{0}$ , $F^{0}$ , $\epsilon^{0}$ , $\sigma^{0}$ are known and $\epsilon_{vp^{0}} = 0$ . The time marching scheme described in Section 8.2.4 can then be employed to advance the solution by one timestep at a time. The solution sequence adopted is as follows. + +Stage 1 Suppose at time $t = t_{n}$ we have an equilibrium situation and $d^{n}, \sigma^{n}$ , $\epsilon^{n}, \epsilon_{vp}^{n}$ , $F^{n}$ are known. The following quantities are assembled: + +(a) $\pmb{B}^n = \pmb{B}_0 + \pmb{B}_{NL}(\pmb{d}^n),$ + +(b) $C^n = C^n (\sigma^n,\Delta t_n),$ + +(c) $\hat{D}^n = (D^{-1} + C^n)^{-1},$ + +(d) $K_{T^n} = \int_{\Omega} [B^n]^T \hat{D}^n B^n d\Omega,$ + +(e) $\dot{\epsilon}_{vp}^{n} = \gamma \langle \Phi \rangle a^{n}$ . + +Stage 2 i) Compute the displacement increments $\Delta d^n$ according to (8.23) as + +$$ +\Delta d ^ {n} = [ K _ {T} ^ {n} ] ^ {- 1} \Delta V ^ {n}, +$$ + +where + +$$ +\Delta V ^ {n} = \int_ {\Omega} [ B ^ {n} ] ^ {T} \hat {D} ^ {n} \dot {\epsilon} _ {v p} ^ {n} \Delta t _ {n} d \Omega + \Delta f ^ {n}. +$$ + +ii) Calculate the stress increment $\Delta\sigma^{n}$ as + +$$ +\Delta \sigma^ {n} = \hat {D} ^ {n} (B ^ {n} \Delta d ^ {n} - \dot {\epsilon} _ {v p} ^ {n} \Delta t _ {n}). +$$ + +Stage 3 Determine the total displacements and stresses + +$$ +\boldsymbol {d} ^ {n + 1} = \boldsymbol {d} ^ {n} + \Delta \boldsymbol {d} ^ {n} +$$ + +$$ +\sigma^ {n + 1} = \sigma^ {n} + \Delta \sigma^ {n}. +$$ + +Stage 4 Calculate the viscoplastic strain rate + +$$ +\dot {\epsilon} _ {v p} ^ {n + 1} = \gamma \langle \Phi \rangle a ^ {n + 1}. +$$ + +Stage 5 Apply the equilibrium correction. First calculate $B^{n+1}$ using dis- + + + +placements $d^{n+1}$ . Substitute stresses $\sigma^{n+1}$ into the equilibrium equations and evaluate the residual forces $\psi^{n+1}$ as + +$$ +\psi^ {n + 1} = \int_ {\Omega} [ B ^ {n + 1} ] ^ {T} \sigma^ {n + 1} d \Omega + f ^ {n + 1}. +$$ + +Add these to the vector of incremental pseudo loads for use in the next time step + +$$ +\Delta \boldsymbol {V} ^ {n + 1} = \int_ {\Omega} [ \boldsymbol {B} ^ {n + 1} ] ^ {T} \hat {\boldsymbol {D}} ^ {n + 1} \dot {\boldsymbol {\epsilon}} _ {v p} ^ {n + 1} \Delta t _ {n + 1} d \Omega + \Delta \boldsymbol {f} ^ {n + 1} + \boldsymbol {\psi} ^ {n + 1}. \tag {8.34} +$$ + +Stage 6 Check to see if the viscoplastic strain rate $\dot{\epsilon}_{vp}^{n+1}$ is acceptably close to zero at each Gaussian integrating point throughout the structure (i.e. to within a specified tolerance). + +If so, steady state conditions are deemed to have been achieved and the solution is either terminated or the next load increment is applied. If $\dot{\epsilon}_{vp}^{n+1}$ is non-zero return to Stage 1 and repeat the entire procedure for the next time step. + +The above algorithm can be employed with either a constant or variable time step length. For the variable time step option the interval length $\Delta t_{n+1}$ , for the next time step must be calculated according to (8.29) or (8.31) subject to the restriction of (8.32). + +# 8.5 Evaluation of matrix, H + +For solution by the fully implicit or semi-implicit (trapezoidal) time stepping scheme, matrix $C^{n}$ is required which in turn can be expressed in terms of $H^{n}$ as indicated in (8.14). Matrix $H^{n}$ must be explicitly determined for the yield criterion assumed for material behaviour. From (8.7) and (8.12) we have + +$$ +\boldsymbol {H} = \frac {\partial \dot {\boldsymbol {\epsilon}} _ {v p}}{\partial \boldsymbol {\sigma} ^ {n}} = \gamma \left\{\Phi \frac {\partial \boldsymbol {a} ^ {T}}{\partial \boldsymbol {\sigma}} + \frac {d \Phi}{d F} \boldsymbol {a} \boldsymbol {a} ^ {T} \right\}, \tag {8.35} +$$ + +where the symbols $\langle\rangle$ on $\Phi$ and the superscript n are dropped for convenience. Restricting discussion to the Von Mises yield criterion we have, from (7.64), + +$$ +\boldsymbol {a} ^ {\prime} = \frac {\partial F}{\partial \sigma} = \frac {\partial [ (\sqrt {3}) (J _ {2} ^ {\prime}) ^ {1 / 2} ]}{\partial \sigma}, \tag {8.36} +$$ + +or + +$$ +\boldsymbol {a} ^ {\prime} = \frac {\partial F}{\partial J _ {2} ^ {\prime}} \frac {\partial J _ {2} ^ {\prime}}{\partial \sigma} = \frac {\sqrt {3}}{2 \left(J _ {2} ^ {\prime}\right) ^ {1 / 2}} \left\{\sigma_ {x} ^ {\prime}, \sigma_ {y} ^ {\prime}, \sigma_ {z} ^ {\prime}, 2 \tau_ {y z}, 2 \tau_ {z x}, 2 \tau_ {x y} \right\}, \tag {8.37} +$$ + + + +for a three dimensional situation. Thus + +$$ +\boldsymbol {a} \boldsymbol {a} ^ {T} = \frac {3}{4 J _ {2} ^ {\prime}} \boldsymbol {M} _ {2}, \tag {8.38} +$$ + +where + +$$ +\boldsymbol {M} _ {2} = \left[ \begin{array}{c c c c c c} \left(\sigma_ {x} ^ {\prime}\right) ^ {2} & \sigma_ {x} ^ {\prime} \sigma_ {y} ^ {\prime} & \sigma_ {x} ^ {\prime} \sigma_ {z} ^ {\prime} & 2 \sigma_ {x} ^ {\prime} \tau_ {y z} & 2 \sigma_ {x} ^ {\prime} \tau_ {z x} & 2 \sigma_ {x} ^ {\prime} \tau_ {x y} \\ & \left(\sigma_ {y} ^ {\prime}\right) ^ {2} & \sigma_ {y} ^ {\prime} \sigma_ {z} ^ {\prime} & 2 \sigma_ {y} ^ {\prime} \tau_ {y z} & 2 \sigma_ {y} ^ {\prime} \tau_ {z x} & 2 \sigma_ {y} ^ {\prime} \tau_ {x y} \\ & & \left(\sigma_ {z} ^ {\prime}\right) ^ {2} & 2 \sigma_ {z} ^ {\prime} \tau_ {y z} & 2 \sigma_ {z} ^ {\prime} \tau_ {z x} & 2 \sigma_ {z} ^ {\prime} \tau_ {x y} \\ & & & 4 \left(\tau_ {y z}\right) ^ {2} & 4 \tau_ {y z} \tau_ {z x} & 4 \tau_ {y z} \tau_ {x y} \\ & \text {Symmetric} & & & 4 \left(\tau_ {z x}\right) ^ {2} & 4 \tau_ {z x} \tau_ {x y} \\ & & & & & 4 \left(\tau_ {x y}\right) ^ {2} \end{array} \right]. \tag {8.39} +$$ + +Also from (8.37) + +$$ +\frac {\partial \boldsymbol {a} ^ {T}}{\partial \sigma} = \frac {\sqrt {3}}{2 \left(J _ {2} ^ {\prime}\right) ^ {1 / 2}} \boldsymbol {M} _ {1} - \frac {\sqrt {3}}{4 \left(J _ {2} ^ {\prime}\right) ^ {3 / 2}} \boldsymbol {M} _ {2}, \tag {8.40} +$$ + +where + +$$ +\boldsymbol {M} _ {1} = \left[ \begin{array}{c c c c c c} \frac {2}{3} & - \frac {1}{3} & - \frac {1}{3} & 0 & 0 & 0 \\ & \frac {2}{3} & - \frac {1}{3} & 0 & 0 & 0 \\ & & \frac {2}{3} & 0 & 0 & 0 \\ & & & 2 & 0 & 0 \\ & & \text { Symmetric } & 2 & 0 \\ & & & & 2 \end{array} \right]. \tag {8.41} +$$ + +Substituting from (8.38) and (8.40) into (8.35), and restoring the symbols $\langle \rangle$ , we have finally + +$$ +\boldsymbol {H} = p _ {1} \boldsymbol {M} _ {1} + p _ {2} \boldsymbol {M} _ {2}, \tag {8.42} +$$ + +where + +$$ +p _ {1} = \gamma \left\langle \frac {\sqrt {3}}{2 \left(J _ {2} ^ {\prime}\right) ^ {1 / 2}}. \Phi \right\rangle +$$ + +$$ +p _ {2} = \gamma \left\langle \frac {3}{4 J _ {2} ^ {\prime}} \frac {d \Phi}{d F} - \frac {(\sqrt {3}) \Phi}{4 \left(J _ {2} ^ {\prime}\right) ^ {3 / 2}} \right\rangle . \tag {8.43} +$$ + +The form of $d\Phi/dF$ depends on the explicit form of $\Phi$ employed, examples of which were given in (8.8) and (8.9). Matrix $H^{n}$ is then obtained by using stresses $\sigma^{n}$ to evaluate $J_{2}^{\prime}$ and $M_{2}$ . + +For two-dimensional situations (plane stress, plane strain and axial symmetry) the only relevant stress terms are given in (7.72). In this case $M_{1}$ and $M_{2}$ reduce, on deletion of the appropriate terms, to diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_030.md b/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_030.md new file mode 100644 index 00000000..f35b133d --- /dev/null +++ b/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_030.md @@ -0,0 +1,4089 @@ + + +$$ +\boldsymbol {M} _ {1} = \left[ \begin{array}{c c c c} \frac {2}{3} & - \frac {1}{3} & 0 & - \frac {1}{3} \\ & \frac {2}{3} & 0 & - \frac {1}{3} \\ \text { Symmetric } & 2 & 0 \\ & & & \frac {2}{3} \end{array} \right], \tag {8.44} +$$ + +and + +$$ +\boldsymbol {M} _ {2} = \left[ \begin{array}{c c c c} \left(\sigma_ {x} ^ {\prime}\right) ^ {2} & \sigma_ {x} ^ {\prime} \sigma_ {y} ^ {\prime} & 2 \sigma_ {x} ^ {\prime} \tau_ {x y} & \sigma_ {x} ^ {\prime} \sigma_ {z} ^ {\prime} \\ & \left(\sigma_ {y} ^ {\prime}\right) ^ {2} & 2 \sigma_ {y} ^ {\prime} \tau_ {x y} & \sigma_ {y} ^ {\prime} \sigma_ {z} ^ {\prime} \\ \text { Symmetric } & & 4 \left(\tau_ {x y}\right) ^ {2} & 2 \tau_ {x y} \sigma_ {z} ^ {\prime} \\ - & - & - & - & - \\ & & & & \left(\sigma_ {z} ^ {\prime}\right) ^ {2} \end{array} \right], \tag {8.45} +$$ + +and $J_{2}'$ is given by (7.76). For plane stress and plane strain problems only the upper $3 \times 3$ partition is employed while for axisymmetric situations the complete matrices are utilised with $x$ and $y$ being replaced by $r$ and $z$ respectively. + +Similar expressions can be derived for the Tresca, Mohr–Coulomb and Drucker–Prager yield criteria by employing the appropriate expression for F in (8.36) and repeating the above calculations. The form of F is given in (7.63), (7.65) and (7.66) for the Tresca, Mohr–Coulomb and Drucker–Prager laws respectively. + +# 8.6 Program structure + +The computation sequence for the program is shown in Fig. 8.1. The program structure follows closely that for static elasto-plastic analysis described in Chapter 7. In fact, the majority of the subroutines utilised are common to both applications and it is only the additional subroutines required that are described in this chapter. For the viscoplastic program the time stepping loop replaces the nonlinear solution iteration loop for conventional plasticity and subroutine STEPVP, whose main role is to evaluate quantities at the end of a timestep, replaces the plasticity subroutine RESIDU. In this chapter we need to describe in detail subroutines STIFVP, TANGVP, STEPVP, FLOWVP and STEADY. The descriptions of all other subroutines required for assembly of a working viscoplastic program have been given in Chapters 6 and 7. The version described is restricted to the case of infinitesimal strains. The modifications required to include large deformation effects are straightforward and are left as an exercise to the reader. Furthermore, for implicit schemes, only the Von Mises yield criterion is considered. + +The list of material properties accepted in subroutine INPUT described in Section 6.5.1 must be extended beyond those required for elasto-plastic analysis, since additional material parameters are required to define the + + + +![](images/page-292_dd0bd1c5b3d69ce32bca506ff681d815b359be4ee2fc3475d6468efc5840cfb8.jpg) + +
+flowchart + +```mermaid +graph TD + A["START"] --> B["DIMEN
Presents the variables associated with the dynamic dimensioning process"] + B --> C["INPUT
Inputs data defining geometry, boundary conditions and material properties"] + C --> D["LOADPS
Evaluates the equivalent nodal forces for pressure loading, gravity loading, etc."] + D --> E["ZERO
Sets to zero arrays required for accumulation of data"] + E --> F["INCREM
Increments the applied loads according to specified load factors"] + F --> G["STIFVP
Calculates the element stiffnesses as K_T^n(σ^n) (Eq. (8.24))"] + F --> H["TANGVP
Evaluates D̂^n according to (8.18)"] + G --> I["FRONT
Solves the simultaneous equation system by the frontal method, i.e. Δd^n = [K_T^n"]^-1ΔV^n d^{n+1} = d^n + Δd^n] + H --> I + I --> J["STEPVP
Evaluates quantities at the end of the timestep
a) Δσ^n = D̂^n(B^nΔd - ε̇_vp^nΔt_n) b) σ^{n+1} = σ^n + Δσ^n
c) ε̇_vp^{n+1} = ε̇_vp^n + ε̇_vp^nΔt_n d) Δt_{n+1}"] + I --> K["INVAR
Evaluates the effective stress level"] + I --> L["YIELDF & FLOWVP
Determines:—
a) The flow vector, a
b) ε̇_vp^{n+1} = γ(Φ)a^{n+1}"] + J --> M["OUTPUT
Prints the results for the current timestep"] + L --> M + M --> N["END"] + O["LOAD INCREMENT LOOP"] --> F + P["TIME STEPPING LOOP"] --> F + Q["END"] --> M +``` +
+ +Fig. 8.1 Flow sequence for the two-dimensional elasto-viscoplastic stress analysis program. + +viscoplastic flow. This is accomplished by specifying the value of NPROP as 10 in subroutine DIMEN, described in Section 7.8.1, and inputting the following properties for each different material. + + + +
PROPS(NUMAT, 1)Elastic modulus, $E$ .
PROPS(NUMAT, 2)Poissons ratio, $\nu$ .
PROPS(NUMAT, 3)Material thickness, $t$ .
PROPS(NUMAT, 4)Material mass density, $\rho$ .
PROPS(NUMAT, 5)Uniaxial yield stress $\sigma_{Y}$ (Tresca and Von Mises solids); Cohesion $c$ (Mohr–Coulomb and Drucker–Prager materials).
PROPS(NUMAT, 6)Hardening parameter $H'$ for linear strain hardening.
PROPS(NUMAT, 7)Angle of internal friction for Mohr–Coulomb and Drucker–Prager materials only.
PROPS(NUMAT, 8)The fluidity parameter, $\gamma$ .
PROPS(NUMAT, 9)The coefficient $M$ in (8.8) or coefficient $N$ in (8.9).
PROPS(NUMAT, 10)Indicator specifying type of flow function to be employed:0 – Flow function (8.8)1 – Flow function (8.9)
+ +# 8.7 Formulation of the tangential stiffness matrix + +The role of the subroutines described in this section is to calculate the tangential stiffness matrix for each element according to $(8.24)$ . The complete operation is shared between three subroutines which will now be described. + +# 8.7.1 Subroutine STIFVP + +This subroutine controls the overall formulation of the tangential stiffness matrix for each element and is very similar to subroutine STIFFP, described in Section 7.8.5, which performs the same task for conventional plasticity. For the case of small deformations, matrix $B^{n}$ is constant and equal to $B_{0}$ the usual infinitesimal elastic value. Matrix $B_{0}$ is given by subroutine BMATPS described in Section 6.4.7. To evaluate $K_{T^{n}}$ it is necessary to find $\hat{D}^{n}$ whose precise form is given by (8.18). With the normal elastic material matrix D replaced by $\hat{D}^{n}$ , the stiffness evaluation follows the standard procedure described in Section 7.8.5. Subroutine STIFVP can now be presented and described. + +
SUBROUTINE STIFVP(COORD,IINCS,LNODS,MATNO,MEVAB,MMATS,STVP1
MPOIN,MTOTV,NELEM,NEVAB,NGAUS,NNODE,NSTRE,STVP2
NSTR1,POSGP,PROPS,WEIGP,MELEM,MTOTG,STVP3
STRSG,NTYPE,NCRIT,TIMEX,DTIME)STVP4
C***************STVP5
CSTVP6
C**** THIS SUBROUTINE EVALUATES THE STIFFNESS MATRIX FOR EACH ELEMENTSTVP7
C IN TURNSTVP8
CSTVP9
C***************STVP10
DIMENSION BMATX(4,18),CARTD(2,9),COORD(MPOIN,2),DBMAT(4,18),STVP11
DERIV(2,9),DEVIA(4),DMATX(4,4),STVP12
ELCOD(2,9),EPSTN(MTOTG),ESTIF(18,18),LNODS(MELEM,9),STVP13
MATNO(MELEM),POSGP(4),PROPS(MMATS,10),SHAPE(9),STVP14
WEIGP(4),STRES(4),STRSG(4,MTOTG),STVP15
+ + + +```asm +DVECT(4), AVECT(4), GPCOD(2,9) +TWOPI=6.283185308 +REWIND 1 +KGAUS=0 +STVP 16 +STVP 17 +STVP 18 +STVP 19 +STVP 20 +STVP 21 +STVP 22 +STVP 23 +STVP 24 +STVP 25 +STVP 26 +STVP 27 +STVP 28 +STVP 29 +STVP 30 +STVP 31 +STVP 32 +STVP 33 +STVP 34 +STVP 35 +STVP 36 +STVP 37 +STVP 38 +STVP 39 +STVP 40 +STVP 41 +STVP 42 +STVP 43 +STVP 44 +STVP 45 +STVP 46 +STVP 47 +STVP 48 +STVP 49 +STVP 50 +STVP 51 +STVP 52 +STVP 53 +STVP 54 +STVP 55 +STVP 56 +STVP 57 +STVP 58 +STVP 59 +STVP 60 +STVP 61 +STVP 62 +STVP 63 +STVP 64 +STVP 65 +STVP 66 +STVP 67 +STVP 68 +STVP 69 +STVP 70 +STVP 71 +STVP 72 +STVP 73 +STVP 74 +STVP 75 +STVP 76 +STVP 77 +STVP 78 +STVP 79 +STVP 80 +STVP 81 +STVP 82 +STVP 83 +STVP 84 +STVP 85 +STVP 86 +STVP 87 +STVP 88 +STVP 89 +STVP 90 +STVP 91 +STVP 92 +STVP 93 +STVP 94 +STVP 95 +STVP 96 +STVP 97 +STVP 98 +STVP 99 +STVP 100 +STVP 101 +STVP 102 +STVP 103 +STVP 104 +STVP 105 +STVP 106 +STVP 107 +STVP 108 +STVP 109 +STVP 110 +STVP 111 +STVP 112 +STVP 113 +STVP 114 +STVP 115 +STVP 116 +STVP 117 +STVP 118 +STVP 119 +STVP 120 +STVP 121 +STVP 122 +STVP 123 +STVP 124 +STVP 125 +STVP 126 +STVP 127 +STVP 128 +STVP 129 +STVP 130 +STVP 131 +STVP 132 +STVP 133 +STVP 134 +STVP 135 +STVP 136 +STVP 137 +STVP 138 +STVP 139 +STVP 140 +STVP 141 +STVP 142 +STVP 143 +STVP 144 +STVP 145 +STVP 146 +STVP 147 +STVP 148 +STVP 149 +STVP 150 +STVP 151 +STVP 152 +STVP 153 +STVP 154 +STVP 155 +STVP 156 +STVP 157 +STVP 158 +STVP 159 +STVP 160 +STVP 161 +STVP 162 +STVP 163 +STVP 164 +STVP 165 +STVP 166 +STVP 167 +STVP 168 +STVP 169 +STVP 170 +STVP 171 +STVP 172 +STVP 173 +STVP 174 +STVP 175 +STVP 176 +STVP 177 +STVP 178 +STVP 179 +STVP 180 +STVP 181 +STVP 182 +STVP 183 +STVP 184 +STVP 185 +STVP 186 +STVP 187 +STVP 188 +STVP 189 +STVP 190 +STVP 191 +STVP 192 +STVP 193 +STVP 194 +STVP 195 +STVP 196 +STVP 197 +STVP 198 +STVP 199 +STVP 200 +STVP 201 +STVP 202 +STVP 203 +STVP 204 +STVP 205 +STVP 206 +STVP 207 +STVP 208 +STVP 209 +STVP 210 +STVP 211 +STVP 212 +STVP 213 +STVP 214 +STVP 215 +STVP 216 +STVP 217 +STVP 218 +STVP 219 +STVP 220 +STVP 221 +STVP 222 +STVP 223 +STVP 224 +STVP 225 +STVP 226 +STVP 227 +STVP 228 +STVP 229 +STVP 230 +STVP 231 +STVP 232 +STVP 233 +STVP 234 +STVP 235 +STVP 236 +STVP 237 +STVP 238 +STVP 239 +STVP 240 +STVP 241 +STVP 242 +STVP 243 +STVP 244 +STVP 245 +STVP 246 +STVP 247 +STVP 248 +STVP 249 +STVP 250 +STVP 251 +STVP 252 +STVP 253 +STVP 254 +STVP 255 +STVP 256 +STVP 257 +STVP 258 +STVP 259 +STVP 260 +STVP 261 +STVP 262 +STVP 263 +STVP 264 +STVP 265 +STVP 266 +STVP 267 +STVP 268 +STVP 269 +STVP 270 +STVP 271 +STVP 272 +STVP 273 +STVP 274 +STVP 275 +STVP 276 +STVP 277 +STVP 278 +STVP 279 +STVP 280 +STVP 281 +STVP 282 +STVP 283 +STVP 284 +STVP 285 +STVP 286 +STVP 287 +STVP 288 +STVP 289 +STVP 290 +STVP 291 +STVP 292 +STVP 293 +STVP 294 +STVP 295 +STVP 296 +STVP 297 +STVP 298 +STVP 299 +STVP 300 +STVP 301 +STVP 302 +STVP 303 +STVP 304 +STVP 305 +STVP 306 +STVP 307 +STVP 308 +STVP 309 +STVP 310 +STVP 311 +STVP 312 +STVP 313 +STVP 314 +STVP 315 +STVP 316 +STVP 317 +STVP 318 +STVP 319 +STVP 320 +STVP 321 +STVP 322 +STVP 323 +STVP 324 +STVP 325 +STVP 326 +STVP 327 +STVP 328 +STVP 329 +STVP 330 +STVP 331 +STVP 332 +STVP 333 +STVP 334 +STVP 335 +STVP 336 +STVP 337 +STVP 338 +STVP 339 +STVP 340 +STVP 341 +STVP 342 +STVP 343 +STVP 344 +STVP 345 +STVP 346 +STVP 347 +STVP 348 +STVP 349 +STVP 350 +STVP 351 +STVP 352 +STVP 353 +STVP 354 +STVP 355 +STVP 356 +STVP 357 +STVP 358 +STVP 359 +STVP 360 +STVP 361 +STVP 362 +STVP 363 +STVP 364 +STVP 365 +STVP 366 +STVP 367 +STVP 368 +STVP 369 +STVP 370 +STVP 371 +STVP 372 +STVP 373 +STVP 374 +STVP 375 +STVP 376 +STVP 377 +STVP 378 +STVP 379 +STVP 380 +STVP 381 +STVP 382 +STVP 383 +STVP 384 +STVP 385 +STVP 386 +STVP 387 +STVP 388 +STVP 389 +STVP 390 +STVP 391 +STVP 392 +STVP 393 +STVP 394 +STVP 395 +STVP 396 +STVP 397 +STVP 398 +STVP 399 +STVP 400 +STVP 401 +STVP 402 +STVP 403 +STVP 404 +STVP 405 +STVP 406 +STVP 407 +STVP 408 +STVP 409 +STVP 410 +STVP 411 +STVP 412 +STVP 413 +STVP 414 +STVP 415 +STVP 416 +STVP 417 +STVP 418 +STVP 419 +STVP 420 +STVP 421 +STVP 422 +STVP 423 +STVP 424 +STVP 425 +STVP 426 +STVP 427 +STVP 428 +STVP 429 +STVP 430 +STVP 431 +STVP 432 +STVP 433 +STVP 434 +STVP 435 +STVP 436 +STVP 437 +STVP 438 +STVP 439 +STVP 440 +STVP 441 +STVP 442 +STVP 443 +STVP 444 +STVP 445 +STVP 446 +STVP 447 +STVP 448 +STVP 449 +STVP 450 +STVP 451 +STVP 452 +STVP 453 +STVP 454 +STVP 455 +STVP 456 +STVP 457 +STVP 458 +STVP 459 +STVP 460 +STVP 461 +STVP 462 +STVP 463 +STVP 464 +STVP 465 +STVP 466 +STVP 467 +STVP 468 +STVP 469 +STVP 470 +STVP 471 +STVP 472 +STVP 473 +STVP 474 +STVP 475 +STVP 476 +STVP 477 +STVP 478 +STVP 479 +STVP 480 +STVP 481 +STVP 482 +STVP 483 +STVP 484 +STVP 485 +STVP 486 +STVP 487 +STVP 488 +STVP 489 +STVP 490 +STVP 491 +STVP 492 +STVP 493 +STVP 494 +STVP 495 +STVP 496 +STVP 497 +STVP 498 +STVP 499 +STVP 500 +STVP 501 +STVP 502 +STVP 503 +STVP 504 +STVP 505 +STVP 506 +STVP 507 +STVP 508 +STVP 509 +STVP 510 +STVP 511 +STVP 512 +STVP 513 +STVP 514 +STVP 515 +STVP 516 +STVP 517 +STVP 518 +STVP 519 +STVP 520 +STVP 521 +STVP 522 +STVP 523 +STVP 524 +STVP 525 +STVP 526 +STVP 527 +STVP 528 +STVP 529 +STVP 530 +STVP 531 +STVP 532 +STVP 533 +STVP 534 +STVP 535 +STVP 536 +STVP 537 +STVP 538 +STVP 539 +STVP 540 +STVP 541 +STVP 542 +STVP 543 +STVP 544 +STVP 545 +STVP 546 +STVP 547 +STVP 548 +STVP 549 +STVP 550 +STVP 551 +STVP 552 +STVP 553 +STVP 554 +STVP 555 +STVP 556 +STVP 557 +STVP 558 +STVP 559 +STVP 560 +STVP 561 +STVP 562 +STVP 563 +STVP 564 +STVP 565 +STVP 566 +STVP 567 +STVP 568 +STVP 569 +STVP 570 +STVP 571 +STVP 572 +STVP 573 +STVP 574 +STVP 575 +STVP 576 +STVP 577 +STVP 578 +STVP 579 +STVP 580 +STVP 581 +STVP 582 +STVP 583 +STVP 584 +STVP 585 +STVP 586 +STVP 587 +STVP 588 +STVP 589 +STVP 590 +STVP 591 +STVP 592 +STVP 593 +STVP 594 +STVP 595 +STVP 596 +STVP 597 +STVP 598 +STVP 599 +STVP 600 +STVP 601 +STVP 602 +STVP 603 +STVP 604 +STVP 605 +STVP 606 +STVP 607 +STVP 608 +STVP 609 +STVP 610 +STVP 611 +STVP 612 +STVP 613 +STVP 614 +STVP 615 +STVP 616 +STVP 617 +STVP 618 +STVP 619 +STVP 620 +STVP 621 +STVP 622 +STVP 623 +STVP 624 +STVP 625 +STVP 626 +STVP 627 +STVP 628 +STVP 629 +STVP 630 +STVP 631 +STVP 632 +STVP 633 +STVP 634 +STVP 635 +STVP 636 +STVP 637 +STVP 638 +STVP 639 +STVP 640 +STVP 641 +STVP 642 +STVP 643 +STVP 644 +STVP 645 +STVP 646 +STVP 647 +STVP 648 +STVP 649 +STVP 650 +STVP 651 +STVP 652 +STVP 653 +STVP 654 +STVP 655 +STVP 656 +STVP 657 +STVP 658 +STVP 659 +STVP 660 +STVP 661 +STVP 662 +STVP 663 +STVP 664 +STVP 665 +STVP 666 +STVP 667 +STVP 668 +STVP 669 +STVP 670 +STVP 671 +STVP 672 +STVP 673 +STVP 674 +STVP 675 +STVP 676 +STVP 677 +STVP 678 +STVP 679 +STVP 680 +STVP 681 +STVP 682 +STVP 683 +STVP 684 +STVP 685 +STVP 686 +STVP 687 +STVP 688 +STVP 689 +STVP 690 +STVP 691 +STVP 692 +STVP 693 +STVP 694 +STVP 695 +STVP 696 +STVP 697 +STVP 698 +STVP 699 +STVP 700 +STVP 701 +STVP 702 +STVP 703 +STVP 704 +STVP 705 +STVP 706 +STVP 707 +STVP 708 +STVP 709 +STVP 710 +STVP 711 +STVP 712 +STVP 713 +STVP 714 +STVP 715 +STVP 716 +STVP 717 +STVP 718 +STVP 719 +STVP 720 +STVP 721 +STVP 722 +STVP 723 +STVP 724 +STVP 725 +STVP 726 +STVP 727 +STVP 728 +STVP 729 +STVP 730 +STVP 731 +STVP 732 +STVP 733 +STVP 734 +STVP 735 +STVP 736 +STVP 737 +STVP 738 +STVP 739 +STVP 740 +STVP 741 +STVP 742 +STVP 743 +STVP 744 +STVP 745 +STVP 746 +STVP 747 +STVP 748 +STVP 749 +STVP 750 +STVP 751 +STVP 752 +STVP 753 +STVP 754 +STVP 755 +STVP 756 +STVP 757 +STVP 758 +STVP 759 +STVP 760 +STVP 761 +STVP 762 +STVP 763 +STVP 764 +STVP 765 +STVP 766 +STVP 767 +STVP 768 +STVP 769 +STVP 770 +STVP 771 +STVP 772 +STVP 773 +STVP 774 +STVP 775 +STVP 776 +STVP 777 +STVP 778 +STVP 779 +STVP 780 +STVP 781 +STVP 782 +STVP 783 +STVP 784 +STVP 785 +STVP 786 +STVP 787 +STVP 788 +STVP 789 +STVP 790 +STVP 791 +STVP 792 +STVP 793 +STVP 794 +STVP 795 +STVP 796 +STVP 797 +STVP 798 +STVP 799 +STVP 800 +STVP 801 +STVP 802 +STVP 803 +STVP 804 +STVP 805 +STVP 806 +STVP 807 +STVP 808 +STVP 809 +STVP 810 +STVP 811 +STVP 812 +STVP 813 +STVP 814 +STVP 815 +STVP 816 +STVP 817 +STVP 818 +STVP 819 +STVP 820 +STVP 821 +STVP 822 +STVP 823 +STVP 824 +STVP 825 +STVP 826 +STVP 827 +STVP 828 +STVP 829 +STVP 830 +STVP 831 +STVP 832 +STVP 833 +STVP 834 +STVP 835 +STVP 836 +STVP 837 +STVP 838 +STVP 839 +STVP 840 +STVP 841 +STVP 842 +STVP 843 +STVP 844 +STVP 845 +STVP 846 +STVP 847 +STVP 848 +STVP 849 +STVP 850 +STVP 851 +STVP 852 +STVP 853 +STVP 854 +STVP 855 +STVP 856 +STVP 857 +STVP 858 +STVP 859 +STVP 860 +STVP 861 +STVP 862 +STVP 863 +STVP 864 +STVP 865 +STVP 866 +STVP 867 +STVP 868 +STVP 869 +STVP 870 +STVP 871 +STVP 872 +STVP 873 +STVP 874 +STVP 875 +STVP 876 +STVP 877 +STVP 878 +STVP 879 +STVP 880 +STVP 881 +STVP 882 +STVP 883 +STVP 884 +STVP 885 +STVP 886 +STVP 887 +STVP 888 +STVP 889 +STVP 890 +STVP 891 +STVP 892 +STVP 893 +STVP 894 +STVP 895 +STVP 896 +STVP 897 +STVP 898 +STVP 899 +STVP 900 +STVP 901 +STVP 902 +``` + + + +
50 CONTINUESTVP81
CSTVP82
C*** CONSTRUCT THE LOWER TRIANGLE OF THE STIFFNESS MATRIXSTVP83
CSTVP84
DO 60 IEVAB=1,NEVABSTVP85
DO 60 JEVAB=1,NEVABSTVP86
60 ESTIF(JEVAB,IEVAB)=ESTIF(IEVAB,JEVAB)STVP87
CSTVP88
C*** STORE THE STIFFNESS MATRIX,STRESS MATRIX AND SAMPLING POINTSTVP89
C COORDINATES FOR EACH ELEMENT ON DISC FILESTVP90
CSTVP91
WRITE(1) ESTIFSTVP92
70 CONTINUESTVP93
RETURNSTVP94
ENDSTVP95
+ +
STVP 17Compute the value of $2\pi$ .
STVP 18Rewind the disc file on which the element stiffness matrices will be stored in turn.
STVP 19Set to zero the counter which indicates the overall Gauss point location.
STVP 23Enter the loop over each element in the structure.
STVP 24Identify the material property type of the current element.
STVP 28–33Store the element nodal coordinates in the local array ELCOD for convenient use later.
STVP 34Identify the element thickness.
STVP 38–40Zero the element stiffness array.
STVP 41Set to zero the element Gauss point counter.
STVP 45–48Enter the numerical integration loops and locate the position $(\xi, \eta)$ of the current point.
STVP 49–50Increment the local and global Gauss point counters.
STVP 54Call subroutine MODPS to evaluate the elasticity matrix, $D$ .
STVP 58Evaluate the shape functions $N_i$ and $\partial N_i/\partial \xi$ , $\partial N_i/\partial \eta$ for the current Gauss point.
STVP 59–60Evaluate the Gauss point coordinates, GPCOD(IDIME, KGASP), the determinant of the Jacobian matrix $|J|$ and the Cartesian derivatives of the shape functions $\partial N_i/\partial x$ , $\partial N_i/\partial y$ (or $\partial N_i/\partial r$ , $\partial N_i/\partial z$ for axisymmetric problems).
STVP 61–63Calculate the elemental volume for numerical integration as $|J|W_\xi W_\eta$ taking care to multiply by the appropriate element thickness or by $2\pi r$ for axisymmetric problems.
STVP 67Evaluate the $B$ matrix.
STVP 68–69Store the current stresses in a local array.
STVP 70–71For an implicit or semi-implicit timestepping scheme $(\Theta \neq 0)$ , call subroutine TANGVP to evaluate $\hat{D}^n$ which is stored as DMATX.
STVP 72Evaluate $DB$ (or $\hat{D}^n B$ for implicit schemes).
STVP 76–80Compute the upper triangle of the element stiffness matrix as
+ + + +$$ +\int_ {\Omega} \boldsymbol {B} ^ {T} \hat {\boldsymbol {D}} ^ {n} \boldsymbol {B} d \Omega . +$$ + +STVP 81 End of loop for numerical integration. + +STVP 85–87 Complete the lower triangle of the element stiffness matrix by symmetry. + +STVP 92 Store the element stiffness matrix on disc file 1. + +STVP 93 Return to process the next element. + +# 8.7.2 Subroutine TANGVP + +The function of this subroutine is to evaluate $\hat{D}^{n}$ for use in (8.24). Matrix $\hat{D}^{n}$ , which is defined in (8.18), is stress dependent and therefore must be calculated for each Gaussian integrating point in turn. The computational sequence followed is: + +a) Evaluate $H^n$ according to (8.42) +b) Calculate $C^n$ according to (8.14) +c) Evaluate $\tilde{D}^n$ according to (8.18) + +Two forms of the flow function $\Phi$ are considered as defined in (8.8) and (8.9). Thus, for use in (8.43), we have + +$$ +\frac {d \Phi}{d F} = \frac {M}{F _ {0}} e ^ {M \left(\frac {F - F _ {0}}{F _ {0}}\right)} +$$ + +or + +$$ +\frac {d \Phi}{d F} = \frac {N}{F _ {0}} \left(\frac {F - F _ {0}}{F _ {0}}\right) ^ {N - 1}. \tag {8.46} +$$ + +Array DMATX which originally contains the elastic matrix D is used to finally store $\hat{D}^{n}$ . The matrix inversions required in (8.18) are performed by a separate subroutine, INVERT. + +Subroutine TANGVP is now presented and described. +```csv +SUBROUTINE TANGVP(LPROP,STRES,PROPS,TIMEX,DTIME,NTSTRE,NTYPE,MMATS,NCRIT,DMATX) TGVP 1 +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +D +D +E +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +C +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +1 +T +G +V +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +M +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +II +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +No +S +I +No +S +I +No +S +I +No +S +I +No +S +I +No +S +I +No +S +I +No +S +I +No +S +I +No +S +I +No +S +I +No +S +I +No +S +I +No +S +I +No +S +I +No +S +I +No +S +1 +T +G +V +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +M +S +I +N +S +1 +T +G +V +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +M +S +I +N +S +I +N +S +1 +T +G +V +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +M +S +I +N +S +I +N +S +I +N +S +1 +T +G +V +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +M +S +I +N +S +I +N +S +I +N +S +I +N +S +1 +T +G +V +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +M +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +1 +T +G +V +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +M +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +1 +T +G +V +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +M +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +1 +T +G +V +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +M +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +1 +T +G +V +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +M +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +1 +T +G +V +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +M +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +1 +T +G +V +P +P +P +P +P +P +P +P +P +P +P +P +P +P +M +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +1 +T +G +V +P +P +P +P +P +P +P +P +P +P +P +M +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +1 +T +G +V +P +P +P +P +P +P +P +P +M +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +1 +T +G +V +P +P +P +P +P +M +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +1 +T +G +V +P +P +M +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +1 +T +G +W +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +No +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +C +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +T +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +F +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +M +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +P +N +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +M +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +M +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +N +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +S +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +M +M +M +M +M +M +M +M +M +M +M +M +M +M +M +M +M +M +M +M +M +M +M +M +M +M +M +M +M +M +M +M +M +M +M +M +M +M +M +M +M +M +M +M +M +M +M +M +M +M +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +M +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +S +I +N +s +I +N +s +I +N +s +I +N +s +I +N +s +I +N +s +I +N +s +I +N +s +I +N +s +I +N +s +I +N +s +I +N +s +I +N +s +I +N +s +I +N +s +I +N +s +I +N +s +I +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +N +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +№ +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +0 +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +Yes +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +NO +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No + No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +S +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +No +# +``` + + + +```csv +10 CMULT=FNORM**DELTA TGVP 24 +GRADP=DELTA*(FNORM**(DELTA-1.0))/FDATM TGVP 25 +20 FACT1=GAMMA*ROOT3*CMULT/(2.0*STEFF) TGVP 26 +FACT2=GAMMA*(0.75*GRADP/VARJ2-3.0*CMULT/(4.0*ROOT3*STEFF*VARJ2)) TGVP 27 +C TGVP 28 +C*** MATRICES M1 AND M2 FOR A VON MISES MATERIAL TGVP 29 +C TGVP 30 +TRIX1(1,1)=0.666666667 TGVP 31 +TRIX1(1,2)=-0.333333333 TGVP 32 +TRIX1(1,3)=0.0 TGVP 33 +TRIX1(2,2)=0.666666667 TGVP 34 +TRIX1(2,3)=0.0 TGVP 35 +TRIX1(3,3)=2.0 TGVP 36 +IF(NTYPE.NE.3) GO TO 30 TGVP 37 +TRIX1(1,4)=-0.333333333 TGVP 38 +TRIX1(2,4)=-0.333333333 TGVP 39 +TRIX1(3,4)=0.0 TGVP 40 +TRIX1(4,4)=0.666666667 TGVP 41 +30 TRIX2(1,1)=DEVIA(1)*DEVIA(1) TGVP 42 +TRIX2(1,2)=DEVIA(1)*DEVIA(2) TGVP 43 +TRIX2(1,3)=2.0*DEVIA(1)*DEVIA(3) TGVP 44 +TRIX2(2,2)=DEVIA(2)*DEVIA(2) TGVP 45 +TRIX2(2,3)=2.0*DEVIA(2)*DEVIA(3) TGVP 46 +TRIX2(3,3)=4.0*DEVIA(3)*DEVIA(3) TGVP 47 +IF(NTYPE.NE.3) GO TO 40 TGVP 48 +TRIX2(1,4)=DEVIA(1)*DEVIA(4) TGVP 49 +TRIX2(2,4)=DEVIA(2)*DEVIA(4) TGVP 50 +TRIX2(3,4)=2.0*DEVIA(3)*DEVIA(4) TGVP 51 +TRIX2(4,4)=DEVIA(4)*DEVIA(4) TGVP 52 +40 DO 50 ISTRE=1,NSTRE TGVP 53 +DO 50 JSTRE=1,NSTRE TGVP 54 +TRIX1(JSTRE,ISTRE)=TRIX1(ISTRE,JSTRE) TGVP 55 +50 TRIX2(JSTRE,ISTRE)=TRIX2(ISTRE,JSTRE) TGVP 56 +DO 60 ISTRE=1,NSTRE TGVP 57 +DO 60 JSTRE=1,NSTRE TGVP 58 +60 CMATX(ISTRE,JSTRE)=TIMEX*DTIME*(FACT1*TRIX1(ISTRE,JSTRE) TGVP 59 +.+FACT2*TRIX2(ISTRE,JSTRE)) TGVP 60 +CALL INVERT(DMATX,TMATX,NSTRE) TGVP 61 +DO 70 ISTRE=1,NSTRE TGVP 62 +DO 70 JSTRE=1,NSTRE TGVP 63 +70 TMATX(ISTRE,JSTRE)=TMATX(ISTRE,JSTRE)+CMATX(ISTRE,JSTRE) TGVP 64 +CALL INVERT(TMATX,DMATX,NSTRE) TGVP 65 +RETURN TGVP 66 +END TGVP 67 +``` + +TGVP 10 Evaluate $\sqrt{(3)}$ . + +TGVP 11 Identify the yield stress F as FDATM. + +TGVP 12 Identify the fluidity parameter $\gamma$ as GAMMA. + +TGVP 13 For flow law (8.8) store the index $M$ as DELTA, or for flow law (8.9) store the index $N$ as DELTA. + +TGVP 14 Identify the type of flow function to be used as governed by material property PROPS(LPROP,10) supplied as input: + +NFLOW = 0 - Flow function (8.8) to be used, + +NFLOW = 1 - Flow function (8.9) to be used. + +TGVP 15–16 Call subroutine INVAR to evaluate the effective stress components, the effective stress level and $J_{2}'$ . + +TGVP 17-18 Evaluate $F-F_{0}/F_{0}$ as FNORM. + + + +TGVP 21-22 Evaluate $\Phi$ and $d\Phi / dF$ for flow function (8.8). +TGVP 24–25 Evaluate $\Phi$ and $d\Phi/dF$ for flow function (8.9). +TGVP 26–27 Compute $p_{1}$ and $p_{2}$ according to (8.43). +TGVP 31–41 Evaluate $M_{1}$ according to (8.44) taking the full $4 \times 4$ matrix for axisymmetric situations. +TGVP 42–52 Evaluate $M_{2}$ according to (8.45) taking the full $4 \times 4$ matrix for axisymmetric situations. +TGVP 53–56 Complete the lower triangle of $M_{1}$ and $M_{2}$ by symmetry. +TGVP 57–60 Compute matrix $C^{n}$ according to (8.14) and (8.42). +TGVP 61 Call subroutine INVERT to evaluate $D^{-1}$ and store as TMATX. +TGVP 62-64 Compute $D^{-1} + C^n$ +TGVP 65 Call subroutine INVERT to evaluate $(D^{-1}+C^{n})^{-1}$ and store as DMATX. + +# 8.7.3 Subroutine INVERT + +The function of this subroutine is to determine the inverse of any arbitrary square matrix. In particular, the subroutine accepts a matrix AMATX with dimensions NARAY×NARAY and evaluates the inverse as BMATX. The procedure employed is the standard method of reduction in which starting from the original matrix AMATX and assuming an identity matrix for BMATX, an elimination process is followed until AMATX is reduced to an identity form. Then at this stage BMATX is the inverse of AMATX. + +The subroutine is presented below without further comment. + +```txt +SUBROUTINE INVERT(AMATX,BMATX,NARAY) INVT 1 +C******************************* +C INVT 2 +C INVT 3 +C*** TO PROVIDE THE INVERSE OF AMATX AS BMATX INVT 4 +C INVT 5 +C******************************* +INVT 6 +DIMENSION AMATX(4,4),BMATX(4,4) INVT 7 +DO 10 IARAY=1,NARAY INVT 8 +DO 10 JARAY=1,NARAY INVT 9 +BMATX(IARAY,JARAY)=0.0 INVT 10 +10 IF(IARAY.EQ.JARAY) BMATX(IARAY,JARAY)=1.0 INVT 11 +DO 20 IARAY=1,NARAY INVT 12 +DENOM=AMATX(IARAY,IARAY) INVT 13 +DO 30 JARAY=1,NARAY INVT 14 +AMATX(IARAY,JARAY)=AMATX(IARAY,JARAY)/DENOM INVT 15 +30 BMATX(IARAY,JARAY)=BMATX(IARAY,JARAY)/DENOM INVT 16 +KARAY=IARAY+1 INVT 17 +IF(KARAY.GT.NARAY) GO TO 40 INVT 18 +DO 20 JARAY=KARAY,NARAY INVT 19 +CONST=AMATX(JARAY,IARAY) INVT 20 +DO 20 LARAY=IARAY,NARAY INVT 21 +AMATX(JARAY,LARAY)=AMATX(JARAY,LARAY)-AMATX(IARAY,LARAY) INVT 22 +.*CONST INVT 23 +20 BMATX(JARAY,LARAY)=BMATX(JARAY,LARAY)-BMATX(IARAY,LARAY) INVT 24 +.*CONST INVT 25 +40 CONTINUE INVT 26 +DO 50 IARAY=2,NARAY INVT 27 +KARAY=NARAY-IARAY+2 INVT 28 +``` + + + +LIMIT=KARAY-1 INVT 29 +DO 50 LARAY=1,LIMIT INVT 30 +CONST=AMATX(LARAY,KARAY) INVT 31 +DO 50 JARAY=1,KARAY INVT 32 +AMATX(LARAY,JARAY)=AMATX(LARAY,JARAY)-AMATX(KARAY,JARAY) INVT 33 +.*CONST INVT 34 +50 BMATX(LARAY,JARAY)=BMATX(LARAY,JARAY)-BMATX(KARAY,JARAY) INVT 35 +.*CONST INVT 36 +RETURN INVT 37 +END INVT 38 + +# 8.8 Subroutine STEPVP for the evaluation of end of time step quantities and equilibrium correction terms + +With reference to Fig. 8.1, this subroutine evaluates quantities, such as stresses and viscoplastic strains, at the end of the current timestep and also calculates the loading to be applied during the next timestep. The subroutine is structured to perform the following operations sequentially: + +(a) All quantities at the end of timestep $n$ are calculated as $( )^{n+1}$ . +(b) Subroutine INVAR, YIELDF and FLOWVP are called to evaluate the current viscoplastic flow rate, $\dot{\epsilon}_{vp}^{n+1}$ . +(c) The maximum permissible interval length, $\Delta t_{n + 1}$ , for the next timestep as governed by (8.29) and (8.32) is calculated. +(d) The residual forces, $\psi^{n+1}$ , are evaluated and the loads, $\Delta V^{n+1}$ , for the next timestep then calculated. + +In the program presented we restrict ourselves to loads applied in discrete increments. An increment of load is applied and the time stepping process is followed until either steady state conditions are achieved, or a specified number of timesteps is reached. Then a further increment of load is applied and the process repeated. Thus in (8.23), $\Delta f^{n} = 0$ for all stages other than the first timestep of a particular load increment. + +The attainment of steady state conditions can be monitored by accumulating some measure of the viscoplastic strain rate for all Gauss points in the structure. At steady state this quantity will become zero. The degree of total viscoplastic flow at any point is best monitored by evaluating the total effective viscoplastic strain rate at all Gauss points according to + +$$ +\bar {\dot {\epsilon}} _ {v p} = (\sqrt {\frac {2}{3}}) \{(\dot {\epsilon} _ {i j}) _ {v p} (\dot {\epsilon} _ {i j}) _ {v p} \} ^ {1 / 2}. \tag {8.47} +$$ + +Subroutine STEPVP is now presented and described. + +SUBROUTINE STEPVP(ASDIS,COORD,ELOAD,ISTEP,LNODS,LPROP,TIMEX, SPVP 1 +. MATNO,MELEM,MMATS,MPOIN,MTOTG,TAUFT,DTIME, SPVP 2 +. MTOTV,NDOFN,NELEM,NEVAB,NGAUS,NNODE,NSTR1, SPVP 3 +. NTYPE,POSGP,PROPS,NSTRE,NCRIT,STRSG,WEIGP, SPVP 4 +. TDISP,VISTN,VIVEL,TLOAD,FTIME,DTINT,IINCS) SPVP 5 +C*************** SPVP 6 +C +C**** EVALUATES QUANTITIES AT END OF TIME STEP AND CALCULATES THE SPVP 7 +C RESIDUAL FORCES AND PSEUDO FORCES FOR THE NEXT STEP SPVP 8 +C +C*************** SPVP 9 +C +C*************** SPVP 10 +C + + + +```asm +DIMENSION ASDIS(MTOTV), AVECT(4), CARTD(2,9), COORD(MPOIN,2), SPVP 12 +. DEVIA(4), ELCOD(2,9), ELDIS(2,9), ELOAD(MELEM,18), SPVP 13 +. LNODS(MELEM,9), POSGP(4), PROPS(MMATS,10), STRAN(4), SPVP 14 +. STRES(4), STRSG(4, MTOTG), VIVEL(5, MTOTG), SPVP 15 +. VISTN(4, MTOTG), WEIGP(4), DMATX(4,4), TLDIS(2,9), SPVP 16 +. DERIV(2,9), SHAPE(9), GPCOD(2,9), TDISP(MTOTV), SPVP 17 +. MATNO(MELEM), DJACM(2,2), BMATX(4,18), DESTN(4), SPVP 18 +. TLOAD(MELEM,18), SVECT(4) SPVP 19 +TWOPI=6.283185308 SPVP 20 +DO 10 IELEM=1, NELEM SPVP 21 +DO 10 IEVAB=1, NEVAB SPVP 22 +10 ELOAD(IELEM,IEVAB)=0.0 SPVP 23 +KGAUS=0 SPVP 24 +DNEXT=FTIME*DTIME SPVP 25 +DO 80 IELEM=1, NELEM SPVP 26 +LPROP=MATNO(IELEM) SPVP 27 +C +C*** STORE COORDINATES AND INCREMENTAL DISPLACEMENTS OF THE SPVP 29 +C ELEMENT NODAL POINTS SPVP 30 +C +DO 20 INODE=1, NNODE SPVP 31 +LNODE=IABS(LNODS(IELEM,INODE)) SPVP 32 +NPOSN=(LNODE-1)*NDOFN SPVP 33 +DO 20 IDOFN=1, NDOFN SPVP 34 +NPOSN=NPOSN+1 SPVP 35 +ELCOD(IDOFN,INODE)=COORD(LNODE,IDOFN) SPVP 36 +TLDIS(IDOFN,INODE)=TDISP(NPOSN) SPVP 37 +20 ELDIS(IDOFN,INODE)=ASDIS(NPOSN) SPVP 38 +THICK=PROPS(LPROP,3) SPVP 39 +KGASP=0 SPVP 40 +DO 70 IGAUS=1, NGAUS SPVP 41 +DO 70 JGAUS=1, NGAUS SPVP 42 +EXISP=POSGP(IGAUS) SPVP 43 +ETASP=POSGP(JGAUS) SPVP 44 +KGAUS=KGAUS+1 SPVP 45 +KGASP=KGASP+1 SPVP 46 +CALL MODPS(DMATX, LPROP, MMATS, NTYPE, PROPS) SPVP 47 +DO 30 ISTR1=1, NSTR1 SPVP 48 +30 STRES(ISTR1)=STRSG(ISTR1, KGAUS) SPVP 49 +CALL INVAR(DEVIA, LPROP, MMATS, NCRIT, PROPS, SINT3, STEFF, STRES, THETA, SPVP 50 +. VARJ2, YIELD) SPVP 51 +IF(TIMEX.GT.0.0) CALL TANGVP(LPROP, STRES, PROPS, TIMEX, DTIME, SPVP 52 +. NSTRE, NTYPE, MMATS, NCRIT, DMATX) SPVP 53 +CALL SFR2(DERIV, ETASP, EXISP, NNODE, SHAPE) SPVP 54 +CALL JACOB2(CARTD, DERIV, DJACB, ELCOD, GPCOD, IELEM, KGASP, NNODE, SHAPE) SPVP 55 +DVOLU=DJACB*WEIGP(IGAUS)*WEIGP(JGAUS)- SPVP 56 +IF(NTYPE.EQ.3) DVOLU=DVOLU*TWOPI*GPCOD(1, KGASP) SPVP 57 +IF(THICK.NE.0.0) DVOLU=DVOLU*THICK SPVP 58 +CALL STRESS(DMATX, LPROP, NTYPE, PROPS, NDOFN, CARTD, ELDIS, SHAPE, SPVP 59 +. GPCOD, NSTRE, VIVEL, DTIME, STRSG, KGASP, MTOTG, MMATS, SPVP 60 +. SVECT, NNODE, NSTR1, KGAUS, TLDIS) SPVP 61 +DO 60 ISTR1=1, NSTR1 SPVP 62 +DESTN(ISTR1)=VIVEL(ISTR1, KGAUS)*DTIME SPVP 63 +60 VISTN(ISTR1, KGAUS)=VISTN(ISTR1, KGAUS)+DESTN(ISTR1) SPVP 64 +DEBAR=SORT((2.0*(DESTN(1)*DESTN(1)+DESTN(2)*DESTN(2)+DESTN(4)* SPVP 65 +. DESTN(4))+DESTN(3)*DESTN(3))/3.0) SPVP 66 +DO 65 ISTR1=1, NSTR1 SPVP 67 +65 STRES(ISTR1)=STRSG(ISTR1, KGAUS) SPVP 68 +VIVEL(5, KGAUS)=VIVEL(5, KGAUS)+DEBAR SPVP 69 +CALL INVAR(DEVIA, LPROP, MMATS, NCRIT, PROPS, SINT3, STEFF, STRES, THETA, SPVP 70 +. VARJ2, YIELD) SPVP 71 +CALL YIELDF(AVECT, DEVIA, LPROP, MMATS, NCRIT, NSTR1, SPVP 72 +. PROPS, SINT3, STEFF, THETA, VARJ2) SPVP 73 +CALL FLOWVP(AVECT, PROPS, LPROP, STEFF, NSTR1, MTOTG, VIVEL, SPVP 74 +. YIELD, KGAUS, MMATS, NCRIT, FNORM, ALLOW) SPVP 75 +SPVP 76 +``` diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_031.md b/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_031.md new file mode 100644 index 00000000..e434ef01 --- /dev/null +++ b/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_031.md @@ -0,0 +1,682 @@ + + +```txt +IF(FNORM.LT.ALLOW) GO TO 70 SPVP 77 +EPBAR=SQRT((2.0*(AVECT(1)*AVECT(1)+AVECT(2)*AVECT(2)+AVECT(4) SPVP 78 +.*AVECT(4))+AVECT(3)*AVECT(3))/3.0) SPVP 79 +TSBAR=SQRT((2.0*(SVECT(1)*SVECT(1)+SVECT(2)*SVECT(2)+SVECT(4) SPVP 80 +.*SVECT(4))+SVECT(3)*SVECT(3))/3.0) SPVP 81 +DELTM=TAUFT*TSBAR/EPBAR SPVP 82 +IF(DELTM.LT.DNEXT) DNEXT=DELTM SPVP 83 +70 CONTINUE SPVP 84 +80 CONTINUE SPVP 85 +DTIME=DNEXT SPVP 86 +IF(ISTEP.EQ.1) DTIME=DTINT SPVP 87 +KGAUS=0 SPVP 88 +DO 140 IELEM=1,NELEM SPVP 89 +LPROP=MATNO(IELEM) SPVP 90 +DO 90 INODE=1,NNODE SPVP 91 +LNODE=IABS(LNODS(IELEM,INODE)) SPVP 92 +NPOSN=(LNODE-1)*NDOFN SPVP 93 +DO 90 IDOFN=1,NDOFN SPVP 94 +NPOSN=NPOSN+1 SPVP 95 +90 ELCOD(IDOFN,INODE)=COORD(LNODE,IDOFN) SPVP 96 +THICK=PROPS(LPROP,3) SPVP 97 +KGASP=0 SPVP 98 +DO 130 IGAUS=1,NGAUS SPVP 99 +DO 130 JGAUS=1,NGAUS SPVP 100 +EXISP=POSGP(IGAUS) SPVP 101 +ETASP=POSGP(JGAUS) SPVP 102 +KGAUS=KGAUS+1 SPVP 103 +KGASP=KGASP+1 SPVP 104 +CALL SFR2(DERIV,ETASP,EXISP,NNODE,SHAPE) SPVP 105 +CALL JACOB2(CARTD,DERIV,DJACB,ELCOD,GPCOD,IELEM,KGASP,NNODE,SHAPE) SPVP 106 +DVOLU=DJACB*WEIGP(IGAUS)*WEIGP(JGAUS) SPVP 107 +IF(NTYPE.EQ.3) DVOLU=DVOLU*TWOPI*GPCOD(1,KGASP) SPVP 108 +IF(THICK.NE.0.0) DVOLU=DVOLU*THICK SPVP 109 +CALL BMATPS(BMATX,CARTD,NNODE,SHAPE,GPCOD,NTYPE,KGASP) SPVP 110 +CALL MODPS(DMATX,LPROP,MMATS,NTYPE,PROPS) SPVP 111 +DO 100 ISTR1=1,NSTR1 SPVP 112 +100 STRES(ISTR1)=STRSG(ISTR1,KGAUS) SPVP 113 +CALL INVAR(DEVIA,LPROP,MMATS,NCRIT,PROPS,SINT3,STEFF,STRES,THETA, SPVP 114 +.VARJ2,YIELD) SPVP 115 +IF(TIMEX.GT.0.0) CALL TANGVP(LPROP,STRES,PROPS,TIMEX,DTIME, SPVP 116 +.NSTRE,NTYPE,MMATS,NCRIT,DMATX) SPVP 117 +C SPVP 118 +C*** CALCULATE THE RESIDUAL FORCES AND INCREMENTAL PSEUDO LOADS SPVP 119 +C SPVP 120 +DO 110 ISTRE=1,NSTRE SPVP 121 +STRES(ISTRE)=0.0 SPVP 122 +DO 110 JSTRE=1,NSTRE SPVP 123 +110 STRES(ISTRE)=STRES(ISTRE)+DMATX(ISTRE,JSTRE)*VIVEL(JSTRE,KGAUS) SPVP 124 +.*DTIME SPVP 125 +MGASH=0 SPVP 126 +DO 120 INODE=1,NNODE SPVP 127 +DO 120 IDOFN=1,NNODFN SPVP 128 +MGASH=MGASH+1 SPVP 129 +DO 120 ISTRE=1,NSTRE SPVP 130 +120 ELOAD(IELEM,MGASH)=ELOAD(IELEM,MGASH)+BMATX(ISTRE,MGASH) SPVP 131 +.*(STRES(ISTRE)-STRSG(ISTRE,KGAUS))*DVOLU SPVP 132 +130 CONTINUE SPVP 133 +140 CONTINUE SPVP 134 +DO 150 IELEM=1,NELEM SPVP 135 +DO 150 IEVAB=1,NEVAB SPVP 136 +150 ELOAD(IELEM,IEVAB)=ELOAD(IELEM,IEVAB)+TLOAD(IELEM,IEVAB) SPVP 137 +RETURN SPVP 138 +END SPVP 139 +``` + + + +SPVP 20 Compute 2π. + +SPVP 21–23 Zero the array in which the pseudo loads for the next time-step will be stored. + +SPVP 24 Zero the Gauss point counter over all elements. + +SPVP 25 Increase the timestep length from the value used for the previous step by the factor FTIME. If this new value is less than that predicted later in this routine, this step length will be employed for the next time step. + +SPVP 26 Loop over each element. + +SPVP 27 Identify the element material property number. + +SPVP 32-39 Store the element coordinates in array ELCOD, the incremental displacements $\Delta d^{n}$ in ELDIS and the total displacements $d^{n}$ in TLDIS. + +SPVP 40 Identify the element thickness. + +SPVP 41 Zero the local Gauss point counter. + +SPVP 42–45 Enter the loops for numerical integration and evaluate the local coordinates $(\xi, \eta)$ at the sampling point. + +SPVP 46–47 Increment the local and global Gauss point counters. + +SPVP 48 Compute the elasticity matrix, D. + +SPVP 49–50 Store the total current stresses $\sigma^{n}$ locally in STRES. + +SPVP 51–52 Evaluate the deviatoric stresses and $J_{2}^{\prime}$ . + +SPVP 53-54 For the implicit or semi-implicit time stepping scheme evaluate $\hat{D}^n$ . + +SPVP 55 Evaluate the shape functions $N_{i}$ and the derivatives $\partial N_{i}/\partial\xi$ , $\partial N_{i}/\partial\eta$ . + +SPVP 56 Evaluate the Gauss point coordinates GPCOD(IDIME, KGASP), the determinant of the Jacobian matrix $|J|$ and the Cartesian derivatives of the shape functions. + +SPVP 57–59 Calculate the elemental volume for numerical integration as $|J|W_{\xi}W_{\eta}$ taking care to multiply by $2\pi r$ for axisymmetric problems. + +SPVP 60–62 Call subroutine STRESS to evaluate the stress increment $\Delta\sigma^{n}$ according to (8.20) and also $\sigma^{n+1} = \sigma^{n} + \Delta\sigma^{n}$ . + +SPVP 63–65 Evaluate the incremental viscoplastic strain and the total current viscoplastic strain, $\epsilon_{vp}^{n+1}$ . + +SPVP 66–67 Accumulate the absolute value of the viscoplastic strain increment. This will allow us to monitor whether or not steady state conditions are being approached. + +SPVP 70 Also calculate the total current effective viscoplastic strain $\bar{\epsilon}_{vp}^{n+1}$ according to (8.47). + +SPVP 71–76 Evaluate the current viscoplastic flow rate $\dot{\epsilon}_{vp}^{n+1}$ according to (8.7). + +SPVP 77 If the Gauss point is elastic, avoid calculation of the new time step length. + + + +SPVP 78–79 Calculate $\bar{\epsilon}_{vp}^{n+1}$ , the effective value of the viscoplastic strain rate. + +SPVP 80-81 Calculate $\bar{\epsilon}^{n+1}$ , the total effective strain. + +SPVP 82-83 Evaluate the interval length for the next time step according to (8.29) as + +$$ +\Delta t _ {n + 1} = \tau \left[ \frac {\bar {\epsilon} ^ {n + 1}}{\bar {\epsilon} _ {v p} ^ {n + 1}} \right] _ {\min} ^ {1 / 2}, +$$ + +where TFACT is the parameter $\tau$ and the minimum value of $\Delta t_{n+1}$ is taken with respect to all Gauss points throughout the structure. + +SPVP 84–85 Termination of loops over Gauss points and elements respectively. + +SPVP 87 For the first time step of a load increment reset the step length equal to the initial value input. + +SPVP 88 Zero the Gauss point counter over all elements. + +SPVP 89 Loop over each element. + +SPVP 90 Identify the element material property number. + +SPVP 91-96 Store the element coordinates in array ELCOD. + +SPVP 97 Identify the element thickness. + +SPVP 98 Zero the local Gauss point counter. + +SPVP 99–102 Enter the loops for numerical integration and evaluate the local coordinates $(\xi, \eta)$ at the sampling point. + +SPVP 103-104 Increment the local and global Gauss point counters. + +SPVP 105 Evaluate the shape functions and their local derivatives. + +SPVP 106 Evaluate the Gauss point coordinates, determinant of the Jacobian matrix and the Cartesian derivatives of the shape functions. + +SPVP 107-109 Calculate the elemental volume for numerical integration. + +SPVP 110 Evaluate the B matrix. + +SPVP 111 Evaluate the D matrix. + +SPVP 112–113 Store the total current stresses $\sigma^{n+1}$ locally in STRES. + +SPVP 114–115 Calculate the deviatoric stresses and $J_{2}^{\prime}$ . + +SPVP 116–117 For the implicit or semi-implicit time stepping scheme evaluate $\hat{D}^{n+1}$ . + +SPVP 121-125 Calculate $\hat{D}^{n+1}\dot{\epsilon}_{vp}^{n+1}\Delta t_{n+1}$ and store locally in STRES. + +SPVP 126–132 Evaluate the pseudo loads to be applied for the next timestep, $\Delta V^{n+1}$ according to (8.28) and (8.34) as + + + +$$ +\Delta V ^ {n + 1} = \int_ {\Omega} B ^ {T} \left\{\hat {D} ^ {n + 1} \dot {\epsilon} _ {r p} ^ {n + 1} \Delta t _ {n + 1} + \sigma^ {n + 1} \right\} d \Omega + f ^ {n + 1}. +$$ + +SPVP 133-134 Termination of loops over Gauss points and elements respectively. + +SPVP 135–137 Complete the computations of SPVP 126–132 by adding the term $f^{n+1}$ . + +Subroutine INVAR which calculates the deviatoric stresses and $J_{2}^{\prime}$ is identical to that employed in Chapter 7 for elasto-plastic problems and is described in detail in Section 7.8.3. Subroutine YIELDF has been previously described in Section 7.8.4.1. + +# 8.9 Subroutine FLOWVP + +The function of this subroutine is to determine the viscoplastic strain rate according to (8.7). + +Subroutine FLOWVP is now presented and described. +```txt +SUBROUTINE FLOWVP(AVECT, PROPS, LPROP, STEFF, NSTR1, MTOTG, VIVEL, YIELD, KGAUS, MMATS, NCRIT, FNORM, ALLOW) FLVP 1 +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +DIMENSION AVECT(4), PROPS(MMATS, 10), VIVEL(5, MTOTG) FLVP 8 +ALLOW=0.01 FLVP 9 +IF(STEFF.EQ.0.0) GO TO 90 FLVP 10 +YOUNG=PROPS(LPROP, 1) FLVP 11 +POISS=PROPS(LPROP, 2) FLVP 12 +HARDS=PROPS(LPROP, 6) FLVP 13 +FRICT=PROPS(LPROP, 7) FLVP 14 +GAMMA=PROPS(LPROP, 8) FLVP 15 +DELTA=PROPS(LPROP, 9) FLVP 16 +NFLOW=PROPS(LPROP, 10) FLVP 17 +ROOT3=1.73205080757 FLVP 18 +FDATM=PROPS(LPROP, 5) FLVP 19 +FRICT=FRICT*0.017453292 FLVP 20 +IF(NCRIT.EQ.3) FDATM=FDATM*COS(FRICT) FLVP 21 +IF(NCRIT.EQ.4) FDATM=6.0*FDATM*COS(FRICT)/(ROOT3*(3.0-SIN(FRICT)))FLVP 22 +IF(HARDS.GT.0.0) FDATM=FDATM+VIVEL(5, KGAUS)*HARDS FLVP 23 +IF(FDATM.LT.0.001) FDATM=1.0 FLVP 24 +FCURR=YIELD-FDATM FLVP 25 +FNORM=FCURR/FDATM FLVP 26 +IF(FNORM.LT.ALLOW) GO TO 90 FLVP 27 +IF(NFLOW.EQ.1) GO TO 50 FLVP 28 +CMULT=GAMMA*(EXP(DELTA*FNORM)-1.0) FLVP 29 +GO TO 60 FLVP 30 +50 CMULT=GAMMA*(FNORM**DELTA) FLVP 31 +60 DO 70 ISTR1=1, NSTR1 FLVP 32 +70 AVECT(ISTR1)=CMULT*AVECT(ISTR1) FLVP 33 +DO 80 ISTR1=1, NSTR1 FLVP 34 +80 VIVEL(ISTR1, KGAUS)=AVECT(ISTR1) FLVP 35 +RETURN FLVP 36 +90 DO 100 ISTR1=1, NSTR1 FLVP 37 +100 VIVEL(ISTR1, KGAUS)=0.0 FLVP 38 +RETURN FLVP 39 +END FLVP 40 +``` + + + +FLVP 9 Specify ALLOW, the permitted tolerance by which the stress point is allowed to deviate from the yield surface. + +FLVP 10 For the (unlikely) case of a Gauss point with zero stress (identified by $J_2' = J_3' = 0$ ) avoid all viscoplastic calculations. + +FLVP 11 Identify YOUNG as the elastic modulus, E. + +FLVP 12 Identify POISS as the Poissons ratio, v. + +FLVP 13 Identify HARDS as $H'$ for linear strain hardening. + +FLVP 14 Identify FRICT as the friction angle $\phi$ for Mohr-Coulomb and Drucker-Prager materials. + +FLVP 15 Identify GAMMA as the fluidity parameter, $\gamma$ . + +FLVP 16 Identify DELTA as the index $M$ in (8.8) or $N$ in (8.9), according to the flow function specified. + +FLVP 17 Identify NFLOW as the parameter specifying type of flow function: + +NFLOW = 0 - flow function (8.8) to be used, + +NFLOW = 1 - flow function (8.9) to be used. + +FLVP 18 Compute √(3). + +FLVP 19–22 Identify FDATM as the effective yield stress, $\sigma_{Y}^{0}$ , according to Column 4, Table 7.2. + +FLVP 23 Evaluate the current yield stress as $F_0 = \sigma Y^0 + H' \bar{\epsilon}_{vp}$ , where $\bar{\epsilon}_{vp}$ is the current effective viscoplastic strain, according to (8.47). + +FLVP 24 For elastic creep problems, solved by setting $F_{0}=0$ , reset $F_{0}$ as a low value to avoid overflow in (8.8) and (8.9). + +FLVP 25–26 Calculate $(F-F_{0})/F_{0}$ where F is the effective stress value evaluated as YIELD in subroutine INVAR. + +FLVP 27 If $(F-F_{0})/F_{0}$ is less than ALLOW avoid any further visco-plastic calculations, i.e. the stress point is assumed to be sufficiently close to the yield surface. + +FLVP 29 Evaluate $\gamma\langle\phi\rangle$ for flow function (8.8). + +FLVP 31 Evaluate $\gamma\langle\Phi\rangle$ for flow function (8.9). + +FLVP 32-35 Use flow vector a to form $\dot{\epsilon}_{vp}^{n+1} = \gamma\langle\Phi\rangle a^{n+1}$ . + +FLVP 37–38 For elastic points only, set the viscoplastic strain rate to zero. + +# 8.10 Subroutine STRESS + +The function of this subroutine is to evaluate the increment in stress occurring during a time step according to (8.20). + +Subroutine STRESS is presented below: + +SUBROUTINE STRESS(DMATX, LPROP, NTYPE, PROPS, NDOFN, CARTD, ELDIS, STRS 1 +. SHAPE, GPCOD, NSTRE, VIVEL, DTIME, STRSG, KGASP, STRS 2 +. MTOTG, MMATS, SVECT, NNODE, NSTR1, KGAUS, TLDIS) STRS 3 + +CSTRS 4 + +C STRS 5 + +C\*\*\*\* EVALUATE THE INCREMENTS OF STRAIN AND STRESS STRS 6 + +C STRS 7 + +C $^{*}$ SSTRS 8 + + + +```asm +DIMENSION SVECT(4), PROPS(MMATS,10), ELDIS(2,9), CARTD(2,9), STRS 9 +. DMATX(4,4), AGASH(2,2), STRES(4), STRAN(4), STRSG(4,MTOTG), STRS 10 +. SHAPE(9), VIVEL(5,MTOTG), TLDIS(2,9), CGASH(2,2), STRS 11 +. GPCOD(2,9) STRS 12 +POISS=PROPS(LPROP,2) STRS 13 +DO 10 IDOFN=1, NDOFN STRS 14 +DO 10 JDOFN=1, NDOFN STRS 15 +BGASH=0.0 STRS 16 +DGASH=0.0 STRS 17 +DO 20 INODE=1, NNODE STRS 18 +DGASH=DGASH+CARTD(JDOFN, INODE)*TLDIS(IDOFN, INODE) STRS 19 +20 BGASH=BGASH+CARTD(JDOFN, INODE)*ELDIS(IDOFN, INODE) STRS 20 +CGASH(IDOFN, JDOFN)=BGASH STRS 21 +10 AGASH(IDOFN, JDOFN)=DGASH STRS 22 +C STRS 23 +C*** CALCULATE THE TOTAL AND INCREMENTAL STRAINS STRS 24 +C STRS 25 +SVECT(1)=AGASH(1,1) STRS 26 +SVECT(2)=AGASH(2,2) STRS 27 +SVECT(3)=AGASH(1,2)+AGASH(2,1) STRS 28 +IF(NTYPE.NE.3) GO TO 70 STRS 29 +SVECT(4)=0.0 STRS 30 +DO 60 INODE=1, NNODE STRS 31 +SVECT(4)=SVECT(4)+TLDIS(1, INODE)*SHAPE(INODE)/GPCOD(1, KGASP) STRS 32 +60 CONTINUE STRS 33 +70 CONTINUE STRS 34 +STRAN(1)=CGASH(1,1) STRS 35 +STRAN(2)=CGASH(2,2) STRS 36 +STRAN(3)=CGASH(1,2)+CGASH(2,1) STRS 37 +IF(NTYPE.NE.3) GO TO 90 STRS 38 +STRAN(4)=0.0 STRS 39 +DO 80 INODE=1, NNODE STRS 40 +STRAN(4)=STRAN(4)+ELDIS(1, INODE)*SHAPE(INODE)/GPCOD(1, KGASP) STRS 41 +80 CONTINUE STRS 42 +90 CONTINUE STRS 43 +DO 50 ISTRE=1, NSTRE STRS 44 +50 STRAN(ISTRE)=STRAN(ISTRE)-VIVEL(ISTRE, KGAUS)*DTIME STRS 45 +C STRS 46 +C*** AND THE INCREMENTAL STRESSES STRS 47 +C STRS 48 +DO 30 ISTRE=1, NSTRE STRS 49 +STRES(ISTRE)=0.0 STRS 50 +DO 30 JSTRE=1, NSTRE STRS 51 +30 STRES(ISTRE)=STRES(ISTRE)+DMATX(ISTRE, JSTRE)*STRAN(JSTRE) STRS 52 +IF(NTYPE.EQ.1) STRES(4)=0.0 STRS 53 +IF(NTYPE.EQ.2) STRES(4)=POISS*(STRES(1)+STRES(2)) STRS 54 +DO 40 ISTR1=1, NSTR1 STRS 55 +40 STRSG(ISTR1, KGAUS)=STRSG(ISTR1, KGAUS)+STRES(ISTR1) STRS 56 +RETURN STRS 57 +END STRS 58 +``` + + + +STRS 13 Identify POISS as the material Poisson's ratio. + +STRS 14–22 Evaluate the Cartesian derivatives of both the displacement increment and the total displacement. + +STRS 26-33 Evaluate the total and incremental strains $Bd^{n}$ and $B\Delta d^{n}$ . + +STRS 34-45 Calculate the elastic portion of the strains, $B\Delta d^n - \dot{\epsilon}_{vp^n}\Delta t_n$ . + +STRS 49–52 Calculate the stresses according to (8.20). + +STRS 53–54 For plane stress and plane strain problems evaluate the out-of-plane stress component. + +STRS 55-56 Finally calculate the total current stress as $\sigma^{n+1} = \sigma^n + \Delta \sigma^n$ . + +# 8.11 Subroutine ZERO + +This subroutine performs the same task as the subroutine described in Section 7.8.2 for elasto-plastic problems. It merely initializes to zero some arrays required for the accumulation of data. Subroutine ZERO is presented below without further comment. + +```asm +SUBROUTINE ZERO(ELOAD,MELEM,MEVAB,MPOIN,MTOTG,MTOTV,NDOFN,NELEM, ZRO2 1 +. NEVAB,NGAUS,NSTR1,NTOTG,NTOTV,NVFIX,STRSG, ZRO2 2 +. TDISP,VIVEL,VISTN,TTIME,TLOAD,TREAC, ZRO2 3 +. TFACT,MVFIX) ZRO2 4 +C******************************* +C******************************* +C******************************* +C******************************* +C******************************* +DIMENSION ELOAD(MELEM,MEVAB),STRSG(4,MTOTG),TDISP(MTOTV), ZRO2 10 +. TLOAD(MELEM,MEVAB),TREAC(MVFIX,2),VIVEL(5,MTOTG), ZRO2 11 +. VISTN(4,MTOTG) ZRO2 12 +TTIME=0.0 ZRO2 13 +TFACT=0.0 ZRO2 14 +DO 30 IELEM=1,NELEM ZRO2 15 +DO 30 IEVAB=1,NEVAB ZRO2 16 +ELOAD(IELEM,IEVAB)=0.0 ZRO2 17 +30 TLOAD(IELEM,IEVAB)=0.0 ZRO2 18 +DO 40 ITOTV=1,NTOTV ZRO2 19 +40 TDISP(ITOTV)=0.0 ZRO2 20 +DO 50 IVFIX=1,NVFIX ZRO2 21 +DO 50 IDOFN=1,NDOFN ZRO2 22 +50 TREAC(IVFIX,IDOFN)=0.0 ZRO2 23 +DO 60 ITOTG=1,NTOTG ZRO2 24 +VIVEL(5,ITOTG)=0.0 ZRO2 25 +DO 60 ISTR1=1,NSTR1 ZRO2 26 +VISTN(ISTR1,ITOTG)=0.0 ZRO2 27 +VIVEL(ISTR1,ITOTG)=0.0 ZRO2 28 +60 STRSG(ISTR1,ITOTG)=0.0 ZRO2 29 +RETURN ZRO2 30 +END ZRO2 31 +``` + +# 8.12 Subroutine STEADY for monitoring steady state convergence + +The role of this subroutine is to check whether or not steady state conditions have been achieved at the end of each time step. Convergence to a + + + +steady state condition is monitored according to the increment in viscoplastic strain which occurs during the time step. For checking purposes the effective viscoplastic strain rate, $\dot{e}_{vp}^{n+1}$ , defined by (8.47) is employed and steady state conditions are deemed to have been achieved at the end of time step n, if + +$$ +\left(\Delta t _ {n + 1} \sum_ {\text { All Gauss points }} \bar {\dot {\epsilon}} _ {v p} ^ {n + 1} / \Delta t _ {1} \sum_ {\text { All Gauss points }} \bar {\dot {\epsilon}} _ {v p} ^ {1}\right) \times 1 0 0 \leqslant \text { TOLER }, \tag {8.48} +$$ + +where TOLER is a convergence tolerance value prescribed as input in Subroutine INCREM, described in Section 6.5.3. From (8.48) it is seen that a global measure of convergence is taken in the subroutine presented in this section. A local steady state convergence condition could alternatively be enforced by requiring (8.48) to be satisfied for each Gauss point in the structure which is yielding viscoplastically. + +The structure of this subroutine is identical to that of subroutine CONVP, presented in Section 4.9, for one-dimensional structures. + +Subroutine STEADY is now presented. +```csv +SUBROUTINE STEADY(NELEM,NGAUS,NCHEK,VIVEL,ISTEP,FIRST,TOLER,PVALU,STDY 1 +. MTOTG,DTIME,NSTR1,TTIME) STDY 2 +C***** +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +``` + + + +# 8.13 The main, master or controlling segment + +This segment controls the timestepping process and accesses all the other subroutines appropriately. In particular it controls the increment of the applied loads and the output of results at selected time intervals. The frequency of output is controlled by means of two parameters NOUTP(1) and NOUTP(2) which are specified as input data for every load increment in subroutine INCREM described in Section 6.5.3. The precise specification of these parameters is however somewhat different for the present application. In this case NOUTP(1) controls the frequency of output of the displacements and NOUTP(2) the frequency of output of the stresses and viscoplastic strains. In particular, if NOUTP(1) is specified as 7 for a particular load increment, then the displacements will be output every 7th timestep within that increment. This is accomplished by evaluating for every timestep, ISTEP, the quantity + +$$ +(\text { ISTEP } / \text { NOUTP } (1)) ^ {*} \text { NOUTP } (1) +$$ + +and then checking this value against ISTEP. The two will be equal only when ISTEP is an exact multiple of NOUTP(1). A similar check for stress output is undertaken for NOUTP(2). + +The parameter MSTEP specifies the maximum number of timesteps to be considered for the load increment. If steady state conditions are achieved before MSTEP timesteps, the next load increment, is applied immediately condition (8.48) is satisfied. + +The role of the load incrementing factor, FACTO, is identical to that described in Section 6.5.3. + +In this segment input data is also received which controls the timestepping algorithm to be employed. The following information is input: + +TIMEX Parameter, $\Theta$ , which controls the type of timestepping algorithm to be employed: + +$$ +\begin{array}{l} \text { TIMEX } = 0. 0 \text { -Explicit scheme }, \\ = 0. 5 - \text { Semi - implicit or trapezoidal scheme }, \\ = 1. 0 - \text { Fully implicit }. \\ \end{array} +$$ + +TAUFT- The parameter $\tau$ discussed in Section 8.3. + +DTINT The initial time step length. This specifies the step length for the first time step of each load increment. The time step length needs to be readjusted at the beginning of a new load increment since the step length computed as steady state conditions are approached in the previous time step will in general be too large. + +FTIME The factor by which it is attempted to increase the step length from the value used for the previous time step. This parameter is generally input as 1.5 as mentioned in Section 8.3. + +The following channel numbers are employed by the program: 5 (card reader), 6 (line printer), 1, 2, 3, 4, 8 (scratch files). This main segment is now presented and descriptive notes provided where necessary. + + + +```asm +MASTER VISCO +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +DIMENSION ASDIS(120),COORD(60,2),ELOAD(20,18),ESTIF(18,18), +EQRHS(10),EQUAT(40,10),FIXED(120), +GLOAD(40),GSTIF(986), +IFFIX(120),LNODS(20,9),LOCEL(18),MATNO(20), +NACVA(40),NAMEV(10),NDEST(18),NDFRO(20),NOFIX(25), +NOUTP(2),NPIVO(10), +POSGP(4),PRESC(25,2),PROPS(5,10),RLOAD(20,18), +STFOR(120),TREAC(25,2),VECRV(40),WEIGP(4), +STRSG(4,180),TDISP(120), +TLOAD(20,18),VIVEL(5,180),VISTN(4,180) +TLOAD(20,18),VIVEL(5,180),VISTN(4,180) +TLOAD(20,18),VIVEL(5,180),VISTN(4,180) +TLOAD(20,18),VIVEL(5,180),VISTN(4,180) +TLOAD(20,18),VIVEL +TLOAD(20,18),VIVEL +TLOAD(20,18),VIVEL +TLOAD(20,18),VIVEL +TLOAD(20,18),VIVEL +TLOAD(20,18),VIVEL +TLOAD(20,18),VIVEL +TLOAD(20,18),VIVEL +TLOAD(20,18), +TLOAD(20,18),VIVEL +TLOAD(20,18),VIVEL +TLOAD(20,18),VIVEL +TLOAD(20,18),VIVEL +TLOAD(20,18),VIVEL +TLOAD(20,18),VIVEL +TLOAD(20,18),VIVEL +TLOAD(20,18) +TLOAD(20,18),VIVEL +TLOAD(20,18),VIVEL +TLOAD(20,18),VIVEL +TLOAD(20,18),VIVEL +TLOAD(20,18),VIVEL +TLOAD(20,18),VIVEL +TLOAD(20,18),VIVEL +TLOAD(20,12) +TLOAD(20,12) +TLOAD(20,12) +TLOAD(20,12) +TLOAD(20,12) +TLOAD(20,12) +TLOAD(20,12) +TLOAD(20,12) +TLOAD(20,12) +TLOAD(20,12) +TLOAD(20,12) +TLOAD(20,12) +VIVEL +TLOAD(20,12) +TLOAD(20,12) +TLOAD(20,12) +TLOAD(20,12) +TLOAD(20,12) +TLOAD(20,12) +TLOAD(20,12) +TLOAD(20,12) +TLOAD(20,12) +TLOAD(20,12) +TLOAD(20 +TLOAD(20,12) +TLOAD(20,12) +TLOAD(20,12) +TLOAD(20,12) +TLOAD(20,12) +TLOAD(20,12) +TLOAD(20,12) +TLOAD(20,12) +TLOAD(20,12) +TLOAD(20,12) +TLOAD(20,12) +``` diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_032.md b/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_032.md new file mode 100644 index 00000000..e51b190a --- /dev/null +++ b/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_032.md @@ -0,0 +1,384 @@ + + +```csv +TTIME=TTIME+DTIME +C +C*** CALL ROUTINE WHICH SELECTS SOLUTION ALORITHM VARIABLE KRESL +C +CALL ALGOR(FIXED,IINCS,ISTEP,KRESL,TIMEX,MTOTV,NALGO,NTOTV) +C*** CHECK WHETHER A NEW EVALUATION OF THE STIFFNESS MATRIX IS REQUIRED +C +IF(KRESL.EQ.1) CALL STIFVP(COORD,IINCS,LNODS,MATNO, +MEVAB,MMATS,MPOIN,MTOTV,NELEM,NEVAB,NGAUS,NNODE, +NSTRE,NSTR1,POSGP,PROPS,WEIGP,MELEM,MTOTG, +STRSG,NTYPE,NCRIT,TIMEX,DTIME) +C +SOLVE EQUATIONS +C +CALL FRONT(ASDIS,ELOAD,EQRHS,EQUAT,ESTIF,FIXED,IFFIX,IINCS,ISTEP, +GLOAD,GSTIF,LOCEL,LNODS,KRESL,MBUFA,MELEM,MEVAB,MFRON, +MSTIF,MTOTV,MVFIX,NACVA,NAMEV,NDEST,NDOFN,NELEM,NEVAB, +NNODE,NOFIX,NPIVO,NPOIN,NTOTV,TDISP,TLOAD,TREAC, +VECRV) +C +C*** CALCULATE RESIDUAL FORCES +C +CALL STEPVP(ASDIS,COORD,ELOAD,ISTEP,LNODS,LPROP,TIMEX, +MATNO,MELEM,MMATS,MPOIN,MTOTG,TAUFT,DTIME, +MTOTV,NDOFN,NELEM,NEVAB,NGAUS,NNODE,NSTR1, +NTYPE,POSGP,PROPS,NSTRE,NCRIT,STRSG,WEIGP, +TDISP,VISTN,VIVEL,TLOAD,FTIME,DTINT,IINCS) +C +C*** CHECK FOR CONVERGENCE TO STEADY STATE +C +CALL STEADY(NELEM,NGAUS,NCHEK,VIVEL,ISTEP,FIRST,TOLER,PVALU, +MTOTG,DTIME,NSTR1,TTIME) +C +C*** OUTPUT RESULTS IF REQUIRED +C +IF(NOUTP(1).EQ.0) GO TO 110 +KOUTD=(ISTEP/NOUTP(1))*NOUTP(1) +KOUTS=(ISTEP/NOUTP(2))*NOUTP(2) +IF(KOUTD.NE.ISTEP.OR.KOUTS.NE.ISTEP) GO TO 110 +KOUTP=2 +IF(KOUTS.EQ.ISTEP) KOUTP=3 +CALL OUTPUT(ISTEP,MTOTG,MTOTV,MVFIX,NELEM,NGAUS,NOFIX,NOUTP, +NPOIN,NVFIX,STRSG,TDISP,TREAC,NTYPE,NCHEK,VIVEL, +KOUTP) +110 CONTINUE +C +C*** IF SOLUTION HAS CONVERGED STOP ITERATING AND OUTPUT RESULTS +C +IF(NCHEK.EQ.0) GO TO 75 +50 CONTINUE +C +C*** +C +75 CALL OUTPUT(ISTEP,MTOTG,MTOTV,MVFIX,NELEM,NGAUS,NOFIX,NOUTP, +NPOIN,NVFIX,STRSG,TDISP,TREAC,NTYPE,NCHEK,VIVEL, +KOUTP) +100 CONTINUE +STOP +END +VISC 65 +VISC 66 +VISC 67 +VISC 68 +VISC 69 +VISC 70 +VISC 71 +VISC 72 +VISC 73 +VISC 74 +VISC 75 +VISC 76 +VISC 77 +VISC 78 +VISC 79 +VISC 80 +VISC 81 +VISC 82 +VISC 83 +VISC 84 +VISC 85 +VISC 86 +VISC 87 +VISC 88 +VISC 89 +VISC 90 +VISC 91 +VISC 92 +VISC 93 +VISC 94 +VISC 95 +VISC 96 +VISC 97 +VISC 98 +VISC 99 +VISC 100 +VISC 101 +VISC 102 +VISC 103 +VISC 104 +VISC 105 +VISC 106 +VISC 107 +VISC 108 +VISC 109 +VISC 110 +VISC 111 +VISC 112 +VISC 113 +VISC 114 +VISC 115 +VISC 116 +VISC 117 +VISC 118 +VISC 119 +VISC 120 +VISC 121 +VISC 122 +VISC 123 +VISC 124 +``` + + + +
VISC 64For each load increment, initialise the time step length.
VISC 65Enter the time-stepping loop for the current load increment.
VISC 66Compute the total time elapsed.
VISC 70For the first timestep of the first load increment prepare for a full equation solution rather than a resolution for an explicit formulation. For the implicit or semi-implicit algorithm a complete equation solution is required each and every time-step.
VISC 73–85Formulate the element stiffnesses and solve the resulting equations.
VISC 89–94Calculate quantities at the end of the timestep and evaluate the loads for the next timestep.
VISC 98–99Check for convergence of the time stepping process to steady state conditions.
VISC 103–105Check to see if either displacement or stress output is required for this timestep.
VISC 106–107Set KOUTP = 2 for displacement output only and KOUTP = 3 for both stress and displacement output.
VISC 108–110Output the results.
VISC 115If steady state conditions have been reached, output the converged results, increment the loads and proceed with the time-stepping process.
+ +# 8.14 General comparison of implicit and explicit time integration schemes + +Before discussing the general case of a two-dimensional continuum it is instructive to consider the behaviour of a single degree of freedom system. In particular we will consider the response of a simple linear Maxwell model, as illustrated in Fig. 8.2. This situation is equivalent to the uniaxial viscoplastic model when the initial yield or threshold value, $F_{0}$ , is reduced to zero. Figure 8.2 shows the stress relaxation histories for different time integration schemes when the model is subjected to a constant total strain. It is observed that all results obtained using the fully implicit scheme ( $\Theta = 1$ ) lie to one side of the theoretical solution while the semi-implicit method ( $\Theta = \frac{1}{2}$ ) gives results which lie to either side of the true curve. It is also evident that the explicit method ( $\Theta = 0$ ) gives an oscillatory solution with the rate of convergence decreasing as the time step stability limit is approached. However, in each case the steady state solution is eventually correctly predicted. For the solution of elasto-plastic problems by use of the viscoplastic algorithm it is only the steady state solution that is of importance. Similarly in problems of creep, the transient stage may not be of interest in itself, as long as the steady state values are correctly arrived at. + +For problems which are geometrically linear the solution process simplifies considerably. The strain matrix $B^{n}$ is then constant throughout the analysis and from (8.19) it is seen to be equal to $B_{0}$ . For solution by the explicit time + + + +![](images/page-313_1b0145fbe7ff706c01da01ba6a6add2873907fbbda42818f64345546fa823ebb.jpg) + +
+line + +| Time | Θ = 0 Explicit | Θ = 1/2 Semi-implicit (C.N.) | Θ = 1 Fully-implicit | +|------|----------------|------------------------------|------------------------| +| 1 | 0.7 | 0.4 | 0.3 | +| 2 | 0.5 | 0.2 | 0.1 | +| 3 | 0.3 | 0.1 | 0.05 | +| 4 | 0.1 | 0.05 | 0.02 | +| 5 | 0.05 | 0.02 | 0.01 | +| 6 | 0.02 | 0.01 | 0.005 | +| 7 | 0.01 | 0.005 | 0.002 | +| 8 | 0.005 | 0.002 | 0.001 | +| 9 | 0.002 | 0.001 | 0.0005 | +
+ +Fig. 8.2 Characteristics of explicit and implicit time stepping algorithms when applied to a linear Maxwell model. + +marching scheme, $\Theta = 0$ and from (8.14) we have that $C^n = 0$ . Consequently, from (8.18), $\hat{D}^n = D$ and (8.24) implies that the tangential stiffness matrix becomes the linear elastic stiffness matrix and is constant throughout the solution process. Thus for the equation solution demanded by (8.23), a complete reduction and back-substitution is only required for the first time step and subsequent time intervals only require equation resolution. + +Experience to date $^{(2)}$ indicates that solution by the implicit method increases the computation time by approximately a factor of 4–5 in comparison with the explicit approach, for the same solution tolerance factor (or time step length). This cost differential must be balanced against the greater time step lengths permitted by the unconditionally stable implicit method. However, increasing the time step length beyond prescribed limits results in a deterioration in solution accuracy. Where a variable stiffness approach is employed for some other reasons, such as to include geometric nonlinearity effects or time dependent material properties, solution by an implicit scheme entails little or no additional computing effort and such an approach is particularly + + + +advantageous. Modification of the program presented to account for large deformation effects is set as an exercise to the reader in Section 8.17. + +Implicit and explicit time integration schemes are considered further in Chapters 10 and 11 for the solution of dynamic transient problems. + +# 8.15 The overlay method for improved material response + +The viscoplastic model described in the previous sections gives a material response whose general form is in keeping with experimental observations. However the precise strain/time histories (or creep curves) of many real materials cannot be accurately represented by a simple viscoplastic model. This is particularly so for materials whose strain response curves are nonlinear with regard to the applied stress level, so that a doubling of the applied stress does not result in twice the strain at any given time. + +A more elaborate material response can be modelled by use of the so-called overlay or mechanical sublayer method $^{(10-13)}$ in which the solid to be analysed is assumed to be composed of several layers or overlays each of which undergoes the same deformation. The total stress field is obtained by summing the different contributions of each overlay. By introducing a suitable number of overlays and assigning different material characteristics to each, a variety of sophisticated composite actions can be reproduced. In this section it is demonstrated how time-dependent overlay models can be used to simulate some experimentally observed material behaviours. + +![](images/page-314_59f884148dd0d0b3f7e699eac712dbec113631a33cefaf50ba8b1473af904a54.jpg) + +
+flowchart + +```mermaid +graph TD + A["Start"] -->|Primary creep| B["Stage B"] + B -->|Secondary creep| C["Stage C"] + C -->|Permanent set| D["Stage D"] + D -->|Permanent set| E["Stage E"] + E -->|Tertiary creep| F["Stage F"] + F -->|Failure| G["Stage I"] + style A fill:#f9f,stroke:#333 + style B fill:#f9f,stroke:#333 + style C fill:#f9f,stroke:#333 + style D fill:#f9f,stroke:#333 + style E fill:#f9f,stroke:#333 + style F fill:#f9f,stroke:#333 + style G fill:#f9f,stroke:#333 +``` +
+ +Fig. 8.3 Strain/time relationship at constant stress for many typical materials. + +The strain-time relationship at constant stress which most materials exhibit to some degree or other is illustrated in Fig. 8.3. The instantaneous + + + +elastic strain, OA, is followed by a primary creep AB during which if unloading takes place an instantaneous elastic recovery results, followed by delayed elastic recovery, CD. If the load is not removed at time $T_{1}$ secondary creep begins which is accompanied by permanent deformation. Unloading at any time on the curve BE leaves a permanent set in the material. On continued loading past time $T_{2}$ tertiary creep begins, leading almost inevitably to failure. + +![](images/page-315_02b37f7b4de2f1fdeebdcc4930d17649b46a2d60ee51683acf932012aab39cb0.jpg) + +
+text_image + +μ₁ +μ₂ +η +
+ +(a) Standard visco-elastic model + +![](images/page-315_30d39ef1efbbbd9789453281ebc09c608b2b97865ee944f1b866688e1c3a6750.jpg) + +
+text_image + +μ₁ +μ₂ +η₂ +Y +η₁ +
+ +(b) Four parameter model +Fig. 8.4 Material models for simulation of the material behaviour of Fig. 8.3. (a) Standard visco-elastic model. (b) Four parameter model. + +This behaviour can be numerically simulated by use of the rheological models shown in Fig. 8.4. The standard linear solid illustrated in Fig. 8.4(a) provides a visco-elastic response and represents the behaviour of the material up to time $T_{1}$ . After this time the behaviour is closely approximated by the five parameter model shown in Fig. 8.4(b) where a friction slider component in parallel with a viscous dashpot has been added. This component becomes active only if the applied stress exceeds some limiting value, Y and the friction slider provides the permanent deformation or viscoplastic effect. For use in the overlay method it is desirable to consider ‘Maxwell equivalents’ of these models. Figure 8.5(a) shows the equivalent model to that of Fig. 8.4(a) both being governed by the differential equation + +$$ +p _ {1} D \sigma + p _ {0} \sigma = q _ {1} D \epsilon + q _ {0} \epsilon , \tag {8.49} +$$ + +where $p_{i}$ and $q_{i}$ are constants and D denotes the differential operator with respect to time. Similarly Fig. 8.5(b) illustrates the Maxwell equivalent of Fig. 8.4(b), the governing equation for this case being + +$$ +p _ {2} D ^ {2} \sigma + p _ {1} D \sigma + p _ {0} \sigma = q _ {2} D ^ {2} \epsilon + q _ {1} D \epsilon + q _ {0} \epsilon . \tag {8.50} +$$ + + + +![](images/page-316_d4216164d08dccb405f8dd971e27062ccea6c3a6975122da26e0d206464a8ad5.jpg) + +
+text_image + +μM +η +μH +
+ +(a) + +![](images/page-316_e01f2f825c09fb9022deff2043db762b47d12cb8fb8ebcb1f34805ff0cd250b9.jpg) + +
+text_image + +μM +μH +η +Y* +η' +Hookean element +
+ +(b) +Fig. 8.5 Equivalent representation of the models of Fig. 8.4 using Maxwell type components. + +The constants for the various components of the models in Figs. 8.4 and 8.5 are different but unique relationships exist. The configurations of Fig. 8.5 immediately suggest the use of overlay models. By employing at least one viscoplastic overlay and one Maxwell overlay (i.e. setting the threshold uniaxial yield value, $F_{0} = 0$ ) the complete behaviour in the visco-elastic range as well as irrecoverable creep deformation can be generated. The model behaves as a ‘standard linear solid’ until failure of the friction slider in the visco-plastic overlay after which it behaves as a four parameter solid. In fact a fifth parameter, the yield limit of the slider must also be defined. These parameters are material characteristics and their values must be experimentally determined. + +![](images/page-316_9ef581a3a53df1369c058d518686a45feb4b67332d3f3e460ae44f0f09882b4c.jpg) + +
+text_image + +t_i +l +
+ +Fig. 8.6 The overlay model in two-dimensional situations. + + + +# 8.15.1 Basic expressions of the overlay concept + +The overlay model in a two-dimensional situation is illustrated schematically in Fig. 8.6. Each overlay can have a different thickness and material behaviour. With the nodes in each overlay coincidental, the same strain pattern is produced in each component. This results in a different stress field $\sigma_{j}$ in each layer which contribute to the total stress field $\sigma$ according to the overlay thickness, $t_{j}$ , so that + +$$ +\sigma = \sum_ {j = 1} ^ {k} \sigma_ {j} t _ {j}, \tag {8.51} +$$ + +in which $k$ is the total number of overlays in the model, and + +$$ +\sum_ {j = 1} ^ {k} t _ {j} = 1. \tag {8.52} +$$ + +The equilibrium equations (8.21) which must be satisfied at each stage become + +$$ +\int_ {\Omega} [ \boldsymbol {B} ^ {n} ] ^ {T} \sum_ {j = 1} ^ {k} \sigma_ {j} ^ {n} t _ {j} d \Omega + \boldsymbol {f} ^ {n} = \mathbf {0}. \tag {8.53} +$$ + +Also the element stiffnesses (8.24) are the sum of each overlay contribution so that + +$$ +\boldsymbol {K} _ {T} ^ {n} = \sum_ {j = 1} ^ {k} \int_ {\Omega} [ \boldsymbol {B} ^ {n} ] ^ {T} (\boldsymbol {D} ^ {n}) _ {j} \boldsymbol {B} ^ {n} d \Omega , \tag {8.54} +$$ + +where $(\hat{D}^{n})_{j}$ is the value of $\hat{D}^{n}$ for each overlay in turn. Matrix $(\hat{D}^{n})_{j}$ will differ from overlay to overlay according to the material properties of each. The solution process is then identical to that described in the preceding sections with stress and strain terms being calculated for each overlay separately. It should be noted that the viscoplastic strain in each overlay will generally be different due to differences in threshold yield values and flow rates but the total strains must be the same. + +Although the name overlay model arises from the physical interpretation of the two-dimensional situation the technique is essentially a mathematical convenience and can be readily extended to three-dimensional problems. In such cases the thickness can no longer be interpreted as a physical quantity and becomes merely a weighting parameter for combining the contribution of individual overlays. Indeed this is also the case in two-dimensional problems where negative thicknesses can be employed to simulate strain-softening conditions. $^{(12)}$ + + + +# 8.15.2 Overlay models for some standard material behaviours + +In this section we reproduce some standard material responses by combining different viscoplastic components through the overlay concept. $^{(13)}$ + +![](images/page-318_f57e742016482c81eab78301b9ba0811983b5a0337657f16ddf6762d3fe6c3d8.jpg) +Fig. 8.7 Use of the overlay concept for the simulation of some standard material behaviours. + +# (i) Visco-elastic response + +A two overlay model with $F_{0}$ set to zero for one overlay and infinitely large in the other reproduces a standard linear visco-elastic solid (Fig. 8.7). Any higher order time dependent constitutive relation can be simulated by the introduction of more overlays of the Maxwell type (i.e. $F_{0} = 0$ ). Quite generally a stress–strain relationship of the form + +$$ +\sum_ {k = 0} ^ {n} a _ {k} D ^ {k} \sigma = \sum_ {k = 0} ^ {n} a _ {k} D ^ {k} \epsilon , \tag {8.55} +$$ + +in which $a_{k}$ and $b_{k}$ are real valued functions of the spatial coordinates and D denotes the differential time operator, can be modelled by the + + + +use of n Maxwell type overlays. The overlay approach reduces the $n^{th}$ order differential equation (8.55) to n first order equations. + +# (ii) Four parameter viscous model + +Two overlays with $F_{0}$ set to zero in each case provides a four parameter viscous model of the first kind (Fig. 8.7). Three overlays with $F_{0}$ set to (a) zero for one overlay (b) infinitely large for the second unit, (c) zero for the third overlay together with a small prescribed elastic modulus, reproduces a four parameter model of the second kind. + +# (iii) Three element viscous model + +A two overlay model with $F_{0}$ set to zero in both and the elastic modulus assigned to be infinitely large in one reproduces the three element viscous model. + +# (iv) Visco-elastic-plastic four parameter model + +This two overlay model is capable of reproducing the behaviour of most real engineering materials and is achieved by setting the threshold yield value of one overlay to zero. Before yielding of the friction slider, the material behaviour is visco-elastic followed by a viscoplastic response after initial yielding. By choosing the viscosity coefficients of the two dashpots appropriately the rate of straining after first yield can be controlled. + +In order to illustrate how the combination of two simple material responses by the overlay method can simulate a more complex material behaviour the load cycling problem indicated in Fig. 8.8 is presented. One elastic (yield value set very large) and one viscoplastic overlay are considered. A static analysis of the load cycling of this model was performed by allowing steady state conditions to be achieved after application of each increment of load. The results are shown in Fig. 8.8 where the material properties employed are also included. A Bauschinger effect is immediately apparent on reversal of loading with yielding in compression commencing at a reduced value compared with initial yield in tension. Thus although each overlay has been assumed to be non-strain hardening with equal yield stress in tension and compression, the composite model exhibits a kinematic hardening behaviour. + +As a further demonstration of the overlay approach, Fig. 8.9 shows how two overlays can be used to simulate the response of a real engineering material. The solid lines represent experimentally obtained creep curves for a rock salt and it is evident that the material behaviour is highly nonlinear with regard to the strain obtained at any time for a given applied load. The broken lines are the numerical material response obtained by using two overlays with material properties as shown in Fig. 8.9. The agreement obtained is acceptable for engineering purposes but a closer correspondence could be readily achieved by the use of additional overlays. + + + +The main advantage of the overlay technique is that it allows the description of complex material behaviours by the use of components which individually exhibit a simple response. + +All the program changes required to implement the overlay method in the viscoplastic program described earlier in this chapter are of a minor nature. Almost all the changes are associated with the summation process over each overlay demanded by (8.51), (8.53) and (8.54). Several array sizes must also be extended to allow separate storage of quantities for each overlay. Modification of the program is set as an exercise for the reader in Section 8.17. + +# 8.16 Numerical examples + +The first problem considered is the elasto-viscoplastic deformation of a thick tube under the action of internal pressure loading with the exterior surface remaining free. The mesh of Fig. 7.12(a) is employed in analysis with + +![](images/page-320_6577f7f1bb303e7ed4a86524a181efe727dbf1c5bf6f46cad43a40ed3d09509f.jpg) + +
+line + +| ε/ε₀ | σ/Yₐ | +|------|------| +| -1.5 | -1.5 | +| -1.0 | -1.0 | +| -0.5 | -0.5 | +| 0.0 | 0.0 | +| 0.5 | 0.5 | +| 1.0 | 1.0 | +| 1.5 | 1.5 | +| 2.0 | 2.0 | +
+ +Fig. 8.8 Load cycling response of an overlay composite illustrating the Bauschinger effect. diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_033.md b/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_033.md new file mode 100644 index 00000000..d7dd1411 --- /dev/null +++ b/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_033.md @@ -0,0 +1,407 @@ + + +![](images/page-321_9aa9dc685fc07be69dfa6b9770fed2b9d64df5fa3709ad27edb45e6501614fc6.jpg) + +
+line + +| Overlay | Time (days) | Experimental Strain (%) | Finite element Strain (%) | +| ------- | ----------- | ------------------------ | -------------------------- | +| 1 | 0 | 1.0 | 1.2 | +| 1 | 10 | 1.6 | 1.8 | +| 1 | 20 | 2.0 | 2.2 | +| 1 | 30 | 2.4 | 2.6 | +| 1 | 40 | 2.6 | 2.8 | +| 1 | 50 | 2.8 | 3.0 | +| 1 | 60 | 3.0 | 3.2 | +| 2 | 0 | 1.0 | 1.2 | +| 2 | 10 | 1.6 | 1.8 | +| 2 | 20 | 2.0 | 2.2 | +| 2 | 30 | 2.4 | 2.6 | +| 2 | 40 | 2.6 | 2.8 | +| 2 | 50 | 2.8 | 3.0 | +| 2 | 60 | 3.0 | 3.2 | +
+ +Fig. 8.9 Numerical simulation of experimental creep curves by use of the overlay method. + + + +plane strain conditions being assumed in the axial direction. The material properties employed are identical to the case of Fig. 7.12(a) and the fluidity parameter is chosen as $\gamma = 0.001$ . Again a Von Mises yield surface is adopted in solution and the flow function $\Phi(F) = F$ is assumed. An explicit time stepping algorithm ( $\Theta = 0$ ) is initially employed and the radial displacement of the inner surface with time is shown in Fig. 8.10 for two increments of applied pressure. Steady state conditions are allowed to develop for an applied pressure of $12\mathrm{dN/mm}^2$ before a further pressure increment of $2\mathrm{dN/mm}^2$ is added. For each increment the time stepping parameter values $\tau = 0.01$ , $k = 1.5$ were employed, the initial time-step length was chosen as 0.1 days and the steady state convergence tolerance parameter taken as $0.1\%$ . Also shown in Fig. 8.10 are the results for the situation when an internal pressure of $P = 14\mathrm{dN/mm}^2$ is instantaneously applied. The steady state displacement is seen to be in good agreement with that obtained from the two-load + +![](images/page-322_8039fbe3ea7f9baf1e40787c95fefe0aba203e06d87d0dcdbc83cf5a58e85704.jpg) + +
+line + +| time (days) | radial displacement of inner face (mm) - P = 14 dN/mm² | radial displacement of inner face (mm) - P = 12 dN/mm² | +| ----------- | -------------------------------------------------- | -------------------------------------------------- | +| 0.0 | 0.127 | 0.110 | +| 1.0 | 0.132 | 0.110 | +| 2.0 | 0.135 | 0.110 | +| 3.0 | 0.137 | 0.110 | +| 4.0 | 0.138 | 0.110 | +| 5.0 | 0.139 | 0.110 | +| 6.0 | 0.139 | 0.110 | +| 7.0 | 0.139 | 0.110 | +| 8.0 | 0.139 | 0.110 | +| 9.0 | 0.139 | 0.110 | +| 10.0 | 0.139 | 0.110 | +
+ +Fig. 8.10 Displacement of the inner surface with time of an elasto-viscoplastic cylinder subjected to an incrementally applied internal pressure (Mesh of Fig. 7.12(a)). + +increment solution. The problem was reanalysed for an applied pressure, $P = 14 \, dN/mm^{2}$ using larger time-step lengths as governed by $\tau = 0.05$ . The loss of accuracy is immediately apparent, with the larger time steps overestimating the viscoplastic strain rates. + +The problem was then resolved using in turn, the implicit trapezoidal time stepping scheme ( $\Theta = \frac{1}{2}$ ) and the full implicit or backward difference scheme ( $\Theta = 1$ ). Good agreement between the three time integration schemes + + + +![](images/page-323_c616896949227cf527c47b3e75a30a48049ef37fc2fe2a7554defa12e83de6c5.jpg) + +
+line + +| time (days) | radial displacement of inner face (mm) | scheme | +| ----------- | ------------------------------------- | ------ | +| 0.0 | 0.128 | 0 | +| 0.5 | 0.129 | 0 | +| 1.0 | 0.130 | 0 | +| 1.5 | 0.131 | 0 | +| 2.0 | 0.132 | 0 | +| 2.5 | 0.133 | 0 | +| 3.0 | 0.134 | 0 | +| 3.5 | 0.135 | 0 | +| 4.0 | 0.136 | 0 | +| 4.5 | 0.137 | 0 | +| 5.0 | 0.138 | 0 | +| 5.5 | 0.139 | 0 | +| 6.0 | 0.1395 | 0 | +| 6.5 | 0.1398 | 0 | +| 7.0 | 0.1399 | 0 | +| 7.5 | 0.13995 | 0 | +| 8.0 | 0.13998 | 0 | +| 0.0 | 0.128 | 0.5 | +| 0.5 | 0.129 | 0.5 | +| 1.0 | 0.130 | 0.5 | +| 1.5 | 0.131 | 0.5 | +| 2.0 | 0.132 | 0.5 | +| 2.5 | 0.133 | 0.5 | +| 3.0 | 0.134 | 0.5 | +| 3.5 | 0.135 | 0.5 | +| 4.0 | 0.136 | 0.5 | +| 4.5 | 0.137 | 0.5 | +| 5.0 | 0.138 | 0.5 | +| 5.5 | 0.139 | 0.5 | +| 6.0 | 0.1395 | 0.5 | +| 6.5 | 0.1398 | 0.5 | +| 7.0 | 0.1399 | 0.5 | +| 7.5 | 0.13995 | 0.5 | +| 8.0 | 0.13998 | 0.5 | +| 0.0 | 0.128 | 1.0 | +| 0.5 | 0.129 | 1.0 | +| 1.0 | 0.130 | 1.0 | +| 1.5 | 0.131 | 1.0 | +| 2.0 | 0.132 | 1.0 | +| 2.5 | 0.133 | 1.0 | +| 3.0 | 0.134 | 1.0 | +| 3.5 | 0.135 | 1.0 | +| 4.0 | 0.136 | 1.0 | +| 4.5 | 0.137 | 1.0 | +| 5.0 | 0.138 | 1.0 | +| 5.5 | 0.139 | 1.0 | +| 6.0 | 0.1395 | 1.0 | +| 6.5 | 0.1398 | 1.0 | +| 7.0 | 0.1399 | 1.0 | +| 7.5 | 0.13995 | 1.0 | +| 8.0 | 0.13998 | 1.0 | +| 0.0 | 0.128 | 0 | +| 0.5 | 0.129 | 0 | +| 1.0 | 0.130 | 0 | +| 1.5 | 0.131 | 0 | +| 2.0 | 0.132 | 0 | +| 2..5 | 0.133 | 0 | +| 3.0 | 0.134 | 0 | +| 3.5 | 0.135 | 0 | +| 4.0 | 0.136 | 0 | +| 4.5 | 0.137 | 0 | +| 5.0 | 0.0000 | 0 | +| 5.5 | 0.0000 | 0 | +| 6.0 | 0.0000 | 0 | +| 6.5 | 0.0000 | 0 | +| 7.0 | 0.0000 | 0 | +| 7.5 | 0.0000 | 0 | +| 8.0 | 0.0000 | 0 | +
+ +Fig. 8.11 Comparison of various time integration schemes for the internally pressurised cylinder of Fig. 8.10. + +![](images/page-323_770fb6cc0e20b57cd8456b712dabffe6de6698fcdd7e8b4b1b0e86f4cdf7f559.jpg) + +
+line + +| Applied Pressure (mm) | Elastic Solution (dN/mm²) | Elastoplastic Solution (dN/mm²) | +| --------------------- | ------------------------ | ------------------------------- | +| 100 | 22.0 | 14.5 | +| 200 | 18.0 | 17.0 | +| 300 | 16.0 | 15.0 | +| 400 | 14.0 | 13.0 | +| 500 | 12.0 | 11.0 | +| 600 | 10.0 | 9.0 | +| 700 | 8.0 | 7.0 | +
+ +Fig. 8.12 Steady state tangential stress distribution in an elasto-viscoplastic internally pressurised cylinder. + +is evident in Fig. 8.11 with the steady state displacement in each case comparing well with the corresponding elasto-plastic value of Fig. 7.12(b). + +The steady state hoop stress distributions are shown in Fig. 8.12 for the time integration schemes $\Theta = 0$ and $\Theta = 1$ , and the results are compared with the elasto-plastic solution of Fig. 7.13. Excellent agreement is obtained + + + +as required; since theoretically the steady state viscoplastic solution coincides with the corresponding elasto-plastic solution. + +The problem of the stresses induced in the vicinity of an excavated underground storage cavity is illustrated in Fig. 8.13. Applications in this area include oil and gas reservoirs, nuclear waste disposal and geothermal energy problems. The cavity is assumed to be axisymmetric and Fig. 8.13 + +![](images/page-324_31942e6434ee77349c2e2c3abee02d214465daa05f913529c94ddfab5d10ea6e.jpg) + +
+text_image + +ground level +z +0 +-0.2 +-0.4 +-0.6 +-0.8 +800 m +60 m +plastic +zone +B +r +800 m +800 m +40800 KN/m² +
+ +![](images/page-324_640187fe8e37f21319feef2b83631fc3750a1863dddd900e94203d6b5f8c8be2.jpg) + +
+line + +| Point | Value (m) | +|-------|-----------| +| A | 0.0 | +| B | -0.6 | +
+ +gravity and pressure loading +instantaneously applied at time, t = 0 + +$$ +\mathrm{E} = 6. 9 \times 1 0 ^ {5} \mathrm{KN} / \mathrm{m} ^ {2} +$$ + +$$ +\nu = 0. 4 +$$ + +$$ +\rho = 2 5 5 0 \mathrm{Kg} / \mathrm{m} ^ {3} +$$ + +$$ +\mathrm{F} _ {0} = 1 0 0 0 0 \mathrm{KN} / \mathrm{m} ^ {2} +$$ + +$$ +\gamma = 0. 0 7 5 / \text { y e a r } +$$ + +$$ +\Phi (\mathbf {F}) = \mathbf {F} +$$ + +$$ +\mathbf {H} ^ {\prime} = 0. 0 +$$ + +Von mises yield criterion + +explicit time integration, $\tau = 0.05$ + +steady state conditions + +achieved in 0.7 years. + +Fig. 8.13 Elasto-viscoplastic analysis of a subterranean cavity, showing zones of plasticity and steady state radial displacement at mid-height. + +shows the finite element idealisation of a cylindrical portion of the surrounding rock mass. Before excavation of the cavity the tectonic stress field in the rock is assumed to be hydrostatic. This condition is simulated by a gravity loading together with a lateral hydrostatic pressure applied to the cylindrical face of the model. The material properties employed are indicated in Fig. 8.13. The cavity is assumed to be instantaneously excavated at time t = 0 and viscoplastic solution to steady state conditions performed by explicit time integration ( $\Theta = 0$ ). Steady state conditions are achieved in 0.7 years and the zones of viscoplastic deformation at this time are illustrated in Fig. 8.13. It should be emphasised that since the fluidity parameter $\gamma$ only enters the viscoplastic expressions through the product $\gamma.t$ , then solution for different material fluidity values simply necessitates an adjustment of the time scale. Figure 8.13 also shows the radial displacement along section AB at steady state. The displacement distribution is seen to be made up of a + + + +![](images/page-325_3f169859936abc7b5d5f400729aabaa2590f0f81585b7dc48b9a1baf8e6f25e1.jpg) + +
+line + +| radial stress, σr (KN/m²) | Value | +| ------------------------- | --------- | +| 0 | 0 | +| 1000 | -2000 | +| 2000 | -4000 | +| 3000 | -6000 | +| 4000 | -8000 | +| 5000 | -10000 | +| 6000 | -12000 | +| 7000 | -14000 | +| 8000 | -16000 | +| 9000 | -18000 | +| 10000 | -20000 | +| 11000 | -21000 | +| 12000 | -22000 | +| 13000 | -22500 | +| 14000 | -22500 | +| 15000 | -22500 | +
+ +![](images/page-325_2fca247702e791287c06ed7f327f2b84c912902f3acf582ec3a618e8b8fba8e3.jpg) + +
+line + +| tangential stress. σ_H (KN/m²) | Value | +| ----------------------------- | --------- | +| -1000 | 0 | +| -800 | -10000 | +| -600 | -12000 | +| -400 | -15000 | +| -200 | -18000 | +| 0 | -23000 | +| 200 | -23000 | +| 400 | -23000 | +| 600 | -23000 | +| 800 | -23000 | +| 1000 | -23000 | +
+ +Fig. 8.14 Radial and tangential stress distributions for the problem of Fig. 8.13. + +linear field caused by the external applied pressure, superimposed on which is the effect of the cavity presence (the shaded area). + +Finally, Fig. 8.14 shows the steady state radial and tangential stress distributions along the line of Gaussian integration points nearest section AB. It is noted that away from the vicinity of the cavity, the hydrostatic condition $\sigma_{r} = \sigma_{0}$ is reproduced. + +# 8.17 Problems + +8.1 Use program VISCOUNT documented in Appendix II, Section A2.2 to solve the thick sphere considered in Problem 7.5 for the viscoplastic case. Employ the same material properties and load increment sizes as used in the elasto-plastic analysis. Assume the fluidity parameter + + + +$\gamma = 0.001$ and flow function $\Phi(F) = F$ . Use explicit time integration $(\Theta = 0)$ and compare your steady state solutions with the results of Problem 7.5. + +8.2 Repeat Problem 8.1 for different limiting time step lengths employing explicit time integration. Take the factor $\tau$ , described in Section 8.3, in the range $0.01 \leqslant \tau \leqslant 0.5$ . Comment on the accuracy of solution in each case. +8.3 Repeat Problem 8.1 using the flow functions (8.8) and (8.9). Take the indices M and N in the range 2 to 4. Comment on the solutions. +8.4 Repeat Problem 8.1 using (a) Fully implicit method ( $\Theta = 1$ ) and (b) Implicit trapezoidal rule ( $\Theta = \frac{1}{2}$ ). Comment on the accuracy and computational costs of solution. +8.5 Modify program VISCOUNT to include the strain-hardening law considered in Problem 7.4. +8.6 Undertake all the coding changes required to program VISCOUNT to include the overlay concept described in Section 8.15. +8.7 Test the modified program of Problem 8.6 by employing it in the solution of the uniaxial problem of Fig. 8.15. A constant stress of 100 is applied at time t = 0 to the plane stress model shown. Determine the development of strain with time. Verify the numerical solution by noting Figs. 8.4 and 8.5 and hence comparing with the analytical solution of Problem 4.2. + +![](images/page-326_d1283a58adbdc45e2a4cc7406a2116047b3b45e899e9443db89b6598a5d7753c.jpg) + +
+text_image + +100 +100 +
+ +
Overlay 1Overlay 2
E1000.01000.0
$\nu$ 0.00.0
t0.50.5
$\sigma_{\gamma}$ 0.025.0
H'100.0100.0
$\gamma$ 0.010.01
+ +Fig. 8.15 Overlay model example—Problem 8.7. + +8.8 In Section 8.2.3 it was stated that large deformation effects could be included, adopting a Lagrangian formulation, by including both the linear and nonlinear terms of the general quadratic relationship between strains and displacements according to (8.19). Details of geometrically nonlinear expressions can be found in Chapters 10 and 11. Modify program VISCOUNT to include such geometrically nonlinear behaviour. + + + +8.9 Employ the modified program of Problem 8.8 to solve the creep buckling problem illustrated in Fig. 8.16. The creep law employed is indicated in Fig. 8.16 and is a particular form of expression (8.9). Using the finite element mesh shown, apply the eccentric load to the cantilever at time, t = 0, and employ the implicit time integration algorithm ( $\Theta = 1$ ) to determine the deformation with increasing time. At some stage of the solution process the structure will become unstable due to creep buckling. Carry out the analysis for $\lambda = 1.0, 1.5, 2.0$ and 2.5 and compare the lateral deflection/time relationships with those provided in Ref. 6. + +![](images/page-327_d5d97ed0479266601128485696b82d196a64e0e13edfeb460d3826e2ecdf1fc4.jpg) + +
+text_image + +λ10⁴ KN +0.25m +E = 2.0 × 10⁶ KN/m² +ν = 0 +ε̅c = 10⁻¹⁶σ̅³ +10m +1m +
+ +Fig. 8.16 Creep buckling example—Problem 8.9. + +8.10 Modify program VISCOUNT to undertake the elasto-viscoplastic solution of three-dimensional solids. The majority of the subroutines required have been already modified in Problem 7.9. + +8.11 Repeat Problem 7.10 for the elasto-viscoplastic program VISCOUNT. + +# 8.18 References + +1. OLSZAK, W. and PERZYNA, P., Stationary and non-stationary visco-plasticity, In: M. F. Kanninen et al. (Eds), Inelastic Behaviour of Solids, McGraw-Hill, 1970. +2. OLSZAK, W. and PERZYNA, P., On elasto-viscoplastic soils, In: Rheology and Soil Mechanics, IUTAM Symposium, Springer-Verlag, Grenoble, 1966. +3. PERZYNA, P., Fundamental problems in visco-plasticity, In: Recent Advances in Applied Mechanics, Academic Press, New York, 1966. +4. ZIENKIEWICZ, O. C. and CORMEAU, I. C., Visco-plasticity—plasticity and creep in elastic solids—a unified numerical solution approach, Int. J. Num. Meth. Engng. 8, 821–845 (1974). +5. ZIENKIEWICZ, O. C., OWEN, D. R. J. and CORMEAU, I. C., Analysis of viscoplastic effects in pressure vessels by the finite element method, Nuclear Engineering & Design, 28(2), 278–288 (1974). + + + +§. KANCHI, M. B., ZIENKIEWICZ, O. C. and OWEN, D. R. J., The visco-plastic approach to problems of elasticity and creep involving geometric nonlinear effects. Int. J. Num. Meth. Engng. 12, 169–181 (1978). +7. STRICKLIN, J. A., HAISLER, W. and REISEMANN, W., Evaluation of solution procedures of material and/or geometrically non-linear structural analysis, AIAA J. 11, 292–299 (1973). +8. DINIS, L. M. S. and OWEN, D. R. J., Elastic-viscoplastic analysis of plates by the finite element method, Computers & Structures, 8, 207-215 (1978). +9. CORMEAU, I., Numerical stability in quasistatic elasto-visco-plasticity, Int. J. Num. Meth. Engng. 9, 109–127 (1975). +10. Duwez, P., On the Plasticity of Crystals, Physical Review, 47(6), 494–501 (1935). +11. ZIENKIEWICZ, O. C., NAYAK, G. C. and OWEN, D. R. J., Composite and overlay models in numerical analysis of elasto-plastic continua, Int. Symp. Foundations of Plasticity, Warsaw (1972). +12. OWEN, D. R. J., PRAKASH, A. and ZIENKIEWICZ, O. C., Finite element analysis of non-linear composite materials by use of overlay systems, Computers & Structures 4, 1251–1267 (1974). +13. PANDE, G. N., OWEN, D. R. J and ZIENKIEWICZ, O. C., Overlay models in time-dependent nonlinear material analysis. Computers & Structures, 7, 435-443 (1977). +14. HUGHES, T. J. R. and TAYLOR, R. L., Unconditionally stable algorithms for quasi-static elasto/viscoplastic finite element analysis, Int. J. Num. Meth. Engng. (to be published). + + + +# Chapter 9 Elasto-plastic Mindlin plate bending analysis + +Written in collaboration with M. M. Huq + +# 9.1 Introduction + +In Chapter 5 we introduced some elastoplastic Timoshenko beam formulations. In this chapter we introduce some related elasto-plastic Mindlin plate bending formulations. + +There are basically three theories which we could use as a basis for elastic plate bending: + +(i) Kirchhoff classical thin plate theory This theory, which takes no account of transverse shear deformation, is usually favoured by engineers because of its simplicity. It is the plate bending equivalent of Euler–Bernoulli beam theory. Many conforming $C(1)$ and non-conforming $C(0)$ plate elements are available. + +(ii) Mindlin (or Reissner) plate theory. Mindlin and the related Reissner plate theories allow for transverse shear effects. Mindlin plate theory is the plate bending equivalent of Timoshenko beam theory. Several Mindlin plate elements have been presented in the literature and it emerges that the most convenient one is the 'Heterosis' element of Hughes. $^{(1)}$ + +(iii) Full three-dimensional theory. For the greatest accuracy, full three-dimensional theory should be employed. Many 3D hexahedral and tetrahedral elements have been presented. Unfortunately when the aspect ratio of the element is very large as in thin plates, an ill-conditioned stiffness matrix results and roundoff problems predominate. Several schemes for avoiding this difficulty have been presented and undoubtedly an analysis based on this procedure is the most accurate. + +Let us now consider the various possibilities for elasto-plastic analysis. + +(i) We could use a full 3D analysis with a yield function $F(\sigma_x, \sigma_y, \sigma_z, \tau_{xy}, \tau_{xz}, \tau_{yz})$ . + +(ii) In a Mindlin plate formulation we can also use the yield function $F(\sigma_x, \sigma_y, \sigma_z, \tau_{xy}, \tau_{xz}, \tau_{yz})$ . It should be noted that $\sigma_z$ is taken as zero in + + + +Mindlin plates. This approach allows for the spread of plasticity from the extreme fibre over the entire plate thickness. In the evaluation of the internal virtual work integrals we may sample the stresses of the Gauss-Legendre or Lobatto integration points. Alternatively we may divide the plate into layers and use a mid-ordinate rule. + +(iii) In a Mindlin or Kirchhoff formulation we can use a yield function $F(\sigma_x, \sigma_y, \tau_{xy})$ . In Mindlin plate theory we ignore the effect of $\tau_{xz}$ and $\tau_{yz}$ on the plastic behaviour. Since, in the absence of inplane forces, the inplane stresses are a maximum at the extreme fibres where the transverse shear stresses are a minimum and the inplane stresses are a minimum at the mid-plane where the transverse shears are a maximum, this is a reasonable assumption. (There is also further evidence to suggest that it is likely to lead to insignificant errors.) This approach also allows for the spread of plasticity over the depth of the plate. In the evaluation of the internal virtual work integrals we may sample the stresses at the Gauss–Legendre or Lobatto integration points. Alternatively we may divide the plate into layers\* and use a mid-ordinate rule. This ‘layered’ approach has been described in Chapter 5 for a Timoshenko beam element and is a very popular method. + +(iv) In a Mindlin or Kirchhoff formulation we can adopt in the absence of inplane forces a yield function $F(M_x, M_y, M_{xy})$ which is a function of the bending moments. Here it is assumed that at a point the whole plate section becomes plastic simultaneously. A similar approach was described in Chapter 5 for Timoshenko beam elements. + +The elasto-plastic analysis of Mindlin plates is considered in this chapter, where both layered and non-layered approaches are treated in detail. + +Finite elements based on Mindlin's assumptions have one important advantage over elements based on classical thin plate theory. Mindlin plate elements require only $C(0)$ continuity of the lateral displacement $w$ and the two independent nodal rotations $\theta_x$ and $\theta_y$ . However elements based on classical Kirchhoff thin plate theory require $C(1)$ continuity; in other words $\partial w / \partial x$ and $\partial w / \partial y$ as well as $w$ must be continuous across element interfaces. Thus, Mindlin plate elements are simpler to formulate and they have the added advantage of being able to model shear-weak as well as shear-stiff plates. Consequently, if transverse shear deformations are present they are automatically modelled with Mindlin elements. + +Recent research $^{(1)}$ indicates that the use of a ‘Heterosis’ quadrilateral Mindlin plate element with quadratic Lagrangian interpolation for $\theta_{x}$ and $\theta_{y}$ and quadratic Serendipity interpolation for w together with selective integration of the stiffness matrix, gives the best overall performance. It diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_034.md b/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_034.md new file mode 100644 index 00000000..d9553c57 --- /dev/null +++ b/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_034.md @@ -0,0 +1,520 @@ + + +avoids locking and contains no spurious mechanisms. The Heterosis element is implemented here using a hierarchical formulation described later. + +We have already considered elastic Mindlin plate finite element analysis in Chapter 6. Nonlinear Mindlin plate finite element analysis is now considered. + +# 9.2 Equilibrium equations + +# 9.2.1 Three-dimensional equilibrium equations + +Let us begin with the equilibrium equations of three-dimensional stress analysis. We will assume that, for convenience, no tractions are present on the boundary $\Gamma_{t}$ of the three-dimensional domain $\Omega$ . The virtual work equation may be expressed as + +$$ +\int_ {\Omega} \{[ \delta \epsilon ] ^ {T} \sigma - [ \delta u ] ^ {T} b \} d \Omega = 0 \tag {9.1} +$$ + +where the vector of virtual displacements in the x, y and z directions is $\delta u = [\delta u, \delta v, \delta w]^{T}$ , the vector of associated virtual strains is $\delta \epsilon = [\delta \epsilon_{x}, \delta \epsilon_{y}, \delta \epsilon_{z}, \delta \gamma_{xy}, \delta \gamma_{xz}, \delta \gamma_{yz}]^{T}$ , the vector of stress is $\sigma = [\sigma_{x}, \sigma_{y}, \sigma_{z}, \tau_{xy}, \tau_{xz}, \tau_{yz}]^{T}$ and the vector of applied body forces is $b = [b_{x}, b_{y}, b_{z}]^{T}$ . Displacements u are prescribed on boundary $\Gamma_{u}$ of domain $\Omega$ . + +The stress-strain relationships for an isotropic material are given as + +$$ +\boldsymbol {D} = a _ {1} \left[ \begin{array}{c c c c c c} a _ {2} & a _ {3} & a _ {3} & 0 & 0 & 0 \\ a _ {3} & a _ {2} & a _ {3} & 0 & 0 & 0 \\ a _ {3} & a _ {3} & a _ {2} & 0 & 0 & 0 \\ 0 & 0 & 0 & a _ {4} & 0 & 0 \\ 0 & 0 & 0 & 0 & a _ {4} & 0 \\ 0 & 0 & 0 & 0 & 0 & a _ {4} \end{array} \right] \tag {9.2} +$$ + +where $a_1 = E / (1 + \nu)(1 - 2\nu)$ , $a_2 = 1 - \nu$ , $a_3 = \nu$ and $a_4 = (1 - 2\nu) / 2$ . Note that $E$ is the elastic modulus and $\nu$ is Poisson's ratio. + +# 9.2.2 Mindlin plate equilibrium equations + +In Mindlin plate theory, the domain of interest $\Omega$ is of the special form + +$$ +\Omega = \{(x, y, z) \in R ^ {3} \mid z \in [ - t / 2, t / 2 ], (x, y) \in A \in R ^ {2} \} \tag {9.3} +$$ + +where t is the plate thickness which may be a function of x and y and A is the plate area. The boundary of A is denoted by $\Gamma$ . + +We also make the following set of assumptions: + +(i) Normals to the midsurface (i.e., $z = 0$ ) before deformation remain straight but not necessarily normal to the midsurface after deformation. If $\theta_x$ and $\theta_y$ are the rotations of the midsurface normal in the $xz$ - and $yz$ -plane respectively, then + + + +$$ +\mathbf {u} = \left[ \begin{array}{l} u (x, y, z) \\ v (x, y, z) \\ w (x, y, z) \end{array} \right] = \left[ \begin{array}{c} - z \theta_ {x} (x, y) \\ - z \theta_ {y} (x, y) \\ w (x, y) \end{array} \right] \tag {9.4} +$$ + +The sign convention is illustrated in Fig. (9.1). Right hand rotations $\bar{\theta}_{x}$ and $\bar{\theta}_{y}$ are defined by the expression + +$$ +\left[ \begin{array}{l} \theta_ {x} \\ \theta_ {y} \end{array} \right] = \left[ \begin{array}{l l} 0 & 1 \\ - 1 & 0 \end{array} \right] \left[ \begin{array}{l} \bar {\theta} _ {x} \\ \bar {\theta} _ {y} \end{array} \right]. \tag {9.5} +$$ + +It is usually more convenient to develop the theory in terms of $\theta_{x}$ and $\theta_{y}$ rather than $\bar{\theta}_{x}$ and $\bar{\theta}_{y}$ since the resulting algebra is greatly simplified. + +(ii) The normal stress $\sigma_z$ is assumed equal to zero. The virtual work statement may be expressed as + +$$ +\int_ {\Omega} \left[ \delta \boldsymbol {\epsilon} ^ {\prime} \right] ^ {T} \boldsymbol {\sigma} ^ {\prime} d \Omega - \int_ {\Omega} \left[ \delta \boldsymbol {u} \right] ^ {T} \boldsymbol {b} d \Omega = 0 \tag {9.6} +$$ + +in which + +$$ +[ \delta \epsilon^ {\prime} ] = [ \delta \epsilon_ {x}, \delta \epsilon_ {y}, \delta \gamma_ {x y} | \delta \gamma_ {x z}, \delta \gamma_ {y z} ] ^ {T} = [ (\delta \epsilon_ {f}) ^ {T}, (\delta \epsilon_ {s}) ^ {T} ] ^ {T} +$$ + +and + +$$ +\sigma^ {\prime} = \left[ \sigma_ {x}, \sigma_ {y}, \sigma_ {z} \mid \tau_ {x z}, \tau_ {y z} \right] ^ {T} = \left[ \left(\sigma_ {f}\right) ^ {T}, \left(\sigma_ {s}\right) ^ {T} \right] ^ {T}. +$$ + +Note that + +$$ +\delta \epsilon_ {f} = z \left[ - \frac {\partial (\delta \theta_ {x})}{\partial x}, - \frac {\partial (\delta \theta_ {y})}{\partial y}, - \left(\frac {\partial (\delta \theta_ {x})}{\partial y} + \frac {\partial (\delta \theta_ {y})}{\partial x}\right) \right] ^ {T} \doteq z \delta \hat {\epsilon} _ {f} \tag {9.7} +$$ + +• In Mindlin plate theory a reduced form of the constitutive relations is obtained by making $\sigma_{z}=0$ and subsequently eliminating $\epsilon_{z}$ . Thus + +$$ +\sigma^ {\prime} = D ^ {\prime} \epsilon^ {\prime} +$$ + +where for elastic isotropic situations + +$$ +\boldsymbol {D} ^ {\prime} = \frac {E}{(1 - \nu^ {2})} \left[ \begin{array}{c c c c c c} 1 & \nu & 0 & & 0 & 0 \\ \nu & 1 & 0 & & 0 & 0 \\ 0 & 0 & \frac {(1 - \nu)}{2} & & 0 & 0 \\ \hline 0 & 0 & 0 & & \frac {(1 - \nu)}{2} & 0 \\ 0 & 0 & 0 & & 0 & \frac {(1 - \nu)}{2} \end{array} \right] = \left[ \begin{array}{c c} \boldsymbol {D} _ {f ^ {\prime}} & \boldsymbol {0} \\ \boldsymbol {0} & \boldsymbol {D} _ {s ^ {\prime}} \end{array} \right] +$$ + +† Terms symbolised thus ( $^{ }$ ) denote quantities integrated over the thickness. + + + +and + +$$ +\delta \epsilon_ {s} = \left[ \frac {\partial (\delta w)}{\partial x} - \delta \theta_ {x}, \frac {\partial (\delta w)}{\partial y} - \delta \theta_ {y} \right] ^ {T} = \delta \hat {\epsilon} _ {s}. \tag {9.8} +$$ + +Using (9.7) and (9.8) we find that (9.6) can be rewritten as + +$$ +\int_ {A} \int_ {- t / 2} ^ {t / 2} [ z (\delta \hat {\epsilon} _ {f}) ^ {T} \sigma_ {f} + (\delta \hat {\epsilon} _ {s}) ^ {T} \sigma_ {s} - (\delta \boldsymbol {u}) ^ {T} \boldsymbol {b} ] d z d A = 0 \tag {9.9} +$$ + +This equation is adopted in the layered approach. After integration over the thickness of the plate (9.9) can be rewritten in the form + +$$ +\int_ {A} \left[ \left(\delta \hat {\epsilon} _ {f}\right) ^ {T} \hat {\sigma} _ {f} - \left(\delta \epsilon_ {s}\right) ^ {T} \hat {\sigma} _ {s} - (\delta \boldsymbol {u}) ^ {T} \hat {\boldsymbol {b}} \right] d A = 0 \tag {9.10} +$$ + +where + +$$ +\hat {\sigma} _ {f} = \int_ {- t / 2} ^ {t / 2} z \sigma_ {f} d z +$$ + +$$ +\hat {\sigma} _ {s} = \int_ {- t / 2} ^ {t / 2} \sigma_ {s} d z +$$ + +and + +$$ +\hat {\boldsymbol {b}} = \int_ {- t / 2} ^ {t / 2} \boldsymbol {b} d z. +$$ + +We interpret $\hat{\sigma}_{f} = [M_{x}, M_{y}, M_{xy}]^{T}$ as the bending moments and $\hat{\sigma}_{s} = [Q_{x}, Q_{y}]^{T}$ as the shear force. Usually we take $\hat{b} = [q, 0, 0]^{T}$ in which q is the lateral distributed loading acting on the plate. We use (9.10) in the non-layered plate formulation. + +![](images/page-333_48c2408f63c5d49e4f1127fc8fa55e87eb7005477b6734ac98e2859e69ed9126.jpg) + +
+text_image + +z, w +Qx +Myx +Myx +Myx +Myx +Myx +Myx +Myx +Myx +Myx +Myx +Myx +Myx +Myx +Myx +Myx +Myx +Myx +Myx +Myx +Myx +Myx +Myx +Myx +Myx +Myx +Myx +Myx +Myx +Myx +Myx +Myx +Myx +Myx +Myy +Myx +Myx +Myx +Myx +Myx +Myx +Myx +Myx +Myx +Myx +Myx +Myx +Myx +Myx +Myx +Myx +Myx +Myx +Myx +Myx +Myx +Myx +Myx +Myx +Myx +Myx +Myx +Myx +Myx +Myx +Myx +Myx +Myx +
+ +Fig. 9.1 Sign convention for Mindlin plate theory. + + + +# 9.3 Discretisation + +# 9.3.1 Standard representation + +If we adopt a standard $C(0)$ finite element representation then the displacements can be written as + +$$ +u = \sum_ {i = 1} ^ {n} N _ {i} d _ {i} \tag {9.11} +$$ + +in which the shape function matrix is $N_{i} = N_{i}I_{3}$ and the vector of nodal values $d_{i} = [w_{i}, \theta_{xi}, \theta_{yi}]^{T}$ . + +The flexural strain displacement equations are given as + +$$ +\delta \hat {\epsilon} _ {f} = \sum_ {i = 1} ^ {n} B _ {f i} \delta d _ {i} \tag {9.12} +$$ + +in which + +$$ +\boldsymbol {B} _ {f i} = \left[ \begin{array}{c c c} 0 & - \frac {\partial N _ {i}}{\partial x} & 0 \\ 0 & 0 & - \frac {\partial N _ {i}}{\partial y} \\ 0 & - \frac {\partial N _ {i}}{\partial y} & - \frac {\partial N _ {i}}{\partial x} \end{array} \right] +$$ + +The shear strain displacement equations have the form + +$$ +\delta \hat {\epsilon} _ {s} = \sum_ {i = 1} ^ {n} B _ {s i} \delta d _ {i} \tag {9.13} +$$ + +in which + +$$ +\boldsymbol {B} _ {s i} = \left[ \begin{array}{c c c} \frac {\partial N _ {i}}{\partial x} & - N _ {i} & 0 \\ \frac {\partial N _ {i}}{\partial y} & 0 & - N _ {i} \end{array} \right] +$$ + +If we substitute (9.11)-(9.13) in (9.9) we obtain the expression + +$$ +\sum_ {i = 1} ^ {n} \left[ \delta \boldsymbol {d} _ {i} \right] ^ {T} \left\{\int_ {A} \int_ {- t / 2} ^ {t / 2} \left[ \boldsymbol {B} _ {f i} \right] ^ {T} \sigma_ {f} z + \left[ \boldsymbol {B} _ {s i} \right] ^ {T} \sigma_ {s} - \left[ N _ {i} \right] ^ {T} \boldsymbol {b} \right] d z d A \Bigg \} = 0. \tag {9.14} +$$ + + + +Since (9.14) must be true for any set of virtual displacements we obtain the expression + +$$ +\int_ {A} \int_ {- t / 2} ^ {t / 2} \left[ \left[ \boldsymbol {B} _ {f i} \right] ^ {T} \sigma_ {f} z + \left[ \boldsymbol {B} _ {s i} \right] ^ {T} \sigma_ {s} - \left[ N _ {i} \right] ^ {T} \boldsymbol {b} \right] d z d A = 0 \tag {9.15} +$$ + +or $\psi_{i}(d) = 0.$ + +We use (9.15) in the layered approach. If we integrate the terms in square brackets over the thickness of the plate then we obtain the following equation + +$$ +\int_ {A} \left[ \left[ \boldsymbol {B} _ {f t} \right] ^ {T} \hat {\boldsymbol {\sigma}} _ {f} + \left[ \boldsymbol {B} _ {s i} \right] ^ {T} \hat {\boldsymbol {\sigma}} _ {s} - \left[ \boldsymbol {N} _ {i} \right] ^ {T} \hat {\boldsymbol {b}} \right] d A = 0 \tag {9.16} +$$ + +or $\psi_{i}(\pmb {d}) = 0.$ + +Equation (9.16) is used in the nonlayered approach. + +Note that we obtain equations for the residual force vector $\psi_{i}(d)$ for every node in the finite element discretisation. When the stresses are nonlinear then both (9.15) and (9.16) are sets of nonlinear simultaneous equations. + +Contributions to the residual force vector $\psi = [\psi_1^T, \ldots, \psi_n^T]^T$ may be evaluated at the element level and then assembled to form $\psi$ . We may use any standard $C(0)$ two-dimensional isoparametric element. Several elements have been presented in the literature and it emerges that the most convenient one is the 8/9 node 'Heterosis' element of Hughes. $^{(1)}$ In the programs described later we use 4, 8 and 9-noded isoparametric quadrilateral elements (see Chapter 6), as well as the Heterosis element. Selective integration is adopted and this will be described later. + +# 9.3.2 Hierarchical formulation of the Heterosis element + +In the implementation of the Heterosis and the 9-node element a hierarchical formulation is adopted. The first 8 shape functions are borrowed from the 8-noded Serendipity element and the shape function for the central $9^{th}$ node is the bubble function + +$$ +N _ {9} ^ {(e)} (\xi , \eta) = (1 - \xi^ {2}) (1 - \eta^ {2}) \tag {9.17} +$$ + +which is already available from the quadratic Lagrangian element. This means that all variables associated with the central node are hierarchical in nature. In other words, they are departures from the interpolated Serendipity values. The hierarchical representation can be used for geometrical representation as well as for interpolating displacements. + +In order to implement the heterosis element we adopt a hierarchical formulation either by adding a stiff spring (large number) to the leading + + + +diagonal term of the stiffness matrix associated with the lateral displacement parameter for node 9, or by prescribing displacement at this centre node to zero. This has the effect of forcing w to behave as though it was represented by Serendipity quadratic shape functions. Thus the desired effect is achieved. + +It is worth noting that if no spring is added the element obtained is identical to the 9-noded Lagrangian element provided that care is taken in evaluating the consistent nodal forces. Furthermore if stiff springs are added to all the terms of the leading diagonal associated with node 9, then the element reverts to a Serendipity 8-noded element. + +For convenience, in the present case, when representing the geometry of the heterosis element, the x and y coordinate departures from the interpolated Serendipity values are taken as equal to zero. In other words, as Serendipity geometrical representation is adopted this distinction is only of importance when elements with curved boundaries are present. (N.B. This is automatically taken care of by a modified version of Subroutine, NODEXY described in Section 6.4.1). + +# 9.4 Solution of nonlinear equations + +# 9.4.1 Plasticity in layered plates + +For Mindlin plates we may assume that the yield function $F$ is a function of $\sigma_f$ , the direct stresses associated with flexure, but not of the transverse shear stresses $\sigma_s$ . The yield function $F$ is also a function of the hardening parameter, $H$ . When yielding occurs at some point, it is assumed that, unless unloading takes place, the stresses always remain on the yield surface so that + +$$ +F (\sigma_ {f}, H) = 0 \tag {9.18} +$$ + +Thus the incremental stress-strain relationship is given as + +$$ +\left[ \begin{array}{c} d \sigma_ {f} \\ d \sigma_ {s} \end{array} \right] = \left[ \begin{array}{c c} \left(\boldsymbol {D} _ {e p ^ {\prime}}\right) _ {f} & \boldsymbol {0} \\ \boldsymbol {0} & \boldsymbol {D} _ {s ^ {\prime}} \end{array} \right] \left[ \begin{array}{c} d \boldsymbol {\epsilon} _ {f} \\ d \boldsymbol {\epsilon} _ {s} \end{array} \right] \tag {9.19} +$$ + +or $d\pmb{\sigma}^{\prime} = \pmb{D}_{ep}{}^{\prime}d\pmb{\epsilon}^{\prime}$ + +in which $(D_{ep}^{\prime})_{f}$ is identical to $D_{ep}$ given in Chapter 7 for the elasto-plastic plane stress problem. Note that $D_{s}^{\prime}$ always remains elastic. Recall from equation (7.47) that + +$$ +\left(\boldsymbol {D} _ {e p} ^ {\prime}\right) _ {f} = \boldsymbol {D} _ {f} ^ {\prime} - \frac {\boldsymbol {d} _ {D} \boldsymbol {d} _ {D} ^ {T}}{A + \boldsymbol {d} _ {D} ^ {T} \boldsymbol {a} ^ {\prime}} \tag {9.20} +$$ + +where + +$$ +\boldsymbol {a} ^ {\prime} = \left[ \frac {\partial F}{\partial \sigma_ {x}}, \frac {\partial F}{\partial \sigma_ {y}}, \frac {\partial F}{\partial \tau_ {x y}} \right] ^ {T} +$$ + + + +$$ +\begin{array}{l} \boldsymbol {d} _ {D} = \boldsymbol {D} _ {f} ^ {\prime} \boldsymbol {a} ^ {\prime} \\ A = - \frac {1}{\lambda} \frac {\dot {c} F}{\dot {c} H} d H \\ \end{array} +$$ + +in which $\lambda$ is the proportionality constant. Here we cater for Von Mises and Tresca materials only. We can thus use a slightly modified version of the coding described in Chapter 7 when evaluating $(D_{rp'})_f$ and when testing for yielding etc. + +# 9.4.2 Solution of the nonlinear equilibrium equations for layered plates + +The incremental equilibrium equations for the plate can be written at some stage in the solution (i.e., at an iteration during a load increment) as + +$$ +\psi (\boldsymbol {d} ^ {p}) + \boldsymbol {K} _ {T} (\boldsymbol {d} ^ {p}) \Delta \boldsymbol {d} ^ {p} = 0 \tag {9.21} +$$ + +where $\psi$ is obtained from (9.15) and $K_{T}(d^{p})$ is the tangential stiffness matrix which may be approximated as + +$$ +\boldsymbol {K} _ {T} (\boldsymbol {d} ^ {p}) = \int_ {A} \int_ {- t / 2} ^ {t / 2} \{[ \boldsymbol {B} _ {f} ] ^ {T} [ \boldsymbol {D} _ {e p ^ {\prime}} ] _ {f} \boldsymbol {B} _ {f} + [ \boldsymbol {B} _ {s} ] ^ {T} \boldsymbol {D} _ {s ^ {\prime}} \boldsymbol {B} _ {s} \} d z d A. \tag {9.22} +$$ + +Since $[D_{ep}^{\prime}]_{f}$ is a function of z we may employ a numerical integration technique to evaluate the integral over the thickness of the plate. Here, we divide the plate into layers and use a mid-ordinate rule as described in Chapter 5 for the Timoshenko beam. We use a similar method to evaluate $\psi(d^{p})$ . Thus we have + +$$ +\boldsymbol {K} _ {T} (\boldsymbol {d} ^ {p}) = \int_ {A} \{[ \boldsymbol {B} _ {f} ] ^ {T} [ \hat {\boldsymbol {D}} _ {e p} ] _ {f} \boldsymbol {B} _ {f} \dots [ \boldsymbol {B} _ {s} ] ^ {T} \hat {\boldsymbol {D}} _ {s} \boldsymbol {B} _ {s} ] \} d A \tag {9.23} +$$ + +where + +$$ +[ \hat {D} _ {e p} ] _ {f} = \int_ {- t / 2} ^ {t / 2} [ D _ {e p} ^ {\prime} ] _ {f} d z +$$ + +and + +$$ +\hat {\boldsymbol {D}} _ {s} = \int_ {- t / 2} ^ {t / 2} \boldsymbol {D} _ {s} ^ {\prime} d z. +$$ + +We now use the standard procedure to solve (9.21). Instead of using $K_{T}(d^{p})$ we may use some previously calculated value of $K_{T}$ just as in the other applications. + +# 9.4.3 Plasticity in nonlayered plates + +In Chapter 5 we considered the elasto-plastic nonlayered analysis of Timoshenko beams in which we assumed that when the bending moment + + + +reaches the yield moment $M_{0}$ , the whole cross-section of the beam becomes plastic instantaneously. We noted that this is a convenient fiction as in reality there is always a gradual spread of plasticity over the depth of the beam. In elasto-plastic nonlayered Mindlin plate analysis we make a similar approximation. Here we assume that the yield function $\hat{F}$ is expressed as a function of the bending moments $\hat{\sigma}_{f}$ , but not of the shear forces $\hat{\sigma}_{s}$ . The yield function is also assumed to be a function of a hardening parameter $\hat{H}$ . During yield it is assumed that the stress resultants $\hat{\sigma}_{f}$ must remain on the yield surface so that + +$$ +\hat {F} (\hat {\sigma} _ {f}, \hat {H}) = 0 \tag {9.24} +$$ + +where for the Tresca and Von Mises materials under consideration + +$$ +\hat {F} (\hat {\sigma} _ {f}, \hat {H}) = \int_ {- t / 2} ^ {t / 2} F (\sigma_ {f}, H) d z. \tag {9.25} +$$ + +Therefore, although $\hat{F}$ replaces $F$ , $(M_x, M_y, M_{xy})$ replace $(\sigma_x, \sigma_y, \tau_{xy})$ and $M_0 = \sigma_0 t^2 / 4$ replaces $\sigma_0$ , everything else remains unchanged and we can again make use of the coding given in Chapter 7. + +The incremental stress-strain resultant relationships are given as + +$$ +\left[ \begin{array}{c} d \hat {\boldsymbol {\sigma}} _ {f} \\ d \hat {\boldsymbol {\sigma}} _ {s} \end{array} \right] = \left[ \begin{array}{c c} [ \hat {\boldsymbol {D}} _ {e p} ] _ {f} & \mathbf {0} \\ \mathbf {0} & \hat {\boldsymbol {D}} _ {s} \end{array} \right] \left[ \begin{array}{c} d \hat {\boldsymbol {\epsilon}} _ {f} \\ d \hat {\boldsymbol {\epsilon}} _ {s} \end{array} \right] \tag {9.26} +$$ + +in which + +$$ +[ \hat {\boldsymbol {D}} _ {e p} ] _ {f} = \hat {\boldsymbol {D}} _ {f} - \frac {\hat {\boldsymbol {d}} _ {D} \hat {\boldsymbol {d}} _ {D} ^ {T}}{\hat {\boldsymbol {A}} + \hat {\boldsymbol {d}} _ {D} ^ {T} \hat {\boldsymbol {a}}} \tag {9.27} +$$ + +in which + +$$ +\hat {\boldsymbol {a}} = \left[ \frac {\partial \hat {F}}{\partial M _ {x}}, \frac {\partial \hat {F}}{\partial M _ {y}}, \frac {\partial \hat {F}}{\partial M _ {x y}} \right] ^ {T} +$$ + +$$ +\hat {\boldsymbol {d}} _ {D} = \hat {\boldsymbol {D}} _ {f} \hat {\boldsymbol {a}} +$$ + +$$ +\hat {A} = - \frac {1}{\lambda} \frac {\partial \hat {F}}{\partial \hat {H}} d \hat {H} +$$ + +and + +$$ +\hat {\boldsymbol {D}} _ {f} = \int_ {- t / 2} ^ {t / 2} \boldsymbol {D} _ {f} ^ {\prime} z d z. +$$ + +Note also that + +$$ +\hat {\boldsymbol {D}} _ {s} = \int_ {- t / 2} ^ {t / 2} \boldsymbol {D} _ {s} ^ {\prime} d z. +$$ + + + +# 9.4.4 Solution of nonlinear equilibrium equations for nonlayered Mindlin plates + +For the nonlayered plates the equilibrium equations are identical to (9.21). Here, however, the tangential stiffness matrix is given as + +$$ +\boldsymbol {K} _ {T} = \int_ {A} \left\{\left[ \boldsymbol {B} _ {f} \right] ^ {T} \left[ \hat {\boldsymbol {D}} _ {e p} \right] _ {f} \boldsymbol {B} _ {f} + \left[ \boldsymbol {B} _ {s} \right] ^ {T} \hat {\boldsymbol {D}} _ {s} \boldsymbol {B} _ {s} \right\} d A. \tag {9.28} +$$ + +Apart from this modification the solution procedure is unchanged. + +# 9.4.5 Summary of solution procedures + +The solution procedures for elasto-plastic Mindlin plate analysis are summarised in Tables 9.1–9.3. The overall process is given in Table 9.1. The iteration loop is shown for the nonlayered and layered plates in Tables 9.2 and 9.3 respectively. + +Table 9.1 Equation solving technique for layered and nonlayered Mindlin plates +
1Begin new load increment, $f = f + \Delta f$ .
2Set $\Delta f$ equal to the current load increment vector.
3Set $d^{0}$ equal to 0 for the first increment or equal to the total displacement vector at the end of the last load increment.
4Set $\psi^{0}$ equal to the residual force vector at the end of the last increment or equal to 0 for the first load increment.
5Set $\psi^{0} = \psi^{0} + \Delta f$ .
6Solve $\Delta d^{0} = -[K_{T}]^{-1} \psi^{0}$ .Use old or updated value $K_{T}$ .
7Set $d^{1} = d^{0} + \Delta d^{0}$ .
8Evaluate $\psi^{1}(d^{1})$ .
9If solution has converged go to 11; otherwise continue.
10Iterate until solution has converged.
11If this is not the last increment go to 1; otherwise stop.
+ +Table 9.2 The iteration loop for elasto-plastic nonlayered Mindlin plates. + +
1Set iteration number i = 1.
2Solve Δdi = - [KT]−1ψi. Use old or updated KT.
3Set di+1 = di + Δdi.
4For each Gauss point, evaluate the increments in strain resultants
$$ +\Delta \hat {\epsilon} _ {f} ^ {i} = B _ {f} \Delta d ^ {i} +$$ + +$$ +\Delta \hat {\boldsymbol {\epsilon}} _ {s} ^ {i} = \boldsymbol {B} _ {s} \Delta \boldsymbol {d} ^ {i}. +$$ + + + +Table 9.2—continued + +5 Using the elastic rigidities estimate, at each Gauss point, the increments in stress resultants and hence the total stress resultants + +$$ +\Delta \hat {\boldsymbol {\sigma}} _ {f} ^ {i} = \hat {\boldsymbol {D}} _ {f} \Delta \hat {\boldsymbol {\epsilon}} _ {f} ^ {i} \quad \text { hence } \quad \hat {\boldsymbol {\sigma}} _ {f} ^ {i + 1} = \hat {\boldsymbol {\sigma}} _ {f} ^ {i} + \Delta \hat {\boldsymbol {\sigma}} _ {f} ^ {i} +$$ + +$$ +\Delta \hat {\boldsymbol {\sigma}} _ {s} ^ {i} = \hat {\boldsymbol {D}} _ {s} \Delta \hat {\boldsymbol {\epsilon}} _ {s} ^ {i} \quad \text { hence } \quad \hat {\boldsymbol {\sigma}} _ {s} ^ {i + 1} = \hat {\boldsymbol {\sigma}} _ {s} ^ {i} + \Delta \hat {\boldsymbol {\sigma}} _ {s} ^ {i}. +$$ + +6 At each Gauss point, depending on the states of $\hat{\sigma}_{f}^{i}$ and $\hat{\sigma}_{f}^{i+1}$ , adjust $\hat{\sigma}_{f}^{i+1}$ to satisfy the yield criterion and preserve the normality condition. + +7 Evaluate the residual force vector + +$$ +\psi^ {i + 1} = \int_ {A} \left\{\left[ B _ {f} \right] ^ {T} \hat {\sigma} _ {f} + \left[ B _ {s} \right] ^ {T} \hat {\sigma} _ {s} \right\} d A - f. +$$ + +8 If the solution has converged, continue, otherwise set $i = i + 1$ and go to 2. + +9 Move to next load increment. + +Table 9.3 The iteration loop for elasto-plastic layered Mindlin plates. + +1 Set iteration number $i = 1$ . +2 Solve $\Delta d^i = -[K_T]^{-1}\psi^i$ . +Use old or updated $K_T$ . +3 Set $d^{i+1} = d^i + \Delta d^i$ . +4 For each Gauss point in each layer evaluate the increment in strain + +$$ +\Delta \epsilon_ {f} ^ {i} = z B _ {f} \Delta d ^ {i} +$$ + +$$ +\Delta \epsilon_ {s} ^ {i} = B _ {s} \Delta d ^ {i}. +$$ + +5 Estimate the increments in stress at each Gauss point in each layer using the elastic stress-strain matrix. Hence evaluate the total stress value. + +$$ +\Delta \sigma_ {f} ^ {i} = \mathbf {D} _ {f} ^ {\prime} \Delta \epsilon_ {f} ^ {\prime}, \quad \sigma_ {f} ^ {i + 1} = \sigma_ {f} ^ {i} + \Delta \sigma_ {f} ^ {i} +$$ + +$$ +\Delta \sigma_ {s} ^ {i} = \mathbf {D} _ {s} ^ {\prime} \Delta \epsilon_ {s} ^ {\prime}, \quad \sigma_ {s} ^ {i + 1} = \sigma_ {s} ^ {i} + \Delta \sigma_ {s} ^ {i}. +$$ + +6 Depending on the states of $\sigma_f^i$ and $\sigma_f^{i+1}$ , adjust $\sigma_f^{i+1}$ to satisfy the yield criterion and preserve the normality condition. + +7 Evaluate the stress resultants $\hat{\sigma}_{f}^{i+1}$ and $\hat{\sigma}_{s}^{i+1}$ at each Gauss point. + +8 Evaluate the residual force vector + +$$ +\psi^ {i + 1} = \int_ {A} \left\{\left[ \boldsymbol {B} _ {f} \right] ^ {T} \hat {\sigma} _ {f} + \left[ \boldsymbol {B} _ {s} \right] ^ {T} \hat {\sigma} _ {s} \right\} d A - \boldsymbol {f}. +$$ + +9 If the solution has converged continue, otherwise set $i = i + 1$ and go to 2. + +10 Move to next load increment. diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_035.md b/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_035.md new file mode 100644 index 00000000..4ba315ee --- /dev/null +++ b/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_035.md @@ -0,0 +1,577 @@ + + +In this application we recommended the following convergence criteria. Let + +$$ +E _ {\delta} = \frac {[ \sum_ {j} (\Delta \delta_ {j} ^ {(i)}) ^ {2} ] ^ {1 / 2}}{[ \sum_ {j} (\delta_ {j} ^ {(i + 1)}) ^ {2} ] ^ {1 / 2}} \tag {9.29} +$$ + +where $\delta_j$ may equal $w_j$ , $\theta_{xj}$ or $\theta_{yj}$ . We take in any combination + +$$ +E _ {w}, E _ {\theta x}, E _ {\theta y}, \left(E _ {w} + E _ {\theta x} + E _ {\theta y}\right) \leqslant \text { TOLER } \tag {9.30} +$$ + +where TOLER is a specified tolerance. We can also take the residual force equivalents of $w_{j}$ , $\theta_{xj}$ or $\theta_{yj}$ in (9.29) and (9.30). + +# 9.5 Software for the non-layered approach + +# 9.5.1 Overall program structure + +The overall program structure for the elasto-plastic Mindlin plate bending analysis program MINDLIN using a nonlayered approach is given in Fig. 9.2. + +The dimensions given in subroutine FEMP agree with those given in subroutine DIMMP and limit the program to the following maximum size problems in the present form + +$$ +\begin{array}{l} \text { MELEM - maximum number of elements } = 2 5 \\ \text { MEVAB } - \text { maximum number of variables per element } = 2 7 \\ \text { MFRON - maximum front width } = 4 0 \\ \text { MMATS - maximum number of material sets } = 1 0 \\ \text { MPOIN } - \text { maximum number of nodal points } = 8 0 \\ \text { MTOTV } - \text { maximum total number of degrees of freedom } = 2 4 0 \\ \text { MVFIX } - \text { maximum number of prescribed boundary nodes } = 4 0 \\ \end{array} +$$ + +To modify these values the DIMENSION statement in FEMP and the appropriate statements in DIMMP should be carefully changed and checked. All new routines are now documented and these include: FEMP, CONVMP, DIMMP, FLOWMP, GRADMP, INVMP, MINDPB, OUTMP, SFR2,\* RESMP, STIFMP, STRMP, SUBMP, VZERO and ZEROMP. The other routines, which have been described earlier, include ALGOR, BMATPB, CHECK1,† CHECK2, ECHO, FRONT, INCREM, INPUT, JACOB2, MODPB and NODEXY.\* + +The files which are used in the program are 5 (cardreader), 6 (lineprinter) and 1, 2, 3, 4, 8 (scratch files). + +\- Note we include the modified versions of SFR2 and NODEXY to allow for hierarchical representation. + +† We include a very slightly modified version of CHECK 1. Note also that for 4-node Mindlin plate elements, GAUSSQ is modified to allow for a single point Gauss rule. See Section 6.4.2. + + + +![](images/page-342_129442c8b5be6e374b8ea47c6bff19b053a98b376cfb0fd9b276ba404cdb0c43.jpg) + +
+flowchart + +```mermaid +graph TD + A["START"] --> B["DIMMP
Presents the variables associated with the dynamic dimensioning process"] + B --> C["INPUT
Inputs data defining geometry, boundary conditions and material properties"] + C --> D["ZEROMP
Sets to zero arrays required for accumulation of data"] + D --> E["MINDPB
Inputs additional data required for Mindlin plate analysis"] + E --> F["LOADPB
Reads loading data and evaluates the equivalent nodal forces for distributed loading"] + F --> G["INCREM
Increments the applied load according to specified load factors"] + G --> H["ALGOR
Sets indicator to identify the type of solution algorithm, i.e., initial or tangential stiffness etc."] + H --> I["A"] +``` +
+ + + +![](images/page-343_5bd99d9564044dfd1a2729d74574247f41e35c758c3e6f473fcb9e2cfe1f9a70.jpg) + +
+flowchart + +```mermaid +graph TD + A["A"] --> B{Is it necessary to recalculate stiffness matrix with present algorithm?} + B -->|Yes| C["STIFMP
Calculate element stiffness matrices for nonlayered elastoplastic Mindlin plate"] + B -->|No| D["END"] + C --> E["FRONT
Solve the simultaneous equation system by the frontal method"] + E --> F["RESMP
Evaluate the residual force vector for the nonlayered elastoplastic Mindlin plate"] + F --> G["CONVMP
Check whether solution has converged using a residual force or displacement norm"] + G --> H["OUTMP
Prints out the displacements, reactions and stress resultants for the current load increment"] + H --> I["END"] + J["LOAD INCREMENT LOOP"] --> C + K["LOAD ITERATION LOOP"] --> C +``` +
+ +Fig. 9.2 Overall structure of program MINDLIN. + + + +# 9.5.2 Subroutine FEMP + +This routine controls the calling sequence of all of the other main routines as indicated in Fig. 9.2. + +```csv +PROGRAM FEMP(INPUT, OUTPUT, TAPE5=INPUT, TAPE6=OUTPUT, FEMP 1 +.TAPE1, TAPE2, TAPE3, TAPE4, TAPE8, TAPE9) FEMP 2 +C +C*** ELASTO-PLASTIC ANALYSIS OF NON-LAYERED MINDLIN PLATES USING FEMP 3 +C*** 4-, 8-, 9-NODED OR HETEROSIS ISOPARAMETRIC QUADRILATERALS FEMP 4 +C +C +DIMENSION ASDIS(240), COORD(80, 2), EFFST(225), ELOAD(25, 27), FEMP 9 +EPSTN(225), ESTIF(27, 27), FEMP 10 +EQRHS(10), EQUAT(40, 10), FIXED(240), FEMP 11 +IFFIX(240), GLOAD(40), GSTIF(860), LNODS(25, 9), LOCEL(27), FEMP 12 +MATNO(25), NACVA(40), NAMEV(10), NCDIS(4), NCRES(4), FEMP 13 +NDEST(27), NDFRO(25), NOFIX(40), NOUTP(2), NPIVO(10), FEMP 14 +POSGP(4), PRESC(40, 3), PROPS(10, 8), REFOR(240), FEMP 15 +RLOAD(25, 27), STRSG(5, 225), TOFOR(240), FEMP 16 +TDISP(240), TLOAD(25, 27), TREAC(40, 3), VECRV(40), FEMP 17 +WEIGP(4) FEMP 18 +C +PRESET VARIABLES ASSOCIATED WITH DYNAMIC DIMENSIONS FEMP 19 +C +CALL DIMMP (MBUFA, MELEM, MEVAB, MFRON, MMATS, MPOIN, FEMP 20 +MSTIF, MTOTG, MTOTV, MVFIX, NDIME, NDOFN, FEMP 21 +NPROP, NSTRE) FEMP 21 +C +CALL THE SUBROUTINE WHICH READS MOST OF THE PROBLEM DATA FEMP 22 +C +CALL INPUT (COORD, IFFIX, LNODS, MATNO, MELEM, MEVAB, FEMP 23 +MFRON, MMATS, MPOIN, MTOTV, MVFIX, NALGO, FEMP 24 +NCRIT, NDFRO, NDIME, NDOFN, NELEM, NEVAB, FEMP 25 +NGAUS, NLAPS, NINCS, NMATS, NNODE, NOFIX, FEMP 26 +NPOIN, NPROP, NSTRE, NSTR1, NSWIT, NTOTG, FEMP 27 +NTOTV, NTYPE, NVFIX, POSGP, PRESC, PROPS, FEMP 28 +WEIGP) FEMP 28 +C +INITIALIZE ARRAYS TO ZERO FEMP 29 +C +CALL ZEROMP (EFFST, ELOAD, EPSTN, MELEM, MEVAB, MTOTG, FEMP 30 +MTOTV, MVFIX, NDOFN, NELEM, NEVAB, NGAUS, FEMP 31 +NTOTG, NTOTV, NVFIX, STRSG, TDISP, TFACT, FEMP 32 +TLOAD, TREAC) FEMP 33 +C +CALL MINDPB (IFDIS, IFFIX, IFRES, LNODS, MELEM, MTOTV, FEMP 34 +NCDIS, NCRES, NELEM, NTYPE) FEMP 35 +C +COMPUTE LOAD AFTER READING RELEVANT EXTRA DATA FEMP 36 +C +CALL LOADPB (COORD, LNODS, MATNO, MELEM, MMATS, MPOIN, FEMP 37 +NELEM, NEVAB, NGAUS, NNODE, NPOIN, PROPS, FEMP 38 +RLOAD) FEMP 39 +C +LOOP OVER EACH INCREMENT FEMP 40 +C +``` + + + +```asm +DO 70 IINCS=1,NINCS +FEMP 58 +C +FEMP 59 +C*** READ DATA FOR CURRENT INCREMENT +FEMP 60 +C +FEMP 61 +CALL INCREM (ELOAD,FIXED,IINCS,MELEM,MEVAB,MITER, MTOTV,MVFIX,NDOFN,NELEM,NEVAB,NOUTP, NOFIX,NTOTV,NVFIX,PRES,RCLOAD,TFACT, TLOAD,TOLER) +FEMP 62 +FEMP 63 +FEMP 64 +FEMP 65 +C +FEMP 66 +C*** LOOP OVER EACH ITERATION +FEMP 67 +C +FEMP 68 +DO 90 IITER=1,MITER +FEMP 69 +C +FEMP 70 +C*** CALL ROUTINE WHICH SELECTS SOLUTION ALGORITHM VARIABLE KRESL +FEMP 71 +C +FEMP 72 +CALL ALGOR (FIXED,IINCS,IITER,KRESL,MTOTV,NALGO, NTOTV) +FEMP 73 +FEMP 74 +C +FEMP 75 +C*** CHECK WHETHER A NEW EVALUATION OF THE STIFFNESS MATRICES IS NEEDED +FEMP 76 +C +FEMP 77 +IF(KRESL.EQ.1) +FEMP 78 +.CALL STIFMP (COORD,EPSTN,IINCS,LNODS,MATNO,MELEM, MEVAB,MMATS,MPOIN,MTOTG,NCRIT,NELEM, NEVAB,NGAUS,NNODE,PROPS,STRSG) +FEMP 79 +FEMP 80 +FEMP 81 +C +FEMP 82 +C*** SOLVE EQUATIONS +FEMP 83 +C +FEMP 84 +CALL FRONT (ASDIS,ELOAD,EQRHS,EQUAT,ESTIF,FIXED, IFFIX,IINCS,IITER,GLOAD,GSTIF,KRESL, LNODS,LOCEL,MBUFA,MELEM,MEVAB,MFRON, MSTIF,MTOTV,MVFIX,NACVA,NAMEV,NDEST, NDOFN,NELEM,NEVAB,NNODE,NOFIX,NPIVO, NPOIN,NTOTV,TDISP,TLOAD,TREAC,VECRV) +FEMP 85 +FEMP 86 +FEMP 87 +FEMP 88 +FEMP 89 +FEMP 90 +C +FEMP 91 +C*** CALCULATE RESIDUAL FORCES +FEMP 92 +C +FEMP 93 +CALL RESMP (ASDIS,COORD,EFFST,ELOAD,EPSTN,LNODS, MATNO,MELEM,MMATS,MPOIN,MTOTG,MTOTV, NCRIT,NELEM,NEVAB,NGAUS,NNODE,PROPS, STRSG) +FEMP 94 +FEMP 95 +FEMP 96 +FEMP 97 +C +FEMP 98 +C*** CHECK FOR CONVERGENCE +FEMP 99 +C +FEMP 100 +CALL CONVMP (ASDIS,ELOAD,IITER,IFDIS,IFRES,LNODS, MELEM,MEVAB,MTOTV,NCHEK,NCDIS,NCRES, NDOFN,NELEM,NEVAB,NNODE,NPOIN,NTOTV, REFOR,TOFOR,TDISP,TLOAD,TOLER) +FEMP 101 +FEMP 102 +FEMP 103 +FEMP 104 +C +FEMP 105 +C*** OUTPUT RESULTS IF REQUIRED +FEMP 106 +C +FEMP 107 +C +FEMP 108 +IF(IITER.EQ.1.AND.NOUTP(1).GT.0) +FEMP 109 +.CALL OUTMP (EPSTN,IITER,MTOTG,MTOTV,MVFIX,NELEM, NGAUS,NOFIX,NUOTP,NPOIN,NVFIX,STRSG, TDISP,TREAC) +FEMP 110 +FEMP 111 +FEMP 112 +C +FEMP 113 +C*** IF SOLUTION HAS CONVERGED STOP ITERATING AND OUTPUT RESULTS +FEMP 114 +C +FEMP 115 +IF(NCHEK.EQ.0) GO TO 100 +FEMP 116 +90 CONTINUE +FEMP 117 +C +FEMP 118 +C*** +FEMP 119 +C +FEMP 120 +IF(NALGO.EQ.2) GO TO 100 +FEMP 121 +``` + + + +
STOPFEMP 122
100 CALLOUTMP(EPSTN,IITER,MTOTG,MTOTV,MVFIX,NELEM,NGAUS,NOFIX,NOUTP,NPOIN,NVFIX,STRSG,TDISP,TREAC)FEMP 123
.FEMP 124
.FEMP 125
70 CONTINUEFEMP 126
20 CONTINUEFEMP 127
10 CONTINUEFEMP 128
STOPFEMP 129
ENDFEMP 130
+ +# 9.5.3 Subroutine CONVMP + +This routine establishes whether a solution has converged with reference to some displacement or residual force norm. + +
SUBROUTINE CONVMP(ASDIS,ELOAD,IITER,IFDIS,IFRES,LNODS,CONV1
.MELEM,MEVAB,MTOTV,NCHEK,NCDIS,NCRES,CONV2
.NDOFN,NELEM,NEVAB,NNODE,NPOIN,NTOTV,CONV3
.REFOR,TOFOR,TDISP,TLOAD,TOLER)CONV4
C**********CONV5
CCONV6
C***ESTABLISHES WHETHER A SOLUTION HAS CONVERGED WITHCONV7
C***REFERENCE TO SOME DISPLACEMENT OR RESIDUAL FORCE NORMCONV8
CCONV9
C**********CONV10
DIMENSION ADIDF(3),ASDIS(MTOTV),ELOAD(MELEM,MEVAB),LNODS(MELEM,9),CONV11
.NCDIS(4),NCRES(4),REFDF(3),REFOR(MTOTV),TDIDF(3),CONV12
.TDISP(MTOTV),TLOAD(MELEM,MEVAB),TOFDF(3),TOFOR(MTOTV)CONV13
WRITE(6,606) IITERCONV14
606FORMAT(///,' IN CONVER',10X,'ITERATION NUMBER',I3,/)CONV15
C***COMPUTE ELEMENT RESIDUAL FORCESCONV16
DO 10 IELEM=1,NELEMCONV17
DO 10 IEVAB=1,NEVABCONV18
10ELOAD(IELEM,IEVAB)=TLOAD(IELEM,IEVAB)-ELOAD(IELEM,IEVAB)CONV19
C***SET CONVERGENCE CODE TO ZEROCONV20
NCHEK=0CONV21
C***DISPLACEMENT CONVERGENCE CHECKCONV22
IF(IFDIS.EQ.0) GOTO 1000CONV23
C***COMPUTE TOTAL AND DIRECTIONAL NORMS OF DISPLACEMENTSCONV24
ADITO=0.0CONV25
TDITO=0.0CONV26
CALL VZERO (NDOFN,ADIDF)CONV27
CALL VZERO (NDOFN,TDIDF)CONV28
NPOSI=0CONV29
DO:20 IPOIN=1,NPOINCONV30
DO^20 IDOFN=1,NDOFNCONV31
NPOSI=NPOSI+1CONV32
ADIDF(IDOFN)=ADIDF(IDOFN)+ASDIS(NPOSI)*ASDIS(NPOSI)CONV33
20TDIDF(IDOFN)=TDIDF(IDOFN)+TDISP(NPOSI)*TDISP(NPOSI)CONV34
DO 30 IDOFN=1,NDOFNCONV35
ADITO=ADITO+ADIDF(IDOFN)CONV36
TDITO=TDITO+TDIDF(IDOFN)CONV37
ADIDF(IDOFN)=SQRT(ADIDF(IDOFN))CONV38
30TDIDF(IDOFN)=SQRT(TDIDF(IDOFN))CONV39
ADITO=SQRT(ADITO)CONV40
TDITO=SQRT(TDITO)CONV41
C***CHECK FOR CONVERGENCE AND PRINT ERRORS PER CENTCONV42
DO 40 IDOFN=1,NDOFNCONV43
IF(TDIDF(IDOFN).EQ.0.0) GOTO 40CONV44
+ + + +```csv +TDIDF(IDOFN)=100.*ADIDF(IDOFN)/TDIDF(IDOFN) CONV 45 +IF(NCDIS(IDOFN).NE.0.AND.TDIDF(IDOFN).GT.TOLER) NCHEK=1 CONV 46 +IF(NCDIS(IDOFN).EQ.0) TDIDF(IDOFN)=-TDIDF(IDOFN) CONV 47 +40 CONTINUE CONV 48 +IF(TDITO.EQ.0.0) GOTO 50 CONV 49 +TDITO=100.*ADITO/TDITO CONV 50 +IF(NCDIS(4).NE.0.AND.TDITO.GT.TOLER) NCHEK=1 CONV 51 +IF(NCDIS(4).EQ.0) TDITO=-TDITO CONV 52 +50 CONTINUE CONV 53 +WRITE(6,600) CONV 54 +WRITE(6,601) (TDIDF(IDOFN), IDOFN=1, NDOFN) CONV 55 +600 FORMAT(1X,'DISPLACEMENT CHANGE NORM',//) CONV 56 +601 FORMAT(1X,5(E10.3,5X)) CONV 57 +WRITE(6,602) CONV 58 +602 FORMAT(5X,'TOTAL') CONV 59 +WRITE(6,603) TDITO CONV 60 +603 FORMAT(3X,E10.3) CONV 61 +C*** RESIDUAL CONVERGENCE CHECK CONV 62 +1000 IF(IFRES.EQ.0) GOTO 2000 CONV 63 +C*** ASSEMBLE TOTAL AND RESIDUAL FORCE VECTORS CONV 64 +DO 1 ITOTV=1,NTOTV CONV 65 +REFOR(ITOTV)=0.0 CONV 66 +1 TOFOR(ITOTV)=0.0 CONV 67 +DO 60 IELEM=1,NELEM CONV 68 +KEVAB=0 CONV 69 +DO 60 INODE=1,NNODE CONV 70 +LOCNO=IABS(LNODS(IELEM,INODE)) CONV 71 +DO 60 IDOFN=1,NDOFN CONV 72 +KEVAB=KEVAB+1 CONV 73 +NPOSI=(LOCNO-1)*NDOFN+IDOFN CONV 74 +TOFOR(NPOSI)=TOFOR(NPOSI)+TLOAD(IELEM,KEVAB) CONV 75 +60 REFOR(NPOSI)=REFOR(NPOSI)+ELOAD(IELEM,KEVAB) CONV 76 +C*** COMPUTE TOTAL AND DIRECTIONAL NORMS OF RESIDUAL AND TOTAL FORCE CONV 77 +REFTO=0.0 CONV 78 +TOFTO=0.0 CONV 79 +CALL VZERO (NDOFN,REFDF) CONV 80 +CALL VZERO (NDOFN,TOFDF) CONV 81 +NPOSI=0 CONV 82 +DO 70 IPOIN=1,NPOIN CONV 83 +DO 70 IDOFN=1,NDOFN CONV 84 +NPOSI=NPOSI+1 CONV 85 +REFDF(IDOFN)=REFDF(IDOFN)+REFOR(NPOSI)*REFOR(NPOSI) CONV 86 +70 TOFDF(IDOFN)=TOFDF(IDOFN)+TOFOR(NPOSI)*TOFOR(NPOSI) CONV 87 +DO 80 IDOFN=1,NDOFN CONV 88 +REFTO=REFTO+REFDF(IDOFN) CONV 89 +TOFTO=TOFTO+TOFDF(IDOFN) CONV 90 +REFDF(IDOFN)=SQRT(REFDF(IDOFN)) CONV 91 +80 TOFDF(IDOFN)=SQRT(TOFDF(IDOFN)) CONV 92 +REFTO=SQRT(REFTO) CONV 93 +TOFTO=SQRT(TOFTO) CONV 94 +C*** CHECK FOR CONVERGENCE AND PRINT ERRORS PER CENT CONV 95 +DO 90 IDOFN=1,NDOFN CONV 96 +IF(TOFDF(IDOFN).EQ.0.0) GOTO 90 CONV 97 +TOFDF(IDOFN)=100.*REFDF(IDOFN)/TOFDF(IDOFN) CONV 98 +IF(NCRES(IDOFN).NE.0.AND.TOFDF(IDOFN).GT.TOLER) NCHEK=1 CONV 99 +IF(NCRES(IDOFN).EQ.0) TOFDF(IDOFN)=-TOFDF(IDOFN) CONV 100 +90 CONTINUE CONV 101 +IF(TOFTO.EQ.0.0) GOTO 100 CONV 102 +TOFTO=100.*REFTO/TOFTO CONV 103 +IF(NCRES(4).NE.0.AND.TOFTO.GT.TOLER) NCHEK=1 CONV 104 +IF(NCRES(4).EQ.0) TOFTO=-TOFTO CONV 105 +100 CONTINUE CONV 106 +WRITE(6,604) CONV 107 +WRITE(6,601) (TOFDF(IDOFN), IDOFN=1, NDOFN) CONV 108 +``` + + + +```txt +WRITE(6,602) CONV 109 +WRITE(6,603) TOFTO CONV 110 +604 FORMAT(1X,'RESIDUAL NORM',//) CONV 111 +C*** PRINT CONVERGENCE CODE CONV 112 +2000 WRITE(6,605) NCHEK CONV 113 +605 FORMAT(1X,'CONVERGENCE CODE',I4,//) CONV 114 +RETURN CONV 115 +END CONV 116 +``` + +# 9.5.4 Subroutine DIMMP + +This subroutine sets up the dimensions which must agree with the size of the arrays in subroutine FEMP. + +```asm +SUBROUTINE DIMMP (MBUFA,MELEM,MEVAB,MFRON,MMATS,MPOIN, DIMP 1 +. MSTIF,MTOTG,MTOTV,MVFIX,NDIME,NDOFN, DIMP 2 +. NPROP,NSTRE) DIMP 3 +C**********DIMP 4 +C DIMP 5 +C*** SETS UP DYNAMIC DIMENSIONS - MUST AGREE WITH DIMENSIONS DIMP 6 +C*** IN FEMP DIMP 7 +C DIMP 8 +C**********DIMP 9 +MBUFA = 10 DIMP 10 +MELEM = 25 DIMP 11 +MFRON = 40 DIMP 12 +MMATS = 10 DIMP 13 +MPOIN = 80 DIMP 14 +MSTIF=(MFRON*MFRON-MFRON)/2.0+MFRON DIMP 15 +MTOTG = MELEM*9 DIMP 16 +NDOFN = 3 DIMP 17 +MTOTV = MPOIN*NDOFN DIMP 18 +MVFIX = 40 DIMP 19 +NDIME=2 DIMP 20 +NPROP = 8 DIMP 21 +NSTRE = 5 DIMP 22 +MEVAB = NDOFN*9 DIMP 23 +RETURN DIMP 24 +END DIMP 25 +``` + +# 9.5.5 Subroutine FLOWMP + +This subroutine determines the yield function derivatives $[\partial F/\partial M_{x}, \partial F/\partial M_{y}, \partial F/\partial M_{xy}]^{T}$ for nonlayered Mindlin plates of Von Mises or Tresca material. This routine is almost identical to the corresponding one given in Chapter 7 for plane stress, plane strain and axisymmetric problems. + +```c +SUBROUTINE FLOWMP (ABETA, AVECT, DEVIA, DMATX, DVECT, HARDS, NCRIT, SINT3, STEFF, THETA, VARJ2) FLOW 1 +C**********FLOW 3 +C FLOW 4 +C*** DETERMINES YIELD FUNCTION DERIVATIVES FOR MINDLIN PLATES FLOW 5 +C*** 1 VON MISES FLOW 6 +C*** 2 TRESCA FLOW 7 +C FLOW 8 +C**********FLOW 9 +``` + + + +```csv +C +DIMENSION AVECT(5),DEVIA(4),DMATX(3,3),DVECT(5), +VECA1(3),VECA2(3),VECA3(3) +C +C*** DETERMINE THE VECTOR DERIVATIVE OF F FOR VON-MISES +SINTH=SIN(THETA) +COSTH=COS(THETA) +ROOT3=1.73205080757 +C +C*** CALCULATE VECTOR A1 +VECA1(1)=0.333333333333 +VECA1(2)=0.333333333333 +VECA1(3)=0.0 +C +C*** CALCULATE VECTOR A2 +DO 10 ISTRE=1,3 +10 VECA2(ISTRE)=DEVIA(ISTRE)/(2.0*STEFF) +VECA2(3)=DEVIA(3)/STEFF +C +C*** CALCULATE VECTOR A3 +VECA3(1)=DEVIA(2)*DEVIA(4)+VARJ2/3.0 +VECA3(2)=DEVIA(1)*DEVIA(4)+VARJ2/3.0 +VECA3(3)=-2.0*DEVIA(3)*DEVIA(4) +GO TO (1,2) NCRIT +C +C*** VON MISES +1 CONS1=0.0 +CONS2=ROOT3 +CONS3=0.0 +GO TO 40 +C +C*** TRESCA +2 CONS1=0.0 +ABTHE=ABS(THETA*57.29577951308) +IF(ABTHE.LT.29.0) GO TO 20 +CONS2=ROOT3 +CONS3=0.0 +GO TO 40 +20 CONS2=2.0*(COSTH+SINTH*SINT3/SQRT(1.0-SINT3*SINT3)) +CONS3=ROOT3*SINTH/(VARJ2*SQRT(1.0-SINT3*SINT3)) +40 CONTINUE +DO 50 ISTRE=1,3 +50 AVECT(ISTRE)=CONS1*VECA1(ISTRE)+CONS2*VECA2(ISTRE)+CONS3* +.VECA3(ISTRE) +C +C*** DETERMINE THE VECTOR D +DENOM=HARDS +DO 120 ISTRE=1,3 +DVECT(ISTRE)=0.0 +DO 110 JSTRE=1,3 +110 DVECT(ISTRE)=DVECT(ISTRE)+DMATX(ISTRE,JSTRE)*AVECT(JSTRE) +120 DENOM=DENOM+AVECT(ISTRE)*DVECT(ISTRE) +ABETA=1.0 DENOM +RETURN +END +FLOW 10 +FLOW 11 +FLOW 12 +FLOW 13 +FLOW 14 +FLOW 15 +FLOW 16 +FLOW 17 +FLOW 18 +FLOW 19 +FLOW 20 +FLOW 21 +FLOW 22 +FLOW 23 +FLOW 24 +FLOW 25 +FLOW 26 +FLOW 27 +FLOW 28 +FLOW 29 +FLOW 30 +FLOW 31 +FLOW 32 +FLOW 33 +FLOW 34 +FLOW 35 +FLOW 36 +FLOW 37 +FLOW 38 +FLOW 39 +FLOW 40 +FLOW 41 +FLOW 42 +FLOW 43 +FLOW 44 +FLOW 45 +FLOW 46 +FLOW 47 +FLOW 48 +FLOW 49 +FLOW 50 +FLOW 51 +FLOW 52 +FLOW 53 +FLOW 54 +FLOW 55 +FLOW 56 +FLOW 57 +FLOW 58 +FLOW 59 +FLOW 60 +FLOW 61 +FLOW 62 +FLOW 63 +FLOW 64 +FLOW 65 +FLOW 66 +FLOW 67 +FLOW 68 +FLOW 69 +FLOW 70 +``` + + + +# 9.5.6 Subroutine GRADMP + +This subroutine evaluates displacement gradients $\partial w/\partial x$ , $\partial w/\partial y$ , $\partial \theta_{x}/\partial x$ , $\partial \theta_{x}/\partial y$ , $\partial \theta_{y}/\partial x$ and $\partial \theta_{y}/\partial y$ . + +```fortran +SUBROUTINE GRADMP (CARTD,DGRAD,ELDIS,NDOFN,NNODE) GRAD 1 +C**************************GRAD 2 +C GRAD 3 +C*** FORM TOTAL DISPLACEMENTS GRADIENTS GRAD 4 +C GRAD 5 +C**************************GRAD 6 +DIMENSION CARTD(2,9),DGRAD(6),ELDIS(3,9) GRAD 7 +C*** ZERO DGRAD GRAD 8 +CALL VZERO(6,DGRAD) GRAD 9 +C*** FORM TOTAL DISPLACEMENTS GRADIENTS GRAD 10 +DO 10 INODE=1,NNODE GRAD 11 +DNIDX=CARTD(1,INODE) GRAD 12 +DNIDY=CARTD(2,INODE) GRAD 13 +DO 10 IDOFN=1,NDOFN GRAD 14 +IPOSN=NDOFN+IDOFN GRAD 15 +CONST=ELDIS(IDOFN,INODE) GRAD 16 +DGRAD(IDOFN)=DGRAD(IDOFN)+DNIDX*CONST GRAD 17 +10 DGRAD(IPOSN)=DGRAD(IPOSN)+DNIDY*CONST GRAD 18 +RETURN GRAD 19 +END GRAD 20 +``` + +# 9.5.7 Subroutine INVMP + +This subroutine evaluates the Mindlin plate bending moment invariants. It also evaluates the effective moment for the Tresca and Von Mises materials. + +```txt +SUBROUTINE INVMP (DEVIA, NCRIT, SINT3, STEFF, STEMP, THETA, INVR 1 +. VARJ2, YIELD) INVR 2 +C************************** INVR 3 +C INVR 4 +C*** CALCULATE MINDLIN PLATE STRESS RESULTANT INVARIANTS INVR 5 +C INVR 6 +C************************** INVR 7 +DIMENSION STEMP(5), DEVIA(4) INVR 8 +SMEAN=(STEMP(1)+STEMP(2))/3.0 INVR 9 +DEVIA(1)=STEMP(1)-SMEAN INVR 10 +DEVIA(2)=STEMP(2)-SMEAN INVR 11 +DEVIA(3)=STEMP(3) INVR 12 +DEVIA(4)=-SMEAN INVR 13 +VARJ2=DEVIA(3)*DEVIA(3)+0.5*(DEVIA(1)*DEVIA(1)+DEVIA(2)*DEVIA(2) INVR 14 +.+DEVIA(4)*DEVIA(4)) INVR 15 +VARJ3=DEVIA(4)*(DEVIA(4)*DEVIA(4)-VARJ2) INVR 16 +STEFF=SQRT(VARJ2) INVR 17 +SINT3=-2.5980762113*VARJ3/(VARJ2*STEFF) INVR 18 +THETA=ASIN(SINT3)/3.0 INVR 19 +GO TO (1,2) NCRIT INVR 20 +C*** VON MISES INVR 21 +1 YIELD=1.73205080757*STEFF INVR 22 +RETURN INVR 23 +C*** TRESCA INVR 24 +2 YIELD=2.0*COS(THETA)*STEFF INVR 25 +RETURN INVR 26 +END INVR 27 +``` diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_036.md b/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_036.md new file mode 100644 index 00000000..88d7015c --- /dev/null +++ b/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_036.md @@ -0,0 +1,586 @@ + + +# 9.5.8 Subroutine MINDPB + +This subroutine simply reads some additional information required for controlling the convergence check and inserting additional constraints for the Heterosis element. + +```asm +SUBROUTINE MINDPB (IFDIS, IFFIX, IFRES, LNODS, MELEM, MTOTV, MIND 1 +. NCDIS, NCRES, NELEM, NTYPE) MIND 2 +C*****************************************************************************************MIND 3 +C MIND 4 +C*** READS ADDITIONAL DATA FOR MINDLIN PLATE ANALYSIS MIND 5 +C MIND 6 +C*****************************************************************************************MIND 7 +DIMENSION DERIV(2,9), IFFIX(MTOTV), MIND 8 +. LNODS(MELEM,9), NCDIS(4), NCRES(4), SHAPE(9) MIND 9 +C MIND 10 +C*** READ DATA CONTROLLING CONVERGENCE CHECK MIND 11 +C MIND 12 +10 READ(5,900) IFDIS, (NCDIS(I), I=1,4) MIND 13 +. ,IFRES, (NCRES(I), I=1,4) MIND 14 +900 FORMAT(5I1) MIND 15 +WRITE(6,901) IFDIS, (NCDIS(I), I=1,4) MIND 16 +. ,IFRES, (NCRES(I), I=1,4) MIND 17 +901 FORMAT(/,23H CONVERGENCE PARAMETERS,/, MIND 18 +. 8H IFDIS =, I2,5X, 8H NCDIS =, 4I1,/, MIND 19 +. 8H IFRES =, I2,5X, 8H NCRES =, 4I1, //) MIND 20 +C*** INSERT ADDITIONAL CONSTRAINT FOR HETEROSIS ELEMENT MIND 21 +IF(NTYPE.NE.5) GO TO 30 MIND 22 +DO 20 IELEM=1, NELEM MIND 23 +LNODE=LNODS(IELEM,9) MIND 24 +NLOCA=LNODE*3-2 MIND 25 +20 IFFIX(NLOCA)=-1 MIND 26 +30 CONTINUE MIND 27 +RETURN MIND 28 +END MIND 29 +``` + +# 9.5.9 Subroutine NODEXY + +This subroutine evaluates midside nodes for straight sided 8 and 9-node quadrilateral elements. In the original subroutine described in Section 6.4.1 this routine also evaluated the coordinates of the central node. Here, as we are choosing a hierarchical formulation, the values at the central node and the departures from the interpolated Serendipity values are always taken as zero. + +Thus the revised subroutine NODEXY is almost identical to its namesake given earlier in Section 6.4.1 and is listed below. + +```c +SUBROUTINE NODEXY (COORD, LNODS, MELEM, MPOIN, NDIME, NELEM, NNODE) NODE 1 +C***** NODE 2 +C***** NODE 3 +C***** NODE 4 +C*** INTERPOLATES MIDSIDE NODE COORDINATES FOR 8-NODED ELEMENTS NODE 5 +C*** INTERPOLATES CENTRAL AND MIDSIDE NODE COORDINATES FOR NODE 6 +C*** 9-NODE ELEMENTS PROVIDED THAT THE SIDES ARE STRAIGHT NODE 7 +C***** NODE 8 +C***** NODE 9 +``` + + + +```fortran +DIMENSION COORD(MPOIN,2),LNODS(MELEM,9) NODE 10 +IF(NNODE.EQ.4) GO TO 60 NODE 11 +C NODE 12 +C*** LOOP OVER EACH ELEMENT NODE 13 +C NODE 14 + DO 30 IELEM=1,NELEM NODE 15 +C NODE 16 +C*** LOOP OVER EACH ELEMENT EDGE NODE 17 +C NODE 18 + NNOD1=NNODE NODE 19 + IF(NNODE.EQ.8).NNOD1=9 NODE 20 + DO 20 INODE=1,NNOD1,2 NODE 21 + IF(INODE.EQ.9.AND.NNODE.EQ.8) GO TO 30 NODE 22 + IF(INODE.EQ.9) GO TO 50 NODE 23 +C NODE 24 +C*** COMPUTE THE NODE NUMBER OF THE FIRST NODE NODE 25 +C NODE 26 + NODST=LNODS(IELEM,INODE) NODE 27 + IGASH=INODE+2 NODE 28 + IF(IGASH.GE.NNODE) IGASH=1 NODE 29 +C NODE 30 +C*** COMPUTE THE NODE NUMBER OF THE LAST NODE NODE 31 +C NODE 32 + NODFN=LNODS(IELEM,IGASH) NODE 33 + MIDPT=INODE+1 NODE 34 +C NODE 35 +C*** COMPUTE THE NODE NUMBER OF THE INTERMEDIATE NODE NODE 36 +C NODE 37 + NODMD=LNODS(IELEM,MIDPT) NODE 38 + TOTAL=ABS(COORD(NODMD,1))+ABS(COORD(NODMD,2)) NODE 39 +C NODE 40 +C*** IF THE COORDINATES OF THE INTERMEDIATE NODE ARE BOTH ZERO NODE 41 +C INTERPOLATE BY A STRAIGHT LINE NODE 42 +C NODE 43 + IF(TOTAL.GT.0.0) GO TO 20 NODE 44 + KOUNT=1 NODE 45 + 10 COORD(NODMD,KOUNT)=(COORD(NODST,KOUNT)+COORD(NODFN,KOUNT))/2.0 NODE 46 + KOUNT=KOUNT+1 NODE 47 + IF(KOUNT.EQ.2) GO TO 10 NODE 48 + 20 CONTINUE NODE 49 + 50 LNODE=LNODS(IELEM,INODE) NODE 50 + 30 CONTINUE NODE 51 + 60 CONTINUE NODE 52 + RETURN NODE 53 + END NODE 54 +``` + +# 9.5.10 Subroutine OUTMP + +This subroutine outputs nodal displacements and reactions and also the Gauss point stress resultants. + +```csv +SUBROUTINE OUTMP (EPSTN,IITER,MTOTG,MTOTV,MVFIX,NELEM, OUTP 1 +. NGAUS,NOFIX,NOUTP,NPOIN,NVFIX,STRSG, OUTP 2 +. TDISP,TREAC) OUTP 3 +C**********OUTP 4 +C OUTP 5 +C*** OUTPUT DISPLACEMENTS,REACTIONS AND GAUSS POINT STRESS OUTP 6 +C*** RESULTANTS FOR EP MINDLIN PLATE ANALYSIS OUTP 7 +C OUTP 8 +C**********OUTP 9 +``` + + + +```csv +DIMENSION EPSTN(MTOTG), GPCOD(2,9), NOFIX(MVFIX), NOUTP(2), OUTP 10 +. STRSG(5, MTOTG), TDISP(MTOTV), TREAC(MVFIX, 3) OUTP 11 +KOUTP=NOUTP(1) OUTP 12 +IF(IITER.GT.1) KOUTP=NOUTP(2) OUTP 13 +C +C*** OUTPUT DISPLACEMENTS OUTP 15 +C +IF(KOUTP.LT.1) GO TO 10 OUTP 17 +WRITE(6,900) OUTP 18 +900 FORMAT(1H0,5X,13HDISPLACEMENTS) OUTP 19 +WRITE(6,950) OUTP 20 +950 FORMAT(1H0,6X,4HNODE,6X,5HDISP.,8X,7HXZ-ROT.,7X,7HYZ-ROT.) OUTP 21 +DO 20 IPOIN=1,NPOIN OUTP 22 +NGASH=IPOIN*3 OUTP 23 +NGISH=NGASH-3+1 OUTP 24 +20 WRITE(6,910) IPOIN,(TDISP(IGASH),IGASH=NGISH,NGASH) OUTP 25 +910 FORMAT(I10,3E14.6) OUTP 26 +10 CONTINUE OUTP 27 +C +C*** OUTPUT REACTIONS OUTP 28 +C +IF(KOUTP.LT.2) GO TO 30 OUTP 30 +WRITE(6,920) OUTP 31 +920 FORMAT(1H0,5X,9HREACTIONS) OUTP 32 +WRITE(6,960) OUTP 33 +960 FORMAT(1H0,6X,4HNODE,6X,5HFORCE,3X,9HXZ-MOMENT,5X,9HYZ-MOMENT) OUTP 34 +DO 40 IVFIX=1,NVFIX OUTP 35 +40 WRITE(6,910) NOFIX(IVFIX),(TREAC(IVFIX,IDOFN),IDOFN=1,3) OUTP 36 +30 CONTINUE OUTP 37 +C +C*** OUTPUT STRESSES OUTP 38 +C +IF(KOUTP.LT.3) GO TO 50 OUTP 39 +REWIND 3 OUTP 40 +WRITE(6,970) OUTP 41 +970 FORMAT(1H0,5X,8HSTRESSES) OUTP 42 +WRITE(6,980) OUTP 43 +980 FORMAT(1H0,4HG.P.,2X,8HX-COORD.,2X,8HY-COORD.,3X,8HX-MOMENT,4X,OUTP 44 +.8HY-MOMENT,3X,9HXY-MOMENT,3X,OUTP 45 +.13HEFF.PL.STRAIN) OUTP 46 +KGAUS=0 OUTP 47 +DO 60 IELEM=1,NELEM OUTP 48 +READ(3)GPCOD OUTP 49 +KELGS=0 OUTP 50 +WRITE(6,930)IELEM OUTP 51 +930 FORMAT(1H0,5X,13HELEMENT NO. =,I5) OUTP 52 +DO 60 IGAUS=1,NGAUS OUTP 53 +DO 60 JGAUS=1,NGAUS OUTP 54 +KGAUS=KGAUS+1 OUTP 55 +KELGS=KELGS+1 OUTP 56 +WRITE(6,940)KELGS,(GPCOD(IDIME,KELGS),IDIME=1,2), OUTP 57 +.(STRSG(ISTRE,KGAUS),ISTRE=1,3),EPSTN(KGAUS) OUTP 58 +940 FORMAT(I5,2F10.4,6E12.5) OUTP 59 +60 CONTINUE OUTP 60 +50 CONTINUE OUTP 61 +RETURN OUTP 62 +END OUTP 63 +OUTP 64 +OUTP 65 +OUTP 66 +``` + + + +# 9.5.11 Subroutine RESMP + +This subroutine evaluates the residual nodal forces. The structure of this routine is similar to that given in Chapter 7 for the other two dimensional elasto-plastic applications and it is illustrated in Fig. 9.3. + +```fortran +SUBROUTINE RESMP (ASDIS,COORD,EFFST,ELOAD,EPSTN,LNODS, RESP 1 +MATNO,MELEM,MMATS,MPOIN,MTOTG,MTOTV, RESP 2 +NCRIT,NELEM,NEVAB,NGAUS,NNODE,PROPS, RESP 3 +STRSG) RESP 4 +C**************************RESP 5 +C +C*** EVALUATES EQUIVALENT NODAL FORCES FOR THE STRESS RESULTANTS RESP 7 +C*** IN MINDLIN PLATES DURING EP ANALYSIS RESP 8 +C +C**************************RESP 9 +DIMENSION ASDIS(MTOTV),AVECT(5),CARTD(2,9), RESP 11 +COORD(MPOIN,2),DERIV(2,9),DESIG(5),DEVIA(4), RESP 12 +DVECT(5), RESP 13 +EFFST(MTOTG),ELCOD(2,9), RESP 14 +ELDIS(3,9),ELOAD(MELEM,27),EPSTN(MTOTG),GPCOD(2,9), RESP 15 +LNODS(MELEM,9),MATNO(MELEM),POSGP(4), RESP 16 +PROPS(MMATS,8),SGTOT(5),SHAPE(9),SIGMA(5), RESP 17 +STRES(5),STRSG(5,MTOTG),WEIGP(4), RESP 18 +DFLEX(3,3),DSHER(2,2),BFLEI(3,3),BSHEI(2,3), RESP 19 +DUMMY(3,3),FORCE(3),DGRAD(6) RESP 20 +NTIME=1 RESP 21 +DO 10 IELEM=1,NELEM RESP 22 +DO 10 IEVAB=1,NEVAB RESP 23 +10 ELOAD(IELEM,IEVAB)=0.0 RESP 24 +KGAUS=0 RESP 25 +LGAUS=0 RESP 26 +DO 20 IELEM=1,NELEM RESP 27 +LPROP=MATNO(IELEM) RESP 28 +C +C*** COMPUTE COORDINATE AND INCREMENTAL DISPLACEMENTS OF THE RESP 30 +ELEMENT NODAL POINTS RESP 31 +C +DO 190 INODE =1,NNODE RESP 32 +LNODE=IABS(LNODS(IELEM,INODE)) RESP 33 +NPOSN=(LNODE-1)*3 RESP 34 +DO 30 IDOFN=1,3 RESP 35 +NPOSN=NPOSN+1 RESP 36 +30 ELDIS(IDOFN,INODE)=ASDIS(NPOSN) RESP 37 +DO 180 IDIME=1,2 RESP 38 +180 ELCOD(IDIME,INODE)=COORD(LNODE,IDIME) RESP 39 +190 CONTINUE RESP 40 +KGASP=0 RESP 41 +CALL MODPB (DFLEX,DUMMY,DSHER,LPROP,MMATS,PROPS, RESP 42 +0, 1, 1) RESP 43 +CALL GAUSSQ (NGAUS,POSGP,WEIGP) RESP 44 +DO 40 IGAUS=1,NGAUS RESP 45 +DO 40 JGAUS=1,NGAUS RESP 46 +BRING=1.0 RESP 47 +KGAUS=KGAUS+1 RESP 48 +EXISP=POSGP(IGAUS) RESP 49 +ETASP=POSGP(JGAUS) RESP 50 +CALL SFR2 (DERIV,ETASP,EXISP,NNODE,SHAPE) RESP 51 +KGASP=KGASP+1 RESP 52 +CALL JACOB2 (CARTD,DERIV,DJACB,ELCOD,GPCOD,IELEM, RESP 53 +KGASP,NNODE,SHAPE) RESP 54 +C +``` + + + +```csv +DAREA=DJACB*WEIGP(IGAUS)*WEIGP(JGAUS) +CALL GRADMP (CARTD,DGRAD,ELDIS, 3,NNODE) +CALL STRMP (CARTD,DFLEX,DGRAD,DSHER,ELDIS,NNODE, SHAPE,STRES, 1, 0) +PREYS=PROPS(LPROP,6)+EPSTN(KGAUS)*PROPS(LPROP,7) +DO 150 ISTRE=1,3 +DESIG(ISTRE)=STRES(ISTRE) +150 SIGMA(ISTRE)=STRSG(ISTRE,KGAUS)+STRES(ISTRE) +CALL INVMP (DEVIA,NCRIT,SINT3,STEFF,SIGMA,THETA, VARJ2,YIELD) +ESPRE=EFFST(KGAUS)-PREYS +IF(ESPRE.GE.0.0) GO TO 50 +ESCUR=YIELD-PREYS +IF(ESCUR.LE.0.0) GO TO 60 +RFACT=ESCUR/(YIELD-EFFST(KGAUS)) +GO TO 70 +50 ESCUR=YIELD-EFFST(KGAUS) +IF(ESCUR.LE.0.0) GO TO 60 +RFACT=1.0 +70 MSTEP=ESCUR*8.0/PROPS(LPROP,6)+1.0 +ASTEP=MSTEP +REDUC=1.0-RFACT +DO 80 ISTRE=1,3 +SGTOT(ISTRE)=STRSG(ISTRE,KGAUS)+REDUC*STRES(ISTRE) +80 STRES(ISTRE)=RFACT*STRES(ISTRE)/ASTEP +DO 90 ISTEP=1,MSTEP +CALL INVMP (DEVIA,NCRIT,SINT3,STEFF,SGTOT,THETA, VARJ2,YIELD) +HARDS=PROPS(LPROP,7) +CALL FLOWMP (ABETA,AVECT,DEVIA,DFLEX,DVECT,HARDS, NCRIT,SINT3,STEFF,THETA,VARJ2) +AGASH=0.0 +DO 100 ISTRE=1,3 +100 AGASH=AGASH+AVECT(ISTRE)*STRES(ISTRE) +DLAMD=AGASH*ABETA +IF(DLAMD.LT.0.0) DLAMD=0.0 +BGASH=0.0 +DO 110 ISTRE=1,3 +BGASH=BGASH+AVECT(ISTRE)*SGTOT(ISTRE) +110 SGTOT(ISTRE)=SGTOT(ISTRE)+STRES(ISTRE)-DLAMD*DVECT(ISTRE) +90 EPSTN(KGAUS)=EPSTN(KGAUS)+DLAMD*BGASH/YIELD +DO 120 ISTRE=1,3 +120 DESIG(ISTRE)=SGTOT(ISTRE)-STRSG(ISTRE,KGAUS) +CALL INVMP (DEVIA,NCRIT,SINT3,STEFF,SGTOT,THETA, VARJ2,YIELD) +CURYS=PROPS(LPROP,6)+EPSTN(KGAUS)*PROPS(LPROP,7) +IF(YIELD.GT.CURYS) BRING=CURYS/YIELD +60 DO 130 ISTRE=1,3 +SGTOT(ISTRE)=BRING*(STRSG(ISTRE,KGAUS)+DESIG(ISTRE)) +130 STRSG(ISTRE,KGAUS)=SGTOT(ISTRE) +EFFST(KGAUS)=BRING*YIELD +*** CALCULATE THE EQUIVALENT NODAL FORCES AND ASSOCIATE WITH THE ELEMENT NODES +DO 140 INODE=1,NNODE +*** ZERO FORCE VECTOR +CALL VZERO (3,FORCE) +CALL BMATPB (BFLEI,DUMMY,BSHEI,CARTD,INODE,SHAPE, 0, 1, 0) +FORCE(2)=(BFLEI(1,2)*SGTOT(1)+BFLEI(3,2)*SGTOT(3))*DAREA ++FORCE(2) +FORCE(3)=(BFLEI(2,3)*SGTOT(2)+BFLEI(3,3)*SGTOT(3))*DAREA ++FORCE(3) +IPOSN=(INODE-1)*3+1 +RESP 56 +RESP 57 +RESP 58 +RESP 59 +RESP 60 +RESP 61 +RESP 62 +RESP 63 +RESP 64 +RESP 65 +RESP 66 +RESP 67 +RESP 68 +RESP 69 +RESP 70 +RESP 71 +RESP 72 +RESP 73 +RESP 74 +RESP 75 +RESP 76 +RESP 77 +RESP 78 +RESP 79 +RESP 80 +RESP 81 +RESP 82 +RESP 83 +RESP 84 +RESP 85 +RESP 86 +RESP 87 +RESP 88 +RESP 89 +RESP 90 +RESP 91 +RESP 92 +RESP 93 +RESP 94 +RESP 95 +RESP 96 +RESP 97 +RESP 98 +RESP 99 +RESP 100 +RESP 101 +RESP 102 +RESP 103 +RESP 104 +RESP 105 +RESP 106 +RESP 107 +RESP 108 +RESP 109 +RESP 110 +RESP 111 +RESP 112 +RESP 113 +RESP 114 +RESP 115 +RESP 116 +RESP 117 +RESP 118 +RESP 119 +``` + + + +```csv +DO 135 IDOFN=2,3 +IPOSN=IPOSN+1 +135 ELOAD(IELEM,IPOSN)=ELOAD(IELEM,IPOSN)+FORCE(IDOFN) +140 CONTINUE +40 CONTINUE +C +C*** CALCULATE FORCES ASSOCIATED WITH SHEAR DEFORMATION +C +NGAUM=NGAUS-1 +CALL GAUSSQ (NGAUM,POSGP,WEIGP) +C +C*** ENTER LOOPS FOR AREA NUMERICAL INTEGRATION +C +KGASP=0 +DO 300 IGAUS=1,NGAUM +DO 300 JGAUS=1,NGAUM +LGAUS=LGAUS+1 +EXISP=POSGP(IGAUS) +ETASP=POSGP(JGAUS) +CALL SFR2 (DERIV,ETASP,EXISP,NNODE,SHAPE) +KGASP=KGASP+1 +CALL JACOB2 (CARTD,DERIV,DJACB,ELCOD,GPCOD,IELEM, KGASP,NNODE,SHAPE) +DAREA=DJACB*WEIGP(IGAUS)*WEIGP(JGAUS) +CALL GRADMP (CARTD,DGRAD,ELDIS, 3,NNODE) +CALL STRMP (CARTD,DFLEX,DGRAD,DSHER,ELDIS,NNODE, SHAPE,STRES, 0, 1) +DO 310 ISTRE=4,5 +SGTOT(ISTRE)=STRSG(ISTRE,LGAUS)+STRES(ISTRE) +310 STRSG(ISTRE,LGAUS)=SGTOT(ISTRE) +C +C*** CALCULATE THE EQUIVALENT NODAL FORCES +C +DO 320 INODE=1,NNODE +C*** ZERO FORCE VECTOR +CALL VZERO(3,FORCE) +CALL BMATPB (BFLEI,DUMMY,BSHEI,CARTD,INODE,SHAPE, 0, 0, 1) +FORCE(1)=(BSHEI(1,1)*SGTOT(4)+BSHEI(2,1)*SGTOT(5))*DAREA ++FORCE(1) +FORCE(2)=(BSHEI(1,2)*SGTOT(4))*DAREA+FORCE(2) +FORCE(3)=(BSHEI(2,3)*SGTOT(5))*DAREA+FORCE(3) +IPOSN=(INODE-1)*3 +DO 315 IDOFN=1,3 +IPOSN=IPOSN+1 +315 ELOAD(IELEM,IPOSN)=ELOAD(IELEM,IPOSN)+FORCE(IDOFN) +320 CONTINUE +300 CONTINUE +20 CONTINUE +RETURN +END +RESP 120 +RESP 121 +RESP 122 +RESP 123 +RESP 124 +RESP 125 +RESP 126 +RESP 127 +RESP 128 +RESP 129 +RESP 130 +RESP 131 +RESP 132 +RESP 133 +RESP 134 +RESP 135 +RESP 136 +RESP 137 +RESP 138 +RESP 139 +RESP 140 +RESP 141 +RESP 142 +RESP 143 +RESP 144 +RESP 145 +RESP 146 +RESP 147 +RESP 148 +RESP 149 +RESP 150 +RESP 151 +RESP 152 +RESP 153 +RESP 154 +RESP 155 +RESP 156 +RESP 157 +RESP 158 +RESP 159 +RESP 160 +RESP 161 +RESP 162 +RESP 163 +RESP 164 +RESP 165 +RESP 166 +RESP 167 +RESP 168 +RESP 169 +RESP 170 +``` + +# 9.5.12 Subroutine SFR2 + +This subroutine evaluates the shape functions and their derivatives for 4, 8 and 9-node quadrilateral isoparametric elements. The 9-node element is treated as a hierarchical element as described in Section 9.3.2. This enables the Heterosis element to be easily incorporated. + + + +![](images/page-357_44388d2b9273b6860372b0273fb3f6333303e4d33acbec0c8c5bcda22cc55164.jpg) + +
+flowchart + +```mermaid +graph TD + A["START"] --> B["Set to zero ELOAD ( . . . )"] + B --> C["Extract local element material property set number, displacements and coordinates"] + C --> D["Call MODPB to evaluate Df, Ds"] + D --> E["Call GAUSSQ to evaluate n-point Gauss-Legendre sampling positions and weights"] + E --> F["Call SFR2, JACOB2, GRADMP and STRMP to evaluate elastic stress increment dσf"] + F --> G["Calculate the effective stress necessary for yielding to occur"] + G --> H["Calculate the total bending moments at the current Gauss points"] + H --> I["If the current bending moments are outside of the yield surface bring them back to the yield surface taking into account unloading if it has taken place"] + I --> J["A"] + K["ELEMENT LOOP"] --> L["GAUSS LOOPS"] + L --> M["Calculate the effective stress necessary for yielding to occur"] + M --> N["Calculate the total bending moments at the current Gauss points"] + N --> O["If the current bending moments are outside of the yield surface bring them back to the yield surface taking into account unloading if it has taken place"] +``` +
+ +Fig. 9.3 Overall structure of subroutine RESMP. + + + +![](images/page-358_645ab8ddf88be9537076f6e1702d87b898be2015b9067ee3985bab44743eca73.jpg) + +
+flowchart + +```mermaid +graph TD + A["A"] --> B["Evaluate [Bf"]T σ̂f × Gauss weights × det J and add into ELOAD ( , ). Use routines VZERO and BMATPB] + B --> C["Call GAUSSQ to evaluate (n - 1) point Gauss-Legendre sampling positions and weights"] + C --> D["Call SFR2, JACOB2, GRADMP and STRMP to evaluate elastic stress increment dσ̂s"] + D --> E["Evaluate [Bs"]T σ̂s × Gauss weights × det J and add into ELOAD ( , ). Use routines VZERO and BMATPB] + E --> F["RETURN"] + G["GAUSS LOOPS"] --> D + G --> E +``` +
+ +Fig. 9.3 Overall structure of subroutine RESMP (continued). + +Subroutine SFR2 is identical to its namesake given earlier in Section 6.4.3 except that SFR2 72–118 are replaced by SFRH 67–73. + +
IF(NNODE.EQ.8) RETURNSFR267
C*** BUBBLE FUNCTION FOR HIERARCHICAL AND HETEROSIS ELEMENTSSFRH68
SHAPE(9)=(1.0-SS)*(1.0-TT)SFRH69
DERIV(1,9)=-S2*(1.0-TT)SFRH70
DERIV(2,9)=-T2*(1.0-SS)SFRH71
RETURNSFRH72
ENDSFRH73
+ +# 9.5.13 Subroutine STIFMP + +This routine evaluates the stiffness matrix for the nonlayered elasto-plastic Mindlin plate elements. The overall structure is shown in Fig. 9.4. + + + +![](images/page-359_a28640cd410af76ff85758e9d9c2ee9a3b8a687c6fad77fd20db7cc907376fdd.jpg) + +
+flowchart + +```mermaid +graph TD + A["START"] --> B["Rewind tapes 1 and 3"] + B --> C["Extract local element material property set number and coordinates"] + C --> D["Initialise array used to store element stiffness matrices"] + D --> E["Call GAUSSQ to evaluate n-point Gauss-Legendre sampling positions and weights"] + E --> F["Call SFR2 and JACOB2 to evaluate N_i^(e) / ∂N_i^(e)/∂x, ∂N_i^(e)/∂y and det J^(e)"] + F --> G["Call MODPB to evaluate D_f"] + G --> H{Is this the first load increment?} + H -->|Yes| I["GAUSS LOOPS"] + H -->|No| J{Was this Gauss point plastic in the last load increment?} + J -->|No| I + J -->|Yes| K["A"] +``` +
+ + + +![](images/page-360_f39271c68a18bd4b50dc64082d2d75c8db47b2eae4026920e44148685f0acc51.jpg) + +
+flowchart + +```mermaid +graph TD + A["A"] --> B["Call INVMP and FLOWMP to evaluate a' and d_D and hence calculate D_ep"] + B --> C["Call BMATPB and SUBMP to add [B_f^(e)"]^T D_f B_f^(e) det J^(e) × Gauss weights into K_ij^(e)] + C --> D["Call GAUSSQ to evaluate (n-1)-point Gauss-Legendre sampling positions and weights"] + D --> E["Call SFR2 and JACOB2 to evaluate N_i^(e), ∂N_i^(e)/∂x, ∂N_i^(e)/∂y and det J^(e)"] + E --> F["Call MODPB to evaluate D_s"] + F --> G["Call BMATPB and SUBMP to add [B_δi^(e)"]^T D_δ B_δj^(e) det J × Gauss weights into K_ij^(e)] + G --> H["Store stiffness matrix K^(e) and Gauss point coordinates on files 1 and 3 respectively"] + H --> I["RETURN"] + I --> J["GAUSS LOOPS"] +``` +
+ +Fig. 9.4 Overall structure of subroutine STIFMP (continued). diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_037.md b/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_037.md new file mode 100644 index 00000000..8a47d976 --- /dev/null +++ b/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_037.md @@ -0,0 +1,645 @@ + + +```fortran +SUBROUTINE STIFMP (COORD,EPSTN,IINCS,LNODS,MATNO,MELEM, STIF 1 +. MEVAB,MMATS,MPOIN,MTOTG,NCRIT,NELEM, STIF 2 +. NEVAB,NGAUS,NNODE,PROPS,STRSG) STIF 3 +C**************************STIF 4 +C STIF 5 +C*** EVALUATE STIFFNESS MATRICES FOR NON-LAYERED STIF 6 +C*** ELASTO-PLASTIC MINDLIN PLATE ELEMENTS STIF 7 +C STIF 8 +C**************************STIF 9 +DIMENSION AVECT(5), STIF 10 +. CARTD(2,9),COORD(MPOIN,2), STIF 11 +. DERIV(2,9),DEVIA(4),DVECT(5),ELCOD(2,9), STIF 12 +. EPSTN(MTOTG),ESTIF(27,27),GPCOD(2,9),LNODS(MELEM,9), STIF 13 +. MATNO(MELEM),POSGP(4),PROPS(MMATS,8),SHAPE(9),STRES(5), STIF 14 +. STRSG(5,MTOTG),WEIGP(4), STIF 15 +. DFLEX(3,3),DSHER(2,2),BFLEI(3,3),BFLEJ(3,3), STIF 16 +. BSHEI(2,3),BSHEJ(2,3),DUMMY(3,3) STIF 17 +REWIND 1 STIF 18 +REWIND 3 STIF 19 +KGAUS=0 STIF 20 +C STIF 21 +C*** LOOP OVER EACH ELEMENT STIF 22 +C STIF 23 +DO 70 IELEM=1,NELEM STIF 24 +LPROP=MATNO(IELEM) STIF 25 +C STIF 26 +C*** EVALUATE THE COORDINATES OF THE ELEMENT NODAL POINTS STIF 27 +C STIF 28 +DO 10 INODE=1,NNODE STIF 29 +LNODE=LNODS(IELEM,INODE) STIF 30 +LNODE=IABS(LNODE) STIF 31 +DO 10 IDIME=1,2 STIF 32 +10 ELCOD(IDIME,INODE)=COORD(LNODE,IDIME) STIF 33 +C STIF 34 +C*** INITIALIZE THE ELEMENT STIFFNESS MATRIX STIF 35 +C STIF 36 +DO 20 IEVAB=1,NEVAB STIF 37 +DO 20 JEVAB=1,NEVAB STIF 38 +20 ESTIF(IEVAB,JEVAB)=0.0 STIF 39 +C STIF 40 +C*** EVALUATE PART OF STIFFNESS MATRIX STIF 41 +C ASSOCIATED WITH BENDING DEFORMATION STIF 42 +C STIF 43 +KGASP=C STIF 44 +C STIF 45 +C*** ENTER LOOPS FOR AREA NUMERICAL INTEGRATION STIF 46 +C STIF 47 +C STIF 48 +C*** SET UP GAUSSIAN INTEGRATION CONSTANTS STIF 49 +C STIF 50 +CALL GAUSSQ (NGAUS,POSGP,WEIGP) STIF 51 +DO 50 IGAUS=1,NGAUS STIF 52 +DO 50 JGAUS=1,NGAUS STIF 53 +KGASP=KGASP+1 STIF 54 +EXISP=POSGP(IGAUS) STIF 55 +ETASP=POSGP(JGAUS) STIF 56 +C STIF 57 +C*** EVALUATE THE SHAPE FUNCTIONS, ELEMENTAL AREA,ETC STIF 58 +C STIF 59 +CALL SFR2 (DERIV,ETASP,EXISP,NNODE,SHAPE) STIF 60 +CALL JACOB2 (CARTD,DERIV,DJACB,ELCOD,GPCOD,IELEM, STIF 61 +KGASP,NNODE,SHAPE) STIF 62 +DAREA=DJACB*WEIGP(IGAUS)*WEIGP(JGAUS) STIF 63 +``` + + + +```csv +C +C*** EVALUATE THE B AND DB MATRICES +C +CALL MODPB (DFLEX,DUMMY,DSHER,LPROP,MMATS,PROPS, +0, 1, 0) +IF(IINCS.EQ.1) GO TO 80 +KGAUS=KGAUS+1 +IF(EPSTN(KGAUS).EQ.0.0) GO TO 80 +DO 90 ISTRE=1,3 +90 STRES(ISTRE)=STRSG(ISTRE,KGAUS) +HARDS=PROPS(LPROP,7) +CALL INVMP (DEVIA,NCRIT,SINT3,STEFF,STRES,THETA, +VARJ2,YIELD) +CALL FLOWMP (ABETA,AVECT,DEVIA,DFLEX,DVECT,HARDS, +NCRIT,SINT3,STEFF,THETA,VARJ2) +DO 100 ISTRE=1,3 +DO 100 JSTRE=1,3 +100 DFLEX(ISTRE,JSTRE)=DFLEX(ISTRE,JSTRE)-ABETA*DVECT(ISTRE)* +.DVECT(JSTRE) +80 CONTINUE +C +C*** CALCULATE THE ELEMENT STIFFNESSES +C +DO 30 INODE=1,NNODE +CALL BMATPB (BFLEI,DUMMY,BSHEI,CARTD,INODE,SHAPE, +0, 1, 0) +DO 30 JNODE=INODE,NNODE +CALL BMATPB (BFLEJ,DUMMY,BSHEJ,CARTD,JNODE,SHAPE, +0, 1, 0) +30 CALL SUBMP (BFLEI,BFLEJ,DAREA,DFLEX,ESTIF,INODE, +JNODE, 3, 3, 3) +50 CONTINUE +C +C*** EVALUATE PART OF STIFFNESS MATRIX +C ASSOCIATED WITH SHEAR DEFORMATION +C +KGASP=0 +NGAUM=NGAUS-1 +C +C*** ENTER LOOPS FOR AREA INTEGRATION +C +C*** SET UP GAUSSIAN INTEGRATION CONSTANTS +C +CALL GAUSSQ (NGAUM,POSGP,WEIGP) +DO 51 IGAUS=1,NGAUM +DO 51 JGAUS=1,NGAUM +KGASP=KGASP+1 +EXISP=POSGP(IGAUS) +ETASP=POSGP(JGAUS) +C +C*** EVALUATE THE SHAPE FUNCTIONS,ELEMENTAL AREA,ETC +C +CALL SFR2 (DERIV,ETASP,EXISP,NNODE,SHAPE) +CALL JACOB2 (CARTD,DERIV,DJACB,ELCOD,GPCOD,IELEM, +KGASP,NNODE,SHAPE) +DAREA=DJACB*WEIGP(IGAUS)*WEIGP(JGAUS) +C +C*** EVALUATE THE B AND DB MATRICES +C +CALL MODPB (DFLEX,DUMMY,DSHER,LPROP,MMATS,PROPS, +0, 0, 1) +C +C*** EVALUATE ELEMENT STIFFNESSES +``` + + + +```csv +C +DO 31 INODE=1,NNODE +CALL BMATPB (BFLEI,DUMMY,BSHEI,CARTD,INODE,SHAPE, +0, 0, 1) +DO 31 JNODE=INODE,NNODE +CALL BMATPB (BFLEJ,DUMMY,BSHEJ,CARTD,JNODE,SHAPE, +0, 0, 1) +31 CALL SUBMP (BSHEI,BSHEJ,DAREA,DSHER,ESTIF,INODE, +JNODE, 3, 2, 3) +51 CONTINUE +C +C*** CONSTRUCT THE LOWER TRIANGLE OF THE STIFFNESS MATRIX +C +DO 60 IEVAB=1,NEVAB +DO 60 JEVAB=IEVAB,NEVAB +60 ESTIF(JEVAB,IEVAB)=ESTIF(IEVAB,JEVAB) +C +C*** STORE THE STIFFNESS MATRIX,STRESS MATRIX AND SAMPLING POINT +C COORDINATES FOR EACH ELEMENT ON DISC FILE +C +WRITE(1) ESTIF +WRITE(3) GPCOD +70 CONTINUE +RETURN +END +STIF 129 +STIF 130 +STIF 131 +STIF 132 +STIF 133 +STIF 134 +STIF 135 +STIF 136 +STIF 137 +STIF 138 +STIF 139 +STIF 140 +STIF 141 +STIF 142 +STIF 143 +STIF 144 +STIF 145 +STIF 146 +STIF 147 +STIF 148 +STIF 149 +STIF 150 +STIF 151 +STIF 152 +STIF 153 +STIF 154 +``` + +# 9.5.14 Subroutine STRMP + +This subroutine evaluates the bending moments and shear forces for Mindlin plates. + +```csv +SUBROUTINE STRMP (CARTD,DFLEX,DGRAD,DSHER,ELDIS,NNODE, STRP 1 +. SHAPE,STRES,IFFLE,IFSHE) STRP 2 +C**************************STRP 3 +C STRP 4 +C*** EVALUATES STRESS RESULTANTS FOR MINDLIN PLATE STRP 5 +C STRP 6 +C**************************STRP 7 +DIMENSION CARTD(2,9),DFLEX(3,3),DGRAD(6),DSHER(2,2), STRP 8 +. ELDIS(3,9),SHAPE(9),STRES(5) STRP 9 +C*** ZERO STRESS VECTOR STRP 10 +CALL VZERO (5,STRES) STRP 11 +C*** EVALUATE ROTATIONS AT GAUSS POINT, IF NEEDED STRP 12 +IF(IFSHE.EQ.0) GOTO 50 STRP 13 +XZROT=0.0 STRP 14 +YZROT=0.0 STRP 15 +DO 30 INODE=1,NNODE STRP 16 +XZROT=XZROT+SHAPE(INODE)*ELDIS(2,INODE) STRP 17 +30 YZROT=YZROT+SHAPE(INODE)*ELDIS(3,INODE) STRP 18 +C*** EVALUATE BENDING STRESS RESULTANTS STRP 19 +50 IF(IFFLE.EQ.0) GOTO 60 STRP 20 +. EFLXX=-DGRAD(2) STRP 21 +EFLYY=-DGRAD(6) STRP 22 +EFLXY=-(DGRAD(3)+DGRAD(5)) STRP 23 +STRES(1)=DFLEX(1,1)*EFLXX+DFLEX(1,2)*EFLYY STRP 24 +STRES(2)=DFLEX(2,1)*EFLXX+DFLEX(2,2)*EFLYY STRP 25 +STRES(3)=DFLEX(3,3)*EFLXY STRP 26 +``` + + + +
C*** EVALUATE SHEAR STRESS RESULTANTS60 IF(IFSHE.EQ.0) RETURNSTRP27
STRP28
ESHXX=DGRAD(1)-XZROTSTRP29
ESHYY=DGRAD(4)-YZROTSTRP30
STRES(4)=DSHER(1,1)*ESHXXSTRP31
STRES(5)=DSHER(2,2)*ESHYYSTRP32
RETURNSTRP33
ENDSTRP34
+ +# 9.5.15 Subroutine SUBMP + +This subroutine evaluates $[B_{i}]^{T}D[B_{j}]detJ\times Gauss$ weights and is used in the evaluation of the element stiffness matrices. + +```csv +SUBROUTINE SUBMP (BIMAT,BJMAT,DAREA,DMATX,ESTIF,INODE, SUBP 1 +. JNODE,NCOLI,NROIJ,NCOLJ) SUBP 2 +C**************************SUBP 3 +C SUBP 4 +C*** CARRY OUT MATRIX MULTIPLICATION SUBP 5 +C SUBP 6 +C**************************SUBP 7 +DIMENSION BIMAT(NROIJ,NCOLI),BJMAT(NROIJ,NCOLJ), SUBP 8 +. DMATX(NROIJ,NROIJ),DBMAT(3,3), SUBP 9 +. ESTIF(27,27),SBSTF(3,3) SUBP 10 +C*** EVALUATE D*BJ SUBP 11 +DO 10 J=1,NCOLJ SUBP 12 +DO 10 I=1,NROIJ SUBP 13 +DBMAT(I,J)=0.0 SUBP 14 +DO 10 K=1,NROIJ SUBP 15 +10 DBMAT(I,J)=DBMAT(I,J)+DMATX(I,K)*BJMAT(K,J) SUBP 16 +C*** EVALUATE BIT*(D*BJ) SUBP 17 +DO 20 J=1,NCOLJ SUBP 18 +DO 20 I=1,NCOLI SUBP 19 +SBSTF(I,J)=0.0 SUBP 20 +DO 20 K=1,NROIJ SUBP 21 +20 SBSTF(I,J)=SBSTF(I,J)+BIMAT(K,I)*DBMAT(K,J) SUBP 22 +C*** ASSEMBLE SBSTF INTO ELEMENT STIFFNESS MATRIX SUBP 23 +IFROW=0 SUBP 24 +JFCOL=0 SUBP 25 +IFROW=(INODE-1)*3+IFROW SUBP 26 +JFCOL=(JNODE-1)*3+JFCOL SUBP 27 +DO 30 I=1,NCOLI SUBP 28 +IRSUB=IFROW+I SUBP 29 +DO 30 J=1,NCOLJ SUBP 30 +JCSUB=JFCOL+J SUBP 31 +30 ESTIF(IRSUB,JCSUB)=ESTIF(IRSUB,JCSUB)+SBSTF(I,J)*DAREA SUBP 32 +RETURN SUBP 33 +END SUBP 34 +``` + +# 9.5.16 Subroutines VZERO and ZEROMP + +These routines simply set to zero the components of various vectors and arrays. + +```txt +SUBROUTINE VZERO (NCOMP, VECTO) ZERO 1 +C************************** ZERO 2 +C ZERO 3 +C*** ZEROES VECTOR VECTO ZERO 4 +C ZERO 5 +C************************** ZERO 6 +DIMENSION VECTO(NCOMP) ZERO 7 +DO 10 ICOMP=1,NCOMP ZERO 8 +10 VECTO(ICOMP)=0.0 ZERO 9 +RETURN ZERO 10 +END ZERO 11 +``` + + + +```txt +SUBROUTINE ZEROMP (EFFST,ELOAD,EPSTN,MELEM,MEVAB,MTOTG, ZERP 1 +. MTOTV,MVFIX,NDOFN,NELEM,NEVAB,NGAUS, ZERP 2 +. NTOTG,NTOTV,NVFIX,STRSG,TDISP,TFACT, ZERP 3 +. TLOAD,TREAC) ZERP 4 +C**************************ZERP 5 +C ZERP 6 +C*** ZERO EFFST,ELOAD,EPSTN,STRSG,TDISP,TFACT,TLOAD,TREAC ZERP 7 +C ZERP 8 +C**************************ZERP 9 +DIMENSION ELOAD(MELEM,MEVAB),STRSG(5,MTOTG),TDISP(MTOTV), ZERP 10 +. TLOAD(MELEM,MEVAB),TREAC(MVFIX,3),EPSTN(MTOTG), ZERP 11 +. EFFST(MTOTG) ZERP 12 +TFACT=0.0 ZERP 13 +DO 30 IELEM=1,NELEM ZERP 14 +DO 30 IEVAB=1,NEVAB ZERP 15 +ELOAD(IELEM,IEVAB)=0.0 ZERP 16 +30 TLOAD(IELEM,IEVAB)=0.0 ZERP 17 +DO 40 ITOTV=1,NTOTV ZERP 18 +40 TDISP(ITOTV)=0.0 ZERP 19 +DO 50 IVFIX=1,NVFIX ZERP 20 +DO 50 IDOFN=1,NDOFN ZERP 21 +50 TREAC(IVFIX,IDOFN)=0.0 ZERP 22 +DO 60 ITOTG=1,NTOTG ZERP 23 +EPSTN(ITOTG)=0.0 ZERP 24 +EFFST(ITOTG)=0.0 ZERP 25 +DO 60 ISTR1=1,5 ZERP 26 +60 STRSG(ISTR1,ITOTG)=0.0 ZERP 27 +RETURN ZERP 28 +END ZERP 29 +``` + +# 9.6 Software for the layered approach + +# 9.6.1 Overall program structure. + +The overall program structure for the elasto-plastic Mindlin plate bending analysis program using the layered approach is given in Fig. 9.5. This program is named MINDLAY. + +The program can solve problems of the same size as those solved by program MINDLIN. A maximum of 26 layers is allowed. + +All new routines are now documented and these include: FEAM, DEPMPA, LAYMPA, MDMPA, OUTMPA, RESMPA, STIMPA and STRMPA. The outer routines, which have been described earlier, include ALGOR, BMATPB, CHECK1, CHECK2, ECHO, FRONT, INCREM, INPUT, JACOB2 and NODEXY. + +The files which are used in the program are 5 (cardreader), 6 (lineprinter) and 1, 2, 3, 4, 8 (scratch files). + +# 9.6.2 Subroutine FEAM + +This routine organises the calling of the main routines in sequence. + + + +![](images/page-366_4957bf2f81a024b454061e697cda2961788c966d555c1fc548ea6b98418d01e2.jpg) + +
+flowchart + +```mermaid +graph TD + A["START"] --> B["DIMMP
Presents the variables associated with the dynamic dimensioning process"] + B --> C["INPUT
Inputs data defining geometry, boundary conditions and material properties"] + C --> D["ZEROMP
Sets to zero arrays required for accumulation of data"] + D --> E["MINDPB
Inputs additional data required for Mindlin plate analysis"] + E --> F["LOADPB
Reads loading data and evaluate the equivalent nodal forces for distributed loading"] + F --> G["INCREM
Increments the applied load according to specified load factors"] + G --> H["ALGOR
Sets indicator to identify the type of solution algorithm, i.e., initial or tangential stiffness etc."] + H --> I["A"] +``` +
+ +Fig. 9.5 Overall program structure of program MINDLAY. + + + +![](images/page-367_7282858d1fec3c9bb621c16a1971c2eb4af44eb16bfaa5d614cb96f3b24826d7.jpg) + +
+flowchart + +```mermaid +graph TD + A["A"] --> B{Is it necessary to recalculate stiffness matrix with present algorithm?} + B -->|No| C["END"] + B -->|Yes| D["STIFMPA
Calculate element stiffness matrices for layered elasto-plastic Mindlin plate"] + D --> E["FRONT
Solve the simultaneous equation system by the frontal method"] + E --> F["RESMPA
Evaluate the residual force vector for the layered elasto-plastic Mindlin plate"] + F --> G["CONVMP
Check whether solution has converged using a residual force or displacement norm"] + G --> H["OUTMPA
Prints out the displacements, reactions and stresses and stress resultants for the current load increment"] + H --> I["LOAD INCREMENT LOOP"] + H --> J["LOAD ITERATION LOOP"] +``` +
+ +Fig. 9.5 Overall program structure of program MINDLAY (continued). + + + +```txt +PROGRAM FEAM(INPUT, OUTPUT, TAPE5=INPUT, TAPE6=OUTPUT, TAPE1, TAPE2, TAPE3, TAPE4, TAPE8, TAPE9) +C +C +C*** ELASTO-PLASTIC ANALYSIS OF LAYERED MINDLIN PLATES USING +C*** 4-, 8-, 9-NODED OR HETEROSIS ISOPARAMETRIC QUADRILATERALS +C +C +DIMENSION ASDIS(240), COORD(80, 2), EFFST(225), ELOAD(25, 27), +EPSTN(225), ESTIF(27, 27), +EQRHS(10), EQUAT(40, 10), FIXED(240), +IFFIX(240), GLOAD(40), GSTIF(860), LNODS(25, 9), LOCEL(27), +MATNO(25), NACVA(40), NAMEV(10), NCDIS(4), NCRES(4), +NDEST(27), NDFRO(25), NOFIX(40), NOUTP(2), NPIVO(10), +POSGP(4), PRESC(40, 3), PROPS(10, 8), REFOR(240), +RLOAD(25, 27), STRSG(5, 225), TOFOR(240), +TDISP(240), TLOAD(25, 27), TREAC(40, 3), VECRV(40), +WEIGP(4) +C +C*** PRESET VARIABLES ASSOCIATED WITH DYNAMIC DIMENSIONS +C +CALL DIMMP (MBUFA, MELEM, MEVAB, MFRON, MMATS, MPOIN, +MSTIF, MTOTG, MTOTV, MVFIX, NDIME, NDOFN, +NPROP, NSTRE) +C +C*** CALL THE SUBROUTINE WHICH READS MOST OF THE PROBLEM DATA +C +CALL INPUT (COORD, IFFIX, LNODS, MATNO, MELEM, MEVAB, +MFRON, MMATS, MPOIN, MTOTV, MVFIX, NALGO, +NCRIT, NDFRO, NDIME, NDOFN, NELEM, NEVAB, +NGAUS, NLAPS, NINCS, NMATS, NNODE, NOFIX, +NPOIN, NPROP, NSTRE, NSTR1, NSWIT, NTOTG, +NTOTV, NTYPE, NVFIX, POSGP, PRESC, PROPS, +WEIGP) +C +C*** INITIALIZE ARRAYS TO ZERO +C +CALL ZEROMP (EFFST, ELOAD, EPSTN, MELEM, MEVAB, MTOTG, +MTOTV, MVFIX, NDOFN, NELEM, NEVAB, NGAUS, +NTOTG, NTOTV, NVFIX, STRSG, TDISP, TFACT, +TLOAD, TREAC) +C +C*** CALL MINDPB (IFDIS, IFFIX, IFRES, LNODS, MELEM, MTOTV, +NCDIS, NCRES, NELEM, NTYPE) +C +C*** COMPUTE LOAD AFTER READING RELEVANT EXTRA DATA +C +CALL LOADPB (COORD, LNODS, MATNO, MELEM, MMATS, MPOIN, +NELEM, NEVAB, NGAUS, NNODE, NPOIN, PROPS, +RLOAD) +C +C*** LOOP OVER EACH INCREMENT +C +DO 70 IINCS=1, NINCS +C +C*** READ DATA FOR CURRENT INCREMENT +C +CALL INCREM (ELOAD, FIXED, IINCS, MELEM, MEVAB, MITER, +MTOTV, MVFIX, NDOFN, NELEM, NEVAB, NOUTP, +NOFIX, NTOTV, NVFIX, PRESC, RLOAD, TFACT, +TLOAD, TOLER) +``` + + + +```asm +C +C*** LOOP OVER EACH ITERATION +C +DO 90 IITER=1,MITER +C +C*** CALL ROUTINE WHICH SELECTS SOLUTION ALGORITHM VARIABLE KRESL +C +CALL ALGOR (FIXED,IINCS,IITER,KRESL,MTOTV,NALGO, +NTOTV) +C +C*** CHECK WHETHER A NEW EVALUATION OF THE STIFFNESS MATRICES IS NEEDED +C +IF(KRESL.EQ.1) +.CALL STIMPA (COORD,EPSTN,IINCS,LNODS,MATNO,MELEM, +MEVAB,MMATS,MPOIN,MTOTG,NCRIT,NELEM, +NEVAB,NGAUS,NNODE,NLAPS,PROPS,STRSG) +C +C*** SOLVE EQUATIONS +C +CALL FRONT (ASDIS,ELOAD,EQRHS,EQUAT,ESTIF,FIXED, +IFFIX,IINCS,IITER,GLOAD,GSTIF,KRESL, +LNODS,LOCEL,MBUFA,MELEM,MEVAB,MFRON, +MSTIF,MTOTV,MVFIX,NACVA,NAMEV,NDEST, +NDOFN,NELEM,NEVAB,NNODE,NOFIX,NPIVO, +NPOIN,NTOTV,TDISP,TLOAD,TREAC,VECRV) +C +C*** CALCULATE RESIDUAL FORCES +C +CALL RESMPA (ASDIS,COORD,EFFST,ELOAD,EPSTN,LNODS, +MATNO,MELEM,MMATS,MPOIN,MTOTG,MTOTV, +NCRIT,NELEM,NEVAB,NGAUS,NNODE,NLAPS, +PROPS,STRSG) +C +C*** CHECK FOR CONVERGENCE +C +CALL CONVMP (ASDIS,ELOAD,IITER,IFDIS,IFRES,LNODS, +MELEM,MEVAB,MTOTV,NCHEK,NCDIS,NCRES, +NDOFN,NELEM,NEVAB,NNODE,NPOIN,NTOTV, +REFOR,TOFOR,TDISP,TLOAD,TOLER) +C +C*** OUTPUT RESULTS IF REQUIRED +C +IF(IITER.EQ.1.AND.NOUTP(1).GT.0) +.CALL OUTMPA (EPSTN,IITER,MTOTG,MTOTV,MVFIX,NELEM, +NGAUS,NLAPS,NOFIX,NOUTP,NPOIN,NVFIX, +STRSG,TDISP,TREAC) +C +C*** IF SOLUTION HAS CONVERGED STOP ITERATING AND OUTPUT RESULTS +C +IF(NCHEK.EQ.0) GO TO 100 +90 CONTINUE +C +C*** IF(NALGO.EQ.2) GO TO 100 +STOP +100 CALL OUTMPA (EPSTN,IITER,MTOTG,MTOTV,MVFIX,NELEM, +NGAUS,NLAPS,NOFIX,NOUTP,NPOIN,NVFIX, +STRSG,TDISP,TREAC) +70 CONTINUE +20 CONTINUE +10 CONTINUE +STOP +END +FEAM 66 +FEAM 67 +FEAM 68 +FEAM 69 +FEAM 70 +FEAM 71 +FEAM 72 +FEAM 73 +FEAM 74 +FEAM 75 +FEAM 76 +FEAM 77 +FEAM 78 +FEAM 79 +FEAM 80 +FEAM 81 +FEAM 82 +FEAM 83 +FEAM 84 +FEAM 85 +FEAM 86 +FEAM 87 +FEAM 88 +FEAM 89 +FEAM 90 +FEAM 91 +FEAM 92 +FEAM 93 +FEAM 94 +FEAM 95 +FEAM 96 +FEAM 97 +FEAM 98 +FEAM 99 +FEAM 100 +FEAM 101 +FEAM 102 +FEAM 103 +FEAM 104 +FEAM 105 +FEAM 106 +FEAM 107 +FEAM 108 +FEAM 109 +FEAM 110 +FEAM 111 +FEAM 112 +FEAM 113 +FEAM 114 +FEAM 115 +FEAM 116 +FEAM 117 +FEAM 118 +FEAM 119 +FEAM 120 +FEAM 121 +FEAM 122 +FEAM 123 +FEAM 124 +FEAM 125 +FEAM 126 +FEAM 127 +FEAM 128 +FEAM 129 +FEAM 130 +``` + + + +# 9.6.3 Subroutine CHECK1 (revised) + +In program MINDLAY we remove card CEK1 25 from subroutine CHECK1 because NLAPS (the number of layers) replaces NSTRE in subroutine INPUT. The variable NSTRE is set in subroutine DIMMP (see Section 9.5.4). + +# 9.6.4 Subroutine DEPMPA + +This subroutine sets up the layered discretisation. + +```fortran +SUBROUTINE DEPMPA (DEPTH,LPROP,MMATS,NLAYR,PROPS) DEPT 1 +C**************************DEPT 2 +C DEPT 3 +C*** SET UP LAYRED DISCRETIZATION DEPT 4 +C DEPT 5 +C**************************DEPT 6 +DIMENSION PROPS(MMATS,8),DEPTH(26) DEPT 7 +C DEPT 8 +C DEPT 9 +NLAY1=NLAYR+1 DEPT 10 +ALAYR=NLAYR DEPT 11 +THICK=PROPS(LPROP,3) DEPT 12 +CONS1=THICK/ALAYR DEPT 13 +CONS2=-THICK/2.0 DEPT 14 +KOUNT=0 DEPT 15 +DO 10 ILAYR=1,NLAY1 DEPT 16 +DEPTH(ILAYR)=CONS2+CONS1*KOUNT DEPT 17 +10 KOUNT=KOUNT+1 DEPT 18 +RETURN DEPT 19 +END DEPT 20 +``` + +# 9.6.5 Subroutine LAYMPA + +This subroutine evaluates $\hat{D}_{f}$ and $\hat{D}_{s}$ using the mid-ordinate rule. + +```txt +SUBROUTINE LAYMPA (DEPTH,DFLEF,DSHES,EPSTN,IINCS,KGAUS, LAYR 1 +, LPROP,MMATS,MTOTG,NCRIT,NLAYR,PROPS, LAYR 2 +, STRSG,JFFLE) LAYR 3 +C****************************************************************************************** +C +C*** CALCULATES THE D-MATRIX INTEGRATED OVER LAYR 6 +C*** THE DEPTH LAYR 7 +C +C****************************************************************************************** +DIMENSION AVECT(3),DEPTH(26),DEVIA(4),DFLEF(3,3), LAYR 10 +, DPLAN(3,3),DVECT(3), LAYR 11 +, DSHER(2,2),DSHES(2,2),EPSTN(MTOTG),PROPS(MMATS,8), LAYR 12 +, SGTOT(5),STRSG(5,MTOTG) LAYR 13 +C +C +IF(JFFLE.EQ.0) GO TO 100 LAYR 16 +HARDS=PROPS(LPROP,7) LAYR 17 +C +C*** ZERO D MATRIX FOR FLEXURE LAYR 19 +C +DO 20 ISTRE=1,3 LAYR 21 +DO 20 JSTRE=1,3 LAYR 22 +20 DFLEF(ISTRE,JSTRE)=0.0 LAYR 23 +C +C*** LOOP AROUND LAYERS LAYR 24 +C +``` diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_038.md b/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_038.md new file mode 100644 index 00000000..0821616a --- /dev/null +++ b/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_038.md @@ -0,0 +1,975 @@ + + +```csv +DO 30 ILAYR=1,NLAYR +KGAUS=KGAUS+1 +JLAYR=ILAYR+1 +C +C*** EVALUATE Z-COORDINATES FOR CURRENT LAYER +C +DEPT1=DEPTH(ILAYR) +DEPT2=DEPTH(JLAYR) +CONS3=(DEPT2+DEPT1)*(DEPT2**2-DEPT1**2)/4.0 +C +C*** EVALUATE ELASTO-PLASTIC D MATRIX FOR CURRENT LAYER +C +CALL MDMPA(DPLAN,DSHER,LPROP,MMATS,PROPS,1,0) +IF(IINCS.EQ.1)GO TO 40 +IF(EPSTN(KGAUS).EQ.0.0)GO TO 40 +DO 50 ISTRE=1,5 +50 SGTOT(ISTRE)=STRSG(ISTRE,KGAUS) +CALL INVMP(DEVIA,NCrit,SINT3,STEFF,SGTOT,THETA,VARJ2,YIELD) +CALL FLOWMP(ABETA,AVECT,DEVIA,DPLAN,DVECT,HARDS,NCrit,SINT3, +STEFF,THETA,VARJ2) +DO 60 ISTRE=1,3 +DO 60 JSTRE=1,3 +60 DPLAN(ISTRE,JSTRE)=DPLAN(ISTRE,JSTRE)-ABETA*DVECT(ISTRE)* +.DVECT(JSTRE) +40 CONTINUE +C +C*** SUM D MATRIX OVER ELEMENT DEPTH +C +DO 70 ISTRE=1,3 +DO 70 JSTRE=1,3 +70 DFLEF(ISTRE,JSTRE)=DFLEF(ISTRE,JSTRE)+CONS3*DPLAN(ISTRE,JSTRE) +30 CONTINUE +GO TO 200 +C +C*** ZERO D MATRIX FOR SHEAR +C +100 DO 80 ISTRE=1,2 +DO 80 JSTRE=1,2 +80 DSHES(ISTRE,JSTRE)=0.0 +C +C*** EVALUATE ELASTIC D MATRIX +C +CALL MDMPA(DPLAN,DSHER,LPROP,MMATS,PROPS,0,1) +C +C*** LOOP AROUND LAYERS +C +DO 90 ILAYR=1,NLAYR +JLAYR=ILAYR+1 +C +C*** EVALUATE Z-COORDINATES FOR CURRENT LAYER +C +DEPT1=DEPTH(ILAYR) +DEPT2=DEPTH(JLAYR) +CONS4=DEPT2-DEPT1 +C +C*** SUM D MATRIX OVER ELEMENT DEPTH +C +DO 110 ISTRE=1,2 +DO 110 JSTRE=1,2 +110 DSHES(ISTRE,JSTRE)=DSHES(ISTRE,JSTRE)+CONS4*Dsher(ISTRE,JSTRE) +90 CONTINUE +200 CONTINUE +RETURN +END +LAYR 27 +LAYR 28 +LAYR 29 +LAYR 30 +LAYR 31 +LAYR 32 +LAYR 33 +LAYR 34 +LAYR 35 +LAYR 36 +LAYR 37 +LAYR 38 +LAYR 39 +LAYR 40 +LAYR 41 +LAYR 42 +LAYR 43 +LAYR 44 +LAYR 45 +LAYR 46 +LAYR 47 +LAYR 48 +LAYR 49 +LAYR 50 +LAYR 51 +LAYR 52 +LAYR 53 +LAYR 54 +LAYR 55 +LAYR 56 +LAYR 57 +LAYR 58 +LAYR 59 +LAYR 60 +LAYR 61 +LAYR 62 +LAYR 63 +LAYR 64 +LAYR 65 +LAYR 66 +LAYR 67 +LAYR 68 +LAYR 69 +LAYR 70 +LAYR 71 +LAYR 72 +LAYR 73 +LAYR 74 +LAYR 75 +LAYR 76 +LAYR 77 +LAYR 78 +LAYR 79 +LAYR 80 +LAYR 81 +LAYR 82 +LAYR 83 +LAYR 84 +LAYR 85 +LAYR 86 +LAYR 87 +LAYR 88 +LAYR 89 +LAYR 90 +``` + + + +LAYR 10 If JFFLE is zero $D_f'$ is not evaluated. If it is one $D_s'$ is not evaluated. + +LAYR 15-17 Initializes $D_{f}'$ . + +LAYR 21 Starts the summation loop to form DFLEF, i.e. + +$$ +\hat {D} _ {f} = \sum_ {i = 1} ^ {n} \frac {1}{4} (z _ {i + 1} + z _ {i}) (z _ {i + 1} ^ {2} - z _ {i} ^ {2}) D _ {f} ^ {\prime}. +$$ + +LAYR 22 Increases the counter for Gauss points in each layer by 1. It is needed to use the effective plastic strain (EPSTN) stresses (STRSG) calculated in RESMPA. + +LAYR 27-29 Forms $\frac{1}{4} (z_{i + 1} + z_i)(z_{i + 1}^2 -z_i^2)$ . + +LAYR 33-45 Calls MDMPA to get DPLAN and $D_{ep'}$ is formed using INVMP and FLOWMP. + +LAYR 49–51 DFLEF is formed. + +LAYR 57-59 DSHES is initialised. + +LAYR 63 Calls MDMPA to form DSHER. + +LAYR 67–74 Starts the summation loop and the integrating constant for DSHES is evaluated, i.e. + +$$ +\hat {\boldsymbol {D}} _ {s} = \sum_ {i = 1} ^ {n} (z _ {i + 1} - z _ {i}) \boldsymbol {D} _ {s}. +$$ + +LAYR 78-81 DSHES is formed. + +# 9.6.6 Subroutine MDMPA + +This subroutine evaluates $D_{f}'$ and $D_{s}'$ . +```txt +SUBROUTINE MDMPA (DPLAN,DSHER,LPROP,MMATS,PROPS, MODL 1 +IFPLA,IFSHE) MODL 2 +C**************************MODL 3 +C +C*** CALCULATES MATRIX OF ELASTIC RIGIDITIES FOR EACH LAYER +C*** OF MINDLIN PLATE +C +C**************************MODL 8 +DIMENSION DPLAN(3,3),DSHER(2,2), MODL 9 +PROPS(MMATS,8) MODL 10 +YOUNG=PROPS(LPROP,1) MODL 11 +POISS=PROPS(LPROP,2) MODL 12 +THICK=PROPS(LPROP,3) MODL 13 +C*** FORM DPLAN +IF(IFPLA.EQ.0) GO TO 10 +DO 1 IROWS=1,3 +DO 1 JCOLS=1,3 +1 DPLAN(IROWS,JCOLS)=0.0 +CONST=YOUNG/(1.0-POISS*POISS) MODL 18 +DPLAN(1,1)=CONST +DPLAN(2,2)=CONST +DPLAN(1,2)=CONST*POISS +MODL 20 +MODL 21 +MODL .22 +``` + + + +```csv +DPLAN(2,1)=CONST*POISS +DPLAN(3,3)=CONST*(1.0-POISS)/2.0 +C*** FORM DSHER +10 IF(IFSHE.EQ.0) RETURN +DO 3 IROWS=1,2 +DO 3 JCOLS=1,2 +3 DSHER(IROWS,JCOLS)=0.0 +DSHER(1,1)=YOUNG/(2.4+2.4*POISS) +DSHER(2,2)=YOUNG/(2.4+2.4*POISS) +RETURN +END +MODL 23 +MODL 24 +MODL 25 +MODL 26 +MODL 27 +MODL 28 +MODL 29 +MODL 30 +MODL 31 +MODL 32 +MODL 33 +``` + +# 9.6.7 Subroutine OUTMPA + +This subroutine outputs nodal displacements and reactions and also the Gauss point stress resultants and the stresses within each layer. It is very similar to subroutine OUTMP which was described in Section 9.5.7. Statements OUTP 1-3 are replaced by OUTL 1-3 and statements OUTP 56-66 are replaced by statements OUTL 56-67. + +```fortran +SUBROUTINE OUTMPA (EPSTN,IITER,MTOTG,MTOTV,MVFIX,NELEM, OUTL 1 +. NGAUS,NLAPS,NOFIX,NUOTP,NPOIN,NVFIX, OUTL 2 +. STRSG,TDISP,TREAC) OUTL 3 +C******************************************************************************************OUTL 4 +C OUTL 5 +C*** OUTPUT DISPLACEMENTS,REACTIONS AND GAUSS POINT STRESSES OUTL 6 +C*** IN EACH LAYER FOR EP MINDLIN PLATE ANALYSIS OUTL 7 +C OUTL 8 +C******************************************************************************************OUTL 9 +DIMENSION EPSTN(MTOTG),GPCOD(2,9),NOFIX(MVFIX),NUOTP(2), OUTL 10 +. STRSG(5,MTOTG),TDISP(MTOTV),TREAC(MVFIX,3) OUTL 11 +KOUTP=NOUTP(1) OUTL 12 +IF(IITER.GT.1) KOUTP=NOUTP(2) OUTL 13 +C OUTL 14 +C*** OUTPUT DISPLACEMENTS OUTL 15 +C OUTL 16 +IF(KOUTP.LT.1) GO TO 10 OUTL 17 +WRITE(6,900) OUTL 18 +900 FORMAT(1H0,5X,13HDISPLACEMENTS) OUTL 19 +WRITE(6,950) OUTL 20 +950 FORMAT(1H0,6X,4HNODE,6X,5HDISP.,8X,7HXZ-ROT.,7X,7HYZ-ROT.) OUTL 21 +DO 20 IPOIN=1,NPOIN OUTL 22 +NGASH=IPOIN*3 OUTL 23 +NGISH=NGASH-3+1 OUTL 24 +20 WRITE(6,910) IPOIN,(TDISP(IGASH),IGASH=NGISH,NGASH) OUTL 25 +910 FORMAT(I10,3E14.6) OUTL 26 +10 CONTINUE OUTL 27 +C OUTL 28 +C*** OUTPUT REACTIONS OUTL 29 +C OUTL 30 +IF(KOUTP.LT.2) GO TO 30 OUTL 31 +WRITE(6,920) OUTL 32 +920 FORMAT(1H0,5X,9HREACTIONS) OUTL 33 +WRITE(6,960) OUTL 34 +960 FORMAT(1H0,6X,4HNODE,6X,5HFORCE,3X,9HXZ-MOMENT,5X,9HYZ-MOMENT) OUTL 35 +DO 40 IVFIX=1,NVFIX OUTL 36 +40 WRITE(6,910) NOFIX(IVFIX),(TREAC(IVFIX,IDOFN),IDOFN=1,3) OUTL 37 +30 CONTINUE OUTL 38 +C OUTL 39 +C*** OUTPUT STRESSES OUTL 40 +``` + + + +```csv +C +IF(KOUTP.LT.3) GO TO 50 +REWIND 3 +WRITE(6,970) +970 FORMAT(1H0,5X,8HSTRESSES) +WRITE(6,980) +980 FORMAT(1H0,4HG.P.,2X,8HX-COORD.,2X,8HY-COORD.,3X,8HX-MOMENT,4X, +.8HY-MOMENT,3X,9HXY-MOMENT,3X, +.13HEFF.PL.STRAIN) +KGAUS=0 +DO 60 IELEM=1,NELEM +READ(3)GPCOD +KELGS=0 +WRITE(6,930)IELEM +930 FORMAT(1H0,5X,13HELEMENT NO. =,I5) +DO 60 IGAUS=1,NGAUS +DO 60 JGAUS=1,NGAUS +KELGS=KELGS+1 +DO 60 ILAYR=1,NLAPS +KGAUS=KGAUS+1 +WRITE(6,940)KELGS,(GPCOD(IDIME,KELGS),IDIME=1,2), +.(STRSG(ISTRE,KGAUS),ISTRE=1,3),EPSTN(KGAUS) +940 FORMAT(I5,2F10.4,6E12.5) +60 CONTINUE +50 CONTINUE +RETURN +END +OUTL 41 +OUTL 42 +OUTL 43 +OUTL 44 +OUTL 45 +OUTL 46 +OUTL 47 +OUTL 48 +OUTL 49 +OUTL 50 +OUTL 51 +OUTL 52 +OUTL 53 +OUTL 54 +OUTL 55 +OUTL 56 +OUTL 57 +OUTL 58 +OUTL 59 +OUTL 60 +OUTL 61 +OUTL 62 +OUTL 63 +OUTL 64 +OUTL 65 +OUTL 66 +OUTL 67 +``` + +# 9.6.8 Subroutine RESMPA + +This routine evaluates the residual forces for the layered Mindlin plate. It is very similar to RESMP described in Section 9.5.10. + +```csv +SUBROUTINE RESMPA (ASDIS,COORD,EFFST,ELOAD,EPSTN,LNODS, RESL 1 +. MATNO,MELEM,MMATS,MPOIN,MTOTG,MTOTV, RESL 2 +. NCRIT,NELEM,NEVAB,NGAUS,NNODE,NLAPS, RESL 3 +. PROPS,STRSG) RESL 4 +C************************** RESL 5 +C RESL 6 +C*** EVALUATES EQUIVALENT NODAL FORCES FOR THE STRESSES RESL 7 +C*** IN LAYERED MINDLIN PLATES DURING EP ANALYSIS RESL 8 +C RESL 9 +C************************** RESL 10 +DIMENSION ASDIS(MTOTV),AVECT(5),CARTD(2,9), RESL 11 +. COORD(MPOIN,2),DERIV(2,9),DESIG(5),DEVIA(4), RESL 12 +. DEPTH(26),DVECT(5), RESL 13 +. EFFST(MTOTG),ELCOD(2,9), RESL 14 +. ELDIS(3,9),ELOAD(MELEM,27),EPSTN(MTOTG),GPCOD(2,9), RESL 15 +. LNODS(MELEM,9),MATNO(MELEM),POSGP(4), RESL 16 +. PROPS(MMATS,8),SGTOT(5),SHAPE(9),SIEMA(5), RESL 17 +. STRES(5),STRSG(5,MTOTG),TOSPB(5),WEIGP(4), RESL 18 +. DPLAN(3,3),DSHER(2,2),BFLEI(3,3),BSHEI(2,3), RESL 19 +. DUMMY(3,3),FORCE(3),DGRAD(6) RESL 20 +NTIME=1 RESL 21 +DO 10 IELEM=1,NELEM RESL 22 +DO 10 IEVAB=1,NEVAB RESL 23 +10 ELOAD(IELEM,IEVAB)=0.0 RESL 24 +KGAUS=0 RESL 25 +LGAUS=0 RESL 26 +DO 20 IELEM=1,NELEM RESL 27 +LPROP=MATNO(IELEM) RESL 28 +``` + + + +```txt +C +C*** COMPUTE COORDINATE AND INCREMENTAL DISPLACEMENTS OF THE +C ELEMENT NODAL POINTS +C +DO 190 INODE =1, NNODE +LNODE=IABS(LNODS(IELEM, INODE)) +NPOSN=(LNODE-1)*3 +DO 30 IDOFN=1,3 +NPOSN=NPOSN+1 +30 ELDIS(IDOFN, INODE)=ASDIS(NPOSN) +DO 180 IDIME=1,2 +180 ELCOD(IDIME, INODE)=COORD(LNODE, IDIME) +190 CONTINUE +KGASP=0 +CALL DEMPA(DEPTH, LPROP, MMATS, NLAPS, PROPS) +CALL MDMPA (DPLAN, DSHER, LPROP, MMATS, PROPS, 1, 1) +CALL GAUSSQ (NGAUS, POSGP, WEIGP) +DO 40 IGAUS=1, NGAUS +DO 40 JGAUS=1, NGAUS +EXISP=POSGP(IGAUS) +ETASP=POSGP(JGAUS) +CALL SFR2 (DERIV, ETASP, EXISP, NNODE, SHAPE) +KGASP=KGASP+1 +CALL JACOB2 (CARTD, DERIV, DJACB, ELCOD, GPCOD, IELEM, KGASP, NNODE, SHAPE) +DAREA=DJACB*WEIGP(IGAUS)*WEIGP(JGAUS) +DO 400 ISTRE=1,3 +400 TOSPB(ISTRE)=0.0 +DO 410 ILAYR=1, NLAPS +BRING=1.0 +KGAUS=KGAUS+1 +JLAYR=ILAYR+1 +DEPT1=DEPTH(ILAYR) +DEPT2=DEPTH(JLAYR) +CONST=0.5*(DEPT2+DEPT1) +CALL GRADMP (CARTD, DGRAD, ELDIS, 3, NNODE) +CALL STRMPA (CARTD, CONST, DPLAN, DGRAD, DSHER, ELDIS, NNODE, SHAPE, STRES, 1, 0) +PREYS=PROPS(LPROP, 6)+EPSTN(KGAUS)*PROPS(LPROP, 7) +DO 150 ISTRE=1,3 +DESIG(ISTRE)=STRES(ISTRE) +150 SIGMA(ISTRE)=STRSG(ISTRE, KGAUS)+STRES(ISTRE) +CALL INVMP (DEVIA, NCRIT, SINT3, STEFF, SIGMA, THETA, VARJ2, YIELD) +ESPRE=EFFST(KGAUS)-PREYS +IF(ESPRE.GE.0.0) GO TO 50 +ESCUR=YIELD-PREYS +IF(ESCUR.LE.0.0) GO TO 60 +RFACT=ESCUR/(YIELD-EFFST(KGAUS)) +GO TO 70 +50 ESCUR=YIELD-EFFST(KGAUS) +IF(ESCUR.LE.0.0) GO TO 60 +RFACT=1.0 +70 MSTEP=ESCUR*8.0/PROPS(LPROP, 6)+1.0 +ASTEP=MSTEP +REDUC=1.0-RFACT +DO 80 ISTRE=1,3 +SGTOT(ISTRE)=STRSG(ISTRE, KGAUS)+REDUC*STRES(ISTRE) +80 STRES(ISTRE)=RFACT*STRES(ISTRE)/ASTEP +DO 90 ISTEP=1, MSTEP +CALL INVMP (DEVIA, NCRIT, SINT3, STEFF, SGTOT, THETA, VARJ2, YIELD) +HARDS=PROPS(LPROP, 7) +CALL FLOWMP (ABETA, AVECT, DEVIA, DPLAN, DVECT, HARDS, +``` + + + +```csv +NCRIT,SINT3,STEFF,THETA,VARJ2) RESL 94 +AGASH=0.0 RESL 95 +DO 100 ISTRE=1,3 RESL 96 +100 AGASH=AGASH+AVECT(ISTRE)*STRES(ISTRE) RESL 97 +DLAMD=AGASH*ABETA RESL 98 +IF(DLAMD.LT.0.0) DLAMD=0.0 RESL 99 +BGASH=0.0 RESL 100 +DO 110 ISTRE=1,3 RESL 101 +BGASH=BGASH+AVECT(ISTRE)*SGTOT(ISTRE) RESL 102 +110 SGTOT(ISTRE)=SGTOT(ISTRE)+STRES(ISTRE)-DLAMD*DVECT(ISTRE) RESL 103 +90 EPSTN(KGAUS)=EPSTN(KGAUS)+DLAMD*BGASH/YIELD RESL 104 +DO 120 ISTRE=1,3 RESL 105 +120 DESIG(ISTRE)=SGTOT(ISTRE)-STRSG(ISTRE,KGAUS) RESL 106 +CALL INVMP (DEVIA,NCRIT,SINT3,STEFF,SGTOT,THETA,VARJ2,YIELD) RESL 107 +CURYS=PROPS(LPROP,6)+EPSTN(KGAUS)*PROPS(LPROP,7) RESL 108 +IF(YIELD.GT.CURYS) BRING=CURYS/YIELD RESL 109 +60 DO 130 ISTRE=1,3 RESL 110 +SGTOT(ISTRE)=BRING*(STRSG(ISTRE,KGAUS)+DESIG(ISTRE)) RESL 111 +130 STRSG(ISTRE,KGAUS)=SGTOT(ISTRE) RESL 112 +EFFST(KGAUS)=BRING*YIELD RESL 113 +CONSA=(DEPT2**2-DEPT1**2)/2.0 RESL 114 +DO 440 ISTRE=1,3 RESL 115 +440 TOSPB(ISTRE)=TOSPB(ISTRE)+SGTOT(ISTRE)*CONSA RESL 116 +410 CONTINUE RESL 117 +DO 430 ISTRE=1,3 RESL 118 +430 SGTOT(ISTRE)=TOSPB(ISTRE) RESL 119 +C RESL 120 +C*** CALCULATE THE EQUIVALENT NODAL FORCES AND ASSOCIATE WITH THE RESL 121 +C ELEMENT NODES RESL 122 +DO 140 INODE=1,NNODE RESL 123 +C*** ZERO FORCE VECTOR RESL 124 +CALL VZERO (3,FORCE) RESL 125 +CALL BMATPB (BFLEI,DUMMY,BSHEI,CARTD,INODE,SHAPE, RESL 126 +0, 1, 0) RESL 127 +FORCE(2)=(BFLEI(1,2)*SGTOT(1)+BFLEI(3,2)*SGTOT(3))*DAREA RESL 128 ++FORCE(2) RESL 129 +FORCE(3)=(BFLEI(2,3)*SGTOT(2)+BFLEI(3,3)*SGTOT(3))*DAREA RESL 130 ++FORCE(3) RESL 131 +IPOSN=(INODE-1)*3+1 RESL 132 +DO 135 IDOFN=2,3 RESL 133 +IPOSN=IPOSN+1 RESL 134 +135 ELOAD(IELEM,IPOSN)=ELOAD(IELEM,IPOSN)+FORCE(IDOFN) RESL 135 +140 CONTINUE RESL 136 +40 CONTINUE RESL 137 +C RESL 138 +C*** CALCULATE FORCES ASSOCIATED WITH SHEAR DEFORMATION RESL 139 +C RESL 140 +NGAUM=NGAUS-1 RESL 141 +CALL GAUSSQ (NGAUM,POSGP,WEIGP) RESL 142 +C RESL 143 +C*** ENTER LOOPS FOR AREA NUMERICAL INTEGRATION RESL 144 +C RESL 145 +KGASP=0 RESL 146 +DO 300 IGAUS=1,NGAUM RESL 147 +DO 300 JGAUS=1,NGAUM RESL 148 +EXISP=POSGP(IGAUS) RESL 149 +ETASP=POSGP(JGAUS) RESL 150 +CALL SFR2 (DERIV,ETASP,EXISP,NNODE,SHAPE) RESL 151 +KGASP=KGASP+1 RESL 152 +CALL JACOB2 (CARTD,DERIV,DJACB,ELCOD,GPCOD,IELEM, RESL 153 +KGASP,NNODE,SHAPE) RESL 154 +DAREA=DJACB*WEIGP(IGAUS)*WEIGP(JGAUS) RESL 155 +DO 610 ISTRE=4,5 RESL 156 +610 TOSPB(ISTRE)=0.0 RESL 157 +RESL 158 +``` + + + +```csv +C +C*** LOOP AROUND LAYRS +C +DO 600 ILAYR=1,NLAPS +LGAUS=LGAUS+1 +JLAYR=ILAYR+1 +DEPT1=DEPTH(ILAYR) +DEPT2=DEPTH(JLAYR) +CONST=1.0 +CALL GRADMP (CARTD,DGRAD,ELDIS, 3,NNODE) +CALL STRMPA (CARTD,CONST,DPLAN,DGRAD,DSHER,ELDIS, NNODE,SHAPE,STRES, 0, 1) +DO 310 ISTRE=4,5 +SGTOT(ISTRE)=STRSG(ISTRE,LGAUS)+STRES(ISTRE) +310 STRSG(ISTRE,LGAUS)=SGTOT(ISTRE) +CONSB=DEPT2-DEPT1 +DO 620 ISTRE=4,5 +620 TOSPB(ISTRE)=TOSPB(ISTRE)+SGTOT(ISTRE)*CONSB +600 CONTINUE +DO 605 ISTRE=4,5 +605 SGTOT(ISTRE)=TOSPB(ISTRE) +C +C*** CALCULATE THE EQUIVALENT NODAL FORCES +C +DO 320 INODE=1,NNODE +C*** ZERO FORCE VECTOR +CALL VZERO(3,FORCE) +CALL BMATPB (BFLEI,DUMMY,BSHEI,CARTD,INODE,SHAPE, 0, 0, 1) +FORCE(1)=(BSHEI(1,1)*SGTOT(4)+BSHEI(2,1)*SGTOT(5))*DAREA ++FORCE(1) +FORCE(2)=(BSHEI(1,2)*SGTOT(4))*DAREA+FORCE(2) +FORCE(3)=(BSHEI(2,3)*SGTOT(5))*DAREA+FORCE(3) +IPOSN=(INODE-1)*3 +DO 315 IDOFN=1,3 +IPOSN=IPOSN+1 +315 ELOAD(IELEM,IPOSN)=ELOAD(IELEM,IPOSN)+FORCE(IDOFN) +320 CONTINUE +300 CONTINUE +20 CONTINUE +RETURN +END +RESL 159 +RESL 160 +RESL 161 +RESL 162 +RESL 163 +RESL 164 +RESL 165 +RESL 166 +RESL 167 +RESL 168 +RESL 169 +RESL 170 +RESL 171 +RESL 172 +RESL 173 +RESL 174 +RESL 175 +RESL 176 +RESL 177 +RESL 178 +RESL 179 +RESL 180 +RESL 181 +RESL 182 +RESL 183 +RESL 184 +RESL 185 +RESL 186 +RESL 187 +RESL 188 +RESL 189 +RESL 190 +RESL 191 +RESL 192 +RESL 193 +RESL 194 +RESL 195 +RESL 196 +RESL 197 +RESL 198 +RESL 199 +RESL 200 +``` + +# 9.6.9 Subroutine STIFMPA + +This routine evaluates the stiffness matrices for layered elasto-plastic Mindlin plate elements. + +```csv +SUBROUTINE STIMPA (COORD, EPSTN, IINCS, LNODS, MATNO, MELEM, STFL 1 +. MEVAB, MMATS, MPOIN, MTOTG, NCRIT, NELEM, STFL 2 +. NEVAB, NGAUS, NNODE, NLAPS, PROPS, STRSG) STFL 3 +C**************************STFL 4 +C STFL 5 +C*** EVALUATE STIFFNESS MATRICES FOR LAYREED ELASTO-PLASTIC STFL 6 +C*** MINDLIN PLATE ELEMENTS STFL 7 +C STFL 8 +C**************************STFL 9 +DIMENSION CARTD(2,9), COORD(MPOIN,2), STFL 10 +. DERIV(2,9), DEPTH(26), ELCOD(2,9), STFL 11 +. EPSTN(MTOTG), ESTIF(27,27), GPCOD(2,9), LNODS(MELEM,9), STFL 12 +. MATNO(MELEM), POSGP(4), PROPS(MMATS,8), SHAPE(9), STFL 13 +. STRSG(5, MTOTG), WEIGP(4), STFL 14 +. DFLEX(3,3), DSHER(2,2), BFLEI(3,3), BFLEJ(3,3), STFL 15 +``` + + + +```asm +BSHEI(2,3),BSHEJ(2,3),DUMMY(3,3) +STFL 16 +REWIND 1 +STFL 17 +REWIND 3 +STFL 18 +KGAUS=0 +STFL 19 +C +C*** LOOP OVER EACH ELEMENT +STFL 20 +C +DO 70 IELEM=1,NELEM +STFL 21 +LPROP=MATNO(IELEM) +STFL 22 +C +C*** EVALUATE THE COORDINATES OF THE ELEMENT NODAL POINTS +STFL 23 +C +DO 10 INODE=1,NNODE +STFL 24 +LNODE=LNODS(IELEM,INODE) +STFL 25 +LNODE=IABS(LNODE) +STFL 26 +DO 10 IDIME=1,2 +STFL 27 +10 ELCOD(IDIME,INODE)=COORD(LNODE,IDIME) +STFL 28 +C +C*** INITIALIZE THE ELEMENT STIFFNESS MATRIX +STFL 29 +C +DO 20 IEVAB=1,NEVAB +STFL 30 +DO 20 JEVAB=1,NEVAB +STFL 31 +20 ESTIF(IEVAB,JEVAB)=0.0 +STFL 32 +CALL DEMPA(DEPTH,LPROP,MMATS,NLAPS,PROPS) +STFL 33 +C +C*** EVALUATE PART OF STIFFNESS MATRIX +STFL 34 +C +ASSOCIATED WITH BENDING DEFORMATION +STFL 35 +C +KGASP=0 +STFL 36 +C +C*** ENTER LOOPS FOR AREA NUMERICAL INTEGRATION +STFL 37 +C +C +C*** SET UP GAUSSIAN INTEGRATION CONSTANTS +STFL 38 +C +STFL 39 +C +STFL 40 +C +C*** ENTER LOOPS FOR AREA NUMERICAL INTEGRATION +STFL 41 +C +C +C*** ENTER LOOPS FOR AREA NUMERICAL INTEGRATION +STFL 42 +C +C +C*** ENTER LOOPS FOR AREA NUMERICAL INTEGRATION +STFL 43 +C +C +C*** ENTER LOOPS FOR AREA NUMERICAL INTEGRATION +STFL 44 +C +C*** ENTER LOOPS FOR AREA NUMERICAL INTEGRATION +STFL 45 +C +C*** ENTER LOOPS FOR AREA NUMERICAL INTEGRATION +STFL 46 +C +C*** ENTER LOOPS FOR AREA NUMERICAL INTEGRATION +STFL 47 +C +C*** ENTER LOOPS FOR AREA NUMERICAL INTEGRATION +STFL 48 +C +C*** ENTER LOOPS FOR AREA NUMERICAL INTEGRATION +STFL 49 +C +C*** ENTER LOOPS FOR AREA NUMERICAL INTEGRATION +STFL 50 +C +CALL GAUSSQ (NGAUS,POSGP,WEIGP) +STFL 51 +C +DO 50 IGAUS=1,NGAUS +STFL 52 +DO 50 JGAUS=1,NGAUS +STFL 53 +KGASP=KGASP+1 +STFL 54 +EXISP=POSGP(IGAUS) +STFL 55 +ETASP=POSGP(JGAUS) +STFL 56 +C +C*** EVALUATE THE SHAPE FUNCTIONS,ELEMENTAL AREA,ETC +STFL 57 +C +C*** EVALUATE THE SHAPE FUNCTIONS,ELEMENTAL AREA,ETC +STFL 58 +C +C*** EVALUATE THE SHAPE FUNCTIONS,ELEMENTAL AREA,ETC +STFL 59 +C +C*** EVALUATE THE SHAPE FUNCTIONS,ELEMENTAL AREA,ETC +STFL 60 +C +CALL SFR2 (DERIV,ETASP,EXISP,NNODE,SHAPE) +STFL 61 +CALL JACOB2 (CARTD,DERIV,DJACB,ELCOD,GPCOD,IELEM, +KGASP,NNODE,SHAPE) +STFL 62 +DAREA=DJACB*WEIGP(IGAUS)*WEIGP(JGAUS) +STFL 63 +C +C*** EVALUATE THE SHAPE FUNCTIONS,ELEMENTAL AREA,ETC +STFL 64 +C +C*** EVALUATE THE SHAPE FUNCTIONS,ELEMENTAL AREA,ETC +STFL 65 +C +C*** EVALUATE THE SHAPE FUNCTIONS,ELEMENTAL AREA,ETC +STFL 66 +C +C*** EVALUATE THE SHAPE FUNCTIONS,ELEMENTAL AREA,ETC +STFL 67 +C +C*** EVALUATE THE SHAPE FUNCTIONS,ELEMENTAL AREA,ETC +STFL 68 +C +C*** EVALUATE THE SHAPE FUNCTIONS,ELEMENTAL AREA,ETC +STFL 69 +C +C*** EVALUATE THE SHAPE FUNCTIONS,ELEMENTAL AREA,ETC +STFL 70 +C +C*** EVALUATE THE SHAPE FUNCTIONS,ELEMENTAL AREA,ETC +STFL 71 +C +C*** EVALUATE THE SHAPE FUNCTIONS,ELEMENTAL AREA,ETC +STFL 72 +C +C*** EVALUATE THE SHAPE FUNCTIONS,ELEMENTAL AREA,ETC +STFL 73 +C +C*** EVALUATE THE SHAPE FUNCTIONS,ELEMENTAL AREA,ETC +STFL 74 +C +C*** EVALUATE THE SHAPE FUNCTIONS,ELEMENTAL AREA,ETC +STFL 75 +C +C*** EVALUATE THE SHAPE FUNCTIONS,ELEMENTAL AREA,ETC +STFL 76 +C +C*** EVALUATE THE SHAPE FUNCTIONS,ELEMENTAL AREA,ETC +STFL 77 +C +C*** EVALUATE THE SHAPE FUNCTIONS,ELEMENTAL AREA,ETC +STFL 78 +C +C*** EVALUATE THE SHAPE FUNCTIONS,ELEMENTAL AREA,ETC +STFL 79 +C +C*** EVALUATE THE SHAPE FUNCTIONS,ELEMENTAL AREA,ETC +STFL 80 +``` + + + +```txt +50 CONTINUE +C +C*** EVALUATE PART OF STIFFNESS MATRIX +C ASSOCIATED WITH SHEAR DEFORMATION +C +KGASP=0 +NGAUM=NGAUS-1 +C +C*** ENTER LOOPS FOR AREA INTEGRATION +C +C +C*** SET UP GAUSSIAN INTEGRATION CONSTANTS +C +CALL GAUSSQ (NGAUM,POSGP,WEIGP) +DO 51 IGAUS=1,NGAUM +DO 51 JGAUS=1,NGAUM +KGASP=KGASP+1 +EXISP=POSGP(IGAUS) +ETASP=POSGP(JGAUS) +C +C*** EVALUATE THE SHAPE FUNCTIONS, ELEMENTAL AREA,ETC +C +CALL SFR2 (DERIV,ETASP,EXISP,NNODE,SHAPE) +CALL JACOB2 (CARTD,DERIV,DJACB,ELCOD,GPCOD,IELEM, +KGASP,NNODE,SHAPE) +DAREA=DJACB*WEIGP(IGAUS)*WEIGP(JGAUS) +C +C*** EVALUATE THE B AND DB MATRICES +C +CALL LAYMPA (DEPTH,DFLEX,DSHER,EPSTN,IINCS,KGAUS,LPROP, +MMATS,MTOTG,NCRIT,NLAPS,PROPS,STRSG,0) +C +C*** EVALUATE ELEMENT STIFFNESSES +C +DO 31 INODE=1,NNODE +CALL BMATPB (BFLEI,DUMMY,BSHEI,CARTD,INODE,SHAPE, +0, 0, 1) +DO 31 JNODE=INODE,NNODE +CALL BMATPB (BFLEJ,DUMMY,BSHEJ,CARTD,JNODE,SHAPE, +0, 0, 1) +31 CALL SUBMP (BSHEI,BSHEJ,DAREA,DSHER,ESTIF,INODE, +JNODE, 3, 2, 3) +51 CONTINUE +C +C*** CONSTRUCT THE LOWER TRIANGLE OF THE STIFFNESS MATRIX +C +DO 60 IEVAB=1,NEVAB +DO 60 JEVAB=IEVAB,NEVAB +60 ESTIF(JEVAB,IEVAB)=ESTIF(IEVAB,JEVAB) +C +C*** STORE THE STIFFNESS MATRIX,STRESS MATRIX AND SAMPLING POINT +C COORDINATES FOR EACH ELEMENT ON DISC FILE +C +WRITE(1) ESTIF +WRITE(3) GPCOD +70 CONTINUE +RETURN +END +STFL 81 +STFL 82 +STFL 83 +STFL 84 +STFL 85 +STFL 86 +STFL 87 +STFL 88 +STFL 89 +STFL 90 +STFL 91 +STFL 92 +STFL 93 +STFL 94 +STFL 95 +STFL 96 +STFL 97 +STFL 98 +STFL 99 +STFL 100 +STFL 101 +STFL 102 +STFL 103 +STFL 104 +STFL 105 +STFL 106 +STFL 107 +STFL 108 +STFL 109 +STFL 110 +STFL 111 +STFL 112 +STFL 113 +STFL 114 +STFL 115 +STFL 116 +STFL 117 +STFL 118 +STFL 119 +STFL 120 +STFL 121 +STFL 122 +STFL 123 +STFL 124 +STFL 125 +STFL 126 +STFL 127 +STFL 128 +STFL 129 +STFL 130 +STFL 131 +STFL 132 +STFL 133 +STFL 134 +STFL 135 +STFL 136 +STFL 137 +STFL 138 +STFL 139 +``` + +# 9.6.10 Subroutine STRMPA + +This subroutine evaluates the stresses within each layer. + + + +```csv +SUBROUTINE STRMPA (CARTD,CONST,DFLEX,DGRAD,DSHER,ELDIS,NNODE, STRL 1 +SHAPE,STRES,IFFLE,IFSHE) STRL 2 +C**********STRRL 3 +C STRL 4 +C*** EVALUATES STRESSES FOR MINDLIN PLATE STRL 5 +C STRL 6 +C**********STRRL 7 +DIMENSION CARTD(2,9),DFLEX(3,3),DGRAD(6),DSHER(2,2), STRL 8 +ELDIS(3,9),SHAPE(9),STRES(5) STRL 9 +C*** ZERO STRESS VECTOR STRL 10 +CALL VZERO (5,STRES) STRL 11 +C*** EVALUATE ROTATIONS AT GAUSS POINT, IF NEEDED STRL 12 +IF(IFSHE.EQ.0) GOTO 50 STRL 13 +XZROT=0.0 STRL 14 +YZROT=0.0 STRL 15 +DO 30 INODE=1,NNODE STRL 16 +XZROT=XZROT+SHAPE(INODE)*ELDIS(2,INODE) STRL 17 +30 YZROT=YZROT+SHAPE(INODE)*ELDIS(3,INODE) STRL 18 +C*** EVALUATE BENDING STRESS RESULTANTS STRL 19 +50 IF(IFFLE.EQ.0) GOTO 60 STRL 20 +EFLXX=-DGRAD(2)*CONST STRL 21 +EFLYY=-DGRAD(6)*CONST STRL 22 +EFLXY=-(DGRAD(3)+DGRAD(5))*CONST STRL 23 +STRES(1)=DFLEX(1,1)*EFLXX+DFLEX(1,2)*EFLYY STRL 24 +STRES(2)=DFLEX(2,1)*EFLXX+DFLEX(2,2)*EFLYY STRL 25 +STRES(3)=DFLEX(3,3)*EFLXY STRL 26 +C*** EVALUATE SHEAR STRESS RESULTANTS STRL 27 +60 IF(IFSHE.EQ.0) RETURN STRL 28 +ESHXX=DGRAD(1)-XZROT STRL 29 +ESHYY=DGRAD(4)-YZROT STRL 30 +STRES(4)=DSHER(1,1)*ESHXX STRL 31 +STRES(5)=DSHER(2,2)*ESHYY STRL 32 +RETURN STRL 33 +END STRL 34 +``` + +# 9.7 Examples + +To test the program, the elasto-plastic analysis of a simply supported plate is performed and 9 noded and Heterosis elements are used. The geometry, material properties of the plate are shown in Fig. 9.6. + +![](images/page-380_949931b2d11daf9bfc7dc0bcf8a47e8fa08c56f5adfa3422e63f364437764ab0.jpg) + +
+text_image + +y +L +x +L += 1.0, E = 10.92, ν = 0.3, t = 0.01, q = 1.0, σ₀ = 1600.0 +
+ +Fig. 9.6 Geometry and material properties of simply supported square plate. diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_039.md b/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_039.md new file mode 100644 index 00000000..515032dd --- /dev/null +++ b/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_039.md @@ -0,0 +1,266 @@ + + +Typical input for the nonlayered approach is given in Appendix IV together with lineprinter output of results. Figures 9.7 and 9.8 show the load displacement curves for both layered and nonlayered approaches. + +![](images/page-381_b3c2f09d0e369b0e1b8120d8c03abe56c39443e5d0fc5930acdcbccfcc9c41ba.jpg) + +
+line + +| wD / (MpL²) | 4-node element | 8-node element | 9-node element | heterosis element | +| ----------- | -------------- | -------------- | -------------- | ----------------- | +| 0 | 0 | 0 | 0 | 0 | +| 7 | 17.5 | 17.5 | 17.5 | 17.5 | +| 8 | 20.0 | 20.0 | 20.0 | 20.0 | +| 9 | 22.0 | 22.0 | 22.0 | 22.0 | +| 10 | 23.5 | 23.5 | 23.5 | 23.5 | +| 11 | 24.5 | 24.5 | 24.5 | 24.5 | +| 12 | 24.8 | 24.8 | 24.8 | 24.8 | +| 13 | 24.9 | 24.9 | 24.9 | 24.9 | +| 14 | 24.95 | 24.95 | 24.95 | 24.95 | +| 15 | 24.98 | 24.98 | 24.98 | 24.98 | +| 16 | 24.99 | 24.99 | 24.99 | 24.99 | +| 17 | 24.995 | 24.995 | 24.995 | 24.995 | +| 18 | 24.998 | 24.998 | 24.998 | 24.998 | +| 19 | 24.999 | 24.999 | 24.999 | 24.999 | +| 20 | 25.0 | 25.0 | 25.0 | 25.0 | +
+ +Fig. 9.7 Load displacement curves for nonlayered approach. +![](images/page-381_b700d1010a27f57feb7a76efafee3a8789cef5047e33dcdf6efdc9bf7ab49249.jpg) + +
+line + +| wD / (Mp L²) | 4-node element | 8-node element | 9-node element | heterosis element | +| ------------ | -------------- | -------------- | -------------- | ----------------- | +| 0 | 0 | 0 | 0 | 0 | +| 5 | 17 | 17 | 18 | 17 | +| 10 | 20 | 20 | 20 | 20 | +| 15 | 22 | 22 | 22 | 22 | +| 20 | 24 | 24 | 23 | 23 | +| 25 | 24 | 24 | 23 | 23 | +| 30 | 24 | 24 | 23 | 23 | +| 35 | 24 | 24 | 23 | 23 | +| 40 | 24 | 24 | 23 | 23 | +
+ +Fig. 9.8 Load displacement curves for layered approach. + + + +![](images/page-382_73cfd3bcb53394c280cf719f78885a56207e394c6592af1168750d74661a3a24.jpg) + +
+text_image + +y +L +-∞← +-∞ +x +
+ +Fig. 9.9 Infinite clamped plate strip under uniform lateral load q. + +# 9.8 Problems + +9.1 Consider the uniformly loaded, clamped plate shown in Fig. 9.9. Using programs MINDLIN and MINDLAY find the collapse load for the plate which has the following properties: + +Elastic modulus $E = 10000.0$ , Poisson's ratio $\nu = 0.3$ , thickness $t = 0.01$ , length $L = 1.00$ and yield stress $\sigma_0 = 1000.0$ . Check your solution using program PLANET. + +9.2 Use program MINDLIN to find the value of the uniformly distributed load intensity q at which yielding first occurs for rectangular, simply supported plates of aspect ratios 1.0, 1.2, 1.4, 1.6, 2.0 and 2.2. Assume a thickness/span ratio of 0.05 and locate also the position of first yielding. Compare your results with those of Turvey $^{(9)}$ for a Von Mises material. + +9.3 Modify program MINDLAY to allow for in-plane deformation of the plate mid-plane. Use a displacement pattern of the form + +$$ +u (x, y, z) = u _ {0} (x, y) - z \theta_ {x} (x, y) \tag {9.31} +$$ + +$$ +v (x, y, z) = v _ {0} (x, y) - z \theta_ {y} (x, y) \tag {9.32} +$$ + +in which $u_{0}$ and $v_{0}$ are the in-plane deflections of the plate mid-plane in the x and y directions respectively. + +9.4 Modify programs MINDLIN and MINDLAY to allow for an elastic Winkler foundation of modulus K. The appropriate virtual work term is + +$$ +\int_ {\Omega} \delta w K w d \Omega +$$ + +in which $\delta w$ is the virtual lateral displacement. + +9.5 Solve the beam problem in Example 5.1 of Chapter 5 using programs MINDLIN and MINDLAY. + +9.6 Develop a program for the nonlayered elastoplastic analysis of axisymmetric Mindlin plates using 2-node radial finite elements. The + + + +virtual work expression for an annular plate of internal and external radii $r_{0}$ and $r_{1}$ respectively is given as + +$$ +\begin{array}{l} 2 \pi \int_ {r _ {0}} ^ {r _ {1}} \left[ - \frac {d (\delta \theta)}{d r} M _ {r} - \frac {\delta \theta}{r} M _ {\theta} + \left(\frac {d (\delta w)}{d r} - \theta\right) Q \right] r d r \\ - 2 \pi \int_ {r _ {0}} ^ {r _ {1}} \delta w q r d r \tag {9.33} \\ \end{array} +$$ + +in which the radial bending moment $M_{r} = -D[d\theta/dr + \nu\theta/r]$ the circumferential bending moment $M_{\theta} = -D[\theta/r + \nu d\theta/dr]$ the shear force $Q = [Gt(dw/dr - \theta)]/1.2$ , $\theta$ is the normal rotation in the radial rz plane and w is the lateral displacement in the z direction. + +# 9.9 References + +1. HUGHES, T. J. R. and COHEN, M., The 'Heterosis' finite element for plate bending, Computers and Structures 9, 445–450 (1978). +2. BHAUMIK, A. K. and HANLEY, J. T., Elasto-plastic plate analysis by finite differences, J. Struct. Div. ASCE, 93, 575 (1967). +3. ARMEN, H., PIFKO, A. and LEVINE, H. S., A finite element method for the plastic bending analysis of structures, Proc. of the Second Conf. on Matrix Methods in Struct. Mech., Wright-Patterson Air Force Base, Dayton, Ohio, 1301-1339 (1968). +4. LOPEZ, L. A. and ANG, A. H. I., Flexural analysis of elastic-plastic rectangular plates, Civil Engineering Studies, Structural Research Series No. 305, University of Illinois (1966). +5. McNIECE, G. M. and KEMP, K. O., Comparison of finite element and unique limit analysis solutions for certain reinforced concrete slabs. Proc. Instn. Civ. Engrs. 43, 629–640 (1969). +6. BACKLUND, J., Mixed finite element analysis of elastic and elasto-plastic plates in bending, Chalmers University of Technology, Department of Structural Mechanics, Publication 71: 1, 30, Göteborg (1971). +7. WEGMULLER, A. W. and KOSTEM, C. N., Finite element analysis of elastic-plastic plates and eccentrically stiffened plates, Fritz Engineering Laboratory Report No. 376A4, Lehigh University, Bethlehem, Pennsylvania (1973). +8. HINTON, E. and OWEN, D. R. J., Finite Element Programming, Academic Press (1977). +9. TURVEY, G. J., First yield analysis of laterally loaded, rectangular, Levy plates with unsymmetric, side boundary conditions, Proc. Instn. Civ. Engrs., 65, 199–206 (1978). + + + + + +# Part III + + + + + +# Chapter 10 Explicit transient dynamic analysis + +Written in collaboration with D. K. Paul and N. Bicanic + +# 10.1 Introduction + +Earlier, in Parts I and II, we considered static (or pseudostatic) applications. However, many structures are subjected to time-varying loads such as impulse, blast, impact or earthquake loading. Here in Part III we consider finite element based methods for dealing with such problems. + +Although a form of mode-superposition has been adopted in nonlinear transient dynamic stress analysis, $^{(1)}$ it is general practice to use a time stepping procedure. Such direct integration schemes may be broadly classified as either explicit or implicit methods. + +In the present chapter, we consider the very popular and easily implemented, explicit, central difference scheme. During each time step, relatively little computational effort is required since no formal matrix factorisation is necessary. Unfortunately, the method is conditionally stable and very small time steps are often needed. + +In implicit schemes, a matrix factorisation is required but we can select an unconditionally stable implicit algorithm in which the time step length is governed by considerations of accuracy alone. In Chapter 11 we consider the Newmark family $^{(2)}$ of time stepping schemes. We then present a program for nonlinear transient dynamic stress analysis in which we may select any of the following algorithms: + +(i) an implicit solution +(ii) an explicit solution +(iii) a combined implicit/explicit solution + +The programs in Chapters 10 and 11 deal with plane stress, plane strain and axisymmetric applications using 4, 8 and 9-node, isoparametric quadrilaterals. Geometrically nonlinear behaviour is taken into account using a Total Lagrangian formulation. In Chapter 10 the material behaviour is assumed to be elasto-viscoplastic, whereas an elasto-plastic model is used in Chapter 11. Test examples are presented for both programs. + + + +# 10.2 Dynamic equilibrium equations + +For dynamic equilibrium of a body in motion we can use the Principle of Virtual Work to write the following equations at time station $t_{n}$ irrespective of material behaviour + +$$ +\begin{array}{l} \int_ {\Omega} \left[ \delta \boldsymbol {\epsilon} _ {n} \right] ^ {T} \boldsymbol {\sigma} _ {n} d \Omega - \int_ {\Omega} \left[ \delta \boldsymbol {u} _ {n} \right] ^ {T} \left[ \boldsymbol {b} _ {n} - \rho_ {n} \ddot {\boldsymbol {u}} _ {n} - c _ {n} \dot {\boldsymbol {u}} _ {n} \right] d \Omega \\ - \int_ {\Gamma_ {t}} \left[ \delta \boldsymbol {u} _ {n} \right] ^ {T} \boldsymbol {t} _ {n} d \Gamma = 0 \tag {10.1} \\ \end{array} +$$ + +where $\delta u_{n}$ is the vector of virtual displacements, $\delta \epsilon_{n}$ is the vector of associated virtual strains, $b_{n}$ is the vector of applied body forces, $t_{n}$ is the vector of surface tractions, $\sigma_{n}$ is the vector of stresses, $\rho_{n}$ is the mass density, $c_{n}$ is the damping parameter and a dot refers to differentiation with respect to time. The domain of interest $\Omega$ has two boundaries: $\Gamma_{t}$ on which boundary tractions $t_{n}$ are specified and $\Gamma_{u}$ on which displacements $u_{n}$ are specified. For plane stress, plane strain and axisymmetric problems all of these terms were defined in Chapter 6. + +Recall that in Chapter 6 we noted that, for a finite element representation, the displacements and strains and also their virtual counterparts are given by the relationships + +$$ +\pmb {u} _ {n} = \sum_ {i = 1} ^ {m} N _ {i} [ \pmb {d} _ {i} ] _ {n}, \qquad \delta \pmb {u} _ {n} = \sum_ {i = 1} ^ {m} N _ {i} [ \delta \pmb {d} _ {i} ] _ {n} \tag {10.2} +$$ + +$$ +\epsilon_ {n} = \sum_ {i = 1} ^ {m} B _ {i} [ d _ {i} ] _ {n}, \quad \delta \epsilon_ {n} = \sum_ {i = 1} ^ {m} B _ {i} [ \delta d _ {i} ] _ {n} \tag {10.3} +$$ + +where at time station $t_{n}$ for node i, $[d_{i}]_{n}$ is the vector of nodal displacements, $[\delta d_{i}]_{n}$ is the vector of virtual nodal variables, $N_{i} = N_{i}I_{2}$ is the matrix of global shape functions and $B_{i}$ is the global strain-displacement matrix. $^{\dagger}$ The total number of nodes is m. + +If (10.2) and (10.3) are substituted into (10.1), and if we note that the resulting equation is true for any set of virtual displacements $[\delta d]_{n}$ then we obtain for each node i the equations. + + + +$$ +[ p _ {i} ] _ {n} - [ f _ {B i} ] _ {n} + [ f _ {I i} ] _ {n} + [ f _ {D i} ] _ {n} - [ f _ {T i} ] _ {n} = 0 \tag {10.4} +$$ + +where the internal resisting forces are + +$$ +[ \pmb {p} _ {i} ] _ {n} = \int_ {\Omega} [ \pmb {B} _ {i} ] ^ {T} \pmb {\sigma} _ {n} d \Omega , \tag {10.5} +$$ + +the consistent forces for the applied body forces are + +$$ +[ \pmb {f} _ {B i} ] _ {n} = \int_ {\Omega} [ N _ {i} ] ^ {T} \pmb {b} _ {n} d \Omega , \tag {10.6} +$$ + +the inertia forces are + +$$ +\begin{array}{l} \left[ \boldsymbol {f} _ {I i} \right] _ {n} = \int_ {\Omega} \left[ \boldsymbol {N} _ {i} \right] ^ {T} \rho_ {n} \left[ \boldsymbol {N} _ {1}, \boldsymbol {N} _ {2}, \dots , \boldsymbol {N} _ {m} \right] d \Omega \left[ \begin{array}{c} \left[ \ddot {\boldsymbol {d}} _ {1} \right] _ {n} \\ \left[ \ddot {\boldsymbol {d}} _ {2} \right] _ {n} \\ \vdots \end{array} \right] \tag {10.7} \\ = \sum_ {j = 1} ^ {m} [ \boldsymbol {M} _ {i j} ] _ {n} [ \ddot {\boldsymbol {d}} _ {j} ] _ {n}, \quad \left\lfloor \quad [ \ddot {\boldsymbol {d}} _ {m} ] _ {n} \right. \\ \end{array} +$$ + +(N.B. $[M_{ij}]_{n}$ is a submatrix of the mass matrix $M_{n}$ ) The damping forces are + +$$ +\begin{array}{l} \left[ \boldsymbol {f} _ {D i} \right] _ {n} = \int_ {\Omega} \left[ \boldsymbol {N} _ {i} \right] ^ {T} c _ {n} \left[ \boldsymbol {N} _ {1}, \boldsymbol {N} _ {2}, \dots , \boldsymbol {N} _ {m} \right] d \Omega \\ = \sum_ {j = 1} ^ {m} \left[ \boldsymbol {C} _ {i j} \right] _ {n} \left[ \dot {\boldsymbol {d}} _ {j} \right] _ {n} \end{array} \left[ \begin{array}{c} \left[ \dot {\boldsymbol {d}} _ {1} \right] \\ \left[ \dot {\boldsymbol {d}} _ {2} \right] \\ \vdots \\ \left[ \dot {\boldsymbol {d}} _ {m} \right] \end{array} \right] \tag {10.8} +$$ + +(N.B. $[C_{ij}]_{n}$ is a submatrix of the damping matrix $C_{n}$ ) and the consistent forces for the traction boundary forces are + +$$ +[ f _ {T i} ] _ {n} = \int_ {\Gamma_ {r}} [ N _ {i} ] ^ {T} t _ {n} d \Gamma . \tag {10.9} +$$ + +If we use $C(0)$ isoparametric finite element representations we can evaluate contributions to (10.4) separately from each element and then assemble them into the appropriate vectors in (10.4). As noted in Chapter 6 the displacements can be expressed in the usual way as + +$$ +[ \boldsymbol {u} ^ {(e)} ] _ {n} = \sum_ {i = 1} ^ {r} N _ {i} ^ {(e)} [ \boldsymbol {d} _ {i} ^ {(e)} ] _ {n} \tag {10.10} +$$ + +where for local node i of element e, $N_{i}^{(e)} = N_{i}^{(e)}I_{2}$ is the local shape function matrix and $[d_{i}^{(e)}]_{n}$ is the vector of nodal displacements. As described in + + + +Chapter 6 we use 4, 8 and 9 noded isoparametric quadrilateral elements and therefore $r = 4, 8$ and 9 respectively for these cases. + +The strain displacement relationships are expressed as + +$$ +[ \epsilon^ {(e)} ] _ {n} = \sum_ {i = 1} ^ {r} B _ {i} ^ {(e)} [ d _ {i} ^ {(e)} ] _ {n} \tag {10.11} +$$ + +in which $\boldsymbol{B}_{t}^{(e)}$ is the local element strain matrix which has been defined for the various applications in Table 6.1. + +The discretised elemental volume is given as + +$$ +d \Omega^ {(e)} = h ^ {(e)} \det J ^ {(e)} d \xi d \eta \tag {10.12} +$$ + +in which $\det J^{(e)}$ is the determinant of the Jacobian matrix and $h^{(e)}$ is defined in Chapter 6. + +Thus the element contributions to the terms in (10.4) may be evaluated using numerical integration based on Gauss-Legendre product rules. These contributions now take the form + +$$ +[ p _ {i} ^ {(e)} ] _ {n} = \int_ {- 1} ^ {+ 1} \int_ {- 1} ^ {+ 1} [ B _ {i} ^ {(e)} ] ^ {T} \sigma_ {n} ^ {(e)} h ^ {(e)} \det J ^ {(e)} d \xi d \eta \tag {10.13} +$$ + +$$ +[ f _ {B i} ^ {(e)} ] _ {n} = \int_ {- 1} ^ {+ 1} \int_ {- 1} ^ {+ 1} [ N _ {i} ^ {(e)} ] ^ {T} b _ {n} h ^ {(e)} \det J ^ {(e)} d \xi d \eta \tag {10.14} +$$ + +$$ +\begin{array}{l} \left[ f _ {I t} ^ {(e)} \right] _ {n} = \int_ {- 1} ^ {+ 1} \int_ {- 1} ^ {+ 1} \left[ N _ {t} ^ {(e)} \right] ^ {T} \rho_ {n} ^ {(e)} \left[ N _ {1} ^ {(e)}, N _ {2} ^ {(e)}, \dots , N _ {r} ^ {(e)} \right] h ^ {(e)} \det J ^ {(e)} d \xi d \eta \left[ \begin{array}{c} \left[ \ddot {d} _ {1} ^ {(e)} \right] _ {n} \\ \cdot \\ \cdot \\ \left[ \ddot {d} _ {r} ^ {(e)} \right] _ {n} \end{array} \right] \\ = \sum_ {j = 1} ^ {r} \left[ M _ {i j} ^ {(e)} \right] _ {n} \left[ \ddot {d} _ {j} ^ {(e)} \right] _ {n} \tag {10.15} \\ \end{array} +$$ + +$$ +\left[ f _ {D i} ^ {(e)} \right] _ {n} = \int_ {- 1} ^ {+ 1} \int_ {- 1} ^ {+ 1} \left[ N _ {i} ^ {(e)} \right] ^ {T} c _ {n} ^ {(e)} \left[ N _ {1} ^ {(e)}, N _ {2} ^ {(e)}, \dots , N _ {r} ^ {(e)} \right] h ^ {(e)} \det J ^ {(e)} d \xi d \eta \left[ \begin{array}{c} \left[ \dot {d} _ {1} ^ {(e)} \right] _ {n} \\ \cdot \\ \cdot \\ \cdot \\ \left[ \dot {d} _ {r} ^ {(e)} \right] _ {n} \end{array} \right] +$$ + +$$ += \sum_ {j = 1} ^ {r} \left[ C _ {i j} ^ {(e)} \right] _ {n} \left[ \dot {\boldsymbol {d}} _ {j} ^ {(e)} \right] _ {n} \tag {10.16} +$$ + +$$ +[ f _ {T t ^ {(e)}} ] _ {n} = \int_ {\Gamma_ {t} ^ {(e)}} [ N _ {t ^ {(e)}} ] ^ {T} t _ {n} ^ {(e)} d \Gamma \tag {10.17} +$$ + +where $\Gamma_{t}^{(e)}$ (if it exists) is that part of $\Gamma_{t}$ which coincides with the boundary of element domain $\Omega^{(e)}$ . diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_040.md b/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_040.md new file mode 100644 index 00000000..4ef72f3c --- /dev/null +++ b/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_040.md @@ -0,0 +1,363 @@ + + +We will assume for simplicity that the mass and damping matrices do not vary with time. + +# 10.3 Modelling of nonlinearities + +# 10.3.1 Introduction + +Dynamic loading of structures often causes excursions of stresses well into the inelastic range and the influence of geometry changes on the response is also significant in many cases. Therefore both material and geometric nonlinear effects should be considered. + +Although material behaviour under dynamic loading is very complex and experimental information is scarce, for most structural materials, some general statements can be made. + +For example, it has frequently been demonstrated that the instantaneous yield stress is significantly influenced by the rate of straining. Also, the value of the elasticity modulus $E_{0}$ is found to be dependent on the strain rate. For structural materials with limited ductility, such as concrete or rock-like materials, the rate of straining can completely change the material response from elasto-plastic behaviour under low rates to brittle elastic behaviour under high rates of straining. For many structural materials there is still an urgent need for a better understanding of the observed phenomena and underlying microscopic behaviour. However, in attempting to perform an analysis of a dynamically-loaded engineering structure, we must look for an idealized material model, where possibly some compromises have to be made. Furthermore, the model parameters should readily be measurable and easily obtained from reliable experimental data. + +For transient dynamic analysis, an elasto-viscoplastic model, as developed in earlier chapters, presents a very good approximation of the true behaviour for many structural materials. The predominant phenomenon of variable instantaneous yield stress is adequately modelled. + +In the following, we shall develop the algorithm for the elasto-viscoplastic transient dynamic analysis of plane stress, plane strain and axisymmetric problems. The computer program DYNPAK will be documented and explained and finally, some illustrative examples are given. + +# 10.3.2 Material model + +Here we adopt the elasto-viscoplastic material model developed in Chapter 8, where the constitutive relationship is given in the form + +$$ +\begin{array}{l} \dot {\epsilon} _ {n} = [ \dot {\epsilon} _ {e} ] _ {n} + [ \dot {\epsilon} _ {v p} ] _ {n} \\ = [ D ] ^ {- 1} \dot {\sigma} _ {n}: \gamma \langle \Phi_ {n} (F) \rangle \frac {\dot {\epsilon} F}{\dot {\epsilon} \sigma_ {n}} \tag {10.18} \\ \end{array} +$$ + +where D is the elasticity matrix, $\gamma$ is the fluidity parameter, F is the yield + + + +function and $\dot{\epsilon}_{n}$ , $[\dot{\epsilon}_{e}]_{n}$ and $[\dot{\epsilon}_{vp}]_{n}$ denote the total, elastic and viscoplastic strain rates at time station $t_{n}$ . We also have the relationships + +$$ +\sigma_ {n} = D [ \epsilon_ {e} ] _ {n} +$$ + +$$ +\epsilon_ {n} = \left[ \epsilon_ {e} \right] _ {n} + \left[ \epsilon_ {v p} \right] _ {n} \tag {10.19} +$$ + +and + +$$ +\langle \Phi_ {n} (F) \rangle = 0 \quad \text { if yield has not occurred. } +$$ + +$$ += 1 \quad \text { if yield has occurred. } \tag {10.20} +$$ + +Thus we can rewrite the internal resisting forces as + +$$ +\boldsymbol {p} _ {n} = \int_ {\Omega} [ \boldsymbol {B} ] ^ {T} \boldsymbol {D} \left\{\epsilon_ {n} - \left[ \epsilon_ {v p} \right] _ {n} \right\} d \Omega \tag {10.21} +$$ + +The temporal discretization of the equations which govern viscoplastic straining is also based on the assumption that the relationship + +$$ +[ \dot {\epsilon} _ {v p} ] _ {n} = \gamma \langle \Phi_ {n} (F) \rangle \frac {\partial F}{\partial \sigma_ {n}} \tag {10.22} +$$ + +is known only for discrete time stations $\Delta t$ apart. The simplest, Euler, integration scheme will here be employed, i.e., + +$$ +[ \epsilon_ {v p} ] _ {n + 1} = [ \epsilon_ {v p} ] _ {n} + [ \dot {\epsilon} _ {v p} ] _ {n} \Delta t. \tag {10.23} +$$ + +The stability limit for the time increment $\Delta t$ , which depends on the specific form of the viscoplastic potential employed in the flow rule, has already been discussed in earlier chapters. + +When we adopt the central difference scheme and the viscoplastic material model that we have just described, the algorithm at a particular time station $t_{n}$ follows the sequence shown in Fig. 10.1. + +# 10.3.3 Geometric nonlinearity + +If we wish to cater for geometrically nonlinear elastic behaviour we can choose either a total or updated Lagrangian coordinate system. Here we choose a total Lagrangian coordinate system which coincides with the initial undeformed position of the body. $^{(3)}$ + +It transpires that, with the central difference scheme, the only changes required to account for geometrically nonlinear effects are + +(i) The modification of the strain-displacement matrix $B(d_n)$ , + +and + +(ii) The evaluation of the strains using a deformation Jacobian matrix $J_{D}(d_{n})$ . + + + +![](images/page-393_cf1633a78dd2a2fb70b11573d186998e6de386dc1528764ad16c61ef3c4b530b.jpg) + +
+flowchart + +```mermaid +graph TD + A["d_{n+1} = \left[ M + \frac{\Delta t}{2} C \right"]^{-1} \left\{ (\Delta t)^2["-p(d_n) + f_n"] + 2M d_n - \left[" M - \frac{\Delta t}{2} C \right"] d_{n-1} \right\} ] --> B["p(d_n) = \int_Ω [B(d_n)"]^T \sigma_n dΩ] + B --> C["ε_n = [B(d_n)"]d_n\n["ε_e"]_n = ε_n_-["ε_vp"]_n] + C --> D["σ_n = D[ε_e"]_n] + D --> E{Φ_n(F) > 0} + E -->|Yes| F["[\dot{\epsilon}_vp"]_n = γ<Φ_n(F) > \frac{\partial F}{\partial \sigma_n}] + F --> G["[\epsilon_vp"]_n + [ε_vp]_n + [\dot{\epsilon}_vp]_n Δt] + G --> H["[\epsilon_vp"]_{n+1} = [ε_vp]_n + [\dot{\epsilon}_vp]_n Δt] + H --> I["[\epsilon_vp"]_{n-1} = [ε_vp]_n + [D(d_n)]^T σ_n dΩ] + I --> J["ε_n = [B(d_n)"]d_n\n["ε_e"]_n = ε_n_-["ε_vp"]_n] + J --> K["σ_n = D[ε_e"]_n] + K --> L["No"] + L --> M["[\dot{\epsilon}_vp"]_n = 0] + M --> N["End"] + subgraph Time steps + A + B + C + D + E + F + G + H + I + end + subgraph Time steps + J + K + L + M + end +``` +
+ +Fig. 10.1 Algorithm for elasto viscoplastic straining during a time step. + +We will now describe briefly the relevant background theory. All vectors and matrices are given explicitly for the plane stress, plane strain and axisymmetric applications in Table 10.1. + +If the initial undeformed position of a particle of material is $x_{0}$ and the total displacement vector at time station $t_{n}$ is $u_{n}$ then the coordinates of the particle are + +$$ +\boldsymbol {x} _ {n} = \boldsymbol {x} _ {0} + \boldsymbol {u} _ {n} \tag {10.24} +$$ + +In a total Lagrangian formulation we use Green's strains. The matrix of Green's strains is given as + +$$ +\boldsymbol {E} _ {n} = \frac {1}{2} \left[ \left[ \boldsymbol {J} _ {D} \right] _ {n} ^ {T} \left[ \boldsymbol {J} _ {D} \right] _ {n} - \boldsymbol {I} \right] \tag {10.25} +$$ + + + +Table 10.1 Vectors and matrices used in a total Lagrangian formulation + +
VariablesPlane stress/strainAxisymmetric
Coordinates of particle in undeformed initial configuration $x = x_0$ $[x_0, y_0]^T$ $[r_0, z_0]^T$
Displacements $u_n$ $[u_n, v_n]^T$ $[u_n, w_n]^T$
Coordinates of particle in deformed configuration $x_n$ $[x_n, y_n]^T = [x_0 + u_n, y_0 + v_n]^T$ $[r_n, z_n]^T = [r_0 + u_n, z_0 + w_n]$
Vector of Green's strains $\epsilon_n$ $\begin{bmatrix} \epsilon_x \\ \epsilon_y \\ \gamma_{xy} \end{bmatrix}_n = \begin{bmatrix} \frac{\partial u_n}{\partial x} + \frac{1}{2} \left( \frac{\partial u_n}{\partial x} \right)^2 + \frac{1}{2} \left( \frac{\partial v_n}{\partial x} \right)^2 \\ \frac{\partial v_n}{\partial y} + \frac{1}{2} \left( \frac{\partial u_n}{\partial y} \right)^2 + \frac{1}{2} \left( \frac{\partial v_n}{\partial y} \right)^2 \\ \frac{\partial u_n}{\partial y} + \frac{\partial v_n}{\partial x} + \frac{\partial u_n}{\partial x} \frac{\partial u_n}{\partial y} + \frac{\partial v_n}{\partial x} \frac{\partial v_n}{\partial y} \end{bmatrix}$ $\begin{bmatrix} \epsilon_r \\ \epsilon_z \\ \gamma_{rz} \\ \epsilon_0 \end{bmatrix}_n = \begin{bmatrix} \frac{\partial u_n}{\partial r} + \frac{1}{2} \left( \frac{\partial u_n}{\partial r} \right)^2 + \frac{1}{2} \left( \frac{\partial w_n}{\partial r} \right)^2 \\ \frac{\partial w_n}{\partial z} + \frac{1}{2} \left( \frac{\partial u_n}{\partial z} \right)^2 + \frac{1}{2} \left( \frac{\partial w_n}{\partial z} \right)^2 \\ \frac{\partial u_n}{\partial z} + \frac{\partial w_n}{\partial r} + \frac{\partial u_n}{\partial r} \frac{\partial u_n}{\partial z} + \frac{\partial w_n}{\partial r} \frac{\partial w_n}{\partial z} \\ \frac{u_n}{r} + \frac{1}{2} \left( \frac{u_n}{r} \right)^2 \end{bmatrix}$
Deformation Jacobian matrix $J_D(u_n) = [J_D]_n$ $\begin{bmatrix} \frac{\partial x_n}{\partial x} & \frac{\partial x_n}{\partial y} \\ \frac{\partial y_n}{\partial x} & \frac{\partial y_n}{\partial y} \end{bmatrix}$ $\begin{bmatrix} \frac{\partial r_n}{\partial r} & \frac{\partial r_n}{\partial z} \\ \frac{\partial z_n}{\partial r} & \frac{\partial z_n}{\partial z} \end{bmatrix}$
Matrix of Green's strains $E_n = \frac{1}{2} \{ [J_D]_n^T [J_D]_n - I \}$ $\begin{bmatrix} \epsilon_{xx} & \epsilon_{xy} \\ \epsilon_{yx} & \epsilon_{yy} \end{bmatrix}_n$ $\begin{bmatrix} \epsilon_{rr} & \epsilon_{rz} \\ \epsilon_{zr} & \epsilon_{zz} \end{bmatrix}_n$
Linear strains $[\epsilon_L]_n$ $\left[ \frac{\partial u_n}{\partial x}, \frac{\partial v_n}{\partial y}, \left( \frac{\partial u_n}{\partial y} + \frac{\partial v_n}{\partial x} \right) \right]^T$ $\left[ \frac{\partial u_n}{\partial r}, \frac{\partial w_n}{\partial r}, \frac{\partial u_n}{\partial z} + \frac{\partial w_n}{\partial r}, \frac{u_n}{r} \right]^T$
+ + + +Table 10.1 (Cont.) + +
VariablePlane stress/strainAxisymmetric
Nonlinear strains $[\epsilon_{NL}]_n = \frac{1}{2}[A_\theta]_n \theta_n$ where $[A_\theta]_n$ is $\begin{bmatrix} \frac{\partial u_n}{\partial x} & \frac{\partial v_n}{\partial x} & 0 & 0 \\ 0 & 0 & \frac{\partial u_n}{\partial y} & \frac{\partial v_n}{\partial y} \\ \frac{\partial u_n}{\partial y} & \frac{\partial v_n}{\partial y} & \frac{\partial u_n}{\partial x} & \frac{\partial v_n}{\partial x} \end{bmatrix}$ $\begin{bmatrix} \frac{\partial u_n}{\partial r} & \frac{\partial w_n}{\partial r} & 0 & 0 & 0 \\ 0 & 0 & \frac{\partial u_n}{\partial z} & \frac{\partial w_n}{\partial z} & 0 \\ \frac{\partial u_n}{\partial z} & \frac{\partial w_n}{\partial z} & \frac{\partial u_n}{\partial r} & \frac{\partial w_n}{\partial r} & 0 \\ 0 & 0 & 0 & 0 & \frac{u_n}{r} \end{bmatrix}$
and displacement gradients $\theta_n$ $\begin{bmatrix} \frac{\partial u_n}{\partial x} & 0 & \frac{\partial u_n}{\partial x} \\ \frac{\partial v_n}{\partial x} & 0 & \frac{\partial v_n}{\partial x} \\ 0 & \frac{\partial u_n}{\partial y} & \frac{\partial u_n}{\partial y} \\ 0 & \frac{\partial v_n}{\partial y} & \frac{\partial v_n}{\partial y} \end{bmatrix}$ $\begin{bmatrix} \frac{\partial u_n}{\partial r} & 0 & \frac{\partial u_n}{\partial r} & 0 \\ \frac{\partial w_n}{\partial r} & 0 & \frac{\partial w_n}{\partial r} & 0 \\ 0 & \frac{\partial u_n}{\partial z} & \frac{\partial u_n}{\partial z} & 0 \\ 0 & \frac{\partial w_n}{\partial z} & \frac{\partial w_n}{\partial z} & 0 \\ 0 & 0 & 0 & \frac{u_n}{r} \end{bmatrix}$
Elastic Piola-Kirchoff stresses $\sigma_n = D_n \epsilon_n$ $[\sigma_x, \sigma_y, \tau_{xy}]_n^T$ $[\sigma_r, \sigma_z, \tau_{rz}, \sigma_\theta]_n^T$
+ + + +where $[J_D]_n$ is the deformation Jacobian matrix at time station $t_n$ . + +The Green's strains can be written as + +$$ +\epsilon_ {n} = \left[ \epsilon_ {L} \right] _ {n} + \left[ \epsilon_ {N L} \right] _ {n} \tag {10.26} +$$ + +where $[\epsilon_{L}]_{n}$ are the linear strains given earlier in Chapter 6 and $[\epsilon_{NL}]_{n}$ , the nonlinear strain terms are given as + +$$ +[ \epsilon_ {N L} ] _ {n} = \frac {1}{2} [ A _ {\theta} ] _ {n} \theta_ {n}. \tag {10.27} +$$ + +For a set of virtual displacements, the corresponding virtual Green's strains are given as + +$$ +[ \delta \epsilon ] _ {n} = [ \delta \epsilon_ {L} ] _ {n} + [ A _ {\theta} ] _ {n} \delta \theta_ {n}. \tag {10.28} +$$ + +Thus the virtual work statement of (10.1) can be rewritten as + +$$ +\begin{array}{l} \int_ {\Omega} \left[ \delta \boldsymbol {\epsilon} _ {n} \right] ^ {T} \boldsymbol {\sigma} _ {n} d \Omega - \int_ {\Omega} \left[ \delta \boldsymbol {u} _ {n} \right] ^ {T} \left[ \boldsymbol {b} _ {n} - \rho \dot {\boldsymbol {u}} _ {n} - c \dot {\boldsymbol {u}} _ {n} \right] d \Omega \\ - \int_ {\Gamma_ {t}} \left[ \delta \boldsymbol {u} _ {n} \right] ^ {T} \boldsymbol {t} _ {n} d \Gamma = 0 \tag {10.29} \\ \end{array} +$$ + +where $\sigma_{n}$ are the Piola–Kirchhoff stresses. + +As mentioned earlier, all relevant terms are given in Table 10.1. + +If we adopt the finite element discretization scheme described earlier, then the displacement gradients $\theta_{n}$ are given in terms of the nodal displacements $[d_{i}]_{n}$ by the linear relation + +$$ +\boldsymbol {\theta} _ {n} = \sum_ {i = 1} ^ {m} \boldsymbol {G} _ {i} [ \boldsymbol {d} _ {i} ] _ {n} \tag {10.30} +$$ + +where $G_{i}$ contains Cartesian shape function derivatives as indicated in Table 10.2 for the various applications. + +Similarly we have + +$$ +\delta \theta_ {n} = \sum_ {i = 1} ^ {m} G _ {i} [ \delta d _ {i} ] _ {n}. \tag {10.31} +$$ + +The linear strain-displacement relationship can be expressed as + +$$ +[ \boldsymbol {\epsilon} _ {L} ] _ {n} = \sum_ {i = 1} ^ {m} [ \boldsymbol {B} _ {L i} ] _ {n} [ \boldsymbol {d} _ {i} ] _ {n} \tag {10.32} +$$ + +where $[B_{Li}]_{n}$ is the linear strain displacement matrix introduced earlier. + + + +Table 10.2 The nonlinear strain displacement matrix evaluation in a total Lagrangian finite element formulation + +
VariablePlane stress/strainAxisymmetric
Strain displacement matrix associated with node i $[B_i]_n = [B_{Li}]_n + [A_0]_n G_i$ $\begin{bmatrix} \frac{\partial x_n}{\partial x} & \frac{\partial N_i}{\partial x} & \frac{\partial y_n}{\partial x} & \frac{\partial N_i}{\partial x} \\ \frac{\partial x_n}{\partial y} & \frac{\partial N_i}{\partial y} & \frac{\partial y_n}{\partial y} & \frac{\partial N_i}{\partial y} \\ \left( \frac{\partial x_n}{\partial y} \frac{\partial N_i}{\partial x} + \frac{\partial x_n}{\partial x} \frac{\partial N_i}{\partial y} \right) & \left( \frac{\partial y_n}{\partial y} \frac{\partial N_i}{\partial x} + \frac{\partial y_n}{\partial x} \frac{\partial N_i}{\partial y} \right) \end{bmatrix}$ $\begin{bmatrix} \frac{\partial r_n}{\partial r} & \frac{\partial N_i}{\partial r} & \frac{\partial z_n}{\partial r} & \frac{\partial N_i}{\partial r} \\ \frac{\partial r_n}{\partial z} & \frac{\partial N_i}{\partial z} & \frac{\partial z_n}{\partial z} & \frac{\partial N_i}{\partial z} \\ \left( \frac{\partial r_n}{\partial z} \frac{\partial N_i}{\partial r} + \frac{\partial r_n}{\partial r} \frac{\partial N_i}{\partial z} \right) & \frac{\partial z_n}{\partial z} \frac{\partial N_i}{\partial r} + \frac{\partial z_n}{\partial r} \frac{\partial N_i}{\partial z} \\ \left( \frac{r_n}{r} \right) \frac{N_i}{r} & 0 \end{bmatrix}$
where $G_i$ is $\begin{bmatrix} \frac{\partial N_i}{\partial x} & 0 & \frac{\partial N_i}{\partial y} & 0 \\ 0 & \frac{\partial N_i}{\partial x} & 0 & \frac{\partial N_i}{\partial y} \end{bmatrix}$ $\begin{bmatrix} \frac{\partial N_i}{\partial r} & 0 & \frac{\partial N_i}{\partial z} & 0 & \frac{N_i}{r} \\ 0 & \frac{\partial N_i}{\partial r} & 0 & \frac{\partial N_i}{\partial z} & 0 \end{bmatrix}$
+ + + +Similarly, we have + +$$ +[ \delta \epsilon_ {N L} ] _ {n} = \sum_ {i = 1} ^ {m} [ B _ {N L i} ] _ {n} [ \delta d _ {i} ] _ {n} \tag {10.33} +$$ + +The components of the vector of Green's strains $\epsilon_{n}$ can be written as + +$$ +\epsilon_ {n} = \sum_ {i = 1} ^ {m} \left[ \left[ B _ {L i} \right] _ {n} + \frac {1}{2} \left[ B _ {N L i} \right] _ {n} \right] \left[ d _ {i} \right] _ {n} \tag {10.34} +$$ + +where the nonlinear strain-displacement matrix $[B_{NLt}]_n$ is given as + +$$ +[ \boldsymbol {B} _ {N L i} ] _ {n} = [ \boldsymbol {A} _ {\theta} ] _ {n} \boldsymbol {G} _ {i}. \tag {10.35} +$$ + +Furthermore it can be shown that the virtual strains can be expressed as + +$$ +\delta \epsilon_ {n} = \sum_ {i = 1} ^ {m} [ B _ {i} ] _ {n} [ \delta d _ {i} ] _ {n} \tag {10.36} +$$ + +where + +$$ +[ \pmb {B} _ {i} ] _ {n} = [ \pmb {B} _ {L i} ] _ {n} + [ \pmb {B} _ {N L i} ] _ {n} +$$ + +is given in Table 10.2 for the various applications. + +If we substitute for $\delta \epsilon_{n}$ and $\delta d_{n}$ in (10.29) and note that the result is true for arbitrary virtual displacements, then we obtain an expression which is identical to (10.4). In the present case we only need to remember that $[B_i]_n$ is defined by (10.36). + +We again note that contributions to $(10.4)$ from each element can be obtained separately and assembled appropriately. + +Note that we now may evaluate $[p_{i}]_{n}$ as + +$$ +\int_ {\Omega} [ B _ {i} ] _ {n} ^ {T} \sigma_ {n} d \Omega \quad \text { rather than } \quad \int_ {\Omega} [ B _ {i} ] ^ {T} \sigma_ {n} d \Omega +$$ + +where $[B_i]_n$ is given by (10.36). + +# 10.4 Explicit time integration scheme + +# 10.4.1 Central difference approximation + +We can write the equations (10.4) in matrix form so that at time station $t_n$ we have + +$$ +\boldsymbol {M} \ddot {\boldsymbol {d}} _ {n} + \boldsymbol {C} \dot {\boldsymbol {d}} _ {n} + \boldsymbol {p} _ {n} = \boldsymbol {f} _ {n} \tag {10.37} +$$ + +\- Note that the body force term $-\mathbf{M}\ddot{\mathbf{u}}_g$ , due to seismic excitation, is included in the body forces which are taken into account in $\mathbf{f}_n$ . Note also that $\mathbf{M}$ and $\mathbf{C}$ may be assembled from the element mass matrices $\mathbf{M}^{(e)}$ and damping matrices $\mathbf{C}^{(e)}$ . + + + +where M and C are the global mass and damping matrices respectively, $p_{n}$ is the global vector of internal resisting nodal forces, $f_{n}$ is the vector of consistent nodal forces for the applied body and surfaces traction forces grouped together, $\ddot{d}_{n}$ is the global vector of nodal accelerations and $\dot{d}_{n}$ is the global vector of nodal velocities. + +So far, only spatial discretization has been introduced. We now employ a temporal discretization of the dynamic equilibrium equations by approximating the accelerations and velocities using finite difference expressions. + +In particular we adopt a central difference approximation $^{(2)}$ so that the accelerations can be written as + +$$ +\ddot {\boldsymbol {d}} _ {n} \simeq \boldsymbol {a} _ {n} = \frac {1}{(\Delta t) ^ {2}} \left\{\boldsymbol {d} _ {n + 1} - 2 \boldsymbol {d} _ {n} + \boldsymbol {d} _ {n - 1} \right\} \tag {10.38} +$$ + +and the velocities are written as + +$$ +\dot {\boldsymbol {d}} _ {n} \simeq \boldsymbol {v} _ {n} = \frac {1}{2 \Delta t} \left\{\boldsymbol {d} _ {n - 1} - \boldsymbol {d} _ {n - 1} \right\} \tag {10.39} +$$ + +in which $\Delta t$ is the time step or interval so that we are sampling the displacements at time stations $t_n - \Delta t$ , $t_n$ and $t_n + \Delta t$ . If we substitute (10.38) and (10.39) into (10.37) we obtain + +$$ +M \left\{\frac {d _ {n + 1} - 2 d _ {n} - d _ {n - 1}}{(\Delta t) ^ {2}} \right\} - C \left\{\frac {d _ {n + 1} - d _ {n - 1}}{2 \Delta t} \right\} - p _ {n} = f _ {n} \tag {10.40} +$$ + +which can be rearranged to give + +$$ +\begin{array}{l} \boldsymbol {d} _ {n + 1} = \left[ \boldsymbol {M} + \frac {\Delta t}{2} \boldsymbol {C} \right] ^ {- 1} \\ \times \left\{(\Delta t) ^ {2} \left[ - p _ {n} + f _ {n} \right] + 2 M d _ {n} - \left[ M - \frac {\Delta t}{2} C \right] d _ {n - 1} \right\}. \tag {10.41} \\ \end{array} +$$ + +Thus we have + +$$ +\boldsymbol {d} _ {n + 1} = \boldsymbol {g} \left(\boldsymbol {d} _ {n}, \boldsymbol {d} _ {n - 1}\right). \tag {10.42} +$$ + +In other words the displacements at time station $t_{n} = \Delta t$ are given explicitly in terms of the displacements at time stations $t_{n}$ and $t_{n} = \Delta t$ . + +If the mass matrix M and the damping matrix C are diagonal then the solution of (10.41) becomes trivial and we have for plane stress and plane strain applications the following equations: + +$$ +\begin{array}{l} \left(d _ {u i}\right) _ {n + 1} = \left(m _ {i i} + \frac {\Delta t}{2} C _ {i i}\right) ^ {- 1} \left[ (\Delta t) ^ {2} \left\{- \left(p _ {u i}\right) _ {n} \dots \left(f _ {u i}\right) _ {n} \right\} \right. \\ \left. + 2 m _ {i i} \left(d _ {u i}\right) _ {n} - \left(m _ {i i} - \frac {\Delta t}{2} c _ {i i}\right) \left(d _ {u i}\right) _ {n - 1} \right] \tag {10.43} \\ \end{array} +$$ + + + +and + +$$ +\begin{array}{l} (d _ {v i}) _ {n + 1} = \left(m _ {i i} + \frac {\Delta t}{2} c _ {i i}\right) ^ {- 1} \left[ (\Delta t) ^ {2} \{- (p _ {v i}) _ {n} + (f _ {v i}) _ {n} \} \right. \\ \left. + 2 m _ {i i} \left(d _ {v i}\right) _ {n} - \left(m _ {i i} - \frac {\Delta t}{2} c _ {i i}\right) \left(d _ {v i}\right) _ {n - 1} \right] \tag {10.44} \\ \end{array} +$$ + +in which at node i, $d_{ut}$ and $d_{vi}$ are the u and v displacement components in the x and y directions, $f_{ut}$ and $f_{vt}$ are the components of the applied nodal forces in the x and y directions, $p_{ut}$ and $p_{vt}$ are the internal resisting nodal forces in the x and y directions and $m_{it}$ and $c_{it}$ are the diagonal terms of the mass and damping matrices. For axisymmetric problems replace v by w. + +From (10.43) and (10.44) we see that for each displacement degree of freedom at time $t_{n}+\Delta t$ we have a separate equation involving information regarding the degree of freedom at times $t_{n}$ and $t_{n}-\Delta t$ . No matrix factorisation or sophisticated equation solving is therefore necessary. + +# 10.4.2 Starting algorithm + +As we have seen the governing equilibrium equation at time station $t_{n} + \Delta t$ in the central difference method involves information at the two previous time stations $t_{n}$ and $t_{n} - \Delta t$ . A starting algorithm is therefore necessary and from the initial conditions the values $d(0 - \Delta t)$ may be obtained. We have from (10.39) the condition that + +$$ +\dot {\boldsymbol {d}} (0) \simeq \boldsymbol {v} (0) = \frac {\boldsymbol {d} (0 + \Delta t) - \boldsymbol {d} (0 - \Delta t)}{2 \Delta t} \tag {10.45} +$$ + +or + +$$ +\boldsymbol {d} (0 - \Delta t) = - 2 \Delta t v (0) + \boldsymbol {d} (0 + \Delta t). +$$ + +If this approximation is substituted in (10.43) then we can write the expression + +$$ +\begin{array}{l} \left(d _ {u i}\right) _ {1} = \left(m _ {i i} + \frac {\Delta t}{2} c _ {i i}\right) ^ {- 1} \left[ (\Delta t) ^ {2} \{- \left(p _ {u i}\right) _ {0} + \left(f _ {u i}\right) _ {0} \} \right. \\ \left. + 2 m _ {i i} \left(d _ {u i}\right) _ {0} - \left(m _ {i i} - \frac {\Delta^ {t}}{2} c _ {i i}\right) \left\{- 2 \Delta t \left(\dot {d} _ {u i}\right) _ {0} + d _ {u i}\right) _ {1} \right\} \Bigg ] \tag {10.46} \\ \end{array} +$$ + +or + +$$ +(d _ {u i}) _ {1} = \frac {(\Delta t) ^ {2}}{2 m _ {i i}} \{- (p _ {u i}) _ {0} + (f _ {u i}) _ {0} \} + (\dot {d} _ {u i}) _ {0} + B \Delta t (d _ {u i}) _ {0} +$$ + +where + +$$ +B = 1 - \frac {c _ {i i} \Delta t}{2 m _ {i i}}. +$$ + +# 10.4.3 Damping + +Very limited information is available on damping in linear solid mechanics problems and there is even less data available for damping in nonlinear situations. It is therefore customary to assume that the damping diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_041.md b/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_041.md new file mode 100644 index 00000000..e0ad57b4 --- /dev/null +++ b/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_041.md @@ -0,0 +1,1446 @@ + + +matrix is proportional to the mass and stiffness matrix. This is known as Rayleigh damping and we have + +$$ +\boldsymbol {C} = \alpha \boldsymbol {M} + \beta \boldsymbol {K} \tag {10.47} +$$ + +In the central difference method we can make the approximation that $\beta = 0$ so that + +$$ +\boldsymbol {C} = \alpha \boldsymbol {M} \tag {10.48} +$$ + +or + +$$ +c _ {i i} = a m _ {i i} +$$ + +where + +$$ +a = 2 \xi_ {r} \omega_ {r} +$$ + +in which $\xi_{r}$ and $\omega_{r}$ are the damping factor and circular frequency for the $r^{th}$ mode. This modelling of damping is rather poor since a is fixed for all modes of vibration. Thus if we take r = 1 then the higher modes will be less damped whereas the opposite would be more desirable. This is the price we pay for an otherwise convenient and efficient solution. + +# 10.5 Critical time step + +In explicit and implicit time integration schemes the selection of an appropriate time step is crucially important. Small time steps are required for accurate and stable solutions whereas for reasons of economy we would prefer large time steps. The analysis of the stability and accuracy characteristics $^{(2)}$ allows us to decide on a suitable time step for the various time stepping schemes. On this basis for the conditionally stable, central difference scheme, the stability considerations are of prime importance and the time step length is limited by the expression + +$$ +\Delta t \leqslant \frac {2}{\omega_ {\max}} \tag {10.49} +$$ + +where $\omega_{max}$ is the highest circular frequency of the finite element mesh. This severe time step limit, required for stability, ensures accuracy in practically all modes of vibration. Providing that $\omega_{max}$ represents the maximum nonlinear frequency, (10.49) holds for nonlinear problems. The estimate of the critical time step for conditionally stable schemes apparently necessitates the solution of the eigenvalue problem for the whole system. This is not so. The bound on the highest eigenvalue can be simply obtained by the consideration of an individual element. This is established by an important theorem proposed by Irons $^{(4)}$ which proves that the highest system eigenvalue must always be less than the highest eigenvalue of the individual elements. This allows a very easy estimate of critical time steps (by the above theorem) which will err on the safe side. To avoid the exact evaluation of the highest + + + +finite element mesh frequency approximate expressions are usually employed. The most common form for plane strain is + +$$ +\Delta t \leqslant \mu L \left(\frac {\rho (1 + \nu) (1 - 2 \nu)}{E (1 - \nu)}\right) ^ {1 / 2} \tag {10.50} +$$ + +where L is the smallest length between any two nodes and $\mu$ is a coefficient dependent on the type of element employed. $^{(5)}$ For problems in which many time steps are used it may be beneficial to calculate the exact highest linear frequency of the finite element mesh prior to the time stepping analysis. + +Recall that when an elasto-viscoplastic model is adopted care must be taken not to exceed the critical time step for the Euler scheme in evaluating the viscoplastic strains. (See Section 8.3). + +# 10.6 Program DYNPAK + +# 10.6.1 Overall structure of DYNPAK + +We now present program DYNPAK for the elasto-viscoplastic or geometrically nonlinear, transient dynamic analysis of plane stress, plane strain and axisymmetric problems. The basic structure of the program is shown in Fig. 10.2. Many of the subroutines used in DYNPAK have already been described in earlier chapters. + +The algorithm adopted has been presented schematically in Fig. 10.1. The program is written in a dynamically dimensioned form. Efficiency has sometimes been sacrificed for clarity of presentation and the reader may consider ways of making the program more efficient when reviewing this chapter. + +Isoparametric 4, 8 and 9-noded quadrilateral elements are included in the program. A special mass lumping procedure $^{(6)}$ has been adopted and separate Gauss–Legendre rules may be adopted in the evaluation of the stiffness and the lumped mass matrices. + +Impact and seismic loading can easily be specified. Material nonlinearity is based on elasto-viscoplastic models with Von Mises, Tresca, Mohr-Coulomb or Drucker-Prager yield criteria with isotropic hardening. A total Lagrangian formulation is used to allow for the geometric nonlinear behaviour. + +Subroutines GAUSSQ, SFR2 and JACOB2 have already been dealt with and only the remaining routines will be listed and described. + +# 10.6.2 Master routine DYNPAK + +The master routine organises the calling of the main routines as outlined in Fig. 10.2. In subroutine CONTOL the variables required for dynamic dimensioning are read and a check is made on the maximum available dimensions. Note that the values given in the DIMENSION statement in + + + +![](images/page-403_016a82b195acd55b89f16ccfae044bcc87d3595172e60184b80669e135f061e5.jpg) + +
+flowchart + +```mermaid +graph TD + A["MASTER DYNPAK"] --> B["CONTOL"] + A --> C["INPUTD"] + C --> D["NODXYR"] + C --> E["GAUSSQ"] + A --> F["INTIME"] + A --> G["PREVOS"] + A --> H["LOADPLf(t)"] + H --> I["MODPS"] + H --> J["SFR2"] + H --> K["JACOB2"] + A --> L["LUMASS"] + L --> M["GAUSSQ"] + L --> N["SFR2"] + L --> O["JACOB2"] + A --> P["RESVPL"] + P --> Q["MODPS"] + P --> R["SFR2"] + P --> S["JACOB2"] + P --> T["JACOBD"] + A --> U["EXPLIT"] + U --> V["FUNCTS"] + U --> W["FUNCTA"] + U --> X["FUNCTA"] + A --> Y["RESVPLp(t)"] + Y --> Z["FWVVP"] + Y --> AA["YIELDF"] + A --> AB["OUTDYN"] + A --> AC["DO LOOP 1, TO NSTEP"] +``` +
+ +Fig. 10.2 Flow diagram for program DYNPAK. + + + +DYNPAK should agree with the values specified in CONTOL. Subroutines INPUTD, INTIME and PREVOS read the mesh data, the time integration data and data for the previous state of the structure. Subroutines LUMASS and LOADPL generate the lumped mass and applied force vectors respectively. FIXITY deals with fixed boundary nodes. In the time step do loop, EXPLIT performs the direct time integration and RESVPL calculates + +$$ +\int_ {\Omega} [ \boldsymbol {B} ] _ {n} ^ {T} \boldsymbol {\sigma} _ {n} d \Omega +$$ + +when an elasto-viscoplastic material model is adopted. + +In this version of DYNPAK it should be noted that the maximum dimensions imply that we can solve problems with no more than 50 elements, 200 nodal points, 50 fixed boundary nodes and 600 acceleration ordinates. + +Of course, larger problems can be accommodated by increasing the values in CONTOL and also the appropriate dimensions in the DIMENSION statement in the main routine DYNPAK. +
PROGRAM DYNPAK (INPUT, TAPE5=INPUT, TAPE4, TAPE10, TAPE12, TAPE3, OUTPUT, TAPE6=OUTPUT, TAPE7, TAPE11, TAPE13)DYNK1
CDYNAMIC TRANSIENT ELASTO - VISCOPLASTIC PROGRAMDYNK2
CDYNAMIC TRANSIENT ELASTO - VISCOPLASTIC PROGRAMDYNK3
CDYNAMIC TRANSIENT ELASTO - VISCOPLASTIC PROGRAMDYNK4
CDYNAMIC TRANSIENT ELASTO - VISCOPLASTIC PROGRAMDYNK5
CDYNAMIC TRANSIENT ELASTO - VISCOPLASTIC PROGRAMDYNK6
CDYNAMIC TRANSIENT ELASTO - VISCOPLASTIC PROGRAMDYNK7
CDIMENSION ACCEH(600), ACCEV(600), COORD(200,2), DISPL(400), DYNKDYNK8
CFORCE(400), IFPRE(2,200), LNODS(50,9), MATNO(50), DYNKDYNK9
CINTGR(50), NPRQD(10), NGRQS(10), POSGP(4), DYNKDYNK10
CPROPS(10,13), RESID(400), RLOAD(50,18), STRIN(4,450), DYNKDYNK11
CSTRSG(4,450), TDISP(400), TEMPE(100), VELOC(400), DYNKDYNK12
CVISTN(4,450), VIVEL(5,450), WEIGP(4), YMASS(400)DYNK13
CDYNKDYNK14
CCALLCONTOL(NDOFN, NELEM, NMATS, NPOIN)DYNK15
CDYNKDYNK16
CCALLINPUTD(COORD, IFPRE, LNODS, MATNO, NCONM, NCRIT, DYNKDYNK17
CDYNKDYNK18
CCALLINTIME(NDIME, NDOFN, NELEM, NGAUM, NGAUS, NLAPS, DYNKDYNK19
CDYNKDYNK20
CCALLINTIME(NMATS, NNODE, NPOIN, NPREV, NSTRE, NTYPE, DYNKDYNK21
CDYNKDYNK22
CCALLINTIME(POSGP, PROPS, WEIGP)DYNK23
CDYNKDYNK24
CCALLINTIME(AALFA, ACCEH, ACCEV, AFACT, AZERO, BEETA, DYNKDYNK25
CDYNKDYNK26
CCALLINTIME(BZERO, DELTA, DTIME, DTEND, GAAMA, IFIXD, DYNKDYNK27
CDYNKDYNK28
CCALLPREVOSIFUNC, INTGR, KSTEP, MITER, NDOFN, NELEM, DYNKDYNK29
CDYNKDYNK30
CCALLLOADPLNGRQS, NOUTD, NOUTP, NPOIN, NPRQD, NREQD, DYNKDYNK31
CDYNKDYNK32
CCALLLOADPL(NREQS, NSTEP, OMEGA, TDISP, TOLER, VELOC, DYNKDYNK33
CDYNKDYNK34
CCALLLOADPL(NREQS, NSTEP, OMEGA, TDISP, TOLER, VELOC, DYNKDYNK35
CDYNKDYNK36
CCALLLOADPL(NREQS, NSTEP, OMEGA, TDISP, TOLER, VELOC, DYNKDYNK37
CDYNKDYNK38
CCALLLOADPL(NREQS, NSTEP, OMEGA, TDISP, TOLER, VELOC, DYNKDYNK39
+ + + +```txt +C +CALL +FIXITY +(IFPRE, NDOFN, NPOIN, YMASS) +DYNK +40 +DYNK +41 +DYNK +42 +IF(NPREV.NE.O) +.CALL +RESVPL +(COORD, DTIME, LNODS, MATNO, NCRIT, NDIME, +NDOFN, NELEM, NGAUS, NLAPS, NNODE, NMATS, +NPOIN, NSTRE, NTYPE, POSGP, PROPS, RESID, +RLOAD, STRIN, STRSG, TDISP, VISTN, VIVEL, +WEIGP) +DYNK +43 +DYNK +44 +DYNK +45 +DYNK +46 +DYNK +47 +DYNK +48 +DYNK +49 +DYNK +50 +DYNK +51 +CALL +EXPLIT +(ACCEH, ACCEV, AFACT, AZERO, AALFA, BZERO, +DTIME, DTEND, FORCE, IFIXD, IFPRE, IFUNC, +ISTEP, NDOFN, NPOIN, OMEGA, RESID, TDISP, +VELOC, YMASS) +DYNK +52 +DYNK +53 +DYNK +54 +DYNK +55 +DYNK +56 +CALL +RESVPL +(COORD, DTIME, LNODS, MATNO, NCRIT, NDIME, +NDOFN, NELEM, NGAUS, NLAPS, NNODE, NMATS, +NPOIN, NSTRE, NTYPE, POSGP, PROPS, RESID, +RLOAD, STRIN, STRSG, TDISP, VISTN, VIVEL, +WEIGP) +DYNK +57 +DYNK +58 +DYNK +59 +DYNK +60 +DYNK +61 +DYNK +62 +CALL +OUTDYN +(DISPL, DTIME, ISTEP, NDOFN, NELEM, NGAUS, +NGRQS, NOUTD, NOUTP, NPOIN, NPRQD, NREQD, +NREQS, NTYPE, STRSG, TDISP, VIVEL) +DYNK +63 +DYNK +64 +DYNK +65 +DYNK +66 +500 CONTINUE +STOP +END +DYNK +67 +DYNK +68 +DYNK +69 +``` + +# 10.6.3 Subroutine BLARGE + +This subroutine evaluates the strain-displacement matrix for geometrically nonlinear displacements using the deformation Jacobian matrix $[J_{D}]_{n}$ . Note that for small displacement analysis we pre-set NLAPS = 0. + +```txt +SUBROUTINE BLARGE (BMATX, CARTD, DJACM, DLCOD, GPCOD, KGASP, BLAR 1 +NLAPS, NNODE, NTYPE, SHAPE) BLAR 2 +C****************************************************************************************** +BLAR 3 +C BLAR 4 +C*** LARGE DISPLACEMENT B MATRIX BLAR 5 +C BLAR 6 +C****************************************************************************************** +BLAR 7 +DIMENSION BMATX(4,18), CARTD(2,9), DJACM(2,2), DLCOD(2,9), BLAR 8 +GPCOD(2,9), SHAPE(9) BLAR 9 +NGASH=0 BLAR 10 +DO 10 INODE=1, NNODE BLAR 11 +MGASH=NGASH+1 BLAR 12 +NGASH=MGASH+1 BLAR 13 +BMATX(1, MGASH)=CARTD(1, INODE)*DJACM(1,1) BLAR 14 +BMATX(1, NGASH)=CARTD(1, INODE)*DJACM(2,1) BLAR 15 +BMATX(2, MGASH)=CARTD(2, INODE)*DJACM(1,2) BLAR 16 +BMATX(2, NGASH)=CARTD(2, INODE)*DJACM(2,2) BLAR 17 +BMATX(3, MGASH)=CARTD(2, INODE)*DJACM(1,1)+CARTD(1, INODE)*DJACM(1,2)BLAR 18 +BMATX(3, NGASH)=CARTD(1, INODE)*DJACM(2,2)+CARTD(2, INODE)*DJACM(2,1)BLAR 19 +10 CONTINUE BLAR 20 +IF(NTYPE.NE.3) RETURN BLAR 21 +FMULT=1. BLAR 22 +IF(NLAPS.LT.2) GO TO 40 BLAR 23 +FMULT=0.0 BLAR 24 +``` + + + +```asm +DO 20 JNODE=1, NNODE BLAR 25 +20 FMULT=FMULT+DLCOD(1, JNODE)*SHAPE(JNODE) BLAR 26 +FMULT=FMULT/GPCOD(1, KGASP) BLAR 27 +40 NGASH=0 BLAR 28 +DO 30 INODE=1, NNODE BLAR 29 +MGASH=NGASH+1 BLAR 30 +NGASH=MGASH+1 BLAR 31 +BMATX(4, MGASH)=SHAPE(INODE)*FMULT/GPCOD(1, KGASP) BLAR 32 +30 BMATX(4, NGASH)=0.0 BLAR 33 +RETURN BLAR 34 +END BLAR 35 +``` + +BLAR 10-20 Evaluate the complete strain matrix for plane stress/strain problems and the first three rows of the strain matrix for axisymmetric problems. + +BLAR 21-33 Evaluate the remainder of the strain matrix for axisymmetric problems, if applicable. + +# 10.6.4 Subroutine CONTOL + +The purpose of this subroutine is to set the values of variables for the dynamic dimensions which are used elsewhere in the program. If any change in the DIMENSION statement in the master routine is made, then a corresponding change in this subroutine should also be made. + +```txt +SUBROUTINE CONTOL (NDOFN, NELEM, NMATS, NPOIN) CONT 1 +C******************************* +C +C*** READ CONTROL DATA AND CHECK FOR DIMENSIONS CONT 2 +C +C******************************* +C +READ(5,110) NPOIN, NELEM, NDOFN, NMATS CONT 3 +IF(NELEM.GT. 50) GO TO 200 CONT 4 +IF(NPOIN.GT.200) GO TO 200 CONT 5 +IF(NMATS.GT. 10) GO TO 200 CONT 6 +GO TO 210 CONT 7 +200 WRITE(6,120) CONT 8 +STOP CONT 9 +120 FORMAT(/'SET DIMENSION EXCEEDED - CONTOL CHECK') CONT 10 +110 FORMAT(16I5) CONT 11 +210 CONTINUE CONT 12 +RETURN CONT 13 +END CONT 14 +CONT 15 +CONT 16 +CONT 17 +CONT 18 +``` + +# 10.6.5 Subroutine EXPLIT + +This subroutine performs the direct time integration using expressions (10.43) and (10.44) to evaluate the nodal displacements at every time step. Special provisions are made for the first time step. + +```txt +SUBROUTINE EXPLIT (ACCEH, ACCEV, AFACT, AZERO, AALFA, BZERO, EXPL 1 +. DTIME, DTEND, FORCE, IFIXD, IFPRE, IFUNC, EXPL 2 +. ISTEP, NDOFN, NPOIN, OMEGA, RESID, TDISP, EXPL 3 +. VELOC, YMASS) EXPL 4 +C*************** EXPL 5 +C EXPL 6 +C *** TIME STEPPING ROUTINE EXPL 7 +C EXPL 8 +C*************** EXPL 9 +``` + + + +```csv +DIMENSION YMASS(1), ACCEH(1), TDISP(1), RESID(1), EXPL 10 +FORCE(1), ACCEV(1), VELOC(1), IFPRE(2,1) +CFACT=1.0+0.5*AALFA*DTIME +CFACT=1./CFACT +CONS1=2.*CFACT +RCONS=1./CONS1 +CONS2=CONS1-1 +CONS3=DTIME*DTIME*CFACT +CONS4=-2.0*CONS2*DTIME +IF(ISTEP.GT.1) CONS4=CONS2 +NPOSN=0 +FACTS=FUNCTS (AZERO, BZERO, DTEND, DTIME, IFUNC, ISTEP, OMEGA) +FACTH=FUNCTA (ACCEH, AFACT, DTEND, DTIME, IFUNC, ISTEP) +FACTV=FUNCTA (ACCEV, AFACT, DTEND, DTIME, IFUNC, ISTEP) +DO 500 IPOIN=1, NPOIN +DO 510 IDOFN=1, NDOFN +FACTT=0.0 +IF(IFUNC.NE.0) GO TO 200 +IF(IFIXD.EQ.0.AND.IDOFN.EQ.1) FACTT=FACTH +IF(IFIXD.EQ.0.AND.IDOFN.EQ.2) FACTT=FACTV +IF(IFIXD.EQ.1.AND.IDOFN.EQ.2) FACTT=FACTV +IF(IFIXD.EQ.2.AND.IDOFN.EQ.1) FACTT=FACTH +IF(IFPRE(IDOFN, IPOIN).EQ.0) GO TO 200 +FACTT=0.0 +FACTS=1.0 +200 CONTINUE +NPOSN=NPOSN+1 +DCURR=TDISP(NPOSN) +RESID(NPOSN)=RESID(NPOSN)-FORCE(NPOSN)*FACTS +TDISP(NPOSN)=-RESID(NPOSN)*CONS3/YMASS(NPOSN) +.-FACTT*CONS3+DCURR*CONS1-VELOC(NPOSN)*CONS4 +IF(ISTEP.EQ.1) TDISP(NPOSN)=TDISP(NPOSN)*RCONS +VELOC(NPOSN)=DCURR +510 CONTINUE +500 CONTINUE +RETURN +END +EXPL 10 +EXPL 11 +EXPL 12 +EXPL 13 +EXPL 14 +EXPL 15 +EXPL 16 +EXPL 17 +EXPL 18 +EXPL 19 +EXPL 20 +EXPL 21 +EXPL 22 +EXPL 23 +EXPL 24 +EXPL 25 +EXPL 26 +EXPL 27 +EXPL 28 +EXPL 29 +EXPL 30 +EXPL 31 +EXPL 32 +EXPL 33 +EXPL 34 +EXPL 35 +EXPL 36 +EXPL 37 +EXPL 38 +EXPL 39 +EXPL 40 +EXPL 41 +EXPL 42 +EXPL 43 +EXPL 44 +EXPL 45 +EXPL 46 +``` + +EXPL 12–19 Evaluate the various time integration constants. After the first time step modify variable CONS4. + +EXPL 21 Evaluate the value of the time varying Heaviside or harmonic function for a particular time step. + +EXPL 22–23 Evaluate the acceleration ordinates (FACTH for horizontal and FACTV for vertical acceleration respectively) at a particular time step. + +EXPL 24–31 The seismic force is only applied for particular degrees of freedom. For IFIXD = 1 only vertical, IFIXD = 2 only horizontal or radial and IFIXD = 0 both components of the acceleration are considered. + +EXPL 32-35 Assign appropriate values for restrained boundary nodes. + +EXPL 36–40 Evaluate displacements. + +EXPL 41 For the first time step modify the displacement. + +EXPL 42 Store the current displacements for the next time step. + +# 10.6.6 Subroutine FIXITY + +This subroutine deals with the restrained degrees of freedom (boundary points). The diagonal mass vector, XMASS, is modified—for restrained + + + +degrees of freedom. The component of the XMASS vector is set to a large value such as 1.E30, which artificially makes the displacement zero. + +```fortran +SUBROUTINE FIXITY (IFPRE, NDOFN, NPOIN, YMASS) FIXY 1 +C********** 1 +C 2 +C 3 +C *** DEALS WITH FIXED BOUNDARY NODES 4 +C 5 +C********** 6 +DIMENSION IFPRE(2,1), YMASS(1) 7 +NTOTV=NDOFN*NPOIN 8 +IPOSN=0 9 +DO 500 IPOIN=1, NPOIN 10 +DO 500 IDOFN=1, NDOFN 11 +IPOSN=IPOSN+1 12 +500 IF(IFPRE(IDOFN, IPOIN).EQ.1) YMASS(IPOSN)=1.E30 13 +WRITE(6,900) 14 +900 FORMAT(/5X,19HNODAL LUMPED MASSES/) 15 +WRITE(6,910) (ITOTV, YMASS(ITOTV), ITOTV=1, NTOTV) 16 +910 FORMAT(6(1X, I5, E13.5)) 17 +RETURN 18 +END 19 +``` + +# 10.6.7 Subroutine FLOWVP + +This routine evaluates the viscoplastic strain rate. + +```txt +SUBROUTINE FLOWVP (AVECT, KGAUS, LPROP, NCRIT, NMATS, PROPS, STEFF, VIVEL, YIELD) FLOV 1 +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +DIMENSION AVECT(4), PROPS(NMATS,1), VIVEL(5,1) +IF(STEFF.EQ.0.0) GO TO 90 +NSTR1=4 +TOLOR=0.01 +FDATM=PROPS(LPROP, 6) +HARDS=PROPS(LPROP, 7) +FRICT=PROPS(LPROP, 8) +GAMMA=PROPS(LPROP, 9) +DELTA=PROPS(LPROP,10) +NFLOW=PROPS(LPROP,11) +FRICT=FRICT*0.017453292 +IF(NCRIT.EQ.3) FDATM=FDATM*COS(FRICT) +IF(NCRIT.EQ.4) FDATM=6.0*FDATM*COS(FRICT)/ +(1.73205080757*(3.0-SIN(FRICT))) +IF(HARDS.GT.0.) FDATM=FDATM+VIVEL(5,KGAUS)*HARDS +IF(FDATM.LT.0.001) FDATM=1.0 +FCURR=YIELD-FDATM +FNORM=FCURR/FDATM +IF(FNORM.LT.TOLOR) GO TO 90 +IF(NFLOW.EQ.1) GO TO 50 +CMULT=GAMMA*(EXP(DELTA*FNORM)-1.0) +GO TO 60 +50 CMULT=GAMMA*(FNORM**DELTA) +60 CONTINUE +DO 70 ISTR1=1,NSTR1 +70 AVECT(ISTR1)=CMULT*AVECT(ISTR1) +DO 80 ISTR1=1,NSTR1 +80 VIVEL(ISTR1,KGAUS)=AVECT(ISTR1) +RETURN +90 DO 100 ISTR1=1,NSTR1 +100 VIVEL(ISTR1,KGAUS)=0. +RETURN +FLOV 10 +``` + + + +# 10.6.8 Function FUNCTA + +This function interpolates the accelerogram data for a particular time step. AFACT is the ratio of the accelerogram record time step length to the computational time step length. + +```txt +FUNCTION FUNCTA (ACCER, AFACT, DTEND, DTIME, IFUNC, JSTEP) FUNA 1 +C******************************* +C +C*** ACCELEROGRAM INTERPOLATION FUNA 2 +C +C******************************* +DIMENSION ACCER(1) FUNA 3 +IF(IFUNC.NE.0) RETURN FUNA 4 +FUNCTA=0.0 FUNA 5 +IF(JSTEP.EQ.0.OR.FLOAT(JSTEP)*DTIME.GT.DTEND) RETURN FUNA 6 +XGASH=(FLOAT(JSTEP)-1.0)/AFACT+1.0 FUNA 7 +MGASH=XGASH FUNA 8 +NGASH=MGASH+1 FUNA 9 +XGASH=XGASH-FLOAT(MGASH) FUNA 10 +FUNCTA=ACCER(MGASH)*(1.0-XGASH)+XGASH*ACCER(NGASH) FUNA 11 +RETURN FUNA 12 +END FUNA 13 +FUNA 14 +FUNA 15 +FUNA 16 +FUNA 17 +``` + +# 10.6.9 Function FUNCTS + +This function sets the value of the time varying function for a particular time step. Heaviside functions $(f(t) = 1.0 \ H(t))$ or harmonic functions, $(f(t) = a - b \sin \omega t)$ can be specified. + +```txt +FUNCTION FUNCTS (AZERO,BZERO,DTEND,DTIME,IFUNC,JSTEP,OMEGA) FUNS 1 +C************************** FUNS 2 +C FUNS 3 +C*** HEAVISIDE AND HARMONIC TIME FUNCTION FUNS 4 +C FUNS 5 +C************************** FUNS 6 +IF(IFUNC.EQ.0) RETURN FUNS 7 +FUNCTS=0.0 FUNS 8 +IF(JSTEP.EQ.0.OR uses FLOAT(JSTEP)*DTIME.GT.DTEND) RETURN FUNS 9 +IF(IFUNC.EQ.1) FUNCTS = 1.0 FUNS 10 +IF(IFUNC.EQ.2) ARGUM=OMEGA*JSTEP*DTIME FUNS 11 +IF(IFUNC.EQ.2) FUNCTS = AZERO + BZERO*SIN(ARGUM) FUNS 12 +RETURN FUNS 13 +END FUNS 14 +``` + +# 10.6.10 Subroutine INPUTD + +This subroutine reads and writes most of the control parameters, nodal point coordinates, element connectivities, boundary conditions and material properties. It also writes the geometric data onto file 13 for deformation plotting. A similar routine was described in Chapter 6. + +```c +SUBROUTINE INPUTD (COORD, IFPRE, LNODS, MATNO, NCONM, NCRIT, NPUT 1 +. NDIME, NDOFN, NELEM, NGAUM, NGAUS, NLAPS, NPUT 2 +. NMATS, NNODE, NPOIN, NPREV, NSTRE, NTYPE, NPUT 3 +. POSGP, PROPS, WEIGP) NPUT 4 +C******************************* +C******************************* +C******************************* +C*** DYNPAK INPUT ROUTINE +C +C******************************* +DIMENSION COORD(NPOIN, 1), IFPRE(NDOFN, 1), WEIGP(1), MATNO(1), NPUT 10 +``` + + + +```csv +READ(5,913) TITLE NPUT 12 +913 FORMAT(10A4) NPUT 13 +WRITE(6,914) TITLE NPUT 14 +914 FORMAT(//,5X,10A4) NPUT 15 +C NPUT 16 +C*** READ THE FIRST DATA CARD, AND ECHO IT IMMEDIATELY. NPUT 17 +C NPUT 18 +READ (5,900) NVFIX,NTYPE,NNODE,NPROP,NGAUS,NDIME,NSTRE,NCRIT, NPREV,NCONM,NLAPS,NGAUM,NRADS NPUT 19 +WRITE(6,901) NPOIN,NELEM,NVFIX,NTYPE,NNODE,NDOFN,NMATS,NPROP, NGAUS,NDIME,NSTRE,NCRIT,NPREV,NCONM,NLAPS,NGAUM, NRADS NPUT 20 +901 FORMAT (/5X,18HCONTROL PARAMETERS/ NPUT 21 +/5X,8H NPOIN =,I10,5X,8H NELEM =,I10,5X,8H NVFIX =,I10/ NPUT 22 +/5X,8H NTYPE =,I10,5X,8H NNODE =,I10,5X,8H NDOFN =,I10/ NPUT 23 +/5X,8H NMATS =,I10,5X,8H NPROP =,I10,5X,8H NGAUS =,I10/ NPUT 24 +/5X,8H NDIME =,I10,5X,8H NSTRE =,I10,5X,8H NCRIT =,I10/ NPUT 25 +/5X,8H NPREV =,I10,5X,8H NCONM =,I10,5X,8H NLAPS =,I10/ NPUT 26 +/5X,8H NGAUM =,I10,5X,8H NRADS =,I10/) NPUT 27 +900 FORMAT(16I5) NPUT 28 +NPUT 29 +NPUT 30 +NPUT 31 +NPUT 32 +NPUT 33 +NPUT 34 +NPUT 35 +NPUT 36 +NPUT 37 +NPUT 38 +NPUT 39 +NPUT 40 +NPUT 41 +NPUT 42 +NPUT 43 +NPUT 44 +NPUT 45 +NPUT 46 +NPUT 47 +NPUT 48 +NPUT 49 +NPUT 50 +NPUT 51 +NPUT 52 +NPUT 53 +NPUT 54 +NPUT 55 +NPUT 56 +NPUT 57 +NPUT 58 +NPUT 59 +NPUT 60 +NPUT 61 +NPUT 62 +NPUT 63 +NPUT 64 +NPUT 65 +NPUT 66 +NPUT 67 +NPUT 68 +NPUT 69 +NPUT 70 +NPUT 71 +NPUT 72 +NPUT 73 +NPUT 74 +NPUT 75 +NPUT 76 +NPUT 77 +NPUT 78 +NPUT 79 +NPUT 80 +NPUT 81 +NPUT 82 +NPUT 83 +NPUT 84 +NPUT 85 +NPUT 86 +NPUT 87 +NPUT 88 +NPUT 89 +NPUT 90 +NPUT 91 +NPUT 92 +NPUT 93 +NPUT 94 +NPUT 95 +NPUT 96 +NPUT 97 +NPUT 98 +NPUT 99 +NPUT 100 +NPUT 101 +NPUT 102 +NPUT 103 +NPUT 104 +NPUT 105 +NPUT 106 +NPUT 107 +NPUT 108 +NPUT 109 +NPUT 110 +NPUT 111 +NPUT 112 +NPUT 113 +NPUT 114 +NPUT 115 +NPUT 116 +NPUT 117 +NPUT 118 +NPUT 119 +NPUT 120 +NPUT 121 +NPUT 122 +NPUT 123 +NPUT 124 +NPUT 125 +NPUT 126 +NPUT 127 +NPUT 128 +NPUT 129 +NPUT 130 +NPUT 131 +NPUT 132 +NPUT 133 +NPUT 134 +NPUT 135 +NPUT 136 +NPUT 137 +NPUT 138 +NPUT 139 +NPUT 140 +NPUT 141 +NPUT 142 +NPUT 143 +NPUT 144 +NPUT 145 +NPUT 146 +NPUT 147 +NPUT 148 +NPUT 149 +NPUT 150 +NPUT 151 +NPUT 152 +NPUT 153 +NPUT 154 +NPUT 155 +NPUT 156 +NPUT 157 +NPUT 158 +NPUT 159 +NPUT 160 +NPUT 161 +NPUT 162 +NPUT 163 +NPUT 164 +NPUT 165 +NPUT 166 +NPUT 167 +NPUT 168 +NPUT 169 +NPUT 170 +NPUT 171 +NPUT 172 +NPUT 173 +NPUT 174 +NPUT 175 +NPUT 176 +NPUT 177 +NPUT 178 +NPUT 179 +NPUT 180 +NPUT 181 +NPUT 182 +NPUT 183 +NPUT 184 +NPUT 185 +NPUT 186 +NPUT 187 +NPUT 188 +NPUT 189 +NPUT 190 +NPUT 191 +NPUT 192 +NPUT 193 +NPUT 194 +NPUT 195 +NPUT 196 +NPUT 197 +NPUT 198 +NPUT 199 +NPUT 200 +NPUT 201 +NPUT 202 +NPUT 203 +NPUT 204 +NPUT 205 +NPUT 206 +NPUT 207 +NPUT 208 +NPUT 209 +NPUT 210 +NPUT 211 +NPUT 212 +NPUT 213 +NPUT 214 +NPUT 215 +NPUT 216 +NPUT 217 +NPUT 218 +NPUT 219 +NPUT 220 +NPUT 221 +NPUT 222 +NPUT 223 +NPUT 224 +NPUT 225 +NPUT 226 +NPUT 227 +NPUT 228 +NPUT 229 +NPUT 230 +NPUT 231 +NPUT 232 +NPUT 233 +NPUT 234 +NPUT 235 +NPUT 236 +NPUT 237 +NPUT 238 +NPUT 239 +NPUT 240 +NPUT 241 +NPUT 242 +NPUT 243 +NPUT 244 +NPUT 245 +NPUT 246 +NPUT 247 +NPUT 248 +NPUT 249 +NPUT 250 +NPUT 251 +NPUT 252 +NPUT 253 +NPUT 254 +NPUT 255 +NPUT 256 +NPUT 257 +NPUT 258 +NPUT 259 +NPUT 260 +NPUT 261 +NPUT 262 +NPUT 263 +NPUT 264 +NPUT 265 +NPUT 266 +NPUT 267 +NPUT 268 +NPUT 269 +NPUT 270 +NPUT 271 +NPUT 272 +NPUT 273 +NPUT 274 +NPUT 275 +NPUT 276 +NPUT 277 +NPUT 278 +NPUT 279 +NPUT 280 +NPUT 281 +NPUT 282 +NPUT 283 +NPUT 284 +NPUT 285 +NPUT 286 +NPUT 287 +NPUT 288 +NPUT 289 +NPUT 290 +NPUT 291 +NPUT 292 +NPUT 293 +NPUT 294 +NPUT 295 +NPUT 296 +NPUT 297 +NPUT 298 +NPUT 299 +NPUT 300 +NPUT 301 +NPUT 302 +NPUT 303 +NPUT 304 +NPUT 305 +NPUT 306 +NPUT 307 +NPUT 308 +NPUT 309 +NPUT 310 +NPUT 311 +NPUT 312 +NPUT 313 +NPUT 314 +NPUT 315 +NPUT 316 +NPUT 317 +NPUT 318 +NPUT 319 +NPUT 320 +NPUT 321 +NPUT 322 +NPUT 323 +NPUT 324 +NPUT 325 +NPUT 326 +NPUT 327 +NPUT 328 +NPUT 329 +NPUT 330 +NPUT 331 +NPUT 332 +NPUT 333 +NPUT 334 +NPUT 335 +NPUT 336 +NPUT 337 +NPUT 338 +NPUT 339 +NPUT 340 +NPUT 341 +NPUT 342 +NPUT 343 +NPUT 344 +NPUT 345 +NPUT 346 +NPUT 347 +NPUT 348 +NPUT 349 +NPUT 350 +NPUT 351 +NPUT 352 +NPUT 353 +NPUT 354 +NPUT 355 +NPUT 356 +NPUT 357 +NPUT 358 +NPUT 359 +NPUT 360 +NPUT 361 +NPUT 362 +NPUT 363 +NPUT 364 +NPUT 365 +NPUT 366 +NPUT 367 +NPUT 368 +NPUT 369 +NPUT 370 +NPUT 371 +NPUT 372 +NPUT 373 +NPUT 374 +NPUT 375 +NPUT 376 +NPUT 377 +NPUT 378 +NPUT 379 +NPUT 380 +NPUT 381 +NPUT 382 +NPUT 383 +NPUT 384 +NPUT 385 +NPUT 386 +NPUT 387 +NPUT 388 +NPUT 389 +NPUT 390 +NPUT 391 +NPUT 392 +NPUT 393 +NPUT 394 +NPUT 395 +NPUT 396 +NPUT 397 +NPUT 398 +NPUT 399 +NPUT 400 +NPUT 401 +NPUT 402 +NPUT 403 +NPUT 404 +NPUT 405 +NPUT 406 +NPUT 407 +NPUT 408 +NPUT 409 +NPUT 410 +NPUT 411 +NPUT 412 +NPUT 413 +NPUT 414 +NPUT 415 +NPUT 416 +NPUT 417 +NPUT 418 +NPUT 419 +NPUT 420 +NPUT 421 +NPUT 422 +NPUT 423 +NPUT 424 +NPUT 425 +NPUT 426 +NPUT 427 +NPUT 428 +NPUT 429 +NPUT 430 +NPUT 431 +NPUT 432 +NPUT 433 +NPUT 434 +NPUT 435 +NPUT 436 +NPUT 437 +NPUT 438 +NPUT 439 +NPUT 440 +NPUT 441 +NPUT 442 +NPUT 443 +NPUT 444 +NPUT 445 +NPUT 446 +NPUT 447 +NPUT 448 +NPUT 449 +NPUT 450 +NPUT 451 +NPUT 452 +NPUT 453 +NPUT 454 +NPUT 455 +NPUT 456 +NPUT 457 +NPUT 458 +NPUT 459 +NPUT 460 +NPUT 461 +NPUT 462 +NPUT 463 +NPUT 464 +NPUT 465 +NPUT 466 +NPUT 467 +NPUT 468 +NPUT 469 +NPUT 470 +NPUT 471 +NPUT 472 +NPUT 473 +NPUT 474 +NPUT 475 +NPUT 476 +NPUT 477 +NPUT 478 +NPUT 479 +NPUT 480 +NPUT 481 +NPUT 482 +NPUT 483 +NPUT 484 +NPUT 485 +NPUT 486 +NPUT 487 +NPUT 488 +NPUT 489 +NPUT 490 +NPUT 491 +NPUT 492 +NPUT 493 +NPUT 494 +NPUT 495 +NPUT 496 +NPUT 497 +NPUT 498 +NPUT 499 +NPUT 500 +NPUT 501 +NPUT 502 +NPUT 503 +NPUT 504 +NPUT 505 +NPUT 506 +NPUT 507 +NPUT 508 +NPUT 509 +NPUT 510 +NPUT 511 +NPUT 512 +NPUT 513 +NPUT 514 +NPUT 515 +NPUT 516 +NPUT 517 +NPUT 518 +NPUT 519 +NPUT 520 +NPUT 521 +NPUT 522 +NPUT 523 +NPUT 524 +NPUT 525 +NPUT 526 +NPUT 527 +NPUT 528 +NPUT 529 +NPUT 530 +NPUT 531 +NPUT 532 +NPUT 533 +NPUT 534 +NPUT 535 +NPUT 536 +NPUT 537 +NPUT 538 +NPUT 539 +NPUT 540 +NPUT 541 +NPUT 542 +NPUT 543 +NPUT 544 +NPUT 545 +NPUT 546 +NPUT 547 +NPUT 548 +NPUT 549 +NPUT 550 +NPUT 551 +NPUT 552 +NPUT 553 +NPUT 554 +NPUT 555 +NPUT 556 +NPUT 557 +NPUT 558 +NPUT 559 +NPUT 560 +NPUT 561 +NPUT 562 +NPUT 563 +NPUT 564 +NPUT 565 +NPUT 566 +NPUT 567 +NPUT 568 +NPUT 569 +NPUT 570 +NPUT 571 +NPUT 572 +NPUT 573 +NPUT 574 +NPUT 575 +NPUT 576 +NPUT 577 +NPUT 578 +NPUT 579 +NPUT 580 +NPUT 581 +NPUT 582 +NPUT 583 +NPUT 584 +NPUT 585 +NPUT 586 +NPUT 587 +NPUT 588 +NPUT 589 +NPUT 590 +NPUT 591 +NPUT 592 +NPUT 593 +NPUT 594 +NPUT 595 +NPUT 596 +NPUT 597 +NPUT 598 +NPUT 599 +NPUT 600 +NPUT 601 +NPUT 602 +NPUT 603 +NPUT 604 +NPUT 605 +NPUT 606 +NPUT 607 +NPUT 608 +NPUT 609 +NPUT 610 +NPUT 611 +NPUT 612 +NPUT 613 +NPUT 614 +NPUT 615 +NPUT 616 +NPUT 617 +NPUT 618 +NPUT 619 +NPUT 620 +NPUT 621 +NPUT 622 +NPUT 623 +NPUT 624 +NPUT 625 +NPUT 626 +NPUT 627 +NPUT 628 +NPUT 629 +NPUT 630 +NPUT 631 +NPUT 632 +NPUT 633 +NPUT 634 +NPUT 635 +NPUT 636 +NPUT 637 +NPUT 638 +NPUT 639 +NPUT 640 +NPUT 641 +NPUT 642 +NPUT 643 +NPUT 644 +NPUT 645 +NPUT 646 +NPUT 647 +NPUT 648 +NPUT 649 +NPUT 650 +NPUT 651 +NPUT 652 +NPUT 653 +NPUT 654 +NPUT 655 +NPUT 656 +NPUT 657 +NPUT 658 +NPUT 659 +NPUT 660 +NPUT 661 +NPUT 662 +NPUT 663 +NPUT 664 +NPUT 665 +NPUT 666 +NPUT 667 +NPUT 668 +NPUT 669 +NPUT 670 +NPUT 671 +NPUT 672 +NPUT 673 +NPUT 674 +NPUT 675 +NPUT 676 +NPUT 677 +NPUT 678 +NPUT 679 +NPUT 680 +NPUT 681 +NPUT 682 +NPUT 683 +NPUT 684 +NPUT 685 +NPUT 686 +NPUT 687 +NPUT 688 +NPUT 689 +NPUT 690 +NPUT 691 +NPUT 692 +NPUT 693 +NPUT 694 +NPUT 695 +NPUT 696 +NPUT 697 +NPUT 698 +NPUT 699 +NPUT 700 +NPUT 701 +NPUT 702 +NPUT 703 +NPUT 704 +NPUT 705 +NPUT 706 +NPUT 707 +NPUT 708 +NPUT 709 +NPUT 710 +NPUT 711 +NPUT 712 +NPUT 713 +NPUT 714 +NPUT 715 +NPUT 716 +NPUT 717 +NPUT 718 +NPUT 719 +NPUT 720 +NPUT 721 +NPUT 722 +NPUT 723 +NPUT 724 +NPUT 725 +NPUT 726 +NPUT 727 +NPUT 728 +NPUT 729 +NPUT 730 +NPUT 731 +NPUT 732 +NPUT 733 +NPUT 734 +NPUT 735 +NPUT 736 +NPUT 737 +NPUT 738 +NPUT 739 +NPUT 740 +NPUT 741 +NPUT 742 +NPUT 743 +NPUT 744 +NPUT 745 +NPUT 746 +NPUT 747 +NPUT 748 +NPUT 749 +NPUT 750 +NPUT 751 +NPUT 752 +NPUT 753 +NPUT 754 +NPUT 755 +NPUT 756 +NPUT 757 +NPUT 758 +NPUT 759 +NPUT 760 +NPUT 761 +NPUT 762 +NPUT 763 +NPUT 764 +NPUT 765 +NPUT 766 +NPUT 767 +NPUT 768 +NPUT 769 +NPUT 770 +NPUT 771 +NPUT 772 +NPUT 773 +NPUT 774 +NPUT 775 +NPUT 776 +NPUT 777 +NPUT 778 +NPUT 779 +NPUT 780 +NPUT 781 +NPUT 782 +NPUT 783 +NPUT 784 +NPUT 785 +NPUT 786 +NPUT 787 +NPUT 788 +NPUT 789 +NPUT 790 +NPUT 791 +NPUT 792 +NPUT 793 +NPUT 794 +NPUT 795 +NPUT 796 +NPUT 797 +NPUT 798 +NPUT 799 +NPUT 800 +NPUT 801 +NPUT 802 +NPUT 803 +NPUT 804 +NPUT 805 +NPUT 806 +NPUT 807 +NPUT 808 +NPUT 809 +NPUT 810 +NPUT 811 +NPUT 812 +NPUT 813 +NPUT 814 +NPUT 815 +NPUT 816 +NPUT 817 +NPUT 818 +NPUT 819 +NPUT 820 +NPUT 821 +NPUT 822 +NPUT 823 +NPUT 824 +NPUT 825 +NPUT 826 +NPUT 827 +NPUT 828 +NPUT 829 +NPUT 830 +NPUT 831 +NPUT 832 +NPUT 833 +NPUT 834 +NPUT 835 +NPUT 836 +NPUT 837 +NPUT 838 +NPUT 839 +NPUT 840 +NPUT 841 +NPUT 842 +NPUT 843 +NPUT 844 +NPUT +``` diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_042.md b/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_042.md new file mode 100644 index 00000000..ec655eb7 --- /dev/null +++ b/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_042.md @@ -0,0 +1,957 @@ + + +```csv +540 IFPRE(IDOFN,IPOIN)=0 NPUT 76 +DO 550 IVFIX=1,NVFIX NPUT 77 +550 READ (5,908) IPOIN,(IFPRE(IDOFN,IPOIN),IDOFN=1,NDOFN) NPUT 78 +DO 560 IPOIN=1,NPOIN NPUT 79 +560 WRITE(6,909) IPOIN,(IFPRE(IDOFN,IPOIN),IDOFN=1,NDOFN) NPUT 80 +908 FORMAT(1X,I4,3X,2I1) NPUT 81 +909 FORMAT(6X,I5,3X,2I1) NPUT 82 +C NPUT 83 +C*** READ THE AVAILABLE SELECTION OF ELEMENT PROPERTIES. NPUT 84 +C NPUT 85 +WRITE(6,910) NPUT 86 +910 FORMAT(//5X,19HATERIAL PROPERTIES) NPUT 87 +DO 520 IMATS=1,NMATS NPUT 88 +READ(5,900) NUMAT NPUT 89 +READ (5,917) (PROPS(NUMAT,IPROP),IPROP=1,NPROP) NPUT 90 +WRITE(6,911) NUMAT NPUT 91 +911 FORMAT(/5X,11HATERIAL NO,I5) NPUT 92 +520 WRITE(6,912) (PROPS(NUMAT,IPROP),IPROP=1,NPROP) NPUT 93 +912 FORMAT(/5X,13HYOUNG MODULUS,G12.4/5X,13HPOISSON RATIO,G12.4/5X,13HTHICKNESS ,G12.4/5X,13HMASS DENSITY ,G12.4/5X,13HALPHA TEMPR ,G12.4/5X,13HREFERENCE FO ,G12.4/5X,13HHARDENING PAR,G12.4/5X,13HFRICT ANGLE ,G12.4/5X,13HFLUIDITY PAR ,G12.4/5X,13HEXP DELTA ,G12.4/5X,13HNFLOW CODE ,G12.4) NPUT 99 +917 FORMAT(8E10.4) NPUT 100 +C NPUT 101 +C*** SET UP GAUSSIAN INTEGRATION CONSTANTS NPUT 102 +C NPUT 103 +CALL GAUSSQ (NGAUS,POSGP,WEIGP) NPUT 104 +RETURN NPUT 105 +END NPUT 106 +``` + +# 10.6.11 Subroutine INTIME + +This routine reads and writes all data required for time integration and plotting stress and displacement histories. + +```fortran +SUBROUTINE INTIME (AALFA, ACCEH, ACCEV, AFACT, AZERO, BEETA, TIME 1 +BZERO, DELTA, DTIME, DTEND, GAAMA, IFIXD, TIME 2 +IFUNC, INTGR, KSTEP, MITER, NDOFN, NELEM, TIME 3 +NGRQS, NOUTD, NOUTP, NPOIN, NPRQD, NREQD, TIME 4 +NREQS, NSTEP, OMEGA, TDISP, TOLER, VELOC, TIME 5 +IPRED) TIME 6 +C******************************************************************************************TIME 7 +C TIME 8 +C ** INITIAL VALUES AND TIME INTEGRATION DATA TIME 9 +C TIME 10 +C******************************************************************************************TIME 11 +DIMENSION TDISP(1), ACCEH(1), NPRQD(1), INTGR(1), TIME 12 +VELOC(1), ACCEV(1), NGRQS(1) TIME 13 +C TIME 14 +C*** READ TIME STEPPING AND SELECTIVE OUTPUT PARAMETERS TIME 15 +C TIME 16 +READ (5,902) NSTEP, NOUTD, NOUTP, NREQD, NREQS, NACCE, IFUNC, TIME 17 +IFIXD, MITER, KSTEP, IPRED TIME 18 +READ (5,190) DTIME, DTEND, DTREC, AALFA, BEETA, DELTA, GAAMA, TIME 19 +AZERO, BZERO, OMEGA, TOLER TIME 20 +WRITE(6,950) NSTEP, NOUTD, NOUTP, NREQD, NREQS, NACCE, IFUNC, TIME 21 +IFIXD, MITER, KSTEP, IPRED TIME 22 +WRITE(6,960) DTIME, DTEND, DTREC, AALFA, BEETA, DELTA, GAAMA, TIME 23 +AZERO, BZERO, OMEGA, TOLER TIME 24 +950 FORMAT(/5X, 'TIME STEPPING PARAMETERS'/ TIME 25 +/5X, 'NSTEP=', I5, 12X, 'NOUTD=', I5, 12X, 'NOUTP=', I5,/ +/5X, 'NREQD=', I5, 12X, 'NREQS=', I5, 12X, 'NACCE=', I5,/ +/5X, 'IFUNC=', I5, 12X, 'IFIXD=', I5, 12X, 'MITER=', I5,/ +/5X, 'KSTEP=', I5, 12X, 'IPRED=', I5) TIME 29 +``` + + + +```csv +960 FORMAT(/5X,'DTIME=',G12.4,5X,'DTEND=',G12.4,5X,'DTREC=',G12.4,/ + /5X,'AALFA=',G12.4,5X,'BEETA=',G12.4,5X,'DELTA=',G12.4,/ + /5X,'GAAMA=',G12.4,5X,'AZERO=',G12.4,5X,'BZERO=',G12.4,/ + /5X,'OMEGA=',G12.4,5X,'TOLER=',G12.4) +C +C*** SELECTED NODES AND GAUSS POINTS FOR OUTPUT +C +READ(5,902) (NPRQD(IREQD),IREQD=1,NREQD) +READ(5,902) (NGRQS(IREQS),IREQS=1,NREQS) +WRITE(6,909) +909 FORMAT(/5X,41H SELECTIVE OUTPUT REQUESTED FOR FOLLOWING ) +WRITE(6,910) (NPRQD(IREQD),IREQD=1,NREQD) +910 FORMAT(/,5X,6H NODES,10I5) +WRITE(6,911) (NGRQS(IREQS),IREQS=1,NREQS) +911 FORMAT(5X,6H G.P.,10I5) +902 FORMAT(16I5) +190 FORMAT(8F10.4) +C +C*** READ THE INDICATOR FOR EXPLICIT OR IMPLICIT ELEMENT +C +READ (5,902) (INTGR(IELEM),IELEM=1,NELEM) +WRITE(6,930) +WRITE(6,902) (INTGR(IELEM),IELEM=1,NELEM) +930 FORMAT(/5X,'TYPE OF ELEMENT, IMPLICIT=1,EXPLICIT=2 '/) +C +C*** INITIAL DISPLACEMENTS +C +JPOIN=0 +DO 500 IPOIN=1,NPOIN +DO 500 IDOFN=1,NDOFN +JPOIN=JPOIN+1 +TDISP(JPOIN)=0. +500 VELOC(JPOIN)=0. +WRITE(6,903) +200 READ(5,904) NGASH,XGASH,YGASH +NPOSN=(NGASH-1)*NDOFN+1 +TDISP(NPOSN)=XGASH +NPOSN=NPOSN+1 +TDISP(NPOSN)=YGASH +WRITE(6,905) NGASH,XGASH,YGASH +IF(NGASH.NE.NPOIN) GO TO 200 +C +C*** INITIAL VELOCITIES +C +WRITE(6,906) +210 READ(5,904) NGASH,XGASH,YGASH +NPOSN=(NGASH-1)*NDOFN+1 +VELOC(NPOSN)=XGASH +NPOSN=NPOSN+1 +VELOC(NPOSN)=YGASH +WRITE(6,905) NGASH,XGASH,YGASH +IF(NGASH.NE.NPOIN) GO TO 210 +904 FORMAT(I5,2F10.5) +903 FORMAT(/5X,5H NODE,2X,16H INITIAL X-DISP.,2X, +.16H INITIAL Y-DISP./) +905 FORMAT(I10,2E18.5) +906 FORMAT(/5X,5H NODE,2X,16H INITIAL X-VELO.,2X, +.16H INITIAL Y-VELO./) +IF (IFUNC.NE.0) GO TO 250 +C +C*** READ ACCELEROGRAM DATA ,X-DIREC FROM TAPE 7,Y-DIREC FROM TAPE 12 +C +AFACT=DTREC/DTIME +IF(IFIXD-1) 220,230,240 +220 READ (7,907)(ACCEH(I),I=1,NACCE) +``` + + + +
WRITE(6,912) DTRECTIME 95
WRITE(6,907)(ACCEH(I),I=1,NACCE)TIME 96
READ(12,907)(ACCEV(I),I=1,NACCE)TIME 97
WRITE(6,913) DTRECTIME 98
WRITE(6,907)(ACCEV(I),I=1,NACCE)TIME 99
GO TO 250TIME 100
230READ(12,907)(ACCEV(I),I=1,NACCE)TIME 101
WRITE(6,913) DTRECTIME 102
WRITE(6,907)(ACCEV(I),I=1,NACCE)TIME 103
GO TO 250TIME 104
240READ(7,907)(ACCEH(I),I=1,NACCE)TIME 105
WRITE(6,912)TIME 106
WRITE(6,907)(ACCEH(I),I=1,NACCE)TIME 107
907FORMAT(7F10.3)TIME 108
912FORMAT(/5X,'HORIZONTAL ACCELERATION ORDINATES AT',F9.4,2X,'SEC')TIME 109
913FORMAT(/5X,'VERTICAL ACCELERATION ORDINATES AT',F9.4,2X,'SEC')TIME 110
250CONTINUETIME 111
RETURNTIME 112
ENDTIME 113
+ +TIME 14-33 Read and write most of the control time integration data. + +TIME 34–46 Read the selective nodal points and integration points for displacement and stress history. + +TIME 54–70 Read initial displacement. + +TIME 71-87 Read initial velocities. + +TIME 89-111 Read appropriate acceleration data. + +# 10.6.12 Subroutine INVAR + +This routine calculates the stress invariants and yield values for the various yield criteria. The choice of yield criterion is governed by the parameter NCRIT. A similar routine was described in Section 7.8.3. + +```fortran +SUBROUTINE INVAR (DEVIA, LPROP, NCRIT, NMATS, PROPS, SINT3, STEFF, STEMP, THETA, VARJ2, YIELD) INVR 1 +C********** INVR 2 +C INVR 3 +C INVR 4 +C** STRESS INVARIANTS INVR 5 +C INVR 6 +C********** INVR 7 +DIMENSION DEVIA(4), PROPS(NMATS, 1), STEMP(4) INVR 8 +C INVR 9 +C*** INVARIANTS INVR 10 +C INVR 11 +ROOT3=1.73205080757 INVR 12 +SMEAN=(STEMP(1)+STEMP(2)+STEMP(4))/3.0 INVR 13 +DEVIA(1)=STEMP(1)-SMEAN INVR 14 +DEVIA(2)=STEMP(2)-SMEAN INVR 15 +DEVIA(3)=STEMP(3) INVR 16 +DEVIA(4)=STEMP(4)-SMEAN INVR 17 +VARJ2=DEVIA(3)*DEVIA(3)+0.5*(DEVIA(1)*DEVIA(1)+ +DEVIA(2)*DEVIA(2)+DEVIA(4)*DEVIA(4)) INVR 18 +VARJ3=DEVIA(4)*(DEVIA(4)*DEVIA(4)-VARJ2) INVR 19 +STEFF=SQRT(VARJ2) INVR 20 +IF (VARJ2.EQ.0.0.OR.STEFF.EQ.0.0) GO TO 5 INVR 21 +SINT3=-2.5980762113*VARJ3/(VARJ2*STEFF) INVR 22 +GO TO 6 INVR 23 +5 SINT3=0.0 INVR 24 +6 CONTINUE INVR 25 +IF(SINT3.LT.-1.0) SINT3=-1.0 INVR 26 +INVR 27 +``` + + + +```csv +IF(SINT3.GT. 1.0) SINT3= 1.0 INVR 28 +THETA=ASIN(SINT3)/3.0 INVR 29 +GO TO (1,2,3,4) NCRIT INVR 30 +C*** TRESCA INVR 31 +1 YIELD=2.0*COS(THETA)*STEFF INVR 32 +RETURN INVR 33 +C*** VON MISES INVR 34 +2 YIELD=ROOT3*STEFF INVR 35 +RETURN INVR 36 +C*** MOHR-COULOMB INVR 37 +3 PHIRA=PROPS(LPROP,8)*0.017453292 INVR 38 +SNPHI=SIN(PHIRA) INVR 39 +YIELD=SMEAN*SNPHI+STEFF*(COS(THETA)-SIN(THETA)*SNPHI/ROOT3) INVR 40 +RETURN INVR 41 +C*** DRUCKER-PRAGER INVR 42 +4 PHIRA=PROPS(LPROP,8)*0.017453292 INVR 43 +SNPHI=SIN(PHIRA) INVR 44 +YIELD=6.0*SMEAN*SNPHI/(ROOT3*(3.0-SNPHI))+STEFF INVR 45 +RETURN INVR 46 +END INVR 47 +``` + +# 10.6.13 Subroutine JACOBD + +This subroutine evaluates the deformation Jacobian matrix $[J_{D}]_{n}$ for a particular sampling point within an element. + +```fortran +SUBROUTINE JACOBD (CARTD, DLCOD, DJACM, NDIME, NLAPS, NNODE) JACD 1 +C******************************* +C******************************* +C*** DEFORMATION JACOBIAN JACD 2 +C JACD 3 +C******************************* +DIMENSION CARTD(2,9), DLCOD(2,9), DJACM(2,2) JACD 4 +IF(NLAPS.GT.1) GO TO 10 JACD 5 +C JACD 6 +C*** FOR SMALL DISPLACEMENT JACD 7 +C JACD 8 +C JACD 9 +DJACM(1,1)=1.0 JACD 10 +DJACM(2,2)=1.0 JACD 11 +DJACM(1,2)=0.0 JACD 12 +DJACM(2,1)=0.0 JACD 13 +RETURN JACD 14 +RETURN JACD 15 +RETURN JACD 16 +C JACD 17 +C*** FOR LARGE DISPLACEMENT JACD 18 +C JACD 19 +10 CONTINUE JACD 20 +DO 20 IDIME=1, NDIME JACD 21 +DO 20 JDIME=1, NDIME JACD 22 +DJACM(IDIME, JDIME)=0.0 JACD 23 +DO 20 INODE=1, NNODE JACD 24 +DJACM(IDIME, JDIME)=DJACM(IDIME, JDIME) JACD 25 +.+DLCOD(IDIME, INODE)*CARTD(JDIME, INODE) JACD 26 +20 CONTINUE JACD 27 +RETURN JACD 28 +END JACD 29 +``` + +# 10.6.14 Subroutine LINGNL + +This routine calculates the total elastic strain and corresponding elastic stresses at a particular integration point. In this calculation the strains are evaluated using the deformation Jacobian matrix if geometric nonlinear behaviour is to be taken into account. + + + +```fortran +SUBROUTINE LINGNL (CARTD, DJACM, DMATX, ELDIS, GPCOD, KGASP, LINR 1 +KGAUS, NDOFN, NLAPS, NNODE, NSTRE, NTYPE, LINR 2 +POISS, SHAPE, STRAN, STRES, STRIN) LINR 3 +C**************************************************************************************** +C +C*** ELASTIC STRAIN AND STRESSES +C +C**************************************************************************************** +DIMENSION CARTD(2,9), STRAN(4), DMATX(4,4), STRIN(4,1), LINR 9 +ELDIS(2,9), STRES(4), DJACM(2,2), AGASH(2,2), LINR 10 +GPCOD(2,9), SHAPE(9) LINR 11 +C +C*** CALCULATE STRAINS FROM DEFORMATION JACOBIAN +C +IF(NLAPS.LT.2) GO TO 15 +STRAN(1)=0.5*(DJACM(1,1)*DJACM(1,1)+DJACM(2,1)*DJACM(2,1)-1.) +STRAN(2)=0.5*(DJACM(1,2)*DJACM(1,2)+DJACM(2,2)*DJACM(2,2)-1.) +STRAN(3)=DJACM(1,1)*DJACM(1,2)+DJACM(2,1)*DJACM(2,2) LINR 18 +C +C *** FOR SMALL DISPLACEMENTS +C +GO TO 25 +15 CONTINUE +DO 10 IDOFN=1, NDOFN +DO 10 JDOFN=1, NDOFN +BGASH=0.0 +DO 20 INODE=1, NNODE +20 BGASH=BGASH+CARTD(JDOFN, INODE)*ELDIS(IDOFN, INODE) LINR 28 +10 AGASH(IDOFN, JDOFN)=BGASH +STRAN(1)=AGASH(1,1) LINR 29 +STRAN(2)=AGASH(2,2) LINR 30 +STRAN(3)=AGASH(1,2)+AGASH(2,1) LINR 31 +25 CONTINUE +IF(NTYPE.LT.3) GO TO 90 +STRAN(4)=0.0 +DO 70 INODE=1, NNODE +70 STRAN(4)=STRAN(4)+ELDIS(1, INODE)*SHAPE(INODE)/GPCOD(1, KGASP) LINR 32 +EXTRA=0.0 +DO 80 INODE=1, NNODE +80 EXTRA=EXTRA+ELDIS(1, INODE)*SHAPE(INODE)/GPCOD(1, KGASP) LINR 33 +STRAN(4)=STRAN(4)+0.5*EXTRA*EXTRA LINR 34 +90 DO 50 ISTRE=1,4 +STRAN(ISTRE)=STRAN(ISTRE)-STRIN(ISTRE, KGAUS) LINR 35 +50 CONTINUE +C +C*** AND THE CORRESPONDING STRESSES +C +DO 30 ISTRE=1, NSTRE +STRES(ISTRE)=0.0 +DO 30 JSTRE=1, NSTRE +30 STRES(ISTRE)=STRES(ISTRE)+DMATX(ISTRE, JSTRE)*STRAN(JSTRE) LINR 40 +IF(NTYPE.EQ.1) STRES(4)=0.0 +IF(NTYPE.EQ.2) STRES(4)=POISS*(STRES(1)+STRES(2)) LINR 41 +RETURN +END +LINR 42 +LINR 43 +LINR 44 +LINR 45 +LINR 46 +LINR 47 +LINR 48 +LINR 49 +LINR 50 +LINR 51 +LINR 52 +LINR 53 +LINR 54 +LINR 55 +``` + +# 10.6.15 Subroutine LOADPL + +This routine reads load data and evaluates the consistent nodal forces associated with thermal loading. A similar routine was described in Section 6.4.5. The additions which are included here have been discussed in detail in the authors' earlier text Finite Element Programming. $^{(7)}$ + + + +```csv +SUBROUTINE LOADPL (COORD, FORCE, LNODS, MATNO, NDIME, NDOFN, LOAD 1 +NELEM, NGAUS, NMATS, NNODE, NPOIN, NSTRE, LOAD 2 +NTYPE, POSGP, PROPS, RLOAD, STRIN, TEMPE, LOAD 3 +WEIGP) LOAD 4 +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +TWOPI=6.283185307179586 +NEVAB=NNODE*NDOFN +DO 10 IELEM=1,NELEM +DO 10 IEVAB=1,NEVAB +10 RLOAD(IELEM,IEVAB)=0.0 +READ(5,901) TITLE +901 FORMAT (10A4) +WRITE(6,903) TITLE +903 FORMAT(/5X,17HLOAD CASE TITLE -,10A4) +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +``` + + + +```txt +THETA=THETA/57.295779514 +LOAD 65 +C +DO 90 IELEM=1,NELEM +LOAD 66 +C +C*** SET UP PRELIMINARY CONSTANTS +LOAD 69 +C +LPROP=MATNO(IELEM) +LOAD 70 +THICK=PROPS(LPROP,3) +LOAD 71 +DENSE=PROPS(LPROP,4) +LOAD 72 +IF(DENSE.EQ.0.0) GO TO 90 +LOAD 73 +GXCOM=DENSE*GRAVY*SIN(THETA) +LOAD 74 +GYCOM=-DENSE*GRAVY*COS(THETA) +LOAD 75 +C +C*** COMPUTE COORDINATES OF THE ELEMENT NODAL POINTS +LOAD 76 +C +DO 60 INODE=1,NNODE +LOAD 77 +LNODE=IABS(LNODS(IELEM,INODE)) +LOAD 78 +DO 60 IDIME=1,NDIME +LOAD 80 +60 ELCOD(IDIME,INODE)=COORD(LNODE,IDIME) +LOAD 81 +C +C*** ENTER LOOPS FOR AREA NUMERICAL INTEGRATION +LOAD 82 +C +KGASP=0 +LOAD 83 +DO 80 IGAUS=1,NGAUS +LOAD 84 +DO 80 JGAUS=1,NGAUS +LOAD 85 +KGASP=KGASP+1 +LOAD 86 +EXISP=POSGP(IGAUS) +LOAD 87 +ETASP=POSGP(JGAUS) +LOAD 88 +C +C*** COMPUTE THE SHAPE FUNCTIONS AT THE SAMPLING POINTS AND ELEMENTAL +C VOLUME +LOAD 89 +C +CALL SFR2 (DERIV,NNODE,SHAPE,EXISP,ETASP) +LOAD 90 +CALL JACOB2 (CARTD,DERIV,DJACB,ELCOD,GPCOD,IELEM, +KGASP,NNODE,SHAPE) +LOAD 91 +DVOLU=DJACB*WEIGP(IGAUS)*WEIGP(JGAUS) +LOAD 92 +IF(NTYPE.EQ.1) DVOLU=DVOLU*THICK +LOAD 93 +IF(NTYPE.EQ.3) DVOLU=DVOLU*TWOPI*GPCOD(1,KGASP) +LOAD 94 +C +C*** CALCULATE LOADS AND ASSOCIATE WITH ELEMENT NODAL POINTS +LOAD 95 +C +DO 70 INODE=1,NNODE +LOAD 96 +NGASH=(INODE-1)*NDOFN+1 +LOAD 97 +MGASH=(INODE-1)*NDOFN+2 +LOAD 98 +RLOAD(IELEM,NGASH)=RLOAD(IELEM,NGASH)+GXCOM*SHAPE(INODE)*DVOLU +LOAD 100 +70 RLOAD(IELEM,MGASH)=RLOAD(IELEM,MGASH)+GYCOM*SHAPE(INODE)*DVOLU +LOAD 101 +80 CONTINUE +LOAD 102 +90 CONTINUE +LOAD 103 +600 CONTINUE +LOAD 104 +IF(IEDGE.EQ.0) GO TO 700 +LOAD 105 +C +C*** DISTRIBUTED EDGE LOADS SECTION +LOAD 106 +C +READ(5,932) NEDGE +LOAD 107 +932 FORMAT(I5) +LOAD 108 +WRITE(6,912) NEDGE +LOAD 109 +912 FORMAT(1H0,5X,21HNO. OF LOADED EDGES =,I5) +LOAD 110 +WRITE(6,915) +LOAD 111 +915 FORMAT(1H0,5X,38HLIST OF LOADED EDGES AND APPLIED LOADS) +LOAD 112 +NODEG=3 +LOAD 113 +NCODE=NNODE +LOAD 114 +IF(NNODE.EQ.4) NODEG=2 +LOAD 115 +IF(NNODE.EQ.9) NCODE=8 +LOAD 116 +C +LOAD 117 +LOAD 118 +LOAD 119 +LOAD 120 +LOAD 121 +LOAD 122 +LOAD 123 +LOAD 124 +LOAD 125 +LOAD 126 +LOAD 127 +C +LOAD 128 +``` + + + +```csv +C*** LOOP OVER EACH LOADED EDGE +C +DO 160 IEDGE=1,NEDGE +C +C*** READ DATA LOCATING THE LOADED EDGE AND APPLIED LOAD +C +READ (5,902) NEASS,(NOPRS(IODEG),IODEG=1,NODEG) +902 FORMAT(4I5) +WRITE(6,913) NEASS,(NOPRS(IODEG),IODEG=1,NODEG) +913 FORMAT(I10,5X,3I5) +READ (5,914)((PRESS(IODEG,IDOFN),IODEG=1,NODEG),IDOFN=1,NDOFN) +WRITE(6,914)((PRESS(IODEG,IDOFN),IODEG=1,NODEG),IDOFN=1,NDOFN) +914 FORMAT(6F10.3) +ETASP=-1.0 +C +C*** CALCULATE THE COORDINATES OF THE NODES OF THE ELEMENT EDGE +C +DO 100 IODEG=1,NODEG +LNODE=NOPRS(IODEG) +DO 100 IDIME=1,NDIME +100 ELCOD(IDIME,IODEG)=COORD(LNODE,IDIME) +C +C*** ENTER LOOP FOR LINEAR NUMERICAL INTEGRATION +DO 150 IGAUS=1,NGAUS +EXISP=POSGP(IGAUS) +C +C*** EVALUATE THE SHAPE FUNCTIONS AT THE SAMPLING POINTS +C +CALL SFR2 (DERIV,NNODE,SHAPE,EXISP,ETASP) +C +C*** CALCULATE COMPONENTS OF THE EQUIVALENT NODAL LOADS +C +DO 110 IDOFN=1,NDOFN +PGASH(IDOFN)=0.0 +DGASH(IDOFN)=0.0 +DO 110 IODEG=1,NODEG +PGASH(IDOFN)=PGASH(IDOFN)+PRESS(IODEG,IDOFN)*SHAPE( IODEG) +110 DGASH(IDOFN)=DGASH(IDOFN)+ELCOD(IDOFN,IODEG)*DERIV(1,IODEG) +DVOLU=WEIGP(IGAUS) +PXCOM=DGASH(1)*PGASH(2)-DGASH(2)*PGASH(1) +PYCOM=DGASH(1)*PGASH(1)+DGASH(2)*PGASH(2) +IF(NTYPE.NE.3) GO TO 115 +RADUS=0.0 +DO 125 IODEG=1,NODEG +125 RADUS=RADIUS+SHAPE(IODEG)*ELCOD(1,IODEG) +DVOLU=DVOLU*TWOPI*RADIUS +115 CONTINUE +C +C*** ASSOCIATE THE EQUIVALENT NODAL EDGE LOADS WITH AN ELEMENT +C +DO 120 INODE=1,NNODE +NLOCA=IABS(LNODS(NEASS,INODE)) +120 IF(NLOCA.EQ.NOPRS(1)) GO TO 130 +130 JNODE=INODE+NODEG-1 +KOUNT=0 +DO 140 KNODE=INODE,JNODE +KOUNT=KOUNT+1 +NGASH=(KNODE-1)*NDOFN+1 +MGASH=(KNODE-1)*NDOFN+2 +IF(KNODE.GT.NCODE) NGASH=1 +IF(KNODE.GT.NCODE) MGASH=2 +RLOAD(NEASS,NGASH)=RLOAD(NEASS,NGASH)+SHAPE(KOUNT)*PXCOM*DVOLU +140 RLOAD(NEASS,MGASH)=RLOAD(NEASS,MGASH)+SHAPE(KOUNT)*PYCOM*DVOLU +150 CONTINUE +160 CONTINUE +LOAD 129 +LOAD 130 +LOAD 131 +LOAD 132 +LOAD 133 +LOAD 134 +LOAD 135 +LOAD 136 +LOAD 137 +LOAD 138 +LOAD 139 +LOAD 140 +LOAD 141 +LOAD 142 +LOAD 143 +LOAD 144 +LOAD 145 +LOAD 146 +LOAD 147 +LOAD 148 +LOAD 149 +LOAD 150 +LOAD 151 +LOAD 152 +LOAD 153 +LOAD 154 +LOAD 155 +LOAD 156 +LOAD 157 +LOAD 158 +LOAD 159 +LOAD 160 +LOAD 161 +LOAD 162 +LOAD 163 +LOAD 164 +LOAD 165 +LOAD 166 +LOAD 167 +LOAD 168 +LOAD 169 +LOAD 170 +LOAD 171 +LOAD 172 +LOAD 173 +LOAD 174 +LOAD 175 +LOAD 176 +LOAD 177 +LOAD 178 +LOAD 179 +LOAD 180 +LOAD 181 +LOAD 182 +LOAD 183 +LOAD 184 +LOAD 185 +LOAD 186 +LOAD 187 +LOAD 188 +LOAD 189 +LOAD 190 +LOAD 191 +LOAD 192 +LOAD 193 +``` + + + +```fortran +700 CONTINUE + IF(ITEMP.EQ.0) GO TO 800 +C +C*** INITIALIZE AND INPUT THE NODAL TEMPERATURES +C + DO 170 IPOIN=1,NPOIN + 170 TEMPE(IPOIN)=0.0 + WRITE(6,917) + 917 FORMAT(1HO,5X,29HPREScribed NODAL TEMPERATURES) + 180 READ (5,916) NODPT,TEMPE(NODPT) + WRITE(6,916) NODPT,TEMPE(NODPT) + 916 FORMAT(I5,F10.3) + IF(NODPT.LT.NPOIN) GO TO 180 + KGAST=0 +C +C*** LOOP OVER EACH ELEMENT +C + DO 280 IELEM=1,NELEM + LPROP=MATNO(IELEM) + DO 200 INODE=1,NNODE + LNODE=IABS(LNODS(IELEM,INODE)) +C +C*** IDENTIFY THE COORDINATES AND TEMPERATURE OF EACH ELEMENT NODE POINTLOAD +C + DO 190 IDIME=1,NDIME + 190 ELCOD(IDIME,INODE)=COORD(LNODE,IDIME) + 200 ELCOD(2,INODE)=TEMPE(LNODE) +C +C*** SET UP MATERIAL PROPERTIES +C + CALL MODPS (DMATX,LPROP,NMATS,NSTRE,NTYPE,PROPS) + YOUNG=PROPS(LPROP,1) + POISS=PROPS(LPROP,2) + THICK=PROPS(LPROP,3) + ALPHA=PROPS(LPROP,5) +C +C*** ENTER LOOPS FOR AREA NUMERICAL INTEGRATION +C + KGASP=0 + DO 270 IGAUS=1,NGAUS + DO 270 JGAUS=1,NGAUS + KGAST=KGAST+1 + KGASP=KGASP+1 + EXISP=POSGP(IGAUS) + ETASP=POSGP(JGAUS) +C +C*** EVALUATE THE SHAPE FUNCTIONS AND TEMPERATURE AT THE SAMPLING POINTSLOAD +C + ,ELEMENTAL VOLUME AND CARTESIAN DERIVATIVES +C + CALL SFR2 (DERIV,NNODE,SHAPE,EXISP,ETASP) + CALL JACOB2 (CARTD,DERIV,DJACB,ELCOD,GPCOD,IELEM, + KGASP,NNODE,SHAPE) + THERM=0.0 + DO 210 INODE=1,NNODE + 210 THERM=THERM+ELCOD(2,INODE)*SHAPE(INODE) + DVOLU=DJACB*WEIGP(IGAUS)*WEIGP(JGAUS) + IF(NTYPE.EQ.1) DVOLU=DVOLU*THICK + IF(NTYPE.EQ.3) DVOLU=DVOLU*TWOPI*GPCOD(1,KGASP) +C +C*** EVALUATE THE INITIAL THERMAL STRAINS +C + EIGEN=THERM*ALPHA + IF(NTYPE.EQ.2) GO TO 220 + STRAN(1)=-EIGEN + STRAN(2)=-EIGEN + LOAD 194 + LOAD 195 + LOAD 196 + LOAD 197 + LOAD 198 + LOAD 199 + LOAD 200 + LOAD 201 + LOAD 202 + LOAD 203 + LOAD 204 + LOAD 205 + LOAD 206 + LOAD 207 + LOAD 208 + LOAD 209 + LOAD 210 + LOAD 211 + LOAD 212 + LOAD 213 + LOAD 214 + LOAD 215 + LOAD 216 + LOAD 217 + LOAD 218 + LOAD 219 + LOAD 220 + LOAD 221 + LOAD 222 + LOAD 223 + LOAD 224 + LOAD 225 + LOAD 226 + LOAD 227 + LOAD 228 + LOAD 229 + LOAD 230 + LOAD 231 + LOAD 232 + LOAD 233 + LOAD 234 + LOAD 235 + LOAD 236 + LOAD 237 + LOAD 238 + LOAD 239 + LOAD 240 + LOAD 241 + LOAD 242 + LOAD 243 + LOAD 244 + LOAD 245 + LOAD 246 + LOAD 247 + LOAD 248 + LOAD 249 + LOAD 250 + LOAD 251 + LOAD 252 + LOAD 253 + LOAD 254 + LOAD 255 + LOAD 256 + LOAD 257 + LOAD 258 +``` + + + +```csv +STRAN(3)=0.0 +GO TO 230 +220 STRAN(1)=-(1.0+POISS)*EIGEN +STRAN(2)=-(1.0+POISS)*EIGEN +STRAN(3)=0.0 +C +C*** AND THE CORRESPONDING INITIAL STRESSES +C +230 DO 250 ISTRE=1,NSTRE +STRES(ISTRE)=0.0 +DO 240 JSTRE=1,NSTRE +240 STRES(ISTRE)=STRES(ISTRE)+DMATX(ISTRE,JSTRE)*STRAN(JSTRE) +250 STRIN(ISTRE,KGAST)=STRES(ISTRE) +IF(NTYPE.EQ.2) STRIN(4,KGAST)=-YOUNG*EIGEN +IF(NTYPE.EQ.1) STRIN(4,KGAST)=0.0 +C +C*** CALCULATE THE EQUIVALENT NODAL FORCES AND ASSOCIATE WITH THE +C ELEMENT NODES +C +EXTRA=0.0 +DO 260 INODE=1,NNODE +IF(NTYPE.EQ.3) EXTRA=DVOLU*SHAPE(INODE)*STRES(4)/GPCOD(1,KGASP) +NGASH=(INODE-1)*NDOFN+1 +MGASH=(INODE-1)*NDOFN+2 +RLOAD(IELEM,NGASH)=RLOAD(IELEM,NGASH)+EXTRA +-(CARTD(1,INODE)*STRES(1)+CARTD(2,INODE)*STRES(3))*DVOLU +260 RLOAD(IELEM,MGASH)=RLOAD(IELEM,MGASH) +-(CARTD(1,INODE)*STRES(3)+CARTD(2,INODE)*STRES(2))*DVOLU +270 CONTINUE +280 CONTINUE +800 CONTINUE +C WRITE(6,907) +C 907 FORMAT(1H0,5X,36H TOTAL NODAL FORCES FOR EACH ELEMENT) +C DO 290 IELEM=1,NELEM +C 290 WRITE(6,905) IELEM,(RLOAD(IELEM,IEVAB),IEVAB=1,NEVAB) +C 905 FORMAT(1X,14,5X,8E12.4/(10X,8E12.4)) +DO 5 IELEM=1,NELEM +KEVAB=0 +DO 5 INODE=1,NNODE +LNODE=LNODS(IELEM,INODE) +NPOSN=(LNODE-1)*NDOFN +DO 5 IDOFN=1,NDOFN +KEVAB KEVAB+1 +NPOSN=NPOSN+1 +FORCE(NPOSN)=FORCE(NPOSN)+RLOAD(IELEM,KEVAB) +5 CONTINUE +RETURN +END +LOAD 259 +LOAD 260 +LOAD 261 +LOAD 262 +LOAD 263 +LOAD 264 +LOAD 265 +LOAD 266 +LOAD 267 +LOAD 268 +LOAD 269 +LOAD 270 +LOAD 271 +LOAD 272 +LOAD 273 +LOAD 274 +LOAD 275 +LOAD 276 +LOAD 277 +LOAD 278 +LOAD 279 +LOAD 280 +LOAD 281 +LOAD 282 +LOAD 283 +LOAD 284 +LOAD 285 +LOAD 286 +LOAD 287 +LOAD 288 +LOAD 289 +LOAD 290 +LOAD 291 +LOAD 292 +LOAD 293 +LOAD 294 +LOAD 295 +LOAD 296 +LOAD 297 +LOAD 298 +LOAD 299 +LOAD 300 +LOAD 301 +LOAD 302 +LOAD 303 +LOAD 304 +LOAD 305 +LOAD 306 +``` + +# 10.6.16 Subroutine LUMASS + +This subroutine evaluates the lumped mass vector and consistent mass matrix for the finite element mesh. If INTGR(I)=1, it generates the consistent mass matrix and if INTGR(I)=2, it generates a special lumped mass vector. In the special mass lumping scheme which is employed, the diagonal terms of the consistent mass matrix are scaled to preserve the total mass. The element consistent mass matrices are written on tape 3. The consistent mass matrix is not used in DYNPAK. + +This subroutine also reads concentrated masses and assembles them into the global diagonal mass vector. diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_043.md b/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_043.md new file mode 100644 index 00000000..62930e0f --- /dev/null +++ b/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_043.md @@ -0,0 +1,821 @@ + + +```txt +SUBROUTINE LUMASS (COORD, INTGR, LNODS, MATNO, NCONM, NDIME, NDOFN, NELEM, NGAUM, NMATS, NNODE, NPOIN, NTYPE, PROPS, YMASS) +``` + +```txt +C +``` + +```txt +C +``` + +```txt +C *** CALCULATES LUMPED MASS FOR 4, 8 AND 9 NODED ELEMENT +``` + +```txt +C +``` + +```txt +C****************************************************************************************** +``` + +```txt +DIMENSION COORD(NPOIN,1),ELCOD(2,9),DIAGM(9),POSGP(4), +. LNODS(NELEM,1),CARTD(2,9),SHAPE(9),WEIGP(4), +. PROPS(NMATS,1),GPCOD(2,9),MATNO(1),YMASS(1), +. EMASS(171),DERIV(2,9),INTGR(1) +``` + +```txt +C +``` + +```csv +REWIND 3 +TWOPI=6.283185307179586 +NEVAB=NNODE*NDOFN +NTOTV=NPOIN*NDOFN +DO 500 ITOTV =1,NTOTV +``` + +```txt +500 YMASS(ITOTV)=0.0 +CALL GAUSSQ (NGAUM, POSGP, WEIGP) +DO 100 IELEM=1, NELEM +DO 5 IEVAB=1, 171 +``` + +```txt +5 EMASS(IEVAB)=0.0 +IMASS=INTGR(IELEM) +KGASP=0 +TAREA=0.0 +LPROP=MATNO(IELEM) +THICK=PROPS(LPROP,3) +RHOEL=PROPS(LPROP,4) +DO 10 INODE=1,NNODE +DIAGM(INODE)=0.0 +LNODE=LNODS(IELEM,INODE) +DO 10 IDIME=1,NDIME +ELCOD(IDIME,INODE)=COORD(LNODE,IDIME) +``` + +```csv +10 CONTINUE +DO 70 IGAUS=1,NGAUM +EXISP=POSGP(IGAUS) +DO 70 JGAUS=1,NGAUM +KGASP=KGASP+1 +ETASP=POSGP(JGAUS) +CALL SFR2 (DERIV,NNODE,SHAPE,EXISP,ETASP) +CALL JACOB2 (CARTD,DERIV,DJACB,ELCOD,GPCOD,IELEM,KGASP,NNODE,SHAPE) +``` + +```txt +DVOLU=DJACB*WEIGP(IGAUS)*WEIGP(JGAUS) +IF(NTYPE.EQ.1) DVOLU=DVOLU*THICK +IF(NTYPE.EQ.3) DVOLU=DVOLU*TWOPI*GPCOD(1,KGASP) +IF(IMASS.EQ.1) GO TO 210 +DO 20 INODE=1,NNODE +SHAPI=SHAPE(INODE) +``` + +```txt +20 DIAGM(INODE)=DIAGM(INODE)+SHAPI*SHAPI*DVOLU +TAREA=TAREA+DVOLU +``` + +```matlab +210 IF(IMASS.EQ.2) GO TO 70 +DVOLU=DVOLU*RHOEL +IEVAB=1 +KOUNT=NEVAB +DO 30 INODE=1,NNODE +SHAPI=SHAPE(INODE) +DO 60 JNODE=INODE,NNODE +DMASS=DVOLU*SHAPI*SHAPE(JNODE) +EMASS(IEVAB)=EMASS(IEVAB)+DMASS +JEVAB=IEVAB+KOUNT +EMASS(JEVAB)=EMASS(JEVAB)+DMASS +60 IEVAB=IEVAB+2 +KOUNT=KOUNT-2 +``` + + + +```fortran +IEVAB=JEVAB+1 MASS 65 +30 CONTINUE MASS 66 +70 CONTINUE MASS 67 +C MASS 68 +C*** WRITES CONSISTENT MASS MATRIX ON TAPE 3 MASS 69 +C MASS 70 +IF(IMASS.EQ.2) GO TO 200 MASS 71 +WRITE(3) EMASS MASS 72 +WRITE(6,90) (EMASS(I),I=1,171) MASS 73 +200 IF(IMASS.EQ.1) GO TO 100 MASS 74 +C MASS 75 +C *** GENERATES LUMPED MASS MATRIX PROPORTIONAL TO DIAGONAL MASS 76 +C MASS 77 +SUMAS=0. MASS 78 +DO 40 INODE=1,NNODE MASS 79 +40 SUMAS=SUMAS+DIAGM(INODE) MASS 80 +TAREA=TAREA*RHOEL MASS 81 +SUMAS=TAREA/SUMAS MASS 82 +DO 50 INODE=1,NNODE MASS 83 +LNODE=LNODS(IELEM,INODE) MASS 84 +IPOSN=(LNODE-1)*NDOFN MASS 85 +DO 50 IDOFN=1,NDOFN MASS 86 +IPOSN=IPOSN+1 MASS 87 +YMASS(IPOSN)=YMASS(IPOSN)+DIAGM(INODE)*SUMAS MASS 88 +50 CONTINUE MASS 89 +90 FORMAT(2X,9E12.3) MASS 90 +100 CONTINUE MASS 91 +C MASS 92 +C CONCENTRATED MASSES MASS 93 +C MASS 94 +IF(NCONM.EQ.0) RETURN MASS 95 +WRITE(6,900) MASS 96 +DO 520 ICONM=1,NNOM MASS 97 +READ(5,910) IPOIN,XCMAS,YCMAS MASS 98 +900 FORMAT(5X,19HCONCENTRATED MASSES) MASS 99 +WRITE(6,910) IPOIN,XCMAS,YCMAS MASS 100 +NPOSN=(IPOIN-1)*NDOFN+1 MASS 101 +YMASS(NPOSN)=YMASS(NPOSN)+XCMAS MASS 102 +NPOSN=NPOSN+1 MASS 103 +YMASS(NPOSN)=YMASS(NPOSN)+YCMAS MASS 104 +520 CONTINUE MASS 105 +C WRITE(6,90) (YMASS(I),I=1,NTOTV) MASS 106 +910 FORMAT(I5,2F10.3) MASS 107 +RETURN MASS 108 +END MASS 109 +``` + +# MASS 24 + +Sets indicator for mass matrix evaluation. INTGR(I) = 1 for the consistent mass matrix and INTGR(I) = 2 for the special lumped mass vector. + +# MASS 35-52 + +Evaluate the diagonal element of the consistent mass matrix DIAGM. + +# MASS 53-63 + +Evaluates the element consistent mass matrix. + +# MASS 72 + +Writes element consistent mass matrix on tape 3. + +# MASS 78-80 + +Evaluates ELMAS, the sum of the diagonal elements. + +# MASS 81 + +Determines the total element mass from the element volume TAREA and mass density RHOEL. + + + +MASS 83–89 Scales the diagonal terms using the factor TAREA/ELMAS to preserve element mass and assembles the result into diagonal mass vector YMASS. + +MASS 95–107 Reads the concentrated masses and assembles them into YMASS. + +# 10.6.17 Subroutine MODPS + +This subroutine evaluates the elasticity matrix and has been described earlier in Chapter 6. The only changes involved are given below. + +```csv +SUBROUTINE MODPS (DMATX ,LPROP ,NMATS ,NSTRE ,NTYPE ,PROPS ) MODP 1 +C************************** +C +C ** ELASTICITY D MATRIX +C +C************************** +DIMENSION DMATX(4,4),PROPS(NMATS,1) +``` + +# 10.6.18 Subroutine NODXYR + +It calculates $(r, z)$ coordinates from $(R, \Theta)$ coordinates for axisymmetric problems. If coordinates of midside nodes are not read, it evaluates them by linear interpolation. An almost identical subroutine was described in Chapter 6. + +```fortran +SUBROUTINE NODXYR (COORD, LNODS, NELEM, NNODE, NPOIN, NRADS, NTYPE) NODX 1 +C*************** NODX 2 +C NODX 3 +C*** INTERPOLATION OF MIDSIDE AND CENTER NODES NODX 4 +C NODX 5 +C*************** NODX 6 +DIMENSION COORD(NPOIN, 1), LNODS(NELEM, 1) NODX 7 +C NODX 8 +IF(NTYPE.NE.3.OR.NRADS.EQ.0) GO TO 40 NODX 9 +C NODX 10 +C*** CHANGE POLAR COORDINATES TO CARTISIAN NODX 11 +DO 50 IPOIN=1, NPOIN NODX 12 +RADDI=COORD(IPOIN, 1) NODX 13 +THETA=COORD(IPOIN, 2) NODX 14 +THETA=0.017453292*THETA NODX 15 +COORD(IPOIN, 1)=RADDI*SIN(THETA) NODX 16 +50 COORD(IPOIN, 2)=RADDI*COS(THETA) NODX 17 +C NODX 18 +40 IF(NNODE.EQ.4) RETURN NODX 19 +C NODX 20 +LNODE = NNODE - 1 NODX 21 +DO 30 Ielem=1, NELEM NODX 22 +C*** LOOP OVER EACH ELEMENT EDGE NODX 23 +DO 20 INODE=1, NNODE, 2 NODX 24 +IF(INODE.EQ.9) GO TO 20 NODX 25 +C*** COMPUTE THE NODE NUMBER OF THE FIRST NODE NODX 26 +NODST=LNODS(IELEM, INODE) NODX 27 +IGASH=INODE+2 NODX 28 +IF(IGASH.GT.LNODE) IGASH=1 NODX 29 +C*** COMPUTE THE NODE NUMBER OF THE LAST NODE NODX 30 +NODFN=LNODS(IELEM, IGASH) NODX 31 +MIDPT=INODE+1 NODX 32 +``` + + + +```fortran +C*** COMPUTE THE NODE NUMBER OF THE INTERMEDIATE NODE NODX 33 + NODMD=LNODS(IELEM,MIDPT) NODX 34 + TOTAL=ABS(COORD(NODMD,1))+ABS(COORD(NODMD,2)) NODX 35 +C*** IF THE COORDINATES OF THE INTERMEDIATE NODE ARE BOTH ZERO NODX 36 +C INTERPOLATE BY A STRAIGHT LINE NODX 37 + IF(TOTAL.GT.0.0) GO TO 20 NODX 38 + KOUNT=1 NODX 39 + 10 COORD(NODMD,KOUNT)=(COORD(NODST,KOUNT)+COORD(NODFN,KOUNT))/2.0 NODX 40 + KOUNT=KOUNT+1 NODX 41 + IF(KOUNT.EQ.2) GO TO 10 NODX 42 + 20 CONTINUE NODX 43 + 30 CONTINUE NODX 44 + RETURN NODX 45 + END NODX 46 +``` + +# 10.6.19 Subroutine OUTDYN + +This routine writes out most of the output on the line printer and on various tapes for plotting purposes. It outputs the displacements and stresses every NOUTP steps. It also writes the displacement and stress histories of specified nodal and integration points at every NOUTP steps. The complete state of displacements is also written on tape 13 for a deformation plot. The complete state of the stresses is written on tape 4. The principal stresses and their directions are also calculated and output. + +```txt +SUBROUTINE OUTDYN (DISPL, DTIME, ISTEP, NDOFN, NELEM, NGAUS, OUTP 1 +NGRQS, NOUTD, NOUTP, NPOIN, NPRQD, NREQD, OUTP 2 +NREQS, NTYPE, STRSG, TDISP, VIVEL) OUTP 3 +C******************************* +OUTP 4 +C +C** OUTPUT ROUTINE +OUTP 5 +C +C******************************* +OUTP 6 +OUTP 7 +OUTP 8 +DIMENSION STRSG(4,1), DISPL(1), NPRQD(1), STRSP(3), OUTP 9 +VIVEL(5,1), TDISP(1), NGRQS(1) OUTP 10 +NSTR1=4 +OUTP 11 +KSTEP=ISTEP +OUTP 12 +MGAUS=NELEM*NGAUS*NGAUS +OUTP 13 +IF(ISTEP.EQ.1) WRITE(10,925) OUTP 14 +TTIME=TTIME+DTIME OUTP 15 +C +OUTP 16 +C *** WRITES DISPLACEMENT HISTORY AT REQUESTED NODAL POINTS ON TAPE 10 OUTP 17 +C *** AND STRESS HISTORY AT REQUESTED GAUSS POINTS AT EVERY NOUTD STEPSOUTP 18 +C +OUTP 19 +KOUNT=0 +OUTP 20 +KOUTD=(ISTEP/NOUTD)*NOUTD +OUTP 21 +IF(KOUTD.NE.ISTEP) GO TO 510 +OUTP 22 +DO 500 IPOIN=1,NPOIN +OUTP 23 +DO 500 IREQD=1,NREQD +OUTP 24 +IF(IPOIN.NE.NPRQD(IREQD)) GO TO 500 +OUTP 25 +NPOSN=(IPOIN-1)*NDOFN+1 +OUTP 26 +NPOSM=NPOSN+1 +OUTP 27 +KOUNT=KOUNT+1 +OUTP 28 +DISPL(KOUNT)=TDISP(NPOSN) +OUTP 29 +KOUNT=KOUNT+1 +OUTP 30 +DISPL(KOUNT)=TDISP(NPOSM) +OUTP 31 +500 CONTINUE +OUTP 32 +WRITE(10,960) (DISPL(IKOUN), IKOUN=1,KOUNT), TTIME OUTP 33 +``` + + + +```csv +DO 520 IGAUS=1,MGAUS +DO 520 IREQS=1,NREQS +IF(IGAUS.NE.NGRQS(IREQS)) GO TO 520 +WRITE(11,950) (STRSG(ISTR1,IGAUS),ISTR1=1,NSTR1) +520 CONTINUE +510 KOUTD=(KSTEP/NOUTP)*NOUTP +IF(KOUTD.NE.KSTEP) RETURN +XTIME=FLOAT(KSTEP)*DTIME +WRITE(6,604) KSTEP,XTIME +604 FORMAT(//5X,28H DISPLACEMENTS AT TIME STEP ,I10,5X,5HTIME ,E20.11) +C +C *** REARRANGE DISPLACEMENT VECTOR +C +NODEI=0 +DO 550 IPOIN=1,NPOIN +DO 550 IDOFN=1,NDOFN +NODEI=NODEI+1 +DISPL(NODEI)=TDISP(NODEI) +550 CONTINUE +C +C*** OUTPUT DISPLACEMENTS +C +925 FORMAT(5X,' DISPLACEMENTS ') +WRITE(6,990) +990 FORMAT(/3(1X,'NNODE',3X,'X-DISP',6X,'Y-DISP',3X)/) +DO 560 IPOIN=1,NPOIN,3 +NGASI=NDOFN*IPOIN-1 +NGASJ=NGASI+NDOFN +NGASK=NGASJ+NDOFN +MGASI=NGASI+1 +MGASJ=NGASJ+1 +MGASK=NGASK+1 +JPOIN=IPOIN+1 +KPOIN=JPOIN+1 +C +C *** WRITES DISPLACEMENTS ON TAPE 13 FOR DEFORMATION PLOT +C +WRITE(13,910) IPOIN ,(DISPL(IGASI),IGASI=NGASI,MGASI) +IF(JPOIN.GT.NPOIN) GO TO 200 +WRITE(13,910) JPOIN ,(DISPL(IGASJ),IGASJ=NGASJ,MGASJ) +IF(KPOIN.GT.NPOIN) GO TO 200 +WRITE(13,910) KPOIN ,(DISPL(IGASK),IGASK=NGASK,MGASK) +200 CONTINUE +C +C *** WRITES DISPLACEMENTS ON OUTPUT FILE +C +560 WRITE(6,920) IPOIN,DISPL(NGASI),DISPL(MGASI), +JPOIN,DISPL(NGASJ),DISPL(MGASJ), +KPOIN,DISPL(NGASK),DISPL(MGASK) +C +C *** WRITES STRESSES ON OUTPUT FILE +C +WRITE(6,900) +IF(NTYPE.NE.3) WRITE(6,970) +970 FORMAT(1HO,1X,4HG.P.,6X,9HXX-STRESS,5X,9HYY-STRESS,5X,9HXY-STRESS,OUTP +.5X,9HZZ-STRESS,6X,8HMAX P.S.,6X,8HMIN P.S.,3X,5HANGLE,3X,6H P.S.)OUTP +IF(NTYPE.EQ.3) WRITE(6,975) +975 FORMAT(1HO,1X,4HG.P.,6X,9HRR-STRESS,5X,9HZZ-STRESS,5X,9HRZ-STRESS,OUTP +.5X,9HTT-STRESS,6X,8HMAX P.S.,6X,8HMIN P.S.,3X,5HANGLE,3X,6H P.S.)OUTP +KGAUS=0 +DO 570 IELEM=1,NELEM +KELGS=0 +WRITE(6,930) IELEM +930 FORMAT(1HO,5X,13HELEMENT NO. =,I5) +``` + + + +```csv +DO 570 IGAUS=1,NGAUS +DO 570 JGAUS=1,NGAUS +KGAUS=KGAUS+1 +KELGS=KELGS+1 +XGASH=(STRSG(1,KGAUS)+STRSG(2,KGAUS))*0.5 +XGISH=(STRSG(1,KGAUS)-STRSG(2,KGAUS))*0.5 +XGESH=STRSG(3,KGAUS) +XGOSH=SQRT(XGISH*XGISH+XGESH*XGESH) +STRSP(1)=XGASH+XGOSH +STRSP(2)=XGASH-XGOSH +IF(XGISH.EQ.0.0) XGISH=0.1E-20 +STRSP(3)=ATAN(XGESH/XGISH)*28.647889757 +C +C *** WRITES COMPLETE STRESS STATE ON TAPE 4 +C +WRITE(4,950) (STRSG(ISTR1,KGAUS),ISTR1=1,NSTR1), +.(STRSP(ISTRE),ISTRE=1,3) +570 WRITE(6,940) KELGS,(STRSG(ISTR1,KGAUS),ISTR1=1,NSTR1), +.(STRSP(ISTRE),ISTRE=1,3),VIVEL(5,KGAUS) +980 FORMAT(1X,60I2) +960 FORMAT(1X,10E11.4) +950 FORMAT(7E10.4) +940 FORMAT(I5,2X,6E14.6,F8.3,E14.6) +900 FORMAT(/,10X,8HSTRESSES,/) +920 FORMAT(3(1X,I5,2E12.5)) +910 FORMAT(I5,2E15.6) +RETURN +END +OUTP 98 +OUTP 99 +OUTP 100 +OUTP 101 +OUTP 102 +OUTP 103 +OUTP 104 +OUTP 105 +OUTP 106 +OUTP 107 +OUTP 108 +OUTP 109 +OUTP 110 +OUTP 111 +OUTP 112 +OUTP 113 +OUTP 114 +OUTP 115 +OUTP 116 +OUTP 117 +OUTP 118 +OUTP 119 +OUTP 120 +OUTP 121 +OUTP 122 +OUTP 123 +OUTP 124 +OUTP 125 +``` + +# 10.6.20 Subroutine PREVOS + +This routine reads and write the initial forces and stresses. + +```csv +SUBROUTINE PREVOS (FORCE ,NDOFN ,NELEM ,NGAUS ,NPOIN ,NPREV , STRIN ) +C*************** PREV 2 +C +C*** GRAVITY LOADS AND STRESSES PREV 3 +C +C*************** PREV 4 +C +DIMENSION FORCE(1) ,STRIN(4,1) PREV 5 +C +IF(NPREV.EQ.0) RETURN PREV 6 +C +NSTR1=4 PREV 7 +NGAU2=NGAUS*NGAUS PREV 8 +C +C*** READ GRAVITY LOADS PREV 9 +C +WRITE(6,920) PREV 10 +920 FORMAT(//4X,6H NODE ,17H GRAVITY X-LOAD: ,17H GRAVITY Y-LOAD: /) PREV 11 +200 READ (5,900) NGASH,XGASH,YGASH PREV 12 +900 FORMAT(I5,4F10.3) PREV 13 +910 FORMAT(I10,4E18.5) PREV 14 +NPOSN=(NGASH-1)*NDOFN+1 PREV 15 +FORCE(NPOSN)=XGASH PREV 16 +NPOSN=NPOSN+1 PREV 17 +FORCE(NPOSN)=YGASH PREV 18 +WRITE(6,910) NGASH,XGASH,YGASH PREV 19 +IF (NGASH.NE.NPOIN) GO TO 200 PREV 20 +C +C*** READ GRAVITY STRESS PREV 21 +C PREV 22 +PREV 23 +PREV 24 +PREV 25 +PREV 26 +PREV 27 +PREV 28 +PREV 29 +PREV 30 +``` + + + +WRITE(6,930) PREV 31 +930 FORMAT(//2X,9HGAUSS PT.,17H GRAVITY X-STRESS,17H GRAVITY Y-STRESS,PREV 32 +.18H GRAVITY XY-STRESS,17H GRAVITY Z-STRESS/) PREV 33 +DO 500 IELEM=1,NELEM PREV 34 +DO 500 IGAUS=1,NGAU2 PREV 35 +READ(5,900) KGAUS,(STRIN(ISTRI,KGAUS),ISTRI=1,NSTR1) PREV 36 +500 WRITE(6,910)KGAUS,(STRIN(ISTRI,KGAUS),ISTRI=1,NSTR1) PREV 37 +RETURN PREV 38 +END PREV 39 + +# 10.6.21 Subroutine RESVPL + +This routine evaluates the internal resisting force vector + +$$ +\boldsymbol {p} _ {n} = \int_ {\Omega} [ \boldsymbol {B} ] _ {n} ^ {T} \boldsymbol {\sigma} _ {n} d \Omega . +$$ + +It is very similar to the routine described in Section 8.8. + +SUBROUTINE RESVPL (COORD, DTIME, LNODS, MATNO, NCRIT, NDIME, RESD 1 +NDOFN, NELEM, NGAUS, NLAPS, NNODE, NMATS, RESD 2 +NPOIN, NSTRE, NTYPE, POSGP, PROPS, RESID, RESD 3 +RLOAD, STRIN, STRSG, TDISP, VISTN, VIVEL, RESD 4 +WEIGP) +C****************************************************************************************** +C +C*** EVALUATION OF INTEGRAL (B)**T*(SIGMA) +C +C****************************************************************************************** +DIMENSION COORD(NPOIN,1), DERIV(2,9), DJACM(2,2), AVECT(4), MATNO(1), RESD 11 +PROPS(NMATS,1), DLCOD(2,9), STRIN(4,1), DEVIA(4), TDISP(1), RESD 12 +LNODS(NELEM,1), GPCOD(2,9), STRSG(4,1), STRAN(4), POSGP(1), RESD 13 +RLOAD(NELEM,1), CARTD(2,9), VISTN(4,1), STRES(4), WEIGP(1), RESD 14 +DMATX(4,4), ELCOD(2,9), VIVEL(5,1), SHAPE(9), RESID(1), RESD 15 +BMATX(4,18), ELDIS(2,9), DESTN(4) RESD 16 +KGAUS=0 +NSTR1=4 +NEVAB=NNODE*NDOFN +NTOTV=NPOIN*NDOFN +TWOPI=6.283185307179586 +DO 530 IELEM=1, NELEM +DO 540 IEVAB=1, NEVAB +540 RLOAD(IELEM, IEVAB)=0.0 +530 CONTINUE +DO 510 ITOTV=1, NTOTV +510 RESID(ITOTV)=0.0 +C +C*** LOOP OVER ALL THE ELEMENTS +C +DO 20 IELEM=1, NELEM +LPROP=MATNO(IELEM) +THICK=PROPS(LPROP,3) +POISS=PROPS(LPROP,2) +FRICT=PROPS(LPROP,8) +C +C*** COMPUTE NEW COORDINATES AND DISPLACEMENTS OF THE +C ELEMENT NODAL POINTS +C +DO 30 INODE =1, NNODE +LNODE=IABS(LNODS(IELEM, INODE)) +NPOSN=(LNODE-1)*NDOFN +RESD 1 +RESD 2 +RESD 3 +RESD 4 +RESD 5 +RESD 6 +RESD 7 +RESD 8 +RESD 9 +RESD 10 +RESD 11 +RESD 12 +RESD 13 +RESD 14 +RESD 15 +RESD 16 +RESD 17 +RESD 18 +RESD 19 +RESD 20 +RESD 21 +RESD 22 +RESD 23 +RESD 24 +RESD 25 +RESD 26 +RESD 27 +RESD 28 +RESD 29 +RESD 30 +RESD 31 +RESD 32 +RESD 33 +RESD 34 +RESD 35 +RESD 36 +RESD 37 +RESD 38 +RESD 39 +RESD 40 +RESD 41 +RESD 42 + + + +```csv +DO 30 IDOFN=1,NDOFN RESD 43 +NPOSN=NPOSN+1 RESD 44 +ELCOD(IDOFN,INODE)=COORD(LNODE,IDOFN) RESD 45 +DLCOD(IDOFN,INODE)=COORD(LNODE,IDOFN)+TDISP(NPOSN) RESD 46 +30 ELDIS(IDOFN,INODE)=TDISP(NPOSN) RESD 47 +CALL MODPS (DMATX,LPROP,NMATS,NSTRE,NTYPE,PROPS) RESD 48 +KGASP=0 RESD 49 +DO 40 IGAUS=1,NGAUS RESD 50 +DO 40 JGAUS=1,NGAUS RESD 51 +KGAUS=KGAUS+1 RESD 52 +KGASP=KGASP+1 RESD 53 +EXISP=POSGP(IGAUS) RESD 54 +ETASP=POSGP(JGAUS) RESD 55 +C +CALL SFR2 (DERIV,NNODE,SHAPE,EXISP,ETASP) RESD 57 +CALL JACOB2 (CARTD,DERIV,DJACB,ELCOD,GPCOD, RESD 58 +IELEM,KGASP,NNODE,SHAPE) RESD 59 +CALL JACOBD (CARTD,DLCOD,DJACM,NDIME,NLAPS,NNODE) RESD 60 +DVOLU=DJACB*WEIGP(IGAUS)*WEIGP(JGAUS) RESD 61 +IF(NTYPE.EQ.1) DVOLU=DVOLU*THICK RESD 62 +IF(NTYPE.EQ.3) DVOLU=DVOLU*TWOPI*GPCOD(1,KGASP) RESD 63 +CALL BLARGE (BMATX,CARTD,DJACM,DLCOD,GPCOD, RESD 64 +KGASP,NLAPS,NNODE,NTYPE,SHAPE) RESD 65 +CALL LINGNL (CARTD,DJACM,DMATX,ELDIS,GPCOD,KGASP, RESD 66 +KGAUS,NDOFN,NLAPS,NNODE,NSTRE,NTYPE, RESD 67 +POISS,SHAPE,STRAN,STRES,VISTN) RESD 68 +C +DO 580 ISTR1=1,NSTR1 RESD 69 +580 STRES(ISTR1)=STRES(ISTR1)+STRIN(ISTR1,KGAUS) RESD 70 +DO 570 ISTR1=1,NSTR1 RESD 71 +570 STRSG(ISTR1,KGAUS)=STRES(ISTR1) RESD 72 +C +IF(NLAPS.EQ.2.OR.NLAPS.EQ.0) GO TO 200 RESD 73 +C +CALL INVAR (DEVIA,LPROP,NCrit,NMATS,PROPS,SINT3,STEFF, RESD 74 +STRES,THETA,VARJ2,YIELD) RESD 75 +CALL YIELDF (AVECT,DEVIA,FRICT,NCrit,SINT3,STEFF,THETA,VARJ2) RESD 76 +CALL FLOWVP (AVECT,KGAUS,LPROP,NCrit,NMATS,PROPS, RESD 77 +STEFF,VIVEL,YIELD) RESD 78 +C +C*** VISCOPLASTIC STRAIN INCREMENT AND A MEASURE FOR HARDENING RESD 79 +C +DO 60 ISTR1=1,NSTR1 RESD 80 +DESTN(ISTR1)=VIVEL(ISTR1,KGAUS)*DTIME RESD 81 +60 VISTN(ISTR1,KGAUS)=VISTN(ISTR1,KGAUS)+DESTN(ISTR1) RESD 82 +DEBAR=SQRT((2.0*(DESTN(1)*DESTN(1)+DESTN(2)*DESTN(2)+ RESD 83 +DESTN(4)*DESTN(4))+DESTN(3)*DESTN(3))/3.0) RESD 84 +VIVEL(5,KGAUS)=DEBAR RESD 85 +C +C*** COMPUT INT(B**T*SICMA) ON ELEMENT LEVEL RESD 86 +C +200 CONTINUE RESD 94 +KEVAB=0 RESD 95 +DO 502 INODE=1,NNODE RESD 96 +DO 502 IDOFN=1,NDOFN RESD 97 +KEVAB=KEVAB+1 RESD 98 +DO 501 ISTRE=1,NSTRE RESD 99 +501 RLOAD(IELEM,KEVAB)=RLOAD(IELEM,KEVAB)+ +.BMATX(ISTRE,KEVAB)*STRSG(ISTRE,KGAUS)*DVOLU RESD 100 +502 CONTINUE RESD 101 +40 CONTINUE RESD 102 +20 CONTINUE RESD 103 +C +C*** ASSEMBLY OF RESID VECTOR RESD 104 +C +``` + + + +
CRESD 107
DO 500 IELEM=1,NELEMRESD 108
KEVAB=0RESD 109
DO 500 INODE=1,NNODERESD 110
LNODE=LNODS(IELEM,INODE)RESD 111
NPOSN=(LNODE-1)*NDOFNRESD 112
DO 500 IDOFN=1,NDOFNRESD 113
KEVAB=KEVAB+1RESD 114
NPOSN=NPOSN+1RESD 115
RESID(NPOSN)=RESID(NPOSN)+RLOAD(IELEM,KEVAB)RESD 116
500CONTINUERESD 117
RETURNRESD 118
ENDRESD 119
+ +
RESD 66-68Call LINGNL to determine the state of stress at the current Gauss point.
RESD 77-78Call INVAR to evaluate stress invariants at the current Gauss point.
RESD 79Call YIELDF to select the yield function and calculate the a vector.
RESD 80-81Call FLOWVP to define the rate of viscoplastic straining VIVEL if the stress point is outside the current yield surface.
RESD 86Evaluate the increments of viscoplastic strains DESTN.
RESD 87Evaluate the viscoplastic strains $(\epsilon_{vp})_{n+1}$ for the next time station $t_n + \Delta t$ , VISTN.
RESD 88-90Determine a measure of hardening for the current yield surface.
RESD 95-101Evaluate $p_n^{(e)}$ at the element level, RLOAD.
RESD 108-117Assemble $p_n$ , RESID.
+ +# 10.6.22 Subroutine YIELDF + +This subroutine selects the yield function and calculates the vector a (AVECT) and is almost identical to the version described in Section 7.8.4.1. + +```fortran +SUBROUTINE YIELDF (AVECT, DEVIA, FRICT, NCRIT, SINT3, STEFF, YELD 1 +THETA, VARJ2) YELD 2 +C********** YELD 3 +C YELD 4 +C *** SELECTS YIELD FUNCTION AND CALCULATES VECTOR 'AVECT' YELD 5 +C YELD 6 +C********** YELD 7 +DIMENSION AVECT(4), DEVIA(4), VECA1(4), VECA2(4), VECA3(4) YELD 8 +IF(STEFF.EQ.0.0) RETURN YELD 9 +NSTR1=4 YELD 10 +TANTH=TAN(THETA) YELD 11 +SINTH=SIN(THETA) YELD 12 +COSTH=COS(THETA) YELD 13 +COST3=COS(3.0*THETA) YELD 14 +ROOT3=1.73205080757 YELD 15 +``` + + + +```txt +C*** CALCULATE VECTOR A1 +VECA1(1)=1.0 +VECA1(2)=1.0 +VECA1(3)=0.0 +VECA1(4)=1.0 +C*** CALCULATE VECTOR A2 +DO 10 ISTR1=1,NSTR1 +10 VECA2(ISTR1)=DEVIA(ISTR1)/(2.0*STEFF) +VECA2(3)=DEVIA(3)/STEFF +C*** CALCULATE VECTOR A3 +VECA3(1)=DEVIA(2)*DEVIA(4)+VARJ2/3.0 +VECA3(2)=DEVIA(1)*DEVIA(4)+VARJ2/3.0 +VECA3(3)=-2.0*DEVIA(3)*DEVIA(4) +VECA3(4)=DEVIA(1)*DEVIA(2)-DEVIA(3)*DEVIA(3)+VARJ2/3.0 +GO TO (1,2,3,4) NCRIT +C*** TRESCA +1 CONS1=0.0 +ABTHE=ABS(THETA*57.29577951308) +IF(ABTHE.LT.29.0) GO TO 20 +CONS2=ROOT3 +CONS3=0.0 +GO TO 40 +20 CONS2=2.0*(COSTH+SINTH*TAN(3.0*THETA)) +CONS3=ROOT3*SINTH/(VARJ2*COST3) +GO TO 40 +C*** VON MISES +2 CONS1=0.0 +CONS2=ROOT3 +CONS3=0.0 +GO TO 40 +C*** MOHR-COULOMB +3 CONS1=SIN(FRICT*0.017453292)/3.0 +ABTHE=ABS(THETA*57.29577951308) +IF(ABTHE.LT.29.0) GO TO 30 +CONS3=0.0 +PLUMI=1.0 +IF(THETA.GT.0.0) PLUMI=-1.0 +CONS2=0.5*(ROOT3+PLUMI*CONS1/ROOT3) +GO TO 40 +30 TANT3=TAN(3.0*THETA) +CONS2=COSTH*(1.0+TANTH*TANT3)+CONS1*(TANT3-TANTH)/ROOT3) +CONS3=(ROOT3*SINTH+CONS1*COSTH)/(2.0*VARJ2*COST3) +GO TO 40 +C*** DRUCKER-PRAGER +4 SNPHI=SIN(FRICT*0.017453292) +CONS1=2.0*SNPHI/(ROOT3*(3.0-SNPHI)) +CONS2=1.0 +CONS3=0.0 +40 CONTINUE +DO 50 ISTR1=1,NSTR1 +50 AVECT(ISTR1)=CONS1*VECA1(ISTR1)+CONS2* +.VECA2(ISTR1)+CONS3*VECA3(ISTR1) +RETURN +END +YELD 16 +YELD 17 +YELD 18 +YELD 19 +YELD 20 +YELD 21 +YELD 22 +YELD 23 +YELD 24 +YELD 25 +YELD 26 +YELD 27 +YELD 28 +YELD 29 +YELD 30 +YELD 31 +YELD 32 +YELD 33 +YELD 34 +YELD 35 +YELD 36 +YELD 37 +YELD 38 +YELD 39 +YELD 40 +YELD 41 +YELD 42 +YELD 43 +YELD 44 +YELD 45 +YELD 46 +YELD 47 +YELD 48 +YELD 49 +YELD 50 +YELD 51 +YELD 52 +YELD 53 +YELD 54 +YELD 55 +YELD 56 +YELD 57 +YELD 58 +YELD 59 +YELD 60 +YELD 61 +YELD 62 +YELD 63 +YELD 64 +YELD 65 +YELD 66 +YELD 67 +YELD 68 +YELD 69 +``` + +# 10.7 Examples + +# 10.7.1 Introduction + +To illustrate the use of DYNPAK we now describe the nonlinear transient dynamic analysis of (i) a spherical shell and (ii) a concrete gravity dam. diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_044.md b/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_044.md new file mode 100644 index 00000000..e1615a79 --- /dev/null +++ b/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_044.md @@ -0,0 +1,278 @@ + + +# 10.7.2 Spherical shell example + +The shell, $^{(8)}$ shown in Fig. 10.3, is subjected to a distributed step pressure of 600 lb/in $^{2}$ . The material is assumed to obey the Von Mises yield condition with linear isotropic hardening. The dimensions and properties of the shell are given as follows: + +
Internal radius $R = 22.27 \text{ in}$
Thickness of shell $t = 0.41 \text{ in}$
Semi angle $\alpha = 26.67^{\circ}$
Elastic modulus $E = 10.5 \times 10^{6} \text{ lb/in}^{2}$
Poisson's ratio $\nu = 0.3$
Yield stress $\sigma_{Y} = 0.024 \times 10^{6} \text{ lb/in}^{2}$
Tangent hardening modulus $E_{T} = 0.21 \times 10^{6} \text{ lb/}^{2}$
Mass density $\rho = 2.45 \times 10^{-4} \text{ lb-sec}^{2}/\text{in}^{4}$
Step distributed pressure $p = 600 \text{ lb/in}^{2}$
+ +![](images/page-431_eaa980c1f220dd59324168f134df3624983f2bb7e24a9d7bd12c5a78037df88e.jpg) + +
+text_image + +P +R +α +
+ +Fig. 10.3 Spherical shell and finite element mesh. + +The shell is divided into ten, 8-noded, axisymmetric, isoparametric elements. The fundamental period of the shell is $T_{f} = 0.55 \times 10^{-3}$ sec, (Reference 8). For explicit central difference analysis, the time step is taken as $0.4 \times 10^{-6}$ sec. + + + +In order to illustrate the versatility of program DYNPAK we consider the following three cases: + +(i) Small elastic displacements +(ii) Large elastic displacements +(iii) Small elasto-viscoplastic displacements (with a fluidity parameter value of $\gamma = 100.0$ ). + +![](images/page-432_b577deeefe5e1301ab205a51ab928b9a75a8ea94a5ffe40df56d3a93b899e519.jpg) + +
+line + +| t × 10⁻³ secs | Small elastic displacement | Large elastic displacement | +| ------------- | -------------------------- | -------------------------- | +| 0 | 0.00 | 0.00 | +| 2 | 0.03 | 0.04 | +| 4 | 0.07 | 0.08 | +| 6 | -0.04 | -0.03 | +| 8 | 0.08 | 0.07 | +| 10 | 0.02 | 0.05 | +
+ +Fig. 10.4(a) Results of the transient dynamic analysis of a spherical shell cap. Cases (i) and (ii). + +Figure 10.4(a) shows the vertical displacement of the crown lower point for the analyses based on both small and large elastic displacement assumptions. The results show that the inclusion of geometrically nonlinear effects in the analysis elongates the period. Figure 10.4(b) shows the small displacement, elasto-viscoplastic response (Case (iii)) of the spherical shell cap in which the value of the fluidity parameter is taken as $\gamma = 100.0$ . It should be noted that permanent viscoplastic deflections occur thus providing a completely different response to either of the elastic responses shown in Fig. 10.4(a). + +In Chapter 11 this problem is repeated using an elasto-plastic material model. It should be noted that in order to simulate elasto-plastic behaviour with DYNPAK a high value of the fluidity parameter (say $\gamma = 10000.0$ ) + + + +![](images/page-433_97afa398ebad3d67bb1d4d5a8214769a8f028ae153e2bb3dc2fe78f8d3577fa0.jpg) + +
+line + +| Time (secs) | Vertical displacement | +|-------------|----------------------| +| 0 | 0.00 | +| 2 | 0.06 | +| 4 | 0.08 | +| 6 | 0.03 | +| 8 | 0.07 | +| 10 | 0.06 | +
+ +Fig. 10.4(b) Results of the transient dynamic analysis of a spherical shell cap. Case (iii). + +should be adopted. Interested readers may like to compare DYNPAK and MIXDYN for elasto-plastic behaviour using a high fluidity parameter. However, care should be taken since the use of high fluidity parameter values requires the use of a smaller time step when an Euler scheme is used to evaluate the viscoplastic strains (see Section 8.3). Typical input data for Case (ii) are given in Appendix IV. + +At this stage it is probably worth mentioning the important problem of combining material and geometric nonlinearities. Among the several papers on this topic in the existing literature we suggest that the interested reader could profitably refer to the following as a starting point for further study: + +McMEEKING, R. M. and RICE, J. R., Finite element formulations for problems of large elastic-plastic deformation, Int. J. Solids Structures, 11, 601–616 (1975). +HIBBITT, H. D., MARCAL, P. V. and RICE, J. R., A finite element formulation for problems of large strain and large displacement, Int. J. Solids Structures, 6, 1069–1086 (1970). +BATHE, K. J., RAMM, E. and WILSON, E. L., Finite element formulations for large deformation analysis, Int. J. Num. Meth. Engng., 9, 353–386 (1975). + + + +# 10.7.3 Gravity dam example + +The geometry of the dam, the seismic acceleration history, the water level and material properties for both dam and foundation are arbitrary. + +![](images/page-434_af57c0d6db14c103667713fd1d93e8f6b5c182a5d826e61bfc600eaebefec384.jpg) + +
+other + +| Dimension | Value | +| --------- | ----------- | +| Height | 81.45 | +| Width | 107.00 | +| E (tsec²) | 3164000 t/m² | +| ν (sec²) | 0.20 | +| ρ (sec²) | 0.269 tsec²/m⁴ | +| Width | 15.24 m | +| E (tsec²) | 1800000 t/m² | +| ν (sec²) | 0.20 | +| ρ (sec²) | 0.183 tsec²/m⁴ | +| Width | -50.00 | +
+ +Fig. 10.5(a) Concrete gravity dam. + +![](images/page-434_a25a23f53784a8ec9287724e7fd79a6699c0670ee71d3b945926545e35993013.jpg) + +
+area + +| Added masses | 8 | 9 | 10 | 11 | 12 | 13 | 14 | +|---|---|---|---|---|---|---|---| +| Row 1 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | +| Row 2 | 18 | 19 | 20 | 23 | 26 | 25 | 28 | +| Row 3 | 15 | 16 | 17 | 22 | 23 | 25 | 26 | +| Row 4 | 30 | 31 | 32 | 27 | 28 | 29 | 24 | +The chart displays a stacked area chart with each row representing a mass value. The total added mass is labeled on the left side. +
+ +Fig. 10.5(b) Finite element mesh for concrete gravity dam. + + + +Both the gravity dam and the foundation shown in Fig. 10.5(a) are idealized with two-dimensional, plane-strain, 8-noded isoparametric elements as shown in Fig. 10.5(b), using a $2 \times 2$ Gauss integration rule for the stiffness evaluation, and using a special mass lumping scheme with a $3 \times 3$ Gauss integration rule. The adopted $2 \times 2$ Gauss integration rule for the stiffness terms ensures that no locking behaviour will occur in the mesh, whereas the $3 \times 3$ Gauss integration rule for the lumped mass matrix terms renders better mass representation. The model base is assumed to be fixed, i.e. u = v = 0, and side boundaries are represented by horizontal rollers, i.e. v = 0. + +A short duration analytic earthquake (sinesweep) $^{(9)}$ with a maximum acceleration level 0.33 g (developed as an equivalent to the El Centro NS accelerogram) will be used as a prescribed horizontal acceleration history at the model base level. It is assumed that this signal is the result of the deconvolution process of a prescribed signal at the foundation level. The displacements obtained in the solution process are relative to the model base. + +Both the concrete and rock are assumed to behave as elasto-viscoplastic materials with no hardening. The Mohr–Coulomb yield surface is adopted, and the parameters c and $\phi$ are obtained from the uniaxial properties $f_{cu}$ and $f_{t}$ as indicated in Table 10.3. + +$f_{t}, f_{cu} = \text{tensile, compressive strengths of concrete,}$ + +$$ +\alpha = \frac {f _ {t}}{f _ {c u}} = \frac {1 - \sin \phi}{1 + \sin \phi}, +$$ + +$$ +\phi = \operatorname{arc} \sin \left(\frac {1 - a}{1 + a}\right), +$$ + +$$ +c = \frac {(a) ^ {- 1 / 2}}{2} f _ {c u}, +$$ + +$$ +F _ {0} (\text { Mohr - Coulomb }) = c \cos \phi . +$$
$f_{cu}$ $(t/m^{2})$ $f_{t}$ $(t/m^{2})$ $\alpha$ $c$ $(t/m^{2})$ $\phi$ $F_{0}=c\cos\phi$ $(t/m^{2})$
concrete40005000.125707.1162.73323.94
rock36004000.133547.7261.93257.75
+ +Table 10.3 Mohr–Coulomb yield surface parameters for concrete dam example. + +The values of the fluidity parameters $\gamma$ are considered to be the same for both the concrete and rock materials. Values of $\gamma = 0.00001$ and $\gamma = 0.001$ have been used for the two analyses presented. The stress level in the structure prior to the seismic excitation is assumed to be due to the self-weight and hydrostatic pressure of the water only. + + + +The influence of the reservoir water on the dynamic behaviour of the dam is considered by taking into account the mass of water attached to the upstream face of the dam. The simple representation of 'added mass' with concentrated masses is used. The adopted model could be improved significantly with transmitting boundaries, better 'added mass' representation, a more realistic signal and a finer mesh. + +The choice of the time step length depends on two criteria. For the explicit central difference integration scheme of the dynamic equilibrium equations, the highest mesh frequency defines the critical time step length + +$$ +\Delta t _ {C D} = \frac {2}{\omega_ {\max}} \simeq \mu L \left(\frac {\rho (1 + \nu) (1 - 2 \nu)}{E (1 - \nu)}\right) ^ {1 / 2}. \tag {10.51} +$$ + +For the integration of the equations, which govern viscoplastic straining using the Euler method, the critical time step for the Mohr-Coulomb viscoplastic material is defined as + +$$ +\Delta t _ {M C} = \frac {4 (1 + \nu) (1 - 2 \nu) c \cos \phi}{\gamma (1 - 2 \nu + \sin^ {2} \phi)}. \tag {10.52} +$$ + +For the mathematical model under consideration, $(L = 2.4665 \, \text{m})$ , the choice of the time step is governed by the $\Delta t_{CD}$ criterion for both analyses. Note that since + +$$ +\Delta t _ {C D} = 0. 0 0 0 4 7 8 \mathrm{sec} \tag {10.53} +$$ + +the adopted time step length is $\Delta t = 0.0004$ sec. + +On the basis of the adopted mathematical model, (Fig. 10.5), input data can be prepared following the user notes, given in the Appendix III. + +![](images/page-436_fa943596f9d1a41bc8de58d4364a9e4a3c9c0e91d228f25f8b87fe58795a0670.jpg) + +
+line +| Time (s) | Acceleration Level 0.5 M/SEC2 | +| -------- | ----------------------------- | +| 0 | 0 | +| 1 | 0.5 | +| 2 | 1.5 | +| 3 | 3.0 | +| 4 | 2.0 | +| 5 | -5.0 | +| 6 | 6.0 | +| 7 | -6.0 | +| 8 | 5.0 | +| 9 | -4.0 | +| 10 | 3.0 | +| 11 | -3.0 | +| 12 | 2.0 | +| 13 | -2.0 | +| 14 | 1.0 | +| 15 | -1.0 | +
+ +JOHNSON/EPSTEIN SINESWEEP EARTHQUAKE 0.20 SEC +Fig. 10.6(a) Johnson/Epstein sinesweep earthquake. + + + +
SINESWEEP DT 0.01 SEC 300 ENTRIES
0.00340.00690.01040.01400.01770.02150.0255
0.02960.03390.03850.04330.04840.05390.0597
0.06590.07250.07950.08710.09510.10380.1130
0.12290.13350.14490.15700.17000.18380.1986
0.21440.23120.24910.26810.28840.30980.3326
0.35670.38230.40920.43770.46770.49920.5324
0.56720.60360.64170.68150.72290.76600.8106
0.85680.90460.95371.00421.05581.10861.1622
1.21651.27131.32631.38121.43571.48941.5420
1.59301.64191.68811.73121.77051.80541.8351
1.85891.87611.88591.88741.87971.86211.8337
1.79351.74081.67471.59451.49931.38871.2621
1.11910.95940.78290.58990.38050.1554-0.0845
-0.3381-0.6038-0.8798-1.1638-1.4533-1.7372-1.9899
-2.2286-2.4500-2.6507-2.8273-2.9764-3.0948-3.1793
-3.2271-3.2356-3.2025-3.1262-3.0056-2.8402-2.6303
-2.3768-2.0819-1.7485-1.3804-0.9825-0.5607-0.1220
0.32580.77421.21391.63492.02722.38062.6849
2.93063.10903.21253.23513.17263.02302.7867
2.46702.06981.60411.08180.5176-0.0715-0.6660
-1.2454-1.7879-2.2718-2.6762-2.9821-3.1733-3.2373
-3.1663-2.9582-2.6169-2.1529-1.5831-0.9256-0.2197
0.47961.13751.72072.19882.54612.74392.7810
2.65532.37431.95511.42350.81330.1640-0.4813
-1.0789-1.5873-1.9703-2.2002-2.2599-2.1450-1.8645
-1.4408-0.9084-0.31160.29880.86991.35101.6985
1.88031.87931.69561.34760.87030.3129-0.2659
-0.8041-1.2427-1.5330-1.6417-1.5565-1.2875-0.8674
-0.34820.20470.72011.13071.38171.43901.2948
0.96960.5104-0.0151-0.5283-0.9507-1.2170-1.2848
-1.1436-0.8166-0.35880.15180.62650.98141.1528
1.10940.85960.4508-0.0377-0.5100-0.8715-1.0488
-1.0054-0.7506-0.33920.13890.57780.87840.9720
0.83710.50570.0580-0.3962-0.7437-0.8966-0.8159
-0.5228-0.09560.35020.69220.83520.73890.4312
0.0025-0.4198-0.7084-0.7749-0.5989-0.23640.1960
0.55750.72850.65190.3541-0.0615-0.4488-0.6697
-0.6446-0.38310.01710.40450.63040.60730.3446
-0.0521-0.4214-0.6111-0.5422-0.24440.15390.4789
0.58700.43020.0801-0.3015-0.5364-0.5135-0.2443
0.13980.44900.52900.3396-0.0213-0.3657-0.5121
-0.3826-0.04790.30810.48760.38990.0713-0.2837
-0.4675-0.3716-0.05410.29160.45280.3295
+ +Fig. 10.6(b) Digital form of Johnson/Epstein sinesweep earthquake. + +Prior to the dynamic analysis, the initial stresses $\sigma_{0}$ must be evaluated using some static finite element program. Nodal loads and the stress state for every Gauss integration point are recorded, and added to the input data for the dynamic analysis. The sinesweep accelerogram and 300 readings for $\Delta t = 0.01$ sec are given in Fig. 10.6. The accelerogram information is read in from a separate input unit (here tape 7, the assumed seismic excitation in the horizontal direction). + +The displacement histories for selected nodal points and stress histories for selected Gauss integration points are written on separate output units (tape 10, tape 11) and may be used later for plotting the results. The displacement histories for nodal points 51 (structure base level) and 127 (dam crest) are given in Fig. 10.7. + + + +![](images/page-438_6aa72dadd8aa2806bea974a97982cde45cadc66f50c6e0a6d1bdce8170f386b6.jpg) + +
+line + +| time (sec) | γ = 0.00001 | γ = 0.01 | +| ---------- | ----------- | -------- | +| 0 | 0 | 0 | +| 1 | ~0.5 | ~0.3 | +| 2 | ~1.0 | ~0.8 | +| 3 | ~0.5 | ~0.3 | +| 4 | ~0.8 | ~0.6 | +
+ +Fig. 10.7 Results of transient dynamic analysis of a concerte gravity dam. + +# 10.8 Problems + +10.1 A simply supported beam is subjected to a step uniformly distributed load. The dimensions and material properties of the beam are shown in Fig. 10.8(a). Only one quarter of the beam needs to be analysed as shown in Fig. 10.8(b). Use DYNPAK to find the midspan lateral deflection when the step lateral load is 0.75 $p_{0}$ where $p_{0}$ is the static collapse load. Note that this problem has been solved by Liu and Lin $^{(10)}$ , Bathe et al. $^{(11)}$ and Nagarajan and Popov. $^{(12)}$ Use the Von Mises yield criterion, a high value of the fluidity parameter $\gamma$ and 8-node elements. +10.2 Repeat Problem 10.1 using the Tresca yield criterion. +10.3 Repeat Problem 10.1 using loads of intensity 0.625 $p_{0}$ and 0.50 $p_{0}$ . Compare your results with those of Liu and Lin. $^{(10)}$ +10.4 For a step lateral load of 0.625 $p_{0}$ , repeat Problem 10.1 for various degrees of hardening. Compare your results with those of Liu and Lin. $^{(10)}$ +10.5 Solve the problem given in Chapters 7 and 8 using dynamic relaxation. $^{(13,14)}$ +10.6 Implement an explicit elasto-plastic, transient dynamic, Mindlin plate program based on DYNPAK. Typical examples are given elsewhere. $^{(15,16)}$ + + + +![](images/page-439_e79e0816d8afd26ac9d2521072c05e8557aa912b4222900bd1c5a98ce9d9a4b5.jpg) + +
+text_image + +0.75 p₀ lb/in² +2 in +30 in +1 in +15 in +
+ +Fig. 10.8 Simply supported beam example (a) Geometry and loading, (b) Finite element idealisation. + +# 10.9 References + +1. NICKELL, R. E., Nonlinear dynamics by mode superposition, Comp. Meth. Appl. Mech. Engng. 7, 107–129 (1976). +2. BATHE, K. J. and WILSON, E. L., Numerical Methods in Finite Element Analysis, Prentice-Hall, Englewood Cliffs, New Jersey, 308-344 (1976). +3. NAYAK, G. C., Plasticity and large deformation by the finite element method, Ph.D. Thesis, University College of Swansea (1971). +4. IRONS, B. M. and AHMAD, S., Finite Element Techniques, Ellis Horwood, Chichester (1980). +5. BICANIĆ, N., Nonlinear finite element transient response of concrete structures, Ph.D. Thesis, University College of Swansea (1978). +6. HINTON, E., ROCK, T. A. and ZIENKIEWICZ, O. C., A note on mass lumping and related processes in the finite element method, Int. J. Earthquake Engng. Struct. Dynamics, 4, 246–249 (1976). +7. HINTON, E. and OWEN, D. R. J., Finite Element Programming, Academic Press, London (1977). +8. BATHE, K. J. and OZDEMIR, H., Elasto-plastic large deformation static and dynamic analysis, Computers and Structures, 6, 81-92 (1976). +9. JOHNSON, G. R. and EPSTEIN, H. I., Short duration analytic earthquake, J. ASCE, Struct. Div., 102, No. ST5, 993-1000 (1976). +10. Liu, S. C. and Lin, T. H., Elastic-plastic dynamic analysis of structures using known elastic solutions, Int. J. Earthquake Engng. Struct. Dynamics, 7, 147-159 (1979). +11. BATHE, K. J., OZDEMIR, H. and WILSON, E. L., Static and geometric and material nonlinear analysis, Structures and Materials Research Report No. UC SESM 74-4, University of California, Berkeley (1974). +12. NAGARAJAN, S. and POPOV, E. P., Elastic-plastic dynamic analysis of axisymmetric solid, Computers and Structures, 4, 1117-1134 (1974). +13. PICA, A. and HINTON, E., Transient and pseudo-transient analysis of Mindlin plates, Int. J. Num. Meth. Engng. 15, 189–208 (1980). +14. BREW, J. S. and BROTTON, D. M., Nonlinear structural analysis by dynamic relaxation, Int. J. Num. Meth. Engng. 3, 463–483 (1971). + + + +15. HINTON, E., OWEN, D. R. J. and SHANTARAM, D., Dynamic transient nonlinear behaviour of thick and thin plates, In: The Mathematics of Finite Elements and Applications, MAFELAP II, 1975, Ed. J. R. Whiteman, Academic Press, London, 423–438 (1977). +16. RAO, S. S. and RAGHAVAN, K. S., Dynamic response of inelastic thick plates, AIAA J., 17, 85–90 (1979). diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_045.md b/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_045.md new file mode 100644 index 00000000..9f156880 --- /dev/null +++ b/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_045.md @@ -0,0 +1,313 @@ + + +# Chapter 11 Implicit-explicit transient dynamic analysis + +Written in collaboration with D. K. Paul + +# 11.1 Introduction + +In Chapter 10 we have shown that the explicit, central difference time stepping scheme is a simple and powerful method of time integration. The main drawback of the scheme is that it is conditionally stable. Thus the computational advantages of the central difference scheme are counterbalanced by the very small size of time step necessary when some stiff (and/or small) elements are present. For such problems the unconditionally stable implicit schemes permit the use of larger time steps, the size of which is governed only by accuracy considerations. Unfortunately these schemes which require matrix factorisations involve larger computer core storage and more operations per time step than the central difference scheme. The selection of a suitable time integration scheme is therefore largely a matter of experience. + +In some problems, typified by the one illustrated in Fig. 11.1, we may be confronted with a situation in which there is a 'soft' subregion $\Omega^{E}$ where an + +![](images/page-441_8f75f37bc4258965dd48ad992b15d6e07ae58b7e496007f37154a7493422bbbb.jpg) + +
+text_image + +Implicit subregion, Ω′ +Explicit subregion, Ω^ε +
+ +Fig. 11.1 Implicit-explicit partitioning. + + + +explicit scheme is desirable and a 'stiff' subregion $\Omega^I$ where an implicit scheme is preferable for greater efficiency. In such cases it is possible to simultaneously make use of both implicit and explicit algorithms. Implicit-explicit schemes offer a unified approach to problems of structural transient dynamics and can lead to significant computational advantages. + +Implicit-explicit schemes were first introduced by Belytschko and Mullen $^{(1-3)}$ and were given an alternative form by Hughes and co-workers $^{(4-6)}$ and Park et al. $^{(7-8)}$ It can be shown that the stability of such schemes is governed by the explicit elements. + +In this chapter Implicit and Implicit-Explicit methods for nonlinear transient dynamic analysis are discussed and we follow the element partitioning approach described by Hughes. A program, named MIXDYN, for Implicit-Explicit linear and nonlinear transient dynamic analysis is included. Some numerical examples are solved to show some of the capabilities of the program. The same program could be modified for static analysis by some simple changes. + +# 11.2 Implicit time integration + +# 11.2.1 Newmark's algorithm + +In order to introduce the implicit/explicit algorithm we describe the predictor-corrector form of the Newmark scheme for the integration of the semi-discrete system of equations which govern nonlinear transient dynamic problems. Typically at time station $t_{n} + \Delta t$ these equations take the form + +$$ +\boldsymbol {M} \boldsymbol {a} _ {n + 1} + \boldsymbol {p} _ {n + 1} = \boldsymbol {f} _ {n + 1} \tag {11.1} +$$ + +where $M, a_{n+1}, p_{n+1}$ and $f_{n+1}$ are the mass matrix, acceleration vector, internal force vector (which may depend on the displacements $d_{n+1}$ and velocities $\dot{d}_{n+1}$ and their histories) and applied force vector respectively. Let + +$$ +[ \boldsymbol {K} _ {T} ] _ {n + 1} = \partial \boldsymbol {p} _ {n + 1} / \partial \boldsymbol {d} _ {n + 1} \text { and } [ \boldsymbol {C} _ {T} ] _ {n + 1} = \partial \boldsymbol {p} _ {n + 1} / \partial \dot {\boldsymbol {d}} _ {n + 1} \tag {11.2} +$$ + +denote the tangent stiffness and damping matrices respectively. + +In the Newmark scheme we endeavour to satisfy the following equations + +$$ +\boldsymbol {M} \boldsymbol {a} _ {n + 1} + \boldsymbol {p} _ {n + 1} = \boldsymbol {f} _ {n + 1} \tag {11.3} +$$ + +$$ +\boldsymbol {d} _ {n + 1} = \tilde {\boldsymbol {d}} _ {n + 1} + \Delta t ^ {2} \beta \boldsymbol {a} _ {n + 1} \tag {11.4} +$$ + + + +$$ +\boldsymbol {v} _ {n + 1} = \widetilde {\boldsymbol {v}} _ {n + 1} + \Delta t \gamma \boldsymbol {a} _ {n + 1} \tag {11.5} +$$ + +where + +$$ +\tilde {\boldsymbol {d}} _ {n + 1} = \boldsymbol {d} _ {n} + \Delta t \boldsymbol {v} _ {n} + \Delta t ^ {2} (1 - 2 \beta) \boldsymbol {a} _ {n} / 2 \tag {11.6} +$$ + +$$ +\widetilde {v} _ {n + 1} = v _ {n} + \Delta t (1 - \gamma) a _ {n}. \tag {11.7} +$$ + +Note that $d_{n}$ , $v_{n}$ and $a_{n}$ are the approximations to $d(t_{n})$ , $\dot{d}(t_{n})$ and $\ddot{d}(t_{n})$ and $\beta$ and $\gamma$ are free parameters which control the accuracy and stability of the method. The values $\tilde{d}_{n+1}$ and $\tilde{v}_{n+1}$ are predictor values and $d_{n+1}$ and $v_{n+1}$ are corrector values. + +Initially the displacements $d_{0}$ and velocities $v_{0}$ are provided and we find the accelerations $a_{0}$ from the expression + +$$ +\boldsymbol {M} \boldsymbol {a} _ {0} = \boldsymbol {f} _ {0} - \boldsymbol {p} (\boldsymbol {d} _ {0}, \boldsymbol {v} _ {0}). \tag {11.8} +$$ + +Thus $a_{0}$ may be found by a factorization, forward reduction and back substitution unless M is diagonal in which case the solution is trivial. + +We then solve (11.3) to (11.7) by forming an ‘effective static problem’† which is solved using a Newton Raphson type scheme, as described earlier. The algorithm is summarised in Table 11.1. + +Table 11.1 Newmark's algorithm +
1Set iteration counter i = 0.
2Begin predictor phase in which we set
$d_{n+1}^{[i]} = \tilde{d}_{n+1} = d_n + \Delta tv_n + \Delta t^2(1-2\beta)a_n/2$ (i)
$v_{n+1}^{[i]} = \tilde{v}_{n+1} = v_n + \Delta t(1-\gamma)a_n$ (ii)
$a_{n+1}^{[i]} = [d_{n+1}^{[i]} - \tilde{d}_{n+1}]/(\Delta t^2\beta) = 0.$ (iii)
3Evaluate residual forces using the equation
$\psi^{[i]} = f_{n+1} - Ma_{n+1}^{[i]} - p(d_{n+1}^{[i]}, v_{n+1}^{[i]}).$ (iv)
4If required, form the effective stiffness matrix using the expression
$K^* = M/(\Delta t^2\beta) + \gamma C_T/(\Delta t\beta) + K_T(d_{n+1}^{[i]}).$ (v)
Otherwise use a previously calculated $K^*$ .
5Factorize, forward reduction and backsubstitute as required to solve
$K^* \Delta d^{[i]} = \psi^{[i]}$ .(vi)
6Enter corrector phase in which we set
$d_{n+1}^{[i+1]} = d_{n+1}^{[i]} + \Delta d^{[i]}$ (vii)
$a_{n+1}^{[i+1]} = [d_{n+1}^{[i+1]} - \tilde{d}_{n+1}]/(\Delta t^2\beta)$ (viii)
$v_{n+1}^{[i+1]} = v_{n+1} + \Delta t\gamma a_{n+1}^{[i+1]}$ .(ix)
7If $\Delta d^{[i]}$ and/or $\psi^{[i]}$ do not satisfy the convergence conditions then set i = i+1 and go to step 3, otherwise continue.
8Set(x)
$d_{n+1} = d_{n+1}^{[i+1]}$ (xi)
$v_{n+1} = v_{n+1}^{[i+1]}$ (xii)
$a_{n+1} = a_{n+1}^{[i+1]}$
for use in the next time step. Also set n = n+1, form p and begin next time step.
+ +\* In this chapter $\gamma$ is a Newmark parameter and not the viscoplastic fluidity parameter. $\dagger K^{*}\Delta d^{[i]} = \psi^{[i]}$ . + + + +# 11.2.2 Predictor-corrector algorithm + +Let us now consider an 'explicit' algorithm associated with the Newmark schemes described earlier. In this explicit predictor-corrector algorithm we assume that the mass matrix $M$ is diagonal and we make use of the expression + +$$ +M a _ {n + 1} + p \left(\tilde {d} _ {n + 1}, \tilde {v} _ {n + 1}\right) = f _ {n + 1} \tag {11.9} +$$ + +Notice that the calculation is explicit since we use corrector values obtained from information given in the previous step. + +As we would like to eventually combine the implicit and explicit methods we organise our implementation of this explicit method in a similar fashion to the implementation given of the implicit scheme in the previous section. Table 11.2 summarises the algorithm. + +Table 11.2 Explicit predictor-corrector algorithm +
1Begin predictor phase by setting $d_{n+1}^{[0]} = \tilde{d}_{n+1} = d_n + \Delta t v_n + \Delta t^2 (1 - 2\beta) a_n / 2$ (i) $v_{n+1}^{[0]} = \tilde{v}_{n+1} = v_n + \Delta t (1 - \gamma) a_n$ (ii) $a_{n+1}^{[0]} = 0.$ (iii)
2Evaluate the residual forces using the equation $\psi^{[0]} = f_{n+1} - p(d_{n+1}^{[0]}, v_{n+1}^{[0]}).$ (iv)
3If required, form the ‘effective’ stiffness matrix using the expression $K^* = M / (\Delta t^2 \beta).$ (v)Note that as the mass matrix $M$ does not change $K^*$ will be formed once only.
4Perform factorization, forward reduction and backsubstitution as required to solve $K^* \Delta d^{[0]} = \psi^{[0]}$ (vi)
5Enter the corrector phase in which we set $d_{n+1}^{[1]} = d_{n+1}^{[0]} + \Delta d^{[0]}$ (vii) $a_{n+1}^{[1]} = [d_{n+1}^{[1]} - \tilde{d}_{n+1}] / (\Delta t^2 \beta)$ (viii) $v_{n+1}^{[1]} = v_{n+1} + \Delta t \gamma a_{n+1}^{[1]}$ . (ix)
6Set $d_{n+1} = d_{n+1}^{[1]}$ (x) $v_{n+1} = v_{n+1}^{[1]}$ (xi) $a_{n+1} = a_{n+1}^{[1]}$ (xii)for use in the next time step. Also set $n = n+1$ , form $p$ and begin next time step.
+ +# 11.3 Implicit-explicit algorithm + +# 11.3.1 Introduction + +We now combine the methods described in Sections 11.2.1 and 11.2.2 so that the finite element mesh contains two groups of elements: the implicit group and the explicit group. The superscripts I and E will henceforth refer to the implicit and explicit groups respectively. + +In the implicit-explicit algorithm we iterate within each time step in order to satisfy the equation + + + +$$ +M a _ {n + 1} + p ^ {I} (d _ {n + 1}, v _ {n + 1}) + p ^ {E} (\tilde {d} _ {n + 1}, \tilde {v} _ {n + 1}) = f _ {n + 1} \tag {11.10} +$$ + +in which $M = M^{I} + M^{E}$ and $f_{n+1} = f_{n+1}^{I} + f_{n+1}^{E}$ . Note that we assume $M^{E}$ is diagonal. + +# 11.3.2 The structure of the effective stiffness matrix + +The algorithm, which is summarised in Table 11.3, is very similar to the implicit algorithm given in Section 11.2.2. The profile structure of $K^{*}$ is very interesting. It has diagonal subregions corresponding to the explicit group of elements. Elsewhere, $K^{*}$ has a profile structure which corresponds to the connectivity of the implicit group only. + +Table 11.3 Implicit-explicit algorithm +
1Set iteration counter i = 0.
2Begin predictor phase in which we set
$d_{n+1}^{[i]} = \tilde{d}_{n+1} = d_n + \Delta tv_n + \Delta t^2(1-2\beta)a_n/2$ (i)
$v_{n+1}^{[i]} = \tilde{v}_{n+1} = v_n + \Delta t(1-\gamma)a_n$ (ii)
$a_{n+1}^{[i]} = [d_{n+1}^{[i]}-d_{n+1}]/(\Delta t^2\beta) = 0.$ (iii)
3Evaluate residual forces using the equation
$\psi^{[i]} = f_{n+1} - Ma_{n+1}^{[i]} - p^I(d_{n+1}^{[i]}, v_{n+1}^{[i]}) - p^E(\tilde{d}_{n+1}, \tilde{v}_{n+1}).$ (iv)
4If required, form the effective stiffness matrix using the expression
$K^* = M/(\Delta t^2\beta) + \gamma C_T^I/(\Delta t\beta) + K_T^I(d_{n+1}^{[i]}).$ (v)
Otherwise use a previously calculated $K^*$ .
(Note that $K_T^I = \partial p^I/\partial d$ and $C_T^I = \partial p^I/\partial v$ ).
5Perform factorization, forward reduction and backsubstitution as required to solve
$K^* \Delta d^{[i]} = \psi^{[i]}$ .(vi)
6Enter corrector phase in which we set
$d_{n+1}^{[i+1]} = d_{n+1}^{[i]} + \Delta d^{[i]}$ (vii)
$a_{n+1}^{[i+1]} = [d_{n+1}^{[i+1]} - \tilde{d}_{n+1}]/(\Delta t^2\beta)$ (viii)
$v_{n+1}^{[i+1]} = v_{n+1} + \Delta t\gamma a_{n+1}^{[i+1]}$ .(ix)
7If $\Delta d^{[i]}$ and/or $\psi^{[i]}$ do not satisfy the convergence conditions, then set i = i+1 and go to step 3, otherwise continue.
8Set(x)
$d_{n+1} = d_{n+1}^{[i+1]}$ (xi)
$v_{n+1} = v_{n+1}^{[i+1]}$ (xii)
$a_{n+1} = a_{n+1}^{[i+1]}$
for use in the next time step. Also set n = n+1, form p and begin next time step.
+ +Consider the three meshes and effective stiffness matrices shown in Fig. 11.2(a)–(c): + +(i) When there are only explicit elements, $K^*$ is diagonal. In other words $K^*$ has the same profile structure as $M^E$ (Fig. 11.2(a)). +(ii) For a mesh consisting of only implicit elements $K^*$ has the same profile structure as $K^I$ (Fig. 11.2(b)). +(iii) For the partitioned mesh containing both implicit and explicit groups we see the appropriate combination of parts of both profile structures (Fig. 11.2(c)). + + + +To fully exploit the profile structure of $K^*$ , Hughes et al. (4) have suggested the use of profile solvers. In our implementation of the scheme we adopt a slightly modified version of the in-core profile solver given by Bathe and Wilson. (9) + +![](images/page-446_73ac69057b6675d7707844c33ea3e53f989601f28d8d633f4ff7433df51a5ee7.jpg) + +
+text_image + +堆芯内 +2 4 6 8 10 12 +E E E E E +E explicit element +1 3 5 7 9 11 +x +
+ +(i) Finite element mesh—2 degrees of freedom per node. + +![](images/page-446_0e03a8892b0b392a9b767e2f1e80c60951e1345af7210eb052fb2ad71e405493.jpg) + +
+line + +| Equation number | Value | +| --------------- | ----- | +| 1 | × | +| 2 | × | +| 3 | × | +| 4 | × | +| 5 | × | +| 6 | × | +| 7 | × | +| 8 | × | +| 9 | × | +| 10 | × | +| 11 | × | +| 12 | × | +| 13 | × | +| 14 | × | +| 15 | × | +| 16 | × | +| 17 | × | +| 18 | × | +| 19 | × | +| 20' | × | +| 21 | × | +| 22 | × | +| 23 | × | +| 24 | × | +
+ +Fig. 11.2(a) Two-dimensional finite element mesh and profile structure of the effective stiffness matrix $K^{*}$ (explicit elements only). + +# 11.3.3 Alternative predictor values + +In equations (i)–(iii) in Table 11.3 we gave the approach described by Hughes and Liu. $^{(4)}$ For implicit–explicit problems other predictor values may be adopted. Here we consider two cases: + +1. Hughes and Liu predictor values + +$$ +\boldsymbol {d} _ {n + 1} ^ {[ 0 ]} = \tilde {\boldsymbol {d}} _ {n + 1} = \boldsymbol {d} _ {n} + \Delta t \boldsymbol {v} _ {n} + \Delta t ^ {2} (1 - 2 \beta) \boldsymbol {a} _ {n} / 2 \tag {i} +$$ + +$$ +\boldsymbol {v} _ {n + 1} ^ {[ 0 ]} = \widetilde {\boldsymbol {v}} _ {n + 1} = \boldsymbol {v} _ {n} + \Delta t (1 - \gamma) \boldsymbol {a} _ {n} \tag {ii} +$$ + +$$ +\boldsymbol {a} _ {n + 1} ^ {[ 0 ]} = \left[ \boldsymbol {d} _ {n + 1} ^ {[ 0 ]} - \tilde {\boldsymbol {d}} _ {n + 1} \right] / \left(\Delta t ^ {2} \beta\right) \tag {iii} +$$ + + + + + +![](images/page-448_b37c753784b44147b038362413d309dfe84c50fb22f6d73cad1bdb9ba7eb6d9b.jpg) + +
+text_image + +2 4 6 8 10 12 +y +E E I I E +1 3 5 7 9 11 +x +E explicit element +I implicit element +
+ +(i) Finite element mesh—2 degrees of freedom per node. + +![](images/page-448_b789f654a0a1753921fc63f9661e18d78f2c5d0244652165acdeaad4670a92d0.jpg) +(ii) Profile of $K^{*}$ . +Fig. 11.2(c) Two-dimensional finite element mesh and profile structure of the effective stiffness matrix $K^{*}$ (Implicit and explicit elements). + +If $\gamma\geqslant\frac{1}{2}$ and $\beta=(\gamma+\frac{1}{2})^{2}/4$ , we achieve unconditional stability in the implicit element group. The time step is then restricted by the explicit element group. For the case in which $\gamma=\frac{1}{2}$ , the critical time step may be written as + +$$ +\Delta t _ {\mathrm{crit}} = 2 / \omega_ {\max} \tag {11.13} +$$ + +where $\omega_{max}$ is the maximum frequency of the explicit group. We can estimate $\omega_{max}$ as + +$$ +\omega_ {\max} \leqslant \max _ {e} \left(\omega_ {\max} ^ {(e)}\right) \tag {11.14} +$$ + +where $\omega_{\max}^{(e)}$ is the maximum frequency of the $e^{th}$ element of the explicit group. + +Since $K_{T}$ is changing from step to step, strictly speaking the maximum frequency should be estimated at the beginning of every step. In elastoplastic analysis, the structure generally becomes more flexible and (11.14) + + + +may be used. However, for a better estimate of the critical time step the nonlinear eigenvalues should be evaluated. + +If only implicit elements are used and if $\gamma\geqslant\frac{1}{2}$ and $\beta=(\gamma+\frac{1}{2})^{2}/4$ , then error investigations carried out in terms of period elongation and amplitude decay with the increase of time step indicate that for reasonable accuracy the time step should be limited to 1/100 of the fundamental (largest) period. It is observed that the amplitude decay caused by the numerical integration errors effectively filters the higher mode response out of the solution in the Houbolt and Wilson $\theta$ method. However when we employ the Newmark constant–average–acceleration scheme, which does not introduce amplitude decay, the higher frequency response is retained in the solution. In order to obtain amplitude decay using the Newmark method, it is necessary to employ $\gamma>\frac{1}{2}$ . + +# 11.4 Evaluation of the tangential stiffness matrix + +In program MIXDYN we adopt an elasto-plastic material model and therefore the stresses and the tangential stiffness matrix at any time station $t_{n}+\Delta t$ may be evaluated in the manner outlined in Chapter 7 for static problems. As an alternative geometrically nonlinear elastic effects are considered using a total Lagrangian formulation. + +The internal resisting force vector for the implicit elements at time station $t_{n} + \Delta t$ is given as + +$$ +\boldsymbol {p} _ {n + 1} ^ {I} = \int_ {\Omega^ {I}} [ \boldsymbol {B} ^ {I} ] ^ {T} _ {n + 1} \sigma_ {n + 1} d \Omega \tag {11.15} +$$ + +and therefore the tangential stiffness matrix may be written as + +$$ +\begin{array}{l} \frac {\partial \boldsymbol {p} _ {n + 1} ^ {I}}{\partial \boldsymbol {d} _ {n + 1}} = [ \boldsymbol {K} _ {T} ^ {I} ] _ {n + 1} = \int_ {\Omega^ {I}} [ \boldsymbol {B} ^ {I} ] ^ {T} _ {n + 1} \boldsymbol {D} _ {n + 1} [ \boldsymbol {B} ^ {I} ] _ {n + 1} d \Omega \\ + \int_ {\Omega^ {I}} [ \boldsymbol {G} ] ^ {T} _ {n + 1} \boldsymbol {S} _ {n + 1} \boldsymbol {G} _ {n + 1} d \Omega \tag {11.16} \\ \end{array} +$$ + +in which $D_{n+1}$ is the elasto-plastic modulus matrix defined in Chapter 7, $[B_{NL}^I]_{n+1}$ is the nonlinear strain-displacement matrix defined in Chapter 10, the matrix $S_{n+1}$ is given as + +$$ +\boldsymbol {S} _ {n + 1} = \left[ \begin{array}{l l} \sigma_ {x} \boldsymbol {I} _ {2} & \tau_ {x y} \boldsymbol {I} _ {2} \\ \tau_ {x y} \boldsymbol {I} _ {2} & \sigma_ {y} \boldsymbol {I} _ {2} \end{array} \right] _ {n + 1} \tag {11.17} +$$ + +for plane stress and plane strain problems, and + +$$ +\boldsymbol {S} _ {n + 1} = \left[ \begin{array}{c c c} \sigma_ {r} \boldsymbol {I} _ {2} & \tau_ {r z} \boldsymbol {I} _ {2} & \boldsymbol {0} \\ \tau_ {r z} \boldsymbol {I} _ {2} & \sigma_ {z} \boldsymbol {I} _ {2} & \boldsymbol {0} \\ \boldsymbol {0} & \boldsymbol {0} & \sigma_ {\theta} \end{array} \right] _ {n + 1} \tag {11.18} +$$ + +\* The second matrix is only included for geometrically nonlinear problems. + + + +for axisymmetric problems, and + +$$ +[ G _ {i} ] _ {n + 1} = \left[ \begin{array}{c c c c} \frac {\partial N _ {i}}{\partial x} & 0 & \frac {\partial N _ {i}}{\partial y} & 0 \\ 0 & \frac {\partial N _ {i}}{\partial x} & 0 & \frac {\partial N _ {i}}{\partial y} \end{array} \right] ^ {T} \tag {11.19} +$$ + +for plane stress and plane strain problems, and + +$$ +[ G _ {i} ] _ {n + 1} = \left[ \begin{array}{c c c c c} \frac {\partial N _ {i}}{\partial r} & 0 & \frac {\partial N _ {i}}{\partial z} & 0 & \frac {N _ {i}}{r} \\ 0 & \frac {\partial N _ {i}}{\partial r} & 0 & \frac {\partial N _ {i}}{\partial z} & 0 \end{array} \right] ^ {T} \tag {11.20} +$$ + +for axisymmetric problems. + +Note that all of the yield criteria described in Chapter 7 are included in program MIXDYN. + +# 11.5 Program MIXDYN + +# 11.5.1 Introduction + +The computer program 'MIXDYN' is based on the Implicit-Explicit time integration scheme of Hughes and Liu $^{(4)}$ for two-dimensional plane stress/strain and axisymmetric nonlinear dynamic transient problems. Some of the subroutines are the same as in DYNPAK. The profile solvers DECOMP and REDBAK and a few other subroutines used in this program are based on those given in Reference (9). (These subroutines are rewritten using new variables names). Some new subroutines have also been included in the program. The program considers geometric or elasto-plastic material nonlinearity. A total Lagrangian formulation using four-, eight- and nine-noded quadrilateral isoparametric elements is adopted to model the geometric nonlinear behaviour. The program has several options; it can be used for small or large deformation elastic and small deformation elasto-plastic transient dynamic analysis and the analysis may be carried out using an explicit, implicit or combined implicit-explicit algorithm. Furthermore, four types of elasto-plastic material models can be considered: (i) Tresca, (ii) Von Mises, (iii) Drucker-Prager and (iv) Mohr-Coulomb. + +The flow diagram for MIXDYN is shown in Fig. 11.3. The program is written in modular form and the input and output data representation is identical to that given for DYNPAK. + +The subroutines which have not appeared elsewhere in the book are now described. diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_046.md b/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_046.md new file mode 100644 index 00000000..4582a9bd --- /dev/null +++ b/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_046.md @@ -0,0 +1,1416 @@ + + +![](images/page-451_fc5e6ec025f5d2ad2cc58890a31055639904dac19ea27e926946f675364aa026.jpg) + +
+flowchart + +```mermaid +graph TD + A["CONTROL"] --> B["INPUTD"] + B --> C["INTIME"] + C --> D["PREVOS"] + D --> E["LOADPL"] + E --> F["LUMASS"] + F --> G["LINKIN"] + G --> H["GSTIFF"] + H --> I["IMPEXP"] + I --> J["RESEPL"] + J --> K["ITRATE"] + K --> L["OUTDYN"] + L --> A + B --> M["NODXYR"] + B --> N["GAUSSQ"] + E --> O["SFR2"] + E --> P["JACOB2"] + E --> Q["MODPS"] + F --> R["SFR2"] + F --> S["JACOB2"] + F --> T["ADDBAN"] + G --> U["COLMHT"] + G --> V["ADDRES"] + H --> W["DECOMP"] + H --> X["MULTPY"] + H --> Y["REDBAK"] + H --> Z["FUNCTS"] + H --> AA["FUNCTA"] + I --> AB["MULTPY"] + I --> AC["REDBAK"] + M --> AD["MODPS"] + N --> AE["SFR2"] + N --> AF["JACOB2"] + N --> AG["JACOBD"] + N --> AH["BLARGE"] + O --> AI["INVAR"] + O --> AJ["FLOWPL"] + O --> AK["DINTOB"] + O --> AL["GEOMST"] + O --> AM["ADDBAN"] + O --> AN["YIELDF"] + P --> AO["MODPS"] + P --> AP["SFR2"] + P --> AQ["JACOB2"] + P --> AR["JACOBD"] + P --> AS["BLARGE"] + Q --> AT["INVAR"] + Q --> AU["FLOWPL"] + Q --> AV["LINGNL"] + Q --> AW["YIELDF"] +``` +
+ +Fig. 11.3 Overall structure of program MIXDYN. + + + +# 11.5.2 Master routine MIXDYN + +The master routine organises the calling of the main routines as outlined in the flow diagram (Fig. 11.3). In subroutine CONTOL control parameters are read and a check is made on the maximum control dimensions. Note that the values used for checking in CONTOL should agree with the maximum dimensions in the master routine. Subroutine INPUTD, INTIME and PREVOS read the mesh data, time integration data and data for the previous state of the structure. Subroutine LINKIN links the rest of the program with the profile solver, i.e., it generates all information required for the profile solver. Subroutines LUMASS and LOADPL generate the lumped mass and applied force vectors respectively. GSTIFF calculates the global stiffness matrix in compacted form. In the time step do loop IMPEXP performs the direct time integration using either of the (i) Implicit, (ii) Explicit or (iii) combined Implicit-Explicit schemes. RESEPL calculates the equivalent nodal forces using elasto-plastic material behaviour. The maximum dimension of the program have been set to a maximum of 50 elements, 200 nodes, 10 sets of material properties, 6000 coefficients in the mass and stiffness matrices and 400 acceleration ordinates. For larger problems the dimensions must therefore be changed. + +```csv +PROGRAM MIXDYN (INPUT, TAPE5=INPUT, TAPE4, TAPE10, TAPE12, TAPE3, OUTPUT, TAPE6=OUTPUT, TAPE7, TAPE11, TAPE13) MDYN 1 +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +CALL +CALL +CALL +CALL +CALL +CALL +CALL +CALL +CALL +CALL +CALL +CALL +CALL +CALL +CALL +CALL +CALL +CALL +CALL +CALL +CALL +CALL +CALL +CALL +CALL +CALL +CALL +CALL +CALL +CALL +CALL +CALL +CALL +CALL +CALL +CALL +CALL +CALL +CALL +CALL +CALL +CALL +CALL +CALL +CALL +CALL +CALL +CALL +CALL +CALL +Call +CALL +CALL +CALL +CALL +CALL +CALL +CALL +CALL +CALL +CALL +CALL +CALL +CALL +CALL +CALL +CALL +CALL +CALL +CALL +CALL +CALL +CALL +CALL +CALL +CALL +CALL +CALL +CALL +CALL +CALL +CALL +CALL +CALL +CALL +CALL +CALL +CALL +CALL +CALL +CALL +CALL +CALL +CALL +CALL +CALL +CALL +CALL +CALL +CALL +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +D +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +M +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +P +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +A +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +I +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +O +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +N +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +B +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +IC +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +U +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +3 +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +6 +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +1 +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +``` + + + +
CMDYN36
CALLPREVOS(FORCE STRIN)NDOFN,NELEM,NGAUS,NPOIN,NPREV,MDYN
CMDYN37
MDYN38
MDYN39
CALLLOADPL(COORD NELEM NTYPE WEIGP),FORCE,NGAUS,POSGP,LNODS,NMATS,PROPS,MATNO,NNODE,RLOAD,NDIME,NPOIN,STRIN,NDOFN,NSTRE,TEMPE,MDYN
CMDYN40
MDYN41
MDYN42
MDYN43
MDYN44
CALLLUMASS(COORD NDOFN NTYPE),INTGR,NELEM,PROPS,LNODS,NGAUM,YMASS,MATNO,NMATS,NCONM,NNODE,NDIME,NPOIN,MDYN
CMDYN45
MDYN46
MDYN47
MDYN48
CALLLINKIN(FORCE MAXAJ NPOIN),IFPRE,MHIGH,NWKTL,INTGR,NDOFN,NWMTL,LEQNS,NELEM,XMASS,LNODS,NEQNS,YMASS,MAXAI,NNODE,MDYN
CMDYN50
MDYN51
MDYN52
DO 510ISTEP=1,NSTEPMDYN53
CMDYN54
DO 500IITER=1,MITERMDYN55
CMDYN56
CALLGSTIFF(COORD LNODS NDOFN NPOIN PROPS),EPSTN,MATNO,NELEM,NSTRE,STIFF,INTGR,MAXAI,NGAUS,NTYPE,STIFI,ISTEP,MAXAJ,NLAPS,NWMTL,STRSG,KSTEP,NCRIT,NMATS,NWKTL,DISPT,LEQNS,NDIME,NNODE,POSGP,WEIGP),MDYN
CMDYN57
MDYN58
MDYN59
MDYN60
MDYN61
MDYN62
CALLIMPEXP(AALFA ACCEV CONSF DISPT IFUNC NDOFN FORCE VELOT),ACCEH,FACT,DAMPI,DTEND,IITER,NEQNS,STIFF,XMASS,ACCEI,AZERO,DAMPG,DTIME,ISTEP,NPOIN,STIFI,YMASS,ACCEJ,BEETA,DELTA,GAAMA,KSTEP,NWKTL,STIFS,IPRED,ACCEK,BZERO,DISPI,IFIXD,MAXAI,VELOI,ACCEL,CONSD,DISPL,IFPRE,MAXAJ,OMEGA,VELOL,MDYN
CMDYN63
MDYN64
MDYN65
MDYN66
MDYN67
MDYN68
MDYN69
MDYN70
CMDYN71
CALLRESEPL(COORD INTGR NDOFN NPOIN STRAG),DISPT,LEQNS,NELEM,NSTRE,STRIN,EFFST,LNODS,NGAUS,NTYPE,STRSG,RLOAD,MATNO,NLAPS,POSGP,WEIGP,EPSTN,NCRIT,NMATS,PROPS,IPRED,IITER,NDIME,NNODE,RESID,ISTEP),MDYN
CMDYN72
MDYN73
MDYN74
MDYN75
MDYN76
CMDYN77
CALLITRATE(ACCEI DISPL RESID IITER),ACCEL,DISPT,STIFS,MITER,CONSD,MAXAI,TOLER),CONSF,NCHEK,VELOI,XMASS,NEQNS,NWMTL,VELOL,DISPI,NWMTL,VELOT,MDYN
CMDYN78
MDYN79
MDYN80
MDYN81
CMDYN82
500IF(NCHEK.EQ.1) GO TO 510MDYN83
CMDYN84
510CALLOUTDYN(DISPQ NDOFN NOUTP STRSG),DTIME,NELEM,NPOIN,NPRQD,DISPI),EPSTN,NGAUS,NPRQD,NREQD,IFPRE,NNGRQS,NREQD,NREQD,IITER,NITER,NOUTD,NTYPE,ISTEPMDYN
CMDYN85
MDYN86
MDYN87
MDYN88
CMDYN89
STOPMDYN90
ENDMDYN91
+ + + +# 11.5.3 Subroutine ADDBAN + +This routine $^{(9)}$ assembles the element stiffness matrix into the global stiffness matrix in a compacted form. + +```asm +SUBROUTINE ADDBAN (STIFF, MAXAI, ESTIF, LEQNS, NEVAB) ADDB 1 +C******************************* +C******************************* +C *** ASSEMBLY OF TOTAL STIFFNESS VECTOR ADDB 2 +C +C******************************* +DIMENSION STIFF(1), MAXAI(1), ESTIF(1), LEQNS(1) ADDB 5 +C +KOUNT=0 ADDB 8 +DO 200 IEVAB=1, NEVAB ADDB 9 +IEQNS=LEQNS(IEVAB) ADDB 10 +IF(IEQNS) 200, 200, 100 ADDB 11 +100 IMAXA=MAXAI(IEQNS) ADDB 12 +KEVAB=IEVAB ADDB 13 +DO 220 JEVAB=1, NEVAB ADDB 14 +JEQNS=LEQNS(JEVAB) ADDB 15 +IF(JEQNS) 220, 220, 110 ADDB 16 +110 IJEQN=IEQNS-JEQNS ADDB 17 +IF(IJEQN) 220, 210, 210 ADDB 18 +210 ISIZE=IMAXA+IJEQN ADDB 19 +JSIZE=KEVAB ADDB 20 +IF(JEVAB.GE.IEVAB) JSIZE=JEVAB+KOUNT ADDB 21 +STIFF(ISIZE)=STIFF(ISIZE)+ESTIF(JSIZE) ADDB 22 +220 KEVAB=KEVAB+NEVAB-JEVAB ADDB 23 +200 KOUNT=KOUNT+NEVAB-IEVAB ADDB 24 +RETURN ADDB 25 +END ADDB 26 +ADDB 27 +``` + +# 11.5.4 Subroutine ADDRES + +This routine $^{(9)}$ addresses the diagonal elements of the global matrix using the column heights. + +```csv +SUBROUTINE ADDRES(MAXAI ,MHIGH ,NEQNS ,NWKTL ,MKOUN ) ADDR 1 +C************************** ADDR 2 +C ADDRESS 3 +C *** EVALUATES ADRESSES OF DIAGONAL ELEMENTS ADDR 4 +C ADDRESS 5 +C************************** ADDR 6 +DIMENSION MAXAI(1) ,MHIGH(1) ADDR 7 +NEQNN=NEQNS+1 ADDR 8 +DO 20 IEQNN=1,NEQNN ADDR 9 +20 MAXAI(1)=1 ADDR 10 +MAXAI(2)=2 ADDR 11 +MKOUN=0 ADDR 12 +IF(NEQNS.EQ.1) GO TO 30 ADDR 13 +DO 10 IEQNS=2,NEQNS ADDR 14 +IF(MHIGH(IEQNS).GT.MKOUN) MKOUN=MHIGH(IEQNS) ADDR 15 +10 MAXAI(IEQNS+1)=MAXAI(IEQNS)+MHIGH(IEQNS)+1 ADDR 16 +30 MKOUN=MKOUN+1 ADDR 17 +NWKTL=MAXAI(NEQNS+1)-MAXAI(1) ADDR 18 +RETURN ADDR 19 +END ADDR 20 +``` + + + +# 11.5.5 Subroutine COLMHT + +This routine $^{(9)}$ calculates the vertical column heights above the diagonal of the global matrix using equation numbers and the total number of degrees of freedom of an element (NEVAB). + +```asm +SUBROUTINE COLMHT (MHIGH, NEVAB, LEQNS) COLM 1 +C*************** COLM 2 +C COLM 3 +C*** EVALUATES THE COLUMN HEIGHT OF STIFFNESS MATRIX COLM 4 +C COLM 5 +C*************** COLM 6 +DIMENSION LEQNS(1), MHIGH(1) COLM 7 +MAXAM=100000 COLM 8 +DO 100 IEVAB=1, NEVAB COLM 9 +IF(LEQNS(IEVAB)) 110, 100, 110 COLM 10 +110 IF(LEQNS(IEVAB)-MAXAM) 120, 100, 100 COLM 11 +120 MAXAM=LEQNS(IEVAB) COLM 12 +100 CONTINUE COLM 13 +DO 200 IEVAB=1, NEVAB COLM 14 +IEQNS=LEQNS(IEVAB) COLM 15 +IF(IEQNS.EQ.0) GO TO 200 COLM 16 +JHIGH=IEQNS-MAXAM COLM 17 +IF(JHIGH.GT.MHIGH(IEQNS)) MHIGH(IEQNS)=JHIGH COLM 18 +200 CONTINUE COLM 19 +RETURN COLM 20 +END COLM 21 +``` + +# 11.5.6 Subroutine DECOMP + +This routine $^{(9)}$ factorises a matrix into lower, diagonal and upper matrices $(LDL^{T})$ + +```fortran +SUBROUTINE DECOMP (STIFF, MAXAI, NEQNS, ISHOT) DECM 1 +C*************** DECM 2 +C DECM 3 +C *** FACTORISES (L)*(D)*(L) TRANSPOSE OF STIFFNESS MATRIX DECM 4 +C DECM 5 +C*************** DECM 6 +DIMENSION STIFF(1), MAXAI(1) DECM 7 +C DECM 8 +IF(NEQNS.EQ.1) RETURN DECM 9 +DO 200 IEQNS=1, NEQNS DECM 10 +IMAXA=MAXAI(IEQNS) DECM 11 +LOWER=IMAXA+1 DECM 12 +KUPER=MAXAI(IEQNS+1)-1 DECM 13 +KHIGH=KUPER-LOWER DECM 14 +IF(KHIGH) 304,240,210 DECM 15 +210 KSIZE=IEQNS-KHIGH DECM 16 +ICOUN=0 DECM 17 +JUPER=KUPER DECM 18 +DO 260 JHIGH=1,KHIGH DECM 19 +ICOUN=ICOUN+1 DECM 20 +JUPER=JUPER-1 DECM 21 +KMAXA=MAXAI(KSIZE) DECM 22 +NDIAG=MAXAI(KSIZE+1)-KMAXA-1 DECM 23 +IF(NDIAG) 260,260,270 DECM 24 +``` + + + +```csv +270 NCOLM=MINO(ICOUN,NDIAG) DECM 25 +COUNT=0. DECM 26 +DO 280 ICOLM=1,NCOLM DECM 27 +280 COUNT=COUNT+STIFF(KMAXA+ICOLM)*STIFF(JUPER+ICOLM) DECM 28 +STIFF(JUPER)=STIFF(JUPER)-COUNT DECM 29 +260 KSIZE=KSIZE+1 DECM 30 +240 KSIZE=IEQNS DECM 31 +BSUMM=0. DECM 32 +DO 300 ICOLM=LOWER,KUPER DECM 33 +KSIZE=KSIZE-1 DECM 34 +JMAXA=MAXAI(KSIZE) DECM 35 +RATIO=STIFF(ICOLM)/STIFF(JMAXA) DECM 36 +BSUMM=BSUMM+RATIO*STIFF(ICOLM) DECM 37 +300 STIFF(ICOLM)=RATIO DECM 38 +STIFF(IMAXA)=STIFF(IMAXA)-BSUMM DECM 39 +304 IF(STIFF(IMAXA)) 310,310,200 DECM 40 +310 IF(ISHOT.EQ.0) GO TO 320 DECM 41 +IF(STIFF(IMAXA).EQ.0) STIFF(IMAXA)=-1.E-16 DECM 42 +GO TO 200 DECM 43 +320 WRITE(6,2000) IEQNS,STIFF(IMAXA) DECM 44 +STOP DECM 45 +200 CONTINUE DECM 46 +RETURN DECM 47 +2000 FORMAT(//48H STOP - STIFFNESS MATRIX NOT POSITIVE DEFINITE ,// +.32H NONPOSITIVE PIVOT FOR EQUATION ,I4,//10H PIVOT = ,E20.12 ) DECM 49 +END DECM 50 +``` + +# 11.5.7 Subroutine DINTOB + +This routine multiplies the modulus matrix D with the strain matrix B. + +```fortran +SUBROUTINE DINTOB (BMATX, DBMAT, DMATX, NEVAB, NSTRE) DINT 1 +C******************************* +C DINT 2 +C DINT 3 +C*** CALCULATE D INTO B DINT 4 +C DINT 5 +C******************************* +DIMENSION DBMAT(4,18), DMATX(4,4), BMATX(4,18) DINT 7 +DO 10 ISTRE=1, NSTRE DINT 8 +DO 10 IEVAB=1, NEVAB DINT 9 +DBMAT(ISTRE, IEVAB)=0.0 DINT 10 +DO 10 JSTRE=1, NSTRE DINT 11 +DBMAT(ISTRE, IEVAB)=DBMAT(ISTRE, IEVAB)+ DINT 12 +.DMATX(ISTRE, JSTRE)*BMATX(JSTRE, IEVAB) DINT 13 +10 CONTINUE DINT 14 +RETURN DINT 15 +END DINT 16 +``` + +# 11.5.8 Subroutine GEOMST + +This routine adds the initial stress matrix to the stiffness matrix. + +```asm +SUBROUTINE GEOMST (CARTD, DVOLU, ESTIF, KGAUS, NDOFN, NNODE, STRSG, SHAPE, NTYPE, GPCOD, KGASP) GEOM 1 +C*************** GEOM 2 +C +C +C +ADD INITIAL STRESS STIFFNESS MATRIX TO STIFFNESS MATRIX GEOM 3 +C +C +C*************** GEOM 4 +DIMENSION STRES(4), CARTD(2,9), ESTIF(171), STRSG(4,1), SHAPE(1), GPCOD(2,9) GEOM 5 +NEVAB=NNODE*NDOFN GEOM 6 +DO 300 ISTR1=1,4 GEOM 7 +GEOM 8 +GEOM 9 +GEOM 10 +GEOM 11 +``` + + + +```txt +300 STRES(ISTR1)=STRSG(ISTR1,KGAUS) GEOM 12 +IEVAB=1 GEOM 13 +KOUNT=NEVAB GEOM 14 +DO 200 INODE=1,NNODE GEOM 15 +DO 100 JNODE=INODE,NNODE GEOM 16 +DGASH=STRES(1)*CARTD(1,INODE)*CARTD(1,JNODE)+ +.STRES(3)*(CARTD(1,INODE)*CARTD(2,JNODE)+ +.CARTD(2,INODE)*CARTD(1,JNODE))+ GEOM 19 +.STRES(2)*CARTD(2,INODE)*CARTD(2,JNODE) GEOM 20 +DGASY=DGASH*DVOLU GEOM 21 +DGASX=DGASY GEOM 22 +IF(NTYPE.NE.3) GO TO 400 GEOM 23 +PRODT=SHAPE(INODE)/(GPCOD(1,KGASP)**2) GEOM 24 +DGASX=DGASY+STRES(4)*PRODT*SHAPE(JNODE)*DVOLU GEOM 25 +400 ESTIF(IEVAB)=ESTIF(IEVAB)+DGASX GEOM 26 +JEVAB=IEVAB+KOUNT GEOM 27 +ESTIF(JEVAB)=ESTIF(JEVAB)+DGASY GEOM 28 +IEVAB=IEVAB+2 GEOM 29 +100 CONTINUE GEOM 30 +KOUNT=KOUNT-2 GEOM 31 +IEVAB=JEVAB+1 GEOM 32 +200 CONTINUE GEOM 33 +RETURN GEOM 34 +END GEOM 35 +``` + +# 11.5.9 Subroutine GSTIFF + +This routine generates the compacted geometrically nonlinear stiffness matrix for two-dimensional plane stress/strain and axisymmetric problems from the element stiffness matrices. + +```txt +SUBROUTINE GSTIFF (COORD ,EPSTN ,INTGR ,ISTEP ,KSTEP ,LEQNS , STIF 1 +. LNODS ,MATNO ,MAXAI ,MAXAJ ,NCRIT ,NDIME , STIF 2 +. NDOFN ,NELEM ,NGAUS ,NLAPS ,NMATS ,NNODE , STIF 3 +. NPOIN ,NSTRE ,NTYPE ,NWMTL ,NWKTL ,POSGP , STIF 4 +. PROPS ,STIFF ,STIFI ,STRSG ,TDISP ,WEIGP ) STIF 5 +C*********************************************************************************************** +C +C EVALUATES GEOMETRICALLY NONLINEAR STIFFNESS MATRIX STIF 8 +C FOR 2-D PLANE STRESS/STRAIN 2-D ELEMENT STIF 9 +C +C*********************************************************************************************** +C +DIMENSION COORD(NPOIN,1) ,DMATX(4, 4) ,ELCOD(2,9) ,AVECT(4) , STIF 12 +. LNODS(NELEM,1) ,BMATX(4,18) ,CARTD(2,9) ,DVECT(4) , STIF 13 +. PROPS(NMATS,1) ,DBMAT(4,18) ,GPCOD(2,9) ,DEVIA(4) , STIF 14 +. LEQNS( 18, 1) ,STRSG(4, 1) ,DLCOD(2,9) ,STRES(4) , STIF 15 +. ESTIF( 171) ,DJACM(2, 2) ,DERIV(2,9) ,SHAPE(9) STIF 16 +C +DIMENSION MAXAI(1) ,INTGR(1) ,STIFF(1) ,POSGP(1) ,EPSTN(1) , STIF 18 +. MAXAJ(1) ,TDISP(1) ,STIFI(1) ,WEIGP(1) ,MATNO(1) STIF 19 +C +C +IF(ISTEP.EQ.1) GO TO 200 STIF 21 +KOUNT=(ISTEP/KSTEP)*KSTEP STIF 22 +IF(KOUNT.NE.ISTEP)RETURN STIF 23 +200 CONTINUE STIF 24 +TWOPI=6.283185307179586 STIF 25 +KGAUS=0 STIF 26 +C +C*** LOOP OVER EACH ELEMENT STIF 27 +C +NSTR1=4 STIF 28 +NEVAB=NDOFN*NNODE STIF 29 +DO 500 IWKTL=1,NWKTL STIF 30 +500 STIFF(IWKTL) ,STIFI(IWKTL)=0.0 STIF 31 +STIF 32 +STIF 33 +STIF 34 +``` + + + +```csv +DO 70 IELEM=1,NELEM +LPROP=MATNO(IELEM) +C +C*** EVALUATE THE COORDINATES OF THE ELEMENT NODAL POINTS +C +IPOSN=0 +DO 10 INODE=1,NNODE +LNODE=LNODS(IELEM,INODE) +DO 10 IDIME=1,NDIME +IPOSN=IPOSN+1 +NPOSN=LEQNS(IPOSN,IELEM) +IF(NPOSN.EQ.0) DISPT=0. +IF(NPOSN.NE.0) DISPT=TDISP(NPOSN) +DLCOD(IDIME,INODE)=COORD(LNODE,IDIME)+DISPT +10 ELCOD(IDIME,INODE)=COORD(LNODE,IDIME) +YOUNG=PROPS(LPROP, 1) +POISS=PROPS(LPROP, 2) +THICK=PROPS(LPROP, 3) +HARDS=PROPS(LPROP, 7) +FRICT=PROPS(LPROP, 8) +C +C*** INITIALIZE THE ELEMENT STIFFNESS MATRIX 171=NEVAB*(NEVAB+1)/2 +C +DO 20 ISIZE=1,171 +20 ESTIF(ISIZE)=0.0 +KGASP=0 +C +C*** ENTER LOOPS FOR AREA NUMERICAL INTEGRATION +C +DO 50 IGAUS=1,NGAUS +EXISP=POSGP(IGAUS) +DO 50 JGAUS=1,NGAUS +ETASP=POSGP(JGAUS) +KGASP=KGASP+1 +KGAUS=KGAUS+1 +CALL MODPS (DMATX,LPROP,NMATS,NSTRE,NTYPE,PROPS) +CALL SFR2 (DERIV,NNODE,SHAPE,EXISP,ETASP) +CALL JACOB2 (CARTD,DERIV,DJACB,ELCOD,GPCOD, +IELEM,KGASP,NNODE,SHAPE) +CALL JACOBD (CARTD,DLCOD,DJACM,NDIME,NLAPS,NNODE) +DVOLU=DJACB*WEIGP(IGAUS)*WEIGP(JGAUS) +IF(NTYPE.EQ.3) DVOLU=DVOLU*TWOPI*GPCOD(1,KGASP) +IF(NTYPE.EQ.1) DVOLU=DVOLU*THICK +C +C*** EVALUATE THE B AND DB MATRICES +C +CALL BLARGE (BMATX,CARTD,DJACM,DLCOD,GPCOD, +KGASP,NLAPS,NNODE,NTYPE,SHAPE) +IF(NLAPS.EQ.2.OR.NLAPS.EQ.0) GO TO 80 +IF(ISSTEP.EQ.1) GO TO 80 +IF(EPSTN(KGAUS).EQ.0.0) GO TO 80 +DO 90 ISTR1=1,NSTR1 +90 STRES(ISTR1)=STRSG(ISTR1,KGAUS) +CALL INVAR (DEVIA,LPROP,NCRIT,NMATS,PROPS,SINT3,STEFF, +STRES,THETA,VARJ2,YIELD) +CALL YIELDF (AVECT,DEVIA,FRICT,NCRIT,SINT3,STEFF, +THETA,VARJ2) +CALL FLOWPL (AVECT,ABETA,DVECT,HARDS,NTYPE,POISS,YOUNG) +DO 100 ISTRE=1,NSTRE +DO 100 JSTRE=1,NSTRE +100 DMAIX(ISTRE,JSTRE)=DMATX(ISTRE,JSTRE)-ABETA*DVECT(ISTRE) +DVECT(JSTRE) +80 CONTINUE +CALL DINTOB (BMATX,DBMAT,DMATX,NEVAB,NSTRE) +``` + + + +```csv +C STIF 99 +C ***EVALUATE GEOMETRIC STIFFNESS TERMS STIF 100 +C STIF 101 +IF(NLAPS.LT.2) GO TO 85 STIF 102 +CALL GEOMST (CARTD,DVOLU,ESTIF,KGAUS,NDOFN,NNODE, STIF 103 +STRSG,SHAPE,NTYPE,GPCOD,KGASP) STIF 104 +C STIF 105 +C*** CALCULATE THE ELEMENT STIFFNESSES STIF 106 +C STIF 107 +85 KOUNT=0 STIF 108 +DO 30 IEVAB=1,NEVAB STIF 109 +DO 30 JEVAB=IEVAB,NEVAB STIF 110 +KOUNT=KOUNT+1 STIF 111 +DO 30 ISTRE=1,NSTRE STIF 112 +30 ESTIF(KOUNT)=ESTIF(KOUNT)+BMATX(ISTRE,IEVAB)* STIF 113 +DBMAT(ISTRE,JEVAB)*DVOLU STIF 114 +50 CONTINUE STIF 115 +C STIF 116 +C *** GENERATES GLOBAL STIFFNSS MATRIX IN COMPACTED COLUMN FORM STIF 117 +C STIF 118 +IF(INTGR(IELEM).EQ.2) GO TO 210 STIF 119 +CALL ADDBAN (STIFI,MAXAI,ESTIF,LEQNS(1,IELEM),NEVAB) STIF 120 +210 CALL ADDBAN (STIFF,MAXAJ,ESTIF,LEQNS(1,IELEM),NEVAB) STIF 121 +70 CONTINUE STIF 122 +C WRITE(6,900) (STIFI(I),I=1,NWMTL) STIF 123 +900 FORMAT(10E12.4) STIF 124 +RETURN STIF 125 +END STIF 126 +``` + +# 11.5.10 Subroutine IMPEXP + +This routine generates the partial effective load vector for direct time integration. + +```asm +SUBROUTINE IMPEXP (AALFA, ACCEH, ACCEI, ACCEJ, ACCEK, ACCEL, IMEX 1 +ACCEV, AFACT, AZERO, BEETA, BZERO, CONSD, IMEX 2 +CONSF, DAMPI, DAMPG, DELTA, DISPI, DISPL, IMEX 3 +DISPT, DTEND, DTIME, GAAMA, IFIXD, IFPRE, IMEX 4 +IFUNC, IITER, ISTEP, KSTEP, MAXAI, MAXAJ, IMEX 5 +NDOFN, NSIZE, NPOIN, NWKTL, NWMTL, OMEGA, IMEX 6 +RLOAD, STIFF, STIFI, STIFS, VELOI, VELOL, IMEX 7 +VELOT, XMASS, YMASS, IPRED) IMEX 8 +C*************** IMEX 9 +C +C *** GENERATES PARTIAL EFFECTIVE LOAD VECTOR IMEX 10 +C +C*************** IMEX 11 +C +DIMENSION STIFF(1), DISPI(1), ACCEH(1), DISPL(1), IFPRE(2,1), IMEX 12 +XMASS(1), VELOI(1), ACCEV(1), VELOL(1), ACCEK(1), IMEX 13 +RLOAD(1), ACCEI(1), MAXAI(1), ACCEL(1), DAMPG(1), IMEX 14 +ACCEJ(1), MAXAJ(1), YMASS(1), STIFI(1), DISPT(1), IMEX 15 +STIFS(1), DAMPI(1), VELOT(1) IMEX 16 +C +C +C +IF(ISTEP.GT.1.OR.IITER.GT.1) GO TO 1000 IMEX 17 +CONSA-DTIME*DTIME*(0.5-DELTA) IMEX 18 +CONSB-DTIME*(1.-GAAMA) IMEX 19 +CONSC-DTIME*DTIME*DELTA IMEX 20 +CONSD-DTIME*GAAMA IMEX 21 +CONSF=1./CONSC IMEX 22 +CONSG-BEETA*GAAMA*DTIME IMEX 23 +CONSH-AALFA*GAAMA*DTIME IMEX 24 +CONSE=1.+CONSH IMEX 25 +IMEX 26 +IMEX 27 +IMEX 28 +IMEX 29 +IMEX 30 +``` + + + +```txt +ISHOT=0 IMEX 31 +DO 550 IPOIN=1,NPOIN IMEX 32 +DO 550 IDOFN=1,NDOFN IMEX 33 +ISIZE=IFPRE(IDOFN,IPOIN) IMEX 34 +IF(ISIZE.EQ.0) GO TO 550 IMEX 35 +ACCEI(ISIZE)=1.0 IMEX 36 +ACCEL(ISIZE)=0.0 IMEX 37 +IF(IDOFN.EQ.1) GO TO 550 IMEX 38 +ACCEI(ISIZE)=0.0 IMEX 39 +ACCEL(ISIZE)=1.0 IMEX 40 +550 CONTINUE IMEX 41 +DO 590 ISIZE=1,NSIZE IMEX 42 +IMAXA=MAXAI(ISIZE) IMEX 43 +590 XMASS(IMAXA)=XMASS(IMAXA)+YMASS(ISIZE) IMEX 44 +C IMEX 45 +C *** CALCULATES VECTORS FOR HORIZONTAL AND VERTICAL EXCITATION IMEX 46 +C IMEX 47 +CALL MULTPY (ACCEK,XMASS,ACCEL,MAXAI,NSIZE,NWMTL) IMEX 48 +CALL MULTPY (ACCEJ,XMASS,ACCEI,MAXAI,NSIZE,NWMTL) IMEX 49 +CALL MULTPY (DISPL,STIFF,DISPI,MAXAJ,NSIZE,NWKT L) IMEX 50 +C IMEX 51 +C *** CALCULATES DAMPING MATRIX (AALFA*M+BEETA*K) IMEX 52 +C IMEX 53 +DO 500 ISIZE=1,NSIZE IMEX 54 +IMAXA=MAXAI(ISIZE) IMEX 55 +KMAXA=MAXAI(ISIZE+1)-1 IMEX 56 +JMAXA=MAXAJ(ISIZE) IMEX 57 +DO 500 LMAXA=IMAXA,KMAXA IMEX 58 +DAMPI(JMAXA)=AALFA*XMASS(LMAXA) IMEX 59 +500 JMAXA=JMAXA+1 IMEX 60 +DO 560 IWKTL=1,NWKTL IMEX 61 +560 DAMPI(IWKTL)=DAMPI(IWKTL)+BEETA*STIFF(IWKTL) IMEX 62 +C IMEX 63 +C *** CALCULATES INITIAL ACCELERATION IMEX 64 +C IMEX 65 +CALL MULTPY (VELOL,DAMPI,VELOI,MAXAJ,NSIZE,NWKTL) IMEX 66 +DO 600 IWMTL=1,NWMTL IMEX 67 +600 DAMPG(IWMTL)=XMASS(IWMTL) IMEX 68 +DO 510 ISIZE = 1,NSIZE IMEX 69 +510 ACCEI(ISIZE)=RLOAD(ISIZE)-DISPL(ISIZE)-VELOL(ISIZE) IMEX 70 +CALL DECOMP (DAMPG,MAXAI,NSIZE,ISHOT) IMEX 71 +CALL REDBAK (DAMPG,ACCEI,MAXAI,NSIZE) IMEX 72 +WRITE (6,900) IMEX 73 +WRITE (6,910) (ACCEI(ISIZE),ISIZE=1,NSIZE) IMEX 74 +900 FORMAT(/' INITIAL ACCELERATION '/') IMEX 75 +910 FORMAT(1X,10E12.5) IMEX 76 +1000 CONTINUE IMEX 77 +IF(IITER.GT.1) GO TO 650 IMEX 78 +C IMEX 79 +C *** CALCULATES PREDICTED DISPLACEMENT AND VELOCITY VECTOR IMEX 80 +C IMEX 81 +DO 540 ISIZE=1,NSIZE IMEX 82 +IF(IPRED.EQ.1) GO TO 210 IMEX 83 +DISPT(ISIZE)=DISPI(ISIZE) IMEX 84 +VELOT(ISIZE)=VELOI(ISIZE) IMEX 85 +210 DISPI(ISIZE)=DISPI(ISIZE)+DTIME*VELOI(ISIZE)+CONSA*ACCEI(ISIZE) IMEX 86 +VELOI(ISIZE)=VELOI(ISIZE)+CONSB*ACCEI(ISIZE) IMEX 87 +IF(IPRED.EQ.2) GO TO 220 IMEX 88 +DISPT(ISIZE)=DISPI(ISIZE) IMEX 89 +VELOT(ISIZE)=VELOI(ISIZE) IMEX 90 +220 ACCEI(ISIZE)=CONSF*(DISPT(ISIZE)-DISPI(ISIZE)) IMEX 91 +540 CONTINUE IMEX 92 +C IMEX 93 +C*** CALCULATES LOAD VECTORS IMEX 94 +``` diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_047.md b/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_047.md new file mode 100644 index 00000000..c16b855e --- /dev/null +++ b/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_047.md @@ -0,0 +1,954 @@ + + +```txt +C +FACTS =FUNCTS (AZERO,BZERO,DTEND,DTIME,IFUNC,ISTEP,OMEGA) +FACTH =FUNCTA (ACCEH,AFACT,DTEND,DTIME,IFUNC,ISTEP) +FACTV =FUNCTA (ACCEV,AFACT,DTEND,DTIME,IFUNC,ISTEP) +WRITE(6,910) FACTS,FACTH,FACTV +650 CONTINUE +IF(ISTEP.EQ.1) GO TO 640 +C +C *** CALCULATES DAMPING AND K-STAR MATRICES +C +DO 530 ISIZE=1,NSIZE +IMAXA=MAXAI(ISIZE) +KMAXA=MAXAI(ISIZE+1)-1 +JMAXA=MAXAJ(ISIZE) +DO 530 LMAXA=IMAXA,KMAXA +DAMPI(JMAXA)=AALFA*XMASS(LMAXA) +530 JMAXA=JMAXA+1 +DO 580 IWKTL=1,NWKTL +580 DAMPI(IWKTL)=DAMPI(IWKTL)+BEETA*STIFF(IWKTL) +CALL MULTPY (VELOL ,DAMPI ,VELOT ,MAXAJ ,NSIZE ,NWKTL ) +KOUNT=(ISTEP/KSTEP)*KSTEP +IF(KOUNT.NE.ISTEP) GO TO 660 +640 DO 610 IWMTL=1,NWMTL +610 DAMPG(IWMTL)=CONSE*XMASS(IWMTL) +DO 620 ISIZE=1,NSIZE +IMAXA=MAXAI(ISIZE) +620 DAMPG(IMAXA)=DAMPG(IMAXA)-CONSH*YMASS(ISIZE) +DO 630 IWMTL=1,NWMTL +DAMPG(IWMTL)=DAMPG(IWMTL)+CONSG*STIFI(IWMTL) +630 STIFS(IWMTL)=STIFI(IWMTL)+DAMPG(IWMTL)*CONSF +WRITE(6,900) (STIFS(I),I=1,NWMTL) +CALL DECOMP (STIFS ,MAXAI ,NSIZE ,ISHOT ) +C +C *** CALCULATES PARTIAL EFFECTIVE LOAD VECTOR +C +660 DO 520 ISIZE=1,NSIZE +IF(IFUNC.NE.0) GO TO 570 +IF(IFIXD.EQ.2) DISPL(ISIZE)=-VELOL(ISIZE)-FACTH*ACCEJ(ISIZE) ++RLOAD(ISIZE) +IF(IFIXD.EQ.1) DISPL(ISIZE)=-VELOL(ISIZE)-FACTV*ACCEK(ISIZE) ++RLOAD(ISIZE) +IF(IFIXD.EQ.0) DISPL(ISIZE)=-VELOL(ISIZE)-FACTH*ACCEJ(ISIZE) ++RLOAD(ISIZE)-FACTV*ACCEK(ISIZE) +IF(IFUNC.EQ.0) GO TO 520 +570 DISPL(ISIZE)=-VELOL(ISIZE)+RLOAD(ISIZE)*FACTS +520 CONTINUE +RETURN +END +IMEX 95 +IMEX 96 +IMEX 97 +IMEX 98 +IMEX 99 +IMEX 100 +IMEX 101 +IMEX 102 +IMEX 103 +IMEX 104 +IMEX 105 +IMEX 106 +IMEX 107 +IMEX 108 +IMEX 109 +IMEX 110 +IMEX 111 +IMEX 112 +IMEX 113 +IMEX 114 +IMEX 115 +IMEX 116 +IMEX 117 +IMEX 118 +IMEX 119 +IMEX 120 +IMEX 121 +IMEX 122 +IMEX 123 +IMEX 124 +IMEX 125 +IMEX 126 +IMEX 127 +IMEX 128 +IMEX 129 +IMEX 130 +IMEX 131 +IMEX 132 +IMEX 133 +IMEX 134 +IMEX 135 +IMEX 136 +IMEX 137 +IMEX 138 +IMEX 139 +IMEX 140 +IMEX 141 +IMEX 142 +``` + +# 11.5.11 Subroutine ITRATE + +This routine generates the total effective load vector and solves for the incremental displacements. It then checks for convergence. + +```txt +SUBROUTINE ITRATE (ACCEI, ACCEL, CONSD, CONSF, XMASS, DISPI, ITER 1 +. DISPL, DISPT, MAXAI, NCHEK, NSIZE, NWMTL, ITER 2 +. RESID, STIFS, TOLER, VELOI, VELOL, VELOT, ITER 3 +. IITER, MITER) ITER 4 +C********** C +C +C *** CALCULATES INCREMENT IN DISPLACEMENT AND APPLIES CONVERGENCE +C +C********** C +DIMENSION DISPI(1), VELOI(1), ACCEI(1), RESID(1), MAXAI(1), ITER 10 +ITER 5 +ITER 6 +ITER 7 +ITER 8 +ITER 9 +ITER 10 +``` + + + +```fortran +: DISPL(1), VELOL(1), ACCEL(1), STIFS(1), DISPT(1), ITER 11 +: XMASS(1), VELOT(1) ITER 12 +C +C NCHEK=0 +CALL MULTPY (ACCEL, XMASS, ACCEI, MAXAI, NSIZE, NWMTL) ITER 13 +C +C *** CALCULATES TOTAL EFFECTIVE LOAD VECTOR +C +DO 660 ISIZE=1, NSIZE +660 ACCEL(ISIZE)=DISPL(ISIZE)-ACCEL(ISIZE)-RESID(ISIZE) ITER 14 +C +C *** CALCULATES DELTA DISPLACEMENT +C +210 CALL REDBAK (STIFS, ACCEL, MAXAI, NSIZE) ITER 15 +C +C *** APPLIES CONVERGENCE +C +SUMPP=0. +SUMPQ=0. +DO 670 ISIZE=1, NSIZE +DISPP=ACCEL(ISIZE) +DISPQ=DISPT(ISIZE)+DISPP +DISPT(ISIZE)=DISPQ +SUMPP=SUMPP+DISPP*DISPP +SUMPQ=SUMPQ+DISPQ*DISPQ +670 CONTINUE +DO 530 ISIZE=1, NSIZE +ACCEI(ISIZE)=CONSF*(DISPT(ISIZE)-DISPI(ISIZE)) ITER 16 +530 VELOT(ISIZE)=VELOI(ISIZE)+CONSD*ACCEI(ISIZE) ITER 17 +220 SUMPP=SQRT(SUMPP/SUMPQ) +IF(SUMPP.GT.TOLER) GO TO 550 +NCHEK=1 +GO TO 240 +550 IF(IITER.LT.MITER) GO TO 230 +240 DO 540 ISIZE=1, NSIZE +VELOI(ISIZE)=VELOT(ISIZE) ITER 18 +540 DISPI(ISIZE)=DISPT(ISIZE) ITER 19 +230 CONTINUE +RETURN +END +ITER 20 +ITER 21 +ITER 22 +ITER 23 +ITER 24 +ITER 25 +ITER 26 +ITER 27 +ITER 28 +ITER 29 +ITER 30 +ITER 31 +ITER 32 +ITER 33 +ITER 34 +ITER 35 +ITER 36 +ITER 37 +ITER 38 +ITER 39 +ITER 40 +ITER 41 +ITER 42 +ITER 43 +ITER 44 +ITER 45 +ITER 46 +ITER 47 +ITER 48 +ITER 49 +ITER 50 +ITER 51 +``` + +ITER 20-21 Calculates total effective load vector. + +ITER 25 Solves for incremental displacements. + +ITER 28-37 Calculates norm of displacement increments. + +ITER 38-40 Calculates new and total displacement, velocities and accelerations. + +ITER 41-42 Applies convergence check. + +ITER 46–49 Stores the final velocities and displacements in vectors VELOI and DISPI respectively. + +# 11.5.12 Subroutine LINKIN + +This routine calculates the equation number from the array IFPRE which stores the information about the restrained degrees of freedom. + + + +```fortran +SUBROUTINE LINKIN (FORCE, IFPRE, INTGR, LEQNS, LNODS, MAXAI, LINK 1 +MAXAJ, MHIGH, NDOFN, NELEM, NEQNS, NNODE, LINK 2 +NPOIN, NWKTL, NWMTL, XMASS, YMASS) LINK 3 +C +C +C *** LINKS WITH PROFILE SOLVER +C +C +C +DIMENSION LNODS(NELEM, 1), XMASS(1), MAXAI(1), INTGR(1), LINK 9 +IFPRE(NDOFN, 1), YMASS(1), MAXAJ(1), MHIGH(1), LINK 10 +LEQNS(18, 1), FORCE(1), EMASS(171) LINK 11 +C +IMASS=1 +REWIND 3 +NEVAB=NNODE*NDOFN +C +C +C +NUMBER OF UNKNOWNS +C +NEQNS=0 +DO 100 IPOIN=1, NPOIN +DO 150 IDOFN=1, NDOFN +IF(IFPRE(IDOFN, IPOIN)) 110, 120, 110 +NEQNS=NEQNS+1 +IFPRE(IDOFN, IPOIN)=NEQNS +GO TO 150 +IFPRE(IDOFN, IPOIN)=0 +CONTINUE +WRITE(6, 7) IPOIN, (IFPRE(IDOFN, IPOIN), IDOFN=1, NDOFN) +CONTINUE +MEQNS=1+NEQNS +C +C +C +C +C +CONNECTIVITY ARRAY LEQNS +C +DO 70 IELEM=1, NELEM +DO 70 IEVAB=1, NEVAB +LEQNS(IEVAB, IELEM)=0 +DO 50 IELEM=1, NELEM +IEVAB=1 +DO 80 INODE=1, NNODE +IDENT=LNODS(IELEM, INODE) +DO 80 IDOFN=1, NDOFN +LEQNS(IEVAB, IELEM)=IFPRE(IDOFN, IDENT) +IEVAB=IEVAB+1 +WRITE(6, 6) IELEM, (LEQNS(IEVAB, IELEM), IEVAB=1, NEVAB) +CONTINUE +FORMAT(I10, 24I3) +FORMAT(4I10) +FORMAT(8E12.4) +C +C +C +C +LOOP OVER ALL ELEMENTS +C +250 DO 190 IELEM=1, NELEM +IF(INTGR(IELEM).NE.IMASS) GO TO 190 +CALL COLMHT (MHIGH, NEVAB, LEQNS(1, IELEM)) +CONTINUE +ADDRESES OF DIAGONAL ELEMENTS - MAXA ARRAY +CALL ADDRES(MAXAJ, MHIGH, NEQNS, NWKTL, MKOUN) +IF(IMASS.EQ.2) GO TO 205 +DO 580 IEQNS=1, MEQNS +MAXAI(IEQNS)=MAXAJ(IEQNS) +IMASS=2 +NWMTL=NWKTL +LINK 1 +LINK 5 +LINK 6 +LINK 7 +LINK 8 +LINK 9 +LINK 10 +LINK 11 +LINK 12 +LINK 13 +LINK 14 +LINK 15 +LINK 16 +LINK 17 +LINK 18 +LINK 19 +LINK 20 +LINK 21 +LINK 22 +LINK 23 +LINK 24 +LINK 25 +LINK 26 +LINK 27 +LINK 28 +LINK 29 +LINK 30 +LINK 31 +LINK 32 +LINK 33 +LINK 34 +LINK 35 +LINK 36 +LINK 37 +LINK 38 +LINK 39 +LINK 40 +LINK 41 +LINK 42 +LINK 43 +LINK 44 +LINK 45 +LINK 46 +LINK 47 +LINK 48 +LINK 49 +LINK 50 +LINK 51 +LINK 52 +LINK 53 +LINK 54 +LINK 55 +LINK 56 +LINK 57 +LINK 58 +LINK 59 +LINK 60 +LINK 61 +LINK 62 +LINK 63 +LINK 64 +``` + + + +```csv +GO TO 250 +205 CONTINUE +WRITE(6,920) NEQNS,NWMTL,NWKTL +WRITE(6,930) (MAXAI(I),I=1,MEQNS) +WRITE(6,930) (MAXAJ(I),I=1,MEQNS) +930 FORMAT(5X,20I5) +920 FORMAT(/5X,'NEQNS=',I5,5X,'NWMTL=',I5,5X,'NWKTL=',I5/) +IF(NWKTL.GT.6000) GO TO 210 +GO TO 220 +210 WRITE(6,910) +STOP +220 CONTINUE +910 FORMAT (/'SET DIMENSION EXCEEDED - CHECK LINKIN '/) +C +C*** GLOBAL MASS MATRIX +C +DO 500 IELEM=1,NELEM +IMASS=INTGR(IELEM) +IF(IMASS.EQ.2) GO TO 500 +READ (3) EMASS +CALL ADDBAN (XMASS,MAXAI,EMASS,LEQNS(1,IELEM),NEVAB) +500 CONTINUE +C +C*** GLOBAL MASS VECTOR +C +NPOSM=0 +DO 510 IPOIN =1,NPOIN +DO 510 IDOFN =1,NDOFN +NPOSM=NPOSM+1 +NPOSN=IFPRE(IDOFN,IPOIN) +IF(NPOSN.EQ.0) GO TO 510 +YMASS(NPOSN)=YMASS(NPOSM) +FORCE(NPOSN)=FORCE(NPOSM) +510 CONTINUE +RETURN +END +LINK 65 +LINK 66 +LINK 67 +LINK 68 +LINK 69 +LINK 70 +LINK 71 +LINK 72 +LINK 73 +LINK 74 +LINK 75 +LINK 76 +LINK 77 +LINK 78 +LINK 79 +LINK 80 +LINK 81 +LINK 82 +LINK 83 +LINK 84 +LINK 85 +LINK 86 +LINK 87 +LINK 88 +LINK 89 +LINK 90 +LINK 91 +LINK 92 +LINK 93 +LINK 94 +LINK 95 +LINK 96 +LINK 97 +LINK 98 +LINK 99 +LINK 100 +``` + +LINK 18-29 Reassigns IFPRE vector with equation numbers. If IFPRE is not zero than IFPRE is reassigned as zero. + +LINK 34-45 Evaluates the vector LEQNS on element level for assigning equation number corresponding to each node in an element. + +LINK 52-55 Calculates column height above the diagonal in global matrix. + +LINK 59–62 Assigns location for diagonal elements in global matrix. + +LINK 80-85 IMASS = 1 calculates stiffness matrix for only implicit elements. + +IMASS = 2 calculates stiffness matrix for complete mesh. + +# 11.5.13 Subroutine MULTPY + +This routine $^{(9)}$ evaluates the product of square matrix AMATX and an array START and stores the result in FINAL. + +```txt +SUBROUTINE MULTPY (FINAL, AMATX, START, MAXAI, NEQNS, NWMTL) MULT 1 +C*************** MULT 2 +C MULT 3 +C *** TO EVALUATE PRODUCT OF B TIMES RR AND STORE RESULT IN TT +C MULT 4 +C*************** MULT 5 +DIMENSION FINAL(1), AMATX(1), START(1), MAXAI(1) MULT 6 +C MULT 7 +MULT 8 +``` + + + +```txt +IF(NWMTL.GT.NEQNS) GO TO 20 +DO 10 IEQNS=1,NEQNS +10 FINAL(IEQNS)=AMATX(IEQNS)*START(IEQNS) +RETURN +C +20 DO 40 IEQNS=1,NEQNS +40 FINAL(IEQNS)=0.0 +DO 100 IEQNS=1,NEQNS +LOWER=MAXAI(IEQNS) +KUPER=MAXAI(IEQNS+1)-1 +JEQNS=IEQNS+1 +TERMI=START(IEQNS) +DO 100 ICOLM=LOWER,KUPER +JEQNS=JEQNS-1 +100 FINAL(JEQNS)=FINAL(JEQNS)+AMATX(ICOLM)*TERMI +IF(NEQNS.EQ.1) RETURN +DO 200 IEQNS=2,NEQNS +LOWER=MAXAI(IEQNS)+1 +KUPER=MAXAI(IEQNS+1)-1 +IF(KUPER-LOWER) 200,210,210 +210 JEQNS=IEQNS +SUMAA=0.0 +DO 220 ICOLM=LOWER,KUPER +JEQNS=JEQNS-1 +220 SUMAA=SUMAA+AMATX(ICOLM)*START(JEQNS) +FINAL(IEQNS)=FINAL(IEQNS)+SUMAA +200 CONTINUE +RETURN +END +MULT 9 +MULT 10 +MULT 11 +MULT 12 +MULT 13 +MULT 14 +MULT 15 +MULT 16 +MULT 17 +MULT 18 +MULT 19 +MULT 20 +MULT 21 +MULT 22 +MULT 23 +MULT 24 +MULT 25 +MULT 26 +MULT 27 +MULT 28 +MULT 29 +MULT 30 +MULT 31 +MULT 32 +MULT 33 +MULT 34 +MULT 35 +MULT 36 +MULT 37 +``` + +# 11.5.14 Subroutine REDBAK + +This routine $^{(9)}$ solves the equations after the matrix is decomposed (into the form $LDL^{T}$ ) using forward and backward substitution. + +```txt +SUBROUTINE REDBAK (STIFF ,FORCE ,MAXAI ,NEQNS ) +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C +C + +DIMENSION STIFF(1) ,FORCE(1) ,MAXAI(1) +C +DO 400 IEQNS=1,NEQNS +LOWER =MAXAI(IEQNS)+1 +KUPER=MAXAI(IEQNS+1)-1 +IF(KUPER-LOWER) 400,410,410 +410 JEQNS=IEQNS +SUMCC=0.0 +DO 420 ICOLM=LOWER,KUPER +JEQNS=JEQNS-1 +420 SUMCC=SUMCC+STIFF(ICOLM)*FORCE(JEQNS) +FORCE(IEQNS)=FORCE(IEQNS)-SUMCC +400 CONTINUE +C +DO 480 IEQNS=1,NEQNS +KMAXA=MAXAI(IEQNS) +480 FORCE(IEQNS)=FORCE(IEQNS)/STIFF(KMAXA) +IF(NEQNS.EQ.1) RETURN +JEQNS=NEQNS +DO 500 IEQNS=2,NEQNS +LOWER=MAXAI(JEQNS)+1 +KUPER=MAXAI(JEQNS+1)-1 +RBAK 1 +RBAK 2 +RBAK 3 +RBAK 4 +RBAK 5 +RBAK 6 +RBAK 7 +RBAK 8 +RBAK 9 +RBAK 10 +RBAK 11 +RBAK 12 +RBAK 13 +RBAK 14 +RBAK 15 +RBAK 16 +RBAK 17 +RBAK 18 +RBAK 19 +RBAK 20 +RBAK 21 +RBAK 22 +RBAK 23 +RBAK 24 +RBAK 25 +RBAK 26 +RBAK 27 +RBAK 28 +``` + + + +
IF(KUPER-LOWER) 500,510,510RBAK29
510KEQNS=JEQNSRBAK30
DO 520 ICOLM=LOWER,KUPERRBAK31
KEQNS=KEQNS-1RBAK32
520FORCE(KEQNS)=FORCE(KEQNS)-STIFF(ICOLM)*FORCE(JEQNS)RBAK33
500JEQNS=JEQNS-1RBAK34
RETURNRBAK35
ENDRBAK36
+ +# 11.5.15 Subroutine RESEPL + +This routine evaluates the internal force vector for elasto-plastic materials. (See Section 7.8.7.) + +```fortran +SUBROUTINE RESEPL (COORD ,DISPL ,EFFST ,ELOAD ,EPSTN ,IITER , RESD 1 +INTGR ,LEQNS ,LNODS ,MATNO ,NCRIT ,NDIME , RESD 2 +NDOFN ,NELEM ,NGAUS ,NLAPS ,NMATS ,NNODE , RESD 3 +NPOIN ,NSTRE ,NTYPE ,POSGP ,PROPS ,RESID , RESD 4 +STRAG ,STRIN ,STRSG ,WEIGP ,IPRED ,ISTEP ) RESD 5 +C****************************************************************************************** RESD 6 +C RESD 7 +C *** EVALUATES RESIDUAL FORCES RESD 8 +C RESD 9 +C****************************************************************************************** RESD 10 +DIMENSION COORD(NPOIN,1),DERIV(2,9),DMATX(4, 4),AVECT(4),MATNO(1), RESD 11 +PROPS(NMATS,1),DLCOD(2,9),BMATX(4,18),DEVIA(4),DISPL(1), RESD 12 +LNODS(NELEM,1),GPCOD(2,9),DJACM(2, 2),STRAN(4),POSGP(1), RESD 13 +ELOAD(NELEM,1),CARTD(2,9),SHAPE( 9),STRES(4),WEIGP(1), RESD 14 +STRIN( 4,1),ELCOD(2,9),SIGMA( 4),SGTOT(4),EFFST(1), RESD 15 +STRSG( 4,1),ELDIS(2,9),DESIG( 4),DVECT(4),EPSTN(1), RESD 16 +STRAG( 4,1),RESID( 1),LEQNS(18,1),INTGR(1) RESD 17 +TWOPI=6.283185307179586 RESD 18 +NEVAB=NNODE*NDOFN RESD 19 +NTOTV=NPOIN*NDOFN RESD 20 +NSTR1=4 RESD 21 +DO 530 IELEM=1,NELEM RESD 22 +IF(INTGR(IELEM).EQ.2.AND.IITER.GT.1.AND.IPRED.EQ.1) GO TO 530 RESD 23 +DO 540 IEVAB=1,NEVAB RESD 24 +540 ELOAD(IELEM,IEVAB)=0.0 RESD 25 +530 CONTINUE RESD 26 +DO 510 ITOTV=1,NTOTV RESD 27 +510 RESID(ITOTV)=0.0 RESD 28 +KGAUS=0 RESD 29 +DO 20 IELEM=1,NELEM RESD 30 +IF(INTGR(IELEM).EQ.2.AND.IITER.GT.1.AND.IPRED.EQ.1) GO TO 20 RESD 31 +LPROP=MATNO(IELEM) RESD 32 +YOUNG=PROPS(LPROP,1) RESD 33 +POISS=PROPS(LPROP,2) RESD 34 +THICK=PROPS(LPROP,3) RESD 35 +UNIAX=PROPS(LPROP,6) RESD 36 +HARDS=PROPS(LPROP,7) RESD 37 +FRICT=PROPS(LPROP,8) RESD 38 +FRICT=FRICT*0.017453292 RESD 39 +IF(NCRIT.EQ.3) UNIAX=UNIAX*COS(FRICT) RESD 40 +IF(NCRIT.EQ.4) UNIAX=6.0*UNIAX*COS(FRICT)/ RESD 41 +(1.73205080757*(3.0-SIN(FRICT))) RESD 42 +C RESD 43 +C*** COMPUTE COORDINATE AND INCREMENTAL DISPLACEMENTS OF THE RESD 44 +C ELEMENT NODAL POINTS RESD 45 +C RESD 46 +IPOSN=0 RESD 47 +DO 30 INODE=1,NNODE RESD 48 +LNODE=LNODS(IELEM,INODE) RESD 49 +``` + + + +DO 30 IDIME=1,NDIME RESD 50 + +IPOSN=IPOSN+1 RESD 51 + +NPOSN=LEQNS(IPOSN, IELEM) RESD 52 + +IF(NPOSN.EQ.0) DISPT=0. RESD 53 + +IF(NPOSN.NE.0) DISPT=DISPL(NPOSN) RESD 54 + +DLCOD(IDIME, INODE)=COORD(LNODE, IDIME)+DISPT RESD 55 + +ELCOD(IDIME, INODE)=COORD(LNODE, IDIME) RESD 56 + +30 ELDIS(IDIME, INODE)=DISPT RESD 57 + +CALL MODPS (DMATX, LPROP, NMATS, NSTRE, NTYPE, PROPS) RESD 58 + +KGASP=0 RESD 59 + +DO 40 IGAUS=1,NGAUS RESD 60 + +DO 40 JGAUS=1,NGAUS RESD 61 + +EXISP=POSGP(IGAUS) RESD 62 + +ETASP=POSGP(JGAUS) RESD 63 + +KGAUS=KGAUS+1 RESD 64 + +KGASP=KGASP+1 RESD 65 + +CALL SFR2 (DERIV, NNODE, SHAPE, EXISP, ETASP) RESD 66 + +CALL JACOB2 (CARTD, DERIV, DJACB, ELCOD, GPCOD, RESD 67 + +IELEM, KGASP, NNODE, SHAPE) RESD 68 + +CALL JACOBD (CARTD, DLCOD, DJACM, NDIME, NLAPS, NNODE) RESD 69 + +DVOLU=DJACB\*WEIGP(IGAUS)\*WEIGP(JGAUS) RESD 70 + +IF(NTYPE.EQ.3) DVOLU=DVOLU\*TWOPI\*GPCOD(1,KGASP) RESD 71 + +IF(NTYPE.EQ.1) DVOLU=DVOLU\*THICK RESD 72 + +CALL BLARGE (BMATX, CARTD, DJACM, DLCOD, GPCOD, RESD 73 + +KGASP, NLAPS, NNODE, NTYPE, SHAPE) RESD 74 + +CALL LINGNL (CARTD, DJACM, DMATX, ELDIS, GPCOD, KGASP, RESD 77) + +KGAUS, NDOFN, NLAPS, NNODE, NSTRE, NTYPE, RESD 78 + +POISS, SHAPE, STRAN, STRES, STRAG) RESD 79 + +DO 560 ISTR1=1,NSTR1 RESD 80 + +560 STRAG(ISTR1,KGAUS)=STRAG(ISTR1,KGAUS)+STRAN(ISTR1) RESD 81 + +IF(ISTEP.GT.1.AND.IITER.GT.1) GO TO 160 RESD 82 + +DO 170 ISTR1=1,NSTR1 RESD 83 + +170 STRES(ISTR1)=STRES(ISTR1)+STRIN(ISTR1,KGAUS) RESD 84 + +160 CONTINUE RESD 85 + +PREYS=UNIAX+EPSTN(KGAUS)\*HARDS RESD 86 + +DO 150 ISTR1=1,NSTR1 RESD 87 + +DESIG(ISTR1)=STRES(ISTR1) RESD 88 + +150 SIGMA(ISTR1)=STRSG(ISTR1,KGAUS)+STRES(ISTR1) RESD 89 + +IF(NLAPS.EQ.2.OR.NLAPS.EQ.0) GO TO 60 RESD 90 + +CALL INVAR (DEVIA, LPROP, NCRIT, NMATS, PROPS, SINT3, STEFF, RESD 91 + +SIGMA, THETA, VARJ2, YIELD) RESD 92 + +ESPRE=EFFST(KGAUS)-PREYS RESD 93 + +IF(ESPRE.GE.0.0) GO TO 50 RESD 94 + +ESCUR=YIELD-PREYS RESD 95 + +IF(ESCUR.LE.0.0) GO TO 60 RESD 96 + +RFACT=ESCUR/(YIELD-EFFST(KGAUS)) RESD 97 + +GO TO 70 RESD 98 + +50 ESCUR=YIELD-EFFST(KGAUS) RESD 99 + +IF(ESCUR.LE.0.0) GO TO 60 RESD 100 + +RFACT=1.0 RESD 101 + +70 MSTEP=ESCUR\*8.0/UNIAX+1.0 RESD 102 + +IF(MSTEP.GT.10) MSTEP=10 RESD 103 + +ASTEP=MSTEP RESD 104 + +REDUC=1.0-RFACT RESD 105 + +DO 80 ISTR1=1,NSTR1 RESD 106 + +SGTOT(ISTR1)=STRSG(ISTR1,KGAUS)+REDUC\*STRES(ISTR1) RESD 107 + +80 STRES(ISTR1)=RFACT\*STRES(ISTR1)/ASTEP RESD 108 + +DO 90 JSTEP=1,MSTEP RESD 109 + +CALL INVAR (DEVIA, LPROP, NCRIT, NMATS, PROPS, SINT3, STEFF, RESD 110) + +SGTOT, THETA, VARJ2, YIELD) RESD 111 + +CALL YIELDF (AVECT,DEVIA,FRICT,NCRIT,SINT3,STEFF, RESD 112 + +. THETA,VARJ2) RESD 113 + +CALL FLOWPL (AVECT, ABETA, DVECT, HARDS, NTYPE, POISS, YOUNG) RESD 114 + + + +```csv +AGASH=0.0 RESD 115 +DO 100 ISTR1=1,NSTR1 RESD 116 +100 AGASH=AGASH+AVECT(ISTR1)*STRES(ISTR1) RESD 117 +DLAMD=AGASH*ABETA RESD 118 +IF(DLAMD.LT.0.0) DLAMD=0.0 RESD 119 +BGASH=0.0 RESD 120 +DO 110 ISTR1=1,NSTR1 RESD 121 +BGASH=BGASH+AVECT(ISTR1)*SGTOT(ISTR1) RESD 122 +110 SGTOT(ISTR1)=SGTOT(ISTR1)+STRES(ISTR1)-DLAMD*DVECT(ISTR1) RESD 123 +EPSTN(KGAUS)=EPSTN(KGAUS)+DLAMD*BGASH/YIELD RESD 124 +90 CONTINUE RESD 125 +CALL INVAR (DEVIA,LPROP,NCRIT,NMATS,PROPS,SINT3,STEFF, RESD 126 +.SGTOT,THETA,VARJ2,YIELD) RESD 127 +CURYS=UNIAX+EPSTN(KGAUS)*HARDS RESD 128 +BRING=1.0 RESD 129 +IF(YIELD.GT.CURYS) BRING=CURYS/YIELD RESD 130 +DO 130 ISTR1=1,NSTR1 RESD 131 +130 STRSG(ISTR1,KGAUS)=BRING*SGTOT(ISTR1) RESD 132 +EFFST(KGAUS)=BRING*YIELD RESD 133 +C*** ALTERNATIVE LOCATION OF STRESS REDUCTION LOOP TERMINATION CARD RESD 134 +C 90 CONTINUE RESD 135 +C*** GO TO 190 RESD 136 +60 DO 180 ISTR1=1,NSTR1 RESD 137 +180 STRSG(ISTR1,KGAUS)=STRSG(ISTR1,KGAUS)+DESIG(ISTR1) RESD 138 +EFFST(KGAUS)=YIELD RESD 139 +C RESD 140 +C*** CALCULATE THE EQUIVALENT NODAL FORCES AND ASSOCIATE WITH THE RESD 141 +C ELEMENT NODES RESD 142 +190 MGASH=0 RESD 143 +DO 140 INODE=1,NNODE RESD 144 +DO 140 IDQFN=1,NDOFN RESD 145 +MGASH=MGASH+1 RESD 146 +DO 140 ISTRE=1,NSTRE RESD 147 +140 ELOAD(IELEM,MGASH)=ELOAD(IELEM,MGASH)+BMATX(ISTRE,MGASH)* RESD 148 +.STRSG(ISTRE,KGAUS)*DVOLU RESD 149 +40 CONTINUE RESD 150 +20 CONTINUE RESD 151 +DO 500 IELEM=1,NELEM RESD 152 +DO 500 IEVAB=1,NEVAB RESD 153 +LMVEB=LEQNS(IEVAB,IELEM) RESD 154 +IF(LMVEB.EQ.0) GO TO 550 RESD 155 +RESID(LMVEB)=RESID(LMVEB)+ELOAD(IELEM,IEVAB) RESD 156 +550 CONTINUE RESD 157 +500 CONTINUE RESD 158 +RETURN RESD 159 +END RESD 160 +RESD 161 +``` + +# 11.6 Examples + +# 11.6.1 Spherical shell example + +Some of the capabilities $^{(10)}$ of the program MIXDYN are explained by analysing some simple problems. The spherical shell problem described $^{(11,12)}$ in Chapter 10 is again solved for the following cases: + +(i) Elastic small deformation (all implicit elements) +(ii) Elastic geometrically nonlinear (all implicit elements) +(iii) Elasto-plastic small deformation (all implicit elements) + + + +(iv) Elastic small deformation (all explicit elements) +(v) Elastic geometrically nonlinear (all explicit elements) +(vi) Elasto-plastic small deformation (all explicit elements) + +![](images/page-469_bc51ca27ea45df3b15bf84651b88a05ecaa35ae89a43f3eff4a464b7a4fae73e.jpg) + +
+text_image + +p = 600 lb/in² +Two +stiff +elements +R = 22.27 in +α +
+ +Fig. 11.4 Modified spherical shell example with stiff elements. + +To demonstrate the capabilities of program MIXDYN we also solve a slightly modified version of the spherical shell example. Two stiff and dense elements are added to the finite element mesh at the crown as shown in Fig. 11.4. The stiff elements have the following properties: + +$$ +\text { Elastic modulus } \quad E = 0. 1 0 5 \times 1 0 ^ {9} \text { lb / in } ^ {2} +$$ + +$$ +\text { poisson's ratio } \quad \nu = 0. 3 +$$ + +$$ +\text { mass density } \quad \rho = 0. 7 8 0 \times 1 0 ^ {- 3} \text { lb.sec } ^ {2} / \text { in } ^ {4} +$$ + +$$ +\text { yield stress } \quad \sigma_ {0} = 0. 5 \times 1 0 ^ {5} \mathrm{lb/in} ^ {2} +$$ + +The following modified shell examples are also analysed: + +(vii) Elasto-plastic small deformations (all implicit elements) +(viii) Elasto-plastic small deformations (all explicit elements) +(ix) Elasto-plastic small deformations (stiff elements are implicit elements, the remaining elements are explicit). + +The highest and lowest eigenvalues are evaluated for both the original and the modified spherical shells. For the original spherical shell the fundamental period is $0.547 \times 10^{-3}$ sec and the smallest time period is $1.380 \times 10^{-6}$ + + + +sec. For the modified spherical shell the fundamental period $T_{f}$ is $0.592 \times 10^{-3}$ sec and the smallest time period $T_{h}$ is $0.776 \times 10^{-6}$ sec. Thus the addition of the stiff elements does not significantly change the largest period but it does change the smallest period quite dramatically. For an accurate solution based on implicit time integration the time step length $\Delta t$ is taken as $T_{f}/100 \simeq 0.6 \times 10^{-5}$ sec for both the original and the modified spherical shell. For a stable and accurate solution based on explicit time integration the time step length $\Delta t \leqslant T_{h}/\pi$ which is $0.25 \times 10^{-6}$ sec for the modified spherical shell or $0.40 \times 10^{-6}$ sec for the original spherical shell. Thus the addition of two stiff elements reduces the critical time step length to 1/1.6 of the original critical time step length. Hence the explicit analysis becomes more expensive. However, if the stiff elements are taken as implicit elements in case (ix) for implicit-explicit analysis, then the critical time step is governed by the remaining explicit elements so that the time step must be less than or equal to $0.40 \times 10^{-6}$ sec. + +![](images/page-470_1223531b36a8f36460cb4cf02a18883696b6a76784ae154818e3217a08179ebc.jpg) + +
+line + +| t (x10⁻³ secs) | Linear small displacement formulation | Linear large displacement formulation | +| -------------- | ------------------------------------- | ------------------------------------- | +| 0 | 0.00 | 0.00 | +| 2 | 0.03 | 0.035 | +| 4 | 0.015 | 0.018 | +| 6 | 0.035 | 0.04 | +| 8 | 0.01 | 0.015 | +| 10 | 0.00 | 0.005 | +
+ +Fig. 11.5(a) Spherical shell results. Cases (i), (ii), (iv) and (v). + +Figure 11.5(a) compares the response of the elastic analyses with small and large deformations.\* The results are similar to the results obtained using DYNPAK. The response with the large deformation gives a time period which is elongated. + +\* Note that the implicit and explicit results overlap. diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_048.md b/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_048.md new file mode 100644 index 00000000..075f30a1 --- /dev/null +++ b/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_048.md @@ -0,0 +1,131 @@ + + +![](images/page-471_bfa00acbfeeac6bfe139752f34ac43b4b01b1c8f19da74699d3f161ba0a7052a.jpg) + +
+line + +| t × 10⁻³ secs | Crown deflection (in) | +| ------------- | --------------------- | +| 0 | 0.00 | +| 2 | 0.06 | +| 4 | 0.05 | +| 6 | 0.05 | +| 8 | 0.07 | +| 10 | 0.08 | +
+ +Fig. 11.5(b) Spherical shell results. Cases (iii) and (vi). + +Figure 11.5(b) illustrates the elasto-plastic small deformation response. The time periods are elongated with the inclusion of plasticity effects. + +In Fig. 11.5(c) the results for the problem with the stiff element are presented with explicit, implicit and mixed explicit–implicit analysis (cases (vii)–(ix)). The execution times and results are compared. The relative computer times are: + +(i) all elements considered as explicit - 120.0 sec +(ii) stiff elements as implicit and rest explicit - 80.8 sec +(iii) all elements considered as implicit - 16.4 sec + +![](images/page-471_049925c12591b1ea9a55c65a8bc0937c394bb7029ed3470da9db2664248a9d16.jpg) + +
+line + +| Method | Time (t × 10⁻³ secs) | Crown deflection (in) | +|--------|----------------------|------------------------| +| Explicit | 0.25 × 10⁶ sec | 0.025 | +| Implicit-explicit | 0.40 × 10⁶ sec | 0.040 | +| Implicit | 0.60 × 10⁶ sec | 0.060 | +
+ +Fig. 11.5(c) Spherical shell results. Cases (vii)-(ix). + + + +This shows that by representing the stiff elements implicitly computer time can be saved. The analysis in which all elements are treated implicitly gives the lowest execution time for this small problem. However, with increasing problem size (and band width) the solution time for an implicit solution increases very rapidly because of the large core requirement and the increased number of computer operations. + +Finally it should be noted that Hughes has recently shown how the implicit-explicit schemes may be used in a more general context where there are, for example, nonsymmetric stiffness matrices involved or an implicit-explicit dynamic relaxation solution is required. $^{(13)}$ + +# 11.7 Problems + +11.1 Repeat Problems 10.1–10.4 using program MIXDYN. Use fully explicit, fully implicit and mixed implicit/explicit meshes. + +# 11.8 References + +1. BELYTSCHKO, T. and MULLEN, R., Mesh partitions of explicit-implicit time integration, In: Formulations and Computational Algorithms in Finite Element Analysis, Ed. K. J. Bathe et al., MIT Press (1977). +2. BELYTSCHKO, T. and MULLEN, R., Stability of explicit-implicit mesh partitions in time integration, Int. J. Num. Meth. Engng. 12, 1575–1586 (1978). +3. BELYTSCHKO, T., YEN, H. J. and MULLEN, R., Mixed methods for time integration, Proc. Int. Conf. on Finite Elements in Non-linear Mechanics (FENO-MECH 78), Univ. Stuttgart, Germany (Sept. 1978). +4. HUGHES, T. J. R. and LIU, W. K., Implicit-explicit finite elements in transient analysis: stability theory, J. Appl. Mech. 45, 371–374 (1978). +5. HUGHES, T. J. R. and LIU, W. K., Implicit-explicit finite elements in transient analysis: implementation and numerical examples, J. Appl. Mech. 45, 375-378 (1978). +6. HUGHES, T. J. R., PISTER, K. S. and TAYLOR, R. L., Implicit-explicit finite elements in nonlinear transient analysis, Comp. Meth. Appl. Mech. Engng. 17/18, 159-182 (1979). +7. PARK, K. C., FELIPPA, C. A. and DERUNTZ, H. A., Stabilization of staggered solution procedures for fluid-structure interaction analysis, In: Computational Methods for Fluid-Structure Interaction Problems, Ed. T. Belytschko and T. L. Geers, ASME Applied Mechanics Symposia Series, AMD, 26, 94–124 (1977). +8. PARK, K. C., Partitioned transient analysis procedures for coupled field problems, to be published J. Appl. Mech. (1980). +9. BATHE, K. J. and WILSON, E. L., Numerical Methods in Finite Element Analysis, Prentice-Hall, Englewood Cliffs, New Jersey (1976). +10. PAUL, D. K. and HINTON, E., User guide and report on program MIXDYN for implicit-explicit transient dynamic analysis, Research Report, University College of Swansea (1980). +11. NAGARAJAN, S. and POPOV, E. P., Elastic-plastic dynamic analysis of axisymmetric solids, Computers and Structures, 4, 1117-1134 (1974). + + + +12. BATHE, K. J. and OZDEMIR, H., Elastic-plastic large deformation static and dynamic analysis, Computers and Structures, 6, 81–90 (1976). +13. HUGHES, T. J. R., Implicit-explicit finite element techniques for symmetric and nonsymmetric systems, Proc. Int. Conf. Numerical Methods for Nonlinear Problems, Swansea, 127–139, Pineridge Press, Swansea, U.K. (1980). + + + + + +# Chapter 12 + +# Alternative formulations and further applications + +# 12.1 Introduction + +Throughout this text we have considered several specific elasto-plastic material problems and, apart from Chapter 3, treatment has been limited to the use of elasto-plastic quasi-static incremental theory or an elasto-viscoplastic formulation. These theories and the application areas of solids and plates form, undoubtedly, the area of most interest and importance in nonlinear material analysis and it is for this reason that they have been chosen for study in this text. However, other topics and applications of possibly equal importance have had to be omitted for reasons of space and it is the aim of this chapter to indicate to the reader some areas for future studies. The developments which will be discussed can be categorised into the following classes: + +\- Further applications. The elasto-plastic and elasto-viscoplastic theories described earlier in this text can be extended to cover some alternative structural forms. Of prime importance in this area is the analysis of both thick and thin three-dimensional shell structures and the main changes necessary to the corresponding linear elastic finite element process relate to expressing the yield criterion in terms of the appropriate stress resultants. + +\- Alternative material models. The behaviour of some engineering materials may not be adequately described by the yield criteria presented in Chapter 7. This is particularly true of soils, rocks and concrete, since these materials, for example, have a limited tensile strength which is not accurately reflected in either the Mohr–Coulomb or Drucker–Prager failure laws. For such materials appropriate failure criteria must be developed. Additionally for soils the assumption of associated plasticity leads to excessive dilatency necessitating alternative formulations for accurate material modelling. + +● Further problem classes. Many physical situations exist which are governed by nonlinear equation systems which are not suitable for solution by the techniques described so far in the text. One such + + + +example is the time dependent deformations which take place during a metal forming process. In this application the elastic strains are negligible compared with the plastic components and therefore the stress increments can no longer be expressed by use of (8.15). + +For dynamic situations, coupled media problems frequently have to be solved. This may involve a fluid/structure interaction problem of the seismic analysis of water retaining structures or the impulsive loading of a nuclear containment vessel together with the coolant fluid. All the above problems may be complicated by further nonlinear behaviour due to gross geometrical deformations. + +\- Improved numerical techniques. Since nonlinear solution processes are necessarily expensive with regard to computational time, any savings which can be made in this area are of prime importance. Developments in this area include improved nonlinear equation solution techniques and self-adaptive schemes for optimisation of the finite element mesh and load incrementation. A further enhancement is the use of sub-structuring techniques to separate elastic and elasto-plastic regions leading ultimately to coupled boundary integral/finite element solutions. + +In this chapter we explore the above developments (and others) in more detail and provide the reader with references for future study. Many of the subroutines presented earlier in the text can be employed (possibly in a modified form) in the development of computer codes for these further applications. Therefore the role of each subroutine presented is summarised and its location in the text also listed. + +# 12.2 List of subroutines + +In this section we record details of each subroutine that has been presented in this text. This library of subroutines can be employed to develop computer codes for the further applications discussed later in this chapter. The section of the chapter in which the subroutine is presented is recorded and the codes in which it is used are also indicated, employing the following program names: + +One-dimensional applications + +
QUITERSolution of quasiharmonic problems by direct iteration (Chapter 3).
QUNEWTSolution of quasiharmonic problems by the Newton-Raphson process (Chapter 3).
NONLASSolution of nonlinear elastic problems (Chapter 3),
ELPLASSolution of elasto-plastic problems (Chapter 3).
UNVISSolution of elasto-viscoplastic problems (Chapter 4).
TIMOSHSolution of elasto-plastic nonlayered Timoshenko beams (Chapter 5).
TIMLAYSolution of elasto-plastic layered Timoshenko beams (Chapter 5).
+ + + +Two-dimensional applications + +
PLANETElasto-plastic analysis of plane stress, plane strain and axisymmetric solids (Chapter 7).
VISCOUNTElasto-viscoplastic analysis of plane stress, plane strain and axisymmetric solids (Chapter 8).
MINDLINElasto-plastic analysis of nonlayered Mindlin plates (Chapter 9).
MINDLAYElasto-plastic analysis of layered Mindlin plates (Chapter 9).
DYNPAKElasto-plastic transient dynamic analysis of two dimensional solids (Chapter 10).
MIXDYNImplicit-explicit elasto-viscoplastic transient dynamic analysis of two dimensional solids (Chapter 11).
+ +12.2.1 Subroutines for one-dimensional applications + +
ASSEMBSection 3.4.2 (QUITER, QUNEWT, NONLAS, ELPLAS, TIMOSH, TIMLAY)Assembles the element contributions to form the global stiffness matrix and global load vector. (Simple equation solver).
ASTIF1Section 3.10.1 (QUNEWT)Formulates the stiffness matrix for each element according to (2.25) and (2.29) for the solution of one dimensional quasi-harmonic problems by the Newton Raphson method.
BAKSUBSection 3.4.4 (QUITER, QUNEWT, NONLAS, ELPLAS, TIMOSH, TIMLAY)Performs the backsubstitution phase of the Gaussian reduction process. (Simple equation solver).
BEAMSection 5.4.5 (TIMOSH)The master routine for elasto-plastic nonlayered Timoshenko beam program TIMOSH.
BEMLSection 5.5.5 (TIMLAY)The master routine for elasto-plastic layered Timoshenko beam program TIMLAY.
CONUNDSection 3.10.3 (QUNEWT, NONLAS, ELPLAS, TIMOSH, TIMLAY)Monitors convergence of the nonlinear solution process based on the residual forces according to (3.27).
CONVPSection 4.9 (UNVIS)Monitors convergence to steady state conditions according to (4.41) for one-dimensional elasto-viscoplastic problems.
DATASection 3.2 (QUITER, QUNEWT, NONLAS, ELPLAS, TIMOSH, TIMLAY)Data input subroutine for one-dimensional applications.
+ + + +
GREDUCSection 3.4.3 (QUITER, QUNEWT, NONLAS, ELPLAS, TIMOSH, TIMLAY)Undertakes equation elimination by Gaussian reduction. (Simple equation solver).
INCLODSection 3.7 (QUITER, QUNEWT, NONLAS, ELPLAS, TIMOSH, TIMLAY)Controls the incrementing of the applied loads for one-dimensional applications (modified for viscoplastic problems in Section 4.10).
INCVPSection 4.8 (UNVIS)Evaluates quantities at the end of the time step and the equilibrium correction terms for one-dimensional elastoviscoplastic problems.
INITALSection 3.6 (QUITER, QUNEWT, NONLAS, ELPLAS, TIMOSH, TIMLAY)Initialises to zero some arrays used by other subroutines for one-dimensional applications.
MONITRSection 3.9.2 (QUITER)Monitors convergence of the direct iteration process for one-dimensional quasiharmonic problems.
NONALSection 3.3 (QUITER, QUNEWT, NONLAS, ELPLAS, TIMOSH, TIMLAY)Controls the nonlinear solution process according to the value of NALGO specified, for one-dimensional applications.
REFOR1Section 3.10.2 (QUNEWT)Evaluates the ‘equivalent nodal forces’ according to (3.26) for one-dimensional quasiharmonic problems. (Newton Raphson solution).
REFOR2Section 3.11.2 (NONLAS)Evaluates the equivalent nodal forces according to (3.32) for one-dimensional nonlinear elastic problems.
REFOR3Section 3.12.2 (ELPLAS)Evaluates the equivalent nodal forces for one-dimensional elastoplastic problems.
REFORBSection 5.4.5 (TIMOSH)Evaluates the residual forces for a nonlayered elastoplastic Timoshenko beam.
RFORBLSection 5.5.5 (TIMLAY)Evaluates the residual forces for a layered elastoplastic Timoshenko beam.
RESOLVSection 3.4.5 (QUITER, QUNEWT, NONLAS, ELPLAS, TIMOSH, TIMLAY)Undertakes reduction of the R.H.S. terms for equation resolution (Simple equation solver).
+ + + +
RESULTSection 3.5 (QUITER, QUNEWT, NONLAS, ELPLAS, TIMOSH, TIMLAY)Outputs the results for one-dimensional applications.
STIFF1Section 3.9.1 (QUITER)Formulates the stiffness matrix for each element according to (2.25) for the solution of one-dimensional quasiharmonic problems by direct iteration.
STIFBLSection 5.5.5 (TIMLAY)Evaluates the elasto-plastic stiffness matrix for each element for the solution of layered Timoshenko beams.
STIFFBSection 5.4.5 (TIMOSH)Formulates the elasto-plastic stiffness matrix for each element for the solution of nonlayered Timoshenko beams.
STIFF2Section 3.11.1 (NONLAS)Formulates the stiffness matrix for each element according to (2.33) for nonlinear elastic one-dimensional problems.
STIFF3Section 3.12.1 (ELPLAS)Formulates the stiffness matrix for each element according to either (2.38) or (2.43) for one-dimensional elasto-plastic problems.
STUNVPSection 4.7 (UNVIS)Formulates the stiffness matrix for each element in turn for one-dimensional elasto-viscoplastic applications.
UNDIMSection 3.8 (QUITER, QUNEWT, NONLAS, ELPLAS)The main or master segment for one-dimensional nonlinear problems. See Fig. 3.1 for the small changes in the different applications.
UNVISCSection 4.11 (UNVIS)The main or master segment for one-dimensional visco-plastic problems.
+ +12.2.2 Subroutines for two-dimensional applications + +
ADDBANSection 11.5.3 (MIXDYN)
Generates the global matrix from the element stiffness matrices.
ADDRESSection 11.5.4 (MIXDYN)
Addresses the diagonal term of a matrix.
ALGORSection 6.5.2 (PLANET, VISCOUNT, MINDLIN, MIND-LAY)
Controls the nonlinear solution process according to the value of NALGO specified, for two-dimensional applications.
BLARGESection 10.6.3 (DYNPAK, MIXDYN)
Evaluates the strain matrix B for small and large deformation.
BMATPBSection 6.4.8 (MINDLIN)
Evaluates the strain matrix, B, for plate bending problems.
+ + + +
BMATPSSection 6.4.7 (PLANET, VISCOUNT)
Evaluates the strain matrix, B, for plane and axisymmetric situations.
CHECK1Section 6.4.13 (PLANET, VISCOUNT, MINDLIN, MINDLAY)
Scrutinises the problem control parameters for possible errors (two-dimensional applications).
CHECK2Section 6.4.15 (PLANET, VISCOUNT, MINDLIN, MINDLAY)
Checks the geometric data, boundary conditions and material properties for possible errors (two-dimensional applications).
COLMHTSection 11.5.5 (MIXDYN)
Evaluates the height of column above the diagonal of a matrix from the known addresses of diagonal terms.
CONTOLSection 10.6.4 (DYNPAK, MIXDYN)
Reads control data for dynamic dimensioning and also checks the dimension limits.
CONVERSection 6.5.4 (PLANET)
Monitors convergence of the nonlinear solution iteration process for two-dimensional applications.
CONVMPSection 9.5.3 (MINDLIN, MINDLAY)
Checks for convergence of solution of elasto-plastic layered and nonlayered Mindlin plates.
DBESection 6.4.11 (PLANET, VISCOUNT)
Forms the matrix product DB.
DECOMPSection 11.5.6 (MIXDYN)
Decomposes positive definite matrix into LDL $^T$ .
DEPMPASection 9.6.4 (MINDLAY)
Sets up the layered discretisation for the layered elasto-plastic Mindlin plate.
DIMENSection 7.8.1 (PLANET, VISCOUNT)
Presets the value of variables associated with dynamic dimensioning.
DIMMPSection 9.5.4 (MINDLIN, MINDLAY)
Sets up dynamic dimensions in programs MINDLIN and MINDLAY for the elasto-plastic analysis of layered and nonlayered plates.
DINTOBSection 11.5.7 (MIXDYN)
Multiplies the modulus and strain matrices to give DB.
DYNPAKSection 10.6.2 (DYNPAK)
Organises the explicit viscoplastic transient dynamic analysis.
ECHOSection 6.4.14 (PLANET, VISCOUNT, MINDLIN, MINDLAY)
diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_049.md b/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_049.md new file mode 100644 index 00000000..9faf7cd2 --- /dev/null +++ b/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_049.md @@ -0,0 +1,225 @@ + + +
Echoes the remaining data after input data errors have been diagnosed.
EXPLITSection 10.6.5 (DYNPAK)Carries out explicit time integration.
FEAMSection 9.6.2 (MINDLAY)Organising routine for the elasto-plastic analysis of layered Mindlin plates.
FEMPSection 9.5.2 (MINDLIN)Organising routine for the elasto-plastic analysis of nonlayered Mindlin plates.
FIXITYSection 10.6.6 (DYNPAK)Boundary conditions are inserted.
FLOWMPSection 9.5.5 (MINDLIN, MINDLAY)Determines $\partial F/\partial \sigma_{f}$ (i.e. yield function derivatives) for elastoplastic layered and nonlayered Mindlin plates.
FLOWPLSection 7.8.4.2 (PLANET, MIXDYN)Determines the vector $d_{D}$ for elasto-plastic analysis.
FLOWVPSection 8.9 (VISCOUNT, DYNPAK)Determines the viscoplastic strain rate for each Gauss point according to (8.7).
FRONTSection 6.4.12 (PLANET, VISCOUNT, MINDLIN, MINDLAY)Performs element assembly and equation solution by the frontal method. Contains a facility for efficient resolution of equations.
FUNCTASection 10.6.8 (DYNPAK, MIXDYN)Interpolates acceleration ordinate at $\Delta t$ intervals.
FUNCTSSection 10.6.9 (DYNPAK, MIXDYN)Evaluates factor for Heaviside and Harmonic time function at $\Delta t$ apart.
GAUSSQSection 6.4.2 (PLANET, VISCOUNT, MINDLIN, MINDLAY, DYNPAK, MIXDYN)Evaluates the sampling point positions and weighing factors for numerical integration by Gauss quadrature.
GEOMSTSection 11.5.8 (MIXDYN)Evaluates the stress stiffness matrix.
GRADMPSection 9.5.6 (MINDLIN)Evaluates the total displacement and rotation derivatives ( $\partial w/\partial x$ , $\partial w/\partial y$ , $\partial \theta_{x}/\partial x$ , $\partial \theta_{x}/\partial y$ , $\partial \theta_{y}/\partial x$ , $\partial \theta_{y}/\partial y$ ).
GSTIFFSection 11.5.9 (MIXDYN)Evaluates the global stiffness matrix in compacted profile form.
IMPEXPSection 11.5.10 (MIXDYN)Sets the constants of integration and evaluates partial effective load vector.
+ + + +
INCREMSection 6.5.3 (PLANET, VISCOUNT, MINDLIN, MINDLAY)Controls the incrementing of the applied loads for two-dimensional applications.
INPUTSection 6.5.1 (PLANET, VISCOUNT, MINDLIN, MINDLAY)Data input subroutine for two-dimensional applications.
INPUTDSection 10.6.10 (DYNPAK, MIXDYN)Data input subroutine. Reads the mesh data, properties etc
INTIMESection 10.6.11 (DYNPAK, MIXDYN)Reads the data necessary for time integration.
INVARSection 7.8.3 (PLANET, VISCOUNT, DYNPAK, MIXDYN)Evaluates the effective stress level at a given point for monitoring plastic yielding.
INVERTSection 8.7.3 (VISCOUNT)This subroutine determines the inverse of any arbitrary square matrix.
INVMPSection 9.5.7 (MINDLIN)Evaluates the Mindlin plate stress resultant invariants for nonlayered plates.
ITRATESection 11.5.11 (MIXDYN)Evaluates the total effective load and iterates until convergence is reached.
JACOBDSection 10.6.13 (DYNPAK, MIXDYN)Evaluates the deformation Jacobian matrix.
JACOB2Section 6.4.4 (PLANET, VISCOUNT, MINDLIN, MINDLAY, DYNPAK, MIXDYN)Evaluates the Jacobian matrix, its inverse and the Cartesian derivatives of the element shape functions for two-dimensional applications.
LAYMPASection 9.6.5 (MINDLAY)Evaluates the matrix of flexural rigidities and the matrix of shear rigidities for the layered elastoplastic Mindlin plate.
LINEARSection 7.8.6 (PLANET, MIXDYN)Determines the stresses from given displacements assuming linear elastic behaviour.
LINGNLSection 10.6.14 (DYNPAK, MIXDYN)Evaluates the linear stresses for small and large deformation analysis.
LINKINSection 11.5.12 (MIXDYN)This routine links with the profile solver.
LOADPBSection 6.4.6 (MINDLIN, MINDLAY)Evaluates the consistent nodal forces for plate bending problems.
+ + + +
LOADPLSection 10.6.15 (DYNPAK, MIXDYN)Generates the load vector.
LOADPSSection 6.4.5 (PLANET, VISCOUNT)Evaluates the consistent nodal forces due to gravity and distributed edge loads for two-dimensional problems.
LUMASSSection 10.6.16 (DYNPAK, MIXDYN)Generates the consistent mass matrix for implicit elements and special lumped mass matrix for explicit elements.
MDMPASection 9.6.6 (MINDLAY)Evaluates the constitutive matrices for use in layered Mindlin plate analysis.
MINDPBSection 9.5.8 (MINDLIN, MINDLAY)Reads additional input data for elasto-plastic, layered and nonlayered Mindlin plates.
MIXDYNSection 11.5.2 (MIXDYN)Organises implicit/explicit transient dynamic program.
MODPBSection 6.4.10 (MINDLIN)Evaluates the D matrix for plate bending applications.
MODPSSection 6.4.9 (PLANET, VISCOUNT, DYNPAK, MIXDYN)Evaluates the D matrix for plane and axisymmetric situations.
MULTPYSection 11.5.13 (MIXDYN)Multiplies square matrix to a vector or vector to a vector.
NODEXYSection 6.4.1 (PLANET, VISCOUNT, MINDLIN, MINDLAY)Interpolates the coordinates of midside nodes for elements with straight sides. This routine is modified in MINDLIN and MINDLAY where a hierarchical formulation is adopted for the ninth node. (See Section 9.5).
NODXYRSection 10.6.18 (DYNPAK, MIXDYN)Evaluates the midside node of elements. In case of axisymmetric problems if (R, Θ) coordinates are read r, z coordinates are evaluated within it.
OUTDYNSection 10.6.19 (DYNPAK, MIXDYN)Writes the output on output file and stress and displacement histories of required Gauss points and nodes respectively on specified tapes.
OUTMPSection 9.5.10 (MINDLIN)Outputs displacements, reactions and Gauss point stress resultants for elasto-plastic nonlayered Mindlin plates.
OUTMPASection 9.6.7 (MINDLAY)Outputs displacements, reactions and Gauss point layer stresses for elasto-plastic layered Mindlin plates.
+ + + +
OUTPUTSection 7.8.8 (PLANET, VISCOUNT)Outputs the results for two-dimensional problems at specified intervals.
PLASTSection 7.8.9 (PLANET)The main or master segment for two-dimensional elastoplastic applications.
PREVOSSection 10.6.20 (DYNPAK, MIXDYN)Reads the initial force and stresses.
REDBAKSection 11.5.14 (MIXDYN)Solves equations after matrix decomposition, using forward and backward substitution.
RESEPLSection 11.5.15 (MIXDYN)Evaluates the internal force for different yield criteria in the implicit explicit program.
RESMPSection 9.5.11 (MINDLIN)Evaluates the internal nodal forces
$$ +\boldsymbol {p} = \int_ {\Omega} \boldsymbol {B} _ {f} ^ {T} \sigma_ {f} d \Omega + \int_ {\Omega} \boldsymbol {B} _ {s} ^ {T} \sigma_ {s} d \Omega +$$ + +for the stress resultants $\sigma_{f}$ and $\sigma_{s}$ for elasto-plastic, non-layered Mindlin plates. +
RESMPASection 9.6.8 (MINDLAY)
Evaluates the residual force vector for layered elasto-plastic Mindlin plates.
RESIDUSection 7.8.7 (PLANET)
Evaluates the nodal forces which are statically equivalent to the stress field satisfying elasto-plastic conditions.
+ +
RESVPLSection 10.6.21 (DYNPAK)
Evaluates the internal forces for different yield criteria in the explicit transient dynamic program.
+ +
SFR2Section 6.4.3 (PLANET/ VISCOUNT, MINDLIN, MINDLAY, DYNPAK, MIXDYN)
Evaluates the element shape functions and their local derivatives for 4, 8 and 9 node isoparametric quadrilateral elements. SFR2 is modified in MINDLIN and MINDLAY to allow for a hierarchical representation for the 9th central node.
+ +
STEADYSection 8.12 (VISCOUNT)Monitors convergence to steady state conditions for two-dimensional elasto-viscoplastic problems.
+ +
STEPVPSection 8.8 (VISCOUNT)
Evaluates quantities, such as stresses and viscoplastic strains, at the end of each time step of a viscoplastic solution.
+ +
STIFFPSection 7.8.5 (PLANET)
+ + + +
Evaluates the stiffness matrix for each element for elastoplastic problems employing either D or $D_{ep}$ as appropriate.
STIFMPSection 9.5.13 (MINDLIN)
Evaluates the stiffness matrices for nonlayered elastoplastic Mindlin plate elements.
STIFVPSection 8.7.1 (VISCOUNT)
Evaluates the stiffness matrix for each element in turn for two-dimensional elastoplastic applications.
STIMPASection 9.6.9 (MINDLAY)
Evaluates the stiffness matrices for layered elastoplastic Mindlin plate elements.
STRESSSection 8.10 (VISCOUNT)
Evaluates the increment in stress occurring during a timestep of a viscoplastic analysis according to (8.20).
STRMPSection 9.5.14 (MINDLIN)
Evaluates stress resultants $[M_x, My, M_{xy}, Q_x, Q_y]^T$ for elastoplastic nonlayered Mindlin plates.
STRMPASection 9.6.10 (MINDLAY)
Evaluates the stresses $[\sigma_x, \sigma_y, \tau_{xy}, \tau_{xz}, \tau_{yz}]^T$ for elastoplastic layered Mindlin plates at each layer and each Gauss point.
SUBMPSection 9.5.15 (MINDLIN, MINDLAY)
Carries out matrix multiplications in elastoplastic layered and nonlayered Mindlin plates.
TANGVPSection 8.7.2 (VISCOUNT)
Evaluates the $D^n$ matrix for viscoplastic analysis by implicit time stepping schemes.
VISCOSection 8.13 (VISCOUNT)
The main or master segment for two-dimensional elastopiscoplastic applications.
VZEROSection 9.5.16 (MINDLIN, MINDLAY)
Zeroes a vector in elastoplastic layered and nonlayered Mindlin plates.
YIELDFSection 7.8.4.1 (PLANET, VISCOUNT, MIXDYN, DYN-PAK)
Determines the flow vector a for plastic and viscoplastic applications. (Amended in Section 10.6.22 for dynamic transient problems).
ZEROSection 7.8.2 (PLANET, VISCOUNT)
Sets to zero the contents of several arrays employed in the programs. (Modified for viscoplastic applications in Section 8.11).
ZEROMPSection 9.5.16 (MINDLIN, MINDLAY)
Zeroes various arrays in elastoplastic layered and nonlayered Mindlin plate programs.
+ + + +# 12.3 Alternative material models + +The plastic behaviour of most solids is adequately described by the four yield criteria presented in Chapter 7; namely the Tresca, Von Mises, Mohr-Coulomb and Drucker-Prager yield surfaces. However, for some engineering materials, notably concrete, rocks and soils, some modifications must be made to the above criteria or new yield surfaces postulated if an accurate prediction of the material response is required. + +For soils, the Mohr–Coulomb and Drucker–Prager criteria suffer from two deficiencies. Firstly, the assumption of an associated flow rule leads to excessive dilatency and secondly it is seen from Fig. 7.4 that both models imply that the material can support an unlimited hydrostatic compression. These deficiencies can be removed by use of the so-called critical state model, which assumes that the yield surface comprises two distinct parts. $^{(1-3)}$ The surface is shown plotted in terms of deviatoric $\sigma_{d}$ and hydrostatic stress, $\sigma_{s}$ , in Fig. 12.1. In the subcritical region yielding is stable due to strain hardening of the material whilst the supercritical region exhibits strain softening so that this portion of the yield surface forms a failure criterion. + +![](images/page-486_19635f7795d9aede192e38a4eece7eb92701a0ce9103390b1be021eb42e099c9.jpg) + +
+text_image + +supercritical +region +subcritical +region +C +critical state line +F SUPER = 0 +non-associated +flow rule +F SUB = 0 +(elliptical section) +associated +flow rule +B +1 +Scs +A A' 0 σc 2σc σ, D +
+ +Fig. 12.1 Critical state model for the behaviour of soil, $[\sigma_d = |\sigma_1 - \sigma_3|, \sigma_s = \frac{1}{2} (\sigma_1 + \sigma_3)]$ . + +A nonassociative flow rule is adopted in the supercritical region and the conical yield surface implied in Fig. 12.1 may be circular or hexagonal in form corresponding to a Mohr–Coulomb behaviour. In the subcritical region, the two most common shapes for the so-called cap is a log spiral or an ellipse and an associated flow rule is assumed to be obeyed. The yield surface can be expressed in the form + + + +$$ +F _ {\mathrm{SUPER}} = \sigma_ {d} - 2 \sin \phi \sigma_ {s} - 2 c \cos \phi = 0 +$$ + +$$ +F _ {\mathrm{SUB}} = \frac {\sigma_ {d} ^ {2} - S _ {c s} ^ {2} \sigma_ {s} (2 \sigma_ {c} - \sigma_ {s})}{\sigma_ {d} + S _ {c s} \sigma_ {c}} = 0, \tag {12.1} +$$ + +in which $S_{cs}$ is the slope of the critical state line. + +In the tensile zone, various options are open for modelling the limited tensile strength of the soil. The curved line $BA'$ can be employed or, more simply the vertical intercept OB (implying zero tensile strength) may be assumed. Complete details of the critical state model for soils can be found in Refs. 1–3 including its application to the numerical solution of practical problems. + +The Mohr–Coulomb and Drucker–Prager criteria exhibit the same deficiencies for modelling concrete behaviour as occur in the case of soils. In particular they overestimate the tensile strength of the material and also allow the material to support an unlimited hydrostatic compression. Many models have been proposed to more accurately predict the behaviour of concrete; a review of which can be found in Ref. 4. + +The most common method of predicting the tensile behaviour of concrete (and rocks) is by use of the no-tension model (or limited tension model). $^{(5)}$ In this, the tensile principal stresses are monitored throughout the structure and as soon as the value at any point exceeds the specified limiting tensile strength of the concrete, the material is assumed to crack in a plane normal to the principal direction. The tensile stress must then be reduced to zero by evaluating its nodal force equivalent and regarding these as residual forces to be applied and redistributed in an iterative process. Should the crack close on load reversal a frictional behaviour between the surfaces of the crack can be modelled. It is worth recording that the numerical stability of such solution processes is relatively poor since on initiation of tensile cracking the existing stress must be eliminated by redistribution, whereas for elasto-plastic problems, yielding merely necessitates that the existing stress level be maintained. + +An example of this type of analysis is illustrated in Fig. 12.2 where a cylindrical prestressed concrete reactor vessel is shown. The geometry of the vessel, together with the location of the prestressing system is indicated and the finite element mesh employed in solution is also shown. The concrete is assumed to behave as a limited tension material and the steel components as a Von Mises elasto-plastic solid. The effects of prestressing are included as an initial stress system and the vessel is incrementally loaded by a progressively increasing internal pressure. Figure 12.3 shows the vertical deflection of the centre point of the end slab with increasing load and good agreement is observed with both the experimental results and numerical analysis of Ref. 6. The zones of tensile cracking are shown in Fig. 12.4 for various applied pressure values and again good agreement with the results of Ref. 6 is evident. + + + +![](images/page-488_e361e30b130b4b0921051fd7e92cb43aa705116f9957defade214558f683dff3.jpg) + +
+text_image + +circumferential +prestressing +40 in +20 in +9 in +5 in +15 in +longitudinal +prestressing +cable +cros section +through vessel +longitudinal prestressing +load +internal +pressure +pressure +equivalent +to prestressing +forces +parabolic isoparametric element +mesh and loading system +
+ +Fig. 12.2 Finite element idealisation of a prestressed concrete reactor vessel by quadratic isoparametric elements. +![](images/page-488_6345e44501381ef122005e669b3427933fa2b544420633ab76ad7a52d0ebef7e.jpg) + +
+line + +| displacement (in) | internal pressure (psi) | Series | +| ----------------- | ------------------------ | ------ | +| 0.00 | 0 | 31 parabolic elements, ν=0 | +| 0.01 | ~450 | 31 parabolic elements, ν=0.15, hoop pressure=620 psi | +| 0.02 | ~550 | 31 parabolic elements, ν=0 | +| 0.03 | ~600 | 31 parabolic elements, ν=0.15, hoop pressure=510 psi | +| 0.04 | ~650 | 31 parabolic elements, ν=0.15, hoop pressure=510 psi | +| 0.05 | ~680 | 31 parabolic elements, ν=0 | +| 0.06 | ~700 | 31 parabolic elements, ν=0.15, hoop pressure=620 psi | +| 0.04 | ~650 | experimental ref (6) | +| 0.03 | ~600 | experimental ref (6) | +| 0.02 | ~550 | experimental ref (6) | +| 0.01 | ~500 | experimental ref (6) | +| 0.00 | 0 | experimental ref (6) | +| 0.04 | ~650 | experimental ref (6) | +| 0.05 | ~680 | experimental ref (6) | +| 0.06 | ~700 | experimental ref (6) | +| 0.04 | ~650 | 31 parabolic elements, ν=0.15 | +| 0.03 | ~600 | 31 parabolic elements, ν=0.15 | +| 0.02 | ~550 | 31 parabolic elements, ν=0.15 | +| 0.01 | ~500 | 31 parabolic elements, ν=0.15 | +| 0.00 | 0 | 31 parabolic elements, ν=0 | +| 0.04 | ~650 | experimental ref (6) | +| 0.05 | ~680 | experimental ref (6) | +| 0.06 | ~700 | experimental ref (6) | +| 0.04 | ~650 | 31 parabolic elements, ν=0.15 | +| 0.03 | ~620 | 31 parabolic elements, ν=0.15 | +| 0.02 | ~580 | 31 parabolic elements, ν=0.15 | +| 0.01 | ~550 | 31 parabolic elements, ν=0.15 | +| 0.00 | 0 | 31 parabolic elements, ν=0 | +| 0.04 | ~650 | experimental ref (6) | +| 0.05 | ~680 | experimental ref (6) | +| 0.06 | ~700 | experimental ref (6) | +
+ +Fig. 12.3 Load/deflection curves for the vessel of Fig. 12.2 failing in slab flexural mode. + + + +![](images/page-489_eeaddfe97b435074858ef1db3520aa8821f81c5efd2601da37c0bd3032793c74.jpg) + +trace of circumferential cracks + +zones of radial cracking + +Fig. 12.4 Zones of tensile cracking for the vessel of Fig. 12.2 failing in slab flexural mode. + +For predicting the compressive behaviour of concrete as well as the tensile response many failure surfaces have been proposed and a typical model is illustrated in Fig. 12.5. In addition to a brittle behaviour in tension, the model allows a viscoplastic range of behaviour before material failure. For further details the reader is directed to Ref. 4. + +A final approach to concrete behaviour which is worthy of mention is afforded by the so-called endochronic theory pioneered by Valanis $^{(7,8)}$ and generalised to concrete structures by Bazant. $^{(9,10)}$ To account for the strain history dependence of materials (in addition to their strain rate dependence) the concept of intrinsic time z is introduced which is related to the Newtonian time scale, t according to + +$$ +d z ^ {2} = \alpha^ {2} (d \zeta^ {2} + \beta^ {2} d t ^ {2}), \tag {12.2} +$$ + +where $d\zeta$ is effectively a measure of the deformation path length, $\beta$ is a + + + +material parameter and $\alpha$ depends on $\dot{\zeta}$ . Bazant has generalised the endochronic model to account for inelastic dilatancy, hydrostatic and shear compaction and fracture behaviour. $^{(10)}$ + +![](images/page-490_12c3427298f98ddc264437b4b8c5d1704b59bed0311132e9e7d9e51d283c71cb.jpg) + +
+text_image + +initial viscoplastic loading surface +elastic region +plastic yield surface +elasto—viscoplastic region +brittle failure surface +-σ₂ +-σ₃ +-σ₁ +
+ +Fig. 12.5 Typical yield and failure surfaces for concrete. + +# 12.4 Further applications + +# 12.4.1 Flow problems + +In this class of problem we are concerned with the continuing viscous flow of materials under steady state conditions. Typical examples include the extrusion of material through a die and flow of lubricating muds in oil drilling applications. In each case the problem is characterized by the fact that the elastic strains are negligible in comparison to the plastic components. For this reason, the viscoplastic numerical process described in Chapter 8 is unsuitable, since the increment of stress occurring during a timestep was based on the elastic strain increment according to (8.15). Thus an alternative formulation is clearly necessary and in fact a considerable simplification is achieved if the elastic components of strain are neglected in solution. $^{(11)}$ + +The plastic strain rate, $\dot{\epsilon}_{vp}$ , which is now assumed to be the total strain rate, $\dot{\epsilon}$ , is given from (8.7) to be + +$$ +\dot {\epsilon} = \dot {\epsilon} _ {v p} = \gamma \langle \Phi (F) \rangle a, \tag {12.3} +$$ + +and we recall that a is the flow vector defined by (7.42), $\Phi$ is an appropriate flow function (given for example by (8.8) or (8.9)) and $\gamma$ is a fluidity parameter. For the particular case of a Von Mises yield surface we have from (7.11) that + +$$ +F (\sigma , \kappa) = \sqrt {3} (J _ {2} ^ {\prime}) ^ {1 / 2} - \sigma_ {Y} (\kappa), \tag {12.4} +$$ + +where $J_{2}^{\prime}$ is the second deviatoric stress invariant and $\sigma_{Y}$ is the uniaxial yield stress of the material which may be a function of the strain hardening diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_050.md b/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_050.md new file mode 100644 index 00000000..f19897ba --- /dev/null +++ b/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_050.md @@ -0,0 +1,310 @@ + + +parameter $\kappa$ . Substituting from (12.4) into (12.3), and using (7.42) to express $a$ , results in + +$$ +\dot {\epsilon} = \gamma \langle \Phi (\sqrt {3} (J _ {2} ^ {\prime}) ^ {1 / 2} - \sigma_ {Y}) \rangle \sqrt {3 / 2} (J _ {2} ^ {\prime}) ^ {1 / 2} \sigma^ {\prime} = \Gamma (\sigma^ {\prime}) \sigma^ {\prime}, \tag {12.5} +$$ + +in which $\sigma'$ are the deviatoric stresses and $\Gamma(\sigma')$ is a symmetric viscoplastic compliance matrix whose form can be explicitly determined on prescription of the appropriate flow function $\Phi$ . Thus a relationship has been established between the total strain rate and the deviatoric stresses. + +The strain rate can be expressed in terms of the displacement velocities v by taking the differential form of the standard strain/displacement relationship, to give + +$$ +\dot {\epsilon} = B v. \tag {12.6} +$$ + +We assume, as for the viscoplastic case of Chapter 8, that the flow velocities are sufficiently slow to neglect inertia effects and that the following standard static equilibrium equations therefore hold. + +$$ +\int_ {V} \pmb {B} ^ {T} \pmb {\sigma} d V + \pmb {f} = 0, \tag {12.7} +$$ + +in which f are the applied forces comprising body forces b and boundary tractions, t. Thus a complete analogy exists between the above problem and the case of an elastic material in which the relationship between stress and strain is nonlinear according to + +$$ +\sigma = D (\sigma) \epsilon . \tag {12.8} +$$ + +Table 12.1 Correspondence between small strain nonlinear elastic problems and viscoplastic flow situations +
Small strain nonlinear elasticityFlow problem
Displacements, dVelocities, v
Stresses, σStresses, σ
Strains, εStrain rates, ε
Applied forces, fApplied forces, f
Nonlinear elastic compliance matrix, [D(σ)]-1Viscoplastic compliance matrix, Γ(σ)
+ +This analogy is indicated in Table 12.1. Therefore flow problems, in which the elastic components of deformation are negligible, can be solved by use of a linear elastic computer code which includes a facility for dealing with a stress dependent D matrix. Obviously the steady state solution to the flow problem must be arrived at in an iterative manner and a similar procedure must be employed in the corresponding elastic solution. The simplest approach + + + +is to proceed by the method of direct iteration, as described in Chapters 2 and 3, and to base the value of the compliance matrix $\Gamma$ on the current value of $\sigma$ . This solution procedure can be summarised as follows: + +(1) From the stresses $\sigma^n$ at iteration $n$ evaluate the viscoplastic compliance matrix $\Gamma(\sigma^n) = \Gamma^n$ . +(2) Compute the element stiffness matrix of each element as + +$$ +\int_ {V} \boldsymbol {B} ^ {T} [ \boldsymbol {\Gamma} ^ {n} ] ^ {- 1} \boldsymbol {B} d V +$$ + +and also the consistent nodal applied forces, $f^{(e)}$ . + +(3) Assemble and solve the stiffness equations to give the improved velocity estimate, $v^{n+1}$ . +(4) Compute the strain rates, $\dot{\mathbf{e}}^{n + 1} = \mathbf{B}\mathbf{v}^{n + 1}$ . +(5) Compute the stresses, $\sigma^{n + 1} = [\Gamma^n ]^{-1}\dot{\epsilon}^{n + 1}$ . +(6) Return to Step 1 and repeat the process until convergence takes place (i.e. $v^{n+1} \approx v^n$ ). + +The procedure described above is most suitable when boundary and body forces produce the forcing action. For the case when the problem is defined in terms of prescribed boundary velocities the compliance matrix $\Gamma$ must be expressed in terms of the current strain rate, $\dot{\epsilon}$ . $^{(12)}$ + +For metal forming problems, the situation is complicated by the fact that the geometry of the deforming solid is continually varying throughout the process. For such problems the transient form of the flow equations must be used and an incremental procedure can be adopted by which the coordinates of the finite element mesh are sequentially updated during solution. $^{(13)}$ + +It should be noted that no volumetric strain rate exists for some viscoplastic flow laws, as generally defined by (12.3), and this is indeed the case for the Von Mises criterion employed in (12.5). Consequently the viscoplastic compliance matrix $\Gamma$ cannot be inverted as required by Step 2 above and the same numerical difficulties that exist in incompressible elastic problems are encountered. However these can be readily overcome by the use of selective integration techniques whereby the element stiffness matrix is separated into volumetric and deviatoric components. $^{(14)}$ The near singularity arising in the former term as incompressible behaviour is approached is then numerically removed by employing a low order Gaussian integration rule. + +An important application of the above solution process is to the flow of non-Newtonian fluids, in which the material viscosity depends nonlinearly on the shear strain rate. Practical examples of such flow can be found in Refs. 15 and 16. Deviations from Newton's law of viscosity are best illustrated by means of flow curves and some of the most important cases are shown in Fig. 12.6. The effective stress, $\bar{\sigma}$ , and effective strain rate, $\bar{\epsilon}$ , are defined by (7.12) and (7.22) respectively. + + + +![](images/page-493_48d298f4acc01d368e24ca8604feafb1faf7d1f156001fae0877dc693ff4ce92.jpg) + +
+line + +| Model | Effective Stress (σ̄) | Effective Strain Rate (ε̅) | +|-------|----------------------|---------------------------| +| Bingham plastic | γ(σ̄ - σy) | Not labeled | +| Eyring pseudoplastic | B sinh(σ̄/A) | Not labeled | +| power law pseudoplastic | γ/ε̅M⁻¹ σ̄ | Not labeled | +| ideal plasticity | γσ̅ | Not labeled | +| Newtonian ε̅ | γσ̅ | Not labeled | +
+ +Fig. 12.6 Various flow curves for non-Newtonian fluids. + +The Bingham fluid is seen to be a particular form of viscoplastic relation (12.3) or (12.5). Writing in terms of the effective stress and strain rate, (12.5) can be expressed as + +$$ +\bar {\sigma} = \mu \dot {\bar {\epsilon}}, \tag {12.9} +$$ + +where the apparent viscosity $\mu$ is given by + +$$ +\frac {1}{\mu} = \frac {\sqrt {(3)} \gamma}{2 (J _ {2} ^ {\prime}) ^ {1 / 2}} \langle \Phi [ (\sqrt {3}) (J _ {2} ^ {\prime}) ^ {1 / 2} - \sigma_ {Y} ] \rangle . \tag {12.10} +$$ + +For the Bingham plastic we can write from the expression given in Fig. 12.6 and using (12.9) that + +$$ +\mu = \frac {\overline {{{\dot {\epsilon}}}} / \gamma + \sigma_ {Y}}{\overline {{{\dot {\epsilon}}}}}. \tag {12.11} +$$ + +As $\gamma \to \infty$ , ideal plasticity behaviour is approached resulting in + +$$ +\mu = \frac {\sigma_ {Y}}{\dot {\bar {\epsilon}}}. \tag {12.12} +$$ + +Similarly for a Power Law pseudoplastic we have from Fig. 12.6 + +$$ +\mu = \frac {\overline {{{\dot {\epsilon}}}} ^ {M - 1}}{\gamma}. \tag {12.13} +$$ + +Thus for each case the problem again reduces to an elastic problem in which the shear modulus is dependent on the current strain rate and can be solved + + + +by use of the analogy indicated in Table 12.1. Solution can be achieved by use of the method of direct iteration or by the Newton–Raphson process described in Chapters 2 and 3. + +As an example of viscous flow analysis $^{(17)}$ the problem of the flow of a Bingham fluid in a cylindrical annulus is illustrated in Fig. 12.7, where the geometry and finite element mesh employed are also indicated. Steady state flow is induced parallel to the axis of the cylinder by the application of an axial pressure gradient. The finite element velocity distributions obtained by a direct iteration solution scheme are shown in Fig. 12.8 for different values of the pressure gradient. The flow velocities are in good agreement with the theoretical solution of Ref. 18. + +![](images/page-494_02588c174fc3c93ce1bb152838e5fddb743d8e420eba2c4ef3e23e3e8f8d0506.jpg) + +
+text_image + +s o +d +p +b +u +g +R₂ +a +R₁ +π/8 +section shown below +∂v/∂n = 0 +finite element +mesh +v = 0 +v = 0 +10 parabolic elements with 2×2 gauss +integration +
+ +Fig. 12.7 Flow of Bingham fluid in an annulus under an axial pressure gradient showing finite element mesh idealisation. + +# 12.4.2 Nonlinear fracture mechanics + +A class of elasto-plastic problems which require special attention is that of crack propagation in ductile materials. Figure 12.9 illustrates the types of problem which demand solution and it is seen that a geometrical singularity exists at the crack tip. The numerical techniques presented in Chapter 7 allows the elasto-plastic stress field to be determined in the vicinity of the crack tip (for Modes I and II at least) but a criterion for propagation of the crack must be established in some way. + + + +![](images/page-495_d5e42f18cba26f5898bed6d0738e30c27101bdf377acd768e85075baf55ffcec.jpg) + +
+line + +| r/R₁ | v/v_AVG (f.e.m. nodal) | +|------|------------------------| +| 1.0 | 0.0 | +| 1.1 | 0.3 | +| 1.2 | 0.6 | +| 1.3 | 0.9 | +| 1.4 | 1.1 | +| 1.5 | 1.2 | +| 1.6 | 1.2 | +| 1.7 | 1.1 | +| 1.8 | 0.9 | +| 1.9 | 0.5 | +| 2.0 | 0.0 | +
+ +Fig. 12.8 Steady state velocity profile for the problem of Fig. 12.7 for various applied pressure gradients. + +For linear elastic fracture problems crack advance can be monitored by specifying a critical value of a quantity, K, termed the stress intensity factor\* which characterises the stress field in the vicinity of the crack tip according to $^{(20)}$ + +$$ +\sigma = K f (\theta) / \sqrt {(2 \pi r)} + \text { terms of order } r ^ {0}. \tag {12.14} +$$ + +A separate K parameter exists for each fracture mode, designated by $K_{I}$ , $K_{II}$ and $K_{III}$ respectively and they are functions only of geometry and loading conditions. A crack in any mode is then assumed to propagate when K attains a critical value $K_{c}$ which is treated as a material parameter. + +We now seek a similar criterion for elasto-plastic material behaviour. The most widely accepted principle in present use is the so-called J contour integral attributed to Rice $^{(21)}$ and which was originally formulated for nonlinear elastic applications. The J integral is defined to be + +$$ +J = \int_ {\Gamma} \omega d y - T _ {i} \frac {d u _ {i}}{d x} d S, \tag {12.15} +$$ + +for a crack aligned in the x direction. Here $\Gamma$ is any contour from the lower crack face leading anticlockwise around the crack tip to the upper face, S is the path length around this contour and $T_{i}du_{i}$ is the work contribution + + + +![](images/page-496_414724e906b3dcdb8e1ee21e29a13116e3165b0e5eaf176a77140d5a630a1975.jpg) + +
+text_image + +y +r +θ +x +Γ +mode I +mode II +mode III +
+ +Fig. 12.9 Basic modes of fracture. + +of traction components $T_{i}$ on $\Gamma$ moving through displacements $du_{i}$ . The term $\omega$ is the strain energy density defined as + +$$ +\omega = \int_ {0} ^ {\epsilon} \sigma_ {i j} d \epsilon_ {i j}. \tag {12.16} +$$ + +The J integral is independent of the choice of path $\Gamma$ provided that the faces of the crack are stress free. + +For Mode I opening in a strain-hardening nonlinear elastic material the near tip solution for the stress, strain and displacement can be shown to be of the form $^{(22-24)}$ + +$$ +\sigma = C \frac {1}{r ^ {1 / (N + 1)}} \sigma (\theta) +$$ + +$$ +\epsilon_ {p} = C \frac {1}{r ^ {N / (N + 1)}} \epsilon (\theta) +$$ + +$$ +u = C r ^ {1 / (N + 1)} u (\theta), \tag {12.17} +$$ + +where + +$$ +C = \left(\frac {J E}{\sigma_ {Y} {} ^ {2} I}\right) ^ {1 / (N + 1)}. \tag {12.18} +$$ + + + +The term N is a constant which measures the strain hardening of the material, E the elastic modulus, $\sigma_{Y}$ the stress denoting the limit of linearity and I is a tabulated constant whose value depends on N. + +For loading situations, nonlinear elastic behaviour is identical to that of a material obeying the laws of 'deformation' plasticity $^{(25)}$ in which the current stiffness is a function only of the current state of deformation and not of the loading path by which this condition has been reached. Furthermore for monotonic loading, experience indicates that there is no significant difference between solutions obtained by use of 'deformation' theories and the incremental theory adopted in Chapter 7. By this argument it is concluded that expressions (12.17) and (12.18) are applicable to elasto-plastic solids. Consequently crack propagation in elasto-plastic materials is governed by a critical value of the J integral. + +One of the difficulties of numerical fracture studies is that a reasonably accurate prediction of the stress field in the vicinity of the crack tip is required. This is a computationally expensive process for elasto-plastic problems and in some instances economies can be made by use of special crack tip elements. For example, in Mode II deformation under plastic conditions, a shear strain singularity of order 1/r develops, which has been modelled by Levy et al. $^{(26)}$ by coalescing two nodes of a linear quadrilateral isoparametric element and treating their displacements independently. This approach has also been employed by Rice et al. $^{(27)}$ + +# 12.4.3 Coupled-field problems + +The transient analysis of many engineering systems involves the formulation of the semi-discrete coupled-field equations of motion which are then solved by a time-stepping procedure. $^{(28)}$ Coupled-field equations involving plasticity arise in the modelling of structure–fluid interaction, soil–fluid interaction, structure–structure interaction, etc. There are two main sources of difficulty in solving such problems: + +(i) The isolated fields may display quite different response characteristics which may only be analysed efficiently by different time integration algorithms and/or different time steps. + +(ii) Most engineering software has been developed for the treatment of single-field problems. The term ‘partitioned transient analysis procedures’ has been used to describe methods which allow the direct time integration of the entire equations to be performed by either sequential or parallel execution of single-field analyzers. + +We have discussed partitioned procedures for structural dynamic problems in Chapter 11. We described an implicit-explicit partition through which meshes that exhibit high (low) frequency response characteristics are treated by implicit (explicit) integration formulae. Park $^{(29)}$ has recently extended the approach described in Chapter 11. + + + +Park et al. $^{(30)}$ have studied implicit–implicit partitions in certain types of fluid-structure interaction problems. The solution of these coupled-field equations was obtained by a sequential execution of fluid and structural analyzers which gave rise to the term ‘staggered solution procedures.’ + +Hughes $^{(31)}$ has summarised recent work on transient fluid-structure interaction problems. In particular he mentions work on procedures known as mixed, or arbitrary, Lagrangian–Eulerian methods. + +In recent work on soil liquefaction problems, Zienkiewicz et al. $^{(32)}$ have devised a model which couples the soil and pore-fluid behaviour during earthquakes. Pore pressure build up and pore water migration are both accurately modelled. + +Many other coupled-field problems involving elasto-plastic behaviour have been reported in the literature. It should however be emphasised that care should be taken in considering the stability of such schemes. + +# 12.4.4 Elasto-plastic and geometrically nonlinear analyses of plates and shells + +The linear and nonlinear finite element analysis of plates and shells has attracted much attention in the last decade. Two basic approaches have been adopted: + +# (i) The classical procedure + +Here a plate or shell theory is used as a basis for the finite element formulation. Let us briefly summarise such an approach. We begin with the field equations of the three-dimensional theory and make various assumptions which lead to the plate or shell theory. In the reduction from three to two dimensions we include an analytical integration over the thickness. We then base our finite element discretisation process on the plate or shell theory. The surface geometry (in the case of shells) and the field variables are approximated using discrete nodal values and suitable interpolation functions. Integration of the various element stiffness and force terms is carried out over the reference surface. Stresses may then be obtained from the stress resultants. Examples of such an approach include the simple facet element and the many elements derived from classical thin plate theory, Mindlin/Reissner plate theory, shallow shell theory or even higher order shell theories. $^{(33,34)}$ There are very many examples of the application of the classical procedures in nonlinear finite element analysis of plates and shells. We include a brief sample in the list of references to this chapter. $^{(35-38)}$ For elasto-plastic problems many research workers express the yield function in terms of the stress resultants (cf. the non-layered approach in Chapter 9). For example, Crisfield $^{(39-44)}$ uses a + + + +modified Ilyushin yield criterion expressed in terms of the bending moments $[M_{x}, M_{y}, M_{xy}]^{T}$ and the membrane forces $[N_{x}, N_{y}, N_{xy}]^{T}$ . To allow for the gradual spread of plasticity over the plate or shell thickness, a modified classical procedure may be adopted in which integration through the thickness is performed numerically during the finite element stiffness and force evaluation rather than analytically prior to the finite element discretisation. Gauss–Legendre, Lobatto and the mid-ordinate rules are frequently used for this purpose. To allow for geometrically nonlinear effects, total or updated Lagrangian approaches are adopted. $^{(45-55)}$ + +# (ii) Ahmad and related elements + +Here isoparametric elements with independent rotational and displacement degrees of freedom are used. This concept originally introduced by Ahmad et al. $^{(56)}$ was later extended to allow for the linear analysis of thin as well as moderately thick shells by Zienkiewicz et al. $^{(57)}$ by the use of the reduced integration technique.\* + +Ahmad elements were originally developed because of the computational difficulties encountered in the use of the usual three-dimensional elements for the analysis of plates and shells. In the three-dimensional elements the stiffness coefficients corresponding to the transverse displacement degrees of freedom are very much larger than those corresponding to the longitudinal displacements. Erroneous strain energy corresponding to the normal stresses in the thickness direction are also introduced. Both of these difficulties are overcome in Ahmad elements. Normals to the plate or shell reference surface before deformation are assumed to remain straight but not necessarily normal to the reference surface after deformation. Furthermore, the normal stresses in the direction of the shell thickness are ignored and suitably modified constitutive equations are adopted. + +Various nonlinear problems have been solved using Ahmad shell elements by Ramm $^{(67)}$ , Krakeland $^{(68)}$ , Bathe and Bolourchi $^{(69)}$ and others $^{(70-73)}$ . As in the modified classical procedures, to allow for the gradual spread of plasticity over the plate or shell thickness, numerical integration techniques are adopted. For geometrically nonlinear behaviour both total and updated + +\- The Mindlin plate elements described in Chapters 6 and 9 are simply plate versions of the Ahmad elements in which integration has been carried out analytically through the plate thickness. Much work on reduced and selective integration techniques $^{(58-65)}$ eventually led to the recognition that the use of selective integration techniques is equivalent to the use of a special type of mixed formulation. $^{(66)}$ Defects in the Ahmad elements have now been widely acknowledged and the use of the 9-node heterosis Mindlin plate element and the 16-node cubic Ahmad element are usually recommended. Other Ahmad/Mindlin $C(0)$ elements should be used with caution as they are known to give overstiff solutions for thin plates and shells and to develop mechanisms (zero energy modes) or near mechanisms (artificially low energy modes) when reduced or selective integration is used. + + + +Lagrangian schemes have been used. Special techniques have been incorporated to allow for large rotations in the total Lagrangian formulations. $^{(67-69)}$ + +The Ahmad shell concept has been developed further by its originator Irons with the introduction of the Semiloof element. $^{(90)}$ Irons adopted a convenient nodal configuration involving rotational degrees of freedom at 'Loof' nodes on the curved boundaries of the element. By imposing a series of constraints to eliminate transverse shear effects (reminiscent of the discrete Kirchhoff hypothesis), a highly effective thin shell element is obtained. Various research workers $^{(74-76)}$ have successfully extended this work into the nonlinear range. + +Both classical and Ahmad procedures may be used as a basis for the nonlinear analysis of reinforced concrete plates and shells using the layering concept described in Chapter 9. Special constitutive relationships are required to represent the concrete and steel reinforcing bars are treated as a 'smeared' layer with uni-directional elasto-plastic properties. Much work has been completed in this area. $^{(77-85)}$ + +Elasto-viscoplastic plates and shells are easily developed using the concepts described in Chapters 8 and 9. $^{(86,87)}$ + +# 12.5 Equation solving techniques + +# 12.5.1 Standard and modified Newton method + +Before considering some alternative nonlinear solution procedures which may be used in elastoplastic finite element analysis we review the techniques described earlier. + +As we have already seen, most elasto-plastic finite element programs are simply extensions of elastic finite element programs with linearised load increments. Some form of iterative procedure is usually adopted to dissipate the out-of-balance nodal forces. + +The standard and variety of modified Newton methods were described earlier in Part I. Recall that the standard Newton method involves iterations in which + +$$ +\boldsymbol {K} ^ {(i)} \left[ \boldsymbol {d} ^ {(i + 1)} - \boldsymbol {d} ^ {(i)} \right] = \psi \left(\boldsymbol {d} ^ {(i)}\right), \tag {12.19} +$$ + +where d is the vector of nodal displacements and the equations $\psi(d)=0$ express a force balance (internal forces = external forces; either for an increment of loading or for the whole applied load). The matrix K in the standard Newton method is the Jacobian of $\psi$ ; which is the tangential stiffness matrix $K_{T}=[\partial\psi(d^{(t)}/\partial d]$ evaluated at the displacements described by $d^{(t)}$ . + +The modified Newton method works with a variety of approximations to K, the most simple of which is the initial elastic stiffness matrix $K_{0}$ evaluated at the first iteration of the first load increment. diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_051.md b/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_051.md new file mode 100644 index 00000000..2af7ad79 --- /dev/null +++ b/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_051.md @@ -0,0 +1,284 @@ + + +We have adopted standard and modified Newton methods throughout this text as they are the most widely used approaches. Though they work well they do have certain disadvantages. The initial stiffness method is slow to converge in cases in which there is a high degree of nonlinearity. The modified Newton methods provide better convergence properties but they diverge during elastic unloading and they can lead to ill-conditioned or singular Jacobian matrices K near the limit load. + +Newton methods are sometimes employed with a slight modification during an iteration in which + +$$ +\boldsymbol {K} ^ {(i)} \Delta \boldsymbol {d} ^ {(i)} = \psi^ {(i)}, \tag {12.20} +$$ + +and in which the new displacement vector is given as + +$$ +\boldsymbol {d} ^ {(i + 1)} = \boldsymbol {d} ^ {(i)} + a ^ {(i)} \Delta \boldsymbol {d} ^ {(i)}, \tag {12.21} +$$ + +where we could take $a^{(i)}$ as much less than 1 for safety or more than 1 for more rapid convergence. Nayak $^{(88)}$ introduced an acceleration technique in which $a^{(i)}$ is replaced by a diagonal matrix. Basu $^{(89)}$ later simplified this technique. + +Although the modified Newton methods with fixed values of $\alpha^{(t)}$ is employed by certain analysts, it has been suggested $^{(90)}$ that we should reject it in favour of a modified Newton with a line search which involves finding a value of $\alpha^{(t)}$ which minimises the total potential energy $\pi(\boldsymbol{d}^{(t+1)})$ or the value of + +$$ +Q = \left| \left[ \boldsymbol {d} ^ {(i)} \right] ^ {T} \psi \left(\boldsymbol {d} ^ {(i + 1)}\right) \right|. \tag {12.22} +$$ + +# 12.5.2 Quasi-Newton method + +Over the past twenty years there has been a rapid development of computer-oriented, sequential search methods in the fields of optimisation and mathematical programming. Of these techniques, the variable metric (Quasi-Newton) method and the method of conjugate gradients show the greatest potential in nonlinear finite element analysis. + +The Quasi–Newton method was introduced to finite element computations by Matthies and Strang. $^{(91)}$ The main idea is to update the matrix K in a simple way after each iteration, rather than to recompute it entirely as in the standard Newton method or leave it unchanged as in the modified Newton method. Here we consider the update, known as the Broyden–Fletcher–Goldfarb–Shanno (BFGS). It is most conveniently written in terms of $K^{(t+1)}$ rather than $K^{(t)}$ and has the form + +$$ +[ K ^ {(i)} ] ^ {- 1} = [ I + w ^ {(i)} \{v ^ {(i)} \} ^ {T} ] [ K ^ {(i - 1)} ] ^ {- 1} [ I + v ^ {(i)} \{w ^ {(i)} \} ^ {T} ]. \tag {12.23} +$$ + +The indicated matrix multiplications are never carried out in the computer implementation; instead $v^{(i)}$ and $w^{(i)}$ are stored and used only in computing the new search direction + +$$ +\Delta \boldsymbol {d} ^ {(i)} = [ \boldsymbol {K} ^ {(i)} ] ^ {- 1} \psi (\boldsymbol {d} ^ {(i)}). \tag {12.24} +$$ + + + +A line search of the form given in (12.21) is adopted. The BFGS formulae for $v^{(i)}$ and $w^{(i)}$ are + +$$ +\boldsymbol {v} ^ {(t)} = \psi (\boldsymbol {d} ^ {(t)}) \left(1 + \alpha^ {(t - 1)} \left[ \frac {\{\Delta \boldsymbol {d} ^ {(t - 1)} \} ^ {T} \gamma^ {(t)}}{\{\delta^ {(t)} \} ^ {T} \{\psi (\boldsymbol {d} ^ {(t - 1)}) \}} \right] ^ {1 / 2}\right) - \psi (\boldsymbol {d} ^ {(t)}), \tag {12.25} +$$ + +and + +$$ +w ^ {(i)} = \frac {\delta^ {(i)}}{\{\delta^ {(i)} \} ^ {T} \gamma^ {(i)}}, \tag {12.26} +$$ + +where + +$$ +\delta^ {(i)} = d ^ {(i)} - d ^ {(i - 1)} = a ^ {(i - 1)} \Delta d ^ {(i - 1)}, +$$ + +and + +$$ +\gamma^ {(i)} = \psi (d ^ {(i)}) - \psi (d ^ {(i - 1)}). +$$ + +The method has been successfully implemented and used by Matthies and Strang $^{(91)}$ and Geradin and Hogge $^{(92)}$ for both static and transient dynamic nonlinear problems. The stability of BFGS with respect to unloading has been emphasised by Matthies and Strang. $^{(91)}$ A related method by Crisfield $^{(93)}$ also shows much promise. + +Rather than work with the inverse of $K^{(t)}$ as given in (12.23), Geradin and Hogge $^{(92)}$ work with the update formula + +$$ +\boldsymbol {K} ^ {(t)} = \boldsymbol {K} ^ {(t - 1)} + \frac {\gamma^ {(i)} \left\{\gamma^ {(i)} \right\} ^ {T}}{\left\{\gamma^ {(i)} \right\} ^ {T} \delta^ {(i)}} - \frac {\left\{\boldsymbol {K} ^ {(t - 1)} \delta^ {(i)} \right\} \left\{\boldsymbol {K} ^ {(t - 1)} \delta^ {(i)} \right\} ^ {T}}{\left\{\delta^ {(i)} \right\} ^ {T} \boldsymbol {K} ^ {(t - 1)} \delta^ {(i)}}, \tag {12.27} +$$ + +and use a frontal solution scheme. + +# 12.5.3 Conjugate gradient methods + +In the conjugate gradient $^{(94)}$ algorithm we take + +$$ +\boldsymbol {d} ^ {(i + 1)} = \boldsymbol {d} ^ {(i)} + \alpha^ {(i)} \boldsymbol {\delta} ^ {(i)}, \tag {12.28} +$$ + +where + +$$ +\delta^ {(i)} = \psi (d ^ {(i)}) + \beta^ {(i)} \delta^ {(i - 1)}, \tag {12.29} +$$ + +in which $a^{(i)}$ is chosen using a line search with the criterion that the total potential energy $\pi(d^{(i+1)})$ should be minimised. + +Initially, $\beta^{(0)}$ is set to zero. We list two possible values for $\beta^{(t)}$ : + +(i) The Hestenes-Stiefel $^{(94)}$ (Fletcher-Reeves $^{(95)}$ ) algorithm + +$$ +\beta^ {(i)} = \frac {\{\psi^ {(i)} \} ^ {T} \psi^ {(i)}}{\{\psi^ {(i - 1)} \} ^ {T} \psi^ {(i - 1)}}. \tag {12.30} +$$ + +(ii) The Polak-Ribiere $^{(96)}$ algorithm + +$$ +\beta^ {(i)} = \frac {\{\psi^ {(i)} \} ^ {T} \gamma^ {(i)}}{\{\psi^ {(i - 1)} \} ^ {T} \psi^ {(i - 1)}}. \tag {12.31} +$$ + + + +The method, which requires modest computer core requirements, has been improved by scaling and other techniques. $^{(97-99)}$ The Conjugate–Newton method of Irons $^{(100)}$ is also a development of the basic conjugate gradient algorithm. + +# 12.5.4 Other useful solution techniques + +Among the remaining solution procedures, dynamic relaxation (DR) methods are quite popular. The main idea in DR originated from the observation that with about 90% of critical damping, an equivalent transient dynamic analysis rapidly converges to the steady state, static solution. Recent modifications $^{(101-103)}$ of the method have concentrated on finding improved replacements for the mass matrix M and the damping matrix C which are used in DR. Although DR methods are generally not as powerful as the various Newton and conjugate gradient methods, they require very little computer core storage and explicit transient dynamic programs such as DYNPAK, described in Chapter 10, can be rapidly modified to be used as DR solvers for ad hoc static problems when no other static program is available and results are urgently required. + +It is usually difficult to decide on the form of load incrementation to adopt for elasto-plastic problems and exploratory analyses are often required. The work of Bergan and Soreide $^{(104)}$ in this area appears to be quite promising. + +Schemes which work with local and global modes, several meshes or hierarchical representations $^{(105-111)}$ for the displacements may also prove to be of prime importance in nonlinear finite element equation solving. + +# 12.6 Other enhancements in elasto-plastic analysis + +# 12.6.1 Substructuring and boundary element methods + +Economies can be made in the numerical solution of elasto-plastic problems by the use of substructuring techniques. A substructure analysis generally comprises the following steps. $^{(112)}$ + +- Separate groups of elements within the solid are collectively identified as substructures as indicated in Fig. 12.10. +- For each substructure, the element stiffness matrices are assembled to give the global stiffness matrix of the substructure. +- The equations relating to the internal nodal points (i.e. nodes not on the boundary) are eliminated. This process is known as condensation. +- Solution of the system of resulting simultaneous equations is obtained by assembling all the individual substructures and any remaining elements which have not been associated with a substructure. This gives the nodal displacements and reactions for all nodal points on interfaces between substructures and for nodes of elements which are not related to any substructure. + + + +\- Return to the individual substructures to evaluate the displacements at interior nodes and finally obtain the element stresses. + +![](images/page-504_2a3cb4fee0e20c242b8daffa75da0fd0c1fc5026ebebe8fd30e761b5a851f092.jpg) + +
+text_image + +substructure 1 +II +III +
+ +Fig. 12.10 Substructure analysis of elasto-plastic problems. + +The very nature of the frontal equation solution process described in Section 6.4.12 makes the use of substructure techniques a simple affair, since, when the front has advanced into a structure to a certain position, the reduced frontal equations are essentially the condensed equations for a substructure corresponding to the part of the structure already considered. + +For elasto-plastic problems, the part of the structure which (by physical considerations or experience!) is known to remain elastic during the deformation process can be defined as one substructure and the remaining elements considered individually. Thus during incremental/iterative solution the substructure stiffness will remain unaltered, for solution by the tangential stiffness method, and the substructure assembly and condensation process described above need be performed only once with an equation resolution process, necessitating only reduction of the R.H.S. terms being followed thereafter. The individual elements not associated with the substructure (and which model the elasto-plastic behaviour) are treated in the normal way as described in Chapter 7. + +This approach can result in considerable computational economies, particularly if the mesh subdivision within the substructure is a fine one. It can be argued that a fine mesh subdivision is not warranted for regions where elastic behaviour is anticipated, but for structures which are to be subjected to more than one type of loading such an optimal mesh grading may not be possible. For example, with reference to Fig. 12.10, two separate loadings may cause plastic yielding in substructures II and III respectively and consequently a fine mesh grading within each of these regions cannot be avoided. + +An extension of the above process is afforded by the use of the boundary integral method. $^{(113-115)}$ The boundary integral procedure requires trial functions which satisfy the governing equations directly and then attempt to satisfy the boundary conditions by a collocation, least-squares or Galerkin + + + +procedure. In order to find trial functions which satisfy the governing equations we are, at present, generally confined to linear elastic situations. Thus for the solution of elasto-plastic problems a coupled approach can be employed $^{(113,115)}$ with the elastic region of the structure being modelled by boundary elements and conventional finite elements employed to treat the elasto-plastic zones. Such direct coupling leads to nonsymmetric matrices which is acceptable if the equation set is dominated by the boundary integral equations. + +This approach promises efficient numerical solutions particularly for cases of limited yielding in three-dimensional solids where the surface area/volume ratio is relatively small. The process can also be used to advantage in infinite domain structures such as rock mass problems or soil/structure interaction problems with boundary elements being employed to model the exterior domain. + +# 12.6.2 Interactive computing + +The solution of elasto-plastic problems inevitably requires some degree of insight into the structural behaviour before choice of solution parameters, such as load increment sizes, can be made. Even then it is difficult, if not impossible, to specify the most suitable values of load increments, tolerance factors for each load case and also choice of the optimal solution process (e.g. initial stiffness, tangential stiffness or some combined algorithm) is equally difficult to arrive at. + +To this end, the developments which are currently taking place in interactive computing will become increasingly important. Here we envisage the situation where the results for a particular load increment are held in core while the solution is scrutinized. Depending on the convergence characteristics, etc., the load increment size and convergence tolerance factor are then input and solution continued for a further increment. If required the nonlinear solution process can be redefined at this stage changing, for example, from a tangential stiffness to an initial stiffness algorithm if collapse conditions are being approached. Furthermore if the numerical process did not converge in the previous increment, the calculations could be repeated for a smaller load increment size or a different solution algorithm. + +# 12.6.3 Computational techniques + +Many new and improved programming strategies are developing in connection with finite element software and the interested reader is directed to the work of Schrem $^{(116,117)}$ and others $^{(118)}$ who are active in this area. + +# 12.7 Concluding remarks + +Throughout this text we have described numerical techniques and computer codes for a variety of engineering applications. Treatment has been limited to situations where the finite element method can be used to provide + + + +nonlinear solutions with a measure of confidence. In this final chapter we have attempted to indicate some areas of further study and here the applicability to design problems is not so clear. For example, for soils and concrete some divergence of opinion still exists as to selection of an appropriate material model. Indeed at the present time it is true to say that numerical solution capabilities are in advance of the knowledge of fundamental material behaviour. This is particularly true for dynamic problems where there is a scarcity of information on material response under transient conditions. In this respect it would appear that nonlinear finite element methods offer the possibility of conducting ‘numerical experiments’ to provide insight on material behaviour which could not be obtained by experiment alone. + +# 12.8 References + +1. NAYLOR, D. J., Stress-strain laws for soil, In: Developments in Soil Mechanics, Ed. C. R. Scott, Applied Science Publishers (1978). +2. SCHOFIELD, A. N. and WROTH, C. P., Critical State Soil Mechanics, McGraw-Hill, New York (1968). +3. ATKINSON, J. H. and BRANSBY, P., Mechanics of Soil, McGraw-Hill, New York (1979). +4. NILSSON, L., Impact loading on concrete structures, Publication 79: 1, Dept. of Structural Mechanics, Chalmers University of Technology, Sweden (1979). +5. ZIENKIEWICZ, O. C., OWEN, D. R. J., PHILLIPS, D. V. and NAYAK, G. C., Finite element methods in the analysis of reactor vessels, Nuclear Engng. and Design, 20, 507–542 (1972). +6. MOHRAZ, B., SCHNOBRICH, W. C. and GOMEZ, A. E., Crack development in a prestressed concrete reactor vessel as determined by a lumped parameter method, Nuclear Engng. and Design, 11, 286 (1970). +7. VALANIS, K. C., A theory of viscoplasticity without a yield surface, Archiwum Mechaniki Stossowanej, 23, 4, 517–533 (Warsaw, 1971). +8. VALANIS, K. C., On the foundations of the endochronic theory of viscoplasticity, Archiwum Mechaniki Stossowanej, 27, 5–6, 857–869 (Warsaw, 1975). +9. BAZANT, Z. P., A new approach to inelasticity and failure of concrete, sand and rock: Endochronic theory, Proc. Soc. of Engng. Science, Eleventh Annual Meeting, Ed. G. J. Dvorak, Duke University, Durham, North Carolina (1974). +10. BAZANT, Z. P., BHAT, P. D. and SHIEH, C. L., Endochronic theory for inelasticity and failure analysis of concrete structures, Struct. Engng. Report No. 1976-12/259, Department of Civil Engineering, Northwestern University (1976). +11. ZIENKIEWICZ, O. C., The Finite Element Method, McGraw-Hill, London (1977). +12. ZIENKIEWICZ, O. C., Viscoplasticity, plasticity, creep and viscoplastic flow, Proc. Int. Conf. on Computational Methods in Nonlinear Mechanics, University of Texas, Austin (1974). +13. PRICE, J. W. H. and ALEXANDER, J. M., The finite element analysis of two high temperature metal deformation processes, Proc. Second Int. Symp. on Finite Element Methods in Flow Problems, Santa Margharita Ligure, Rappelo, Italy (June 1976). +14. HINTON, E., SALONEN, E. M. and BICANIC, N., A study of locking phenomena in isoparametric elements, In: The Mathematics of Finite Elements and Applications III, 437–447, Ed. J. R. Whiteman, Academic Press, London (1979). + + + +15. SKELLAND, A. N. P., Non-Newtonian Flow and Heat Transfer, John Wiley, New York (1967). +16. BIRD, R. B., STEWART, W. E. and LIGHTFOOT, E. N., Transport Phenomena, John Wiley, New York (1960). +17. LYNESS, J. F., OWEN, D. R. J. and ZIENKIEWICZ, O. C., The finite element analysis of engineering systems governed by a nonlinear quasi-harmonic equation, Computers and Structures, 5, 65–79 (1975). +18. LAIRD, W. M., Slurry and suspension transport, Ind. Engng. Chem. 49, 1, 138–141 (1957). +19. Journal of Strain Analysis (Special Issue), Vol. 10, No. 4 (October, 1975). +20. KNOTT, J. F., Fundamentals of Fracture Mechanics, Butterworths, London (1973). +21. RICE, J. R., A path independent integral and the approximate analysis of strain concentration by notches and cracks, J. Appl. Mechanics, 379–386 (1968). +22. HUTCHINSON, J. W., Singular behaviour at the end of a tensile crack in a hardening material, J. Mech. Phys. Solids, 16, 13–31 (1968). +23. RICE, J. R. and ROSENGRUN, G. F., Plane strain deformation near a crack tip in a power-law hardening material, J. Mech. Phys. Solids, 16, 1–12 (1968). +24. McCLINTOCK, F., Fracture, An Advanced Treatise, Chapter 2, Ed. H. Liebowitz, Academic Press, New York (1971). +25. HILL, R., The Mathematical Theory of Plasticity, Oxford University Press (1950). +26. LEVY, N., MARCAL, P. V., OSTERGREN, W. J. and RICE, J. R., Small scale yielding near a crack in plane strain: a finite element analysis, Int. J. Fract. Mech. 9, 98–100 (1973). +27. RICE, J. R. and TRACEY, D. M., Computational fracture mechanics, In: Numerical and Computer Methods in Structural Mechanics, Eds. S. J. Fenves et al., Academic Press (1973). +28. ZIENKIEWICZ, O. C., Finite elements in the time domain, Contribution for 'State-of-Art' Survey by the Committee in Computing in Applied Mechanics of ASME (to be published). +29. PARK, K. C., Partitional transient analysis procedures for coupled field problems, Report No. LMSC-D633955, Lockheed Palo Alto Laboratory (1979). +30. PARK, K. C., FELLIPA, C. A. and DERUNTZ, J. A., Stabilization of staggered solution procedures for fluid-structure interaction analysis, In: Computer Methods for Fluid-Structure Interaction Problems—AMD, Vol. 26, Eds. T. Belytschko and T. L. Geers (1977). +31. HUGHES, T. J. R., Recent developments in computer methods for structural analysis, Division M Principal Lecture, Fifth SMIRT Conference, Berlin (to appear in Nuclear Engng. and Design). +32. ZIENKIEWICZ, O. C., CHANG, C. T., HINTON, E. and LEUNG, K. H., Effective stress dynamic modelling for soil structures including drainage and liquefaction, In: Proc. of Int. Symp. on Soils Under Cyclic and Transient Loading, Eds. G. N. Pande and O. C. Zienkiewicz, 551–554, A. A. Balkema, Rotterdam (1980). +33. FREDRIKSSON, B. and MACKERLE, J., Finite Element Review, Publication No. AEC-L-003, AEC, Box 3944, S-58903, Linköping, Sweden. +34. ASHWELL, D. G. and GALLAGHER, R. H. (Eds.), Finite Elements for Thin Shells and Curved Members, Wiley, London (1976). +35. ARMEN, H., PIFKO, A., and LEVINE, H. S., A finite element method for the plastic bending analysis of structures, Proc. Second Conf. on Matrix Methods + + + +in Structural Mechanics, DOC.AFFDL-TR-69-150, Wright-Patterson Air Force Base, Ohio, 1969. +36. BUSHNELL, D., Buckling of elastic-plastic complex shells of revolution including large deflections and creep, Computers and Structures, 6, 221-239 (1976). +37. WEGMULLER, A. W. and KOSTEM, C. N., Finite element analysis of elastoplastic plates and eccentrically stiffened plates, Report No. 378A.4, Fritz Engg. Laboratory, Lehigh University (1973). +38. BERGAN, P. G., Application of finite element method to nonlinear problems, Finite Element Congress, Baden-Baden, FRG, 18–19 Nov. (1974). +39. CRISFIELD, M. A., Large deflection elasto-plastic buckling analysis of plates using finite elements, TRRL Report LR 593, Crowthorne (1973). +40. CRISFIELD, M. A., Some approximations in the nonlinear analysis of rectangular plates using finite elements, TRRL Suppl. Report 51 UC, Crowthorne (1974). +41. CRISFIELD, M. A., On an approximate yield criterion for thin steel shells, TRRL Report 658, Crowthorne (1974). +42. CRISFIELD, M. A., Large deflection elasto-plastic buckling analysis of eccentrically stiffened plates using finite elements, TRRL Report 725, Crowthorne (1976). +43. CRISFIELD, M. A. and PUTALI, R. S., Approximations in the nonlinear analysis of thin plated structures, Int. Conf. Finite Elements in Nonlinear Solid and Structural Mechanics, Geilo, Norway (1977). +44. CRISFIELD, M. A., Ivanov's yield criterion for thin plates and shells using finite elements, TRRL Lab. Report 919, Crowthorne (1979). +45. BERGAN, P. G., Nonlinear analysis of plates considering geometric and material effects, Ph.D. Thesis, University of California, Berkeley, SESM Report No. 71-7 (1971). +46. HORRIGMOE, G. and BERGAN, P. G., Incremental variational principles and finite element models for nonlinear problems, Comp. Meths. Appl. Mech. and Engng. 7, 201-217 (1976). +47. HORRIGMOE, G., Large displacement analysis of shells by a hybrid stress finite element method, IASS World Congress on Space Enclosures (NCOSE-76, Montreal, 489–499 (1976). +48. HORRIGMOE, G., Finite element instability analysis of free-form shells, Report No. 77-2, The Norwegian Inst. Techn., University of Trondheim (1977). +49. Wood, R. D., The application of finite element methods to geometrically nonlinear structural analysis, Ph.D. Thesis, University College of Swansea (1973). +50. PICA, A., Geometrically nonlinear analysis of Mindlin plates by finite element method, M.Sc. Thesis, University College of Swansea (1978). +51. PICA, A., WOOD, R. D. and HINTON, E., Finite element analysis of geometrically nonlinear plate behaviour using a Mindlin formulation, Computers and Structures, 11, 203–215 (1980). +52. GALLAGHER, R. H., Geometrically nonlinear shell analysis, In: Advances in Computational Methods in Structural Mechanics and Design, Eds. J. T. Oden, R. W. Clough and Y. Yamamoto, Univ. Alabama Press, 641–678 (Proc. Second US–Japan Seminar) (1972). +53. FREY, F. and CESCOTTO, S., Some new aspects of the incremental total Lagrangian description in nonlinear analysis, Proc. Int. Conf. Finite Elements in Nonlinear Solid Mechanics, Geilo, Norway, Paper CO5 (1977). +54. BATOZ, J. L., CATTOPADHYAY, A. and DHATT, G., Finite element large deflection analysis of shallow shells, Int. J. Num. Meth. Engng. 10, 39-58 (1976). + + + +55. ARGYRIS, J. H. and DUNNE, P. C., Non-linear and post-buckling analysis of structures, In: Formulations and Computational Algorithms in Finite Element Analysis, Eds. K. J. Bathe, J. T. Oden and W. Wunderlich, 525–571, MIT Press, Massachusetts (1977). +56. AHMAD, S., IRONS, B. M. and ZIENKIEWICZ, O. C., Curved thick shell and membrane elements with particular reference to axisymmetric problems, Proc. Second Conf. Matrix Method Struct. Mech., Wright-Patterson Air Force Base, Ohio (1968). +57. ZIENKIEWICZ, O. C., TAYLOR, R. L. and TOO, J. M., Reduced integration technique in general analysis of plates and shells, Int. J. Num. Meth. Engng. 3, 275–290 (1971). +58. HINTON, E., RAZZAQUE, A., ZIENKIEWICZ, O. C. and DAVIES, J. D., A simple finite element solution for plates of homogeneous, sandwich and cellular construction, Proc. Inst. Civ. Engrs. Part 2, 59, 43–65 (1975). +59. ZIENKIEWICZ, O. C. and HINTON, E., Reduced integration, function smoothing and non-conformity in finite element analysis (with special reference to thick plates), J. of the Franklin Inst., 302, 443–461 (1976). +60. HINTON, E., OWEN, D. R. J. and SHANTARAM, D., Dynamic transient nonlinear behaviour of thick and thin plates, In: The Mathematics of Finite Elements and Applications II MAFELAP, Ed. J. R. Whiteman, 423-438, Academic Press, London (1977). +61. HINTON, E. and PUGH, E. D. L., Some quadrilateral isoparametric finite elements based on Mindlin plate theory, Proc. of Symp. on Applications of Computer Methods in Engineering, Univ. of S. California, Vol. II, 851-858 (1977). +62. BICANIC, N. and HINTON, E., Spurious modes in two-dimensional isoparametric elements, Int. J. Num. Meths. Engng. 14, 1545–1557 (1979). +63. HINTON, E. and BICANIC, N., A comparison of Lagrangian and Serendipity Mindlin plate elements for free vibration analysis, Computers and Structures, 10, 483–493 (1979). +64. HUGHES, T. J. R., COHEN, M. and HAROUN, M., Reduced and selective integration techniques in the finite element analysis of plates, Nuclear Engng. Design, 46, 203–222 (1978). +65. HUGHES, T. J. R. and COHEN, M., The 'Heterosis' finite element for plate bending, Computers and Structures, 9, 445–450 (1978). +66. MALKUS, D. and HUGHES, T. J. R., Mixed finite element methods—reduced selective integration techniques: a unification of concept, Comp. Appl. Mech. Engng. 15, 63–81 (1978). +67. RAMM, E., A plate/shell element for large deflection and rotations, In: Formulations and Computational Algorithms in Finite Element Analysis, Eds. K. J. Bathe, J. T. Oden and W. Wunderlich, 264–293, MIT Press, Massachusetts (1977). +68. KRAKELAND, B., Large displacement analysis of shells considering elastoplastic and elasto-viscoplastic materials, Report No. 776, The Norwegian Institute of Technology, The University of Trondheim, Norway (1977). +69. BATHE, K. J. and BOLOURCHI, S., A geometric and material nonlinear plate and shell element (to be published). +70. LARSEN, P. K. and POPOV, E. P., Large displacement analysis of viscoelastic shells of revolution, Comp. Meth. Appl. Mech. Engng. 3, 237–253 (1974). +71. NOOR, A. K. and HARLEY, S. J., Nonlinear shell analysis via mixed isoparametric elements, Computers and Structures, 7, 615–626 (1977). + + + +72. FREY, F., L'analyse statique non linéaire des structures par la méthode des éléments finis et son application à la construction metallique, Ph.D. Thesis, University of Liège, Belgium (1978). +73. BOLOURCHI, S., On finite element nonlinear analysis of general shell structures, Ph.D. Thesis, Mechanical Engineering Department, MIT (1979). +74. MARTINS, R. A. F. and OWEN, D. R. J., Elastoplastic and geometrically nonlinear thin shell analysis by the Semiloof element, Computers and Structures (to be published). +75. DINIS, L. M. S., MARTINS, R. A. F. and OWEN, D. R. J., Analysis of material and geometrically nonlinear thin plates and arbitrary shells, Proc. Int. Conf. Numerical Methods for Nonlinear Problems, Swansea, 425–442, Pineridge Press, Swansea, U.K. (1980). +76. OWEN, D. R. J. and DINIS, L. M. S., Transient dynamic analysis of thin shells including viscoplastic and large displacement effect, In: The Mathematics of Finite Elements and Applications III, Ed. J. R. Whiteman, Academic Press (1978). +77. WEGNER, R., Finite element models for reinforced concrete, In: Formulation and Computational Algorithms in Finite Element Analysis, Eds. K. J. Bathe, J. T. Oden and W. Wunderlich, 393–439, MIT Press, Massachusetts (1977). +78. JOFREIT, J. C. and MCNIECE, G. M., Finite element analysis of reinforced concrete slabs, J. Struct. Div. ASCE, 97, 785–806 (1971). +79. HAND, F. R., PENCOLD, D. A. and SCHONBRICH, W. C., Non-linear layered analysis of reinforced concrete plates and shells, Univ. Illinois, Struct. Research Report No. 389 (1972). +80. LIN, C. S., Nonlinear analysis of reinforced concrete slabs and shells, Report UC SESM 73-7, Univ. of California, Berkeley (1973). +81. COPE, R. J. and RAO, P. U., Non-linear finite element analysis of concrete slab structures, Proc. Inst. Civ. Engrs. Part 2, 63, 149–179 (1977). +82. SCHNOBRICH, W. C., Behaviour of reinforced concrete structures predicted by the finite element method, Computers and Structures, 7, 365–376 (1977). +83. BASHUR, F. K. and DARWIN, D., Non-linear model for reinforced concrete slabs, J. Struct. Divn., ASCE, 104, 157–170 (1978). +84. GILBERT, R. F. and WARNER, R. F., Tension stiffening in reinforced concrete slabs, J. Struct. Divn. ASCE, 104, 1885-1900 (1978). +85. DUNCAN, W. and JOHNARRY, T., Further studies on the constant stiffness method of non-linear analysis of concrete structures, Proc. Inst. Civ. Engrs., Part 2, 67, 951–969 (1979). +86. CORMEAU, I. C., Viscoplasticity and plasticity in the finite element method, Ph.D. Thesis, University College of Swansea (1976). +87. DINIS, L. M. S., Finite element viscoplastic analyses of plates, M.Sc. Thesis, University College of Swansea (1975). +88. NAYAK, G. C., Plasticity and large deformation problems by finite element method, Ph.D. Thesis, University of Wales, Swansea (1971). +89. BASU, A. K., Letter to the Editor, 'New light on the Nayak alpha technique', Int. J. Num. Meth. Engng. 6, 152 (1973). +90. IRONS, B. M. and AHMAD, S., Techniques of Finite Elements, Ellis Horwood, Chichester (1980). +91. MATTHIES, H. and STRANG, G., The solution of nonlinear finite element equations, Int. J. Num. Meth. Engng. 14, 1613-1626 (1979). +92. GERADIN, M. and HOGGE, M. A., Quasi-Newton iteration in non-linear structural dynamics, Paper M7/1, Trans. Fifth Int. Conf. on SMIRT, Berlin, North-Holland, Amsterdam (1979). diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_052.md b/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_052.md new file mode 100644 index 00000000..f899d5df --- /dev/null +++ b/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_052.md @@ -0,0 +1,284 @@ + + +93. CRISFIELD, M. A., Iterative solution procedure for linear and non-linear structural analysis, TRRL Lab. Report 900, Transport and Road Research Laboratory, Crowthorne, Berkshire, U.K. (1979). +94. HESTENES, M. and STIEFEL, E., Method of conjugate gradients for solving linear systems, J. Res. Nat. Bur. Standards, 49, 409–436 (1952). +95. FLETCHER, R. and REEVES, C. M., Function minimisation by conjugate gradients, The Computer Journal, 7, 149–154 (1964). +96. POLAK, E., Computational methods in optimisation: a unified approach, Academic Press, London (1971). +97. JENNINGS, A. and MALIK, G. M., The solution of sparse linear equations by the conjugate gradient method, Int. J. Num. Meth. Engng. 12, 141-158 (1978). +98. FRIED, I. and METZLER, J., SOR v. conjugate gradients in a finite element discretization, Int. J. Num. Meth. Engng. 12, 1329-1342 (1978). +99. KERSHAW, D. S., The incomplete Cholesky-conjugate gradient method for the iterative solution of systems of linear equations, J. Comp. Phys., 26, 43–65 (1978). +100. IRONS, B. M. and ELSAWAF, A. F., The conjugate Newton algorithm for solving finite element equations, In: Formulations and Computational Algorithms in Finite Element Analysis, Eds. K. J. Bathe, J. T. Oden and W. Wunderlich, MIT Press, 655–672 (1977). +101. BREW, J. S. and BROTTON, D. M., Non-linear structural analysis by dynamic relaxation, Int. J. Num. Meth. Engng. 3, 463–483 (1971). +102. PICA, A. and HINTON, E., Transient and pseudo-transient analyses of Mindlin plates: Theory and applications, Int. J. Num. Meth. Engng. 15, 189–208 (1980). +103. UNDERWOOD, P., An adaptive dynamic relaxation technique for nonlinear structural analysis, Draft Report, Lockheed Palo Alto Research Laboratory. +104. BERGAN, P. G. and SOREIDE, T., Solution of large displacement and instability problems using current stiffness parameter, Proc. Conf. on Finite Elements in Nonlinear Solid and Structural Mechanics, Geilo, Norway (1977). +105. MOTE, C. D. J., Global-local finite element, Int. J. Num. Meth. Engng. 3, 565–574 (1971). +106. CRISFIELD, M. A., A combined Rayleigh-Ritz/finite element method for the non-linear analysis of stiffened plate structures, Computers and Structures, 8, 679–689 (1978). +107. WILSON, E. L., Special numerical and computer techniques for the analysis of finite element systems, Proc. US-German Symposium on Formulations and Algorithms in Finite Element Analysis, Eds. K. J. Bathe, T. J. Oden and W. Wunderlich, 3–25, MIT Press, Boston (1977). +108. WACHSPRESS, E. L., Two-level finite element computation, Proc. US-German Symposium on Formulations and Algorithms in Finite Element Analysis, Eds. K. J. Bathe, T. J. Oden and W. Wunderlich, 877-913, MIT Press, Boston (1977). +109. BRANDT, A., Multi-level adaptive solutions to boundary-value problems, Math. Comptr., 31, 333-390 (1977). +110. FELIPPA, C. A., Procedures for computer analysis of large nonlinear structural analysis, Int. Symp. on Large Engineering Systems, The University of Manitoba, Winnipeg, Canada, 1976, Pergamon Press, Oxford (1977). +111. PEANO, A. M., FANELLI, M., RICCIONI, R. and SANDELLA, L., Self-adaptive convergence at the crack tip of a dam buttress, Proc. of First Int. Conf. on Numerical Methods in Fracture Mechanics. Eds. A. R. Luxmoore and D. R. J. Owen, 268–280, Pineridge Press, Swansea, U.K. (1978). + + + +112. Row, D. G. and POWELL, G. H., A substructure technique for nonlinear static and dynamic analysis, Report No. UCB/EER/C-78/15, University of California, Berkeley (1978). +113. ZIENKIEWICZ, O. C., KELLY, D. W. and BETTESS, P., 'Marriage à la mode', Finite elements and boundary integrals, Proc. Conf. Innovative Numerical Analysis in Engineering Science, CETIM, Paris (1977). +114. BANERJEE, P. K., CATHIE, D. N. and DAVIES, T. G., Two- and three-dimensional problems of elastoplasticity, In: Developments in Boundary Integral Methods, Chapter 4, Eds. P. K. Banerjee and R. Butterfield, Applied Science Publishers (1979). +115. KELLY, D.W., MUSTOE, G.G.W. and ZIENKIEWICZ, O.C., Coupling boundary element methods with other numerical methods, In: Developments in Boundary Integral Methods, Chapter 10, Eds. P. K. Banerjee and R. Butterfield, Applied Science Publishers (1979). +116. SCHREM, E., Trends and aspects of the development of large finite element software systems, Computers and Structures, 10, 419–425 (1979). +117. SCHREM, E., From program systems to programming systems for finite element analysis, In: Formulations and Computational Algorithms in Finite Element Analysis, Eds. K. J. Bathe, J. T. Oden and W. Wunderlich, MIT Press, 163-189 (1977). +118. ZAVE, P. and RHEINBOLDT, W. C., Design of an adaptive, parallel finite-element system, ACM Transactions on Mathematical Software, 5, No. 1, 1-17 (1979). + + + +# Appendix I + +# Instructions for preparing input data for one-dimensional problems + +In Part I of this text computer codes have been presented for the nonlinear analysis of several classes of one-dimensional problems. In Chapter 3 the data structure for the following applications was discussed: + +- Direct iteration solution of nonlinear quasiharmonic problems. +- Use of the Newton–Raphson process for the solution of nonlinear quasiharmonic problems. +● Nonlinear elastic applications. +- Elasto-plastic material behaviour. + +In Chapter 4 the time transient phenomenon of one-dimensional visco-plasticity was discussed. In Chapter 5 solution techniques were presented for elasto-plastic beam bending problems. In this appendix user instructions for preparing input data for each of these applications are provided. + +# A.1.1 Program QUITER for the solution of nonlinear one-dimensional quasiharmonic problems by direct iteration + +CARD SET 1 TITLE CARD (12A6)—One card + +Cols. 1-72 Title of the problem—limited to 72 alphanumeric characters. + +CARD SET 2 CONTROL CARD (915)—One card + +Cols. 1-5 NPOIN + +6-10 NELEM + +11-15 NBOUN + +Total number of nodal points. + +Total number of elements. + +Total number of restrained boundary points—nodes at which the value of the unknown (e.g. temperature) is prescribed. + +Total number of different materials. + +16-20 NMATS + +21-25 NPROP + +Number of independent properties per material (= 1). + +26-30 NNODE + +Number of nodes per element (= 2). + +31-35 NINCS + +Number of increments in which the total 'loading' is to be applied. + + + +36-40 NALGO Nonlinear solution process indicator (= 1, for solution by direct iteration). + +41-45 NDOFN Number of degrees of freedom per node $(= 1)$ . + +CARD SET 3 MATERIAL CARDS (I5, F15.5)—One card for each different material. Total of NMATS cards (See Card Set 2). + +Cols. 1–5 JMATS Material identification number. +6–20 PROPS(JMATS,1) The material coefficient, $K_{0}$ in (2.27). + +CARD SET 4 ELEMENT CARDS (4I5)—One card for each element. Total of NELEM cards (See Card Set 2). + +Cols. 1–5 JELEM Element number. +6–10 LNODS(JELEM,1) 1st nodal connection number. +11–15 LNODS(JELEM,2) 2nd nodal connection number. +16–20 MATNO(JELEM) Material property number. + +NOTE: The two nodal connection numbers for an element can be taken in any order. + +CARD SET 5 NODAL COORDINATE CARDS (I10,F15.5)—One card for each node. Total of NPOIN cards (See Card Set 2). + +Cols. 1–10 JPOIN Node number. +11–25 COORD(JPOIN) The x coordinate of the node. + +Note: The origin of the coordinate system may be arbitrarily located. + +CARD SET 6 RESTRAINED NODE CARDS (I10,I5,F10.5)—One card for each restrained node. Total of NBOUN cards (See Card Set 2). + +Cols. 1-10 NODFX Restrained node number. +11-15 ICODE(1) Condition of restraint(=1). +16-25 PRESC(1) The prescribed value of the nodal variable. + +CARD SET 7 APPLIED 'LOAD' CARDS (I10,2F15.5)—One card for each loaded element. + +Cols. 1–10 IELEM The element-number. +11–25 RLOAD(IELEM,1) The applied load at the 1st node of the element. +26–40 RLOAD(IELEM,2) The applied load at the 2nd node of the element. + +Notes: 1) The 1st and 2nd nodes must be taken in the order listed in Card Set 4. +2) This card set must terminate with data for the highest numbered element whether it is loaded or not. + + + +CARD SET 8 LOAD INCREMENT CONTROL CARDS (2I5,2F15.5)—One card for each load increment. Total of NINCS cards (See Card Set 2). + +
Cols. 1-5 NITERMaximum number of iterations allowed for the ‘load’ increment.
6-10 NOUTPOutput control parameter:1—Results output only after the first iteration and after convergence,2—Results output after each iteration.
11-25 FACTOApplied ‘load’ factor for the increment—specified as a factor of the loading input in Card Set 7.
26-40 TOLERConvergence tolerance factor.—The term TOLER in (3.21).
+ +Note: The applied loading factors are accumulative. If FACTO is specified as 0.6, 0.3, 0.3 for the first three 'load' increments, then the total loading acting during the third increment is 1.2 times that specified in Card Set 7. + +If the form of the material nonlinearity is to be changed, then FUNCTION VARIA must be modified in accordance with the process described in Section 3.9.1. + +# A.1.2 Program QUNEWT for the solution of nonlinear one-dimensional quasiharmonic problems by the Newton–Raphson process + +Data input for this application is identical to that described in Section A.1.1 above with the following exceptions: + +# CARD SET 2 CONTROL CARD + +
Cols. 21–25 NPROPNumber of independent properties per material (= 2).
36–40 NALGONonlinear solution process parameter (= 2, for Newton–Raphson solution technique).
+ +CARD SET 3 MATERIAL CARDS (I5,2F15.5)—One card for each different material. + +
Cols.1-5JMATSMaterial identification number.
6-20PROPS(JMATS,1)The material coefficient $K_0$ in (2.27).
21-35PROPS(JMATS,2)The term $b$ in (2.27).
+ + + +# A.1.3 Program NONLAS for the solution of one-dimensional nonlinear elastic problems + +The input data for this application is again identical to that described in Section A.1.1 with the following exceptions. The basic nodal variable is now the axial displacement. + +CARD SET 2 CONTROL CARD + +
Cols. 21–25 NPROPNumber of independent properties per material(= 2).
36–40 NALGONonlinear solution process indicator:1 or 2 Tangential stiffness algorithm. The element stiffnesses are recalculated for each iteration of the solution process.3 Initial stiffness method. The stiffnesses are calculated at the beginning of the solution process and maintained constant thereafter.4 Combined algorithm (Version I). The element stiffnesses are recomputed for the first iteration of each load increment.5 Combined algorithm (Version II). The element stiffnesses are recomputed for the second iteration of each load increment.
+ +CARD SET 3 MATERIAL CARDS (I5,2F15,5)—One card for each different material. + +
Cols.1-10JMATSMaterial identification number.
6-20PROPS(JMATS,1)Elastic modulus, $E$ .
21-35PROPS(JMATS,2)Cross-sectional area, $A$ .
+ +# A.1.4 Program ELPLAS for the solution of one-dimensional elastoplastic problems + +The input data for this application is again identical to that described in Section A.1.1 with the following exceptions. The basic nodal variable is the axial displacement. + +CARD SET 2 CONTROL CARD (9I5) + +
Cols. 21-25 NPROPNumber of independent properties per material (= 4).
36-40 NALGONonlinear solution process indicator: 1 or 2 Tangential stiffness algorithm.
+ + + +3 Initial stiffness method. +4 Combined algorithm with stiffnesses recomputed for the 1st iteration. +5 Combined algorithm with stiffnesses recomputed for the 2nd iteration. + +CARD SET 3 MATERIAL CARDS (15,4F15.5)—One card for each different material. + +
Cols.1-5JMATSMaterial identification number.
6-20PROPS(JMATS,1)Elastic modulus, $E$ .
21-35PROPS(JMATS,2)Cross-sectional area, $A$ .
36-50PROPS(JMATS,3)Uniaxial yield stress, $\sigma_{Y}$ .
51-65PROPS(JMATS,4)Linear strain-hardening parameter, $H'$ .
+ +# A.1.5 Program UNVIS for the solution of one-dimensional elastoviscoplastic problems + +The input data for this application is once again identical to that described in Section A.1.1 with the following exceptions. The basic nodal variable is the axial displacement. + +# CARD SET 2 CONTROL CARD + +
Cols. 21-25 NPROPNumber of independent properties per material (= 5).
36-40 NALGONonlinear solution process indicator (= 1, for Euler time stepping scheme).
+ +CARD SET 3 MATERIAL CARDS (I5,5F15.5)—One card for each different material. + +
Cols.1-5JMATSMaterial identification number.
6-20PROPS(JMATS,1)Elastic modulus, $E$ .
21-35PROPS(JMATS,2)Cross-sectional area, $A$ .
36-50PROPS(JMATS,3)Uniaxial yield stress, $\sigma_{Y}$ .
51-65PROPS(JMATS,4)Linear strain-hardening parameter, $H'$ .
66-80PROPS(JMATS,5)Fluidity parameter, $\gamma$ .
+ +CARD SET 8 TIMESTEPPING PARAMETER CARD (3F15.5)—One card. + +
Cols. 1–15 TAUFTThe factor $\tau$ employed to limit the time-step length according to (4.38).
16–30 DTINTThe initial time step length (required to initiate the time stepping process.
31–45 FTIMEThe factor $k$ in (4.39).
+ + + +# CARD SET 9 LOAD INCREMENT CONTROL CARDS + +This card set is identical to Card Set 8, Section A.1.1 where the term 'iteration' is now replaced by 'timestep'. + +# A.1.6 Program TIMOSH for the nonlayered elasto-plastic analysis of Timoshenko beams + +The input data for this application is identical to that described in Section A.1.1 with the following exceptions. + +# CARD SET 2 CONTROL CARD (9I5) + +
Cols.21-25NPROPNumber of independent properties per material (=4)
36-40NALGONonlinear solution process indicator:1 or 2 Tangential stiffness algorithm.3 Initial stiffness method.4 Combined algorithm with stiffnesses recomputed for the 1st iteration.5 Combined algorithm with stiffnesses recomputed for the 2nd iteration.
41-45NDOFNNumber of degrees of freedom per node (=2).
+ +CARD SET 3 MATERIAL CARDS (I5, 4F15.5)—One card for each different material. + +Cols. 6–20 PROPS(JMATS, 1) Flexural rigidity, EI. + +21-35 PROPS(JMATS,2) Shear constant, GA/1.5. + +36-50 PROPS(JMATS, 3) Yield moment, $M_0$ . + +51-65 PROPS(JMATS, 4) Strain hardening parameter, $H'$ . + +CARD SET 6 RESTRAINED NODE CARDS (I10, 2(I5, F10.5))—One card for each restrained node. Total of NBOUN cards. + +
Cols. 11-15ICODE(1)Condition of restraint on nodal displacement, w.
0-No displacement restraint.
1-Nodal displacement restrained.
16-25VALUE(1)The prescribed value of nodal displacement, w.
26-30ICODE(2)Condition of restraint on nodal rotation, θ.
0-No rotation restraint.
1-Nodal rotation restrained.
31-40VALUE(2)The prescribed value of nodal rotation, θ.
+ + + +CARD SET 7 APPLIED LOAD CARDS (I10, 4FI5.5)—One card for each loaded element. + +Cols. 1-10 JELEM Element number. + +11-25 RLOAD(JELEM,1) Transverse load applied at the first node. + +26-40 RLOAD(JELEM,2) Couple applied at the first node. + +41-55 RLOAD(JELEM,3) Transverse load applied at the second node. + +56-70 RLOAD(JELEM,4) Couple applied at the second node. + +Note: The last card should be that for the highest numbered element whether it is loaded or not. + +# A.1.7 Program TIMLAY for the layered elasto-plastic analysis of Timoshenko beams + +The input data for this application is identical to that described in Section A.1.6 with the following exceptions. + +# CARD SET 2 CONTROL CARD (10I5) + +Cols. 21–25 NPROP + +Number of independent properties per material ( $=4+2\times$ Total number of layers). + +46-50 NLAYR + +Total number of layers. + +# CARD SET 3 MATERIAL CARDS + +1st Card (I5, 4F15.5) + +Cols. 1–5 NUMAT Material identification number. + +6-20 PROPS(NUMAT,1) Young's modulus, $E$ . + +21-35 PROPS(NUMAT,2)Modified shear modulus, $G / 1.5$ . + +36-50 PROPS(NUMAT,3) Yield stress, $\sigma_{Y}$ . + +51-65 PROPS(NUMAT,4)Strain hardening parameter, $H'$ . + +# 2nd and subsequent cards (4F15.5) + +Cols. 1–15 BRDTH(1) Breadth of the 1st layer. + +16-30 THICK(1) Thickness of the 1st layer. + +31-45 BRDTH(2) Breadth of the 2nd layer. + +• • +• • +• • + +BRDTH(NLAYR) Breadth of the last layer. + +• THICK(NLAYR) Thickness of the last layer. + + diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_053.md b/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_053.md new file mode 100644 index 00000000..53f776ba --- /dev/null +++ b/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_053.md @@ -0,0 +1,477 @@ + + +# Appendix II + +# Instructions for preparing input data for plane, axisymmetric and plate bending problems + +In this appendix user instructions are provided for the computer programs developed in Part II of this text. Chapter 7 dealt with elasto-plastic problems in two dimensions and in Chapter 8 the corresponding time-dependent situation of elasto-viscoplasticity was discussed. The elasto-plastic behaviour of plates in bending was considered in Chapter 9. + +# A.2.1 Program PLANET for the elasto-plastic analysis of plane and axisymmetric solids + +CARD SET 1 TITLE CARD (12A6)—One card. + +Cols. 1–72 Title of the problem—limited to 72 alphanumeric characters. + +CARD SET 2 CONTROL CARD (1115)—One card. + +
Cols.1-5 NPOINTotal number of nodal points.
6-10 NELEMTotal number of elements.
11-15 NVFIXTotal number of restrained boundary points—where one or more degrees of freedom are restrained.
16-20 NTYPEProblem type parameter:
1—Plane stress,
2—Plane strain,
3—Axial symmetry.
21-25 NNODENumber of nodes per element:
4—Linear quadrilateral element,
8—Quadratic Serendipity element,
9—Quadratic Lagrangian element.
26-30 NMATSTotal number of different materials.
31-35 NGAUSOrder of integration formula for numerical integration:
2—Two point Gauss quadrature rule,
3—Three point Gauss quadrature rule.
+ + + +36-40 NALGO + +Nonlinear solution parameter: + +1 Initial stiffness method. The element stiffnesses are calculated at the beginning of the solution process and remain unchanged thereafter. +2 Tangential stiffness method. The element stiffnesses are recalculated for every iteration of each load increment. +3 Combined algorithm (Version I). The element stiffnesses are recalculated for the first iteration of each load increment only. +4 Combined algorithm (Version II). The element stiffnesses are recalculated for the second iteration of each load increment only. + +41-45 NCRIT + +Yield criterion parameter: + +1—Tresca, +2—Von Mises, +3—Mohr-Coulomb, +4—Drucker–Prager. + +46-50 NINCS + +Number of increments in which the total loading is to be applied. + +51-55 NSTRE + +Number of stress components at a point: +3—Plane stress or plane strain, +4—Axial symmetry. + +CARD SET 3 ELEMENT CARDS (1115)—One card for each element. Total of NELEM cards (See Card Set 2). + +Cols. 1-5 NUMEL + +Element number. + +6-10 MATNO(NUMEL) Material property number. + +11-15 LNODS(NUMEL,1) 1st Nodal connection number. + +16-20 LNODS(NUMEL,2) 2nd Nodal connection number. + +51-55 LNODS(NUMEL,9) 9th Nodal connection number. + +Notes: 1) Columns 31–55 remain blank for linear 4-noded elements. +2) Columns 51–55 remain blank for 8-noded elements. +3) The nodal connection numbers must be listed in an anti-clockwise sequence, starting from any corner node. + +CARD SET 4 NODE CARDS (I5,2F10.5)—One card for each node whose coordinates are to be input. + + + +Cols. 1-5 IPOIN + +Nodal point number. + +6-15 COORD(IPOIN,1) + +x (or r) coordinate of the node. + +16-25 COORD(IPOIN,2) + +$y$ (or $z$ ) coordinate of the node. + +Notes: 1) The total number of cards in this set will generally differ from NPOIN (see Card Set 2) since for quadratic elements whose sides are linear, it is only necessary to specify data for corner nodes, intermediate nodal coordinates being automatically interpolated if on a straight line. +2) For Lagrangian elements the coordinates of the 9th (central) node are never input. +3) The coordinates of the highest numbered node must be input regardless of whether it is a midside node or not. + +CARD SET 5 RESTRAINED NODE CARDS (1X,14,5X,15,5X,2F10.5)—One card for each restrained node. Total of NVFIX cards (See Card Set 2). + +Cols. 2-5 NOFIX(IVFIX) + +11-15 IFPRE + +Restrained node number. + +Restraint code: + +01 Nodal displacement restrained in the $x$ (or $r$ ) direction, +10 Nodal displacement restrained in the $y$ (or $z$ ) direction, +11 Nodal displacement restrained in both coordinate directions. + +21-30 PRESC(IVFIX,1) + +The prescribed value of the x (or r) component of nodal displacement. + +31-40 PRESC(IVFIX,2) + +The prescribed value of the y (or z) component of nodal displacement. + +CARD SET 6 MATERIAL CARDS + +6(a) CONTROL CARD (I5)—One card. + +Cols. 1-5 NUMAT + +Material identification number. + +6(b) PROPERTIES CARDS (7F10.5)—One card for each different material. + +Cols. 1-10 PROPS(NUMAT,I) + +Elastic modulus, E. + +11-20 PROPS(NUMAT,2) + +Poisson's ratio, v. + +21-30 PROPS(NUMAT,3) + +Material thickness, t (leave blank for plane strain and axisymmetric problems). + +31-40 PROPS(NUMAT,4) + +Mass density, ρ. + +41-50 PROPS(NUMAT,5) + +Uniaxial yield stress, $\sigma_{Y}$ (or cohesion $c$ for Mohr-Coulomb or Drucker-Prager materials). + +51-60 PROPS(NUMAT,6) + +Strain hardening parameter, $H'$ . + + + +61-70 PROPS(NUMAT,7) Friction angle $\phi$ (measured in degrees) for Mohr-Coulomb and Drucker-Prager materials only). + +Note: This card set to be repeated for each different material. Total of NMATS card sets (See Card Set 2). + +CARD SET 7 LOAD CASE TITLE CARD (12A6)—One card. + +Cols. 1–72 TITLE Title of the load case—limited to 72 alphanumeric characters. + +CARD SET 8 LOAD CONTROL CARD (315)—One card. + +Cols. 1-5 IPLOD Applied point load control parameter: 0 No applied nodal loads to be input, 1 Applied nodal loads to be input. + +6-10 IGRAV Gravity loading control parameter: +0 No gravity loads to be considered, +1 Gravity loading to be considered. + +11-15 IEDGE Distributed edge load control parameter: 0 No distributed edge loads to be input, 1 Distributed edge loads to be input. + +CARD SET 9 APPLIED LOAD CARDS (15,2F10.3)—One card for each loaded nodal point. + +Cols. 1-5 LODPT Node number. + +6-15 POINT(1) Load component in $x$ (or $r$ ) direction. + +16-25 POINT(2) Load component in $y$ (or $z$ ) direction. + +Notes: 1) The last card should be that for the highest numbered node whether it is loaded or not. + +2) For axisymmetric problems, the loads input should be the total loading on the circumferential ring passing through the nodal point concerned. + +3) If IPLOD = 0 in Card Set 8, omit this set. + +CARD SET 10 GRAVITY LOADING CARD (2F10.3)—One card. + +Cols. 1–10 THETA Angle of gravity axis measured from the positive y axis (see Fig. 6.7). + +11-20 GRAVY Gravity constant—specified as a multiple of the gravitational acceleration, g. + +Note: If IGRAV = 0 in Card Set 8, omit this set. + +CARD SET 11 DISTRIBUTED EDGE LOAD CARDS + +11(a) CONTROL CARD (I5)—One card. + +Cols. 1-5 NEDGE Number of element edges on which distributed loads are to be applied. + + + +# 11(b) ELEMENT FACE TOPOLOGY CARD (415) + +Cols. 1–5 NEASS + +$$ +\left. \begin{array}{l l} 6 - 1 0 & \text { NOPRS } (1) \\ 1 1 - 1 5 & \text { NOPRS } (2) \\ 1 6 - 2 0 & \text { NOPRS } (3) \end{array} \right\} +$$ + +The element number with which the element edge is associated. + +List of nodal points, in an anticlockwise sequence, of the nodes forming the element face on which the distributed load acts. + +Note: For linear 4-noded elements, Cols. 16–20 remain blank. + +# 11(c) DISTRIBUTED LOAD CARDS (6F10.3) + +Cols. 1-10 PRESS(1,1) + +11-20 PRESS(1,2) + +21-30 PRESS(2,1) + +31-40 PRESS(2,2) + +41-50 PRESS(3,1) + +51-60 PRESS(3,2) + +Value of normal component of distributed load at node NOPRS(1). + +Value of tangential component of distributed load at node NOPRS(1). + +Value of normal component of distributed load at node NOPRS(2). + +Value of tangential component of distributed load at node NOPRS(2). + +Value of normal component of distributed load at node NOPRS(3). + +Value of tangential component of distributed load at node NOPRS(3). + +Notes: 1) For linear 4-noded elements, Cols. 41–60 remain blank. + +2) Subsets 11(b) and 11(c) must be repeated in turn for every element edge on which a distributed load acts. The element edges can be considered in any order. + +3) If IEDGE = 0 in Card Set 8, omit this card set. + +# CARD SET 12 LOAD INCREMENT CONTROL CARDS (2F10.5,3I5)—One card for each load increment. Total of NINCS cards (see Card Set 2). + +Cols. 1-10 FACTO + +11-20 TOLER + +21-25 MITER + +26-30 NOUTP(1) + +Applied load factor for this increment—specified as a factor of the loading input in Card Sets 8 to 11. + +Convergence tolerance factor.—The term TOLER in (3.27). + +Maximum number of iterations allowed for the load increment. + +Parameter controlling output of results after 1st iteration: + +0—No output, + +1—Output displacements, + +2—Output displacements and reactions, + + + +31-35 NOUTP(2) + +3—Output displacements, reactions and stresses. + +Parameter controlling output of the converged results: + +0—No output, + +1—Output displacements, + +2—Output displacements and reactions, + +3—Output displacements, reactions and stresses. + +Note: The applied loading factors are accumulative. If FACTO is specified as 0.6, 0.3, 0.2 for the first three load increments, then the total loading acting during the third increment is 1.1 times that specified in Card Sets 8 to 11. + +# A.2.2 Program VISCOUNT for the elasto-viscoplastic analysis of plane and axisymmetric solids + +The input data for this application is identical to that described in Section A.2.1, for elasto-plastic problems, with the following exceptions. + +# CARD SET 2 CONTROL CARD (1115) + +Cols. 36–40 NALGO + +Equation solution parameter: + +1 Explicit time stepping scheme (i.e. TIMEX = 0—See Card Set 12), + +2 Implicit or Semi-implicit schemes (TIMEX ≠ 0). + +CARD SET 6(b) PROPERTIES CARDS (8F10.5)—Two cards for each different material. + +1st Card + +Cols. 1–70 + +Identical to Card Set 6(b), Section A.2.1. + +71-80 PROPS(NUMAT,8) + +Fluidity parameter, γ. + +2nd Card + +Cols. 1-10 PROPS(NUMAT,9) + +The constant $M$ in (8.8) or constant $N$ in (8.9). + +11-20 PROPS(NUMAT,10) + +Parameter controlling choice of the flow function: + +0 Expression (8.8) to be used, + +1 Expression (8.9) to be used. + +CARD SET 12 TIMESTEPPING PARAMETER CARD (4F10.3)—One card. + +Cols. 1-10 TIMEX + +Timestepping algorithm parameter, $\Theta$ in (8.10). + + + +
11-20 TAUFTThe factor $\tau$ employed to limit the time step length according to (8.29).
21-30 DTINTThe initial time step length (required to initiate the time stepping process).
31-40 FTIMEThe factor $k$ in (8.32).
+ +# CARD SET 13 LOAD INCREMENT CONTROL CARDS + +This card set is identical to Card Set 12, Section A.2.1 where the term 'iteration' is now replaced by 'timestep'. + +# A.2.3 Programs MINDLIN and MINDLAY for the nonlayered and layered elasto-plastic analysis of Mindlin plates + +The input data for this application is identical to that described in Section A.2.1, for elasto-plastic plane and axisymmetric solids, with the following exceptions. + +CARD SET 2 (1115)—One card + +
Cols.16-20NTYPEProblem type parameter:5—for Heterosis element,0—for 4- or 8-node elements.
21-25NNODENumber of nodes per element:4—Linear 4-node quadrilateral element.8—Quadratic 8-node Serendipity element.9—Quadratic 9-node Lagrangian element or Heterosis element.
31-35NGAUS2 for 4-node element,3 for 8-, 9-node and Heterosis element.(N.B. This is the integration rule to evaluate the flexural contribution to the element stiffness matrix. Since selective integration is adopted a (NGAUS-1) integration is automatically used to evaluate the transverse shear contribution to the element stiffness matrix.)
41-45NCRITYield criterion parameter:1—Tresca,2—Von-Mises.(Mohr-Coulomb and Drucker-Prager yield criteria are not included.)
51-55NLAPSTotal number of layers.(for program MINDLAY only—in program MINDLIN leave blank.)
+ + + +CARD SET 5 RESTRAINED NODE CARDS (1X, I4, 5X, I5, 5X, 3F10.5) One card for each restrained node. Total of NVFIX cards. + +Cols.11-15 IFPRE + +Restraint code: + +100 Lateral displacement w restrained. + +010 Rotation $\theta_{x}$ restrained. + +001 Rotation $\theta_{y}$ restrained. + +110 Lateral displacement w and rotation $\theta_{x}$ restrained, etc. + +21-30 PRESC(IVFIX,1) + +The prescribed value of the lateral nodal displacement w. + +31-40 PRESC(IVFIX,2) + +The prescribed value of the nodal rotation $\theta_{x}$ . + +41-50 PRESC(IVFIX.3) + +The prescribed value of the nodal rotation $\theta_{y}$ . + +# CARD SET 6 MATERIAL CARDS + +6(b) PROPERTIES CARDS (7F10.5)—One card for each different material. + +Cols.31-40 PROPS(NUMAT,4) + +Uniform distributed loading value. + +41-50 PROPS(NUMAT,5) + +Blank. + +51-60 PROPS(NUMAT,6) + +Uniaxial yield stress, $\sigma_0$ . + +61-70 PROPS(NUMAT,7) + +Strain hardening parameter $H'$ . + +# CARD SET 6X CONVERGENCE CHECK CARDS + +6X(a) DISPLACEMENT CHECK CARD (511)—One card. + +Cols. 1 IFDIS + +1 The displacement check is to be employed. + +2 NCDIS(1) + +1 Check based on norm involving w. + +3 NCDIS(2) + +1 Check based on norm involving $\theta_{x}$ . + +4 NCDIS(3) + +1 Check based on norm involving $\theta_y$ . + +5 NCDIS(4) + +1 Check based on $w$ , $\theta_x$ and $\theta_y$ . + +6X(b) RESIDUAL FORCE CHECK + +CARD (511)—One card. + +Cols. 1 IFRES + +1 The residual force check is to be employed. + +2 NCRES(1) + +1 Check based on norm involving residual forces associated with w. + +3 NCRES(2) + +1 Check based on norm involving residual forces associated with $\theta_{x}$ . + +4 NCRES(3) + +1 Check based on norm involving residual forces associated with $\theta_{y}$ . + +5 NCRES(4) + +1 Check based on norm involving residual forces associated with $\dot{w}$ , $\theta_{x}$ and $\theta_{y}$ . + +Note: A zero value for any item implies that the check is not being used. + + + +CARD SET 8 LOAD CONTROL CARD (I5)—One card. + +Cols. 1–5 IPLOD + +Applied point load control parameter: + +0 No applied nodal loads to be input. + +1 Applied nodal loads to be input. + +6-15 + +Blank. + +CARD SET 9 APPLIED LOAD CARDS (I5, 3F10.3)—One card for each loaded nodal point. + +Cols. 1-5 LODPT + +Node number. + +6-15 POINT(1) + +Lateral nodal load. + +16-25 POINT(2) + +Nodal couple in xz plane. + +26-35 POINT(3) + +Nodal couple in yz plane. + +Omit CARD SETS 10, 11(a), 11(b) and 11(c). + + diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_054.md b/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_054.md new file mode 100644 index 00000000..44d95f8f --- /dev/null +++ b/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_054.md @@ -0,0 +1,365 @@ + + +# Appendix III + +# Instructions for preparing input data for dynamic transient problems + +The program DYNPAK has been described in Section 10.6 and MIXDYN in Section 11.5. These programs perform large displacement or viscoplastic or elasto-plastic, transient dynamic analysis of plane stress/strain or axisymmetric problems respectively. The format of the input data is identical for both programs. In this appendix user instructions for preparing input data are provided. + +CARD SET 1 DYNAMIC DIMENSIONING (4I5)—One card. + +Cols. 1-5 NPOIN Total number of nodal points. + +6-10 NELEM Total number of elements. + +11-15 NDOFN Number of degrees of freedom per node $(= 2)$ . + +16-20 NMATS Number of different material sets. + +CARD SET 2 TITLE CARD (10A4)—One card. + +Cols. 1–40 Title of the problem—limited to 40 alphanumeric characters. + +CARD SET 3 CONTROL CARD (13I5)—One card. + +Cols. 1–5 NVFIX Total number of nodal points with fixed degrees of freedom. + +$$ +\begin{array}{l} = 1, \text { Plane stress }, \\ = 2, \text { Plane strain }, \\ = 3, \text { Axisymmetric problem. } \\ \end{array} +$$ + +6-10 NTYPE Type of problem: + +11-15 NNODE Number of nodes per element. + +16-20 NPROP Number of material properties $(= 11)$ . + +21-25 NGAUS Integration rule for stiffness matrix. + +26-30 NDIME Number of coordinate dimensions (=2). + +31-35 NSTRE Number of stress components (= 3 for plane stress/strain, = 4 for axisymmetric). + + + +
36-40NCRITYield criterion: = 1 — Tresca, = 2 — Von Mises, = 3 — Mohr-Coulomb, = 4 — Drucker-Prager.
41-45NPREVIndicator for the previous state to be read (= 1 for previous state, otherwise, = 0).
46-50NCONMNumber of concentrated masses (≥1 if concentrated mass present, otherwise, = 0).
51-55NLAPSIndicator for large displacement analysis: = 0—Elastic analysis, = 1—Elasto-plastic small displacement analysis, = 2—Elastic large displacement analysis,
56-60NGAUMIntegration rule for mass matrix.
61-65NRADS= 0, Read (r, z) coordinates for nodes, = 1, Read (R, Θ) coordinates for nodes for axisymmetric analysis.
+ +CARD SET 4 ELEMENT CARDS (11I5)—One card for each element, total of NELEM cards. The node numbers are read in anticlockwise sequence. The number of nodes depends upon the type of element. For four and eight noded elements read only four and eight nodes respectively. + +
Cols.1-5IELEMElement number.
6-10MATNOMaterial identification number.
11-15LNODS(IELEM,1)
16-20LNODS(IELEM,2)
21-25LNODS(IELEM,3)
26-30LNODS(IELEM,4)
31-35LNODS(IELEM,5)Nodal connection numbers.
36-40LNODS(IELEM,6)
41-45LNODS(IELEM,7)
46-50LNODS(IELEM,8)
51-55LNODS(IELEM,9)
+ +CARD SET 5 NODAL COORDINATE CARDS (I5,2F10.5)—One card for each node. Last nodal point (IPOIN=NPOIN) must be read at the end. Only corner and central nodes need to be specified. Midside nodes are interpolated if not specified. For axisymmetric cases, $(R, \Theta)$ values are read for NRADS = 1, and $(r, z)$ coordinates are calculated in the program. + + + +Cols. 1-5 IPOIN Current nodal point. + +6-15 COORD(IPOIN,1) x-coordinate.\* + +16-25 COORD(IPOIN,2) y-coordinate. + +CARD SET 6 RESTRAINED NODE CARDS (1X,14,3X,211)—One card for each restrained node. Total of NVFIX cards. + +Cols. 2-5 IPOIN Restrained node number. + +9 IFPRE(IVFIX,1) Fixity in $x$ -direction $(= 0$ , Free; $= 1$ , Fixed). + +10 IFPRE(IVFIX,2) Fixity in y-direction (= 0, Free; = 1, Fixed). + +CARD SET 7 MATERIAL CARDS—Three cards for each different material, a total of NMATS\*3 cards. + +1st Card MATERIAL IDENTIFICATION CARD (I5) + +Cols. 1-5 NUMAT Material identification number. + +2nd Card MATERIAL PROPERTIES CARD—(a) (8E10.4) + +Cols. 1-10 PROPS(NUMAT,1) Young's Modulus, $E$ . + +11-20 PROPS(NUMAT,2) Poisson's ratio, $\nu$ . + +21-30 PROPS(NUMAT,3) Thickness for plane stress problem, t. + +31-40 PROPS(NUMAT,4) Mass density per unit volume, $\rho$ . + +41-50 PROPS(NUMAT,5) Temperature coefficient, $\alpha_{t}$ . + +51-60 PROPS(NUMAT,6) Reference yield value $F_{0}$ : + +Von Mises, $F_0 = \sigma Y,$ + +Tresca, $F_{0} = \sigma_{Y},$ + +Mohr-Coulomb, $F_0 = c\cos \phi$ + +Drucker-Prager, $F_{0} = 6c\cos \phi /$ + +$(\sqrt{3}(3 - \sin \phi)).$ + +61-70 PROPS(NUMAT,7) Hardening parameter, $H'$ : + +$$ +H ^ {\prime} = \frac {E _ {T}}{1 - E _ {T} / E}, +$$ + +where $E_{T}$ is the hardening tangent modulus, + +$E$ is the tangent modulus, + +$\sigma_{Y}$ is the yield stress, + +$c$ is the cohesion, + +$\phi$ is the friction angle. + +71-80 PROPS(NUMAT,8) Friction angle ‘ $\phi$ ’. + + + +3rd Card MATERIAL PROPERTIES CARD—(b) (3E10.4) + +Cols. 1–10 PROPS(NUMAT,9) Fluidity parameter, γ. + +11-20 PROPS(NUMAT,10)Exponent, $\delta$ . + +21-30 PROPS(NUMAT,11)NFLOW code + +(NFLOW = 1—Power law, + +NFLOW ≠ 1—Exponential law). + +CARD SET 8 TIME INTEGRATION CONTROL CARD (1115)—One card. + +Cols. 1–5 NSTEP Total number of time steps. + +6-10 NOUTD Writes displacement and stress history of required points on tapes 10 and 11 respectively at NOUTD timesteps. + +11-15 NOUTP Output for displacements and stresses at every NOUTP step (NOUTP $\leqslant 500$ ). + +16-20 NREQD Number of nodes for selective output of displacements at NOUTD steps. + +21-25 NREQS Number of integration points for selective output of stresses at every NOUTP step. + +26-30 NACCE Number of acceleration ordinates (If IFUNC $\neq 0$ , NACCE is not used, then leave blank). + +31-35 IFUNC Time function code: + +IFUNC = 0 Acceleration time history, IFUNC = 1 Heaviside function, $f(t) = 1.0$ , + +IFUNC = 2 Harmonic excitation, $f(t)$ $= a_{0} + b_{0} \sin \omega t.$ + +36-40 IFIXD Indicator for excitation: + +IFIXD = 0, Horizontal acceleration read from tape 7, Vertical acceleration read from tape 12. + +IFIXD = 1, Vertical acceleration read from tape 12, + +IFIXD = 2, Horizontal acceleration read from tape 7. (If IFUNC ≠ 0 IFIXD is not used, then leave blank.) + +41-45 MITER Maximum number of iterations. This variable is not used in DYNPAK, so leave blank. + + + +46-50 KSTEP + +Number of steps after which the stiffness matrix is reformed. Not used in DYN-PAK, leave blank. + +51-55 IPRED + += 1 Standard algorithm, + += 2 Modified algorithm. + +CARD SET 9 TIME INTEGRATION PARAMETERS CARD (8F10.3)—Two cards. + +1st Card + +Cols. 1-10 DTIME + +11-20 DTEND + +21-30 DTREC + +31-40 AALFA + +41-50 BEETA + +51-60 DELTA + +61-70 GAAMA + +71-80 AZERO +2nd Card +1-10 BZERO +11-20 OMEGA +21-30 TOLER + +Time step length. + +Time at the end of the excitation force. + +Time step of acceleration records. + +$\alpha = \text{Damping} \quad \text{parameter}, \quad C = \alpha M,$ $\alpha = 2\xi_i\omega_i.$ + +$\beta = \text{Damping parameter}, C = \beta K.$ $(\alpha + \beta \omega_{i}^{2} = 2\omega_{i}\xi_{i}, \text{not used in DYNPAK})$ + +Newmark's integration parameter $(\delta = 0.25 (\gamma + 0.5)^2$ , not used in DYN-PAK). + +Newmark's integration parameter ( $\gamma \geqslant 0.5$ for stable solution, not used in DYN-PAK). + +Constants for harmonic excitation $f(t) = a_{0} + b_{0} \sin \omega t$ . + +Specified tolerance (Not used in DYN-PAK). + +CARD SET 10 CARD FOR NODAL POINTS FOR WHICH DISPLACEMENT HISTORY IS REQUIRED (16I5)—Total of NREQD nodes. + +Cols. 1-5 NPRQD(1) + +6-10 NPRQD(2) + +First nodal point at which displacement history is required. + +Second nodal point at which displacement history is required. + +11-15 . + + + +CARD SET 11 CARD FOR INTEGRATION POINTS FOR WHICH STRESS HISTORY IS REQUIRED (16I5)—Total of NREQS integration points. + +Cols. 1-5 NGRQS(1) + +First integration point at which stress history is required. + +6-10 NGRQS(2) + +Second integration point at which stress history is required. + +11-15 . +. . +. . +. . +. . + +CARD SET 12 IMPLICIT-EXPLICIT ELEMENT INDICATOR CARDS (16I5). Number of cards depends on number of elements. For each 16 elements one card is needed. In DYNPAK, INTGR(IELEM) is 2 for every element. + +INTGR(IELEM) = 1, Implicit element. + +INTGR(IELEM) = 2, Explicit element. + +CARD SET 13 INITIAL DISPLACEMENT CARDS (I5,2F10.5)—One card for each node. If all displacements are zero, read data for last node. + +Cols. 1–5 NGASH + +Nodal point. + +6-15 XGASH + +Initial $x$ -displacement. + +16-25 YGASH + +Initial y-displacement. + +CARD SET 14 INITIAL VELOCITY CARDS (15,2F10.5)—One card for each node. If all velocities are zero, read data for last node. + +Cols. 1–5 NGASH + +Nodal point. + +6-15 XGASH + +Initial $x$ -velocity. + +16-25 YGASH + +Initial y-velocity. + +CARD SET 15 PREVIOUS LOAD STATE CARDS (15,2F10.3)—One card for one node, a total of NNODE cards. Data for the last nodal point should always be read even when it is not loaded. If NPREV = 0 then omit this set of data. + +Cols. 1-5 NGASH + +Nodal point. + +6-15 XGASH + +Equivalent nodal load in $x$ direction. + +16-25 YGASH + +Equivalent nodal load in $y$ direction. + +CARD SET 16 PREVIOUS STRESS STATE CARD (I5,4F10.3)—One card for one integration point. Total of (NELEM\*NGAUS\*NGAUS) cards. If NPREV = 0 omit this set of data. + + + +
Cols.1-5KGAUSIntegration point.
6-15STRESS(1)Initial stress, $\sigma_x$ or $\sigma_r$ .
16-25STRESS(2)Initial stress, $\sigma_y$ or $\sigma_z$ .
26-35STRESS(3)Initial stress, $\gamma_{xy}$ or $\gamma_{rz}$ .
36-45STRESS(4)Initial stress, $\sigma_z$ or $\sigma_\theta$ .
+ +CARD SET 17 LOAD TITLE CARD (10A4)—One card. + +Cols. 1–40 Title of load applied—limited to 40 alphanumeric characters. + +CARD SET 18 LOAD INDICATOR CARD (415)—One card. + +
Cols.1-5IPLODPoint load indicator.
6-10IGRAVGravity load indicator.
11-15IEDGEEdge load indicator.
16-20ITEMPTemperature load indicator.
+ +CARD SET 19 POINT LOAD CARD (I5,2F10.3)—One card for each node. Data for the last node must be specified at the end. If IPLOD = 0 then omit this set of data. + +
Cols.1-5LODPTNode number.
6-15POINT(1)Load in x-direction.
16-25POINT(2)Load in y-direction.
+ +CARD SET 20 GRAVITY LOAD CARD (2F10.3)—One card only. If IGRAV = 0 then omit this set of data. + +
Cols. 1-10 THETAAngle of gravity axis to the positive y axis.
11-20 GRAVYGravity constant.
+ +CARD SET 21 NUMBER OF PRESSURE EDGE CARD (I5)—One card. If IEDGE = 0, then omit card sets 21 and 22. + +Cols. 1-5 NEDGE Number of loaded edges. + +CARD SET 22 PRESSURE CARDS—Two cards for each pressure loaded edge. + +1st Card PRESSURE NODES CARD (415)—One card for each edge. Total of NEDGE cards. + +
Cols. 1-5 NEASSElement number with edge load.
Cols. 6-10 NOPRS(1)Edge nodes read in anticlockwise sequence.
11-15 NOPRS(2)
16-20 NOPRS(3)
+ + + +2nd Card PRESSURE CARD (6F10.3)—One card for each edge. Total of NEDGE cards. A pressure normal to a face is assumed to be positive if it acts in a direction into the element. A tangential load is assumed to be positive if it acts in an anticlockwise direction with respect to the loanedWW positive if it acts in an anticlockwise direction with respect to the loaded element. + +Cols. 1-10 PRESS(1,1) } Normal component of edge load for each node. +11-20 PRESS(2,1) +21-30 PRESS(3,1) +31-40 PRESS(1,2) +41-50 PRESS(2,2) +51-60 PRESS(3,2) } Tangential component of edge load for each node. + +CARD SET 24 TEMPERATURE CARDS (I5, F10.3)—One card for each node. The last card must be for the highest numbered node. If $\text{ITEMP} = 0$ , omit this set of data. + +Cols. 1–5 NODPT Node number. +6–15 TEMPE Nodal temperature. + +CARD SET 25 CONCENTRATED MASSES (I5,2F10.3)—One card for each node. Total of NCONM cards. If NCONM = 0, omit this set of data. + +Cols. 1–5 IPOIN Current nodal point with concentrated mass. +6–15 XCMAS Concentrated mass associated with the x-direction. +16–25 YCMAS Concentrated mass associated with the y-direction. + + + +# Appendix IV + +# Sample input data and line printer output for one – and two-dimensional applications + +In this appendix input data and line printer output are provided for a selection of the numerical examples presented in the text. This information will be of assistance to readers who wish to implement the programs contained in the book on their own computer. For economy of space, presentation is limited to one example from each area of application. Also in some cases the line printer output is edited for the same reason. + +# A.4.1 Solution of one-dimensional quasiharmonic problem by direct iteration. Example of Section 3.9.3, Fig. 3.3 + +Input data + +
1-D QUASIHARMONIC EXAMPLE, SECTION 3.9.3, FIG. 3.3
11102112111
110.0
1121
2231
3341
4451
5561
6671
7781
8891
99101
1010111
10.0
21.0
32.0
43.0
54.0
65.0
76.0
87.0
98.0
109.0
1110.0
110.0
1111.0
100.00.0
2011.00.5
+ + + +Line printer output + +
1-D QUASIHARMONIC EXAMPLE, SECTION 3.9.3, FIG. 3.3
NPOIN = 11 NELEM = 10 NBOUN = 2 NMATS = 1
NPROP = 1 NNODE = 2 NINCS = 1 NALGO = 1
NDOFN = 1
MATERIAL PROPERTIES
1 10.00000
EL NODES MAT.
1 1 2 1
2 2 3 1
3 3 4 1
4 4 5 1
5 5 6 1
6 6 7 1
7 7 8 1
8 8 9 1
9 9 10 1
10 10 11 1
NODE COORD.
1 0.00000
2 1.00000
3 2.00000
4 3.00000
5 4.00000
6 5.00000
7 6.00000
8 7.00000
9 8.00000
10 9.00000
11 10.00000
RES. NODE CODE PRES. VALUES
1 1 0.00000
11 1 1.00000
ELEMENT NODAL LOADS
1 0.00000 0.00000
2 0.00000 0.00000
3 0.00000 0.00000
4 0.00000 0.00000
5 0.00000 0.00000
6 0.00000 0.00000
7 0.00000 0.00000
8 0.00000 0.00000
9 0.00000 0.00000
10 0.00000 0.00000
IINCS = 1 NITER = 20 NOUTP = 1 FACTO = 0.100000E 01 TOLER = 0.500000E 00
CONVERGENCE CODE = 1 NORM OF RESIDUAL SUM RATIO = 0.000000E 00
NODE DISPL. REACTIONS
1 0.000000E 00 -0.100000E 01
2 0.100000E 00 0.000000E 00
3 0.200000E 00 0.000000E 00
4 0.300000E 00 0.000000E 00
5 0.400000E 00 0.000000E 00
6 0.500000E 00 0.000000E 00
7 0.600000E 00 0.000000E 00
8 0.700000E 00 0.000000E 00
9 0.800000E 00 0.000000E 00
10 0.900000E 00 0.000000E 00
11 0.100000E 01 0.100000E 01
ELEMENT STRESSES PL.STRAIN
1 0.000000E 00 0.000000E 00
2 0.000000E 00 0.000000E 00
3 0.000000E 00 0.000000E 00
4 0.000000E 00 0.000000E 00
5 0.000000E 00 0.000000E 00
diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_055.md b/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_055.md new file mode 100644 index 00000000..b330bef3 --- /dev/null +++ b/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_055.md @@ -0,0 +1,282 @@ + + +
60.000000E 000.000000E 00
70.000000E 000.000000E 00
80.000000E 000.000000E 00
90.000000E 000.000000E 00
100.000000E 000.000000E 00
CONVERGENCE CODE =1NORM OF RESIDUAL SUM RATIO = 0.706275E 02
CONVERGENCE CODE =1NORM OF RESIDUAL SUM RATIO = 0.393376E 02
CONVERGENCE CODE =1NORM OF RESIDUAL SUM RATIO = 0.983804E 01
CONVERGENCE CODE =1NORM OF RESIDUAL SUM RATIO = 0.801219E 01
CONVERGENCE CODE =1NORM OF RESIDUAL SUM RATIO = 0.472308E 01
CONVERGENCE CODE =1NORM OF RESIDUAL SUM RATIO = 0.127390E 01
CONVERGENCE CODE =1NORM OF RESIDUAL SUM RATIO = 0.974302E 00
CONVERGENCE CODE =1NORM OF RESIDUAL SUM RATIO = 0.574815E 00
CONVERGENCE CODE =0NORM OF RESIDUAL SUM RATIO = 0.153335E 00
NODEDISPL.REACTIONS
10.000000E 00-0.600000E 01
20.260555E 000.000000E 00
30.399999E 000.000000E 00
40.508276E 000.000000E 00
50.599999E 000.000000E 00
60.681025E 000.000000E 00
70.754400E 000.000000E 00
80.821954E 000.000000E 00
90.884886E 000.000000E 00
100.944031E 000.000000E 00
110.100000E 010.600000E 01
ELEMENTSTRESSESPL.STRAIN
10.000000E 000.000000E 00
20.000000E 000.000000E 00
30.000000E 000.000000E 00
40.000000E 000.000000E 00
50.000000E 000.000000E 00
60.000000E 000.000000E 00
70.000000E 000.000000E 00
80.000000E 000.000000E 00
90.000000E 000.000000E 00
100.000000E 000.000000E 00
+ +# A.4.2 Solution of one-dimensional elasto-plastic problem. Example of Section 3.12.3, Fig. 3.9 + +Input data + +
1-D ELASTO-PLASTIC EXAMPLE, SECTION 3.12.3,FIG. 3.9
111022421631
110000.01.05.01000.0
210000.02.07.52000.0
1121
2231
3341
4451
5561
6672
7782
8892
99102
1010112
10.0
21.0
32.0
43.0
+ + + +
54.0
65.0
74.0
83.0
92.0
101.0
110.0
110.0
1110.0
50.010.0
100.00.0
3021.250.5
3020.250.5
3020.250.5
3020.250.5
3020.250.5
3020.250.5
302-3.50.5
302-0.250.5
302-0.250.5
302-0.250.5
302-0.250.5
302-0.250.5
302-0.250.5
312-0.250.5
302-0.250.5
302-0.250.5
302-0.250.5
+ +Line printer output + +
1-D ELASTO-PLASTIC EXAMPLE, SECTION 3.12.3,FIG. 3.9
NPOIN = 11NELEM = 10NBOUN = 2NMATS = 2
NPROP = 4NNODE = 2NINCS = 16NALGO = 3
NDOFN = 1
MATERIAL PROPERTIES
110000.000001.000005.000001000.00000
210000.000002.000007.500002000.00000
ELNODESMAT.
1121
2231
3341
4451
5561
6672
7782
8892
99102
1010112
NODECOORD.
10.00000
21.00000
32.00000
43.00000
54.00000
65.00000
74.00000
83.00000
92.00000
101.00000
110.00000
RES.NODECODEPRES.VALUES
110.00000
1110.00000
+ + + +
ELEMENTNODAL LOADS
10.000000.00000
20.000000.00000
30.000000.00000
40.000000.00000
50.0000010.00000
60.000000.00000
70.000000.00000
80.000000.00000
90.000000.00000
100.000000.00000
IINCS = 1 NITER = 30 NOUTP = 2 FACTO = 0.125000E 01 TOLER = 0.500000E 00
ITERATION NUMBER = 1
CONVERGENCE CODE = 0 NORM OF RESIDUAL SUM RATIO = 0.629197E-08
NODEDISPL.REACTIONS
10.000000E 00-0.416667E 01
20.416667E-030.000000E 00
30.833333E-030.000000E 00
40.125000E-020.000000E 00
50.166667E-020.000000E 00
60.208333E-020.000000E 00
70.166667E-020.000000E 00
80.125000E-020.000000E 00
90.833333E-030.000000E 00
100.416667E-030.000000E 00
110.000000E 00-0.833333E 01
ELEMENTSTRESSESPL.STRAIN
10.416667E 010.000000E 00
20.416667E 010.000000E 00
30.416667E 010.000000E 00
40.416667E 010.000000E 00
50.416667E 010.000000E 00
60.416667E 010.000000E 00
70.416667E 010.000000E 00
80.416667E 010.000000E 00
90.416667E 010.000000E 00
100.416667E 010.000000E 00
....
....
....
....
IINCS = 3 NITER = 30 NOUTP = 2 FACTO = 0.250000E 00 TOLER = 0.500000E 00
ITERATION NUMBER = 1
CONVERGENCE CODE = 1 NORM OF RESIDUAL SUM RATIO = 0.490863E 01
NODEDISPL.REACTIONS
10.000000E 00-0.583333E 01
20.583333E-030.000000E 00
30.116667E-020.000000E 00
40.175000E-020.000000E 00
50.233333E-020.000000E 00
60.291667E-020.000000E 00
70.233333E-020.000000E 00
80.175000E-020.000000E 00
90.116667E-020.000000E 00
100.583333E-030.000000E 00
110.000000E 00-0.116667E 02
ELEMENTSTRESSESPL.STRAIN
10.507576E 010.757576E-04
20.507576E 010.757576E-04
30.507576E 010.757576E-04
40.507576E 010.757576E-04
+ + + +```txt +5 0.507576E 01 0.757576E-04 +6 0.583333E 01 0.000000E 00 +7 0.583333E 01 0.000000E 00 +8 0.583333E 01 0.000000E 00 +9 0.583333E 01 0.000000E 00 +10 0.583333E 01 0.000000E 00 +ITERATION NUMBER = 2 +CONVERGENCE CODE = 1 NORM OF RESIDUAL SUM RATIO = 0.147757E 01 +NODE DISPL. REACTIONS +1 0.000000E 00 -0.532828E 01 +2 0.608586E-03 0.000000E 00 +3 0.121717E-02 0.000000E 00 +4 0.182576E-02 0.000000E 00 +5 0.243434E-02 0.000000E 00 +6 0.304293E-02 0.000000E 00 +7 0.243434E-02 0.000000E 00 +8 0.182576E-02 0.000000E 00 +9 0.121717E-02 0.000000E 00 +10 0.608586E-03 0.000000E 00 +11 0.000000E 00 -0.121717E 02 +ELEMENT STRESSES PL.STRAIN +1 0.509871E 01 0.987144E-04 +2 0.509871E 01 0.987144E-04 +3 0.509871E 01 0.987144E-04 +4 0.509871E 01 0.987144E-04 +5 0.509871E 01 0.987144E-04 +6 0.608586E 01 0.000000E 00 +7 0.608586E 01 0.000000E 00 +8 0.608586E 01 0.000000E 00 +9 0.608586E 01 0.000000E 00 +10 0.608586E 01 0.000000E 00 +ITERATION NUMBER = 3 +CONVERGENCE CODE = 0 NORM OF RESIDUAL SUM RATIO = 0.446758E 00 +NODE DISPL. REACTIONS +1 0.000000E 00 -0.517524E 01 +2 0.616238E-03 0.000000E 00 +3 0.123248E-02 0.000000E 00 +4 0.184871E-02 0.000000E 00 +5 0.246495E-02 0.000000E 00 +6 0.308119E-02 0.000000E 00 +7 0.246495E-02 0.000000E 00 +8 0.184871E-02 0.000000E 00 +9 0.123248E-02 0.000000E 00 +10 0.616238E-03 0.000000E 00 +11 0.000000E 00 -0.123248E 02 +ELEMENT STRESSES PL.STRAIN +1 0.510567E 01 0.105671E-03 +2 0.510567E 01 0.105671E-03 +3 0.510567E 01 0.105671E-03 +4 0.510567E 01 0.105671E-03 +5 0.510567E 01 0.105671E-03 +6 0.616238E 01 0.000000E 00 +7 0.616238E 01 0.000000E 00 +8 0.616238E 01 0.000000E 00 +9 0.616238E 01 0.000000E 00 +10 0.616238E 01 0.000000E 00 +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . etc. +``` + + + +# A.4.3 Solution of one-dimensional elasto-viscoplastic problem. Example of Section 4.12, Fig. 4.6 + +Input data +```txt +1-D ELASTO VISCO-PLASTIC EXAMPLE, SECTION 4.12, FIG. 4.6 +2 1 1 1 5 2 1 3 1 +1 10000.0 1.0 10.0 5000.0 0.001 +1 1 2 1 +1 0.0 +2 10.0 +1 1 0.0 +1 0.0 15.0 +0.05 0.025 1.5 +90 2 1.0 0.1 +``` + +Line printer output +```txt +1-D ELASTO VISCO-PLASTIC EXAMPLE, SECTION 4.12, FIG. 4.6 +NPOIN = 2 NELEM = 1 NBOUN = 1 NMATS = 1 +NPROP = 5 NNODE = 2 NINCS = 1 NALGO = 3 +NDOFN = 1 +MATERIAL PROPERTIES +1 10000.00000 1.00000 10.00000 5000.00000 0.00100 +EL NODES MAT. +1 1 2 1 +NODE COORD. +1 0.00000 +2 10.00000 +RES.NODE CODE PRES.VALUES +1 1 0.00000 +ELEMENT NODAL LOADS +1 0.00000 15.00000 +TAUFT = 0.500000E-01 DTINT = 0.250000E-01 FTIME = 0.150000E 01 +IINCS = 1 NSTEP = 90 NOUTP = 2 FACTO = 0.100000E 01 TOLER = 0.100000E 00 +TOTAL TIME = 0.000000E 00 +CONVERGENCE CODE = 999 NORM OF RESIDUAL SUM RATIO = 0.100000E 03 +NODE DISPL. REACTIONS +1 0.000000E 00 -0.150000E 02 +2 0.150000E-01 0.000000E 00 +ELEMENT STRESSES PL.STRAIN +1 0.150000E 02 0.000000E 00 +TOTAL TIME = 0.250000E-01 +CONVERGENCE CODE = 1 NORM OF RESIDUAL SUM RATIO = 0.650000E 02 +NODE DISPL. REACTIONS +1 0.000000E 00 -0.150000E 02 +2 0.162500E-01 0.000000E 00 +ELEMENT STRESSES PL.STRAIN +1 0.150000E 02 0.125000E-03 +TOTAL TIME = 0.435714E-01 +CONVERGENCE CODE = 999 NORM OF RESIDUAL SUM RATIO = 0.682500E 02 +NODE DISPL. REACTIONS +1 0.000000E 00 -0.150000E 02 +2 0.170625E-01 0.000000E 00 +ELEMENT STRESSES PL.STRAIN +1 0.150000E 02 0.206250E-03 +TOTAL TIME = 0.650675E-01 +CONVERGENCE CODE = 999 NORM OF RESIDUAL SUM RATIO = 0.716625E 02 +NODE DISPL. REACTIONS +1 0.000000E 00 -0.150000E 02 +2 0.179156E-01 0.000000E 00 +``` + + + +```csv +ELEMENT STRESSES PL.STRAIN +1 0.150000E 02 0.291562E-03 +TOTAL TIME = 0.903564E-01 +CONVERGENCE CODE = 999 NORM OF RESIDUAL SUM RATIO = 0.752456E 02 +NODE DISPL. REACTIONS +1 0.000000E 00 -0.150000E 02 +2 0.188114E-01 0.000000E 00 +ELEMENT STRESSES PL.STRAIN +1 0.150000E 02 0.381141E-03 +TOTAL TIME = 0.120753E 00 +CONVERGENCE CODE = 999 NORM OF RESIDUAL SUM RATIO = 0.790079E 02 +NODE DISPL. REACTIONS +1 0.000000E 00 -0.150000E 02 +2 0.197520E-01 0.000000E 00 +ELEMENT STRESSES PL.STRAIN +1 0.150000E 02 0.475198E-03 +TOTAL TIME = 0.158390E 00 +CONVERGENCE CODE = 999 NORM OF RESIDUAL SUM RATIO = 0.829583E 02 +NODE DISPL. REACTIONS +1 0.000000E 00 -0.150000E 02 +2 0.207396E-01 0.000000E 00 +ELEMENT STRESSES PL.STRAIN +1 0.150000E 02 0.573958E-03 +TOTAL TIME = 0.207070E 00 +CONVERGENCE CODE = 999 NORM OF RESIDUAL SUM RATIO = 0.871062E 02 +NODE DISPL. REACTIONS +1 0.000000E 00 -0.150000E 02 +2 0.217766E-01 0.000000E 00 +ELEMENT STRESSES PL.STRAIN +1 0.150000E 02 0.677655E-03 +TOTAL TIME = 0.274627E 00 +CONVERGENCE CODE = 1 NORM OF RESIDUAL SUM RATIO = 0.865247E 02 +NODE DISPL. REACTIONS +1 0.000000E 00 -0.150000E 02 +2 0.228654E-01 0.000000E 00 +ELEMENT STRESSES PL.STRAIN +1 0.150000E 02 0.786538E-03 +TOTAL TIME = 0.375962E 00 +CONVERGENCE CODE = 1 NORM OF RESIDUAL SUM RATIO = 0.640271E 02 +NODE DISPL. REACTIONS +1 0.000000E 00 -0.150000E 02 +2 0.239469E-01 0.000000E 00 +ELEMENT STRESSES PL.STRAIN +1 0.150000E 02 0.894694E-03 +TOTAL TIME = 0.527964E 00 +CONVERGENCE CODE = 1 NORM OF RESIDUAL SUM RATIO = 0.230485E 02 +NODE DISPL. REACTIONS +1 0.000000E 00 -0.150000E 02 +2 0.247473E-01 0.000000E 00 +ELEMENT STRESSES PL.STRAIN +1 0.150000E 02 0.974728E-03 +TOTAL TIME = 0.755969E 00 +CONVERGENCE CODE = 0 NORM OF RESIDUAL SUM RATIO = 0.000000E 00 +NODE DISPL. REACTIONS +1 0.000000E 00 -0.150000E 02 +2 0.250354E-01 0.000000E 00 +ELEMENT STRESSES PL.STRAIN +1 0.150000E 02 0.100354E-02 +``` + + + +# A.4.4 Solution of elasto-plastic layered Timoshenko beam. Example of Section 5.5.6, Fig. 5.11 + +Input data + +
1-DEPTIMOSHENKOLAYEREDBEAMEXAMPLE,SECTION 5.5.6,FIG. 5.11
11102117214226
1
210.053.84440.250000.0
200.0200.020.010.0
40.010.040.010.0
40.010.040.0200.0
20.0
1121
2231
3341
4451
5561
6671
7781
8891
99101
1010111
10.0
2300.0
3600.0
4900.0
51200.0
61500.0
71800.0
82100.0
92400.0
102700.0
113000.0
110.010.0
1110.010.0
168.850000.0000068.850000.00000
268.850000.0000068.850000.00000
368.850000.0000068.850000.00000
468.850000.0000068.850000.00000
568.850000.0000068.850000.00000
668.850000.0000068.850000.00000
768.850000.0000068.850000.00000
868.850000.0000068.850000.00000
968.850000.0000068.850000.00000
1068.850000.0000068.850000.00000
10020.300.50
10020.200.50
10020.100.50
10020.100.50
10020.050.50
10020.050.50
10020.050.50
10020.050.50
10020.050,50
10020.020.50
10020.020.50
10020.020.50
10020.020.50
10020.010.50
10020.010.50
+ + + +Line printer output +1-D EP TIMOSHENKO LAYERED BEAM EXAMPLE, SECTION 5.5.6, FIG. 5.11 + +
NPOIN =11NELEM =10NBOUN =2NMATS =1
NPROP =17NNODE =2NINCS =14NALGO =2
+ +NDOFN = 2 NLAYR = 6 +MATERIAL PROPERTIES + +
210.0000053.844400.250000.00000
200.00000200.0000020.0000010.00000
40.0000010.0000040.0000010.00000
40.0000010.0000040.00000200.00000
20.00000
+ +
ELNODESMAT.
1121
2231
3341
4451
5561
6671
7781
8891
99101
1010111
+ +
NODECOORD.
10.00000
2300.00000
3600.00000
4900.00000
51200.00000
61500.00000
71800.00000
82100.00000
92400.00000
102700.00000
113000.00000
+ +
RES.NODECODEPRES.VALUESCODEPRES.VALUES
110.0000010.00000
1110.0000010.00000
ELEMENTNODAL LOADS
168.850000.0000068.850000.00000
268.850000.0000068.850000.00000
368.850000.0000068.850000.00000
468.850000.0000068.850000.00000
568.850000.0000068.850000.00000
668.850000.0000068.850000.00000
768.850000.0000068.850000.00000
868.850000.0000068.850000.00000
968.850000.0000068.850000.00000
1068.850000.0000068.850000.00000
IINCS =1NITER = 100NOUTP =2FACTO = 0.300000E 00TOLER = 0.500000E 00
+ +ITERATION NUMBER = 1 + +CONVERGENCE CODE = 0 NORM OF RESIDUAL SUM RATIO = 0.113611E-07 +NODE DISPLACEMENTS REACTIONS + +
10.000000E 00-0.206550E 030.000000E 00-0.102242E 06
20.342210E 000.000000E 000.156214E-020.000000E 00
30.972874E 000.000000E 000.208286E-020.000000E 00
40.161862E 010.000000E 000.182250E-020.000000E 00
50.208417E 010.000000E 000.104143E-020.000000E 00
+ + + +
60.225237E 010.000000E 00-0.255548E-120.000000E 00
70.208417E 010.000000E 00-0.104143E-020.000000E 00
80.161862E 010.000000E 00-0.182250E-020.000000E 00
90.972874E 000.000000E 00-0.208286E-020.000000E 00
100.342210E 000.000000E 00-0.156214E-020.000000E 00
110.000000E 00-0.206550E 030.000000E 000.102242E 06
+ +ELEMENT STRESSES + +
1-0.743580E050.185895E03
2-0.247860E050.144585E03
30.123930E050.103275E03
40.371790E050.619650E02
50.495720E050.206550E02
60.495720E05-0.206550E02
70.371790E05-0.619650E02
80.123930E05-0.103275E03
9-0.247860E05-0.144585E03
10-0.743580E05-0.185895E03
+ +
....
....
....
....
+ +IINCS = 6 NITER = 100 NOUTP = 2 FACTO = 0.500000E-01 TOLER = 0.500000E 00 + +ITERATION NUMBER = 1 + +CONVERGENCE CODE = 1 NORM OF RESIDUAL SUM RATIO = 0.464588E 01 + +NODE DISPLACEMENTS REACTIONS + +
10.000000E 00-0.550800E 030.000000E 00-0.272646E 06
20.912561E 000.000000E 000.416571E-020.000000E 00
30.259433E 010.000000E 000.555429E-020.000000E 00
40.431631E 010.000000E 000.486000E-020.000000E 00
50.555778E 010.000000E 000.277714E-020.000000E 00
60.600632E 010.000000E 00-0.645258E-130.000000E 00
70.555778E 010.000000E 00-0.277714E-020.000000E 00
80.431631E 010.000000E 00-0.486000E-020.000000E 00
90.259433E 010.000000E 00-0.555429E-020.000000E 00
100.912561E 000.000000E 00-0.416571E-020.000000E 00
110.000000E 00-0.550800E 030.000000E 000.272646E 06
+ +ELEMENT STRESSES + +
1-0.189331E060.495720E03
2-0.660960E050.385560E03
30.330480E050.275400E03
40.991440E050.165240E03
50.132192E060.550800E02
60.132192E06-0.550800E02
70.991440E05-0.165240E03
80.330480E05-0.275400E03
9-0.660960E05-0.385560E03
10-0.189331E06-0.495720E03
+ +ITERATION NUMBER = 2 + +CONVERGENCE CODE = 0 NORM OF RESIDUAL SUM RATIO = 0.210144E-08 + +NODE DISPLACEMENTS REACTIONS + +
10.000000E 00-0.550800E 030.000000E 00-0.265108E 06
20.100758E 010.000000E 000.479915E-020.000000E 00
30.285562E 010.000000E 000.602936E-020.000000E 00
40.469637E 010.000000E 000.517672E-020.000000E 00
50.600911E 010.000000E 000.293550E-020.000000E 00
60.648140E 010.000000E 00-0.118097E-120.000000E 00
70.600911E 010.000000E 00-0.293550E-020.000000E 00
80.469637E 010.000000E 00-0.517672E-020.000000E 00
90.285562E 010.000000E 00-0.602936E-020.000000E 00
+ + + +FINITE ELEMENTS IN PLASTICITY + +
100.100758E 010.000000E 00-0.479915E-020.000000E 00
110.000000E 00-0.550800E 030.000000E 000.265108E 06
ELEMENTSTRESSES
1-0.190750E 060.495720E 03
2-0.585581E 050.385560E 03
30.405859E 050.275400E 03
40.106682E 060.165240E 03
50.139730E 060.550800E 02
60.139730E 06-0.550800E 02
70.106682E 06-0.165240E 03
80.405859E 05-0.275400E 03
9-0.585581E 05-0.385560E 03
10-0.190750E 06-0.495720E 03
...
...
...
...
IINCS = 11 NITER = 100 NOUTP = 2 FACTO = 0.200000E-01 TOLER = 0.500000E 00
ITERATION NUMBER = 1
CONVERGENCE CODE = 1 NORM OF RESIDUAL SUM RATIO = 0.149229E 01
NODEDISPLACEMENTSREACTIONS
10.000000E 00-0.660960E 030.000000E 00-0.287981E 06
20.486620E 010.000000E 000.301397E-010.000000E 00
30.143031E 020.000000E 000.309826E-010.000000E 00
40.235411E 020.000000E 000.293260E-010.000000E 00
50.319556E 020.000000E 000.260032E-010.000000E 00
60.358944E 020.000000E 000.210285E-090.000000E 00
70.319556E 020.000000E 00-0.260032E-010.000000E 00
80.235411E 020.000000E 00-0.293260E-010.000000E 00
90.143031E 020.000000E 00-0.309826E-010.000000E 00
100.486620E 010.000000E 00-0.301397E-010.000000E 00
110.000000E 00-0.660960E 030.000000E 000.287981E 06
ELEMENTSTRESSES
1-0.196000E 060.594864E 03
2-0.401209E 050.462672E 03
30.788519E 050.330480E 03
40.158167E 060.198288E 03
50.196000E 060.660960E 02
60.196000E 06-0.660960E 02
70.158167E 06-0.198288E 03
80.788519E 05-0.330480E 03
9-0.401209E 05-0.462672E 03
10-0.196000E 06-0.594864E 03
ITERATION NUMBER = 2
CONVERGENCE CODE = 999 NORM OF RESIDUAL SUM RATIO = 0.562938E 10
NODEDISPLACEMENTSREACTIONS
10.000000E 00-0.656460E 030.000000E 00-0.284525E 06
2-0.227149E 080.000000E 00-0.151432E 060.000000E 00
3-0.681446E 080.000000E 00-0.151432E 060.000000E 00
4-0.113574E 090.000000E 00-0.151432E 060.000000E 00
5-0.159004E 090.000000E 00-0.151432E 060.000000E 00
6-0.163102E 090.000000E 000.124115E 060.000000E 00
7-0.126424E 090.000000E 000.120404E 060.000000E 00
8-0.903028E 080.000000E 000.120404E 060.000000E 00
9-0.541817E 080.000000E 000.120404E 060.000000E 00
10-0.180606E 080.000000E 000.120404E 060.000000E 00
110.000000E 00-0.656351E 030.000000E 000.284576E 06
ELEMENTSTRESSES
10.719122E 130.589934E 03
2-0.390314E 050.457742E 03
diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_056.md b/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_056.md new file mode 100644 index 00000000..2781d896 --- /dev/null +++ b/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_056.md @@ -0,0 +1,83 @@ + + +
30.784888E050.327522E03
40.156624E060.197302E03
5-0.131161E140.684992E02
60.196000E06-0.616594E02
70.156896E06-0.197302E03
80.787157E05-0.325057E03
9-0.388044E05-0.458235E03
100.573122E13-0.590427E03
ITERATION NUMBER = 3
CONVERGENCE CODE = 999NORM OF RESIDUAL SUM RATIO = 0.247769E 12
NODEDISPLACEMENTSREACTIONS
10.000000E000.386547E110.000000E000.131941E 14
2-0.256689E180.000000E00-0.171126E160.000000E 00
3-0.770066E180.000000E00-0.171126E160.000000E 00
4-0.128344E190.000000E00-0.171126E160.000000E 00
5-0.179682E190.000000E00-0.171126E160.000000E 00
6-0.707142E180.000000E000.897579E160.000000E 00
70.559323E180.000000E00-0.532688E150.000000E 00
80.399516E180.000000E00-0.532688E150.000000E 00
90.239710E180.000000E00-0.532688E150.000000E 00
100.799033E170.000000E00-0.532688E150.000000E 00
110.000000E000.316249E090.000000E00-0.594731E 13
ELEMENTSTRESSES
10.719122E13-0.381105E11
2-0.195980E06-0.169380E11
3-0.195887E06-0.846899E10
4-0.195820E060.197302E03
5-0.131161E140.684992E02
60.196000E06-0.616594E02
70.196011E060.148207E11
80.195954E060.211725E10
90.195971E060.635174E10
10-0.253560E230.211725E10
ITERATION NUMBER = 4
CONVERGENCE CODE = 999NORM OF RESIDUAL SUM RATIO = 0.576146E 14
NODEDISPLACEMENTSREACTIONS
10.000000E000.386547E110.000000E000.131941E 14
20.808314E270.000000E000.538876E250.000000E 00
30.808244E270.000000E00-0.538923E250.000000E 00
4-0.940584E280.000000E00-0.627047E260.000000E 00
5-0.116832E250.000000E000.125402E270.000000E 00
60.679753E250.000000E00-0.125349E270.000000E 00
7-0.395493E260.000000E000.125040E270.000000E 00
80.230105E270.000000E00-0.123243E270.000000E 00
9-0.133880E280.000000E000.112783E270.000000E 00
100.778935E280.000000E00-0.519290E260.000000E 00
110.000000E00-0.198094E210.000000E000.507119E 23
ELEMENTSTRESSES
1-0.255902E33-0.381105E11
2-0.195980E060.241990E18
3-0.195887E06-0.290992E21
4-0.195820E060.197302E03
50.119358E35-0.124894E21
6-0.119186E35-0.254618E21
70.196011E06-0.109122E21
80.195954E060.109122E21
90.195971E060.145496E21
10-0.253560E230.211725E10
+ + + +# A.4.5 Solution of two-dimensional elasto-plastic problem. Example of Section 7.9, Fig. 7.12 + +Input data + +
2-D ELASTO-PLASTIC EXAMPLE, SECTION 7.9, FIG 7.12
51121828122213
1118121314932
21391415161054
315101617181176
411219232425201413
511420252627211615
611621272829221817
712330343536312524
812531363738322726
912732383940332928
1013441454647423635
1113642474849433837
1213843495051444039
1100.00.02770.0121.243
296.59225.8822836.234135.230
386.60250.0290.0140.0
470.71070.71030155.00.0
550.086.60231134.23477.5
625.88296.5923277.5134.234
70.0100.0330.0155.0
8110.00.034170.00.0
995.26355.035164.20743.999
1055.095.26336147.22485.0
110.0110.037120.208120.208
12120.00.03885.0147.224
13115.91131.0583943.999164.207
14103.92360.0400.0170.0
1584.85384.85341185.00.0
1660.0103.92342160.21592.5
1731.058115.9114392.5160.215
180.0120.0440.0185.0
19130.00.045200.00.0
20112.58365.046193.18551.764
2165.0112.58347173.205100.0
220.0130.048141.421141.421
23140.00.049100.0173.205
24135.23036.2345051.764193.185
25121.24370.0510.0200.0
2698.99598.995
1010.00.0
7100.00.0
8010.00.0
11100.00.0
12010.00.0
18100.00.0
19010.00.0
22100.00.0
23010.00.0
29100.00.0
30010.00.0
33100.00.0
34010.00.0
40100.00.0
41310.00.0
44100.00.0
45010.00.0
51100.00.0
+ + + +
1 $\overline{21000.0}$ $\overline{0.3}$ 0.00.024.00.00.0-0.0
INTERNAL PRESSURE
001
3 $\overline{N}$
1321
20.00.020.00.020.00.0
2543
20.00.020.00.020.00.0
3765
20.00.020.00.020.00.0
0.71.0 $\overline{30}$ 33
+ +Line printer output + +
2-D ELASTO-PLASTIC EXAMPLE, SECTION 7.9, FIG 7.12
NPOIN = 51NELEM = 12NVFIX = 18NTYPE = 2NNODE = 8
NMATS = 1NGAUS = 2NEVAB = 16NALGO = 2
NCRIT = 2NINCS = 1NSTRE = 3
ELEMENTPROPERTYNODE NUMBERS
1118121314932
21391415161054
315101617181176
411219232425201413
511420252627211615
611621272829221817
712330343536312524
812531363738322726
912732383940332928
1013441454647423635
1113642474849433837
1213843495051444039
NODEXY
1100.0000.0002770.000121.243
296.59225.8822836.234135.230
386.60250.000290.000140.000
470.71070.71030155.0000.000
550.00086.60231134.23477.500
625.88296.5923277.500134.234
70.000100.000330.000155.000
8110.0000.00034170.0000.000
995.26355.00035164.20743.999
1055.00095.26336147.22485.000
110.000110.00037120.208120.208
12120.0000.0003885.000147.224
13115.91131.0583943.999164.207
14103.92360.000400.000170.000
1584.85384.85341185.0000.000
1660.000103.92342160.21592.500
1731.058115.9114392.500160.215
180.000120.000440.000185.000
19130.0000.00045200.0000.000
20112.58365.00046193.18551.764
2165.000112.58347173.205100.000
220.000130.00048141.421141.421
23140.0000.00049100.000173.205
24135.23036.2345051.764193.185
25121.24370.000510.000200.000
2698.99598.995
+ + + +
NODECODEFIXED VALUES
110.0000000.000000
7100.0000000.000000
810.0000000.0000000.0000000.000000
11100.0000000.0000000.0000000.000000
1210.0000000.0000000.0000000.0000000.000000E0.000000
1910.0000000.000000
22100.0000000.0000000.000000
2310.0000000.000000
29100.0000000.000000
3010.0000000.000000
33100.0000000.000000
3410.0000000.0000000.000000E0.000000E
40100.0000000.000000
4110.0000000.000000
44100.0000000.0000000.000000E0.000000E0.000000E
4510.0000000.000000
+ + + +
0.0000E000.0000E000.0000E000.0000E000.0000E000.0000E000.0000E000.0000E00
50.0000E000.0000E000.0000E000.0000E000.0000E000.0000E000.0000E000.0000E00
0.0000E000.0000E000.0000E000.0000E000.0000E000.0000E000.0000E000.0000E00
60.0000E000.0000E000.0000E000.0000E000.0000E000.0000E000.0000E000.0000E00
0.0000E0.0000E000.0000E000.0000E000.0000E000.0000E000.0000E000.0000E00
70.0000E000.0000E000.0000E000.0000E000.0000E000.0000E000.0000E000.0000E00
0.0000E0.0000E000.0000E000.0000E000.0000E000.0000E000.0000E000.0000E00
80.0000E000.0000E000.0000E000.0000E000.0000E000.0000E000.0000E000.0000E00
0.0000E0.0000E000.0000E000.0000E000.0000E000.0000E000.0000E000.0000E00
90.0000E000.0000E000.0000E000.0000E000.0000E000.0000E000.0000E000.0000E00
0.0000E 000.0000E000.0000E000.0000E000.0000E000.0000E000.0000E000.0000E00
100.0000E000.0000E000.0000E000.0000E000.0000E000.0000E000.0000E000.0000E00
0.0000E000.0000E000.0000E000.0000E000.0000E000.0000E000.0000E000.0000E00
110.0000E000.0000E000.0000E000.0000E000.0000E000.0000E000.0000E000.0000E00
0.0000E.000.0000E000.0000E000.0000E000.0000E000.0000E000.0000E000.0000E00
120.0000E000.0000E000.0000E000.0000E000.0000E000.0000E000.0000E000.0000E00
0.0000E .000.0000E000.0000E000.0000E000.0000E000.0000E000.0000E000.0000E00
+ +INCREMENT NUMBER 1 + +
LOAD FACTOR = 0.70000CONVERGENCE TOLERANCE = 1.00000MAX. NO. OF ITERATIONS = 30
INITIAL OUTPUT PARAMETER = 3FINAL OUTPUT PARAMETER = 3
CONVERGENCE CODE = 1NORM OF RESIDUAL SUM RATIO = 0.336960E 02MAXIMUM RESIDUAL = 0.155988E 03
DISPLACEMENTS
+ +NODE X-DISP. Y-DISP. + +
10.127198E 000.000000E 00180.000000E 000.110185E 00
20.122734E 000.328877E-01190.103925E 000.000000E 00
30.110156E 000.636002E-01200.900022E-010.519632E-01
40.898486E-010.898486E-01210.519632E-010.900022E-01
50.636002E-010.110156E 00220.000000E 000.103925E 00
60.328877E-010.122734E 00230.987474E-010.000000E 00
70.000000E 000.127198E 00240.953363E-010.255449E-01
80.117795E 000.000000E 00250.855186E-010.493745E-01
90.102014E 000.588984E-01260.697915E-010.697915E-01
100.588984E-010.102014E 00270.493745E-010.855186E-01
110.000000E 000.117795E 00280.255449E-010.953363E-01
120.110185E 000.000000E 00290.000000E 000.987474E-01
130.106396E 000.285087E-01300.924750E-010.000000E 00
140.954232E-010.550931E-01310.800863E-010.462379E-01
150.778875E-010.778875E-01320.462379E-010.800863E-01
160.550931E-010.954232E-01330.000000E 000.924750E-01
170.285087E-010.106396E 00340.876445E-010.000000E 00
+ + + +
350.846176E-010.226732E-01
360.759029E-010.438226E-01
370.619449E-010.619449E-01
380.438226E-010.759029E-01
390.226732E-010.846176E-01
400.000000E 000.876445E-01
410.838477E-010.000000E 00
420.726148E-010.419240E-01
430.419240E-010.726148E-01
440.000000E 000.838477E-01
450.808966E-010.000000E 00
460.781269E-010.209340E-01
470.700591E-010.404483E-01
480.571933E-010.571933E-01
490.404483E-010.700591E-01
500.209340E-010.781269E-01
510.000000E 000.808966E-01
+ +
REACTIONS
NODEX-REAC.Y-REAC.
10.000000E00-0.761999E02
7-0.761999E020.000000E00
80.000000E00-0.269921E03
11-0.269921E030.000000E00
120.000000E00-0.116327E03
18-0.116327E030.000000E00
190.000000E00-0.210260E03
22-0.210260E030.000000E00
230.000000E00-0.117156E03
29-0.117156E030.000000E00
300.000000E00-0.250153E03
33-0.250153E030.000000E00
340.000000E00-0.110283E03
40-0.110283E030.000000E00
410.000000E00-0.203189E03
44-0.203189E030.000000E00
450.000000E00-0.465120E02
51-0.465120E020.000000E00
+ +
G.P.XX-STRESSYY-STRESSXY-STRESSZZ-STRESSMAX P.S.MIN P.S.ANGLEE.P.S.
ELEMENT NO. =1
1-0.893805E 010.180284E 02-0.307422E 010.304329E 010.183744E 02-0.928408E 016.4220.240602E-03
2-0.485865E 010.139487E 02-0.101400E 020.304318E 010.183743E 02-0.928420E 0123.5790.240580E-03
3-0.880961E 010.181337E 02-0.306125E 010.280970E 010.184771E 02-0.915305E 016.4010.770100E-05
4-0.472518E 010.140487E 02-0.101362E 020.280953E 010.184768E 02-0.915334E 0123.5990.770430E-05
ELEMENT NO. =2
10.465341E 000.862395E 01-0.132139E 020.304290E 010.183739E 02-0.928461E 0136.4220.240568E-03
20.862395E 010.465341E 00-0.132139E 020.304290E 010.183739E 02-0.928461E 01-36.4220.240568E-03
30.577107E 000.874645E 01-0.131974E 020.280952E 010.184769E 02-0.915330E 0136.4010.768219E-05
40.874645E 010.577107E 00-0.131974E 020.280952E 010.184769E 02-0.915330E 01-36.4010.768219E-05
ELEMENT NO. =3
10.139487E 02-0.485865E 01-0.101400E 020.304318E 010.183743E 02-0.928420E 01-23.5790.240580E-03
20.180284E 02-0.893805E 01-0.307422E 010.304329E 010.183744E 02-0.928408E 01-6.4220.240602E-03
30.140487E 02-0.472518E 01-0.101362E 020.280953E 010.184768E 02-0.915334E 01-23.5990.770431E-05
40.181337E 02-0.880961E 01-0.306125E 010.280970E 010.184771E 02-0.915305E 01-6.4010.770100E-05
ELEMENT NO. =4
1-0.713097E 010.164644E 02-0.267828E 010.280004E 010.167646E 02-0.743116E 016.3950.000000E 00
2-0.355180E 010.128851E 02-0.887785E 010.280000E 010.167646E 02-0.743124E 0123.6040.000000E 00
3-0.520488E 010.145383E 02-0.224680E 010.280002E 010.147907E 02-0.545735E 016.4110.000000E 00
+ + + +
4-0.221523E 010.115483E 02-0.742551E 010.279991E 010.147906E 02-0.545755E 0123.5880.000000E 00
ELEMENT NO. =5
10.108723E 010.824570E 01-0.115562E 020.279988E 010.167643E 02-0.743133E 0136.3950.000000E 00
20.824570E 010.108723E 01-0.115562E 020.279988E 010.167643E 02-0.743133E 01-36.3950.000000E 00
30.167670E 010.765648E 01-0.967249E 010.279995E 010.147906E 02-0.545747E 0136.4110.000000E 00
40.765648E 010.167670E 01-0.967249E 010.279995E 010.147906E 02-0.545747E 01-36.4110.000000E 00
ELEMENT NO. =6
10.128851E 02-0.355180E 01-0.887785E 010.280000E 010.167646E 02-0.743124E 01-23.6040.000000E 00
20.164644E 02-0.713097E 01-0.267828E 010.280004E 010.167646E 02-0.743116E 01-6.3950.000000E 00
30.115483E 02-0.221523E 01-0.742551E 010.279991E 010.147906E 02-0.545755E 01-23.5880.000000E 00
40.145383E 02-0.520488E 01-0.224680E 010.280002E 010.147907E 02-0.545735E 01-6.4110.000000E 00
ELEMENT NO. =7
1-0.383616E 010.131694E 02-0.193148E 010.279998E 010.133861E 02-0.405278E 016.3990.000000E 00
2-0.125760E 010.105909E 02-0.639778E 010.279999E 010.133861E 02-0.405277E 0123.6000.000000E 00
3-0.212632E 010.114596E 02-0.154577E 010.279997E 010.116332E 02-0.229997E 016.4100.000000E 00
4-0.686952E-010.940184E 01-0.510990E 010.279994E 010.116332E 02-0.230005E 0123.5900.000000E 00
ELEMENT NO. =8
10.208787E 010.724522E 01-0.832942E 010.279993E 010.133860E 02-0.405291E 0136.3990.000000E 00
20.724522E 010.208787E 01-0.832942E 010.279993E 010.133860E 02-0.405291E 01-36.3990.000000E 00
30.260888E 010.672438E 01-0.665579E 010.279998E 010.116333E 02-0.229999E 0136.4100.000000E 00
40.672438E 010.260888E 01-0.665579E 010.279998E 010.116333E 02-0.229999E 01-36.4100.000000E 00
ELEMENT NO. =9
10.105909E 02-0.125760E 01-0.639778E 010.279999E 010.133861E 02-0.405277E 01-23.6000.000000E 00
20.131694E 02-0.383616E 01-0.193148E 010.279998E 010.133861E 02-0.405278E 01-6.3990.000000E 00
30.940184E 01-0.686952E-01-0.510990E 010.279994E 010.116332E 02-0.230005E 01-23.5900.000000E 00
40.114596E 02-0.212632E 01-0.154577E 010.279997E 010.116332E 02-0.229997E 01-6.4100.000000E 00
ELEMENT NO. =10
1-0.118841E 010.105216E 02-0.132981E 010.279995E 010.106707E 02-0.133753E 016.3980.000000E 00
20.587478E 000.874580E 01-0.440564E 010.279998E 010.106707E 02-0.133746E 0123.6020.000000E 00
3-0.186150E 000.951929E 01-0.110110E 010.279994E 010.964264E 01-0.309504E 006.3920.000000E 00
40.128661E 010.804648E 01-0.365206E 010.279993E 010.964263E 01-0.309548E 0023.6080.000000E 00
ELEMENT NO. =11
10.289070E 010.644254E 01-0.573552E 010.279997E 010.106708E 02-0.133755E 0136.3980.000000E 00
20.644254E 010.289070E 01-0.573552E 010.279997E 010.106708E 02-0.133755E 01-36.3980.000000E 00
30.319390E 010.613950E 01-0.475323E 010.280002E 010.964288E 01-0.309476E 0036.3920.000000E 00
40.613950E 010.319390E 01-0.475323E 010.280002E 010.964288E 01-0.309476E 00-36.3920.000000E 00
ELEMENT NO. =12
10.874580E 010.587478E 00-0.440564E 010.279998E 010.106707E 02-0.133746E 01-23.6020.000000E 00
20.105216E 02-0.118841E 01-0.132981E 010.279995E 010.106707E 02-0.133753E 01-6.3980.000000E 00
30.804648E 010.128661E 01-0.365206E 010.279993E 010.964263E 01-0.309548E 00-23.6080.000000E 00
+ + + +
40.951929E01-0.186150E00-0.110110E010.279994E010.964264E01-0.309504E00-6.3920.000000E00
CONVERGENCECODE=1NORMOFRESIDUALSUMRATIO=0.118830E02MAXIMUMRESIDUAL=0.416687E02
CONVERGENCECODE=1NORMOFRESIDUALSUMRATIO=0.556571E01MAXIMUMRESIDUAL=0.222848E02
CONVERGENCECODE=1NORMOFRESIDUALSUMRATIO=0.297375E01MAXIMUMRESIDUAL=0.127533E02
CONVERGENCECODE=1NORMOFRESIDUALSUMRATIO=0.165985E01MAXIMUMRESIDUAL=0.728396E01
CONVERGENCECODE=0NORMOFRESIDUALSUMRATIO=0.939223E00MAXIMUMRESIDUAL=0.415713E01
+ +DISPLACEMENTS + +
NODEX-DISP.Y-DISP.
10.139121E 000.000000E 00370.665485E-010.665485E-01
20.134201E 000.359609E-01380.470796E-010.815441E-01
30.120482E 000.695626E-01390.243581E-010.909056E-01
40.982428E-010.982428E-01400.000000E 000.941578E-01
50.695626E-010.120482E 00410.900786E-010.000000E 00
60.359609E-010.134201E 00420.780114E-010.450397E-01
70.000000E 000.139121E 00430.450397E-010.780114E-01
80.127126E 000.000000E 00440.000000E 000.900786E-01
90.110094E 000.635643E-01450.869080E-010.000000E 00
100.635643E-010.110094E 00460.839328E-010.224896E-01
110.000000E 000.127126E 00470.752657E-010.434542E-01
120.118379E 000.000000E 00480.614439E-010.614439E-01
130.114299E 000.306268E-01490.434542E-010.752657E-01
140.102520E 000.591908E-01500.224896E-010.839328E-01
150.836738E-010.836738E-01510.000000E 000.869080E-01
160.591908E-010.102520E 00REACTIONS
170.306268E-010.114299E 00NODEX-REAC.Y-REAC.
180.000000E 000.118379E 0010.000000E 00-0.464276E 02
190.111650E 000.000000E 007-0.464276E 020.000000E 00
200.966928E-010.558264E-0180.000000E 00-0.220459E 03
210.558264E-010.966928E-0111-0.220459E 030.000000E 00
220.000000E 000.111650E 00120.000000E 00-0.125854E 03
230.106084E 000.000000E 0018-0.125854E 030.000000E 00
240.102421E 000.274436E-01190.000000E 00-0.225928E 03
250.918730E-010.530435E-0122-0.225928E 030.000000E 00
260.749788E-010.749788E-01230.000000E 00-0.125859E 03
270.530435E-010.918730E-0129-0.125859E 030.000000E 00
280.274436E-010.102421E 00300.000000E 00-0.268735E 03
290.000000E 000.106084E 0033-0.268735E 030.000000E 00
300.993465E-010.000000E 00340.000000E 00-0.118479E 03
310.860377E-010.496741E-0140-0.118479E 030.000000E 00
320.496741E-010.860377E-01410.000000E 00-0.218290E 03
+ + + +33 0.000000E 00 0.993465E-01 + +34 0.941578E-01 0.000000E 00 + +35 0.909056E-01 0.243581E-01 + +36 0.815441E-01 0.470796E-01 + +44 -0.218290E 03 0.000000E 00 + +45 0.000000E 00 -0.499673E 02 + +51 -0.499673E 02 0.000000E 00 + +
G.P.XX-STRESSYY-STRESSXY-STRESSZZ-STRESSMAX P.S.MIN P.S.ANGLEE.P.S.
ELEMENT NO. = 1
1-0.123717E 020.146473E 02-0.308107E 010.117112E 010.149941E 02-0.127186E 026.4240.451304E-03
2-0.828491E 010.105605E 02-0.101593E 020.117110E 010.149942E 02-0.127186E 0223.5770.451255E-03
3-0.948121E 010.174939E 02-0.306568E 010.257060E 010.178380E 02-0.982523E 016.4030.108534E-03
4-0.539247E 010.134047E 02-0.101479E 020.257044E 010.178377E 02-0.982547E 0123.5980.108528E-03
ELEMENT NO. = 2
1-0.294888E 010.522409E 01-0.132401E 020.117090E 010.149940E 02-0.127188E 0236.4240.451200E-03
20.522409E 01-0.294888E 01-0.132401E 020.117090E 010.149940E 02-0.127188E 02-36.4240.451200E-03
3-0.825393E-010.809511E 01-0.132134E 020.257046E 010.178379E 02-0.982530E 0136.4030.108473E-03
40.809511E 01-0.825394E-01-0.132134E 020.257046E 010.178379E 02-0.982530E 01-36.4030.108473E-03
ELEMENT NO. = 3
10.105605E 02-0.828491E 01-0.101593E 020.117110E 010.149942E 02-0.127186E 02-23.5770.451255E-03
20.146473E 02-0.123717E 02-0.308107E 010.117112E 010.149941E 02-0.127186E 02-6.4240.451304E-03
30.134047E 02-0.539247E 01-0.101479E 020.257044E 010.178377E 02-0.982547E 01-23.5980.108528E-03
40.174939E 02-0.948121E 01-0.306568E 010.257060E 010.178380E 02-0.982523E 01-6.4030.108534E-03
ELEMENT NO. = 4
1-0.766058E 010.176878E 02-0.287878E 010.300817E 010.180106E 02-0.798341E 016.3980.000000E 00
2-0.381672E 010.138438E 02-0.953667E 010.300813E 010.180105E 02-0.798344E 0123.6010.000000E 00
3-0.559170E 010.156189E 02-0.241350E 010.300815E 010.158900E 02-0.586286E 016.4100.000000E 00
4-0.237967E 010.124063E 02-0.797755E 010.300798E 010.158898E 02-0.586315E 0123.5890.000000E 00
ELEMENT NO. = 5
10.116933E 010.885683E 01-0.124153E 020.300785E 010.180098E 02-0.798366E 0136.3990.000000E 00
20.885683E 010.116933E 01-0.124153E 020.300785E 010.180098E 02-0.798366E 01-36.3990.000000E 00
30.180098E 010.822568E 01-0.103912E 020.300800E 010.158897E 02-0.586305E 0136.4110.000000E 00
40.822568E 010.180098E 01-0.103912E 020.300800E 010.158897E 02-0.586305E 01-36.4110.000000E 00
ELEMENT NO. = 6
10.138438E 02-0.381672E 01-0.953667E 010.300813E 010.180105E 02-0.798344E 01-23.6010.000000E 00
20.176878E 02-0.766058E 01-0.287878E 010.300817E 010.180106E 02-0.798341E 01-6.3980.000000E 00
30.124063E 02-0.237967E 01-0.797755E 010.300798E 010.158898E 02-0.586315E 01-23.5890.000000E 00
40.156189E 02-0.559170E 01-0.241350E 010.300815E 010.158900E 02-0.586286E 01-6.4100.000000E 00
ELEMENT NO. = 7
1-0.412127E 010.141482E 02-0.207478E 010.300809E 010.143809E 02-0.435393E 016.3980.000000E 00
2-0.135088E 010.113778E 02-0.687337E 010.300808E 010.143809E 02-0.435393E 0123.6010.000000E 00
3-0.228431E 010.123112E 02-0.166069E 010.300806E 010.124977E 02-0.247088E 016.4100.000000E 00
4-0.738630E-010.101006E 02-0.548958E 010.300802E 010.124977E 02-0.247098E 0123.5890.000000E 00
+ + + +
ELEMENT NO. = 8
10.224272E 010.778385E 01-0.894834E 010.300797E 010.143807E 02-0.435415E 0136.3980.000000E 00
20.778385E 010.224272E 01-0.894834E 010.300797E 010.143807E 02-0.435415E 01-36.3980.000000E 00
30.280277E 010.722406E 01-0.715043E 010.300805E 010.124978E 02-0.247095E 0136.4100.000000E 00
40.722406E 010.280277E 01-0.715043E 010.300805E 010.124978E 02-0.247095E 01-36.4100.000000E 00
ELEMENT NO. = 9
10.113778E 02-0.135088E 01-0.687337E 010.300808E 010.143809E 02-0.435393E 01-23.6010.000000E 00
20.141482E 02-0.412127E 01-0.207478E 010.300809E 010.143809E 02-0.435393E 01-6.3980.000000E 00
30.101006E 02-0.738630E-01-0.548958E 010.300802E 010.124977E 02-0.247098E 01-23.5890.000000E 00
40.123112E 02-0.228431E 01-0.166069E 010.300806E 010.124977E 02-0.247088E 01-6.4100.000000E 00
ELEMENT NO. = 10
1-0.127671E 010.113035E 02-0.142867E 010.300803E 010.114637E 02-0.143691E 016.3980.000000E 00
20.631079E 000.939580E 01-0.473299E 010.300806E 010.114637E 02-0.143686E 0123.6010.000000E 00
3-0.199987E 000.102267E 02-0.118290E 010.300800E 010.103592E 02-0.332502E 006.3920.000000E 00
40.138223E 010.864445E 01-0.392346E 010.300800E 010.103592E 02-0.332548E 0023.6080.000000E 00
ELEMENT NO. = 11
10.310555E 010.692130E 01-0.616178E 010.300805E 010.114638E 02-0.143697E 0136.3980.000000E 00
20.692130E 010.310555E 01-0.616178E 010.300805E 010.114638E 02-0.143697E 01-36.3980.000000E 00
30.343125E 010.659581E 01-0.510649E 010.300812E 010.103595E 02-0.332480E 0036.3920.000000E 00
40.659581E 010.343125E 01-0.510649E 010.300812E 010.103595E 02-0.332480E 00-36.3920.000000E 00
ELEMENT NO. = 12
10.939580E 010.631079E 00-0.473299E 010.300806E 010.114637E 02-0.143686E 01-23.6010.000000E 00
20.113035E 02-0.127671E 01-0.142867E 010.300803E 010.114637E 02-0.143691E 01-6.3980.000000E 00
30.864445E 010.138223E 01-0.392346E 010.300800E 010.103592E 02-0.332548E 00-23.6080.000000E 00
40.102267E 02-0.199987E 00-0.118290E 010.300800E 010.103592E 02-0.332502E 00-6.3920.000000E 00
+ +# A.4.6 Solution of two-dimensional elasto-viscoplastic problem. Example of Section 8.16, Fig. 8.10 + +Input data + +
2-D ELASTO - VISCOPLASTIC EXAMPLE, SECTION 8.16, FIG. 8.10
51121828122213
1118121314932
21391415161054
315101617181176
411219232425201413
diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_057.md b/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_057.md new file mode 100644 index 00000000..704452c1 --- /dev/null +++ b/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_057.md @@ -0,0 +1,63 @@ + + +
511420252627211615
611621272829221817
712330343536312524
812531363738322726
912732383940332928
1013441454647423635
1113642474849433837
1213843495051444039
1100.00.02770.0121.243
296.59225.8822836.234135.230
386.60250.0290.0140.0
470.71070.71030155.00.0
550.086.60231134.23477.5
625.88296.5923277.5134.234
70.0100.0330.0155.0
8110.00.034170.00.0
995.26355.035164.20743.999
1055.095.26336147.22485.0
110.0110.037120.208120.208
12120.00.03885.0147.224
13115.91131.0583943.999164.207
14103.92360.0400.0170.0
1584.85384.85341185.00.0
1660.0103.92342160.21592.5
1731.058115.9114392.5160.215
180.0120.0440.0185.0
19130.00.045200.00.0
20112.58365.046193.18551.764
2165.0112.58347173.205100.0
220.0130.048141.421141.421
23140.00.049100.0173.205
24135.23036.2345051.764193.185
25121.24370.0510.0200.0
2698.99598.995
1010.00.0
7100.00.0
8010.00.0
11100.00.0
12010.00.0
18100.00.0
+ + + +
19010.00.0
22100.00.0
23010.00.0
29100.00.0
30010.00.0
33100.00.0
34010.00.0
40100.00.0
41010.00.0
44100.00.0
45010.00.0
51100.00.0
1
21000.00.30.00.024.00.00.00.001
1.01.0
INTERNAL PRESSURE
001
3
1321
20.00.020.00.020.00.0
2543
20.00.020.00.020.00.0
3765
20.00.020.00.020.00.0
0.00.050.11.5
0.70.15010
+ +Line printer output + +
2-D ELASTO - VISCOPLASTIC EXAMPLE, SECTION 8.16, FIG. 8.10
NPOIN = 51NELEM = 12NVFIX = 18NTYPE = 2NNODE = 8
NMATS = 1NGAUS = 2NEVAB = 16NALGO = 2
NCRIT = 2NINCS = 1NSTRE = 3
ELEMENTPROPERTYNODE NUMBERS
1118121314932
21391415161054
315101617181176
411219232425201413
511420252627211615
611621272829221817
+ + + +
712330343536312524
812531363738322726
912732383940332928
1013441454647423635
1113642474849433837
1213843495051444039
NODEXY
1100.0000.0002770.000121.243
296.59225.8822836.234135.230
386.60250.000290.000140.000
470.71070.71030155.0000.000
550.00086.60231134.23477.500
625.88296.5923277.500134.234
70.000100.000330.000155.000
8110.0000.00034170.0000.000
995.26355.00035164.20743.999
1055.00095.26336147.22485.000
110.000110.00037120.208120.208
12120.0000.0003885.000147.224
13115.91131.0583943.999164.207
14103.92360.000400.000170.000
1584.85384.85341185.0000.000
1660.000103.92342160.21592.500
1731.058115.9114392.500160.215
180.000120.000440.000185.000
19130.0000.00045200.0000.000
20112.58365.00046193.18551.764
2165.000112.58347173.205100.000
220.000130.00048141.421141.421
23140.0000.00049100.000173.205
24135.23036.2345051.764193.185
25121.24370.000510.000200.000
2698.99598.995
NODECODEFIXED VALUES
110.0000000.000000
7100.0000000.000000
810.0000000.000000
11100.0000000.000000
1210.0000000.000000
18100.0000000.000000
+ + + +
1910.0000000.000000
22100.0000000.000000
2310.0000000.000000
29100.0000000.000000
3010.0000000.0000000.000000E000.000000E000.000000E000.000000E020.000000E000.000000E000.000000E000.100000E-020.000000E010.100000E010.240000E020.000000E000.000000E000.000000E000.100000E-020.100000E010.100000E010.240000E020.000000E000.000000E000.000000E000.100000E-020.100000E010.100000E010.100000E010.240000E020.000000E000.000000E000.000000E000.100000E-020.100000E010.100000E000.000000E000.000000E000.000000E000.000000E000.000000E000.100000E-020.100000E010.100000E000.000000E000.000000E000.000000E010.100000E000.000000E000.000000E000.000000E000.000000E000.000000E000.000000E000.000000E000.000000E000.000000E000.0100000E-020.100000E000.000000E000.000000E000.000000E000.000000E000.000000E000.000000E000.000000E000.000000E0010.100000E000.000000E000.000000E000.000000E000.000000E000.000000E000.000000E000.000000E000.000000E-020.100000E000.000000E000.000000E000.000000E000.000000E000.000000E000.000000E000.000000E000.100000E-020.100000E000.000000E000.000000E000.000000E000.000000E000.000000E000.000000E000.000000E0000.000000E000.000000E000.000000E000.000000E000.000000E000.000000E000.000000E000.000000E000.000000E-020.000000E000.000000E000.000000E000.000000E000.000000E000.000000E000.000000E000.000000E000.000000E0010.000000E000.000000E000.000000E000.000000E000.000000E000.000000E000.000000E000.000000E000.000000E00
40100.0000000.0000000.000000E000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000.000
4110.0000000.000000
44100.0000000.000000
4510.0000000.000000
51100.0000000.000000
NUMBER ELEMENT PROPERTIES0.210000E050.300000E000.000000E000.000000E000.000000E000.000000E000.000000E000.000000E000.000000E000.000000E000.0240000E020.000000E000.000000E000.000000E000.000000E000.000000E000.000000E000.000000E000.000000E000.000000E-00.000000E-00.000000E-00.000000E-00.000000E-00.000000E-00.000000E-00.000000E-00.000000E-00.000000E-10.000000E-10.000000E-10.000000E-10.000000E-10.000000E-10.000000E-10.000000E-10.000000E-10.000000E-12.000000E-12.000000E-12.000000E-12.000000E-12.000000E-12.000000E-12.000000E-12.000000E-12.000000E-120.210000E050.300000E000.000000E000.000000E000.000000E000.000000E000.000000E000.000000E000.000000E000.000000E000.00000.000000E000.000000E000.000000E000.000000E000.000000E000.000000E000.000000E000.000000E000.000000E000.000000E000.000000E000.000000E000.000000E000.000000E000.000000E000.000000E000.100000E-000.000000E-000.000000E-000.000000E-000.000000E-000.000000E-000.000000E-000.000000E-000.000000E-020.000000E-020.000000E-020.000000E-020.000000E-020.000000E-020.000000E-020.000000E-020.000000.000000E-020.000000E-020.000000E-020.000000E-020.000000E-020.000000E-020.000000E-020.0000.000000E-000.000000E-000.000000E-000.000000E-000.000000E-000.000000E-000.000000E-000.0000.000000E-000.000000E-000.000000E-000.000000E-000.000000E-000.000000E-000.000000E-0000.000000E-000.000000E-000.000000E-000.000000E-000.000000E-000.000000E-000.000000E-000.00000.000000E-000.000000E-000.000000E-000.000000E-000.000000E-000.000000E-000.000000E-000.000.000000E-000.000000E-000.000000E-000.000000E-000.000000E-000.000000E-000.000000E-000.000
10.210000E050.300000E000.000000E000.000000E000.000000E000.000000E000.000000E000.000000E000.240000E000.000000E000.000000E000.000000E000.000000E000.000000E000.000000E000.000000E000.000000.000000E000.000000E000.000000E000.000000E000.000000E000.000000E000.000000E000.000000E00.000000E000.000000.000000E000.000000E000.000000E000.000000E000.000000E000.000000E000.0000.000000E-000.000000E-000.000000E-000.000000E-000.000000E-000.000000E-000.000000E-00
13210.000000E000.000000E000.000000E000.000000E000.000000E000.000000E000.000000E000.000000E000.210000E000.000000E000.000000E000.000000E000.000000E000.000000E000.000000E000.000000E000.00000E000.000000E000.000000E000.000000E000.000000E000.000000E000.000000E000.000000E000.0000
20.000000E000.000000E000.000000E000.000000E000.000000E000.000000E000.000000E000.000000E000.0854E000.000000E000.0000
+ + + +
0.0000E000.0000E000.0000E000.0000E000.0000E000.0000E000.0000E00
80.0000E000.0000E000.0000E000.0000E000.0000E000.0000E000.0000E00
0.0000E000.0000E000.0000E000.0000E000.0000E000.0000E000.0000E00
90.0000E000.0000E000.0000E000.0000E000.0000E000.0000E000.0000E00
0.0000E000.0000E0.0000E000.0000E000.0000E000.0000E000.0000E00
100.0000E000.0000E000.0000E000.0000E000.0000E000.0000E000.0000E00
0.0000E000.0000E0.0000E000.0000E000.0000E000.0000E000.0000E00
110.0000E000.0000E000.0000E000.0000E000.0000E000.0000E000.0000E00
0.0000E000.0000E0.0000E000.0000E000.0000E000.0000E000.0000E00
120.0000E000.0000E000.0000E000.0000E000.0000E000.0000E000.0000E00
0.0000E000.0000E
0.0000E000.0000E000.0000E000.0000E000.0000E000.0000E000.0000E00
TIME STEPPING PARAMETER =0.000TIME STEP STABILITY FACTOR = 0.05000
INITIAL TIME STEP LENGTH =0.10000TIME STEP INCREMENT PARAMETER = 1.50000
+ +INCREMENT NUMBER 1 + +
LOAD FACTOR = 0.70000CONVERGENCE TOLERANCE = 0.10000MAX. NO. OF ITERATIONS = 50
INITIAL OUTPUT PARAMETER = 10FINAL OUTPUT PARAMETER = 10
TOTAL TIME = 0.000000E 00
CONVERGENCE CODE = 1 NORM OF RESIDUAL SUM RATIO = 0.100000E 03MAXIMUM RESIDUAL = 0.000000E 00
TOTAL TIME = 0.100000E 00
CONVERGENCE CODE = 999 NORM OF RESIDUAL SUM RATIO = 0.148250E 03MAXIMUM RESIDUAL = 0.000000E 00
TOTAL TIME = 0.250000E 00
CONVERGENCE CODE = 999 NORM OF RESIDUAL SUM RATIO = 0.207778E 03MAXIMUM RESIDUAL = 0.000000E 00
TOTAL TIME = 0.475000E 00
CONVERGENCE CODE = 999 NORM OF RESIDUAL SUM RATIO = 0.280997E 03MAXIMUM RESIDUAL = 0.000000E 00
TOTAL TIME = 0.812500E 00
CONVERGENCE CODE = 999 NORM OF RESIDUAL SUM RATIO = 0.313019E 03MAXIMUM RESIDUAL = 0.000000E 00
TOTAL TIME = 0.125353E 01
CONVERGENCE CODE = 999 NORM OF RESIDUAL SUM RATIO = 0.340506E 03MAXIMUM RESIDUAL = 0.000000E 00
TOTAL TIME = 0.184786E 01
CONVERGENCE CODE = 999 NORM OF RESIDUAL SUM RATIO = 0.377261E 03MAXIMUM RESIDUAL = 0.000000E 00
TOTAL TIME = 0.273772E 01
CONVERGENCE CODE = 1 NORM OF RESIDUAL SUM RATIO = 0.345160E 03MAXIMUM RESIDUAL = 0.000000E 00
TOTAL TIME = 0.407250E 01
CONVERGENCE CODE = 1 NORM OF RESIDUAL SUM RATIO = 0.213414E 03MAXIMUM RESIDUAL = 0.000000E 00
TOTAL TIME = 0.607467E 01
CONVERGENCE CODE = 0 NORM OF RESIDUAL SUM RATIO = 0.000000E 00MAXIMUM RESIDUAL = 0.000000E 00
DISPLACEMENTS
+ +NODE X-DISP. Y-DISP. + +
10.139590E000.000000E00
20.134655E000.360826E-01
+ +3 0.120888E 00 0.697974E-01 + +
40.985748E-010.985748E-01
+ + + +
50.697974E-010.120888E 00390.244471E-010.912376E-01
60.360826E-010.134655E 00400.000000E 000.945016E-01
70.000000E 000.139590E 00410.904075E-010.000000E 00
80.127595E 000.000000E 00420.782963E-010.452042E-01
90.110501E 000.637993E-01430.452042E-010.782963E-01
100.637993E-010.110501E 00440.000000E 000.904075E-01
110.000000E 000.127595E 00450.872253E-010.000000E 00
120.118811E 000.000000E 00460.842393E-010.225717E-01
130.114717E 000.307387E-01470.755406E-010.436128E-01
140.102894E 000.594071E-01480.616684E-010.616684E-01
150.839794E-010.839794E-01490.436128E-010.755406E-01
160.594071E-010.102894E 00500.225717E-010.842393E-01
170.307387E-010.114717E 00510.000000E 000.872253E-01
180.000000E 000.118811E 00REACTIONS
190.112058E 000.000000E 00NODEX-REAC.Y-REAC.
200.970459E-010.560303E-0110.000000E 00-0.456968E 02
210.560303E-010.970459E-017-0.456968E 020.000000E 00
220.000000E 000.112058E 0080.000000E 00-0.217851E 03
230.106472E 000.000000E 0011-0.217851E 030.000000E 00
240.102796E 000.275438E-01120.000000E 00-0.125513E 03
250.922085E-010.532372E-0118-0.125513E 030.000000E 00
260.752527E-010.752527E-01190.000000E 00-0.226754E 03
270.532372E-010.922085E-0122-0.226754E 030.000000E 00
280.275438E-010.102796E 00230.000000E 00-0.126319E 03
290.000000E 000.106472E 0029-0.126319E 030.000000E 00
300.997092E-010.000000E 00300.000000E 00-0.269717E 03
310.863519E-010.498555E-0133-0.269717E 030.000000E 00
320.498555E-010.863519E-01340.000000E 00-0.118912E 03
330.000000E 000.997092E-0140-0.118912E 030.000000E 00
340.945016E-010.000000E 00410.000000E 00-0.219087E 03
350.912376E-010.244471E-0144-0.219087E 030.000000E 00
360.818419E-010.472516E-01450.000000E 00-0.501497E 02
370.667910E-010.667916E-0151-0.501497E 020.000000E 00
380.472516E-010.818419E-01
G.P.XX-STRESSYY-STRESSXY-STRESSZZ-STRESSMAX P.S.MIN P.S.ANGLEE.P.S.
ELEMENT NO. = 1
1-0.125015E 020.145585E 02-0.308575E 010.617103E 000.149059E 02-0.128489E 026.4240.452901E-03
2-0.840843E 010.104656E 02-0.101747E 020.617146E 000.149060E 02-0.128488E 0223.5770.452852E-03
3-0.959430E 010.174053E 02-0.306854E 010.234329E 010.177496E 02-0.993866E 016.4030.112244E-03
4-0.550191E 010.133124E 02-0.101570E 020.234314E 010.177494E 02-0.993889E 0123.5980.112237E-03
+ + + +
ELEMENT NO. = 2
1-0.306428E 010.512105E 01-0.132601E 020.617031E 000.149057E 02-0.128490E 0236.4240.452796E-03
20.512105E 01-0.306428E 01-0.132601E 020.617031E 000.149057E 02-0.128490E 02-36.4240.452796E-03
3-0.187011E 000.799786E 01-0.132254E 020.234325E 010.177495E 02-0.993866E 0136.4030.112182E-03
40.799786E 01-0.187011E 00-0.132254E 020.234325E 010.177495E 02-0.993866E 01-36.4030.112182E-03
ELEMENT NO. = 3
10.104656E 02-0.840843E 01-0.101747E 020.617146E 000.149060E 02-0.128488E 02-23.5770.452852E-03
20.145585E 02-0.125015E 02-0.308575E 010.617103E 000.149059E 02-0.128489E 02-6.4240.452901E-03
30.133124E 02-0.550191E 01-0.101570E 020.234314E 010.177494E 02-0.993889E 01-23.5980.112237E-03
40.174053E 02-0.959430E 01-0.306854E 010.234329E 010.177496E 02-0.993866E 01-6.4030.112244E-03
ELEMENT NO. = 4
1-0.768855E 010.177524E 02-0.288931E 010.301916E 010.180764E 02-0.801257E 016.3980.000000E 00
2-0.383066E 010.138944E 02-0.957149E 010.301912E 010.180763E 02-0.801259E 0123.6010.000000E 00
3-0.561211E 010.156759E 02-0.242231E 010.301914E 010.159481E 02-0.588426E 016.4100.000000E 00
4-0.238836E 010.124516E 02-0.800669E 010.301897E 010.159478E 02-0.588457E 0123.5890.000000E 00
ELEMENT NO. = 5
10.117360E 010.888913E 01-0.124607E 020.301882E 010.180756E 02-0.801283E 0136.3990.000000E 00
20.888913E 010.117360E 01-0.124607E 020.301882E 010.180756E 02-0.801283E 01-36.3990.000000E 00
30.180755E 010.825572E 01-0.104291E 020.301898E 010.159477E 02-0.588446E 0136.4110.000000E 00
40.825572E 010.180755E 01-0.104291E 020.301898E 010.159477E 02-0.588446E 01-36.4110.000000E 00
ELEMENT NO. = 6
10.138944E 02-0.383066E 01-0.957149E 010.301912E 010.180763E 02-0.801259E 01-23.6010.000000E 00
20.177524E 02-0.768855E 01-0.288931E 010.301916E 010.180764E 02-0.801257E 01-6.3980.000000E 00
30.124516E 02-0.238836E 01-0.800669E 010.301897E 010.159478E 02-0.588457E 01-23.5890.000000E 00
40.156759E 02-0.561211E 01-0.242231E 010.301914E 010.159481E 02-0.588426E 01-6.4100.000000E 00
ELEMENT NO. = 7
1-0.413632E 010.141999E 02-0.208235E 010.301908E 010.144334E 02-0.436983E 016.3980.000000E 00
2-0.135581E 010.114194E 02-0.689847E 010.301907E 010.144334E 02-0.436983E 0123.6010.000000E 00
3-0.229264E 010.123561E 02-0.166675E 010.301905E 010.125434E 02-0.247990E 016.4100.000000E 00
4-0.741370E-010.101375E 02-0.550962E 010.301901E 010.125434E 02-0.248000E 0123.5890.000000E 00
ELEMENT NO. = 8
10.225090E 010.781227E 01-0.898102E 010.301895E 010.144332E 02-0.437006E 0136.3980.000000E 00
20.781227E 010.225090E 01-0.898102E 010.301895E 010.144332E 02-0.437006E 01-36.3980.000000E 00
30.281300E 010.725044E 01-0.717655E 010.301903E 010.125434E 02-0.247998E 0136.4100.000000E 00
40.725044E 010.281300E 01-0.717655E 010.301903E 010.125434E 02-0.247998E 01-36.4100.000000E 00
ELEMENT NO. = 9
10.114194E 02-0.135581E 01-0.689847E 010.301907E 010.144334E 02-0.436983E 01-23.6010.000000E 00
20.141999E 02-0.413632E 01-0.208235E 010.301908E 010.144334E 02-0.436983E 01-6.3980.000000E 00
30.101375E 02-0.741370E-01-0.550962E 010.301901E 010.125434E 02-0.248000E 01-23.5890.000000E 00
40.123561E 02-0.229264E 01-0.166675E 010.301905E 010.125434E 02-0.247990E 01-6.4100.000000E 00
+ + + +
ELEMENT NO. = 10
1-0.128137E 010.113448E 02-0.143389E 010.301902E 010.115056E 02-0.144216E 016.3980.000000E 00
20.633380E 000.943011E 01-0.475028E 010.301905E 010.115056E 02-0.144210E 0123.6010.000000E 00
3-0.200717E 000.102640E 02-0.118721E 010.301899E 010.103970E 02-0.333716E 006.3920.000000E 00
40.138728E 010.867602E 01-0.393779E 010.301899E 010.103971E 02-0.333762E 0023.6080.000000E 00
ELEMENT NO. = 11
10.311689E 010.694658E 01-0.618428E 010.301904E 010.115057E 02-0.144222E 0136.3980.000000E 00
20.694658E 010.311689E 01-0.618428E 010.301904E 010.115057E 02-0.144222E 01-36.3980.000000E 00
30.344379E 010.661991E 01-0.512514E 010.301911E 010.103974E 02-0.333695E 0036.3920.000000E 00
40.661991E 010.344379E 01-0.512514E 010.301911E 010.103974E 02-0.333695E 00-36.3920.000000E 00
ELEMENT NO. = 12
10.943011E 010.633380E 00-0.475028E 010.301905E 010.115056E 02-0.144210E 01-23.6010.000000E 00
20.113448E 02-0.128137E 01-0.143389E 010.301902E 010.115056E 02-0.144216E 01-6.3980.000000E 00
30.867602E 010.138728E 01-0.393779E 010.301899E 010.103971E 02-0.333762E 00-23.6080.000000E 00
40.102640E 02-0.200717E 00-0.118721E 010.301899E 010.103970E 02-0.333716E 00-6.3920.000000E 00
+ +# A.4.7 Solution of a non-layered elasto-plastic Mindlin plate. Example of Section 9.7, Fig. 9.6 + +Input data + +
MINDLINNON-LAYEREDEXAMPLE,SECTION9.7,FIG.9.6
2541591321390
11123813121167
21345~1015141389
31111213182322211617
41131415202524231819
10.00.0
30.250.0
50.50.0
110.00.25
130.250.25
150.500.25
210.00.50
230.250.50
250.500.50
1111
2110
+ + + +
3110
4110
5110
6101
10010
11101
15010
16101
20010
21101
22001
23001
24001
25011
1
10.920.30.011.00.04
1111
1111
UNIFORMLY DISTRIBUTED LOADING INTENSITY -0.01LB/SQ INCH
0
0.50.160030.0050.16003
0.020.160030.0020.16003
0.020.160030.0020.16003
0.020.160030.0020.16003
0.020.16030.0020.16003
0.020.160030.0020.16003
0.020.160030.0020.16003
0.020.160030.00206003
0.020.160030.0020.16003
0.020.160030.0020.16003
0.020.160030.0020.16003
0.020.160030.0020.16003
0.020.160030.0020.16003
0.020.160030.0020.06003
0.020.160030.0020.16003
0.020.160030.0020.16003
0.020.16003
0.020.160030.0020.16003
0.020.160030.0020.16003
0.020.160030.0026003
0.010.160030.0020.16003
0.010.160030.0020.16003
0.0050.160030.0020.16003
0.0050.160030.0020.16003
0.0050.160030.0020.1603
+ + + +Line printer output + +
MINDLIN NON-LAYERED EXAMPLE, SECTION 9.7, FIG. 9.6
NPOIN = 25NELEM = 4NVFIX = 16NTYPE = 5NNODE = 9
NMATS = 1NGAUS = 3NEVAB = 27NALGO = 2
NCRIT = 1NINCS = 39NLAPS = 0NSWIT = 0
ELEMENTPROPERTYNODE NUMBERS
11123813121167
213451015141389
31111213182322211617
41131415202524231819
NODEXY
10.000000.00000
2.125000.00000
3.250000.00000
4.375000.00000
5.500000.00000
60.00000.12500
70.000000.00000
8.25000.12500
90.000000.00000
10.50000.12500
110.00000.25000
12.12500.25000
13.25000.25000
14.37500.25000
15.50000.25000
160.00000.37500
170.000000.00000
18.25000.37500
190.000000.00000
20.50000.37500
210.00000.50000
22.12500.50000
23.25000.50000
24.37500.50000
25.50000.50000
NODECODEFIXED VALUES
11110.0000000.0000000.000000
21100.0000000.0000000.000000
31100.0000000.0000000.000000
diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_058.md b/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_058.md new file mode 100644 index 00000000..81fa644e --- /dev/null +++ b/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_058.md @@ -0,0 +1,131 @@ + + +
41100.0000000.0000000.000000
51100.0000000.0000000.000000
61010.0000000.0000000.000000
10100.0000000.0000000.000000
111010.0000000.0000000.000000
15100.0000000.0000000.000000
161010.0000000.0000000.000000
20100.0000000.0000000.000000
211010.0000000.0000000.000000
2210.0000000.0000000.000000
2310.0000000.0000000.000000
2410.0000000.0000000.000000
25110.0000000.0000000.000000
+ +NUMBER ELEMENT PROPERTIES + +
3.4472.0528.14846E-01.14812E-01-.67909E-020.
4.4472.1972.20658E-02.21696E-02-.51738E-020.
5.3750.1250.93182E-02.10703E-01-.45473E-020.
6.3750.2218.16282E-01.19151E-01-.37263E-020.
7.4718.0282.32303E-02.51552E-02-.17322E-020.
8.4718.1250.10243E-01.11671E-01-.53073E-030.
9.4718.2218.16015E-01.17815E-01-.10205E-020.
+ +ELEMENT NO. = 3 + +
1.0528.3028.42854E-02.24316E-02-.83085E-020.
2.0528.4472.21696E-02.20658E-02-.51738E-020.
3.1972.3028.51552E-02.32303E-02-.17322E-020.
4.1972.4472.94492E-02.83078E-02-.85898E-020.
5.1250.3750.10703E-01.93182E-02-.45473E-020.
6.1250.4718.11671E-01.10243E-01-.53073E-030.
7.2218.2782.14812E-01.14846E-01-.67909E-020.
8.2218.3750.19151E-01.16282E-01-.37263E-020.
9.2218.4718.17815E-01.16015E-01-.10205E-020.
+ +ELEMENT NO. = 4 + +
1.3028.3028.17182E-01.17182E-01-.45805E-020.
2.3028.4472.18488E-01.18023E-01-.33076E-020.
3.4472.3028.21135E-01.19267E-01-.13109E-020.
4.4472.4472.18023E-01.18488E-01-.33076E-020.
5.3750.3750.20733E-01.20733E-01-.17880E-020.
6.3750.4718.22807E-01.22787E-01-.23644E-030.
7.4718.2782.19267E-01.21135E-01-.13109E-020.
8.4718.3750.22787E-01.22807E-01-.23644E-030.
9.4718.4718.23695E-01.23695E-01.17833E-030.
$$ +\begin{array}{c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c} \cdot & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & \\ & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & \end{array} +$$ + + + +
INCREMENT NUMBER 30
LOAD FACTOR = .85600CONVERGENCE TOLERANCE = .10000
INITIAL OUTPUT PARAMETER = 0FINAL OUTPUT PARAMETER = 3
IN CONVERITERATION NUMBER 1
+ +DISPLACEMENT CHANGE NORM + +
.282E+00.280E+00.280E+00
+ +TOTAL + +
-.281E+00
+ +RESIDUAL NORM + +
.182E-10.896E-06.896E-06
+ +TOTAL + +
-.605E-07
+ +CONVERGENCE CODE 1 + +
IN CONVERITERATION NUMBER2
+ +DISPLACEMENT CHANGE NORM + +
.293E-06.294E-06.294E-06
+ +TOTAL + +
-.294E-06
+ +RESIDUAL NORM + +
.183E-11.266E-11.245E-11
+ +TOTAL + +
-.183E-11
+ +CONVERGENCE CODE O + +DISPLACEMENTS + +
NODEDISP.XZ-ROT.YZ-ROT.
10.0.0.
20.0..455052E+04
30.0..879322E+04
40.0..106738E+05
50.0..118879E+05
60..455052E+040.
70..410180E+03.410180E+03
8.101997E+04.289215E+04.742347E+04
90..984409E+02.623582E+03
10.139568E+040..102627E+05
110..879322E+040.
12.101997E+04.742347E+04.289215E+04
13.183493E+04.557560E+04.557560E+04
14.235881E+04.275674E+04.688004E+04
15.252626E+040..772635E+04
+ +MAX. NO. OF ITERATIONS = 60 + + + +
160..106738E+050.
170..623582E+03.984409E+02
18.235881E+04.688004E+04.275674E+04
190..230744E+03.230744E+03
20.325803E+040..389260E+04
210..118879E+050.
22.139568E+04.102627E+050.
23.252626E+04.772635E+040.
24.325803E+04.389260E+040.
25.349631E+040.0.
+ +REACTIONS + +
NODEFORCEXZ-MOMENTYZ-MOMENT
1.254174E-01-.405413E-03-.405413E-03
2-.704030E-01-.474595E-020.
3.489298E-01-.861086E-030.
4-.130462E+00-.178824E-020.
5.322264E-01-.228435E-020.
6-.704030E-010.-.474595E-02
100.-.368943E-020.
11.489298E-010.-.861086E-03
150.-.181699E-020.
16-.130462E+000.-.178824E-02
200.-.720662E-020.
21.322264E-010.-.228435E-02
220.0.-.368943E-02
230.0.-.181699E-02
240.0.-.720662E-02
250.-.132398E-02-.132398E-02
+ +STRESSES + +
G.P.X-COORD.Y-COORD.X-MOMENTY-MOMENTXY-MOMENTEFF.PL.STRAIN
ELEMENT NO. = 1
1.0528.0528-.99908E-03-.99908E-03-.23087E-01.57698E+04
2.0528.1972.51873E-03.14760E-02-.23082E-01.26193E+04
3.1972.0528.80061E-02.59482E-02-.20218E-010.
4.1972.1972.14760E-02.51873E-03-.23082E-01.26193E+04
5.1250.1250.86235E-02.86235E-02-.20786E-010.
6.1250.2218.15459E-01.16648E-01-.16501E-010.
7.2218.0282.59482E-02.80061E-02-.20218E-010.
8.2218.1250.16648E-01.15459E-01-.16501E-010.
9.2218.2218.21010E-01.21010E-01-.14518E-010.
+ + + +
ELEMENT NO. = 2
1.3028.0528.43677E-02.77768E-02-.14262E-010.
2.3028.1972.14580E-01.16625E-01-.14808E-010.
3.4472.0528.25755E-01.25762E-01-.11744E-010.
4.4472.1972.36231E-02.39462E-02-.88176E-020.
5.3750.1250.16121E-01.18645E-01-.78343E-020.
6.3750.2218.28118E-01.33194E-01-.64873E-020.
7.4718.0282.55108E-02.88899E-02-.29126E-020.
8.4718.1250.17512E-01.20166E-01-.91283E-030.
9.4718.2218.27549E-01.30853E-01-.17952E-020.
ELEMENT NO. = 3
1.0528.3028.77768E-02.43677E-02-.14262E-010.
2.0528.4472.39462E-02.36231E-02-.88176E-020.
3.1972.3028.88899E-02.55108E-02-.29126E-020.
4.1972.4472.16625E-01.14580E-01-.14808E-010.
5.1250.3750.18645E-01.16121E-01-.78343E-020.
6.1250.4718.20166E-01.17512E-01-.91283E-030.
7.2218.2782.25762E-01.25755E-01-.11744E-010.
8.2218.3750.33194E-01.28118E-01-.64873E-020.
9.2218.4718.30853E-01.27549E-01-.17952E-020.
ELEMENT NO. = 4
1.3028.3028.29634E-01.29634E-01-.79751E-020.
2.3028.4472.31762E-01.31040E-01-.57935E-020.
3.4472.3028.36223E-01.33145E-01-.23092E-020.
4.4472.4472.31040E-01.31762E-01-.57935E-020.
5.3750.3750.35776E-01.35776E-01-.31804E-020.
6.3750.4718.39413E-01.39460E-01-.46572E-030.
7.4718.2782.33145E-01.36223E-01-.23092E-020.
8.4718.3750.39460E-01.39413E-01-.46572E-030.
9.4718.4718.39997E-01.39997E-01.26371E-03.19186E+04
1.109200E+02.300000E+00.100000E-01.100000E+010..400000E-01 0.
+ +CONVERGENCE PARAMETERS + +
IFDIS = 1NCDIS =1110
IFRES = 1NCRES =1110
+ +UNIFORMLY DISTRIBUTED LOADING INTENSITY -0.01LB/SQ INCH + +
0TOTAL NODAL FORCES FOR EACH ELEMENT
1-.5208E-020.0..2083E-010.0.-.5208E-020.
0..2083E-010.0.-.5208E-020.0..2083E-01
0.0.-.5208E-020.0..2083E-010.0.
+ + + +
.2778E-010.0.
2-.5208E-020.0..2083E-010.0.-.5208E-020.
0..2083E-010.0.-.5208E-020.0..2083E-01
0.0.-.5208E-020.0..2083E-010.0.
.2778E-010.0.
3-.5208E-020.0..2083E-010.0.-.5208E-020.
0..2083E-010.0.-.5208E-020.0..2083E-01
0.-.5208E-020.0..2083E-010.0.
.2778E-010.0.
4-.5208E-020.0..2083E-010.0.-.5208E-020.
0..2083E-010.0.-.5208E-020.0..2083E-01
0.
.2778E-010.0.
INIncrement NUMBER 1
LOAD FACTOR = .50000 CONVERGENCE TOLERANCE = .10000 MAX. NO. OF ITERATIONS = 60
INITIAL OUTPUT PARAMETER = 0 FINAL OUTPUT PARAMETER = 3
IN CONVER ITERATION NUMBER 1
DISPLACEMENT CHANGE NORM
.100E+03 .100E+03 .100E+03
TOTAL
-.100E+03
RESIDUAL NORM
.845E-08 .662E-08 .628E-08
TOTAL
-.845E-08
CONVERGENCE CODE 1
IN CONVER ITERATION NUMBER 2
DISPLACEMENT CHANGE NORM
.918E-08 .908E-08 .897E-08
TOTAL
-.903E-08
RESIDUAL NORM
.265E-11 .200E-11 .295E-11
TOTAL
-.265E-11
CONVERGENCE CODE 0
DISPLACEMENTS
NODE DISP. XZ-ROT. YZ-ROT.
1 0. 0. 0.
2 0. 0. .261614E+04
+ + + +
30.0..505686E+04
40.0..615962E+04
50.0..687815E+04
60..261614E+040.
70..230157E+03.230157E+03
8.587914E+03.167781E+04.428957E+04
90..639453E+02.362274E+03
10.807234E+030..593553E+04
110..505686E+040.
12.587914E+03.428957E+04.167781E+04
13.105976E+04.323511E+04.323511E+04
14.136395E+04.160213E+04.398710E+04
15.146134E+040..447417E+04
160..615962E+040.
170..362274E+03.639453E+02
18.136395E+04.398710E+04.160213E+04
190..132888E+03.132888E+03
20.188400E+040..224070E+04
210..687815E+040.
22.807234E+03.593553E+040.
23.146134E+04.447417E+040.
24.188400E+04.224070E+040.
25.202089E+040.0.
+ +REACTIONS + +
NODEFORCEXZ-MOMENTYZ-MOMENT
1.124667E-01-.357597E-03-.357597E-03
2-.399935E-01-.292695E-020.
3.280665E-01-.486232E-030.
4-.754754E-01-.103874E-020.
5.186691E-01-.132162E-020.
6-.399935E-010.-.292695E-02
100.-.215061E-020.
11.280665E-010.-.486232E-03
150.-.105925E-020.
16-.754754E-010.-.103874E-02
200.-.417385E-020.
21.186691E-010.-.132162E-02
220.0.-.215061E-02
230.0.-.105925E-02
240.0.-.417385E-02
250.-.783842E-03-.783842E-03
+ + + +
G.P.STRESSESEX-COORD. ELEMENT NO. = 1
1.0528.0528-.61926E-03-.61926E-03-.15278E-010.
2.0528.1972.27063E-03.86852E-03-.14197E-010.
3.1972.0528.44517E-02.33436E-02-.11677E-010.
4.1972.1972.86852E-03.27063E-03-.14197E-010.
5.1250.1250.48794E-02.48794E-02-.12011E-010.
6.1250.2218.87584E-02.94485E-02-.95831E-020.
7.2218.0282.33436E-02.44517E-02-.11677E-010.
8.2218.1250.94485E-02.87584E-02-.95831E-020.
9.2218.2218.11999E-01.11999E-01-.84458E-020.
ELEMENT NO. = 2
1.3028.0528.24316E-02.42854E-02-.83085E-020.
2.3028.1972.83078E-02.94492E-02-.85898E-020.
etc.
+ +A.4.8 Solution of dynamic transient elasto-plastic problem by explicit time integration. Example of Section 10.7.2, Fig. 10.3 + +Input data + +
531021
SPHERICAL CAPEXAMPLE ,DYNPAK , SECTION 10.7.2 ,FIG. 10.3
63811224200231
1114678532
21691112131087
311114161718151312
411619212223201817
512124262728252322
612629313233302827
713134363738353332
813639414243403837
914144464748454342
1014649515253504847
122.270.00003222.47516.0020
422.271.33353722.47518.6690
622.272.66704222.47521.3360
922.274.00054722.47524.0030
+ + + +
1122.275.33405222.47526.6700
1422.276.6675322.6800.0000
16
1622.278.0010522.681.3335
19
1922.279.3345822.682.667021
2122.2710.66801022.684.000534
2422.2712.00151322.685.3340
42
2622.2713.33501522.686.6675
2922.2714.66851822.688.001031
3122.2716.00202022.689.3345
3422.2717.33552322.6810.6680
3622.2718.66902522.6812.0015
3922.2720.00252822.6813.3350
4122.2721.33603022.6814.6685
4422.2722.66953322.6816.0020
4622.2724.00303522.6817.3355
+ + + +DISTRIBUTED STEP PRESSURE P=600LB/IN SQ. + +
0010
10
1853
600.0600.0600.00.00.00.0
213108
600.0600.0600.00.00.00.0
3181513
600.0600.0600.00.00.00.0
4232018
600.0600.0600.00.00.00.0
5282523
600.0600.0600.00.00.00.0
6333028
600.0600.0600.00.00.00.0
7383533
600.0600.0600.00.00.00.0
8434038
600.0600.0600.00.00.00.0
9484543
600.0600.0600.00.00.00.0
10535048
600.0600.0600.00.00.00.0
+ +00000 + +Line printer output + +
SPHERICAL CAP EXAMPLE ,DYNPAK ,SECTION 10.7.2 ,FIG. 10.3
CONTROL PARAMETERS
NPOIN =53NELEM =10NVFIX =
NTYPE =3NNODE =8NDOFN =
NMATS =1NPROP =11NGAUS =
NDIME =2NSTRE =4NCRIT =
NPREV =0NCONM =0NLAPS =
NGAUM =3NRADS =1
+ + + +
ELEMENTPROPERTYNODE NUMBERS
1114678532
21691112131087
311114161718151312
411619212223201817
512124262728252322
612629313233302827
713134363738353332
813639414243403837
914144464748454342
1014649515253504847
122.2700.000NODEXY
422.2701.33410.00022.270
622.2702.66720.00022.475
922.2704.00130.00022.680
1122.2705.3344.51822.264
1422.2706.6685.52822.674
1622.2708.00161.03622.246
1922.2709.33571.04622.451
2122.27010.66881.05522.655
2422.27012.00291.55422.216
2622.27013.335101.58222.625
2922.27014.669112.07022.174
3122.27016.002122.08922.378
3422.27017.336132.10822.582
3622.27018.669142.58622.119
3922.27020.003152.63322.527
4122.27021.336163.10022.053
4422.27022.670173.12822.256
4622.27024.003183.15722.459
4922.27025.337193.61221.975
5122.27026.670203.67922.380
222.4750.000214.12321.885
722.4752.667224.16122.087
1222.4755.334234.19822.288
1722.4758.001244.63121.783
2222.47510.668254.71622.184
2722.47513.335265.13621.670
3222.47516.002275.18421.869
3722.47518.669285.23122.069
diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_059.md b/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_059.md new file mode 100644 index 00000000..611b89b5 --- /dev/null +++ b/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_059.md @@ -0,0 +1,67 @@ + + +
4222.47521.336295.63921.544
4722.47524.003305.74321.941
5222.47526.670316.13921.407
322.6800.000326.19621.604
522.6801.334336.25221.801
822.6802.667346.63621.258
1022.6804.001356.75821.650
1322.6805.334367.12921.098
1522.6806.668377.19421.292
1822.6808.001387.26021.487
2022.6809.335397.61820.927
2322.68010.668407.75821.312
2522.68012.002418.10320.744
2822.68013.335428.17720.935
3022.68014.669438.25221.126
3322.68016.002448.58320.549
3522.68017.336458.74120.928
3822.68018.669469.05920.344
4022.68020.003479.14220.531
4322.68021.336489.22620.719
4522.68022.670499.53020.128
4822.68024.003509.70620.498
5022.68025.337519.99619.901
5322.68026.6705210.08820.084
5310.18020.267
NODECODE
1101300250037004900
2101400260038005000
3101500270039005111
4001600280040005211
5001700290041005311
600180030004200
700190031004300
800200032004400
900210033004500
1000220034004600
1100230035004700
1200240036004800
+ + + +MATERIAL PROPERTIES + +
MATERIAL NO1
YOUNG MODULUS.1050E+08
POISSON RATIO.3000
THICKNESS0.
MASS DENSITY.2450E-03
ALPHA TEMPR0.
REFERENCE FO.2400E+05
HARDENING PAR.2143E+06
FRICT ANGLE0.
FLUIDITY PAR.1000E+05
EXP DELTA1.000
NFLOW CODE1.000
TIME STEPPING PARAMETERS
NSTEP=500NOUTD=10NOUTP=250
NREQD=1NREQS=1NACCE=1
IFUNC=1IFIXD=0MITER=0
KSTEP=0IPRED=1
DTIME=.4000E-06DTEND=.1000E-02DTREC=0.
AALFA=0.BEETA=0.DELTA=0.
GAAMA=0.AZERO=0.BZERO=0.
OMEGA=0.TOLER=0.
SELECTIVE OUTPUT REQUESTED FOR FOLLOWING
NODES1
G.P.1
TYPE OF ELEMENT, IMPLICIT=1,EXPLICIT=2
2222222
NODEINITIALX-DISP.INITIALY-DISP.
530.0.
NODEINITIALX-VELO.INITIALY-VELO.
530.0.
LOAD CASE TITLE - DISTRIBUTED STEP PRESSURE P=600LB/IN
LOAD INPUT PARAMETERS
POINT LOADS0
GRAVITY0
EDGE LOAD1
TEMPERATURE0
NO. OF LOADED EDGES = 10
LIST OF LOADED EDGES AND APPLIED LOADS
1853
+ + + +
600.000600.000600.0000.0000.0000.000
213108
600.000600.000600.0000.0000.0000.000
3181513
600.000600.000600.0000.0000.0000.000
4232018
600.000600.000600.0000.0000.0000.000
5282523
600.000600.000600.0000.0000.0000.000
6333028
600.000600.000600.0000.0000.0000.000
7383533
600.000600.000600.0000.0000.0000.000
8434038
600.000600.000600.0000.0000.0000.000
9484543
600.000600.000600.0000.0000.0000.000
10535048
600.000600.000600.0000.0000.0000.000
NODAL LUMPED MASSES
1.10000E+312.90632E-053.10000E+314.36354E-045.10000E+316.91129E-05
7.72039E-048.72039E-049.73365E-0410.73365E-0411.54175E-0412.54175E-04
13.29072E-0314.29072E-0315.54838E-0416.54838E-0417.21596E-0318.21596E-03
19.21994E-0320.21994E-0321.10823E-0322.10823E-0323.58081E-0324.58081E-03
25.10956E-0326.10956E-0327.35941E-0328.35941E-0329.36603E-0330.36603E-03
31.16206E-0332.16206E-0333.86965E-0334.86965E-0335.16404E-0336.16404E-03
37.50209E-0338.50209E-0339.51133E-0340.51133E-0341.21553E-0342.21553E-03
43.11566E-0244.11566E-0245.21816E-0346.21816E-0347.64368E-0348.64368E-03
49.65553E-0350.65553E-0351.26853E-0352.26853E-0353.14410E-0254.14410E-02
55.27182E-0356.27182E-0357.78387E-0358.78387E-0359.79830E-0360.79830E-03
61.32096E-0362.32096E-0363.17224E-0264.17224E-0265.32488E-0366.32488E-03
67.92236E-0368.92236E-0369.93934E-0370.93934E-0371.37268E-0372.37268E-03
73.20000E-0274.20000E-0275.37724E-0376.37724E-0377.10589E-0278.10589E-02
79.10784E-0280.10784E-0281.42360E-0382.42360E-0383.22732E-0284.22732E-02
85.42879E-0386.42879E-0387.11931E-0288.11931E-0289.12150E-0290.12150E-02
91.47361E-0392.47361E-0393.25415E-0294.25415E-0295.47940E-0396.47940E-03
97.13247E-0298.13247E-0299.13491E-02100.13491E-02101.10000E+31102.10000E+31
103.10000E+31104.10000E+31105.10000E+31106.10000E+31
DISPLACEMENTS AT TIME STEP250TIME.10000000000E-03
NODEX-DISPY-DISPNNODEX-DISPY-DISPNNODEX-DISPY-DISP
+ + + +
1-.80217E-36-.24592E-012.16169E-37-.24444E-013.80603E-36-.24278E-01
4-.48654E-03-.24378E-015-.47049E-03-.24082E-016-.10271E-02-.24452E-01
7-.95811E-03-.24277E-018-.89235E-03-.24136E-019-.16057E-02-.24657E-01
10-.12997E-02-.24370E-0111-.21111E-02-.25054E-0112-.19625E-02-.24907E-01
13-.18173E-02-.24753E-0114-.24674E-02-.25202E-0115-.24494E-02-.24886E-01
16-.27549E-02-.24939E-0117-.29046E-02-.24767E-0118-.30484E-02-.24575E-01
19-.31689E-02-.24317E-0120-.34286E-02-.23956E-0121-.38241E-02-.23993E-01
22-.37338E-02-.23818E-0123-.36581E-02-.23704E-0124-.46693E-02-.24321E-01
25-.39541E-02-.24145E-0126-.55750E-02-.25293E-0127-.50270E-02-.25239E-01
28-.44904E-02-.25223E-0129-.63674E-02-.26563E-0130-.53025E-02-.26494E-01
31-.68354E-02-.27549E-0132-.65796E-02-.27490E-0133-.63050E-02-.27373E-01
34-.68148E-02-.27616E-0135-.72371E-02-.27120E-0136-.61813E-02-.26275E-01
37-.70445E-02-.25897E-0138-.78704E-02-.25402E-0139-.49243E-02-.23242E-01
40-.78642E-02-.21832E-0141-.31809E-02-.18638E-0142-.51813E-02-.17807E-01
43-.71315E-02-.16874E-0144-.13257E-02-.13014E-0145-.55830E-02-.10958E-01
46.15858E-03-.72532E-0247-.16481E-02-.63066E-0248-.35144E-02-.55039E-02
49.71545E-03-.26453E-0250-.14109E-02-.13718E-0251.17196E-33-.61722E-33
52.47545E-33.53008E-3353-.45643E-33-.26357E-33540.0.
+ +STRESSES + +
G.P.RR-STRESSZZ-STRESSRZ-STRESSTT-STRESSMAX P.S.MIN P.S.ANGLEP.S.
ELEMENT NO. =1
1-.142403E+05-.105408E+04.188251E+04-.141343E+05-.790586E+03-.145038E+05-7.9680.
2-.137476E+05-.359556E+03.175780E+04-.138344E+05-.132611E+03-.139746E+05-7.3570.
3-.147863E+05-.306845E+03.360377E+03-.144528E+05-.297881E+03-.147952E+05-1.4250.
4-.130724E+05-.794640E+03.369614E+03-.134549E+05-.783523E+03-.130835E+05-1.7230.
ELEMENT NO. =2
1-.152776E+05-.476674E+03.112943E+04-.149803E+05-.390986E+03-.153633E+05-4.3390.
2-.123991E+05-.419294E+03.854064E+03-.129006E+05-.358713E+03-.124596E+05-4.0570.
3-.139979E+05-.424122E+03.544093E+03-.146986E+05-.402347E+03-.140197E+05-2.2920.
4-.137154E+05-.101627E+03.580061E+03-.133725E+05-.769561E+02-.137401E+05-2.4350.
ELEMENT NO. =3
1-.120725E+05.859252E+02.104211E+04-.137897E+05.174599E+03-.121612E+05-4.8640.
2-.155576E+05-.672336E+03.121601E+04-.144822E+05-.573651E+03-.156563E+05-4.6400.
3-.115157E+05-.754333E+02.151423E+04-.131835E+05.121596E+03-.117127E+05-7.4140.
4-.158138E+05-.995668E+03.202746E+04-.149737E+05-.723272E+03-.160862E+05-7.6520.
ELEMENT NO. =4
1-.133486E+05-.746264E+03.285467E+04-.135909E+05-.129786E+03-.139651E+05-12.1860.
2-.133564E+05-.736688E+03.284774E+04-.140809E+05-.123832E+03-.139692E+05-12.1450.
3-.161873E+05-.716272E+03.322767E+04-.145098E+05-.698999E+02-.168337E+05-11.3240.
4-.105957E+05-.963920E+03.210077E+04-.129019E+05-.525664E+03-.110339E+05-11.7840.
+ + + +
ELEMENT NO. =5
1-.168527E+05-.712967E+03.355630E+04-.151044E+05.358997E+02-.176015E+05-11.8910.
2-.965035E+04-.718838E+03.203275E+04-.122643E+05-.277959E+03-.100912E+05-12.2370.
3-.157929E+05-.601834E+03.309818E+04-.154933E+05.573147E+01-.164005E+05-11.0950.
4-.109494E+05-.556352E+03.189021E+04-.127634E+05-.223251E+03-.112825E+05-9.9940.
ELEMENT NO. =6
1-.136993E+05-.430264E+03.276735E+04-.153917E+05.123755E+03-.142533E+05-11.3210.
2-.130603E+05-.829663E+03.258448E+04-.139319E+05-.305957E+03-.135840E+05-11.4550.
3-.100891E+05-.428639E+03.185225E+04-.146315E+05-.856752E+02-.104321E+05-10.4900.
4-.167170E+05-.106309E+04.358040E+04-.158653E+05-.283044E+03-.174970E+05-12.2910.
ELEMENT NO. =7
1-.748198E+04-.154126E+03.155617E+04-.136080E+05.162653E+03-.779876E+04-11.5060.
2-.188538E+05-.171938E+04.502582E+04-.172146E+05-.354022E+03-.202192E+05-15.1990.
3-.498918E+04-.283243E+03.123802E+04-.118093E+05.225769E+02-.529500E+04-13.8760.
4-.211022E+05-.239729E+04.599980E+04-.181324E+05-.638225E+03-.228613E+05-16.3400.
ELEMENT NO. =8
1-.421590E+04-.332032E+03.167829E+04-.101971E+05.292698E+03-.484063E+04-20.4170.
2-.203671E+05-.292740E+04.739486E+04-.175571E+05-.213976E+03-.230805E+05-20.1500.
3-.579043E+04-.191795E+04.260848E+04-.859256E+04-.605623E+03-.710276E+04-26.7070.
4-.179717E+05-.280583E+04.716447E+04-.151334E+05.434134E+02-.208210E+05-21.6870.
ELEMENT NO. =9
1-.808480E+04-.179792E+04.441523E+04-.705104E+04.478552E+03-.103613E+05-27.2750.
2-.138434E+05-.337445E+04.706677E+04-.122206E+05.185344E+03-.174031E+05-26.7360.
3-.126711E+05-.279604E+04.796913E+04-.613091E+04.164120E+04-.171083E+05-29.1090.
4-.825151E+04-.490425E+04.337893E+04-.813863E+04-.280717E+04-.103486E+05-31.8250.
ELEMENT NO. =10
1-.175308E+05-.460688E+04.983514E+04-.674090E+04.699203E+03-.228369E+05-28.3470.
2-.149784E+04-.237914E+04.298633E+04-.325566E+04.108018E+04-.495715E+0440.8030.
3-.253662E+05-.151577E+05.109573E+05-.118829E+05-.817417E+04-.323498E+05-32.5110.
4.721668E+04.453440E+04.661719E+03.320558E+04.737105E+04.438003E+0413.1310.
$$ +\cdot \cdot \cdot +$$ + +etc. + + + +# A.4.9 Solution of dynamic transient elastoplastic problem by implicit/explicit approach. Example of Section 11.6.1, Fig. 11.4 + +Input data + +
531021
SPHERICALCAPEXAMPLE ,MIXDYN ,SECTION 11.6.1 ,FIG. 11.4
63811224200231
1114678532
21691112131087
311114161718151312
411619212223201817
512124262728252322
612629313233302827
713134363738353332
813639414243403837
914144464748454342
1014649515253504847
122.270.00002622.2713.3350
422.271.33352922.2714.6685
622.272.66703122.2716.0020
922.274.00053422.2717.3355
1122.275.33403622.2718.6690
1422.276.66753922.2720.0025
1622.278.00104122.2721.3360
1922.279.33454422.2722.6695
2122.2710.66804622.2724.0030
2422.2712.00154922.2725.3365
5122.2726.6700
+ + + +
222.47500.00001522.686.6675
722.4752.66701822.688.0010
1222.4755.33402022.689.3345
1722.4758.00102322.6810.6680
2222.47510.66802522.6812.0015
2722.47513.33502822.6813.3350
3222.47516.00203022.6814.6685
3722.47518.66903322.6816.0020
4222.47521.33603522.6817.3355
4722.47524.00303822.6818.6690
5222.47526.67004022.6820.0025
322.6800.00004322.6821.3360
522.681.33354522.6822.6695
822.682.66704822.6824.0030
1022.684.00055022.6825.3365
1322.685.33405322.6826.6700
+ +
110
210
310
5111
5211
5311
1
10500000.00.30.00.0002450.024000.0214285.71 0.0
10000.01.01.0
2001201111052012
0.00000500.0010.00.00.00.25000.50 0.0
0.00.00.00010000
1
1
1111111111
530.0
530.0
DISTRIBUTED STEP PRESSURE P=600LB/IN SQ.
0010
10
1853
600.0600.0600.00.00.00.0
213108
+ + + +
600.0600.0600.00.00.00.0
318 1513
600.0600.0600.00.00.00.0
423 2018
600.0600.0600.00.00.00.0
528 2523
600.0600.0600.00.00.00.0
633 3028
600.0600.0600.00.00.00.0
738 3533
600.0600.0600.00.00.00.0
843 4038
600.0600.0600.00.00.00.0
948 4543
600.0600.0600.00.00.00.0
1053 5048
600.0600.0600.00.00.00.0
+ +Line printer output + +SPHERICAL CAP EXAMPLE, MIXDYN, SECTION 11.6.1, FIG. 11.4 CONTROL PARAMETERS + +
NPOIN =53NELEM =10NVFIX =6
NTYPE =3NNODE =8NDOFN =2
NMATS =1NPROP =11NGAUS =2
NDIME =2NSTRE =4NCRIT =2
NPREV =0NCONM =0NLAPS =2
NGAUM =3NRADS =1
ELEMENTPROPERTYNODE NUMBERS
1114678532
21691112131087
311114161718151312
411619212223201817
512124262728252322
612629313233302827
713134363738353332
813639414243403837
914144464748454342
1014649515253504847
+ + + +
122.2700.000
422.2701.334
622.2702.667
922.2704.001
1122.2705.334
1422.2706.668
1622.2708.001
1922.2709.335
2122.27010.668
2422.27012.002
2622.27013.335
2922.27014.669
3122.27016.002
3422.27017.336
3622.27018.669
3922.27020.003
4122.27021.336
4422.27022.670
4622.27024.003
4922.27025.337
5122.27026.670
222.4750.000
722.4752.667
1222.4755.334
1722.4758.001
2222.47510.668
2722.47513.335
NODEXY
10.00022.270
20.00022.475
30.00022.680
4.51822.264
5.52822.674
61.03622.246
71.04622.451
81.05522.655
91.55422.216
101.58222.625
112.07022.174
122.08922.378
+ +
3222.47516.002
3722.47518.669
4222.47521.336
4722.47524.003
5222.47526.670
322.6800.000
522.6801.334
822.6802.667
1022.6804.001
1322.6805.334
1522.6806.668
1822.6808.001
2022.6809.335
2322.68010.668
2522.68012.002
2822.68013.335
3022.68014.669
3322.68016.002
3522.68017.336
3822.68018.669
4022.68020.003
4322.68021.336
4522.68022.670
4822.68024.003
5022.68025.337
5322.68026.670
132.10822.582
142.58622.119
152.63322.527
163.10022.053
173.12822.256
183.15722.459
193.61221.975
203.67922.380
214.12321.885
224.16122.087
234.19822.288
244.63121.783
+ + + +
254.71622.184407.75821.312
265.13621.670418.10320.744
275.18421.869428.17720.935
285.23122.069438.25221.126
295.63921.544448.58320.549
305.74321.941458.74120.928
316.13921.407469.05920.344
326.19621.604479.14220.531
336.25221.801489.22620.719
346.63621.258499.53020.128
356.75821.650509.70620.498
367.12921.098519.99619.901
377.19421.2925210.08820.084
387.26021.4875310.18020.267
397.61820.927
NODECODE
11019003700
21020003800
31021003900
40022004000
50023004100
60024004200
70025004300
80026004400
90027004500
100028004600
110029004700
120030004800
130031004900
140032005000
150033005111
160034005211
170035005311
18003600
+ +MATERIAL PROPERTIES + +
MATERIAL NO1
YOUNG MODULUS.1050E+08
POISSON RATIO.3000
THICKNESS0.
diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_060.md b/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_060.md new file mode 100644 index 00000000..4f5cfe16 --- /dev/null +++ b/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_060.md @@ -0,0 +1,803 @@ + + +MASS DENSITY .2450E-03 + +ALPHA TEMPR 0. + +REFERENCE FO .2400E+05 + +HARDENING PAR .2143E+06 + +FRICT ANGLE 0. + +FLUIDITY PAR .1000E+05 + +EXP DELTA 1.000 + +NFLOW CODE 1.000 + +TIME STEPPING PARAMETERS + +
NSTEP=200NOUTD=1NOUTP=20
NREQD=1NREQS=1NACCE=1
IFUNC=1IFIXD=0MITER=5
KSTEP=201IPRED=2
DTIME=.5000E-05DTEND=.1000E-02DTREC=0.
AALFA=0.BEETA=0.DELTA=.2500
GAAMA=.5000AZERO=0.BZERO=0.
OMEGA=0.TOLER=.1000E-03
+ +SELECTIVE OUTPUT REQUESTED FOR FOLLOWING + +
NODES1
G.P.1
+ +TYPE OF ELEMENT, IMPLICIT=1, EXPLICIT=2 + +
111111111
$INITIALX-DISP.INITIALY-DISP.
530.0.
NODEINITIALX-VELO.INITIALY-VELO.
530.0.
+ +LOAD CASE TITLE - DISTRIBUTED STEP PRESSURE P=600LB/IN + +LOAD INPUT PARAMETERS + +
POINT LOADS0
GRAVITY0
EDGE LOAD1
TEMPERATURE0
+ +NO. OF LOADED EDGES = 10 + +LIST OF LOADED EDGES AND APPLIED LOADS + +
1853
600.000600.000600.0000.0000.0000.000
213108
600.000600.000600.0000.0000.0000.000
3181513
600.000600.000600.0000.0000.0000.000
4232018
+ + + +
600.000600.000600.0000.0000.0000.000
52825235
600.000600.000600.0000.0000.0000.000
6333028
600.000600.000600.0000.0000.0000.000
7383533
600.000600.000600.0000.0000.0000.0008434038
600.000600.000600.0000.0000.0000.0009484543
600.000600.000600.0000.0000.0000.0000.000
10535048
+ +INITIAL ACCELERATION + +
-.18236E+08.91173E+07-.54133E+08-.35754E+06-.18247E+08-.13920E+07-.54061E+08-.93018E+06-.18228E+08.46432E+06
.91132E+07-.23578E+07-.54047E+08-.12424E+07-.18205E+08-.38329E+07-.53943E+08-.17443E+07-.18168E+08.87161E+06
.90830E+07-.49321E+07-.53867E+08-.21006E+07-.18128E+08-.63175E+07-.53711E+08-.25752E+07-.18076E+08.12872E+07
.90371E+07-.74684E+07-.53581E+08-.29436E+07-.18003E+08-.87955E+07-.53354E+08-.33945E+07-.17917E+08.16969E+07
.89579E+07-.99591E+07-.53151E+08-.37934E+07-.17869E+08-.11275E+08-.52913E+08-.42561E+07-.17812E+08.21275E+07
.89049E+07-.12464E+08-.52692E+08-.45797E+07-.17590E+08-.13681E+08-.52254E+08-.49619E+07-.17373E+08.24809E+07
.86861E+07-.14781E+08-.51805E+08-.55366E+07-.17632E+08-.16247E+08-.51858E+08-.62011E+07-.17977E+08.30987E+07
.89849E+07-.17535E+08-.51741E+08-.58186E+07-.16318E+08-.18180E+08-.50157E+08-.57372E+07-.15178E+08.28705E+07
.75929E+07-.18229E+08-.47544E+08-.82556E+07-.19283E+08-.22510E+08-.53199E+08-.12268E+08-.26298E+08.61105E+07
.13102E+08-.23821E+08-.53089E+08-.20703E+07-.50752E+07-.19969E+08-.42545E+08
2665555554444443444434
+ +DISPLACEMENTS AT TIME STEP 20 TIME .10000000000E-03 + +
NNODEX-DISPY-DISPNNODEX-DISPY-DISPNNODEX-DISPY-DISP
10.-.24848E-0120.-.24695E-0130.-.24531E-01
4-.49085E-03-.24866E-015-.47375E-03-.24549E-016-.10248E-02-.24900E-01
+ + + +
7-.96448E-03-.24721E-018-.90720E-03-.24585E-019-.16044E-02-.25115E-01
10-.13198E-02-.24815E-0111-.20699E-02-.25440E-0112-.19725E-02-.25296E-01
13-.18734E-02-.25137E-0114-.23709E-02-.25414E-0115-.25289E-02-.25071E-01
16-.26710E-02-.24905E-0117-.28704E-02-.24721E-0118-.30683E-02-.24530E-01
19-.31289E-02-.24240E-0120-.33998E-02-.23877E-0121-.37999E-02-.23915E-01
22-.37049E-02-.23738E-0123-.36218E-02-.23626E-0124-.46539E-02-.24268E-01
25-.39278E-02-.24100E-0126-.55678E-02-.25249E-0127-.50092E-02-.25194E-01
28-.44634E-02-.25182E-0129-.63749E-02-.26559E-0130-.52828E-02-.26501E-01
31-.68507E-02-.27577E-0132-.65811E-02-.27518E-0133-.62954E-02-.27401E-01
34-.68161E-02-.27664E-0135-.72519E-02-.27166E-0136-.61665E-02-.26284E-01
37-.70370E-02-.25898E-0138-.78691E-02-.25398E-0139-.49158E-02-.23247E-01
40-.78646E-02-.21836E-0141-.31688E-02-.18623E-0142-.51687E-02-.17788E-01
43-.71196E-02-.16855E-0144-.13242E-02-.13015E-0145-.55782E-02-.10963E-01
46.15487E-03-.72526E-0247-.16405E-02-.62975E-0248-.35098E-02-.54896E-02
49.71525E-03-.26446E-0250-.14080E-02-.13752E-02510.0.
520.0.530.0.540.0.
+ +STRESSES + +
G.P.RR-STRESSZZ-STRESSRZ-STRESSTT-STRESSMAX P.S.MIN P.S.ANGLEP.S.
ELEMENT NO. =1
1-.140401E+05-.297607E+03.102996E+02-.139518E+05-.297599E+03-.140401E+05-.0430.
2-.137677E+05-.163963E+03-.597302E+01-.138441E+05-.163961E+03-.137677E+05.0250.
3-.144323E+05.442655E+03.738033E+03-.141613E+05.479184E+03-.144688E+05-2.8340.
4-.133330E+05-.936998E+03.632685E+03-.136731E+05-.904790E+03-.133652E+05-2.9140.
ELEMENT NO. =2
1-.152286E+05-.445944E+02.890663E+03-.148846E+05.747159E+01-.152807E+05-3.3460.
2-.125683E+05-.504156E+03.703159E+03-.130704E+05-.463311E+03-.126091E+05-3.3240.
3-.135150E+05-.291803E+03.500085E+03-.144796E+05-.272918E+03-.135339E+05-2.1630.
4-.143944E+05-.155896E+03.601210E+03-.137405E+05-.130555E+03-.144197E+05-2.4140.
ELEMENT NO. =3
1-.114077E+05.321093E+03.890750E+03-.133350E+05.388356E+03-.114749E+05-4.3180.
2-.163302E+05-.869776E+03.130850E+04-.149798E+05-.759812E+03-.164402E+05-4.8040.
3-.119282E+05-.302176E+02.202031E+04-.130545E+05.303478E+03-.122619E+05-9.3790.
4-.153434E+05-.105963E+04.231258E+04-.149313E+05-.694548E+03-.157084E+05-8.9710.
ELEMENT NO. =4
1-.140507E+05-.635684E+03.272355E+04-.136079E+05-.103829E+03-.145826E+05-11.0500.
2-.129503E+05-.771685E+03.251263E+04-.139120E+05-.273660E+03-.134483E+05-11.2110.
3-.162765E+05-.534438E+03.331560E+04-.144035E+05.135395E+03-.169463E+05-11.4210.
4-.105661E+05-.107081E+04.217283E+04-.128314E+05-.597214E+03-.110397E+05-12.2960.
ELEMENT NO. =5
1-.169374E+05-.616907E+03.343220E+04-.150478E+05.755061E+02-.176298E+05-11.4060.
+ + + +
2-.985701E+04-.929878E+03.189549E+04-.123230E+05-.544081E+03-.102428E+05-11.5040.
3-.158004E+05-.584241E+03.320362E+04-.154607E+05.627411E+02-.164474E+05-11.4180.
4-.109975E+05-.917806E+03.197734E+04-.128391E+05-.543789E+03-.113715E+05-10.7110.
ELEMENT NO. =6
1-.137487E+05-.310136E+03.270211E+04-.153590E+05.212832E+03-.142716E+05-10.9540.
2-.131377E+05-.977521E+03.241519E+04-.139613E+05-.515392E+03-.135998E+05-10.8320.
3-.100168E+05-.385414E+03.191198E+04-.146078E+05-.197402E+02-.103824E+05-10.8270.
4-.167785E+05-.117197E+04.363810E+04-.159001E+05-.365543E+03-.175849E+05-12.4980.
ELEMENT NO. =7
1-.722734E+04.111484E+03.143221E+04-.134652E+05.381084E+03-.749694E+04-10.6610.
2-.192763E+05-.178155E+04.492010E+04-.173694E+05-.492791E+03-.205651E+05-14.6780.
3-.478449E+04-.226073E+03.140219E+04-.117250E+05.170709E+03-.518127E+04-15.8000.
4-.209663E+05-.258450E+04.619192E+04-.181578E+05-.693317E+03-.228575E+05-16.9840.
ELEMENT NO. =8
1-.437572E+04-.157141E+03.160307E+04-.101811E+05.382897E+03-.491576E+04-18.6180.
2-.205255E+05-.290008E+04.724227E+04-.175949E+05-.306030E+03-.231196E+05-19.7070.
3-.569987E+04-.196356E+04.272300E+04-.856768E+04-.529486E+03-.713394E+04-27.7740.
4-.178667E+05-.303932E+04.725350E+04-.151639E+05-.811257E+02-.208249E+05-22.1870.
ELEMENT NO. =9
1-.820383E+04-.168373E+04.434615E+04-.704395E+04.489173E+03-.103767E+05-26.5630.
2-.140072E+05-.338917E+04.689124E+04-.122639E+05.945125E+00-.173973E+05-26.1950.
3-.124091E+05-.260385E+04.807719E+04-.598991E+04.194216E+04-.169551E+05-29.3720.
4-.832578E+04-.512213E+04.347770E+04-.822056E+04-.289508E+04-.105528E+05-32.6350.
ELEMENT NO. =10
1-.175228E+05-.441212E+04.989584E+04-.667527E+04.902688E+03-.228376E+05-28.2390.
2-.153824E+04-.224010E+04.286990E+04-.322046E+04.100211E+04-.478044E+0441.5140.
3-.253810E+05-.152556E+05.109737E+05-.119169E+05-.823306E+04-.324035E+05-32.6170.
4.708117E+04.431688E+04.684627E+03.310099E+04.724144E+04.415661E+0413.1750.
$$ +\begin{array}{c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c} \end{array} +$$ + + + +# AUTHOR INDEX + +Abel, J. F., 11 + +Ahmad, S., 214, 429, 499, 500 + +Alexander, J. M., 496 + +Ang, A. H. L., 373 + +Argyris, J. H., 31, 499 + +Armen, H., 268, 373, 497 + +Atkinson, J. H., 496 + +Backlund, J., 373 + +Banerjee, P. K., 502 + +Baron, M. L., 94 + +Barrer, R. M., 94 + +Bashur, F. K., 500 + +Basu, A. K., 500 + +Bathe, K. J., 214, 429, 463, 499 + +Batoz, J. L., 498 + +Bhaumik, A. K., 373 + +Bazant, Z. P., 496 + +Belytschko, T., 462 + +Berg, C. A., 269 + +Bergan, P. G., 498, 501 + +Bettess, P., 502 + +Bhat, P. D., 496 + +Bicanic, N., 429, 496, 499 + +Bird, R. B., 497 + +Bishop, A. W., 268 + +Bland, D. R., 268 + +Bolourchi, S., 499, 500 + +Booth, A. D., 31 + +Brandt, A., 501 + +Bransby, P., 496 + +Brew, J. S., 429, 501 + +Bridgman, P. W., 268 + +Brotton, D. M., 429, 501 + +Bushnell, D., 498 + +Cathie, D. N., 502 + +Cattopadhyay, A., 498 + +Cescotto, S., 498 + +Chang, C. T., 497 + +Chang, T. S., 268 + +Cohen, M., 214, 373, 499 + +Cope, R. J., 500 + +Cormeau, I., 119, 317, 318, 500 + +Cowper, G. R., 153 + +Crisfield, M. A., 498, 501 + +Darwin, D., 500 + +Davies, J. D., 499 + +Davies, T. G., 502 + +Davis, E. M., 268 + +Deruntz, J. A., 463, 497 + +Desai, C. S., 11 + +Dhatt, G., 498 + +Dinis, L. M. S., 318, 500 + +Duncan, W., 500 + +Dunne, P. C., 499 + +Duwez, P., 318 + +Dym, C. L., 153 + +Elsawaf, A. F., 501 + +Epstein, H. I., 429 + +Fanelli, M., 501 + +Felippa, C. A., 463, 497, 501 + +Fletcher, R., 501 + +Frederick, D., 268 + +Fredriksson, B., 497 + +Frey, F., 498, 500 + +Fried, I., 501 + +Fröier, M., 269 + +Gallagher, R. H., 11, 498 + +Geradin, M., 500 + +Gilbert, R. F., 500 + +Gomez, A. E., 496 + +Goodier, J. N., 214 + +Haisler, W., 318 + +Hanü, F. R., 500 + + + +Hanley, J. T., 373 + +Harley, S. J., 499 + +Haroun, M., 214, 499 + +Hestenes, M., 501 + +Hill, R., 31, 94, 119, 214, 268, 497 + +Hinton, E., 11, 31, 94, 152, 153, 214, 268, 373, 429, 430, 463, 496, 497, 498, 499, 501 + +Hodge, P. G., 268 + +Hoffman, O., 268 + +Hogge, M. A., 500 + +Horrigmoe, G., 498 + +Hughes, T. J. R., 153, 214, 318, 373, 462, 463, 497, 499 + +Hutchinson, J. W., 497 + +Irons, B. M., 214, 429, 499, 500, 501 + +Jennings, A., 94, 501 + +Jofreit, J. C., 500 + +Johnarry, T., 500 + +Johnson, G. R., 429 + +Kanchi, M. B., 318 + +Kanoknukulchai, S., 153 + +Kelly, D. W., 502 + +Kemp, K. O., 373 + +Kershaw, D. S., 501 + +King, I. P., 31, 268 + +Knott, J. F., 497 + +Koiter, W. T., 268 + +Kostem, C. N., 373, 498 + +Krakeland, B., 499 + +Kramer, J. M., 269 + +Krieg, D. B., 269 + +Krieg, R. D., 269 + +Kulak, R. F., 269 + +Laird, W. M., 497 + +Larsen, P. K., 499 + +Leung, K. H., 497 + +Levine, H. S., 268, 373, 497 + +Levy, N., 497 + +Lightfoot, E. N., 497 + +Lin, C. S., 500 + +Lin, T. H., 429 + +Liu, S. C., 429 + +Liu, W. K., 462 + +Lopez, L. A., 373 + +Lyness, J. F., 31, 497 + +Mackerle, J., 497 + +Malik, G. M., 501 + +Malkus, D., 499 + +Marcal, P. V., 497 + +Martins, R. A. F., 500 + +Matthies, H., 500 + +McClintock, F., 497 + +McNiece, G. M., 373, 500 + +Metzler, J., 501 + +Mohraz, B., 496 + +Mote, C. D. J., 501 + +Mullen, R., 462 + +Mustoe, G. G. W., 502 + +Nagarajan, S., 429, 463 + +Nayak, G. C., 268, 318, 429, 496, 500 + +Naylor, D. J., 496 + +Nickell, R. E., 429 + +Nilsson, L., 496 + +Noor, A. K., 499 + +Oden, J. T., 11, 31 + +Olszak, W., 317 + +Ostergren, W. J., 497 + +Owen, D. R. J., 11, 31, 94, 152, 153, 214, 268, 317, 318, 373, 429, 430, 496, 497, 499, 500 + +Ozdemir, H., 429, 463 + +Pande, G. N., 318 + +Park, K. C., 463, 497 + +Paul, D. K., 463 + +Peano, A. M., 501 + +Pencold, D. A., 500 + +Perzyna, P., 317 + +Phillips, D. V., 496 + +Pica, A., 429, 498, 501 + +Pifko, A., 268, 373, 497 + +Pister, K. S., 463 + +Polak, E., 501 + +Popov, E. P., 429, 463, 499 + +Powell, G. H., 502 + +Prager, W., 268 + +Prakash, A., 94, 318 + +Price, J. W. H., 496 + +Pugh, E. D. L., 499 + +Putali, R. S., 498 + +Raghavan, K. S., 430 + +Ralston, A., 94 + +Ramm, E., 499 + +Rao, P. U., 500 + +Rao, S. S., 430 + +Razzaque, A., 499 + +Reeves, C. M., 501 + + + +Reisemann, W., 318 + +Rheinboldt, W. C., 502 + +Riccioni, R., 501 + +Rice, J. R., 497 + +Rock, T. A., 429 + +Rosengrun, G. F., 497 + +Row, D. G., 502 + +Sachs, G., 268 + +Sakurai, T., 268 + +Salonen, E. M., 496 + +Salvadori, M. G., 94 + +Samuelsson, A., 269 + +Sandella, L., 501 + +Scharpf, D. W., 31 + +Schnobrich, W. C., 496, 500 + +Schofield, A. N., 496 + +Schrem, E., 502 + +Schreyer, H. L., 269 + +Seegerlind, L. J., 31 + +Shames, I. H., 153 + +Shantaram, D., 430, 499 + +Shieh, C. L., 496 + +Skelland, A. N. P., 497 + +Soreide, T., 501 + +Stewart, W. E., 497 + +Stiefel, E., 501 + +Strang, G., 500 + +Stricklin, J. A., 318 + +Taylor, R. L., 153, 318, 463, 499 + +Timoshenko, S. P., 214 + +Too, J. M., 499 + +Tracey, D. M., 497 + +Turvey, G., 373 + +Underwood, P., 501 + +Valanis, K. C., 496 + +Valliappan, S., 31, 268 + +Wachspress, E. L., 501 + +Warner, R. F., 500 + +Wegmuller, A. W., 373, 498 + +Wegner, R., 500 + +White, G. N., 268 + +Wilson, E. L., 214, 429, 463, 501 + +Wood, R. D., 498 + +Wroth, C. P., 496 + +Yamada, Y., 268 + +Yen, H. J., 462 + +Yoshimura, N., 268 + +Zave, P., 502 + +Zienkiewicz, O. C., 11, 31, 94, 268, 317, 318, 429, 496, 497, 499, 502 + + + + + +# SUBJECT INDEX + +Accelerogram, 399, 425, 427 + +Ahmad elements, 489 + +Alternative form of the yield criterion, 229 + +Alternative material models, 465 + +Array initialisation, 238, 297 + +Associated plasticity, 224, 273 + +Axisymmetric Mindlin plates, 372 + +Axisymmetric problems, one dimension, 92 + +Axisymmetric solids, elastic expressions, 165 + +Back substitution, 45, 48 + +Backward difference method, 274 + +Banded equations, 45, 58 + +Bauschinger effect, 90, 222, 309 + +Beams on elastic foundations, 151 + +Berg yield criterion, 265 + +BFGS procedures, 491 + +Bingham plastic, 483 + +Body forces, 164 + +Boundary data, 38, 206 + +Boundary element methods, 493 + +Boundary tractions, 164 + +Buffer area, 195 + +Central difference time stepping scheme, 388 + +Circular plate, elasto-plastic, 264 + +Cohesion, 219 + +Combined initial/tangential algorithm, 21, 41, 206 + +Computational techniques, 495 + +Concrete nonlinearity, 477 + +Conditional stability, 276, 302, 391, 437 + +Conjugate gradient method, 492 + +Consistent load vector, 173, 183, 188, 214 + +Constitutive matrix, D: + +Dynamic applications, 413 + +Elastic, 163, 165, 167, 169, 192, 193, 232, 233 + +Elasto-plastic, 227, 244 + +Elasto-plastic Mindlin plates, 326 + +Visco-plastic, 274, 286 + +Convergence, 14, 21, 65, 72, 109, 212, 267, 297, 336, 451 + +Coupled-field problems, 487 + +Crack tip elements in plasticity, 487 + +Creep buckling, 317 + +Critical state model, 476 + +Cylinder: + +Elasto-plastic, 262 + +Elasto-viscoplastic, 310 + +Damping forces, 379, 390 + +Deformation Jacobian matrix, 382, 404 + +Diagonal mass matrix, 389, 392, 410 + +Distortional strain energy, 219, 265 + +Distributed edge loading, 184 + +Drucker–Prager yield criterion, 220, 230 + +Dynamic dimensioning, 174, 238, 396 + +Dynamic equilibrium equations, 378 + +Dynamic relaxation, 493 + +Dynamic transient analysis; 377 + +Discretisation by isoparametric elements, 379 + +Equilibrium equations, 378 + +Geometric nonlinearity, 382 + +Modelling of nonlinearities, 381 + +Effective, generalised or equivalent plastic strain, 223 + +Effective, generalised or equivalent stress, 218 + +Effective stiffness matrix, implicit dynamic, 435 + +Effective stress level, 239 + + + +Elasto-plastic general solution process, 235 + +Elasto-plastic one-dimensional problems, 26 + +Elasto-plastic stress/strain relation, 224 + +Elasto-plasticity, matrix formulation, 227 + +Elasto-plasticity, two-dimensional, 232 + +Element, one-dimensional, 24, 25, 100 + +Element shape functions, 24, 158, 159, 160, 179 + +Endochronic theory, 479 + +Equation assembly and solution, 42, 194 + +Equation reduction or elimination, 45 + +Equation resolution, 21, 57, 194 + +Equation solution, numerical example, 43 + +Equilibrium correction, 101, 107, 276, 289 + +Equilibrium equations, 13, 236, 275, 321 + +Error diagnostics, 200, 202, 214, 360 + +Euler-Bernoulli beam theory, 121 + +Euler's rule, 99, 273 + +Explicit time stepping, 273, 302, 377, 378, 431 + +Failure criterion, 223 + +Flow problems, 480 + +Flow rule, 224 + +Flow vector, 227, 233, 241, 338, 419 + +Fluidity parameter, 97, 273 + +Forward difference method, 273 + +Fracture mechanics, 484 + +Friction slider, 95 + +Frontal equation solution, 194 + +Further applications, 465 + +Galerkin process, 23, 29 + +Gas diffusion, 22, 68 + +Gaussian direct elimination, 45 + +Gaussian quadrature data, 179 + +Geometric data, 36, 206 + +Geometric nonlinearity, 274, 316, 382 + +Global shape functions, 23 + +Gravity dam, seismic example, 424 + +Gravity loading, 183 + +Green strains, 383 + +Groundwater flow problems, 90 + +Heat conduction, 22, 29, 66 + +Heterosis element, 319, 325, 370 + +Hierarchical formulation, 325 + +Hyperelastic problems, 25 + +Implicit/explicit time stepping, 377, 431, 434 + +Implicit time stepping, 274, 302, 377, 431 + +Implicit trapezoidal time stepping scheme, 274, 302 + +Improved numerical techniques, 466, 490 + +Incrementation of load, 60, 110, 210 + +Inertia forces, 379 + +Initial stiffness method, 20, 29, 41, 206 + +In-plane deformation in plates, 372 + +Input data, 35, 205, 281, 399 + +Instructions for preparing input data for dynamic transient problems: + +Programs DYNPAK and MIXDYN, 521 + +Instructions for preparing input data for one-dimensional problems: + +Program ELPLAS, 506 + +Program NONLAS, 506 + +Program QUITER, 503 + +Program QUNEWT, 505 + +Program TIMLAY, 509 + +Program TIMOSH, 508 + +Program UNVIS, 507 + +Instructions for preparing input data for plane, axisymmetric and plate bending problems: + +Programs MINDLIN and MIND-LAY, 517 + +Program PLANET, 511 + +Program VISCOUNT, 516 + +Interactive computing, 495 + +Internal friction angle, 219 + +Isoparametric elements: + +Lagrangian 9-node, 5, 157 + +Linear 4-node, 5, 157 + +Serendipity, 8-node, 5, 157 + +Isoparametric finite element representation, 169 + +Isotropic hardening, 222 + +J contour integral, 485 + +Jacobian matrix, 17, 24, 171, 181 + +Kinematic hardening, 222, 309 + +Kirchhoff thin plate theory, 319 + +Lagrangian description, 382 + +List of computer programs, 466 + +List of subroutines: + +One-dimensional applications, 467 + +Two-dimensional applications, 469 diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_061.md b/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_061.md new file mode 100644 index 00000000..2253282f --- /dev/null +++ b/.raw/FiniteElementsinPlasticityTheoryandPractice/FiniteElementsinPlasticityTheoryandPractice_061.md @@ -0,0 +1,596 @@ + + +Load factor, 60, 210 + +Load vector, 13, 24, 183, 188, 405 + +Locking material, 30, 92 + +Lumped mass matrix, 389, 392, 410 + +Material properties, 37, 207, 281 + +Mathematical theory of plasticity, 215 + +Matrix inversion, 288 + +Maxwell model, 117, 302, 305 + +Mechanical sublayer method, 304 + +Metal forming problems, 482 + +Method of direct iteration, 14, 24, 40, 63 + +Method of successive approximations, 14 + +Midside nodal coordinate generation, 178, 341, 413 + +Mindlin plates: +Elastic expressions, 167 + +Mindlin plates, elasto-plastic; 319 + +Discretisation, 324 + +Equilibrium equations, 321 + +Mindlin plates, elasto-plastic layered; 326 + +Nonlinear equilibrium equations, 327 + +Program structure, 355 + +Mindlin plates, elasto-plastic non-layered; 327 + +Nonlinear equilibrium equations, 329 + +Program structure, 331 + +Mohr-Coulomb yield criterion, 219, 230, 234 + +Newmark time stepping scheme, 432 + +Newton-Raphson method, 15, 24, 40, 68 + +No-tension model, 477 + +Non-associated flow rule, 476 + +Nonlinear elastic problems, 25, 74 + +Non-Newtonian fluid flow, 482 + +Normality condition, 224 + +Notched bend specimen, 5 + +Numerical integration, 174 + +Octahedral shear stress, 218 + +One-dimensional FORTRAN programs; 33 + +Direct iteration of quasiharmonic problems, 63 + +Elasto-plastic problems, 78 + +Elasto-viscoplastic problems, 104 + +Newton-Raphson solution of quasi-harmonic problems, 68 + +Nonlinear elastic problems, 74 + +One-dimensional nonlinear problems, 13 + +Output of results, 58, 211, 258, 342, 363, 414 + +Overlay method, 90, 304, 316 + +Overlay simulation of: Four parameter viscous model, 309 + +Three element viscous model, 309 + +Visco-elastic model, 308 + +Visco-elastic-plastic four parameter model, 309 + +Pi plane, 217 + +Piecewise linear strain hardening representation, 266 + +Piola-Kirchhoff stresses, 386 + +Plane Strain, elastic expressions, 164 + +Plane Stress, elastic expressions, 162 + +Plastic multiplier, 224 + +Plastic potential, 224, 273 + +Power law pseudoplastic, 483 + +Prandtl-Reuss equations, 225 + +Predictor-corrector algorithm, 434, 436 + +Prescribed displacements in equation solution, 46 + +Principal stress evaluation, 258 + +Profile equation solver, 436, 440 + +Program structure, 8, 34, 104, 134, 235, 281, 331, 355, 392, 440 + +Programming notation, 10 + +Pseudo-loads, viscoplastic, 100, 275 + +Quasi-harmonic equation, 22, 63, 68 + +Quasi-Newton method, 491 + +Rayleigh damping, 391 + +Residual forces, 15, 71, 76, 81, 102, 236, 249, 344, 364 + +Sample input data and line printer output: + +Dynamic transient elasto-plastic explicit time integration example, 567 + +Dynamic transient elasto-plastic implicit/explicit example, 578 + +Elasto-plastic layered Timoshenko beam, 537 + +Non-layered elasto-plastic Mindlin plate problem, 558 + +One-dimensional direct iteration quasi harmonic example, 529 + +One-dimensional elasto-plastic problem, 531 + + + +Sample input data—contd. + +One-dimensional elasto-viscoplastic problem, 535 + +Two-dimensional elasto-plastic problem, 542 + +Two-dimensional elasto-viscoplastic problem, 550 + +Seismic analysis, 377, 399, 424 + +Selective integration, 128, 325, 482 + +Shape function derivatives: + +Cartesian, 171, 182 + +Local, 171 + +Shape function evaluation, 179, 346 + +Shells, elasto-plastic and geometrically nonlinear, 488 + +Singular points on the yield surface, 234 + +Space diagonal, 217 + +Sphere: + +Elasto-plastic, 267 + +Elasto-viscoplastic, 315 + +Spherical shell, dynamic example, 421, 458 + +Starting algorithm for central difference scheme, 390 + +Steady state conditions, 104, 109, 279, 297 + +Stiffness matrix, 13, 24, 28, 100, 127, 142, 173, 244, 283, 348, 367, 439, 447 + +Strain energy function, 25 + +Strain hardening, 26, 222, 223 + +Strain matrix, B, 172, 191 + +Strain matrix, geometric nonlinear, 382, 395 + +Strain softening, 223 + +Stress intensity factor, 485 + +Stress invariants, 216, 233 + +Stress space, 217 + +Subroutines, elasto-plastic (additional): DIMEN, 238 + +FLOWPL, 243 + +INVAR, 239 + +LINEAR, 247 + +Master segment, 260 + +OUTPUT, 258 + +RESIDU, 249 + +STIFFP, 244 + +YIELDF, 241 + +ZERO, 238 + +Subroutines, elasto-plastic layered Mindlin plates (additional): + +CHECK1, 360 + +DEPMPA, 360 + +Subroutines—contd. + +FEAM, 355 + +LAYMPA, 360 + +MDMPA, 362 + +OUTMPA, 363 + +RESMPA, 364 + +STIMPA, 367 + +STRMPA, 369 + +Subroutines, elasto-plastic nonlayered Mindlin plates: + +CONVMP, 336 + +DIMMP, 338 + +FEMP, 334 + +FLOWMP, 338 + +GRADMP, 340 + +INVMP, 340 + +MINDPB, 341 + +NODEXY, 341 + +OUTMP, 342 + +RESMP, 344 + +SFR2, 346 + +STIFMP, 348 + +STRMP, 353 + +SUBMP, 354 + +VZERO, 354 + +ZEROMP, 354 + +Subroutines, elasto-plastic (standard): + +ALGOR, 209 + +CONVER, 212 + +INCREM, 210 + +INPUT, 205 + +Subroutines, elasto-viscoplastic (additional): + +FLOWVP, 294 + +INVERT, 288 + +Master segment, 299 + +STEADY, 297 + +STEPVP, 289 + +STIFVP, 283 + +STRESS, 295 + +TANGVP, 286 + +ZERO, 297 + +Subroutines, elasto-viscoplastic transient dynamic analysis: + +BLARGE, 395 + +CONTOL, 396 + +DYNPAK, 392 + +EXPLIT, 396 + +FIXITY, 397 + +FLOWVP, 398 + +FUNCTA, 399 + +FUNCTS, 399 + +INPUTD, 399 + + + +Subroutines—contd. + +INTIME, 401 + +INVAR, 403 + +JACOBD, 404 + +LINGNL, 404 + +LOADPL, 405 + +LUMASS, 410 + +MODPS, 413 + +NODXYR, 413 + +OUTDYN, 414 + +PREVOS, 416 + +RESVPL, 417 + +YIELDF, 419 + +Subroutines, implicit/explicit transient dynamic analysis (additional): + +ADDBAN, 444 + +ADDRES, 444 + +COLMHT, 445 + +DECOMP, 445 + +DINTOB, 446 + +GEOMST, 446 + +GSTIFF, 447 + +IMPEXP, 449 + +ITRATE, 451 + +LINKIN, 452 + +MIXDYN, 442 + +MULTPY, 454 + +REDBAK, 455 + +RESEPL, 456 + +Subroutines, one-dimensional: + +ASSEMB, 49 + +ASTIF1, 70 + +BAKSUB, 54 + +CONUND, 72 + +CONVP, 109 + +DATA, 35 + +GREDUC, 51 + +INCLOD, 60, 110 + +INCVP, 107 + +INITAL, 59 + +Master segment, 61 + +Master segment (viscoplasticity), 111 + +MONITR, 65 + +NONAL, 40 + +REFOR1, 71 + +REFOR2, 76 + +REFOR3, 81 + +RESOLV, 57 + +RESULT, 58 + +STIFF1, 63 + +STIFF2, 75 + +STIFF3, 78 + +STUNVP, 106 + +Subroutines, Timoshenko beam analysis: + +BEAM, 135 + +BEAML, 144 + +LAYER, 147 + +REFORB, 137 + +RFORBL, 146 + +STIFBL, 145 + +STIFFB, 136 + +Subroutines, two-dimensional (elastic): + +BMATPB, 191 + +BMATPS, 191 + +CHECK1,200 + +CHECK2, 202 + +DBE, 194 + +ECHO, 201 + +FRONT, 194 + +GAUSSQ, 179 + +JACOB2, 181 + +LOADPB, 188 + +LOADPS, 183 + +MODPB, 193 + +MODPS, 192 + +NODEXY, 178 + +SFR2, 179 + +Substructuring, 493 + +Subterranean cavity, viscoplastic, 314 + +Tangent modulus, 26, 225 + +Tangential stiffness, 20, 26, 28, 236, 275, 327, 329 + +Tangential stiffness method, 20, 40, 206 + +Theorem of minimum total potential energy, 44 + +Time-step limitations: + +Dynamic transient, 391, 426, 437 + +Elasto-viscoplastic, 102, 276 + +Timoshenko beam analysis; 121 + +Basic assumptions, 122 + +Element stress resultants, 128 + +Finite element idealisation, 125 + +Formulation of the stiffness matrix, 127, 142 + +Layered approach, 122, 141 + +Non-layered approach, 121, 122 + +Solution of nonlinear equations, 132, 143 + +Timoshenko layered beam program, TIMLAY, 144 + +Timoshenko non-layered beam program, TIMOSH, 135 + +Tolerance value, 65, 72, 298 + +Tresca yield criterion, 217, 230, 234 + + + +# SUBJECT INDEX + +Unconditional stability, 276, 302, 437 + +Uniaxial yield stress, 26, 219 + +Uniaxial tension test, 26, 225 + +Virtual work, 124, 162 + +Viscoelastic behaviour, 305, 308 + +Viscoplastic: + +Strain increment, 273 + +Stress increment, 274, 295 + +Viscoplastic computational procedure: + +One dimension, 103 + +Two dimensions, 278 + +Viscoplastic element stiffness formulation, 283 + +Viscoplastic flow function, 273, 286 + +Viscoplastic strain rate, 100, 272, 294, 398 + +Viscoplastic strain rate derivative matrix, 274, 279 + +Viscoplasticity, basic theoretical response: + +Dynamic application, 381 + +One dimension, 98 + +Two dimensions, 272 + +Viscoplasticity in two dimensions, 271 + +Viscoplasticity in one dimension, 95 + +Viscosity coefficient, 97 + +Volume, elemental, 172 + +Von Mises yield criterion, 218, 230 + +Weighting functions, 23 + +Winkler foundation, 151, 372 + +Work hardening, 222, 223, 228 + +Yield criterion, 26, 216, 272, 326, 328 + +Yield function constants, 231, 234, 235 + +Yield moment, 129 + +Yield surface, 217 diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-015_53ac111761c9ad18723000ebb385b96414593b7d253b0ece3041c9ee3af55422.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-015_53ac111761c9ad18723000ebb385b96414593b7d253b0ece3041c9ee3af55422.jpg new file mode 100644 index 00000000..f4ee56ab Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-015_53ac111761c9ad18723000ebb385b96414593b7d253b0ece3041c9ee3af55422.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-015_82648261cf9624240e89d1e01be65045c13538e5949d4589fbf6efe3f34e2588.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-015_82648261cf9624240e89d1e01be65045c13538e5949d4589fbf6efe3f34e2588.jpg new file mode 100644 index 00000000..5fe34ce0 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-015_82648261cf9624240e89d1e01be65045c13538e5949d4589fbf6efe3f34e2588.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-015_bb83187e781424ff0af7c9ba80c7bf9f8c86b520b442bd48341615f78678fb78.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-015_bb83187e781424ff0af7c9ba80c7bf9f8c86b520b442bd48341615f78678fb78.jpg new file mode 100644 index 00000000..39741a55 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-015_bb83187e781424ff0af7c9ba80c7bf9f8c86b520b442bd48341615f78678fb78.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-016_cc7aad6598747466ef78508754fdf8451309c57a22e71f5b67adbcd31478c8a3.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-016_cc7aad6598747466ef78508754fdf8451309c57a22e71f5b67adbcd31478c8a3.jpg new file mode 100644 index 00000000..60d6bb73 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-016_cc7aad6598747466ef78508754fdf8451309c57a22e71f5b67adbcd31478c8a3.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-016_f27db06b79380f55875db51acee957ac608142384a4dee7cdc5f285783a0b53c.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-016_f27db06b79380f55875db51acee957ac608142384a4dee7cdc5f285783a0b53c.jpg new file mode 100644 index 00000000..61c11b42 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-016_f27db06b79380f55875db51acee957ac608142384a4dee7cdc5f285783a0b53c.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-019_baf612902b02d3f830d3de370553915a03c9f703d975019b9bce9870d5f0fc63.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-019_baf612902b02d3f830d3de370553915a03c9f703d975019b9bce9870d5f0fc63.jpg new file mode 100644 index 00000000..8ff04542 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-019_baf612902b02d3f830d3de370553915a03c9f703d975019b9bce9870d5f0fc63.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-023_90c1c1c7a3835111b2aa257debcf5f57e4781ea1704b7698f2e4156884f241e1.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-023_90c1c1c7a3835111b2aa257debcf5f57e4781ea1704b7698f2e4156884f241e1.jpg new file mode 100644 index 00000000..453d2317 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-023_90c1c1c7a3835111b2aa257debcf5f57e4781ea1704b7698f2e4156884f241e1.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-024_5219866ee218ea81254db6e0ccf1c384d9d15dd7910cbdceb029c1b57c0b8874.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-024_5219866ee218ea81254db6e0ccf1c384d9d15dd7910cbdceb029c1b57c0b8874.jpg new file mode 100644 index 00000000..57833a06 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-024_5219866ee218ea81254db6e0ccf1c384d9d15dd7910cbdceb029c1b57c0b8874.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-024_60359afebdab861594dec56b929519b134307dbc542c0d17002e063f0dda8695.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-024_60359afebdab861594dec56b929519b134307dbc542c0d17002e063f0dda8695.jpg new file mode 100644 index 00000000..a2d97c1c Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-024_60359afebdab861594dec56b929519b134307dbc542c0d17002e063f0dda8695.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-025_0312473a92fe53299c399ee0c805683afb7f56dd00ef8e07844d5dd004f0e83c.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-025_0312473a92fe53299c399ee0c805683afb7f56dd00ef8e07844d5dd004f0e83c.jpg new file mode 100644 index 00000000..e70cf06b Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-025_0312473a92fe53299c399ee0c805683afb7f56dd00ef8e07844d5dd004f0e83c.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-025_54ea7ca4dbe9144ee8dd4251952687ad6c8bfb5f9d5f13829ca596b3d97066b5.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-025_54ea7ca4dbe9144ee8dd4251952687ad6c8bfb5f9d5f13829ca596b3d97066b5.jpg new file mode 100644 index 00000000..9bc707ba Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-025_54ea7ca4dbe9144ee8dd4251952687ad6c8bfb5f9d5f13829ca596b3d97066b5.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-025_d7468ef1bbad5943cd1d9027e0f7fb2ad8ad2a10dd140d407c312ad886f984b7.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-025_d7468ef1bbad5943cd1d9027e0f7fb2ad8ad2a10dd140d407c312ad886f984b7.jpg new file mode 100644 index 00000000..291cf38d Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-025_d7468ef1bbad5943cd1d9027e0f7fb2ad8ad2a10dd140d407c312ad886f984b7.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-025_dbee49abd4c05426368f38041d6147a3e2b726f5e9d7992f072c1e89d03bbb0f.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-025_dbee49abd4c05426368f38041d6147a3e2b726f5e9d7992f072c1e89d03bbb0f.jpg new file mode 100644 index 00000000..d6414383 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-025_dbee49abd4c05426368f38041d6147a3e2b726f5e9d7992f072c1e89d03bbb0f.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-026_24da6b2750dcc375e900d65efed0c23da0cb7df19e14731a9261a1b77aef6a24.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-026_24da6b2750dcc375e900d65efed0c23da0cb7df19e14731a9261a1b77aef6a24.jpg new file mode 100644 index 00000000..40b3ffb2 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-026_24da6b2750dcc375e900d65efed0c23da0cb7df19e14731a9261a1b77aef6a24.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-026_3088db828b5f8b6a5c7ff01743aaca7a35af04268bef0916d1e82e4b3be78222.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-026_3088db828b5f8b6a5c7ff01743aaca7a35af04268bef0916d1e82e4b3be78222.jpg new file mode 100644 index 00000000..990cc44b Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-026_3088db828b5f8b6a5c7ff01743aaca7a35af04268bef0916d1e82e4b3be78222.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-026_56398f8079052d2a958a8b6f27354cd95a0af8a0feab07cf9b2d9bb4d90eba53.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-026_56398f8079052d2a958a8b6f27354cd95a0af8a0feab07cf9b2d9bb4d90eba53.jpg new file mode 100644 index 00000000..e97813d6 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-026_56398f8079052d2a958a8b6f27354cd95a0af8a0feab07cf9b2d9bb4d90eba53.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-026_7ead789637eef39e460fdff244549655e1b5843604fc2a4672e05e0b51b1b124.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-026_7ead789637eef39e460fdff244549655e1b5843604fc2a4672e05e0b51b1b124.jpg new file mode 100644 index 00000000..6a723b58 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-026_7ead789637eef39e460fdff244549655e1b5843604fc2a4672e05e0b51b1b124.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-026_ae6168fa63008149d867a8b6e504c08848e8d9949cc28111276f4866b766b950.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-026_ae6168fa63008149d867a8b6e504c08848e8d9949cc28111276f4866b766b950.jpg new file mode 100644 index 00000000..018efb68 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-026_ae6168fa63008149d867a8b6e504c08848e8d9949cc28111276f4866b766b950.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-027_0f5848899ddc544ee54ec746a5d6a32e14f6869cea5611b1bd65d18e0a430b4d.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-027_0f5848899ddc544ee54ec746a5d6a32e14f6869cea5611b1bd65d18e0a430b4d.jpg new file mode 100644 index 00000000..74e359d9 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-027_0f5848899ddc544ee54ec746a5d6a32e14f6869cea5611b1bd65d18e0a430b4d.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-027_4e20952fab8cf7ad024e32feff679f49b118af5ce8bd04453c37d729413ac1f3.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-027_4e20952fab8cf7ad024e32feff679f49b118af5ce8bd04453c37d729413ac1f3.jpg new file mode 100644 index 00000000..0c841a9f Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-027_4e20952fab8cf7ad024e32feff679f49b118af5ce8bd04453c37d729413ac1f3.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-027_51420e1636e67ee1a6a934276ce5401e140b1f2f4673ec24975027e06877f183.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-027_51420e1636e67ee1a6a934276ce5401e140b1f2f4673ec24975027e06877f183.jpg new file mode 100644 index 00000000..8d675685 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-027_51420e1636e67ee1a6a934276ce5401e140b1f2f4673ec24975027e06877f183.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-027_55af075863c7e8d1b9804adb75e44f85bbee16fe2a11599b9d37e0fccdca3de0.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-027_55af075863c7e8d1b9804adb75e44f85bbee16fe2a11599b9d37e0fccdca3de0.jpg new file mode 100644 index 00000000..c1e8767e Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-027_55af075863c7e8d1b9804adb75e44f85bbee16fe2a11599b9d37e0fccdca3de0.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-027_5d657872ab444c922f090786b6a571baa05125909054138c6acb42abda687ece.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-027_5d657872ab444c922f090786b6a571baa05125909054138c6acb42abda687ece.jpg new file mode 100644 index 00000000..3ed3f1f5 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-027_5d657872ab444c922f090786b6a571baa05125909054138c6acb42abda687ece.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-027_af41382109f4140e95fbcf37fa3d034dcef295601e1d66f45271a9d021a69f80.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-027_af41382109f4140e95fbcf37fa3d034dcef295601e1d66f45271a9d021a69f80.jpg new file mode 100644 index 00000000..32e314a9 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-027_af41382109f4140e95fbcf37fa3d034dcef295601e1d66f45271a9d021a69f80.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-027_bdc48a5d2e1949aba431232374409e4a44e13f4b388fc4b2ca6a1811c539044c.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-027_bdc48a5d2e1949aba431232374409e4a44e13f4b388fc4b2ca6a1811c539044c.jpg new file mode 100644 index 00000000..396a42a9 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-027_bdc48a5d2e1949aba431232374409e4a44e13f4b388fc4b2ca6a1811c539044c.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-027_dca6fa62c5e3e6063b10bba3ab1c4f1bafc8f4a4248d8f2abc53ee9cd2e34e06.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-027_dca6fa62c5e3e6063b10bba3ab1c4f1bafc8f4a4248d8f2abc53ee9cd2e34e06.jpg new file mode 100644 index 00000000..149ae501 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-027_dca6fa62c5e3e6063b10bba3ab1c4f1bafc8f4a4248d8f2abc53ee9cd2e34e06.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-028_3baca9f79ae33c60c648d45da1351c914cf6c42c066af5e9d619c7da022fd2ff.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-028_3baca9f79ae33c60c648d45da1351c914cf6c42c066af5e9d619c7da022fd2ff.jpg new file mode 100644 index 00000000..566dd26b Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-028_3baca9f79ae33c60c648d45da1351c914cf6c42c066af5e9d619c7da022fd2ff.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-028_444bd2f939ff8a254a19317905a100316aa7b2c55de3263b5d2d2a69c6a03235.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-028_444bd2f939ff8a254a19317905a100316aa7b2c55de3263b5d2d2a69c6a03235.jpg new file mode 100644 index 00000000..475ff04c Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-028_444bd2f939ff8a254a19317905a100316aa7b2c55de3263b5d2d2a69c6a03235.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-028_8ae5c1144c8a6ace36ddad4766052730de209baa600743c9bae389170786a1a6.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-028_8ae5c1144c8a6ace36ddad4766052730de209baa600743c9bae389170786a1a6.jpg new file mode 100644 index 00000000..128527e3 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-028_8ae5c1144c8a6ace36ddad4766052730de209baa600743c9bae389170786a1a6.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-028_edaa428362dd011b9e3f3e46f3cb84c50115eeddb676d87217b8095d8122eff9.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-028_edaa428362dd011b9e3f3e46f3cb84c50115eeddb676d87217b8095d8122eff9.jpg new file mode 100644 index 00000000..a948e160 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-028_edaa428362dd011b9e3f3e46f3cb84c50115eeddb676d87217b8095d8122eff9.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-029_27861a5412395508186d6d39343973837af87b73626ddbbe0a31bdeccf14f3d7.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-029_27861a5412395508186d6d39343973837af87b73626ddbbe0a31bdeccf14f3d7.jpg new file mode 100644 index 00000000..3c72afbc Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-029_27861a5412395508186d6d39343973837af87b73626ddbbe0a31bdeccf14f3d7.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-029_43018df1516d06829c8ff76a5339cdf67a8076f95c78a68be2022a41647fb3c9.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-029_43018df1516d06829c8ff76a5339cdf67a8076f95c78a68be2022a41647fb3c9.jpg new file mode 100644 index 00000000..4956ee62 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-029_43018df1516d06829c8ff76a5339cdf67a8076f95c78a68be2022a41647fb3c9.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-029_9436c69d0ca16696952d91187b5d8114d5e1e810029191ae162e45d5fe75a40a.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-029_9436c69d0ca16696952d91187b5d8114d5e1e810029191ae162e45d5fe75a40a.jpg new file mode 100644 index 00000000..a8914413 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-029_9436c69d0ca16696952d91187b5d8114d5e1e810029191ae162e45d5fe75a40a.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-029_d203c40f894967121e8e6cf6b6cc2228a4e2f3c4b54a74a3114746add2bab68c.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-029_d203c40f894967121e8e6cf6b6cc2228a4e2f3c4b54a74a3114746add2bab68c.jpg new file mode 100644 index 00000000..3ab4af57 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-029_d203c40f894967121e8e6cf6b6cc2228a4e2f3c4b54a74a3114746add2bab68c.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-029_ea39d889c334cc2fe733855725891c8e254b647dadff16995241886b8bf89720.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-029_ea39d889c334cc2fe733855725891c8e254b647dadff16995241886b8bf89720.jpg new file mode 100644 index 00000000..46669b86 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-029_ea39d889c334cc2fe733855725891c8e254b647dadff16995241886b8bf89720.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-030_cf16c74e24ddfb1033d623423c691acca4b9427c550f0661ddb5f21f4a0ed03d.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-030_cf16c74e24ddfb1033d623423c691acca4b9427c550f0661ddb5f21f4a0ed03d.jpg new file mode 100644 index 00000000..e9b96dcc Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-030_cf16c74e24ddfb1033d623423c691acca4b9427c550f0661ddb5f21f4a0ed03d.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-031_34963c308dc70333f95678e42d5a568d89cc3cb3618464dce89aeead03982f5f.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-031_34963c308dc70333f95678e42d5a568d89cc3cb3618464dce89aeead03982f5f.jpg new file mode 100644 index 00000000..be90431e Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-031_34963c308dc70333f95678e42d5a568d89cc3cb3618464dce89aeead03982f5f.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-031_e4e8f35e48af9bdd9435f5fd692bc4ee10704e114647db88d08d81bc5387e532.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-031_e4e8f35e48af9bdd9435f5fd692bc4ee10704e114647db88d08d81bc5387e532.jpg new file mode 100644 index 00000000..a4533b3b Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-031_e4e8f35e48af9bdd9435f5fd692bc4ee10704e114647db88d08d81bc5387e532.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-032_05915a37af3005249ef07088cbe5efda27256ff2580f4b6547594b63fc7ca40d.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-032_05915a37af3005249ef07088cbe5efda27256ff2580f4b6547594b63fc7ca40d.jpg new file mode 100644 index 00000000..07d33057 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-032_05915a37af3005249ef07088cbe5efda27256ff2580f4b6547594b63fc7ca40d.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-032_47664b7e9ef4376d22420ea5d35a6c0cda5db3f7d8e46d3821225319fece812e.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-032_47664b7e9ef4376d22420ea5d35a6c0cda5db3f7d8e46d3821225319fece812e.jpg new file mode 100644 index 00000000..5504cab3 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-032_47664b7e9ef4376d22420ea5d35a6c0cda5db3f7d8e46d3821225319fece812e.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-032_a0573cb4c5dacac6feb1c5f31f4fcd20d8d8e72d37b98955769cb5f9d3d164e9.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-032_a0573cb4c5dacac6feb1c5f31f4fcd20d8d8e72d37b98955769cb5f9d3d164e9.jpg new file mode 100644 index 00000000..458eae32 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-032_a0573cb4c5dacac6feb1c5f31f4fcd20d8d8e72d37b98955769cb5f9d3d164e9.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-033_4324d47b1b08d4fc5ec08671a79a624d9141419586a4ce1be0acb20563b0fa5c.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-033_4324d47b1b08d4fc5ec08671a79a624d9141419586a4ce1be0acb20563b0fa5c.jpg new file mode 100644 index 00000000..8d0ad198 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-033_4324d47b1b08d4fc5ec08671a79a624d9141419586a4ce1be0acb20563b0fa5c.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-033_a6a2989dada0af74ba1d81f3df35a0103d75a48567d6400b2735f76ac98adf98.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-033_a6a2989dada0af74ba1d81f3df35a0103d75a48567d6400b2735f76ac98adf98.jpg new file mode 100644 index 00000000..59d47c34 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-033_a6a2989dada0af74ba1d81f3df35a0103d75a48567d6400b2735f76ac98adf98.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-033_aea40b9ab88601c5da7698df44d779be103dfa776bb2ded63fcd27e8f11e9261.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-033_aea40b9ab88601c5da7698df44d779be103dfa776bb2ded63fcd27e8f11e9261.jpg new file mode 100644 index 00000000..43e631ff Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-033_aea40b9ab88601c5da7698df44d779be103dfa776bb2ded63fcd27e8f11e9261.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-033_b3c4805e565eb60e5f89e34fb7bb1090bd29eac175a82b0e3daced8cd1c876f4.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-033_b3c4805e565eb60e5f89e34fb7bb1090bd29eac175a82b0e3daced8cd1c876f4.jpg new file mode 100644 index 00000000..4ca1aa97 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-033_b3c4805e565eb60e5f89e34fb7bb1090bd29eac175a82b0e3daced8cd1c876f4.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-033_c1400bff2b67a05fb1fbbbff6e7088fce1700e1e6995c06351ce7cefb0004d87.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-033_c1400bff2b67a05fb1fbbbff6e7088fce1700e1e6995c06351ce7cefb0004d87.jpg new file mode 100644 index 00000000..29c84140 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-033_c1400bff2b67a05fb1fbbbff6e7088fce1700e1e6995c06351ce7cefb0004d87.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-033_d6837875b0e7406ca1d5e7d8512a1d424c752bd82cc3fc5685e5483632c4c29a.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-033_d6837875b0e7406ca1d5e7d8512a1d424c752bd82cc3fc5685e5483632c4c29a.jpg new file mode 100644 index 00000000..29e1823a Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-033_d6837875b0e7406ca1d5e7d8512a1d424c752bd82cc3fc5685e5483632c4c29a.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-033_f5d550cab3352e169438c83d852eb681528486924d4eec9716cab440cf7a0d1f.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-033_f5d550cab3352e169438c83d852eb681528486924d4eec9716cab440cf7a0d1f.jpg new file mode 100644 index 00000000..a9ea02b4 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-033_f5d550cab3352e169438c83d852eb681528486924d4eec9716cab440cf7a0d1f.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-034_19ae1dadee17c40684b9a142e25955598d975117ecaffb1c0337514f56002447.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-034_19ae1dadee17c40684b9a142e25955598d975117ecaffb1c0337514f56002447.jpg new file mode 100644 index 00000000..38e77da2 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-034_19ae1dadee17c40684b9a142e25955598d975117ecaffb1c0337514f56002447.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-034_344f461eee21a477a3732064f66db6e8f5974b6f95d532a78742e3b2cf7fe59c.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-034_344f461eee21a477a3732064f66db6e8f5974b6f95d532a78742e3b2cf7fe59c.jpg new file mode 100644 index 00000000..8bde3d56 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-034_344f461eee21a477a3732064f66db6e8f5974b6f95d532a78742e3b2cf7fe59c.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-034_557131080c83b83b6541ba4541404b485fb29042cb63f3e082a153d3cc93d1b7.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-034_557131080c83b83b6541ba4541404b485fb29042cb63f3e082a153d3cc93d1b7.jpg new file mode 100644 index 00000000..e9fe9bb8 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-034_557131080c83b83b6541ba4541404b485fb29042cb63f3e082a153d3cc93d1b7.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-034_a249c9f8c44fb2271a7c98ce490e9d1bc08fd6a4759ecdd1dde654f3ceac5d5c.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-034_a249c9f8c44fb2271a7c98ce490e9d1bc08fd6a4759ecdd1dde654f3ceac5d5c.jpg new file mode 100644 index 00000000..5b6162dd Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-034_a249c9f8c44fb2271a7c98ce490e9d1bc08fd6a4759ecdd1dde654f3ceac5d5c.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-034_ef3dbe5297fb2218dfad18a9a85eb82ab6ef41ec6c34297491c43c886eb3dafb.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-034_ef3dbe5297fb2218dfad18a9a85eb82ab6ef41ec6c34297491c43c886eb3dafb.jpg new file mode 100644 index 00000000..80503aa8 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-034_ef3dbe5297fb2218dfad18a9a85eb82ab6ef41ec6c34297491c43c886eb3dafb.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-035_06c435c9adfaaace774404b351ba637da5e75857fe4243883a99fb8e141fd197.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-035_06c435c9adfaaace774404b351ba637da5e75857fe4243883a99fb8e141fd197.jpg new file mode 100644 index 00000000..27fa8106 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-035_06c435c9adfaaace774404b351ba637da5e75857fe4243883a99fb8e141fd197.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-035_27907f441fbc628d0e4805e0fbd8c939d579d3b3a6104b41e06b059d64dfdfe7.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-035_27907f441fbc628d0e4805e0fbd8c939d579d3b3a6104b41e06b059d64dfdfe7.jpg new file mode 100644 index 00000000..110527a0 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-035_27907f441fbc628d0e4805e0fbd8c939d579d3b3a6104b41e06b059d64dfdfe7.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-035_ba743bab495133438c6803ca19836f6d870de3261178dfe5e0c493bfb9bfb447.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-035_ba743bab495133438c6803ca19836f6d870de3261178dfe5e0c493bfb9bfb447.jpg new file mode 100644 index 00000000..351d452c Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-035_ba743bab495133438c6803ca19836f6d870de3261178dfe5e0c493bfb9bfb447.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-035_cd8de8d9195d234dc7acdb89dd1e1c0d9a00476fba5483e93e17f8a55ed0c11b.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-035_cd8de8d9195d234dc7acdb89dd1e1c0d9a00476fba5483e93e17f8a55ed0c11b.jpg new file mode 100644 index 00000000..125f2848 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-035_cd8de8d9195d234dc7acdb89dd1e1c0d9a00476fba5483e93e17f8a55ed0c11b.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-035_cfb75a943225e29d20f04e04b63c63e7673144dd3d7311381908887742d98769.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-035_cfb75a943225e29d20f04e04b63c63e7673144dd3d7311381908887742d98769.jpg new file mode 100644 index 00000000..f8b30729 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-035_cfb75a943225e29d20f04e04b63c63e7673144dd3d7311381908887742d98769.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-036_885944e3163af36ed38a9b05b94e884f5424ff5f7439599fb5be0553a3dd89c7.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-036_885944e3163af36ed38a9b05b94e884f5424ff5f7439599fb5be0553a3dd89c7.jpg new file mode 100644 index 00000000..4c308e2f Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-036_885944e3163af36ed38a9b05b94e884f5424ff5f7439599fb5be0553a3dd89c7.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-036_ed97400804cb6ce5ea306de626211620cf855f08c3d4094be4ff78481659bae5.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-036_ed97400804cb6ce5ea306de626211620cf855f08c3d4094be4ff78481659bae5.jpg new file mode 100644 index 00000000..8c75dfd6 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-036_ed97400804cb6ce5ea306de626211620cf855f08c3d4094be4ff78481659bae5.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-037_0c8349f6b4bfb8553ae288c7cbd67011d9652f79a58e15b6533c191f7a0d2041.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-037_0c8349f6b4bfb8553ae288c7cbd67011d9652f79a58e15b6533c191f7a0d2041.jpg new file mode 100644 index 00000000..0cc28744 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-037_0c8349f6b4bfb8553ae288c7cbd67011d9652f79a58e15b6533c191f7a0d2041.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-037_5d816f7386f3e62ffdd7dd06774ccdebf1eec409e46f8d5976315bc433805f69.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-037_5d816f7386f3e62ffdd7dd06774ccdebf1eec409e46f8d5976315bc433805f69.jpg new file mode 100644 index 00000000..ffc88c81 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-037_5d816f7386f3e62ffdd7dd06774ccdebf1eec409e46f8d5976315bc433805f69.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-037_6a7c8b161b2e4bb541a4c7280aab407ddf0e0b4ebb3c1769fbc390b199ca53b6.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-037_6a7c8b161b2e4bb541a4c7280aab407ddf0e0b4ebb3c1769fbc390b199ca53b6.jpg new file mode 100644 index 00000000..67776cf8 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-037_6a7c8b161b2e4bb541a4c7280aab407ddf0e0b4ebb3c1769fbc390b199ca53b6.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-037_6aa716a07a43e276377770394a49a9be7960a6f4b5424f4c2ffbafe3fb0bafa7.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-037_6aa716a07a43e276377770394a49a9be7960a6f4b5424f4c2ffbafe3fb0bafa7.jpg new file mode 100644 index 00000000..a2a67b5d Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-037_6aa716a07a43e276377770394a49a9be7960a6f4b5424f4c2ffbafe3fb0bafa7.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-037_df55fdd64be5144f6059551d10dfb0df89f2117519cca98c356b2d6454091975.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-037_df55fdd64be5144f6059551d10dfb0df89f2117519cca98c356b2d6454091975.jpg new file mode 100644 index 00000000..6661442c Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-037_df55fdd64be5144f6059551d10dfb0df89f2117519cca98c356b2d6454091975.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-038_08dc6a859198b612dc5ddcab8e0397e5f2cda31fc94afffe89504d2ba1d76224.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-038_08dc6a859198b612dc5ddcab8e0397e5f2cda31fc94afffe89504d2ba1d76224.jpg new file mode 100644 index 00000000..75bb813f Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-038_08dc6a859198b612dc5ddcab8e0397e5f2cda31fc94afffe89504d2ba1d76224.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-038_13b9784bcd93943bbb864dae05f808f16e4aa44fd6ee900ab4c98eea18d1559e.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-038_13b9784bcd93943bbb864dae05f808f16e4aa44fd6ee900ab4c98eea18d1559e.jpg new file mode 100644 index 00000000..a6e39239 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-038_13b9784bcd93943bbb864dae05f808f16e4aa44fd6ee900ab4c98eea18d1559e.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-038_14003a1015b31cff5ba4444cf87f4e58cff199766a26a7d19538ccaa9d73f50f.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-038_14003a1015b31cff5ba4444cf87f4e58cff199766a26a7d19538ccaa9d73f50f.jpg new file mode 100644 index 00000000..97929f35 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-038_14003a1015b31cff5ba4444cf87f4e58cff199766a26a7d19538ccaa9d73f50f.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-038_259046067e58ee50ce51f57824a76619dc606b44624638340b08b2db321f3e00.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-038_259046067e58ee50ce51f57824a76619dc606b44624638340b08b2db321f3e00.jpg new file mode 100644 index 00000000..73556ba5 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-038_259046067e58ee50ce51f57824a76619dc606b44624638340b08b2db321f3e00.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-038_26cffe940b5a7cce63dec1d875d213520b8827968045eb6fb1fa106907da24bb.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-038_26cffe940b5a7cce63dec1d875d213520b8827968045eb6fb1fa106907da24bb.jpg new file mode 100644 index 00000000..3bda6fa8 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-038_26cffe940b5a7cce63dec1d875d213520b8827968045eb6fb1fa106907da24bb.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-038_4f2abdf4c533beb22346c4c75b3bc4e60f857b36bb1ee740f7246fee0f20b27a.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-038_4f2abdf4c533beb22346c4c75b3bc4e60f857b36bb1ee740f7246fee0f20b27a.jpg new file mode 100644 index 00000000..01907b01 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-038_4f2abdf4c533beb22346c4c75b3bc4e60f857b36bb1ee740f7246fee0f20b27a.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-038_81974b91c7400a7086cef0ba57f353fec5f581acb79d2657a96311e9acd3ad63.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-038_81974b91c7400a7086cef0ba57f353fec5f581acb79d2657a96311e9acd3ad63.jpg new file mode 100644 index 00000000..9dc20c4f Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-038_81974b91c7400a7086cef0ba57f353fec5f581acb79d2657a96311e9acd3ad63.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-038_92f062651fc0e1c97f62ab24c26279aedea87c58361dfe11b15a09ccaa9e4d6f.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-038_92f062651fc0e1c97f62ab24c26279aedea87c58361dfe11b15a09ccaa9e4d6f.jpg new file mode 100644 index 00000000..8406476e Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-038_92f062651fc0e1c97f62ab24c26279aedea87c58361dfe11b15a09ccaa9e4d6f.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-038_a2145d8ffe0bb87a80e222e3ee10b95a61dda2acefb080e4e6ef6a2acdc59c18.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-038_a2145d8ffe0bb87a80e222e3ee10b95a61dda2acefb080e4e6ef6a2acdc59c18.jpg new file mode 100644 index 00000000..d8296231 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-038_a2145d8ffe0bb87a80e222e3ee10b95a61dda2acefb080e4e6ef6a2acdc59c18.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-039_d2e4b3a243665662f225d018ffb0a20ac1ab158615ba3da13cece2f3ce25fcf8.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-039_d2e4b3a243665662f225d018ffb0a20ac1ab158615ba3da13cece2f3ce25fcf8.jpg new file mode 100644 index 00000000..fcb01338 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-039_d2e4b3a243665662f225d018ffb0a20ac1ab158615ba3da13cece2f3ce25fcf8.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-040_570a87f73a30efd6200eb13cdf06cbe8584d8ac9969a00692a3b8229dbadb111.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-040_570a87f73a30efd6200eb13cdf06cbe8584d8ac9969a00692a3b8229dbadb111.jpg new file mode 100644 index 00000000..c6fe9852 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-040_570a87f73a30efd6200eb13cdf06cbe8584d8ac9969a00692a3b8229dbadb111.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-040_72a0700e664c0d2f0bae90b7fe3c1146f511f42f202a07bd6b936bf157c34c74.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-040_72a0700e664c0d2f0bae90b7fe3c1146f511f42f202a07bd6b936bf157c34c74.jpg new file mode 100644 index 00000000..8504ccd5 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-040_72a0700e664c0d2f0bae90b7fe3c1146f511f42f202a07bd6b936bf157c34c74.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-040_bb17abb75e7663bdf676020c7a2d035467e6775eb192acb5791e6f7474e361f3.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-040_bb17abb75e7663bdf676020c7a2d035467e6775eb192acb5791e6f7474e361f3.jpg new file mode 100644 index 00000000..fa47ed5e Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-040_bb17abb75e7663bdf676020c7a2d035467e6775eb192acb5791e6f7474e361f3.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-040_df27ad1cb5e3d56e1763fbe5e8a2e28e30e96a31cf420c4064ff88abc48d5e5a.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-040_df27ad1cb5e3d56e1763fbe5e8a2e28e30e96a31cf420c4064ff88abc48d5e5a.jpg new file mode 100644 index 00000000..8abd90a0 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-040_df27ad1cb5e3d56e1763fbe5e8a2e28e30e96a31cf420c4064ff88abc48d5e5a.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-040_e775e54a1601966561bf17fcb23d932d4c2562a143ea4d45f5216c7b7e98035d.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-040_e775e54a1601966561bf17fcb23d932d4c2562a143ea4d45f5216c7b7e98035d.jpg new file mode 100644 index 00000000..842858a6 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-040_e775e54a1601966561bf17fcb23d932d4c2562a143ea4d45f5216c7b7e98035d.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-040_fc3edfd28234b9c5e6ba703473bd68f8818534b6a8f8a6917a72444499b43798.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-040_fc3edfd28234b9c5e6ba703473bd68f8818534b6a8f8a6917a72444499b43798.jpg new file mode 100644 index 00000000..1de1bd44 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-040_fc3edfd28234b9c5e6ba703473bd68f8818534b6a8f8a6917a72444499b43798.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-044_30b762f35ad25fc73a24946ed3ec1b7f05b8cd72cd47c3ff60182ea0b18cf575.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-044_30b762f35ad25fc73a24946ed3ec1b7f05b8cd72cd47c3ff60182ea0b18cf575.jpg new file mode 100644 index 00000000..db049bc7 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-044_30b762f35ad25fc73a24946ed3ec1b7f05b8cd72cd47c3ff60182ea0b18cf575.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-050_3646e8b68a8e6f46ea9a655e65c813f535f1acaada9a7c8bfe343ad02ccacce7.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-050_3646e8b68a8e6f46ea9a655e65c813f535f1acaada9a7c8bfe343ad02ccacce7.jpg new file mode 100644 index 00000000..fc7e45e3 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-050_3646e8b68a8e6f46ea9a655e65c813f535f1acaada9a7c8bfe343ad02ccacce7.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-052_8d2840cd8baf409e9909213bdee50c1eae95eb462636bda997a888e595231441.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-052_8d2840cd8baf409e9909213bdee50c1eae95eb462636bda997a888e595231441.jpg new file mode 100644 index 00000000..c4cca696 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-052_8d2840cd8baf409e9909213bdee50c1eae95eb462636bda997a888e595231441.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-053_20ad846759cda7801b3e17317a59174bcd77dfb904d61a8a2c0d5ccdcdaca633.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-053_20ad846759cda7801b3e17317a59174bcd77dfb904d61a8a2c0d5ccdcdaca633.jpg new file mode 100644 index 00000000..6bdd9194 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-053_20ad846759cda7801b3e17317a59174bcd77dfb904d61a8a2c0d5ccdcdaca633.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-053_3352a64c7fa74a3e00a697f0f9257d8a2372ae65005b57ef1e4b13ea56141e73.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-053_3352a64c7fa74a3e00a697f0f9257d8a2372ae65005b57ef1e4b13ea56141e73.jpg new file mode 100644 index 00000000..3d5628fb Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-053_3352a64c7fa74a3e00a697f0f9257d8a2372ae65005b57ef1e4b13ea56141e73.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-053_49e23763324eda01ac44b5f6b5276a440c8a7f12cade10cf6a79edd875fc88f1.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-053_49e23763324eda01ac44b5f6b5276a440c8a7f12cade10cf6a79edd875fc88f1.jpg new file mode 100644 index 00000000..d7c47e69 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-053_49e23763324eda01ac44b5f6b5276a440c8a7f12cade10cf6a79edd875fc88f1.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-053_4fdbe6252d17494de0f06b7a7743b8816459a9691a365175f241db750259b7c3.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-053_4fdbe6252d17494de0f06b7a7743b8816459a9691a365175f241db750259b7c3.jpg new file mode 100644 index 00000000..f3981b63 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-053_4fdbe6252d17494de0f06b7a7743b8816459a9691a365175f241db750259b7c3.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-053_832db51e9c66f803ee95ffe1bf0b63a4ca56126225accbd2e6c4d7a59750b8d9.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-053_832db51e9c66f803ee95ffe1bf0b63a4ca56126225accbd2e6c4d7a59750b8d9.jpg new file mode 100644 index 00000000..c94610aa Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-053_832db51e9c66f803ee95ffe1bf0b63a4ca56126225accbd2e6c4d7a59750b8d9.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-053_ad6bcd1c513f781ebc0ec9b712b5c995e5bc46631f20c6526dceedea7bcb01f7.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-053_ad6bcd1c513f781ebc0ec9b712b5c995e5bc46631f20c6526dceedea7bcb01f7.jpg new file mode 100644 index 00000000..3845a4ca Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-053_ad6bcd1c513f781ebc0ec9b712b5c995e5bc46631f20c6526dceedea7bcb01f7.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-053_f647de5fd747c15c213fb219eed59d0ae5e5120d4ccaf1693a52e4b3f3cf808d.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-053_f647de5fd747c15c213fb219eed59d0ae5e5120d4ccaf1693a52e4b3f3cf808d.jpg new file mode 100644 index 00000000..657d09b3 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-053_f647de5fd747c15c213fb219eed59d0ae5e5120d4ccaf1693a52e4b3f3cf808d.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-054_2a0c6582e52a8e0cf370727cd27783912ef367cfd9f93573830f03e72628f720.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-054_2a0c6582e52a8e0cf370727cd27783912ef367cfd9f93573830f03e72628f720.jpg new file mode 100644 index 00000000..5c47e8be Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-054_2a0c6582e52a8e0cf370727cd27783912ef367cfd9f93573830f03e72628f720.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-054_7163242404ecde13436ab013cf5cedc8de4417c8aa43577bb1eb873c7860a7c3.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-054_7163242404ecde13436ab013cf5cedc8de4417c8aa43577bb1eb873c7860a7c3.jpg new file mode 100644 index 00000000..01c537ee Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-054_7163242404ecde13436ab013cf5cedc8de4417c8aa43577bb1eb873c7860a7c3.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-054_732e94644f428413fad7cf66c05b1d2aeb208c71210d8164f7a189313d02022b.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-054_732e94644f428413fad7cf66c05b1d2aeb208c71210d8164f7a189313d02022b.jpg new file mode 100644 index 00000000..d5b9c445 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-054_732e94644f428413fad7cf66c05b1d2aeb208c71210d8164f7a189313d02022b.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-054_8db5314f6a885ad87ba365c4a791f5860a8c931e561933e214a9d8592de14ef3.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-054_8db5314f6a885ad87ba365c4a791f5860a8c931e561933e214a9d8592de14ef3.jpg new file mode 100644 index 00000000..2dcacc19 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-054_8db5314f6a885ad87ba365c4a791f5860a8c931e561933e214a9d8592de14ef3.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-054_9328b2c8851381045297850ab860df5d240440debc24d176cbf4e5b93283b587.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-054_9328b2c8851381045297850ab860df5d240440debc24d176cbf4e5b93283b587.jpg new file mode 100644 index 00000000..28679992 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-054_9328b2c8851381045297850ab860df5d240440debc24d176cbf4e5b93283b587.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-054_984363a23673be7148396c2d539e01b7b385d0203ba5bb54c6cf027a2ade64ad.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-054_984363a23673be7148396c2d539e01b7b385d0203ba5bb54c6cf027a2ade64ad.jpg new file mode 100644 index 00000000..540ed20e Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-054_984363a23673be7148396c2d539e01b7b385d0203ba5bb54c6cf027a2ade64ad.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-054_9e217d287767e2ad0bd2cc573d7f47de0fb7ea567406f8c7c982d1df7efa6fb0.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-054_9e217d287767e2ad0bd2cc573d7f47de0fb7ea567406f8c7c982d1df7efa6fb0.jpg new file mode 100644 index 00000000..1c4e3796 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-054_9e217d287767e2ad0bd2cc573d7f47de0fb7ea567406f8c7c982d1df7efa6fb0.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-054_d55ea9e367d9b59522969130a8584a286d078a1b05bf6de195b426724865013e.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-054_d55ea9e367d9b59522969130a8584a286d078a1b05bf6de195b426724865013e.jpg new file mode 100644 index 00000000..b3694466 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-054_d55ea9e367d9b59522969130a8584a286d078a1b05bf6de195b426724865013e.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-055_22a1f96aa2ac0dbafa0d727027332034b1c30c0c531359a4df04ff67a2bc3a14.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-055_22a1f96aa2ac0dbafa0d727027332034b1c30c0c531359a4df04ff67a2bc3a14.jpg new file mode 100644 index 00000000..bc6d5d18 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-055_22a1f96aa2ac0dbafa0d727027332034b1c30c0c531359a4df04ff67a2bc3a14.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-055_2627c7781c2422e61f15dc061530fb61ee895dfe9d597eb198f9b192444e153a.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-055_2627c7781c2422e61f15dc061530fb61ee895dfe9d597eb198f9b192444e153a.jpg new file mode 100644 index 00000000..4df1e402 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-055_2627c7781c2422e61f15dc061530fb61ee895dfe9d597eb198f9b192444e153a.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-055_3436e5dc415f437f69b406e5880841060f0d266225965e1a6efdc7184ba5e9da.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-055_3436e5dc415f437f69b406e5880841060f0d266225965e1a6efdc7184ba5e9da.jpg new file mode 100644 index 00000000..ec9f660f Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-055_3436e5dc415f437f69b406e5880841060f0d266225965e1a6efdc7184ba5e9da.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-055_5c5502e6c7e55f7b85b5bd6fc81e1d5981d9ed2706537e2939d5abbe903d591f.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-055_5c5502e6c7e55f7b85b5bd6fc81e1d5981d9ed2706537e2939d5abbe903d591f.jpg new file mode 100644 index 00000000..2a7f70bb Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-055_5c5502e6c7e55f7b85b5bd6fc81e1d5981d9ed2706537e2939d5abbe903d591f.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-055_90c5a5b0182346096dc069e6dc945b371afeb7cc492dcdc09115b98af5b08629.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-055_90c5a5b0182346096dc069e6dc945b371afeb7cc492dcdc09115b98af5b08629.jpg new file mode 100644 index 00000000..6357e66a Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-055_90c5a5b0182346096dc069e6dc945b371afeb7cc492dcdc09115b98af5b08629.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-055_9d22be8ab13d5e5b7b429709cdd0d5e17f735f7bfe619a384a9add203b09156c.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-055_9d22be8ab13d5e5b7b429709cdd0d5e17f735f7bfe619a384a9add203b09156c.jpg new file mode 100644 index 00000000..fbd09aa3 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-055_9d22be8ab13d5e5b7b429709cdd0d5e17f735f7bfe619a384a9add203b09156c.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-055_d2d617f50410e3c28ef3a0dbee02e19dcd01e5e60db4266b4d4a73674e91c902.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-055_d2d617f50410e3c28ef3a0dbee02e19dcd01e5e60db4266b4d4a73674e91c902.jpg new file mode 100644 index 00000000..31dde76a Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-055_d2d617f50410e3c28ef3a0dbee02e19dcd01e5e60db4266b4d4a73674e91c902.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-055_f6b7a8cdbf2edf68b743983f1290c2ecd8e3402409ca771926de5e386b2363e7.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-055_f6b7a8cdbf2edf68b743983f1290c2ecd8e3402409ca771926de5e386b2363e7.jpg new file mode 100644 index 00000000..8a685050 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-055_f6b7a8cdbf2edf68b743983f1290c2ecd8e3402409ca771926de5e386b2363e7.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-055_f99e1e53175ee4b32bcf29605cc02b060e6e2e3e69ff0a933e893f3cd0ebddb9.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-055_f99e1e53175ee4b32bcf29605cc02b060e6e2e3e69ff0a933e893f3cd0ebddb9.jpg new file mode 100644 index 00000000..47bcfdcd Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-055_f99e1e53175ee4b32bcf29605cc02b060e6e2e3e69ff0a933e893f3cd0ebddb9.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-056_3094c93b7de5825dfd55624e332b78ae5643d821e5116070a5779480c23a7949.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-056_3094c93b7de5825dfd55624e332b78ae5643d821e5116070a5779480c23a7949.jpg new file mode 100644 index 00000000..f0aa22c8 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-056_3094c93b7de5825dfd55624e332b78ae5643d821e5116070a5779480c23a7949.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-056_3fd36f395247a760a94b2095aaefbd11a61b227dfe56b7ab70e23df4a8523acf.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-056_3fd36f395247a760a94b2095aaefbd11a61b227dfe56b7ab70e23df4a8523acf.jpg new file mode 100644 index 00000000..cf858441 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-056_3fd36f395247a760a94b2095aaefbd11a61b227dfe56b7ab70e23df4a8523acf.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-056_5dfe4f63eb3525d1af31243e39f77e693a2e559b05b8d1d53081fda78d499e6b.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-056_5dfe4f63eb3525d1af31243e39f77e693a2e559b05b8d1d53081fda78d499e6b.jpg new file mode 100644 index 00000000..0384a490 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-056_5dfe4f63eb3525d1af31243e39f77e693a2e559b05b8d1d53081fda78d499e6b.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-056_beaf8c50723c5fb7148b94609c1fa02ca4f84760448aa767f5cd7f7390ca57f7.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-056_beaf8c50723c5fb7148b94609c1fa02ca4f84760448aa767f5cd7f7390ca57f7.jpg new file mode 100644 index 00000000..bc9c5fa3 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-056_beaf8c50723c5fb7148b94609c1fa02ca4f84760448aa767f5cd7f7390ca57f7.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-057_02fb3294dbb3aea01b2e476887c52fdb4b8cb593d0a75e8c33e28cdc4c4bfd48.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-057_02fb3294dbb3aea01b2e476887c52fdb4b8cb593d0a75e8c33e28cdc4c4bfd48.jpg new file mode 100644 index 00000000..0ab46156 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-057_02fb3294dbb3aea01b2e476887c52fdb4b8cb593d0a75e8c33e28cdc4c4bfd48.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-057_03c5a72515addb330f6ff04d71944374121aaf6ba9f9aa2ea0ef56ffd08c883a.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-057_03c5a72515addb330f6ff04d71944374121aaf6ba9f9aa2ea0ef56ffd08c883a.jpg new file mode 100644 index 00000000..b384fceb Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-057_03c5a72515addb330f6ff04d71944374121aaf6ba9f9aa2ea0ef56ffd08c883a.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-057_08853a886c511ebb7a6b3c5f3e0ea191e7eab1ef3f2d24e8a6c6dfda98789df6.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-057_08853a886c511ebb7a6b3c5f3e0ea191e7eab1ef3f2d24e8a6c6dfda98789df6.jpg new file mode 100644 index 00000000..ab767786 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-057_08853a886c511ebb7a6b3c5f3e0ea191e7eab1ef3f2d24e8a6c6dfda98789df6.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-057_1b31df55326234830ecd41b1d6386d57827d45ec6834fa2370f738feaab33ffd.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-057_1b31df55326234830ecd41b1d6386d57827d45ec6834fa2370f738feaab33ffd.jpg new file mode 100644 index 00000000..f930fde0 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-057_1b31df55326234830ecd41b1d6386d57827d45ec6834fa2370f738feaab33ffd.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-057_1cb041e50d49e04698a8f8543654b75b6712107d966ecccde01f8be48d102486.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-057_1cb041e50d49e04698a8f8543654b75b6712107d966ecccde01f8be48d102486.jpg new file mode 100644 index 00000000..7f6269eb Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-057_1cb041e50d49e04698a8f8543654b75b6712107d966ecccde01f8be48d102486.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-057_3fce70325d90c19ad8c506236135c912dfce677093ed7b0a973d063663d4d575.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-057_3fce70325d90c19ad8c506236135c912dfce677093ed7b0a973d063663d4d575.jpg new file mode 100644 index 00000000..f5e1b348 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-057_3fce70325d90c19ad8c506236135c912dfce677093ed7b0a973d063663d4d575.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-057_4974a42428657e51ad07c553a7ab611b9351043adac29d25e0ad9cfa2ee666d3.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-057_4974a42428657e51ad07c553a7ab611b9351043adac29d25e0ad9cfa2ee666d3.jpg new file mode 100644 index 00000000..5d868377 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-057_4974a42428657e51ad07c553a7ab611b9351043adac29d25e0ad9cfa2ee666d3.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-057_8d671092e3878c5f10d62c2d79e3643c67c13d952d652f5f80d57377db66f3a9.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-057_8d671092e3878c5f10d62c2d79e3643c67c13d952d652f5f80d57377db66f3a9.jpg new file mode 100644 index 00000000..088cfce7 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-057_8d671092e3878c5f10d62c2d79e3643c67c13d952d652f5f80d57377db66f3a9.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-057_9f706d5ad610c6b2b3617aaa3a4baba33759ceedfbe9492bbdbf21945b053f44.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-057_9f706d5ad610c6b2b3617aaa3a4baba33759ceedfbe9492bbdbf21945b053f44.jpg new file mode 100644 index 00000000..f9944f8b Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-057_9f706d5ad610c6b2b3617aaa3a4baba33759ceedfbe9492bbdbf21945b053f44.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-057_abcce995f08b479a79a4823a223a4c391438ab14eba7bfb5dddf92585d955d89.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-057_abcce995f08b479a79a4823a223a4c391438ab14eba7bfb5dddf92585d955d89.jpg new file mode 100644 index 00000000..b80bee9f Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-057_abcce995f08b479a79a4823a223a4c391438ab14eba7bfb5dddf92585d955d89.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-057_bf18bae7a5bf60a7a07d909642dda4555662edd86508f88f85662158e093adec.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-057_bf18bae7a5bf60a7a07d909642dda4555662edd86508f88f85662158e093adec.jpg new file mode 100644 index 00000000..fdc604f6 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-057_bf18bae7a5bf60a7a07d909642dda4555662edd86508f88f85662158e093adec.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-057_c42518e77ea48a44daba3313808185b63d630bb4c46680a20a05b37c7bfea63b.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-057_c42518e77ea48a44daba3313808185b63d630bb4c46680a20a05b37c7bfea63b.jpg new file mode 100644 index 00000000..6ad79470 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-057_c42518e77ea48a44daba3313808185b63d630bb4c46680a20a05b37c7bfea63b.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-057_de3edb4cc0bf8ab77080d2b953f950dcc5a0f7b6827cca95fe36b9ab30e5c27e.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-057_de3edb4cc0bf8ab77080d2b953f950dcc5a0f7b6827cca95fe36b9ab30e5c27e.jpg new file mode 100644 index 00000000..704a3482 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-057_de3edb4cc0bf8ab77080d2b953f950dcc5a0f7b6827cca95fe36b9ab30e5c27e.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-058_0021c8ae7f23306a48c862ee3de074ec7f140638fa8f1ac34b23195f3cebd63b.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-058_0021c8ae7f23306a48c862ee3de074ec7f140638fa8f1ac34b23195f3cebd63b.jpg new file mode 100644 index 00000000..3abced4d Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-058_0021c8ae7f23306a48c862ee3de074ec7f140638fa8f1ac34b23195f3cebd63b.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-058_0e707b527379ce1e2fbd600eb2b769bfaf38d934cf465532c496c82d4e1056e4.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-058_0e707b527379ce1e2fbd600eb2b769bfaf38d934cf465532c496c82d4e1056e4.jpg new file mode 100644 index 00000000..b2a28795 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-058_0e707b527379ce1e2fbd600eb2b769bfaf38d934cf465532c496c82d4e1056e4.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-058_23ee948803cb3a7f6575b21a4cb6ebe45a492a773fad00e1be51ac67b41fda7c.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-058_23ee948803cb3a7f6575b21a4cb6ebe45a492a773fad00e1be51ac67b41fda7c.jpg new file mode 100644 index 00000000..20248f0b Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-058_23ee948803cb3a7f6575b21a4cb6ebe45a492a773fad00e1be51ac67b41fda7c.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-058_350d1514e25da16659ed3a3903c16dd5dc65d67d5eb9f879868e782b43decb72.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-058_350d1514e25da16659ed3a3903c16dd5dc65d67d5eb9f879868e782b43decb72.jpg new file mode 100644 index 00000000..881f89ea Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-058_350d1514e25da16659ed3a3903c16dd5dc65d67d5eb9f879868e782b43decb72.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-058_48c898c2e5e9c59b6182f47f4ef5436f12eed8dd1ab548b862d1b859c52198bd.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-058_48c898c2e5e9c59b6182f47f4ef5436f12eed8dd1ab548b862d1b859c52198bd.jpg new file mode 100644 index 00000000..4fb91d93 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-058_48c898c2e5e9c59b6182f47f4ef5436f12eed8dd1ab548b862d1b859c52198bd.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-058_4e9599114f0ec932f1352cdea716cb4f4f3e63e920ded151d36aa4e9d2881a0e.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-058_4e9599114f0ec932f1352cdea716cb4f4f3e63e920ded151d36aa4e9d2881a0e.jpg new file mode 100644 index 00000000..7acb14de Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-058_4e9599114f0ec932f1352cdea716cb4f4f3e63e920ded151d36aa4e9d2881a0e.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-058_6325d13eac62f1d59ebedc70d7d70355748e484f93b23ce1768de341fb442b81.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-058_6325d13eac62f1d59ebedc70d7d70355748e484f93b23ce1768de341fb442b81.jpg new file mode 100644 index 00000000..e0b639a9 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-058_6325d13eac62f1d59ebedc70d7d70355748e484f93b23ce1768de341fb442b81.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-058_80975027d9e251faa741cf19c21f5be6de2a6f20139d3ab0b4e6b22943a00065.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-058_80975027d9e251faa741cf19c21f5be6de2a6f20139d3ab0b4e6b22943a00065.jpg new file mode 100644 index 00000000..a0cabd07 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-058_80975027d9e251faa741cf19c21f5be6de2a6f20139d3ab0b4e6b22943a00065.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-058_8b0f8f533b6628e9fa989c741804bc0b73dac07c8d691666ab639744fe3a2126.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-058_8b0f8f533b6628e9fa989c741804bc0b73dac07c8d691666ab639744fe3a2126.jpg new file mode 100644 index 00000000..f26af33c Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-058_8b0f8f533b6628e9fa989c741804bc0b73dac07c8d691666ab639744fe3a2126.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-058_b64ac896d2217416530751e69e0e8545dadee4656fd7c12cd28b8e0ed17ba0cf.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-058_b64ac896d2217416530751e69e0e8545dadee4656fd7c12cd28b8e0ed17ba0cf.jpg new file mode 100644 index 00000000..b3995d7f Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-058_b64ac896d2217416530751e69e0e8545dadee4656fd7c12cd28b8e0ed17ba0cf.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-058_b6d55fea481f15134e232b1d452e78bb095c2859ac8ad927e79cef9e663f5d7b.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-058_b6d55fea481f15134e232b1d452e78bb095c2859ac8ad927e79cef9e663f5d7b.jpg new file mode 100644 index 00000000..3c531859 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-058_b6d55fea481f15134e232b1d452e78bb095c2859ac8ad927e79cef9e663f5d7b.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-058_d04979845dddb0a125ea878b096ac0eff9119daf1daf8d1f8d0c22c7672b2f2d.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-058_d04979845dddb0a125ea878b096ac0eff9119daf1daf8d1f8d0c22c7672b2f2d.jpg new file mode 100644 index 00000000..d3941af1 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-058_d04979845dddb0a125ea878b096ac0eff9119daf1daf8d1f8d0c22c7672b2f2d.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-058_d79c553d1225434d9df3b9666acce49ed0f7df280e803bb9acfe312fbad0ad72.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-058_d79c553d1225434d9df3b9666acce49ed0f7df280e803bb9acfe312fbad0ad72.jpg new file mode 100644 index 00000000..3f2b3674 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-058_d79c553d1225434d9df3b9666acce49ed0f7df280e803bb9acfe312fbad0ad72.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-059_619aa88b28847e1fd2cac6b6cc18aa7915cc7fa5b8292542ac955e0710bf6cde.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-059_619aa88b28847e1fd2cac6b6cc18aa7915cc7fa5b8292542ac955e0710bf6cde.jpg new file mode 100644 index 00000000..59e811cf Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-059_619aa88b28847e1fd2cac6b6cc18aa7915cc7fa5b8292542ac955e0710bf6cde.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-061_0b955f930c7e53b847791197786303bac9555973e1c7d7782141b2d3324bcac3.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-061_0b955f930c7e53b847791197786303bac9555973e1c7d7782141b2d3324bcac3.jpg new file mode 100644 index 00000000..3187ac7c Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-061_0b955f930c7e53b847791197786303bac9555973e1c7d7782141b2d3324bcac3.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-061_70549882fd3543200fe2cdbbd85c8c0fa0cea064e982d6b92772124c9b3504d0.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-061_70549882fd3543200fe2cdbbd85c8c0fa0cea064e982d6b92772124c9b3504d0.jpg new file mode 100644 index 00000000..77811b36 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-061_70549882fd3543200fe2cdbbd85c8c0fa0cea064e982d6b92772124c9b3504d0.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-061_b46a9651f5260ff710b6f018d1fb6aa7b867b456c2ee9e71d9e4b9ae71c0ca88.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-061_b46a9651f5260ff710b6f018d1fb6aa7b867b456c2ee9e71d9e4b9ae71c0ca88.jpg new file mode 100644 index 00000000..c22a84be Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-061_b46a9651f5260ff710b6f018d1fb6aa7b867b456c2ee9e71d9e4b9ae71c0ca88.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-063_19d61d9ef637ebf31af8219b02e9bcfd24084474fb31098473b896b1cfe8ea24.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-063_19d61d9ef637ebf31af8219b02e9bcfd24084474fb31098473b896b1cfe8ea24.jpg new file mode 100644 index 00000000..7d48c27e Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-063_19d61d9ef637ebf31af8219b02e9bcfd24084474fb31098473b896b1cfe8ea24.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-063_bb69558b2545b1ad630bca2d75e3ce2bb22ccfe481d95c98a4289160321aa868.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-063_bb69558b2545b1ad630bca2d75e3ce2bb22ccfe481d95c98a4289160321aa868.jpg new file mode 100644 index 00000000..d4da1465 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-063_bb69558b2545b1ad630bca2d75e3ce2bb22ccfe481d95c98a4289160321aa868.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-063_fd9ed771d4c4c47c6b303905131513b6a3022f3dff758223629bb99c6e08b1aa.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-063_fd9ed771d4c4c47c6b303905131513b6a3022f3dff758223629bb99c6e08b1aa.jpg new file mode 100644 index 00000000..68f3da1c Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-063_fd9ed771d4c4c47c6b303905131513b6a3022f3dff758223629bb99c6e08b1aa.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-064_7896cfdd5bcaf7bcfe4a4bb1b437da348e3c02bff0aeba2d629f175706b97ed9.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-064_7896cfdd5bcaf7bcfe4a4bb1b437da348e3c02bff0aeba2d629f175706b97ed9.jpg new file mode 100644 index 00000000..4e1c43d9 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-064_7896cfdd5bcaf7bcfe4a4bb1b437da348e3c02bff0aeba2d629f175706b97ed9.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-064_c04e383f281563bc83b9efdc69aad72139e687a23094a175c73f962cdf3587f9.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-064_c04e383f281563bc83b9efdc69aad72139e687a23094a175c73f962cdf3587f9.jpg new file mode 100644 index 00000000..32b8c8f7 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-064_c04e383f281563bc83b9efdc69aad72139e687a23094a175c73f962cdf3587f9.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-066_cc287bf1cb36a6192dd5fc2df1d2d1dbb3b0a6b9fdd2adc4c9e1057dc7529ee0.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-066_cc287bf1cb36a6192dd5fc2df1d2d1dbb3b0a6b9fdd2adc4c9e1057dc7529ee0.jpg new file mode 100644 index 00000000..845e1e0f Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-066_cc287bf1cb36a6192dd5fc2df1d2d1dbb3b0a6b9fdd2adc4c9e1057dc7529ee0.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-066_df4a90f508cf1a3f9758b9bc40308e7e87aaefd3b7eab92d06fb480396094a0c.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-066_df4a90f508cf1a3f9758b9bc40308e7e87aaefd3b7eab92d06fb480396094a0c.jpg new file mode 100644 index 00000000..0eb9f316 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-066_df4a90f508cf1a3f9758b9bc40308e7e87aaefd3b7eab92d06fb480396094a0c.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-066_eb13bacf93b07a567a405e54de83b9c6f228354a098f2542d2a3237a2bfe788d.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-066_eb13bacf93b07a567a405e54de83b9c6f228354a098f2542d2a3237a2bfe788d.jpg new file mode 100644 index 00000000..deb09d7e Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-066_eb13bacf93b07a567a405e54de83b9c6f228354a098f2542d2a3237a2bfe788d.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-070_0414350d9ff022106fcdcfe29f245c73f3b6d1e5bcb3abca4e5f6c7a1224584e.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-070_0414350d9ff022106fcdcfe29f245c73f3b6d1e5bcb3abca4e5f6c7a1224584e.jpg new file mode 100644 index 00000000..d14149e8 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-070_0414350d9ff022106fcdcfe29f245c73f3b6d1e5bcb3abca4e5f6c7a1224584e.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-073_3cb152a61fa8a1cc0856f9f602ef69c94221fba4a3701a89ab17ed8764ab6694.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-073_3cb152a61fa8a1cc0856f9f602ef69c94221fba4a3701a89ab17ed8764ab6694.jpg new file mode 100644 index 00000000..72bce89b Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-073_3cb152a61fa8a1cc0856f9f602ef69c94221fba4a3701a89ab17ed8764ab6694.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-075_6b79e379471a87a8bd71de60486fd9b6dabffe9257a0ad29f2ea8417dab06eda.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-075_6b79e379471a87a8bd71de60486fd9b6dabffe9257a0ad29f2ea8417dab06eda.jpg new file mode 100644 index 00000000..391a85ac Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-075_6b79e379471a87a8bd71de60486fd9b6dabffe9257a0ad29f2ea8417dab06eda.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-076_8cb71bff55e951cb9c90a3aceb7ff70d64b723598b6ae7c3270c66c7ab2b5d16.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-076_8cb71bff55e951cb9c90a3aceb7ff70d64b723598b6ae7c3270c66c7ab2b5d16.jpg new file mode 100644 index 00000000..2ed50c42 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-076_8cb71bff55e951cb9c90a3aceb7ff70d64b723598b6ae7c3270c66c7ab2b5d16.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-076_d9b6194410a6e16fd869a5bc9b0c3a4142b9d2a51a27287dc0f409bf3460516b.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-076_d9b6194410a6e16fd869a5bc9b0c3a4142b9d2a51a27287dc0f409bf3460516b.jpg new file mode 100644 index 00000000..dfe9b45c Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-076_d9b6194410a6e16fd869a5bc9b0c3a4142b9d2a51a27287dc0f409bf3460516b.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-077_f02046f242e0154e2d52fbd49794a8baa0a1de8a70a651bb48d7ac6fb386c5f0.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-077_f02046f242e0154e2d52fbd49794a8baa0a1de8a70a651bb48d7ac6fb386c5f0.jpg new file mode 100644 index 00000000..6e773854 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-077_f02046f242e0154e2d52fbd49794a8baa0a1de8a70a651bb48d7ac6fb386c5f0.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-078_882236ee5ef0bde0223cf8848ed41ec4bda2bcf76c6093a4e9e3b24260285327.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-078_882236ee5ef0bde0223cf8848ed41ec4bda2bcf76c6093a4e9e3b24260285327.jpg new file mode 100644 index 00000000..9aed583c Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-078_882236ee5ef0bde0223cf8848ed41ec4bda2bcf76c6093a4e9e3b24260285327.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-078_906bd0acfe713b6d244d0978e20cc4c76b04e8441580d9fa4b38d14463bb9508.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-078_906bd0acfe713b6d244d0978e20cc4c76b04e8441580d9fa4b38d14463bb9508.jpg new file mode 100644 index 00000000..bc329cdb Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-078_906bd0acfe713b6d244d0978e20cc4c76b04e8441580d9fa4b38d14463bb9508.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-078_a2b0cc7d641d6a79140b3ba5f151b4bbf25647a3a73aa3681a41a8553e8c8058.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-078_a2b0cc7d641d6a79140b3ba5f151b4bbf25647a3a73aa3681a41a8553e8c8058.jpg new file mode 100644 index 00000000..3ec86b5a Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-078_a2b0cc7d641d6a79140b3ba5f151b4bbf25647a3a73aa3681a41a8553e8c8058.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-078_e12a52f66d1d2154f88875288e881af591c13975efdd4ff75751e5123054095b.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-078_e12a52f66d1d2154f88875288e881af591c13975efdd4ff75751e5123054095b.jpg new file mode 100644 index 00000000..04477666 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-078_e12a52f66d1d2154f88875288e881af591c13975efdd4ff75751e5123054095b.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-079_ea0ea723b0923e0f34a13e59da0fce4c9fbb2990b48109c91a470d9e1356bcde.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-079_ea0ea723b0923e0f34a13e59da0fce4c9fbb2990b48109c91a470d9e1356bcde.jpg new file mode 100644 index 00000000..0c567de3 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-079_ea0ea723b0923e0f34a13e59da0fce4c9fbb2990b48109c91a470d9e1356bcde.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-081_93e76ff84c571f46ffebedab637e41d765ced18d5d355e9fe83fc412207844f5.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-081_93e76ff84c571f46ffebedab637e41d765ced18d5d355e9fe83fc412207844f5.jpg new file mode 100644 index 00000000..1c24d6fe Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-081_93e76ff84c571f46ffebedab637e41d765ced18d5d355e9fe83fc412207844f5.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-081_febeb502f581cb810710c145bacf417af9dfa036a73e96182aa790b4a8ae6a91.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-081_febeb502f581cb810710c145bacf417af9dfa036a73e96182aa790b4a8ae6a91.jpg new file mode 100644 index 00000000..07f87807 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-081_febeb502f581cb810710c145bacf417af9dfa036a73e96182aa790b4a8ae6a91.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-082_53ff67d6bd170454a73005a5cbef3f941899b79657837599e019997294a9f7c5.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-082_53ff67d6bd170454a73005a5cbef3f941899b79657837599e019997294a9f7c5.jpg new file mode 100644 index 00000000..d2eb984e Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-082_53ff67d6bd170454a73005a5cbef3f941899b79657837599e019997294a9f7c5.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-082_d4f807e837061dfdefd29cba212df26823fd21ad98dcb1f7f5e4af29ba6c82a4.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-082_d4f807e837061dfdefd29cba212df26823fd21ad98dcb1f7f5e4af29ba6c82a4.jpg new file mode 100644 index 00000000..b71ed424 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-082_d4f807e837061dfdefd29cba212df26823fd21ad98dcb1f7f5e4af29ba6c82a4.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-083_40a9ef7a7b2e68a8a20cdf2213f94c22dca230db0b8dfb1cde4ef0716fa9fde4.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-083_40a9ef7a7b2e68a8a20cdf2213f94c22dca230db0b8dfb1cde4ef0716fa9fde4.jpg new file mode 100644 index 00000000..0d2e4da6 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-083_40a9ef7a7b2e68a8a20cdf2213f94c22dca230db0b8dfb1cde4ef0716fa9fde4.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-084_7bb3861fd79ee6ec9bba5d577654aa42373ecc907a18f58dc144f7a13febd3e1.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-084_7bb3861fd79ee6ec9bba5d577654aa42373ecc907a18f58dc144f7a13febd3e1.jpg new file mode 100644 index 00000000..fb49e6e9 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-084_7bb3861fd79ee6ec9bba5d577654aa42373ecc907a18f58dc144f7a13febd3e1.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-085_8386c0878267382ac15315204d1b2c733738d44010d405b44fcf0080e5101f99.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-085_8386c0878267382ac15315204d1b2c733738d44010d405b44fcf0080e5101f99.jpg new file mode 100644 index 00000000..3a48373e Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-085_8386c0878267382ac15315204d1b2c733738d44010d405b44fcf0080e5101f99.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-085_d22349885f86d6f35b7459b85d37124c50cfc8dfcef7d3e5fd6c90a5a99b4f48.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-085_d22349885f86d6f35b7459b85d37124c50cfc8dfcef7d3e5fd6c90a5a99b4f48.jpg new file mode 100644 index 00000000..4e830ace Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-085_d22349885f86d6f35b7459b85d37124c50cfc8dfcef7d3e5fd6c90a5a99b4f48.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-086_1c2ce715771417b390b95692923607e688a7a67a4f790212611d691dc864ebd3.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-086_1c2ce715771417b390b95692923607e688a7a67a4f790212611d691dc864ebd3.jpg new file mode 100644 index 00000000..f43ab8e8 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-086_1c2ce715771417b390b95692923607e688a7a67a4f790212611d691dc864ebd3.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-086_6c890d481157f6c0e32233bb5f5bb0cb87fcb42390480f7f3effd17a221e0c49.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-086_6c890d481157f6c0e32233bb5f5bb0cb87fcb42390480f7f3effd17a221e0c49.jpg new file mode 100644 index 00000000..34d1075c Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-086_6c890d481157f6c0e32233bb5f5bb0cb87fcb42390480f7f3effd17a221e0c49.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-086_6d8d0b7daaedb66be5667df45bba656901ba7f1de331ead8a472d899dafc2022.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-086_6d8d0b7daaedb66be5667df45bba656901ba7f1de331ead8a472d899dafc2022.jpg new file mode 100644 index 00000000..10fb07ca Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-086_6d8d0b7daaedb66be5667df45bba656901ba7f1de331ead8a472d899dafc2022.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-086_7726c03a40c6dd8a7684e77bae0710bef42c2d56bfc2ba49eb83855112933fd4.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-086_7726c03a40c6dd8a7684e77bae0710bef42c2d56bfc2ba49eb83855112933fd4.jpg new file mode 100644 index 00000000..996525d7 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-086_7726c03a40c6dd8a7684e77bae0710bef42c2d56bfc2ba49eb83855112933fd4.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-086_eb3cdeba3292b5b3f0bd0b54ba60ddaddc64d5605c597d4007b53cc509d6499f.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-086_eb3cdeba3292b5b3f0bd0b54ba60ddaddc64d5605c597d4007b53cc509d6499f.jpg new file mode 100644 index 00000000..e1256f1b Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-086_eb3cdeba3292b5b3f0bd0b54ba60ddaddc64d5605c597d4007b53cc509d6499f.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-088_89a76ea89ffbe1acd28680d207755bf880e4942cf46ae5d05a1ac536d103f555.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-088_89a76ea89ffbe1acd28680d207755bf880e4942cf46ae5d05a1ac536d103f555.jpg new file mode 100644 index 00000000..a0ae1ce7 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-088_89a76ea89ffbe1acd28680d207755bf880e4942cf46ae5d05a1ac536d103f555.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-089_ec7b98e64af0773d3912bb1ac4ffbf46497bfdf2c34a18b772aaa9ade9f0a723.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-089_ec7b98e64af0773d3912bb1ac4ffbf46497bfdf2c34a18b772aaa9ade9f0a723.jpg new file mode 100644 index 00000000..285b119a Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-089_ec7b98e64af0773d3912bb1ac4ffbf46497bfdf2c34a18b772aaa9ade9f0a723.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-090_7fa5539778c211b6fd54a3e34d89cf95fbfd18d1b7b8be367bf6dd57a2657758.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-090_7fa5539778c211b6fd54a3e34d89cf95fbfd18d1b7b8be367bf6dd57a2657758.jpg new file mode 100644 index 00000000..64b067db Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-090_7fa5539778c211b6fd54a3e34d89cf95fbfd18d1b7b8be367bf6dd57a2657758.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-090_99c47a2d240be0f0012b3d5c6077199dd388dcec73c1c6a97bb8bea7dce1c6eb.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-090_99c47a2d240be0f0012b3d5c6077199dd388dcec73c1c6a97bb8bea7dce1c6eb.jpg new file mode 100644 index 00000000..00694979 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-090_99c47a2d240be0f0012b3d5c6077199dd388dcec73c1c6a97bb8bea7dce1c6eb.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-091_9504235f79258d35ca2cb7868a3c4a1fcabcb93c1ed8af77f10b582b7cbd8ec7.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-091_9504235f79258d35ca2cb7868a3c4a1fcabcb93c1ed8af77f10b582b7cbd8ec7.jpg new file mode 100644 index 00000000..289c420f Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-091_9504235f79258d35ca2cb7868a3c4a1fcabcb93c1ed8af77f10b582b7cbd8ec7.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-092_8662acd15458d6863d6651da265c0285782701a432ae2785ed027878a9567e06.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-092_8662acd15458d6863d6651da265c0285782701a432ae2785ed027878a9567e06.jpg new file mode 100644 index 00000000..471359eb Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-092_8662acd15458d6863d6651da265c0285782701a432ae2785ed027878a9567e06.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-093_63dc8e93b23d7e7ed7e827cf2c194a21603e7e73778c17820a85ceaded0a161e.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-093_63dc8e93b23d7e7ed7e827cf2c194a21603e7e73778c17820a85ceaded0a161e.jpg new file mode 100644 index 00000000..07fa5374 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-093_63dc8e93b23d7e7ed7e827cf2c194a21603e7e73778c17820a85ceaded0a161e.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-093_cdebab83171b40011b56607ebb0f601021c2123a8a0d28b5ead13013ad6d89ff.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-093_cdebab83171b40011b56607ebb0f601021c2123a8a0d28b5ead13013ad6d89ff.jpg new file mode 100644 index 00000000..f8f93ac1 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-093_cdebab83171b40011b56607ebb0f601021c2123a8a0d28b5ead13013ad6d89ff.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-094_00abdcb4bb43d656426939593c12e57357229fa0beee37e8aa139095603e822a.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-094_00abdcb4bb43d656426939593c12e57357229fa0beee37e8aa139095603e822a.jpg new file mode 100644 index 00000000..c2f933af Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-094_00abdcb4bb43d656426939593c12e57357229fa0beee37e8aa139095603e822a.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-094_628c66eff925ef3970176c4a08bf23121b887a08d44a6a75d853c5424f082607.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-094_628c66eff925ef3970176c4a08bf23121b887a08d44a6a75d853c5424f082607.jpg new file mode 100644 index 00000000..fca8ba25 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-094_628c66eff925ef3970176c4a08bf23121b887a08d44a6a75d853c5424f082607.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-094_c1e9fe58f7df129d9070f847ffb4e862dbf9a90a8bfdf3f6599269e3daa5481b.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-094_c1e9fe58f7df129d9070f847ffb4e862dbf9a90a8bfdf3f6599269e3daa5481b.jpg new file mode 100644 index 00000000..8f1a04e2 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-094_c1e9fe58f7df129d9070f847ffb4e862dbf9a90a8bfdf3f6599269e3daa5481b.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-095_177a23b5c8b85fb65aafeaf7f3baa81c208e3a6e900777f3831812be7e922df4.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-095_177a23b5c8b85fb65aafeaf7f3baa81c208e3a6e900777f3831812be7e922df4.jpg new file mode 100644 index 00000000..2c9353b6 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-095_177a23b5c8b85fb65aafeaf7f3baa81c208e3a6e900777f3831812be7e922df4.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-095_21fe25fa4514e05c6b64ff0da81f0873104edb2e73d11a411d67b2b76a505a6b.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-095_21fe25fa4514e05c6b64ff0da81f0873104edb2e73d11a411d67b2b76a505a6b.jpg new file mode 100644 index 00000000..6ccc12d8 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-095_21fe25fa4514e05c6b64ff0da81f0873104edb2e73d11a411d67b2b76a505a6b.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-095_3277f8c6b302254770300a2e792eb43e6d677b3f27dcc4771e8269ee40b4f6ec.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-095_3277f8c6b302254770300a2e792eb43e6d677b3f27dcc4771e8269ee40b4f6ec.jpg new file mode 100644 index 00000000..70f5d2ba Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-095_3277f8c6b302254770300a2e792eb43e6d677b3f27dcc4771e8269ee40b4f6ec.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-095_4624b77e4669051641dc4ce8d0f6233a9ef41410def8b1c9ddbf7b7c74608d4c.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-095_4624b77e4669051641dc4ce8d0f6233a9ef41410def8b1c9ddbf7b7c74608d4c.jpg new file mode 100644 index 00000000..4db745cd Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-095_4624b77e4669051641dc4ce8d0f6233a9ef41410def8b1c9ddbf7b7c74608d4c.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-095_54b6c09bee36bf4807283ea746ba80af5099e339a0867c5f2de50a172e6bddf1.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-095_54b6c09bee36bf4807283ea746ba80af5099e339a0867c5f2de50a172e6bddf1.jpg new file mode 100644 index 00000000..457eb034 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-095_54b6c09bee36bf4807283ea746ba80af5099e339a0867c5f2de50a172e6bddf1.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-095_9726715bbeeeaea2d8747c6d1cebeb53c51944be14af4334a1f1462fb4092e9a.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-095_9726715bbeeeaea2d8747c6d1cebeb53c51944be14af4334a1f1462fb4092e9a.jpg new file mode 100644 index 00000000..751498ca Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-095_9726715bbeeeaea2d8747c6d1cebeb53c51944be14af4334a1f1462fb4092e9a.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-095_b182f93c5641f215d0dea841f8b16d6fd942366205562aa186c2983dc25e0014.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-095_b182f93c5641f215d0dea841f8b16d6fd942366205562aa186c2983dc25e0014.jpg new file mode 100644 index 00000000..1b83a17e Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-095_b182f93c5641f215d0dea841f8b16d6fd942366205562aa186c2983dc25e0014.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-095_fc0f223d06f789505ca941a3fa36ea4b337ce44f20d0ef8b6a6bd671b9f3439e.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-095_fc0f223d06f789505ca941a3fa36ea4b337ce44f20d0ef8b6a6bd671b9f3439e.jpg new file mode 100644 index 00000000..4e295192 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-095_fc0f223d06f789505ca941a3fa36ea4b337ce44f20d0ef8b6a6bd671b9f3439e.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-097_9db6e1ed6745f0680a95c71b8983c8aeea9890681516ea3f8a1bd567d54fc83d.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-097_9db6e1ed6745f0680a95c71b8983c8aeea9890681516ea3f8a1bd567d54fc83d.jpg new file mode 100644 index 00000000..3ca70597 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-097_9db6e1ed6745f0680a95c71b8983c8aeea9890681516ea3f8a1bd567d54fc83d.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-098_35b8167453b5fa4dd85d33baec6bf6c9892d77b967058a8eb9b87cd3a7e54a79.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-098_35b8167453b5fa4dd85d33baec6bf6c9892d77b967058a8eb9b87cd3a7e54a79.jpg new file mode 100644 index 00000000..894e5998 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-098_35b8167453b5fa4dd85d33baec6bf6c9892d77b967058a8eb9b87cd3a7e54a79.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-099_95b2fcfd25d0a63313102b8df5086fb4443c5b2b9c924020612ac9c2f750c2ee.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-099_95b2fcfd25d0a63313102b8df5086fb4443c5b2b9c924020612ac9c2f750c2ee.jpg new file mode 100644 index 00000000..0293fdb6 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-099_95b2fcfd25d0a63313102b8df5086fb4443c5b2b9c924020612ac9c2f750c2ee.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-100_51d93ea7baa9057f05c6e5c191d33507638cb2436d3c11273434fabc727540c3.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-100_51d93ea7baa9057f05c6e5c191d33507638cb2436d3c11273434fabc727540c3.jpg new file mode 100644 index 00000000..025680de Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-100_51d93ea7baa9057f05c6e5c191d33507638cb2436d3c11273434fabc727540c3.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-101_6da9c68ab204211c7135c0138c01cddcfa08f68d938afdda444e82067f71e73c.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-101_6da9c68ab204211c7135c0138c01cddcfa08f68d938afdda444e82067f71e73c.jpg new file mode 100644 index 00000000..67797c69 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-101_6da9c68ab204211c7135c0138c01cddcfa08f68d938afdda444e82067f71e73c.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-101_cf52779feb491906a1102171de4099983114596c28c816337ddee9e89db974e3.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-101_cf52779feb491906a1102171de4099983114596c28c816337ddee9e89db974e3.jpg new file mode 100644 index 00000000..cd792de4 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-101_cf52779feb491906a1102171de4099983114596c28c816337ddee9e89db974e3.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-102_0489c764bd10a78b7874855be312985168200715610dd7aa80be07f6148d0efa.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-102_0489c764bd10a78b7874855be312985168200715610dd7aa80be07f6148d0efa.jpg new file mode 100644 index 00000000..5900cc43 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-102_0489c764bd10a78b7874855be312985168200715610dd7aa80be07f6148d0efa.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-102_6e2c28946c10449a21f605a5dcc6ebf0eec0be87d6a81c3caa60fbd88447bd7b.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-102_6e2c28946c10449a21f605a5dcc6ebf0eec0be87d6a81c3caa60fbd88447bd7b.jpg new file mode 100644 index 00000000..ae0fea7c Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-102_6e2c28946c10449a21f605a5dcc6ebf0eec0be87d6a81c3caa60fbd88447bd7b.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-102_7e3af0bee07c2fbf02347e4799c018837dd0606008cc1647c5a98cd0237e8fb3.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-102_7e3af0bee07c2fbf02347e4799c018837dd0606008cc1647c5a98cd0237e8fb3.jpg new file mode 100644 index 00000000..c83df856 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-102_7e3af0bee07c2fbf02347e4799c018837dd0606008cc1647c5a98cd0237e8fb3.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-102_91939921a0d296ecb7e0c857641de8738ede488524fff6993ed11425b0053c48.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-102_91939921a0d296ecb7e0c857641de8738ede488524fff6993ed11425b0053c48.jpg new file mode 100644 index 00000000..a91278ac Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-102_91939921a0d296ecb7e0c857641de8738ede488524fff6993ed11425b0053c48.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-103_9be979aebddd3873b96201f1e4ebeaae5e0e82be36e6b7d261faa4ede290e3ed.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-103_9be979aebddd3873b96201f1e4ebeaae5e0e82be36e6b7d261faa4ede290e3ed.jpg new file mode 100644 index 00000000..4fec0ec2 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-103_9be979aebddd3873b96201f1e4ebeaae5e0e82be36e6b7d261faa4ede290e3ed.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-103_f91950ee1fdef5b13f1cd340e16230c51156e0253a1e29e838e96fd9dcdf744d.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-103_f91950ee1fdef5b13f1cd340e16230c51156e0253a1e29e838e96fd9dcdf744d.jpg new file mode 100644 index 00000000..724feb45 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-103_f91950ee1fdef5b13f1cd340e16230c51156e0253a1e29e838e96fd9dcdf744d.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-105_113275304cf7f0b2a20e933e9ed771da2bba6524439d8e158931ea46ebefdd11.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-105_113275304cf7f0b2a20e933e9ed771da2bba6524439d8e158931ea46ebefdd11.jpg new file mode 100644 index 00000000..2e3505d4 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-105_113275304cf7f0b2a20e933e9ed771da2bba6524439d8e158931ea46ebefdd11.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-106_9cd7e55bb7621a786a44fa86fc3abeb6413f7d18cf058b420f53c043b3153014.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-106_9cd7e55bb7621a786a44fa86fc3abeb6413f7d18cf058b420f53c043b3153014.jpg new file mode 100644 index 00000000..556bd67e Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-106_9cd7e55bb7621a786a44fa86fc3abeb6413f7d18cf058b420f53c043b3153014.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-106_b5d17b5fdab269cd4a8f3dc2928428e0fd1c73a677d6a06b7b465f18c80bba4a.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-106_b5d17b5fdab269cd4a8f3dc2928428e0fd1c73a677d6a06b7b465f18c80bba4a.jpg new file mode 100644 index 00000000..662b9e54 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-106_b5d17b5fdab269cd4a8f3dc2928428e0fd1c73a677d6a06b7b465f18c80bba4a.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-106_c5708822cc51fb2460f737d441a5386ba10799a48c6e849cc9ec431e588056f9.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-106_c5708822cc51fb2460f737d441a5386ba10799a48c6e849cc9ec431e588056f9.jpg new file mode 100644 index 00000000..7ed0f454 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-106_c5708822cc51fb2460f737d441a5386ba10799a48c6e849cc9ec431e588056f9.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-106_d38328231494b9e7c06c2892f9b7430520417ab8b5995473bd18e77a3a8b2f8b.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-106_d38328231494b9e7c06c2892f9b7430520417ab8b5995473bd18e77a3a8b2f8b.jpg new file mode 100644 index 00000000..a3b297d6 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-106_d38328231494b9e7c06c2892f9b7430520417ab8b5995473bd18e77a3a8b2f8b.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-107_25004ce6584904948cb05d29653ef67c7966be9b97c250bc83d2c54b4dd73814.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-107_25004ce6584904948cb05d29653ef67c7966be9b97c250bc83d2c54b4dd73814.jpg new file mode 100644 index 00000000..cf7a3f79 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-107_25004ce6584904948cb05d29653ef67c7966be9b97c250bc83d2c54b4dd73814.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-107_4b6d7e0c5de4889040b431028653cb1d419e676b3ca38c2315515db3f7a330ed.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-107_4b6d7e0c5de4889040b431028653cb1d419e676b3ca38c2315515db3f7a330ed.jpg new file mode 100644 index 00000000..d9d9a6bc Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-107_4b6d7e0c5de4889040b431028653cb1d419e676b3ca38c2315515db3f7a330ed.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-107_542d081dc6863984219450384cfc5ba4640f650c75143969f365ed0d813f2645.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-107_542d081dc6863984219450384cfc5ba4640f650c75143969f365ed0d813f2645.jpg new file mode 100644 index 00000000..00460365 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-107_542d081dc6863984219450384cfc5ba4640f650c75143969f365ed0d813f2645.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-107_7a70bd694f9ebb73e7f3643d5e0dadc10001128f6031e52cfab2f026f4c1acd5.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-107_7a70bd694f9ebb73e7f3643d5e0dadc10001128f6031e52cfab2f026f4c1acd5.jpg new file mode 100644 index 00000000..def59b00 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-107_7a70bd694f9ebb73e7f3643d5e0dadc10001128f6031e52cfab2f026f4c1acd5.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-107_8b4e0c2c9df9da72c3ba1236982ad8890fc769283d71b8a9f1bd8e7636b91db0.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-107_8b4e0c2c9df9da72c3ba1236982ad8890fc769283d71b8a9f1bd8e7636b91db0.jpg new file mode 100644 index 00000000..5c0fc12e Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-107_8b4e0c2c9df9da72c3ba1236982ad8890fc769283d71b8a9f1bd8e7636b91db0.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-107_9c409d16372f117a4fc7ba61ee8f41868073e43dec8000c642347ea642c50f3c.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-107_9c409d16372f117a4fc7ba61ee8f41868073e43dec8000c642347ea642c50f3c.jpg new file mode 100644 index 00000000..ebe48b62 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-107_9c409d16372f117a4fc7ba61ee8f41868073e43dec8000c642347ea642c50f3c.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-107_af91c71ae4b1f016d8c8d30a1a32f67a9bc46f8c29b72c1f0f13fc633f2e002a.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-107_af91c71ae4b1f016d8c8d30a1a32f67a9bc46f8c29b72c1f0f13fc633f2e002a.jpg new file mode 100644 index 00000000..5961727f Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-107_af91c71ae4b1f016d8c8d30a1a32f67a9bc46f8c29b72c1f0f13fc633f2e002a.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-107_d14b811bd49192059c14ead97f13eb9e3bec7124e0326638d6d441eb24652ea1.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-107_d14b811bd49192059c14ead97f13eb9e3bec7124e0326638d6d441eb24652ea1.jpg new file mode 100644 index 00000000..1a183018 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-107_d14b811bd49192059c14ead97f13eb9e3bec7124e0326638d6d441eb24652ea1.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-107_f53898b3625b10a78faf7eadb8e0bafd0640fd76d25e82f4cfce27832ce1efe0.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-107_f53898b3625b10a78faf7eadb8e0bafd0640fd76d25e82f4cfce27832ce1efe0.jpg new file mode 100644 index 00000000..b70d1a22 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-107_f53898b3625b10a78faf7eadb8e0bafd0640fd76d25e82f4cfce27832ce1efe0.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-107_f81edf70819fc462bb08448665093ac78af9879aa750fc18284af025bb5a8885.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-107_f81edf70819fc462bb08448665093ac78af9879aa750fc18284af025bb5a8885.jpg new file mode 100644 index 00000000..c522523b Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-107_f81edf70819fc462bb08448665093ac78af9879aa750fc18284af025bb5a8885.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-108_58153e5018a76b200a341cc975b30d3d84297fabfba132182da8b24b6fd777cd.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-108_58153e5018a76b200a341cc975b30d3d84297fabfba132182da8b24b6fd777cd.jpg new file mode 100644 index 00000000..54ca635d Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-108_58153e5018a76b200a341cc975b30d3d84297fabfba132182da8b24b6fd777cd.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-108_bc8c0e239250d8e1a4fa7638f5cbf20fcfcea6921882cc7574985b22263a70d8.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-108_bc8c0e239250d8e1a4fa7638f5cbf20fcfcea6921882cc7574985b22263a70d8.jpg new file mode 100644 index 00000000..db6ee195 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-108_bc8c0e239250d8e1a4fa7638f5cbf20fcfcea6921882cc7574985b22263a70d8.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-108_c015ac0f80e97889a76d3bab7b42b9a0b44e40623c89a8a6689611f133947fc6.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-108_c015ac0f80e97889a76d3bab7b42b9a0b44e40623c89a8a6689611f133947fc6.jpg new file mode 100644 index 00000000..8063970a Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-108_c015ac0f80e97889a76d3bab7b42b9a0b44e40623c89a8a6689611f133947fc6.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-108_faa9e66f10cec4b54119d306b98c7f39ee505e37694df1d2132c1b9e8cc2d2c8.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-108_faa9e66f10cec4b54119d306b98c7f39ee505e37694df1d2132c1b9e8cc2d2c8.jpg new file mode 100644 index 00000000..55d690e8 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-108_faa9e66f10cec4b54119d306b98c7f39ee505e37694df1d2132c1b9e8cc2d2c8.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-109_253d5b707ef423498373d799f761881b8501a72ce694fa8d4f3b91c9d1d26e4e.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-109_253d5b707ef423498373d799f761881b8501a72ce694fa8d4f3b91c9d1d26e4e.jpg new file mode 100644 index 00000000..5920b938 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-109_253d5b707ef423498373d799f761881b8501a72ce694fa8d4f3b91c9d1d26e4e.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-109_8491d83c7d8a58d6282727fb5c0fd00ba5999b328e5d121ed33e23aea5c58b17.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-109_8491d83c7d8a58d6282727fb5c0fd00ba5999b328e5d121ed33e23aea5c58b17.jpg new file mode 100644 index 00000000..ac4f9389 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-109_8491d83c7d8a58d6282727fb5c0fd00ba5999b328e5d121ed33e23aea5c58b17.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-109_95022e9d2859e7b65c7732850fe05294eea78d5be2feb2c0f1be8d48847d385f.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-109_95022e9d2859e7b65c7732850fe05294eea78d5be2feb2c0f1be8d48847d385f.jpg new file mode 100644 index 00000000..11dfe47c Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-109_95022e9d2859e7b65c7732850fe05294eea78d5be2feb2c0f1be8d48847d385f.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-110_0c4a2cf5f94eb21b75a835cfc638647115f65960c1394bc39bff31b024e40a93.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-110_0c4a2cf5f94eb21b75a835cfc638647115f65960c1394bc39bff31b024e40a93.jpg new file mode 100644 index 00000000..00e0a10a Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-110_0c4a2cf5f94eb21b75a835cfc638647115f65960c1394bc39bff31b024e40a93.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-110_128e136950a0bb48cee58d2eb9b4b9489b58d19154aaafa7dc878998dcd367fb.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-110_128e136950a0bb48cee58d2eb9b4b9489b58d19154aaafa7dc878998dcd367fb.jpg new file mode 100644 index 00000000..7d8a9080 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-110_128e136950a0bb48cee58d2eb9b4b9489b58d19154aaafa7dc878998dcd367fb.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-110_5234595dec08daf77edec88e013f7afb9d71600f286cdeb09c57969a39ce4a0a.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-110_5234595dec08daf77edec88e013f7afb9d71600f286cdeb09c57969a39ce4a0a.jpg new file mode 100644 index 00000000..c97d3d06 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-110_5234595dec08daf77edec88e013f7afb9d71600f286cdeb09c57969a39ce4a0a.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-110_8f830a9bcdde2467fed6f08736e3f78027235fc6d912fab491e28feeca4091fc.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-110_8f830a9bcdde2467fed6f08736e3f78027235fc6d912fab491e28feeca4091fc.jpg new file mode 100644 index 00000000..b9e06bee Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-110_8f830a9bcdde2467fed6f08736e3f78027235fc6d912fab491e28feeca4091fc.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-110_907a5f108411c65a333d5f2ba79aa854a790c3ee2b6e400119ef0d41b19e70b4.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-110_907a5f108411c65a333d5f2ba79aa854a790c3ee2b6e400119ef0d41b19e70b4.jpg new file mode 100644 index 00000000..27f5e972 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-110_907a5f108411c65a333d5f2ba79aa854a790c3ee2b6e400119ef0d41b19e70b4.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-110_96db7ffbfe4d5a9f9f2433d08d37fc39b112173d0ae56e158c1f43435e3c2827.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-110_96db7ffbfe4d5a9f9f2433d08d37fc39b112173d0ae56e158c1f43435e3c2827.jpg new file mode 100644 index 00000000..226b0ab5 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-110_96db7ffbfe4d5a9f9f2433d08d37fc39b112173d0ae56e158c1f43435e3c2827.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-110_cb94c3498a925ad88873647dc5cdd3e59a285ab0d4ccd08782260cd0374fb02e.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-110_cb94c3498a925ad88873647dc5cdd3e59a285ab0d4ccd08782260cd0374fb02e.jpg new file mode 100644 index 00000000..52d41fb7 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-110_cb94c3498a925ad88873647dc5cdd3e59a285ab0d4ccd08782260cd0374fb02e.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-111_35d70bafd027e9adef09512dc4a0576a3a2e96f7e317e731704def5500f7eeed.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-111_35d70bafd027e9adef09512dc4a0576a3a2e96f7e317e731704def5500f7eeed.jpg new file mode 100644 index 00000000..66985df6 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-111_35d70bafd027e9adef09512dc4a0576a3a2e96f7e317e731704def5500f7eeed.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-111_4166fd51f04f76880a91a344555006dab46000b3681121db7e32850c9035967a.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-111_4166fd51f04f76880a91a344555006dab46000b3681121db7e32850c9035967a.jpg new file mode 100644 index 00000000..d4c5bd51 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-111_4166fd51f04f76880a91a344555006dab46000b3681121db7e32850c9035967a.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-111_5c20fd2099f0c3d78a86f4ae06103e8a5dbf3f25d88b6dab727ec0431ddc2e98.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-111_5c20fd2099f0c3d78a86f4ae06103e8a5dbf3f25d88b6dab727ec0431ddc2e98.jpg new file mode 100644 index 00000000..81b5f68c Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-111_5c20fd2099f0c3d78a86f4ae06103e8a5dbf3f25d88b6dab727ec0431ddc2e98.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-111_73e0d32e15c52c7a2e392c848d2c67107450892cf476b1a1f1cbeca074b0a012.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-111_73e0d32e15c52c7a2e392c848d2c67107450892cf476b1a1f1cbeca074b0a012.jpg new file mode 100644 index 00000000..680cca4c Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-111_73e0d32e15c52c7a2e392c848d2c67107450892cf476b1a1f1cbeca074b0a012.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-111_7904aeec1340c8a0a99817abbe97d116357baa9b0ef8a641c50375d7eb49343e.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-111_7904aeec1340c8a0a99817abbe97d116357baa9b0ef8a641c50375d7eb49343e.jpg new file mode 100644 index 00000000..25fd4dcb Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-111_7904aeec1340c8a0a99817abbe97d116357baa9b0ef8a641c50375d7eb49343e.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-111_a9d2507d84085439ce84d341b7342c467f335b9621fd18f0071f4daae798a33e.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-111_a9d2507d84085439ce84d341b7342c467f335b9621fd18f0071f4daae798a33e.jpg new file mode 100644 index 00000000..27e3be46 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-111_a9d2507d84085439ce84d341b7342c467f335b9621fd18f0071f4daae798a33e.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-111_ae601c1f61aedba5fb1492beafec833e14b41cd2ed9acf3b91f20c02e9a66294.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-111_ae601c1f61aedba5fb1492beafec833e14b41cd2ed9acf3b91f20c02e9a66294.jpg new file mode 100644 index 00000000..04fe0dee Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-111_ae601c1f61aedba5fb1492beafec833e14b41cd2ed9acf3b91f20c02e9a66294.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-111_d4368f91ea45daed8ebc0ee4fb7ff13d6d30deb7636641063f638788a8a4688e.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-111_d4368f91ea45daed8ebc0ee4fb7ff13d6d30deb7636641063f638788a8a4688e.jpg new file mode 100644 index 00000000..86ebdcd3 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-111_d4368f91ea45daed8ebc0ee4fb7ff13d6d30deb7636641063f638788a8a4688e.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-111_d4d6e7367b8a688584e37726bc2af47a7af71071bb65145c7986e70a6da3fe71.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-111_d4d6e7367b8a688584e37726bc2af47a7af71071bb65145c7986e70a6da3fe71.jpg new file mode 100644 index 00000000..16ba83fa Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-111_d4d6e7367b8a688584e37726bc2af47a7af71071bb65145c7986e70a6da3fe71.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-111_e8ad7e2d41aca6f2d3119d2255fc603f4cdfea8b69e6ce70e08cb472091eebe8.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-111_e8ad7e2d41aca6f2d3119d2255fc603f4cdfea8b69e6ce70e08cb472091eebe8.jpg new file mode 100644 index 00000000..f1f0990f Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-111_e8ad7e2d41aca6f2d3119d2255fc603f4cdfea8b69e6ce70e08cb472091eebe8.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-111_f663b6ca52b222fbf4cc531256cf3919c23262641ce3de082c9d6213193b4315.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-111_f663b6ca52b222fbf4cc531256cf3919c23262641ce3de082c9d6213193b4315.jpg new file mode 100644 index 00000000..27c5f784 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-111_f663b6ca52b222fbf4cc531256cf3919c23262641ce3de082c9d6213193b4315.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-112_115b3a01f9f83b9f60641265a0885558e4978f238a9a9d7b8ce7fd11b5b60deb.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-112_115b3a01f9f83b9f60641265a0885558e4978f238a9a9d7b8ce7fd11b5b60deb.jpg new file mode 100644 index 00000000..5e782a94 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-112_115b3a01f9f83b9f60641265a0885558e4978f238a9a9d7b8ce7fd11b5b60deb.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-112_14e2f99b9789b0f4ba7223e61bab244d90f63ed37755272ebd9cd7329730cd34.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-112_14e2f99b9789b0f4ba7223e61bab244d90f63ed37755272ebd9cd7329730cd34.jpg new file mode 100644 index 00000000..95b7d39b Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-112_14e2f99b9789b0f4ba7223e61bab244d90f63ed37755272ebd9cd7329730cd34.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-112_32afbe420696ebddb543c6825bf437515a343ce8423c370ac18a7b0f4dd75146.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-112_32afbe420696ebddb543c6825bf437515a343ce8423c370ac18a7b0f4dd75146.jpg new file mode 100644 index 00000000..65c45fb9 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-112_32afbe420696ebddb543c6825bf437515a343ce8423c370ac18a7b0f4dd75146.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-112_4d62c8024ef59f8ffccf8260e5afb563bcdf7b1158e2594c4e14a37205985c4e.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-112_4d62c8024ef59f8ffccf8260e5afb563bcdf7b1158e2594c4e14a37205985c4e.jpg new file mode 100644 index 00000000..0f93eb0b Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-112_4d62c8024ef59f8ffccf8260e5afb563bcdf7b1158e2594c4e14a37205985c4e.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-112_52bfa7263b0bf2886953b0f743f5fa18c535050ac71d2ccb6ba298ba55dc96c9.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-112_52bfa7263b0bf2886953b0f743f5fa18c535050ac71d2ccb6ba298ba55dc96c9.jpg new file mode 100644 index 00000000..425e8067 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-112_52bfa7263b0bf2886953b0f743f5fa18c535050ac71d2ccb6ba298ba55dc96c9.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-112_da717f93bfc54905cbc96ef82b0ca399182e5b89823d7f9da19469066dc38d53.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-112_da717f93bfc54905cbc96ef82b0ca399182e5b89823d7f9da19469066dc38d53.jpg new file mode 100644 index 00000000..848ab6df Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-112_da717f93bfc54905cbc96ef82b0ca399182e5b89823d7f9da19469066dc38d53.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-113_03b84272efc02f5a145dedb697f01d6b57d3476b4a4b67c97709c3c8ceaca3c6.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-113_03b84272efc02f5a145dedb697f01d6b57d3476b4a4b67c97709c3c8ceaca3c6.jpg new file mode 100644 index 00000000..0d323158 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-113_03b84272efc02f5a145dedb697f01d6b57d3476b4a4b67c97709c3c8ceaca3c6.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-113_219562c729ec2baa16b03ba071e05cdaee59662a5a6d9e1abae17a8f689ad96a.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-113_219562c729ec2baa16b03ba071e05cdaee59662a5a6d9e1abae17a8f689ad96a.jpg new file mode 100644 index 00000000..3d35433d Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-113_219562c729ec2baa16b03ba071e05cdaee59662a5a6d9e1abae17a8f689ad96a.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-113_2b6600dd446912c6f1afc843d555cf74faa1c07410b4d04883a768cbb2a9ac0a.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-113_2b6600dd446912c6f1afc843d555cf74faa1c07410b4d04883a768cbb2a9ac0a.jpg new file mode 100644 index 00000000..79429460 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-113_2b6600dd446912c6f1afc843d555cf74faa1c07410b4d04883a768cbb2a9ac0a.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-113_5814b3292057f7280c0bb7cb77300d690d0997a223b61bc43ebeffb44da3057d.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-113_5814b3292057f7280c0bb7cb77300d690d0997a223b61bc43ebeffb44da3057d.jpg new file mode 100644 index 00000000..fe538a0b Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-113_5814b3292057f7280c0bb7cb77300d690d0997a223b61bc43ebeffb44da3057d.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-113_6895ed82c479edb5a5828f932599eb10f8992c8b75289669e8e1e30f7f5c298f.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-113_6895ed82c479edb5a5828f932599eb10f8992c8b75289669e8e1e30f7f5c298f.jpg new file mode 100644 index 00000000..a9dd51bf Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-113_6895ed82c479edb5a5828f932599eb10f8992c8b75289669e8e1e30f7f5c298f.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-113_6ca3455f74de85158952fbb3321ba0c417cfff890f689c6e15d26f5ce379ecde.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-113_6ca3455f74de85158952fbb3321ba0c417cfff890f689c6e15d26f5ce379ecde.jpg new file mode 100644 index 00000000..6271b8d9 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-113_6ca3455f74de85158952fbb3321ba0c417cfff890f689c6e15d26f5ce379ecde.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-113_799cfaeaad1c1b8928a09e005468a379bbf74a57d37768f9a6649df4e10ac076.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-113_799cfaeaad1c1b8928a09e005468a379bbf74a57d37768f9a6649df4e10ac076.jpg new file mode 100644 index 00000000..cd2b57fc Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-113_799cfaeaad1c1b8928a09e005468a379bbf74a57d37768f9a6649df4e10ac076.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-113_8770441ed0906733e62fc8828067c05608be9271a3e1d3b5573837b48dc3a80a.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-113_8770441ed0906733e62fc8828067c05608be9271a3e1d3b5573837b48dc3a80a.jpg new file mode 100644 index 00000000..607a5332 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-113_8770441ed0906733e62fc8828067c05608be9271a3e1d3b5573837b48dc3a80a.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-113_9b4c537e1010c6a7bc8aba58bb8db6a945d2451e8cedb72ab2e6d70e07114aba.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-113_9b4c537e1010c6a7bc8aba58bb8db6a945d2451e8cedb72ab2e6d70e07114aba.jpg new file mode 100644 index 00000000..adbaf20e Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-113_9b4c537e1010c6a7bc8aba58bb8db6a945d2451e8cedb72ab2e6d70e07114aba.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-113_c1f23d884a0e1ecb565384cc3079ed4f85b9bf614f70f67001044367ee90b862.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-113_c1f23d884a0e1ecb565384cc3079ed4f85b9bf614f70f67001044367ee90b862.jpg new file mode 100644 index 00000000..01cd4b3d Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-113_c1f23d884a0e1ecb565384cc3079ed4f85b9bf614f70f67001044367ee90b862.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-114_477a7591fe91ba82232c4403165654410fc0922357e9ba4ada8d64a3fbc17888.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-114_477a7591fe91ba82232c4403165654410fc0922357e9ba4ada8d64a3fbc17888.jpg new file mode 100644 index 00000000..c221e33e Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-114_477a7591fe91ba82232c4403165654410fc0922357e9ba4ada8d64a3fbc17888.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-114_58f6eb5f6f6f252543eb39302e16210dc7a4db750bf6faa9f822cee01a2474d7.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-114_58f6eb5f6f6f252543eb39302e16210dc7a4db750bf6faa9f822cee01a2474d7.jpg new file mode 100644 index 00000000..692a697d Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-114_58f6eb5f6f6f252543eb39302e16210dc7a4db750bf6faa9f822cee01a2474d7.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-115_f42e61708dc6ef92588fcf0e5382f1942a78c61947d1e760ed792a6bb99722ed.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-115_f42e61708dc6ef92588fcf0e5382f1942a78c61947d1e760ed792a6bb99722ed.jpg new file mode 100644 index 00000000..22c6a896 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-115_f42e61708dc6ef92588fcf0e5382f1942a78c61947d1e760ed792a6bb99722ed.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-118_91a406fff8da98ee8d59d149854a9d638221e13e7ce7da2a7a45039e10e942ec.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-118_91a406fff8da98ee8d59d149854a9d638221e13e7ce7da2a7a45039e10e942ec.jpg new file mode 100644 index 00000000..70dd92dc Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-118_91a406fff8da98ee8d59d149854a9d638221e13e7ce7da2a7a45039e10e942ec.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-119_634d416c49be0d128fbeec99c4f3d75bcb4c6d932a611fa345533feae2275083.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-119_634d416c49be0d128fbeec99c4f3d75bcb4c6d932a611fa345533feae2275083.jpg new file mode 100644 index 00000000..f3227876 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-119_634d416c49be0d128fbeec99c4f3d75bcb4c6d932a611fa345533feae2275083.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-120_40753f5253c157aa004645246e2e06cdf4c2895e3c47eeb581b37185165964ad.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-120_40753f5253c157aa004645246e2e06cdf4c2895e3c47eeb581b37185165964ad.jpg new file mode 100644 index 00000000..7176bf07 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-120_40753f5253c157aa004645246e2e06cdf4c2895e3c47eeb581b37185165964ad.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-121_6b8bf98330b39e1ab11dbbd1a51e4b2373d8ab1e0201c7240a4be1fa50517efd.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-121_6b8bf98330b39e1ab11dbbd1a51e4b2373d8ab1e0201c7240a4be1fa50517efd.jpg new file mode 100644 index 00000000..9c7ca9d7 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-121_6b8bf98330b39e1ab11dbbd1a51e4b2373d8ab1e0201c7240a4be1fa50517efd.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-121_bb70052015d37e74bdc1d0df7cbd264b4ae7b0eeea868ef07d7d9fda99d63c38.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-121_bb70052015d37e74bdc1d0df7cbd264b4ae7b0eeea868ef07d7d9fda99d63c38.jpg new file mode 100644 index 00000000..3e487a85 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-121_bb70052015d37e74bdc1d0df7cbd264b4ae7b0eeea868ef07d7d9fda99d63c38.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-122_88a938f25903e14ad5d667614a5d6bf38b87e93274e90242d4a24e2473ee3e3e.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-122_88a938f25903e14ad5d667614a5d6bf38b87e93274e90242d4a24e2473ee3e3e.jpg new file mode 100644 index 00000000..3b2890fe Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-122_88a938f25903e14ad5d667614a5d6bf38b87e93274e90242d4a24e2473ee3e3e.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-124_d6bc359a7e3be14348a8d72b414d761503f050dcba1e21b236bc06fdf4802a27.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-124_d6bc359a7e3be14348a8d72b414d761503f050dcba1e21b236bc06fdf4802a27.jpg new file mode 100644 index 00000000..ffdb99f6 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-124_d6bc359a7e3be14348a8d72b414d761503f050dcba1e21b236bc06fdf4802a27.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-125_0e467fe8e943511b3d97600db3c0ed69cd530fa800c840989c462dce1abf2a16.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-125_0e467fe8e943511b3d97600db3c0ed69cd530fa800c840989c462dce1abf2a16.jpg new file mode 100644 index 00000000..f80790bb Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-125_0e467fe8e943511b3d97600db3c0ed69cd530fa800c840989c462dce1abf2a16.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-126_f9dd395efc3ebc04a9504aa429fc51d65121f845b083c5192461b7ab548c4e7e.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-126_f9dd395efc3ebc04a9504aa429fc51d65121f845b083c5192461b7ab548c4e7e.jpg new file mode 100644 index 00000000..d9e9f9a4 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-126_f9dd395efc3ebc04a9504aa429fc51d65121f845b083c5192461b7ab548c4e7e.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-127_b655f4d2763b6d1ade5d91d5be38cfc240ee5f286f33fd615bae7b4f1a4c4b5b.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-127_b655f4d2763b6d1ade5d91d5be38cfc240ee5f286f33fd615bae7b4f1a4c4b5b.jpg new file mode 100644 index 00000000..a46d3322 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-127_b655f4d2763b6d1ade5d91d5be38cfc240ee5f286f33fd615bae7b4f1a4c4b5b.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-127_cd1e38ef3c6c0dde715e75814e6d54f6dc0aef446d5340753e27f5bf5572447b.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-127_cd1e38ef3c6c0dde715e75814e6d54f6dc0aef446d5340753e27f5bf5572447b.jpg new file mode 100644 index 00000000..443d0de1 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-127_cd1e38ef3c6c0dde715e75814e6d54f6dc0aef446d5340753e27f5bf5572447b.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-128_4a52a1e6bcef40c98310b1b9347146781893e0c0759e7d1bb0df0c16908b1ef9.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-128_4a52a1e6bcef40c98310b1b9347146781893e0c0759e7d1bb0df0c16908b1ef9.jpg new file mode 100644 index 00000000..35428357 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-128_4a52a1e6bcef40c98310b1b9347146781893e0c0759e7d1bb0df0c16908b1ef9.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-128_78404c3b1a2a51165251b6982cdb07f24f36f20174969f9ad01abcfe2b860571.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-128_78404c3b1a2a51165251b6982cdb07f24f36f20174969f9ad01abcfe2b860571.jpg new file mode 100644 index 00000000..1e1d631c Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-128_78404c3b1a2a51165251b6982cdb07f24f36f20174969f9ad01abcfe2b860571.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-128_784d9dfeb63e232fd43f2b3a0b3fd1d75493089745d46c277106b7647a18cf26.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-128_784d9dfeb63e232fd43f2b3a0b3fd1d75493089745d46c277106b7647a18cf26.jpg new file mode 100644 index 00000000..5fe17495 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-128_784d9dfeb63e232fd43f2b3a0b3fd1d75493089745d46c277106b7647a18cf26.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-128_a484d99b549cffa740f4623be1e3b557670257a20af9f94cf0afef46ed9f2cc7.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-128_a484d99b549cffa740f4623be1e3b557670257a20af9f94cf0afef46ed9f2cc7.jpg new file mode 100644 index 00000000..2b4e0379 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-128_a484d99b549cffa740f4623be1e3b557670257a20af9f94cf0afef46ed9f2cc7.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-132_6425d8c62315f6c3eca3305db0ae1c932d9bc392ce8026282ea1bacb4ae3ea1b.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-132_6425d8c62315f6c3eca3305db0ae1c932d9bc392ce8026282ea1bacb4ae3ea1b.jpg new file mode 100644 index 00000000..59037544 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-132_6425d8c62315f6c3eca3305db0ae1c932d9bc392ce8026282ea1bacb4ae3ea1b.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-132_75e89e0aadf652eb61cfa3b1e1ad86eb4940ededb01677de06e0fb86b299fe2f.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-132_75e89e0aadf652eb61cfa3b1e1ad86eb4940ededb01677de06e0fb86b299fe2f.jpg new file mode 100644 index 00000000..3e8a5932 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-132_75e89e0aadf652eb61cfa3b1e1ad86eb4940ededb01677de06e0fb86b299fe2f.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-133_021faa65e7f05a2356c1fc2bf9cfec06df1f4db3259e10efd7abc0c448954f27.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-133_021faa65e7f05a2356c1fc2bf9cfec06df1f4db3259e10efd7abc0c448954f27.jpg new file mode 100644 index 00000000..686beec7 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-133_021faa65e7f05a2356c1fc2bf9cfec06df1f4db3259e10efd7abc0c448954f27.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-133_3798e6af00a4613b6dd8a7a0b5f6863c69913e4679d80c7aeb8c336c5a2f4964.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-133_3798e6af00a4613b6dd8a7a0b5f6863c69913e4679d80c7aeb8c336c5a2f4964.jpg new file mode 100644 index 00000000..ad400672 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-133_3798e6af00a4613b6dd8a7a0b5f6863c69913e4679d80c7aeb8c336c5a2f4964.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-133_38b4ca1f72ac65b760ea5aeadc4abbb1ac1d5939a3bb7e5a29d06e6edf7c470f.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-133_38b4ca1f72ac65b760ea5aeadc4abbb1ac1d5939a3bb7e5a29d06e6edf7c470f.jpg new file mode 100644 index 00000000..a363938e Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-133_38b4ca1f72ac65b760ea5aeadc4abbb1ac1d5939a3bb7e5a29d06e6edf7c470f.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-133_3e66c1ae2711cbf98b9c2d3a63d89b1ec88e2e4685c28c928edb960d7b571b5f.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-133_3e66c1ae2711cbf98b9c2d3a63d89b1ec88e2e4685c28c928edb960d7b571b5f.jpg new file mode 100644 index 00000000..61404ddc Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-133_3e66c1ae2711cbf98b9c2d3a63d89b1ec88e2e4685c28c928edb960d7b571b5f.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-133_5a081cd211e9d40196ff879a376c7e2b116cca78ccfa36af04120973c47eb316.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-133_5a081cd211e9d40196ff879a376c7e2b116cca78ccfa36af04120973c47eb316.jpg new file mode 100644 index 00000000..d7a4b122 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-133_5a081cd211e9d40196ff879a376c7e2b116cca78ccfa36af04120973c47eb316.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-133_bed565f35b1418b62cc4d8d05b9a8db37d1e3c978baba65a4f91157d0ec136c3.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-133_bed565f35b1418b62cc4d8d05b9a8db37d1e3c978baba65a4f91157d0ec136c3.jpg new file mode 100644 index 00000000..d6984c20 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-133_bed565f35b1418b62cc4d8d05b9a8db37d1e3c978baba65a4f91157d0ec136c3.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-133_c8bda65719e58221f9cbcb26797e401140149ac571e3ba2e9a511ea291fc309b.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-133_c8bda65719e58221f9cbcb26797e401140149ac571e3ba2e9a511ea291fc309b.jpg new file mode 100644 index 00000000..d7f163c0 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-133_c8bda65719e58221f9cbcb26797e401140149ac571e3ba2e9a511ea291fc309b.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-133_d38900424c1e063825924572bc9b91aacdf5c016563e7403a03c58cb433460a4.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-133_d38900424c1e063825924572bc9b91aacdf5c016563e7403a03c58cb433460a4.jpg new file mode 100644 index 00000000..0f860364 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-133_d38900424c1e063825924572bc9b91aacdf5c016563e7403a03c58cb433460a4.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-134_5b8eb3216b84e74ce0755e5cee83d39149f9bbb0c6435e1cb3617228b594e312.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-134_5b8eb3216b84e74ce0755e5cee83d39149f9bbb0c6435e1cb3617228b594e312.jpg new file mode 100644 index 00000000..d298e6dc Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-134_5b8eb3216b84e74ce0755e5cee83d39149f9bbb0c6435e1cb3617228b594e312.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-134_7ed2d2393f5c686c58ee2067c04fd9d717cc2405d4113a247a48d1aabcfe431b.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-134_7ed2d2393f5c686c58ee2067c04fd9d717cc2405d4113a247a48d1aabcfe431b.jpg new file mode 100644 index 00000000..ef41fd37 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-134_7ed2d2393f5c686c58ee2067c04fd9d717cc2405d4113a247a48d1aabcfe431b.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-134_8c24f88718b2c44861544cec21c358435eb5b1c3b77591d2d96e2cef11e92304.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-134_8c24f88718b2c44861544cec21c358435eb5b1c3b77591d2d96e2cef11e92304.jpg new file mode 100644 index 00000000..e4082cf3 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-134_8c24f88718b2c44861544cec21c358435eb5b1c3b77591d2d96e2cef11e92304.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-134_d20fc9c084b5d8bf374aee5530d9ed65ba01e293a75192b9ce1adcb2371bd331.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-134_d20fc9c084b5d8bf374aee5530d9ed65ba01e293a75192b9ce1adcb2371bd331.jpg new file mode 100644 index 00000000..7acd22ea Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-134_d20fc9c084b5d8bf374aee5530d9ed65ba01e293a75192b9ce1adcb2371bd331.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-134_d7c4f3da6437ab79f3b96dbad0b11e749395cff41074b8b1022aad49b9396030.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-134_d7c4f3da6437ab79f3b96dbad0b11e749395cff41074b8b1022aad49b9396030.jpg new file mode 100644 index 00000000..16655248 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-134_d7c4f3da6437ab79f3b96dbad0b11e749395cff41074b8b1022aad49b9396030.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-134_f87a34680e1e985107c51dcc6d5703102358b6af093ca6ea13cc4a4a303959e7.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-134_f87a34680e1e985107c51dcc6d5703102358b6af093ca6ea13cc4a4a303959e7.jpg new file mode 100644 index 00000000..fc422286 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-134_f87a34680e1e985107c51dcc6d5703102358b6af093ca6ea13cc4a4a303959e7.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-134_fb043904e13ce561cb54f0fae7493823e80f2e123612f8797fd1b2671b217ae7.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-134_fb043904e13ce561cb54f0fae7493823e80f2e123612f8797fd1b2671b217ae7.jpg new file mode 100644 index 00000000..e4d0a764 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-134_fb043904e13ce561cb54f0fae7493823e80f2e123612f8797fd1b2671b217ae7.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-135_017b04b8ae30533b103f02bed9f5006b17ceb429aa3c730a66202861a85688e7.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-135_017b04b8ae30533b103f02bed9f5006b17ceb429aa3c730a66202861a85688e7.jpg new file mode 100644 index 00000000..39572dc4 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-135_017b04b8ae30533b103f02bed9f5006b17ceb429aa3c730a66202861a85688e7.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-135_acaa19e50437d40c1f69ea2eb449267224bc6015d25b5c9b7af2d9c3386d5ace.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-135_acaa19e50437d40c1f69ea2eb449267224bc6015d25b5c9b7af2d9c3386d5ace.jpg new file mode 100644 index 00000000..0fc1bc0d Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-135_acaa19e50437d40c1f69ea2eb449267224bc6015d25b5c9b7af2d9c3386d5ace.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-135_e957350918f168b4b1db60d5b090115f0013b06e10ae27cebe4075b9afd487bd.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-135_e957350918f168b4b1db60d5b090115f0013b06e10ae27cebe4075b9afd487bd.jpg new file mode 100644 index 00000000..c62c01e9 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-135_e957350918f168b4b1db60d5b090115f0013b06e10ae27cebe4075b9afd487bd.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-135_f7a531df2d3e6db50252d47c314dfcfe3335a476ddcca65c4a99cafad2a39110.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-135_f7a531df2d3e6db50252d47c314dfcfe3335a476ddcca65c4a99cafad2a39110.jpg new file mode 100644 index 00000000..37e94621 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-135_f7a531df2d3e6db50252d47c314dfcfe3335a476ddcca65c4a99cafad2a39110.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-136_23258ee259aaf4c1f489cb1c9e0d0fd4013448ea99118ecf120a7b747b95d4fe.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-136_23258ee259aaf4c1f489cb1c9e0d0fd4013448ea99118ecf120a7b747b95d4fe.jpg new file mode 100644 index 00000000..15743b30 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-136_23258ee259aaf4c1f489cb1c9e0d0fd4013448ea99118ecf120a7b747b95d4fe.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-136_2ba6801f22a661582a595a8ed698fa96fb6b2ced8c33e175e22055945c3efb25.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-136_2ba6801f22a661582a595a8ed698fa96fb6b2ced8c33e175e22055945c3efb25.jpg new file mode 100644 index 00000000..09981a84 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-136_2ba6801f22a661582a595a8ed698fa96fb6b2ced8c33e175e22055945c3efb25.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-136_3c5ec44fe779d358484598284e777f25a250adc80154eb7270289f9b44067a6f.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-136_3c5ec44fe779d358484598284e777f25a250adc80154eb7270289f9b44067a6f.jpg new file mode 100644 index 00000000..c8d31df0 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-136_3c5ec44fe779d358484598284e777f25a250adc80154eb7270289f9b44067a6f.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-136_55fd256a5513861bbd19e660637e0663fcfc456e614279bb2c60371c897a3e1a.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-136_55fd256a5513861bbd19e660637e0663fcfc456e614279bb2c60371c897a3e1a.jpg new file mode 100644 index 00000000..6df799af Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-136_55fd256a5513861bbd19e660637e0663fcfc456e614279bb2c60371c897a3e1a.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-136_5ac0762da54e7e9465666a0864c2309dfa87b2a23fc4622c465594cd561bd864.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-136_5ac0762da54e7e9465666a0864c2309dfa87b2a23fc4622c465594cd561bd864.jpg new file mode 100644 index 00000000..1ce8ad13 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-136_5ac0762da54e7e9465666a0864c2309dfa87b2a23fc4622c465594cd561bd864.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-136_7dfb83e860fbd64275f0ecd59652ed4330b66d40aeffa0d6f8527ec453297357.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-136_7dfb83e860fbd64275f0ecd59652ed4330b66d40aeffa0d6f8527ec453297357.jpg new file mode 100644 index 00000000..5124ff88 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-136_7dfb83e860fbd64275f0ecd59652ed4330b66d40aeffa0d6f8527ec453297357.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-136_9bd4c77933da74751870864a02effc4ca091b7cceddc56866e397218c7825a02.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-136_9bd4c77933da74751870864a02effc4ca091b7cceddc56866e397218c7825a02.jpg new file mode 100644 index 00000000..eb966f00 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-136_9bd4c77933da74751870864a02effc4ca091b7cceddc56866e397218c7825a02.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-137_14c486b5563511d722b5fe6db8d13baf0d76da16ef7fdf528aac92034e388361.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-137_14c486b5563511d722b5fe6db8d13baf0d76da16ef7fdf528aac92034e388361.jpg new file mode 100644 index 00000000..e2277817 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-137_14c486b5563511d722b5fe6db8d13baf0d76da16ef7fdf528aac92034e388361.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-137_330effd4555517a3fe13d796e0915362669273e856c2742c32790b9d03004e96.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-137_330effd4555517a3fe13d796e0915362669273e856c2742c32790b9d03004e96.jpg new file mode 100644 index 00000000..b91b880d Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-137_330effd4555517a3fe13d796e0915362669273e856c2742c32790b9d03004e96.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-137_768ea0433796c1d041443f468d1006d71204b106e685e4bae99f3bc084e29067.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-137_768ea0433796c1d041443f468d1006d71204b106e685e4bae99f3bc084e29067.jpg new file mode 100644 index 00000000..1dd8f831 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-137_768ea0433796c1d041443f468d1006d71204b106e685e4bae99f3bc084e29067.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-137_9a065ac93e8657bdbbdf646d016cd63ce42d3b62e797d9c107119894b2b031c9.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-137_9a065ac93e8657bdbbdf646d016cd63ce42d3b62e797d9c107119894b2b031c9.jpg new file mode 100644 index 00000000..4b1c1cb7 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-137_9a065ac93e8657bdbbdf646d016cd63ce42d3b62e797d9c107119894b2b031c9.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-137_9d93d9bd4a6ed0e0dd02469e299454e48292f00e09b43ea52cfb1285ad84db18.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-137_9d93d9bd4a6ed0e0dd02469e299454e48292f00e09b43ea52cfb1285ad84db18.jpg new file mode 100644 index 00000000..4a18247a Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-137_9d93d9bd4a6ed0e0dd02469e299454e48292f00e09b43ea52cfb1285ad84db18.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-137_b5a19a5b6dee40490ffea19f0986420d243592e5961faebc01b2c06c229257b1.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-137_b5a19a5b6dee40490ffea19f0986420d243592e5961faebc01b2c06c229257b1.jpg new file mode 100644 index 00000000..ae3f81ba Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-137_b5a19a5b6dee40490ffea19f0986420d243592e5961faebc01b2c06c229257b1.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-137_c2bfe27cdcf76751a3e8fdc8952114df841c9965432632cd6aa1dea0568bea39.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-137_c2bfe27cdcf76751a3e8fdc8952114df841c9965432632cd6aa1dea0568bea39.jpg new file mode 100644 index 00000000..b75fb71d Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-137_c2bfe27cdcf76751a3e8fdc8952114df841c9965432632cd6aa1dea0568bea39.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-138_03ae01e0f6d24efc17198519e8bf0ae0c4e6400eaf4aa534a857ec26c163ec84.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-138_03ae01e0f6d24efc17198519e8bf0ae0c4e6400eaf4aa534a857ec26c163ec84.jpg new file mode 100644 index 00000000..4f695337 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-138_03ae01e0f6d24efc17198519e8bf0ae0c4e6400eaf4aa534a857ec26c163ec84.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-138_3407508b7040a4b99228f22961ae773dcad51051d888ec86a9d2a7457a683181.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-138_3407508b7040a4b99228f22961ae773dcad51051d888ec86a9d2a7457a683181.jpg new file mode 100644 index 00000000..808325a7 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-138_3407508b7040a4b99228f22961ae773dcad51051d888ec86a9d2a7457a683181.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-138_ca12c954c11802ec2c3fea0d4a73f6361ce7df34d5a79c81a17fdebd911191d3.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-138_ca12c954c11802ec2c3fea0d4a73f6361ce7df34d5a79c81a17fdebd911191d3.jpg new file mode 100644 index 00000000..7012d6ee Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-138_ca12c954c11802ec2c3fea0d4a73f6361ce7df34d5a79c81a17fdebd911191d3.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-139_17cfc7d5ae91f6aaf0e0c08117bc69c871727b4004fed53c699db9d47c4a85fa.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-139_17cfc7d5ae91f6aaf0e0c08117bc69c871727b4004fed53c699db9d47c4a85fa.jpg new file mode 100644 index 00000000..5092790c Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-139_17cfc7d5ae91f6aaf0e0c08117bc69c871727b4004fed53c699db9d47c4a85fa.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-139_44fb79837141b808595d2cf760d2fd5a3942b3e1aecbd4c6ea67f3cf27816715.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-139_44fb79837141b808595d2cf760d2fd5a3942b3e1aecbd4c6ea67f3cf27816715.jpg new file mode 100644 index 00000000..dae0af45 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-139_44fb79837141b808595d2cf760d2fd5a3942b3e1aecbd4c6ea67f3cf27816715.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-139_d8cd9130429c35a9f0735958071215476df0ac430afe6fe8a634144170e7cea5.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-139_d8cd9130429c35a9f0735958071215476df0ac430afe6fe8a634144170e7cea5.jpg new file mode 100644 index 00000000..8d453e86 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-139_d8cd9130429c35a9f0735958071215476df0ac430afe6fe8a634144170e7cea5.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-139_f3da674af6583450293593c9722de06e9406a4f160cb50d396a129f11433520a.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-139_f3da674af6583450293593c9722de06e9406a4f160cb50d396a129f11433520a.jpg new file mode 100644 index 00000000..de27901a Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-139_f3da674af6583450293593c9722de06e9406a4f160cb50d396a129f11433520a.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-140_176321f737bdc80233fd107f8c0745cc1aaa029fdb693665fb759d9c84bda233.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-140_176321f737bdc80233fd107f8c0745cc1aaa029fdb693665fb759d9c84bda233.jpg new file mode 100644 index 00000000..6c7a335b Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-140_176321f737bdc80233fd107f8c0745cc1aaa029fdb693665fb759d9c84bda233.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-140_2abb303e1f28d031afca3678129cc85e374f2565922e1dcf5aacc1e53a9530f0.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-140_2abb303e1f28d031afca3678129cc85e374f2565922e1dcf5aacc1e53a9530f0.jpg new file mode 100644 index 00000000..5e230a94 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-140_2abb303e1f28d031afca3678129cc85e374f2565922e1dcf5aacc1e53a9530f0.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-140_442312107d9fd06572bbf835cca0bc89edb3eca8766e42a991d284481e6227be.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-140_442312107d9fd06572bbf835cca0bc89edb3eca8766e42a991d284481e6227be.jpg new file mode 100644 index 00000000..d5ba20cc Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-140_442312107d9fd06572bbf835cca0bc89edb3eca8766e42a991d284481e6227be.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-140_b5d42403a2df02c3f3ce8c4f6489b9ee78c11c42227d5fad18f035b81b434f5d.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-140_b5d42403a2df02c3f3ce8c4f6489b9ee78c11c42227d5fad18f035b81b434f5d.jpg new file mode 100644 index 00000000..23a0287c Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-140_b5d42403a2df02c3f3ce8c4f6489b9ee78c11c42227d5fad18f035b81b434f5d.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-140_b66357afddd9cf6ac65c0a1684c33069871f22d811be8a6e098132037fd571b5.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-140_b66357afddd9cf6ac65c0a1684c33069871f22d811be8a6e098132037fd571b5.jpg new file mode 100644 index 00000000..3ff6218a Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-140_b66357afddd9cf6ac65c0a1684c33069871f22d811be8a6e098132037fd571b5.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-140_f81a232f80ccb7c927ebf19935c0f3e9ba916822d4442d91b286f4c6dbb79de0.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-140_f81a232f80ccb7c927ebf19935c0f3e9ba916822d4442d91b286f4c6dbb79de0.jpg new file mode 100644 index 00000000..e20dfb38 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-140_f81a232f80ccb7c927ebf19935c0f3e9ba916822d4442d91b286f4c6dbb79de0.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-141_6cc4854876db4334d475e500b9e682ac2fca3b78b085cf8ed2b276fab7ee7cce.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-141_6cc4854876db4334d475e500b9e682ac2fca3b78b085cf8ed2b276fab7ee7cce.jpg new file mode 100644 index 00000000..0a8f5bef Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-141_6cc4854876db4334d475e500b9e682ac2fca3b78b085cf8ed2b276fab7ee7cce.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-141_7da1474f654b953f94a1e390af93627768e3f0d270646cf5b1f5f4ef4d21dd5b.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-141_7da1474f654b953f94a1e390af93627768e3f0d270646cf5b1f5f4ef4d21dd5b.jpg new file mode 100644 index 00000000..7be3a1b2 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-141_7da1474f654b953f94a1e390af93627768e3f0d270646cf5b1f5f4ef4d21dd5b.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-141_90805ef571b969f4d9a426c9384bf391db2fcd2c88091977f5074399e10b88d9.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-141_90805ef571b969f4d9a426c9384bf391db2fcd2c88091977f5074399e10b88d9.jpg new file mode 100644 index 00000000..729789e8 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-141_90805ef571b969f4d9a426c9384bf391db2fcd2c88091977f5074399e10b88d9.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-141_a4611aae36d2541739ef8da9c65709bb8b89850ae1d9eb4877a24828b02aa4de.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-141_a4611aae36d2541739ef8da9c65709bb8b89850ae1d9eb4877a24828b02aa4de.jpg new file mode 100644 index 00000000..c922b6b9 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-141_a4611aae36d2541739ef8da9c65709bb8b89850ae1d9eb4877a24828b02aa4de.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-141_a90bab0b4aeb71fd88573da69bebc1379dfd460ec5cfa86db398358a8fc0d25e.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-141_a90bab0b4aeb71fd88573da69bebc1379dfd460ec5cfa86db398358a8fc0d25e.jpg new file mode 100644 index 00000000..1cdaa1f0 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-141_a90bab0b4aeb71fd88573da69bebc1379dfd460ec5cfa86db398358a8fc0d25e.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-141_b50c92359e315b0303a62d7007527069c6d171612d09f0ce4d95cb4fb520e950.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-141_b50c92359e315b0303a62d7007527069c6d171612d09f0ce4d95cb4fb520e950.jpg new file mode 100644 index 00000000..4640171e Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-141_b50c92359e315b0303a62d7007527069c6d171612d09f0ce4d95cb4fb520e950.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-141_c042553ebb990b23668258faec308a872c2aa80b6406581530e4a7a60cf59d2e.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-141_c042553ebb990b23668258faec308a872c2aa80b6406581530e4a7a60cf59d2e.jpg new file mode 100644 index 00000000..e85cd382 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-141_c042553ebb990b23668258faec308a872c2aa80b6406581530e4a7a60cf59d2e.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-142_0597803b3d8703425978d28bb6a3dbd02afc6f48f0da8c658fd69a2d4f99e811.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-142_0597803b3d8703425978d28bb6a3dbd02afc6f48f0da8c658fd69a2d4f99e811.jpg new file mode 100644 index 00000000..371aa30a Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-142_0597803b3d8703425978d28bb6a3dbd02afc6f48f0da8c658fd69a2d4f99e811.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-142_20101d170161da02d04ae06a0a85602058178c9da08ac1d8e85608a248a2f653.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-142_20101d170161da02d04ae06a0a85602058178c9da08ac1d8e85608a248a2f653.jpg new file mode 100644 index 00000000..1189ae18 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-142_20101d170161da02d04ae06a0a85602058178c9da08ac1d8e85608a248a2f653.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-142_61fb5aed38b2f56cf895b6f8136173e4de6ed82311d057209301db798bd1bef8.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-142_61fb5aed38b2f56cf895b6f8136173e4de6ed82311d057209301db798bd1bef8.jpg new file mode 100644 index 00000000..596e05f3 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-142_61fb5aed38b2f56cf895b6f8136173e4de6ed82311d057209301db798bd1bef8.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-142_b2287ed2f01567b9054dfe0df775e6dfae0f25714a192170f53fa39dfff51182.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-142_b2287ed2f01567b9054dfe0df775e6dfae0f25714a192170f53fa39dfff51182.jpg new file mode 100644 index 00000000..b05c514e Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-142_b2287ed2f01567b9054dfe0df775e6dfae0f25714a192170f53fa39dfff51182.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-142_d28f7930d7c76ef67c5c93af02170c797e8867f8f98b138a172f39ceac286489.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-142_d28f7930d7c76ef67c5c93af02170c797e8867f8f98b138a172f39ceac286489.jpg new file mode 100644 index 00000000..78bfd3d5 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-142_d28f7930d7c76ef67c5c93af02170c797e8867f8f98b138a172f39ceac286489.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-142_dc5760f44ccd35de1c6b61e3062dd71114d07ee5e4f918bef9a0be2cc3937df6.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-142_dc5760f44ccd35de1c6b61e3062dd71114d07ee5e4f918bef9a0be2cc3937df6.jpg new file mode 100644 index 00000000..e5309c79 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-142_dc5760f44ccd35de1c6b61e3062dd71114d07ee5e4f918bef9a0be2cc3937df6.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-142_ff8ccc35de23de21cefde0d0756b6220afa38f0538393ccea79b07517105e498.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-142_ff8ccc35de23de21cefde0d0756b6220afa38f0538393ccea79b07517105e498.jpg new file mode 100644 index 00000000..21d82268 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-142_ff8ccc35de23de21cefde0d0756b6220afa38f0538393ccea79b07517105e498.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-143_066b5d37a0de44c7922bab8dddd82b5aee14b05a7f4a910c9fa8c192107197ad.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-143_066b5d37a0de44c7922bab8dddd82b5aee14b05a7f4a910c9fa8c192107197ad.jpg new file mode 100644 index 00000000..3f2cb082 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-143_066b5d37a0de44c7922bab8dddd82b5aee14b05a7f4a910c9fa8c192107197ad.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-143_4a22f7380c3685b6397d15f203fc964bc4c9a770450489c3baaa65d3078ad6cd.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-143_4a22f7380c3685b6397d15f203fc964bc4c9a770450489c3baaa65d3078ad6cd.jpg new file mode 100644 index 00000000..fb74cb0c Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-143_4a22f7380c3685b6397d15f203fc964bc4c9a770450489c3baaa65d3078ad6cd.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-143_662a46f364c3a02e6827093bce2105c9211d353d375796964cfce7bdb6e65397.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-143_662a46f364c3a02e6827093bce2105c9211d353d375796964cfce7bdb6e65397.jpg new file mode 100644 index 00000000..a55d927e Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-143_662a46f364c3a02e6827093bce2105c9211d353d375796964cfce7bdb6e65397.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-143_68b4792798f47f0f7dbf41becc16f246f8997d67c647cf46f5387b6bcfe38a46.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-143_68b4792798f47f0f7dbf41becc16f246f8997d67c647cf46f5387b6bcfe38a46.jpg new file mode 100644 index 00000000..84d75b3d Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-143_68b4792798f47f0f7dbf41becc16f246f8997d67c647cf46f5387b6bcfe38a46.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-143_7446e9da0d7e56ac6cfba16c9a67655b358f6308c71a0754958a0fe1200c8011.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-143_7446e9da0d7e56ac6cfba16c9a67655b358f6308c71a0754958a0fe1200c8011.jpg new file mode 100644 index 00000000..b4b174ca Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-143_7446e9da0d7e56ac6cfba16c9a67655b358f6308c71a0754958a0fe1200c8011.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-143_fa414aaa4abe7fcc6e8261d52dae1ad2b5ea512ba55642af7b2223e3f2c6e01e.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-143_fa414aaa4abe7fcc6e8261d52dae1ad2b5ea512ba55642af7b2223e3f2c6e01e.jpg new file mode 100644 index 00000000..7b200835 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-143_fa414aaa4abe7fcc6e8261d52dae1ad2b5ea512ba55642af7b2223e3f2c6e01e.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-144_286badf9fdf174f5c6d48de7f661af3aedd8e1501ccaa009f381fe58e33ee0b8.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-144_286badf9fdf174f5c6d48de7f661af3aedd8e1501ccaa009f381fe58e33ee0b8.jpg new file mode 100644 index 00000000..b31e5b82 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-144_286badf9fdf174f5c6d48de7f661af3aedd8e1501ccaa009f381fe58e33ee0b8.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-145_9ea8079c9d07a8043be3c3ace95413d8989a97c256da58d83f579cded8b343ec.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-145_9ea8079c9d07a8043be3c3ace95413d8989a97c256da58d83f579cded8b343ec.jpg new file mode 100644 index 00000000..37f001fc Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-145_9ea8079c9d07a8043be3c3ace95413d8989a97c256da58d83f579cded8b343ec.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-147_bacbbcc5384326abb4176b9ae94b61cefaebebaf8ad5921dc45bbc344d9c68c4.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-147_bacbbcc5384326abb4176b9ae94b61cefaebebaf8ad5921dc45bbc344d9c68c4.jpg new file mode 100644 index 00000000..09ed0038 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-147_bacbbcc5384326abb4176b9ae94b61cefaebebaf8ad5921dc45bbc344d9c68c4.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-147_ea300779a7d139556fa678f78225cb1f47f3604502bd3af674e8f091092e2375.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-147_ea300779a7d139556fa678f78225cb1f47f3604502bd3af674e8f091092e2375.jpg new file mode 100644 index 00000000..48804036 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-147_ea300779a7d139556fa678f78225cb1f47f3604502bd3af674e8f091092e2375.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-148_065c33c36461b2f5b1219334a6c5baa0d45827a8cf1df328f326826d3d7c2b1f.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-148_065c33c36461b2f5b1219334a6c5baa0d45827a8cf1df328f326826d3d7c2b1f.jpg new file mode 100644 index 00000000..20984535 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-148_065c33c36461b2f5b1219334a6c5baa0d45827a8cf1df328f326826d3d7c2b1f.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-148_232d2b7ece2552f9b9a54632972794207013f8fe6509e5bd697efb73e81d2d5a.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-148_232d2b7ece2552f9b9a54632972794207013f8fe6509e5bd697efb73e81d2d5a.jpg new file mode 100644 index 00000000..ffbf6416 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-148_232d2b7ece2552f9b9a54632972794207013f8fe6509e5bd697efb73e81d2d5a.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-148_2a3ab6c2845096255e6d2a335f913699547678e9df60e8d5dda4d62998f12e80.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-148_2a3ab6c2845096255e6d2a335f913699547678e9df60e8d5dda4d62998f12e80.jpg new file mode 100644 index 00000000..5aef3bf5 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-148_2a3ab6c2845096255e6d2a335f913699547678e9df60e8d5dda4d62998f12e80.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-148_2d4c4f2670fdf3adcfa4db69980e52ce3a1710e5d2eda7022a29f6809304ba80.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-148_2d4c4f2670fdf3adcfa4db69980e52ce3a1710e5d2eda7022a29f6809304ba80.jpg new file mode 100644 index 00000000..f794889e Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-148_2d4c4f2670fdf3adcfa4db69980e52ce3a1710e5d2eda7022a29f6809304ba80.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-148_3c1fd22e2db8f1602a625e377428119ec1b8fdd15a69691988b01279bbe3e7a9.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-148_3c1fd22e2db8f1602a625e377428119ec1b8fdd15a69691988b01279bbe3e7a9.jpg new file mode 100644 index 00000000..d621c88a Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-148_3c1fd22e2db8f1602a625e377428119ec1b8fdd15a69691988b01279bbe3e7a9.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-148_e7521c7c93f6185b1431f46578cbf6f6deb56a6f037ff3ee4678de9fadf14541.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-148_e7521c7c93f6185b1431f46578cbf6f6deb56a6f037ff3ee4678de9fadf14541.jpg new file mode 100644 index 00000000..cc0b2619 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-148_e7521c7c93f6185b1431f46578cbf6f6deb56a6f037ff3ee4678de9fadf14541.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-148_e91af475c1fbb725411e2b658c4a93e335319fdf9f716c1a40a06d562919f255.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-148_e91af475c1fbb725411e2b658c4a93e335319fdf9f716c1a40a06d562919f255.jpg new file mode 100644 index 00000000..423b0c76 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-148_e91af475c1fbb725411e2b658c4a93e335319fdf9f716c1a40a06d562919f255.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-149_e74f84438d5463fc913231dfde01b388b0d270c33f82df45ee677c8a957577fb.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-149_e74f84438d5463fc913231dfde01b388b0d270c33f82df45ee677c8a957577fb.jpg new file mode 100644 index 00000000..c6602d0a Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-149_e74f84438d5463fc913231dfde01b388b0d270c33f82df45ee677c8a957577fb.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-150_2ee00c46464c06751da049bcbb4706294fbb447496bb466db1ff9b0b155d464a.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-150_2ee00c46464c06751da049bcbb4706294fbb447496bb466db1ff9b0b155d464a.jpg new file mode 100644 index 00000000..b2598032 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-150_2ee00c46464c06751da049bcbb4706294fbb447496bb466db1ff9b0b155d464a.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-150_6330a2419f3ac9588a0d910ddaebdc2ea7546ec53e6c787a2201931c5f80125e.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-150_6330a2419f3ac9588a0d910ddaebdc2ea7546ec53e6c787a2201931c5f80125e.jpg new file mode 100644 index 00000000..5780bc70 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-150_6330a2419f3ac9588a0d910ddaebdc2ea7546ec53e6c787a2201931c5f80125e.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-150_f09a675eb77b8f34b357f897f1326ccf573ece3513d714eca9a0076183984a4e.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-150_f09a675eb77b8f34b357f897f1326ccf573ece3513d714eca9a0076183984a4e.jpg new file mode 100644 index 00000000..8e05b0e8 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-150_f09a675eb77b8f34b357f897f1326ccf573ece3513d714eca9a0076183984a4e.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-150_f11cebe34b8217798d422cd2889adec23311cada3fcf17a7146ab1a7e61ae8c4.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-150_f11cebe34b8217798d422cd2889adec23311cada3fcf17a7146ab1a7e61ae8c4.jpg new file mode 100644 index 00000000..076b967e Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-150_f11cebe34b8217798d422cd2889adec23311cada3fcf17a7146ab1a7e61ae8c4.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-151_6bfe52495bd45102f795b9dbb822019a54d76af7c74213f351a498f800e122f3.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-151_6bfe52495bd45102f795b9dbb822019a54d76af7c74213f351a498f800e122f3.jpg new file mode 100644 index 00000000..c2d32437 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-151_6bfe52495bd45102f795b9dbb822019a54d76af7c74213f351a498f800e122f3.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-151_9b90c92a45ed30fcdfb217f4a692963c1cd26cecb09caa31b7b8ded73d3c000f.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-151_9b90c92a45ed30fcdfb217f4a692963c1cd26cecb09caa31b7b8ded73d3c000f.jpg new file mode 100644 index 00000000..76841a8a Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-151_9b90c92a45ed30fcdfb217f4a692963c1cd26cecb09caa31b7b8ded73d3c000f.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-152_0323639f7063b3bda7767d26832db5a1500a31114690aff18440c193159c5d54.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-152_0323639f7063b3bda7767d26832db5a1500a31114690aff18440c193159c5d54.jpg new file mode 100644 index 00000000..07046b04 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-152_0323639f7063b3bda7767d26832db5a1500a31114690aff18440c193159c5d54.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-152_0ec578c4c7a4026e464012a48969ae27e72dc98d4ccac76a42748d536fda5d5c.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-152_0ec578c4c7a4026e464012a48969ae27e72dc98d4ccac76a42748d536fda5d5c.jpg new file mode 100644 index 00000000..a4783801 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-152_0ec578c4c7a4026e464012a48969ae27e72dc98d4ccac76a42748d536fda5d5c.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-152_7726e669d0f40d48102d270991afff679f59faa5a3ad10af1ce7c46b266ceb89.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-152_7726e669d0f40d48102d270991afff679f59faa5a3ad10af1ce7c46b266ceb89.jpg new file mode 100644 index 00000000..413fb085 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-152_7726e669d0f40d48102d270991afff679f59faa5a3ad10af1ce7c46b266ceb89.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-152_7ce2772d93df636f14903dfb0cca9a9109385df4f97cdffdd7f63e65d6c910fa.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-152_7ce2772d93df636f14903dfb0cca9a9109385df4f97cdffdd7f63e65d6c910fa.jpg new file mode 100644 index 00000000..399f3ec8 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-152_7ce2772d93df636f14903dfb0cca9a9109385df4f97cdffdd7f63e65d6c910fa.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-152_98a16b745150c2eb9632defddc5ae9b020c1fc0a08da0b9c6e5f5b3a00a44449.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-152_98a16b745150c2eb9632defddc5ae9b020c1fc0a08da0b9c6e5f5b3a00a44449.jpg new file mode 100644 index 00000000..00e63b08 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-152_98a16b745150c2eb9632defddc5ae9b020c1fc0a08da0b9c6e5f5b3a00a44449.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-152_99c783f17cd0d34fff1039087edc75b4e24cc48ea88ad0c0b9d807ae3afcf4c7.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-152_99c783f17cd0d34fff1039087edc75b4e24cc48ea88ad0c0b9d807ae3afcf4c7.jpg new file mode 100644 index 00000000..77eb3e5e Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-152_99c783f17cd0d34fff1039087edc75b4e24cc48ea88ad0c0b9d807ae3afcf4c7.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-152_dc190f8c2c17f756be6a9a6a2a2fb7b2980401a26bc4a34fc8a7d667dd1d9eb9.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-152_dc190f8c2c17f756be6a9a6a2a2fb7b2980401a26bc4a34fc8a7d667dd1d9eb9.jpg new file mode 100644 index 00000000..2321c574 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-152_dc190f8c2c17f756be6a9a6a2a2fb7b2980401a26bc4a34fc8a7d667dd1d9eb9.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-152_e9d44b1de7c350ea29b0ddcbd48e8211af0e02ad6e09e5130789cf87166396e7.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-152_e9d44b1de7c350ea29b0ddcbd48e8211af0e02ad6e09e5130789cf87166396e7.jpg new file mode 100644 index 00000000..ec579788 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-152_e9d44b1de7c350ea29b0ddcbd48e8211af0e02ad6e09e5130789cf87166396e7.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-153_129768b460e75907cc12b4196ef43e72281766cf634b92c6812749aa22c5cb4a.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-153_129768b460e75907cc12b4196ef43e72281766cf634b92c6812749aa22c5cb4a.jpg new file mode 100644 index 00000000..718e53a1 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-153_129768b460e75907cc12b4196ef43e72281766cf634b92c6812749aa22c5cb4a.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-153_3c2221538de811c460b9a85b3e3245f1bcd53a66ba9697dbd88191fd4fa346a3.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-153_3c2221538de811c460b9a85b3e3245f1bcd53a66ba9697dbd88191fd4fa346a3.jpg new file mode 100644 index 00000000..c44858b1 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-153_3c2221538de811c460b9a85b3e3245f1bcd53a66ba9697dbd88191fd4fa346a3.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-153_5d52897ca978e6897f5b8b3258863df375eec024d3804048d73da286acdaf0ed.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-153_5d52897ca978e6897f5b8b3258863df375eec024d3804048d73da286acdaf0ed.jpg new file mode 100644 index 00000000..c00110d0 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-153_5d52897ca978e6897f5b8b3258863df375eec024d3804048d73da286acdaf0ed.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-153_76b2c53443dde307045da8135e3e90877cc30ecacfb78c2951942a4724c79088.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-153_76b2c53443dde307045da8135e3e90877cc30ecacfb78c2951942a4724c79088.jpg new file mode 100644 index 00000000..dcbdb619 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-153_76b2c53443dde307045da8135e3e90877cc30ecacfb78c2951942a4724c79088.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-153_776f86b7cc710bddc16f05aadfbb6f5b468ae66dbcac2c18af76044688c78032.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-153_776f86b7cc710bddc16f05aadfbb6f5b468ae66dbcac2c18af76044688c78032.jpg new file mode 100644 index 00000000..6874f3ca Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-153_776f86b7cc710bddc16f05aadfbb6f5b468ae66dbcac2c18af76044688c78032.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-153_7f6047aa450fac7cdc8fb4669a53773724312133d1d9486c2c075eda15a6199a.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-153_7f6047aa450fac7cdc8fb4669a53773724312133d1d9486c2c075eda15a6199a.jpg new file mode 100644 index 00000000..0b2116bc Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-153_7f6047aa450fac7cdc8fb4669a53773724312133d1d9486c2c075eda15a6199a.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-153_81d32094e6309c5950fad3cfbd0fb78e1cbc44daef0659d27d6f26473ba472b2.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-153_81d32094e6309c5950fad3cfbd0fb78e1cbc44daef0659d27d6f26473ba472b2.jpg new file mode 100644 index 00000000..db797edb Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-153_81d32094e6309c5950fad3cfbd0fb78e1cbc44daef0659d27d6f26473ba472b2.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-153_cce472691a75f45444321e1d34f862a2b9f46673d6a59f79e9e957ea592cb968.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-153_cce472691a75f45444321e1d34f862a2b9f46673d6a59f79e9e957ea592cb968.jpg new file mode 100644 index 00000000..6b15ac0f Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-153_cce472691a75f45444321e1d34f862a2b9f46673d6a59f79e9e957ea592cb968.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-153_de0a09a0ac84e7a0c9b10be23b651ff3c27d478dfc5145dfa91ab0c449efb056.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-153_de0a09a0ac84e7a0c9b10be23b651ff3c27d478dfc5145dfa91ab0c449efb056.jpg new file mode 100644 index 00000000..cff92602 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-153_de0a09a0ac84e7a0c9b10be23b651ff3c27d478dfc5145dfa91ab0c449efb056.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-154_f54d0ed82e2a0460640340f526dcd4ebf0b2869d8c0ef5d68c78edceec977de0.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-154_f54d0ed82e2a0460640340f526dcd4ebf0b2869d8c0ef5d68c78edceec977de0.jpg new file mode 100644 index 00000000..15e04933 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-154_f54d0ed82e2a0460640340f526dcd4ebf0b2869d8c0ef5d68c78edceec977de0.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-155_2c3729de795d2ba70d8634403925df9ddae8563cb566a1e449d47bc9a0d4e795.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-155_2c3729de795d2ba70d8634403925df9ddae8563cb566a1e449d47bc9a0d4e795.jpg new file mode 100644 index 00000000..2c340a87 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-155_2c3729de795d2ba70d8634403925df9ddae8563cb566a1e449d47bc9a0d4e795.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-155_bb8a193166890cc32388079bee998af456687125e8cdb9a86442832334c321bf.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-155_bb8a193166890cc32388079bee998af456687125e8cdb9a86442832334c321bf.jpg new file mode 100644 index 00000000..e26e68fd Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-155_bb8a193166890cc32388079bee998af456687125e8cdb9a86442832334c321bf.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-156_4d5ef26654cbec0209b78d5fc02844a09a3bb311e1435a6d02e849b20203ee56.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-156_4d5ef26654cbec0209b78d5fc02844a09a3bb311e1435a6d02e849b20203ee56.jpg new file mode 100644 index 00000000..6f327c9f Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-156_4d5ef26654cbec0209b78d5fc02844a09a3bb311e1435a6d02e849b20203ee56.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-158_376344301c5c8fdd4924fc60e3b08c9bf3e411b0d24320d0add7a6ba2ddb7478.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-158_376344301c5c8fdd4924fc60e3b08c9bf3e411b0d24320d0add7a6ba2ddb7478.jpg new file mode 100644 index 00000000..3989eb1a Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-158_376344301c5c8fdd4924fc60e3b08c9bf3e411b0d24320d0add7a6ba2ddb7478.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-158_7644cde7ae2e7bb589fe25d4d925b107f1d3fdc43201daf1030259b849929d27.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-158_7644cde7ae2e7bb589fe25d4d925b107f1d3fdc43201daf1030259b849929d27.jpg new file mode 100644 index 00000000..b8ee20d0 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-158_7644cde7ae2e7bb589fe25d4d925b107f1d3fdc43201daf1030259b849929d27.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-158_8b413cc32e0080f3bbeb83f4ceb1b6f30f2c85457b8089190e2015f8bcf61924.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-158_8b413cc32e0080f3bbeb83f4ceb1b6f30f2c85457b8089190e2015f8bcf61924.jpg new file mode 100644 index 00000000..f0fe85d8 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-158_8b413cc32e0080f3bbeb83f4ceb1b6f30f2c85457b8089190e2015f8bcf61924.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-159_8174ae8d5da81596c17c54c9a2288f0fba5472e4689445d81687bb0907911a06.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-159_8174ae8d5da81596c17c54c9a2288f0fba5472e4689445d81687bb0907911a06.jpg new file mode 100644 index 00000000..7f012e2c Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-159_8174ae8d5da81596c17c54c9a2288f0fba5472e4689445d81687bb0907911a06.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-160_16aceee34f32a2e7a46460ee62635c2cc4a1fc931c2a2ccff3156327f37911a6.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-160_16aceee34f32a2e7a46460ee62635c2cc4a1fc931c2a2ccff3156327f37911a6.jpg new file mode 100644 index 00000000..e3fcfaa5 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-160_16aceee34f32a2e7a46460ee62635c2cc4a1fc931c2a2ccff3156327f37911a6.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-161_2855151fabdab6e69869e250fbd0e72d2e57473f90fe1eac8fe96fff7ca54491.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-161_2855151fabdab6e69869e250fbd0e72d2e57473f90fe1eac8fe96fff7ca54491.jpg new file mode 100644 index 00000000..60099c69 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-161_2855151fabdab6e69869e250fbd0e72d2e57473f90fe1eac8fe96fff7ca54491.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-161_436818be7e69148659f6e6a4df2280baa69bcba4c1378fe1c197c7518017313e.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-161_436818be7e69148659f6e6a4df2280baa69bcba4c1378fe1c197c7518017313e.jpg new file mode 100644 index 00000000..e31f10f5 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-161_436818be7e69148659f6e6a4df2280baa69bcba4c1378fe1c197c7518017313e.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-161_4967386e84cdb04b81f878a734caa14072d7083c4836fcb9daf5fa25f152d616.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-161_4967386e84cdb04b81f878a734caa14072d7083c4836fcb9daf5fa25f152d616.jpg new file mode 100644 index 00000000..97c8f885 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-161_4967386e84cdb04b81f878a734caa14072d7083c4836fcb9daf5fa25f152d616.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-161_607ce940208b164730101b8dadbc2b72e51284bf17d6c135e1cf36a4fc64883a.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-161_607ce940208b164730101b8dadbc2b72e51284bf17d6c135e1cf36a4fc64883a.jpg new file mode 100644 index 00000000..9f2aa3b4 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-161_607ce940208b164730101b8dadbc2b72e51284bf17d6c135e1cf36a4fc64883a.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-161_a0818ab1901af57a010d9da982bbfc39e6bd57ef747cb10b1b91321636a3644f.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-161_a0818ab1901af57a010d9da982bbfc39e6bd57ef747cb10b1b91321636a3644f.jpg new file mode 100644 index 00000000..23cde15e Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-161_a0818ab1901af57a010d9da982bbfc39e6bd57ef747cb10b1b91321636a3644f.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-161_b3994baac0e3985ff59ba668350af5fa6f1db21e0b9456195f5512068ea6a3dc.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-161_b3994baac0e3985ff59ba668350af5fa6f1db21e0b9456195f5512068ea6a3dc.jpg new file mode 100644 index 00000000..d6471c08 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-161_b3994baac0e3985ff59ba668350af5fa6f1db21e0b9456195f5512068ea6a3dc.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-161_dcc6fbf1c222ad528e48ac737039d76151f6202868b747527400c6c2c7cafc3f.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-161_dcc6fbf1c222ad528e48ac737039d76151f6202868b747527400c6c2c7cafc3f.jpg new file mode 100644 index 00000000..7106fdcb Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-161_dcc6fbf1c222ad528e48ac737039d76151f6202868b747527400c6c2c7cafc3f.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-162_738a117c0e0b2eb437828c7cb573617fa2a2a3655f2f3568a2ec97cb7f007f33.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-162_738a117c0e0b2eb437828c7cb573617fa2a2a3655f2f3568a2ec97cb7f007f33.jpg new file mode 100644 index 00000000..759d26d6 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-162_738a117c0e0b2eb437828c7cb573617fa2a2a3655f2f3568a2ec97cb7f007f33.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-162_9a973f2b87bc996bb073385e67ac24a62a653af53ec75428328a09621ed472bb.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-162_9a973f2b87bc996bb073385e67ac24a62a653af53ec75428328a09621ed472bb.jpg new file mode 100644 index 00000000..a666e7c4 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-162_9a973f2b87bc996bb073385e67ac24a62a653af53ec75428328a09621ed472bb.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-162_f2964763729d91fe604ac99c5d536423fe48d700a3c2145309fff884d85afe26.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-162_f2964763729d91fe604ac99c5d536423fe48d700a3c2145309fff884d85afe26.jpg new file mode 100644 index 00000000..c0be5401 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-162_f2964763729d91fe604ac99c5d536423fe48d700a3c2145309fff884d85afe26.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-168_43cdc07545b197fc7b960da42f56375a617e4e4dca7dae8c27cc6fb0a6efe5e3.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-168_43cdc07545b197fc7b960da42f56375a617e4e4dca7dae8c27cc6fb0a6efe5e3.jpg new file mode 100644 index 00000000..21757aee Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-168_43cdc07545b197fc7b960da42f56375a617e4e4dca7dae8c27cc6fb0a6efe5e3.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-168_5f5f3f6bd03aa82de913d469087d4ff926d13c9de2d9306faa15c0fe985543c1.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-168_5f5f3f6bd03aa82de913d469087d4ff926d13c9de2d9306faa15c0fe985543c1.jpg new file mode 100644 index 00000000..205efb71 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-168_5f5f3f6bd03aa82de913d469087d4ff926d13c9de2d9306faa15c0fe985543c1.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-168_8b926f43a8a92704fac6ac50c1871a5bcc211882eae6e143c3a38eb54fa7a674.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-168_8b926f43a8a92704fac6ac50c1871a5bcc211882eae6e143c3a38eb54fa7a674.jpg new file mode 100644 index 00000000..ec16b20e Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-168_8b926f43a8a92704fac6ac50c1871a5bcc211882eae6e143c3a38eb54fa7a674.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-168_954cc3ab5d86ad8b7d83e07f8b469a936ef0cf4e0009dff9875f6d542c535354.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-168_954cc3ab5d86ad8b7d83e07f8b469a936ef0cf4e0009dff9875f6d542c535354.jpg new file mode 100644 index 00000000..13123b85 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-168_954cc3ab5d86ad8b7d83e07f8b469a936ef0cf4e0009dff9875f6d542c535354.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-169_7ab9e9dc37ba51601d2d263a8b3a4da56528d7ed685857f6cef25ea669dab60a.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-169_7ab9e9dc37ba51601d2d263a8b3a4da56528d7ed685857f6cef25ea669dab60a.jpg new file mode 100644 index 00000000..3154ca3c Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-169_7ab9e9dc37ba51601d2d263a8b3a4da56528d7ed685857f6cef25ea669dab60a.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-169_80c587331bdb1f72b46095a0b7990abda3a93051a6fc4b163b574f1c0caa734f.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-169_80c587331bdb1f72b46095a0b7990abda3a93051a6fc4b163b574f1c0caa734f.jpg new file mode 100644 index 00000000..d8ebc910 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-169_80c587331bdb1f72b46095a0b7990abda3a93051a6fc4b163b574f1c0caa734f.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-169_84b42657afc0dbb68b9388bda4d07a1a4db91b5da5b0ec5be5d91572708a004d.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-169_84b42657afc0dbb68b9388bda4d07a1a4db91b5da5b0ec5be5d91572708a004d.jpg new file mode 100644 index 00000000..9179377a Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-169_84b42657afc0dbb68b9388bda4d07a1a4db91b5da5b0ec5be5d91572708a004d.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-169_8b4b79e8569ff88d2a6b5c049adcffedce312901ae1698219c421e8646b0503a.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-169_8b4b79e8569ff88d2a6b5c049adcffedce312901ae1698219c421e8646b0503a.jpg new file mode 100644 index 00000000..0eb8e717 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-169_8b4b79e8569ff88d2a6b5c049adcffedce312901ae1698219c421e8646b0503a.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-169_9362d8d13e4c07dba4352a45ef2ce329426d1a77e9685da4afda724225d2cd84.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-169_9362d8d13e4c07dba4352a45ef2ce329426d1a77e9685da4afda724225d2cd84.jpg new file mode 100644 index 00000000..baf36bdf Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-169_9362d8d13e4c07dba4352a45ef2ce329426d1a77e9685da4afda724225d2cd84.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-169_b8a0a934233fb4829ad9be346a3ea18c7104ff2a83705430838af19b0f0c5563.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-169_b8a0a934233fb4829ad9be346a3ea18c7104ff2a83705430838af19b0f0c5563.jpg new file mode 100644 index 00000000..b31db6e2 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-169_b8a0a934233fb4829ad9be346a3ea18c7104ff2a83705430838af19b0f0c5563.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-169_dae32b249ab4548670d88c52a219cdbe5ec5ffa94fadb267c1f00875ab39f61e.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-169_dae32b249ab4548670d88c52a219cdbe5ec5ffa94fadb267c1f00875ab39f61e.jpg new file mode 100644 index 00000000..f6cf29a8 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-169_dae32b249ab4548670d88c52a219cdbe5ec5ffa94fadb267c1f00875ab39f61e.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-169_e3f132bc4ffae0d87b1ce04482c6c9dcb7b99d0c170fd1313ba0e430fc9d9e95.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-169_e3f132bc4ffae0d87b1ce04482c6c9dcb7b99d0c170fd1313ba0e430fc9d9e95.jpg new file mode 100644 index 00000000..c9b0176e Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-169_e3f132bc4ffae0d87b1ce04482c6c9dcb7b99d0c170fd1313ba0e430fc9d9e95.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-169_ffb0341245902808ca295ba03d8d54f28e310baf4dc6065db3e39e76324dffc3.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-169_ffb0341245902808ca295ba03d8d54f28e310baf4dc6065db3e39e76324dffc3.jpg new file mode 100644 index 00000000..4633a7be Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-169_ffb0341245902808ca295ba03d8d54f28e310baf4dc6065db3e39e76324dffc3.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-170_4daa147bf2602233dfa1eebc63666acfde14e09a1d85b18bf5086b2eac076be5.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-170_4daa147bf2602233dfa1eebc63666acfde14e09a1d85b18bf5086b2eac076be5.jpg new file mode 100644 index 00000000..23c754ce Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-170_4daa147bf2602233dfa1eebc63666acfde14e09a1d85b18bf5086b2eac076be5.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-170_74b58659bc7485b8a2ac51147f1b505b130ca9a4a3cba3650caa7fc901a7157e.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-170_74b58659bc7485b8a2ac51147f1b505b130ca9a4a3cba3650caa7fc901a7157e.jpg new file mode 100644 index 00000000..b769c6cb Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-170_74b58659bc7485b8a2ac51147f1b505b130ca9a4a3cba3650caa7fc901a7157e.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-170_d74f70b20cebb472fbeecbef16328d78a89e330381892eb8d21bc312b59d3e92.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-170_d74f70b20cebb472fbeecbef16328d78a89e330381892eb8d21bc312b59d3e92.jpg new file mode 100644 index 00000000..862e2ca0 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-170_d74f70b20cebb472fbeecbef16328d78a89e330381892eb8d21bc312b59d3e92.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-170_e2734ee87400b781b87ff884c6bcd7c21d49653e5600dd00a2607abe69dce0d2.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-170_e2734ee87400b781b87ff884c6bcd7c21d49653e5600dd00a2607abe69dce0d2.jpg new file mode 100644 index 00000000..92cfbd81 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-170_e2734ee87400b781b87ff884c6bcd7c21d49653e5600dd00a2607abe69dce0d2.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-170_ff0e10d9d79fd4ca03da50ae86bc128216d321b2e759adb7586deb07d1d27a6b.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-170_ff0e10d9d79fd4ca03da50ae86bc128216d321b2e759adb7586deb07d1d27a6b.jpg new file mode 100644 index 00000000..e3e3e00c Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-170_ff0e10d9d79fd4ca03da50ae86bc128216d321b2e759adb7586deb07d1d27a6b.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-171_ee53d9f53bb1b85c6e5c506af4eca19b334993509bee8bbb4c13fb70dfbedf13.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-171_ee53d9f53bb1b85c6e5c506af4eca19b334993509bee8bbb4c13fb70dfbedf13.jpg new file mode 100644 index 00000000..9c43cd7c Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-171_ee53d9f53bb1b85c6e5c506af4eca19b334993509bee8bbb4c13fb70dfbedf13.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-172_0e2fbe4426dc39fef7ab5f2ccdd565e9fb14abadf7649473646920f9d765e4fe.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-172_0e2fbe4426dc39fef7ab5f2ccdd565e9fb14abadf7649473646920f9d765e4fe.jpg new file mode 100644 index 00000000..f619d2a7 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-172_0e2fbe4426dc39fef7ab5f2ccdd565e9fb14abadf7649473646920f9d765e4fe.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-172_4ecd8c269bb7245577039ec1a5051bb1ad5fd37eaa955ceb20c74713076733bb.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-172_4ecd8c269bb7245577039ec1a5051bb1ad5fd37eaa955ceb20c74713076733bb.jpg new file mode 100644 index 00000000..fb3e9057 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-172_4ecd8c269bb7245577039ec1a5051bb1ad5fd37eaa955ceb20c74713076733bb.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-172_b466249b4e0344d9e19adf397e473bc64596df923c388cf522c60bcac27c40ed.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-172_b466249b4e0344d9e19adf397e473bc64596df923c388cf522c60bcac27c40ed.jpg new file mode 100644 index 00000000..be899f4d Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-172_b466249b4e0344d9e19adf397e473bc64596df923c388cf522c60bcac27c40ed.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-172_e9519706923060a1a86c74ab7c64bd098be870710376a9252b965a2d03494a12.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-172_e9519706923060a1a86c74ab7c64bd098be870710376a9252b965a2d03494a12.jpg new file mode 100644 index 00000000..21f6466a Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-172_e9519706923060a1a86c74ab7c64bd098be870710376a9252b965a2d03494a12.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-173_324010709cc4fb7582a3ff35d978c946f8077c71b3da15824753f965b40ee85c.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-173_324010709cc4fb7582a3ff35d978c946f8077c71b3da15824753f965b40ee85c.jpg new file mode 100644 index 00000000..bc0e6d65 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-173_324010709cc4fb7582a3ff35d978c946f8077c71b3da15824753f965b40ee85c.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-173_46f658799121f7b0bdcfea7b6ea35511d2e9021a43d557fd136619f7883ffdc9.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-173_46f658799121f7b0bdcfea7b6ea35511d2e9021a43d557fd136619f7883ffdc9.jpg new file mode 100644 index 00000000..18eaa7e1 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-173_46f658799121f7b0bdcfea7b6ea35511d2e9021a43d557fd136619f7883ffdc9.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-173_56124e0dde653011f726e50d1d858f2c7cb6e63949d91a991d59a30ac7be97ef.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-173_56124e0dde653011f726e50d1d858f2c7cb6e63949d91a991d59a30ac7be97ef.jpg new file mode 100644 index 00000000..122479b1 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-173_56124e0dde653011f726e50d1d858f2c7cb6e63949d91a991d59a30ac7be97ef.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-173_568edd4997e47df28008cae82a188e3fb9cee00fc3942dc5e594a27fbedabe01.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-173_568edd4997e47df28008cae82a188e3fb9cee00fc3942dc5e594a27fbedabe01.jpg new file mode 100644 index 00000000..7a143d8e Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-173_568edd4997e47df28008cae82a188e3fb9cee00fc3942dc5e594a27fbedabe01.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-173_7fe5d4d2c8d7f69f5c0357d19c86db55158e0d3a9b921246c1df44155ee60a10.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-173_7fe5d4d2c8d7f69f5c0357d19c86db55158e0d3a9b921246c1df44155ee60a10.jpg new file mode 100644 index 00000000..66d0cd1b Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-173_7fe5d4d2c8d7f69f5c0357d19c86db55158e0d3a9b921246c1df44155ee60a10.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-173_8ad43f6c745dce592fefd2218394264897459e33ea1971362fa100a06bfebf77.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-173_8ad43f6c745dce592fefd2218394264897459e33ea1971362fa100a06bfebf77.jpg new file mode 100644 index 00000000..08fd20b3 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-173_8ad43f6c745dce592fefd2218394264897459e33ea1971362fa100a06bfebf77.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-173_b5ab876ed6890b9f8965f3dc0c60a6a0d024bc4b66934e2cb926302a07126c07.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-173_b5ab876ed6890b9f8965f3dc0c60a6a0d024bc4b66934e2cb926302a07126c07.jpg new file mode 100644 index 00000000..2c069953 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-173_b5ab876ed6890b9f8965f3dc0c60a6a0d024bc4b66934e2cb926302a07126c07.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-173_b924c0acb75853fe962bc9cb58d94032389c472ecfa692e3194070c99e6f5a30.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-173_b924c0acb75853fe962bc9cb58d94032389c472ecfa692e3194070c99e6f5a30.jpg new file mode 100644 index 00000000..52287d4f Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-173_b924c0acb75853fe962bc9cb58d94032389c472ecfa692e3194070c99e6f5a30.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-173_e89be193b846bf51c3610f2802ea9ca94fb2c0679721899068f4c691ca292e78.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-173_e89be193b846bf51c3610f2802ea9ca94fb2c0679721899068f4c691ca292e78.jpg new file mode 100644 index 00000000..7a022476 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-173_e89be193b846bf51c3610f2802ea9ca94fb2c0679721899068f4c691ca292e78.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-174_4ae2a54982d4f218f5ea4fd61ce5f9ba54b1ac3e959e340be58a5abff24e538d.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-174_4ae2a54982d4f218f5ea4fd61ce5f9ba54b1ac3e959e340be58a5abff24e538d.jpg new file mode 100644 index 00000000..08e58bad Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-174_4ae2a54982d4f218f5ea4fd61ce5f9ba54b1ac3e959e340be58a5abff24e538d.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-174_a76a56f706cf2dfcfde10955830f79b0439f789502d1b9b8d66998e0697b5ebf.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-174_a76a56f706cf2dfcfde10955830f79b0439f789502d1b9b8d66998e0697b5ebf.jpg new file mode 100644 index 00000000..bfa17c18 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-174_a76a56f706cf2dfcfde10955830f79b0439f789502d1b9b8d66998e0697b5ebf.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-174_b448d0488010ef21b5c83cb612314109c56f3232e150186eae70c22cef9378e6.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-174_b448d0488010ef21b5c83cb612314109c56f3232e150186eae70c22cef9378e6.jpg new file mode 100644 index 00000000..c6feab8f Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-174_b448d0488010ef21b5c83cb612314109c56f3232e150186eae70c22cef9378e6.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-174_bb1a02d5ac4355c398dd346abeee15d20386a84d35b57baabfa05b951e7d40b4.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-174_bb1a02d5ac4355c398dd346abeee15d20386a84d35b57baabfa05b951e7d40b4.jpg new file mode 100644 index 00000000..d7509c3c Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-174_bb1a02d5ac4355c398dd346abeee15d20386a84d35b57baabfa05b951e7d40b4.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-174_cd6428315c35e5ffd73d2613dc517898758f95e5bab75b7fea44b3874d4b8c7a.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-174_cd6428315c35e5ffd73d2613dc517898758f95e5bab75b7fea44b3874d4b8c7a.jpg new file mode 100644 index 00000000..f36806a6 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-174_cd6428315c35e5ffd73d2613dc517898758f95e5bab75b7fea44b3874d4b8c7a.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-174_d8327c1c71db6131d8926bc0255582796b6008e238b22278280a1d6b89216610.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-174_d8327c1c71db6131d8926bc0255582796b6008e238b22278280a1d6b89216610.jpg new file mode 100644 index 00000000..704e0585 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-174_d8327c1c71db6131d8926bc0255582796b6008e238b22278280a1d6b89216610.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-175_0fc46b0604ff8f762c206b9802374a43fa1d9ac23c1536ae93f672c5ef3490ad.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-175_0fc46b0604ff8f762c206b9802374a43fa1d9ac23c1536ae93f672c5ef3490ad.jpg new file mode 100644 index 00000000..d3aa2d4f Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-175_0fc46b0604ff8f762c206b9802374a43fa1d9ac23c1536ae93f672c5ef3490ad.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-175_25f7088226127c529e7a84d3bcb26c46c09b9cc6b6f71708546e12707a7eb055.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-175_25f7088226127c529e7a84d3bcb26c46c09b9cc6b6f71708546e12707a7eb055.jpg new file mode 100644 index 00000000..7741df79 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-175_25f7088226127c529e7a84d3bcb26c46c09b9cc6b6f71708546e12707a7eb055.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-175_88bc5d76fc8085a6d6383a86b782d4693c69b795c9ba79bcaaf314e9a1b20d1b.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-175_88bc5d76fc8085a6d6383a86b782d4693c69b795c9ba79bcaaf314e9a1b20d1b.jpg new file mode 100644 index 00000000..9432c51f Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-175_88bc5d76fc8085a6d6383a86b782d4693c69b795c9ba79bcaaf314e9a1b20d1b.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-175_905548d77adeab011beea8f39282f97be06c5306078f87e25dc0047449b72f3b.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-175_905548d77adeab011beea8f39282f97be06c5306078f87e25dc0047449b72f3b.jpg new file mode 100644 index 00000000..5804bff8 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-175_905548d77adeab011beea8f39282f97be06c5306078f87e25dc0047449b72f3b.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-175_92f5ad7be0f4bb3a7036bd365cc589df15dcfb2d86fcf6d9751078f0003c7333.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-175_92f5ad7be0f4bb3a7036bd365cc589df15dcfb2d86fcf6d9751078f0003c7333.jpg new file mode 100644 index 00000000..f6dbd6cd Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-175_92f5ad7be0f4bb3a7036bd365cc589df15dcfb2d86fcf6d9751078f0003c7333.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-175_c5a95f2cf0572bc9401df9247db02aab308f383d83e34fb946a9f786986e6d38.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-175_c5a95f2cf0572bc9401df9247db02aab308f383d83e34fb946a9f786986e6d38.jpg new file mode 100644 index 00000000..13a6ffca Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-175_c5a95f2cf0572bc9401df9247db02aab308f383d83e34fb946a9f786986e6d38.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-175_ccd4c6f7b2445b51c70b59acfc0127d066d3d99c1ca6eb303ae430a6cbebaa61.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-175_ccd4c6f7b2445b51c70b59acfc0127d066d3d99c1ca6eb303ae430a6cbebaa61.jpg new file mode 100644 index 00000000..605c05d3 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-175_ccd4c6f7b2445b51c70b59acfc0127d066d3d99c1ca6eb303ae430a6cbebaa61.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-176_103fb16e7b7eab5395cce096cc3713b71140310bcd05e911e7c5de7dfd93cb80.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-176_103fb16e7b7eab5395cce096cc3713b71140310bcd05e911e7c5de7dfd93cb80.jpg new file mode 100644 index 00000000..7f8d2572 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-176_103fb16e7b7eab5395cce096cc3713b71140310bcd05e911e7c5de7dfd93cb80.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-176_118cd54c42500a7e55bb02c9e08ab1863a905708336ed65dc7757e40316c5663.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-176_118cd54c42500a7e55bb02c9e08ab1863a905708336ed65dc7757e40316c5663.jpg new file mode 100644 index 00000000..5c0a0dee Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-176_118cd54c42500a7e55bb02c9e08ab1863a905708336ed65dc7757e40316c5663.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-176_2308376513b1e2de2b7deaf6146465fc83638584261551667e7f79161f6b0dfa.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-176_2308376513b1e2de2b7deaf6146465fc83638584261551667e7f79161f6b0dfa.jpg new file mode 100644 index 00000000..409532b9 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-176_2308376513b1e2de2b7deaf6146465fc83638584261551667e7f79161f6b0dfa.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-176_28b875a3ec70467febd24adcd1a0760d532d9024bacc364d46ced0b535bc5741.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-176_28b875a3ec70467febd24adcd1a0760d532d9024bacc364d46ced0b535bc5741.jpg new file mode 100644 index 00000000..4ec2d36f Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-176_28b875a3ec70467febd24adcd1a0760d532d9024bacc364d46ced0b535bc5741.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-176_426dabab95365ceddfe6de0ff9a8417400ae978497cf1c9247377ad2869fe2c9.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-176_426dabab95365ceddfe6de0ff9a8417400ae978497cf1c9247377ad2869fe2c9.jpg new file mode 100644 index 00000000..356f8ce7 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-176_426dabab95365ceddfe6de0ff9a8417400ae978497cf1c9247377ad2869fe2c9.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-176_5c318e3109693812002a92e57810cd1fee2d0beba8ad69cca1a4d7c90e125694.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-176_5c318e3109693812002a92e57810cd1fee2d0beba8ad69cca1a4d7c90e125694.jpg new file mode 100644 index 00000000..f07fda68 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-176_5c318e3109693812002a92e57810cd1fee2d0beba8ad69cca1a4d7c90e125694.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-176_5ddc17c8ff9020bcac7e7469535f07909c2da6ec8db4ec002d04ce307404336c.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-176_5ddc17c8ff9020bcac7e7469535f07909c2da6ec8db4ec002d04ce307404336c.jpg new file mode 100644 index 00000000..601b6fb2 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-176_5ddc17c8ff9020bcac7e7469535f07909c2da6ec8db4ec002d04ce307404336c.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-176_9544d3279f817b7055be5881bc8c2b2be0d0c2a6a4fa44feeee7abc7b03e2951.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-176_9544d3279f817b7055be5881bc8c2b2be0d0c2a6a4fa44feeee7abc7b03e2951.jpg new file mode 100644 index 00000000..cf900421 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-176_9544d3279f817b7055be5881bc8c2b2be0d0c2a6a4fa44feeee7abc7b03e2951.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-177_20296c55a335a35dbed4ce129a5c802daa663bd61c19a8b86a59f3f27f900b27.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-177_20296c55a335a35dbed4ce129a5c802daa663bd61c19a8b86a59f3f27f900b27.jpg new file mode 100644 index 00000000..40c255a1 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-177_20296c55a335a35dbed4ce129a5c802daa663bd61c19a8b86a59f3f27f900b27.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-177_29512a8c4af6760c7c99c4d9c39a1d4af48eaedadb64805aff4df167e044b740.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-177_29512a8c4af6760c7c99c4d9c39a1d4af48eaedadb64805aff4df167e044b740.jpg new file mode 100644 index 00000000..c5a6ffd6 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-177_29512a8c4af6760c7c99c4d9c39a1d4af48eaedadb64805aff4df167e044b740.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-177_5a188fa5fe505c4e8205c6dd485c4b26897b71f47c5614093561151c7e5ecdc0.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-177_5a188fa5fe505c4e8205c6dd485c4b26897b71f47c5614093561151c7e5ecdc0.jpg new file mode 100644 index 00000000..ffeee58f Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-177_5a188fa5fe505c4e8205c6dd485c4b26897b71f47c5614093561151c7e5ecdc0.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-177_aaf484d311d0f6e05efd52b5cedae7c706caae02eac373ca2a5637ad3f401641.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-177_aaf484d311d0f6e05efd52b5cedae7c706caae02eac373ca2a5637ad3f401641.jpg new file mode 100644 index 00000000..d7fd65b2 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-177_aaf484d311d0f6e05efd52b5cedae7c706caae02eac373ca2a5637ad3f401641.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-177_d215c0f1b8ebe96feed5b5b43a39a29871d17c646481d14a4ea61c29af8d7c1b.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-177_d215c0f1b8ebe96feed5b5b43a39a29871d17c646481d14a4ea61c29af8d7c1b.jpg new file mode 100644 index 00000000..0cb1217b Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-177_d215c0f1b8ebe96feed5b5b43a39a29871d17c646481d14a4ea61c29af8d7c1b.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-177_fc8eb999610f4908f994b2b6cf2c71b7a3411dc8db38595e2e8283df48465853.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-177_fc8eb999610f4908f994b2b6cf2c71b7a3411dc8db38595e2e8283df48465853.jpg new file mode 100644 index 00000000..b7c5f6b7 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-177_fc8eb999610f4908f994b2b6cf2c71b7a3411dc8db38595e2e8283df48465853.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-178_05ed0031da354df9b4ee02c51f5492a60eb6fedaabe2ac7c23ec54155e5d47a7.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-178_05ed0031da354df9b4ee02c51f5492a60eb6fedaabe2ac7c23ec54155e5d47a7.jpg new file mode 100644 index 00000000..345c67d1 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-178_05ed0031da354df9b4ee02c51f5492a60eb6fedaabe2ac7c23ec54155e5d47a7.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-178_251f56c86e9cabbb81d3be7ef1b075a678f29cc2bd810d670c9dbd54d58f6c50.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-178_251f56c86e9cabbb81d3be7ef1b075a678f29cc2bd810d670c9dbd54d58f6c50.jpg new file mode 100644 index 00000000..03c37252 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-178_251f56c86e9cabbb81d3be7ef1b075a678f29cc2bd810d670c9dbd54d58f6c50.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-178_5b82dedf308fe6914f3000ee36af0494fe66bad420a9e9424bdb592b977467a6.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-178_5b82dedf308fe6914f3000ee36af0494fe66bad420a9e9424bdb592b977467a6.jpg new file mode 100644 index 00000000..cf52ea15 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-178_5b82dedf308fe6914f3000ee36af0494fe66bad420a9e9424bdb592b977467a6.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-178_86de4fc4fde1263331b91a101ef1e64885f15d6fe2feea335f7327da919d052e.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-178_86de4fc4fde1263331b91a101ef1e64885f15d6fe2feea335f7327da919d052e.jpg new file mode 100644 index 00000000..2b86ab9b Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-178_86de4fc4fde1263331b91a101ef1e64885f15d6fe2feea335f7327da919d052e.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-178_a5ae519125815c12cc4cb63c0ad5768d8dbf47a6a27be5f604273ab2722c10de.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-178_a5ae519125815c12cc4cb63c0ad5768d8dbf47a6a27be5f604273ab2722c10de.jpg new file mode 100644 index 00000000..115e435a Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-178_a5ae519125815c12cc4cb63c0ad5768d8dbf47a6a27be5f604273ab2722c10de.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-178_c3a6b046628ee3c97f4cbd6c83c7c22db5c8e2a0caeb1243b2494bf263d86cb9.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-178_c3a6b046628ee3c97f4cbd6c83c7c22db5c8e2a0caeb1243b2494bf263d86cb9.jpg new file mode 100644 index 00000000..9e6e3ee7 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-178_c3a6b046628ee3c97f4cbd6c83c7c22db5c8e2a0caeb1243b2494bf263d86cb9.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-178_c9dd8520d4916e0c265df209624868fdd3e4396c0e6282cd1244dd89497ba6b0.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-178_c9dd8520d4916e0c265df209624868fdd3e4396c0e6282cd1244dd89497ba6b0.jpg new file mode 100644 index 00000000..a3ed7339 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-178_c9dd8520d4916e0c265df209624868fdd3e4396c0e6282cd1244dd89497ba6b0.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-178_e6b1c8c1148d6f2951c6cd04df7b0d0250ea63cccc63b17ea402f70803e78509.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-178_e6b1c8c1148d6f2951c6cd04df7b0d0250ea63cccc63b17ea402f70803e78509.jpg new file mode 100644 index 00000000..7f6aa4ba Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-178_e6b1c8c1148d6f2951c6cd04df7b0d0250ea63cccc63b17ea402f70803e78509.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-179_23eff196f6777b2800dfbfd440bf89ad0432d93aeac80ccbd71b7e108e7d2578.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-179_23eff196f6777b2800dfbfd440bf89ad0432d93aeac80ccbd71b7e108e7d2578.jpg new file mode 100644 index 00000000..1a2ef7a0 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-179_23eff196f6777b2800dfbfd440bf89ad0432d93aeac80ccbd71b7e108e7d2578.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-179_367f72844caefdb0ed1e23d193cf6c4ac5ce6c76329139cf697cd3fd158a4b16.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-179_367f72844caefdb0ed1e23d193cf6c4ac5ce6c76329139cf697cd3fd158a4b16.jpg new file mode 100644 index 00000000..708cd51f Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-179_367f72844caefdb0ed1e23d193cf6c4ac5ce6c76329139cf697cd3fd158a4b16.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-179_6e132ad8d6be74939111fec0d87659a74fe7f4399c5d4d6c996f933af8c49fe1.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-179_6e132ad8d6be74939111fec0d87659a74fe7f4399c5d4d6c996f933af8c49fe1.jpg new file mode 100644 index 00000000..34ccf240 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-179_6e132ad8d6be74939111fec0d87659a74fe7f4399c5d4d6c996f933af8c49fe1.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-179_8a15c2209987987853d09bde696d9726fde83074f3bc13347a3915dfd9c785d2.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-179_8a15c2209987987853d09bde696d9726fde83074f3bc13347a3915dfd9c785d2.jpg new file mode 100644 index 00000000..f89cb0e3 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-179_8a15c2209987987853d09bde696d9726fde83074f3bc13347a3915dfd9c785d2.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-179_b18b72c94b04eded6ff571387ace2f5d008590b3a253eebfdf564912a2df464b.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-179_b18b72c94b04eded6ff571387ace2f5d008590b3a253eebfdf564912a2df464b.jpg new file mode 100644 index 00000000..ea09bb97 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-179_b18b72c94b04eded6ff571387ace2f5d008590b3a253eebfdf564912a2df464b.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-179_da841db2ca66aef10fd75cb6c9ef3ad2104f58754ef52bbd53a40cc209180ae3.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-179_da841db2ca66aef10fd75cb6c9ef3ad2104f58754ef52bbd53a40cc209180ae3.jpg new file mode 100644 index 00000000..93633181 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-179_da841db2ca66aef10fd75cb6c9ef3ad2104f58754ef52bbd53a40cc209180ae3.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-179_ee26b333f93fdef891c3fbdcaab5c51cc5a47a02968fea1f162d50277410e1a9.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-179_ee26b333f93fdef891c3fbdcaab5c51cc5a47a02968fea1f162d50277410e1a9.jpg new file mode 100644 index 00000000..bee24503 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-179_ee26b333f93fdef891c3fbdcaab5c51cc5a47a02968fea1f162d50277410e1a9.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-180_41b51f12fe7d44d7959d2f6f9269fbfe0122eaf5d5efc777ecaef6d633025db6.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-180_41b51f12fe7d44d7959d2f6f9269fbfe0122eaf5d5efc777ecaef6d633025db6.jpg new file mode 100644 index 00000000..e8d1b891 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-180_41b51f12fe7d44d7959d2f6f9269fbfe0122eaf5d5efc777ecaef6d633025db6.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-180_bbcdda2e08e5245736bb38ba615a595e8ebf930ebd6815c2d1862883c51fdb56.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-180_bbcdda2e08e5245736bb38ba615a595e8ebf930ebd6815c2d1862883c51fdb56.jpg new file mode 100644 index 00000000..43b654f3 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-180_bbcdda2e08e5245736bb38ba615a595e8ebf930ebd6815c2d1862883c51fdb56.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-180_d89de0913ebc8c5e38691c981501ecea72a336775082b1dc4767cd9a5b82219d.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-180_d89de0913ebc8c5e38691c981501ecea72a336775082b1dc4767cd9a5b82219d.jpg new file mode 100644 index 00000000..45cd6219 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-180_d89de0913ebc8c5e38691c981501ecea72a336775082b1dc4767cd9a5b82219d.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-180_e7cbda51ea6f81e6e7be673d9648cd2dedbe8de61fccb32fed0eeed06675c74f.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-180_e7cbda51ea6f81e6e7be673d9648cd2dedbe8de61fccb32fed0eeed06675c74f.jpg new file mode 100644 index 00000000..d10147e6 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-180_e7cbda51ea6f81e6e7be673d9648cd2dedbe8de61fccb32fed0eeed06675c74f.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-181_02c55a1c578fc9cf03963a75e9462d73e601b36574397c8d3c14bc78edd1e545.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-181_02c55a1c578fc9cf03963a75e9462d73e601b36574397c8d3c14bc78edd1e545.jpg new file mode 100644 index 00000000..e2798b13 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-181_02c55a1c578fc9cf03963a75e9462d73e601b36574397c8d3c14bc78edd1e545.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-181_083fd5bcdaa21d1c40b0b90e6902b206c45cb9bede64720110be1d40419b39c8.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-181_083fd5bcdaa21d1c40b0b90e6902b206c45cb9bede64720110be1d40419b39c8.jpg new file mode 100644 index 00000000..46fe89e1 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-181_083fd5bcdaa21d1c40b0b90e6902b206c45cb9bede64720110be1d40419b39c8.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-181_3681f6559ec9c4842612c1f38165e177f0b22955359cd7bb3be8e9e98edb5e88.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-181_3681f6559ec9c4842612c1f38165e177f0b22955359cd7bb3be8e9e98edb5e88.jpg new file mode 100644 index 00000000..a1963c58 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-181_3681f6559ec9c4842612c1f38165e177f0b22955359cd7bb3be8e9e98edb5e88.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-181_3698de5d851e2a9ab3173859661179a0d4d8c0cf895dc0861bdc8025463b8b3f.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-181_3698de5d851e2a9ab3173859661179a0d4d8c0cf895dc0861bdc8025463b8b3f.jpg new file mode 100644 index 00000000..0b351fc8 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-181_3698de5d851e2a9ab3173859661179a0d4d8c0cf895dc0861bdc8025463b8b3f.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-181_46f4d3dd3ebfdc27146805be25c97c39ae8e894a673c6f1bd616ef2de4c204d7.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-181_46f4d3dd3ebfdc27146805be25c97c39ae8e894a673c6f1bd616ef2de4c204d7.jpg new file mode 100644 index 00000000..7fd18620 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-181_46f4d3dd3ebfdc27146805be25c97c39ae8e894a673c6f1bd616ef2de4c204d7.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-181_f2de409d85d3737278b73712c9ce2a5266e3391aac65147b80688c359a8abc61.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-181_f2de409d85d3737278b73712c9ce2a5266e3391aac65147b80688c359a8abc61.jpg new file mode 100644 index 00000000..c65d7715 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-181_f2de409d85d3737278b73712c9ce2a5266e3391aac65147b80688c359a8abc61.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-182_c24c61742cb590310cb85d9387a347a20ac6a8ca23fa69abc25e32c61977e116.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-182_c24c61742cb590310cb85d9387a347a20ac6a8ca23fa69abc25e32c61977e116.jpg new file mode 100644 index 00000000..7cac1d8d Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-182_c24c61742cb590310cb85d9387a347a20ac6a8ca23fa69abc25e32c61977e116.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-183_301a12d89787c3bd73f0a267765efe8ed49a733d96c406854e24e1333ee7138b.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-183_301a12d89787c3bd73f0a267765efe8ed49a733d96c406854e24e1333ee7138b.jpg new file mode 100644 index 00000000..26b3c8e2 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-183_301a12d89787c3bd73f0a267765efe8ed49a733d96c406854e24e1333ee7138b.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-183_5ad74ae133fd371c166982bfd240b1a7b85fd17f1d4f45c1bbc1174d36ebfdf2.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-183_5ad74ae133fd371c166982bfd240b1a7b85fd17f1d4f45c1bbc1174d36ebfdf2.jpg new file mode 100644 index 00000000..897674b2 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-183_5ad74ae133fd371c166982bfd240b1a7b85fd17f1d4f45c1bbc1174d36ebfdf2.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-183_5fbb2d4c8859b716ee32430f0af77386fa028d3b595f50ced976233598c16b5e.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-183_5fbb2d4c8859b716ee32430f0af77386fa028d3b595f50ced976233598c16b5e.jpg new file mode 100644 index 00000000..338ff965 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-183_5fbb2d4c8859b716ee32430f0af77386fa028d3b595f50ced976233598c16b5e.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-183_90cb500dfa0c3a6a32ff2bb0d5535c8fe3b3c7262f364140b28b777510bf83d3.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-183_90cb500dfa0c3a6a32ff2bb0d5535c8fe3b3c7262f364140b28b777510bf83d3.jpg new file mode 100644 index 00000000..e061adb9 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-183_90cb500dfa0c3a6a32ff2bb0d5535c8fe3b3c7262f364140b28b777510bf83d3.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-183_af72a8ceedec77b075fa09d673152e47602496925cf8ba4eb222697fcca9ee8a.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-183_af72a8ceedec77b075fa09d673152e47602496925cf8ba4eb222697fcca9ee8a.jpg new file mode 100644 index 00000000..44cd6e35 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-183_af72a8ceedec77b075fa09d673152e47602496925cf8ba4eb222697fcca9ee8a.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-183_cb5abec3bc5c488e07634a450eb3f095f34c2c27bc23b1d96a944444f8950bd7.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-183_cb5abec3bc5c488e07634a450eb3f095f34c2c27bc23b1d96a944444f8950bd7.jpg new file mode 100644 index 00000000..db5a41af Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-183_cb5abec3bc5c488e07634a450eb3f095f34c2c27bc23b1d96a944444f8950bd7.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-183_e27f5bf16daccb54cbd84ff0857a3ae7073645a0e61f12f5e1ebbc88e1410fb3.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-183_e27f5bf16daccb54cbd84ff0857a3ae7073645a0e61f12f5e1ebbc88e1410fb3.jpg new file mode 100644 index 00000000..a7314a4d Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-183_e27f5bf16daccb54cbd84ff0857a3ae7073645a0e61f12f5e1ebbc88e1410fb3.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-183_f0cc1bfae7af53e6399673f1fc689bc9864dcebd44503824fe64a7e7740b4285.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-183_f0cc1bfae7af53e6399673f1fc689bc9864dcebd44503824fe64a7e7740b4285.jpg new file mode 100644 index 00000000..11daf7c9 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-183_f0cc1bfae7af53e6399673f1fc689bc9864dcebd44503824fe64a7e7740b4285.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-184_1e613312678b365b8714801b2ddaf6007587afe2c8be7a1cc05545233be1dd84.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-184_1e613312678b365b8714801b2ddaf6007587afe2c8be7a1cc05545233be1dd84.jpg new file mode 100644 index 00000000..a9d262de Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-184_1e613312678b365b8714801b2ddaf6007587afe2c8be7a1cc05545233be1dd84.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-184_51a545a404e99c4652b0a1e39593eebef0c63b2d5ad6c9c4cf3aa8266c7aea5f.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-184_51a545a404e99c4652b0a1e39593eebef0c63b2d5ad6c9c4cf3aa8266c7aea5f.jpg new file mode 100644 index 00000000..ee4bad53 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-184_51a545a404e99c4652b0a1e39593eebef0c63b2d5ad6c9c4cf3aa8266c7aea5f.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-184_6e08711d996e9a7927f857e0622a112b7623600a9f0d27c73cdcad777d6ccead.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-184_6e08711d996e9a7927f857e0622a112b7623600a9f0d27c73cdcad777d6ccead.jpg new file mode 100644 index 00000000..4867467a Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-184_6e08711d996e9a7927f857e0622a112b7623600a9f0d27c73cdcad777d6ccead.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-184_b17d78e7fe3d87f5680ab379f462d7c43e647ca63050b1c6d2cf496392a53387.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-184_b17d78e7fe3d87f5680ab379f462d7c43e647ca63050b1c6d2cf496392a53387.jpg new file mode 100644 index 00000000..7f157065 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-184_b17d78e7fe3d87f5680ab379f462d7c43e647ca63050b1c6d2cf496392a53387.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-184_ef0cb6b6e699013838d0c6ff61f10f77064e562a90805b3297f71146b9edd98f.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-184_ef0cb6b6e699013838d0c6ff61f10f77064e562a90805b3297f71146b9edd98f.jpg new file mode 100644 index 00000000..522a6fba Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-184_ef0cb6b6e699013838d0c6ff61f10f77064e562a90805b3297f71146b9edd98f.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-189_84260831127a7e00b0c2bb171fb670b4b71fba7d0e81a47c0f8124c2340a1bb7.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-189_84260831127a7e00b0c2bb171fb670b4b71fba7d0e81a47c0f8124c2340a1bb7.jpg new file mode 100644 index 00000000..141ce874 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-189_84260831127a7e00b0c2bb171fb670b4b71fba7d0e81a47c0f8124c2340a1bb7.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-189_a7b8d194bb97459d43e9ae92ddc3cdbf3a91aae4a7de8e73681d7d082faf382c.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-189_a7b8d194bb97459d43e9ae92ddc3cdbf3a91aae4a7de8e73681d7d082faf382c.jpg new file mode 100644 index 00000000..047e8dfa Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-189_a7b8d194bb97459d43e9ae92ddc3cdbf3a91aae4a7de8e73681d7d082faf382c.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-193_10bb033449cc3bfe2d517ab735da70677fade3130792d06b41877470595fd75e.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-193_10bb033449cc3bfe2d517ab735da70677fade3130792d06b41877470595fd75e.jpg new file mode 100644 index 00000000..e2084fc1 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-193_10bb033449cc3bfe2d517ab735da70677fade3130792d06b41877470595fd75e.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-193_5ede687edc9f92a1bb5daafeb5e232b4e6cd35cd073abcef73deea8b37639889.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-193_5ede687edc9f92a1bb5daafeb5e232b4e6cd35cd073abcef73deea8b37639889.jpg new file mode 100644 index 00000000..ddad05b4 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-193_5ede687edc9f92a1bb5daafeb5e232b4e6cd35cd073abcef73deea8b37639889.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-194_541dbe32bcb2f59b43275a8e8d978c2afa1c2e322bb2c2bcdf5db183b5edb476.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-194_541dbe32bcb2f59b43275a8e8d978c2afa1c2e322bb2c2bcdf5db183b5edb476.jpg new file mode 100644 index 00000000..3e9e386d Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-194_541dbe32bcb2f59b43275a8e8d978c2afa1c2e322bb2c2bcdf5db183b5edb476.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-194_73fddd4e53f983503cb0c0745f599206e6b691f07bc07863d5f06ddcd032a31d.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-194_73fddd4e53f983503cb0c0745f599206e6b691f07bc07863d5f06ddcd032a31d.jpg new file mode 100644 index 00000000..a6a988b5 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-194_73fddd4e53f983503cb0c0745f599206e6b691f07bc07863d5f06ddcd032a31d.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-194_aec9666facbf979ab613276ea785f0c9551f3e30a1eed0052506e2f4acd01624.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-194_aec9666facbf979ab613276ea785f0c9551f3e30a1eed0052506e2f4acd01624.jpg new file mode 100644 index 00000000..47b5034e Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-194_aec9666facbf979ab613276ea785f0c9551f3e30a1eed0052506e2f4acd01624.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-194_b28b0a4f04e825d4ea42522300fb0f3a86b17849b0b57f31cb33437896d08015.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-194_b28b0a4f04e825d4ea42522300fb0f3a86b17849b0b57f31cb33437896d08015.jpg new file mode 100644 index 00000000..369f0d04 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-194_b28b0a4f04e825d4ea42522300fb0f3a86b17849b0b57f31cb33437896d08015.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-195_f990afdf4a36ce9eff3a1c174c46fa8ecd8ab62c509d30ea5316d3315eae399d.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-195_f990afdf4a36ce9eff3a1c174c46fa8ecd8ab62c509d30ea5316d3315eae399d.jpg new file mode 100644 index 00000000..61318eb9 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-195_f990afdf4a36ce9eff3a1c174c46fa8ecd8ab62c509d30ea5316d3315eae399d.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-199_12f7eebe561571e7d62ec3478907e328773403722387073b60a691fbee292c98.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-199_12f7eebe561571e7d62ec3478907e328773403722387073b60a691fbee292c98.jpg new file mode 100644 index 00000000..46e988ff Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-199_12f7eebe561571e7d62ec3478907e328773403722387073b60a691fbee292c98.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-199_5c45665ecc54338c69541cc77619c56316abdcabb2f8fc05d5fe5c9829742b3b.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-199_5c45665ecc54338c69541cc77619c56316abdcabb2f8fc05d5fe5c9829742b3b.jpg new file mode 100644 index 00000000..99b97c78 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-199_5c45665ecc54338c69541cc77619c56316abdcabb2f8fc05d5fe5c9829742b3b.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-202_f842c7076636ff0a6e2bc6a0b3434715c724da8cd7bdcce7888dfdd104dc7cd6.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-202_f842c7076636ff0a6e2bc6a0b3434715c724da8cd7bdcce7888dfdd104dc7cd6.jpg new file mode 100644 index 00000000..481dec3c Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-202_f842c7076636ff0a6e2bc6a0b3434715c724da8cd7bdcce7888dfdd104dc7cd6.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-211_140ff31cc3cca17cb716e90d7c4b1c07807c6ecb803da13d12dff410f9bf3653.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-211_140ff31cc3cca17cb716e90d7c4b1c07807c6ecb803da13d12dff410f9bf3653.jpg new file mode 100644 index 00000000..18ea1f36 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-211_140ff31cc3cca17cb716e90d7c4b1c07807c6ecb803da13d12dff410f9bf3653.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-211_d522e736fdc624d80febe6e102f6c0d2c625c9ce5d19f8c938392196c9df0512.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-211_d522e736fdc624d80febe6e102f6c0d2c625c9ce5d19f8c938392196c9df0512.jpg new file mode 100644 index 00000000..8d141b4f Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-211_d522e736fdc624d80febe6e102f6c0d2c625c9ce5d19f8c938392196c9df0512.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-211_f6f527087133ae99637f15113fdb765215c306bd394853b0ee9b6d72857eb6ab.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-211_f6f527087133ae99637f15113fdb765215c306bd394853b0ee9b6d72857eb6ab.jpg new file mode 100644 index 00000000..10364f13 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-211_f6f527087133ae99637f15113fdb765215c306bd394853b0ee9b6d72857eb6ab.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-212_08bd41c746f7507211ba096407f80a13332e4b357addbe27f1dd5ed54bb5b892.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-212_08bd41c746f7507211ba096407f80a13332e4b357addbe27f1dd5ed54bb5b892.jpg new file mode 100644 index 00000000..1026dea5 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-212_08bd41c746f7507211ba096407f80a13332e4b357addbe27f1dd5ed54bb5b892.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-214_84f0375eb529f455f1a20b1b54ea3b81df352b281933faeae97eff0c8a203b13.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-214_84f0375eb529f455f1a20b1b54ea3b81df352b281933faeae97eff0c8a203b13.jpg new file mode 100644 index 00000000..6c56f799 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-214_84f0375eb529f455f1a20b1b54ea3b81df352b281933faeae97eff0c8a203b13.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-214_cba42bcefc0692b4fd3af9292d58d9acc5648b3941ca8641ce95b6e0717baf67.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-214_cba42bcefc0692b4fd3af9292d58d9acc5648b3941ca8641ce95b6e0717baf67.jpg new file mode 100644 index 00000000..ecae39ef Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-214_cba42bcefc0692b4fd3af9292d58d9acc5648b3941ca8641ce95b6e0717baf67.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-224_d3f97c480bd4041473202d4659d62449eb2ea1ec28ed91e0b81457a636d5b0fe.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-224_d3f97c480bd4041473202d4659d62449eb2ea1ec28ed91e0b81457a636d5b0fe.jpg new file mode 100644 index 00000000..f6374b90 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-224_d3f97c480bd4041473202d4659d62449eb2ea1ec28ed91e0b81457a636d5b0fe.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-224_d50e629da5a4aab4ba02688e5bcb7389ea8117f30c1ae6a0f63a33af3fe7989b.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-224_d50e629da5a4aab4ba02688e5bcb7389ea8117f30c1ae6a0f63a33af3fe7989b.jpg new file mode 100644 index 00000000..f920bf73 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-224_d50e629da5a4aab4ba02688e5bcb7389ea8117f30c1ae6a0f63a33af3fe7989b.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-226_16a13ac4a8fc105325f5ebdddc31eba03c46d0800a8b12d94c1fce82ac61a664.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-226_16a13ac4a8fc105325f5ebdddc31eba03c46d0800a8b12d94c1fce82ac61a664.jpg new file mode 100644 index 00000000..64440b67 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-226_16a13ac4a8fc105325f5ebdddc31eba03c46d0800a8b12d94c1fce82ac61a664.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-226_3a14edbc8ebef7a7392a11d8bfc9b76b638d65f37499d7041582b152f24ba703.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-226_3a14edbc8ebef7a7392a11d8bfc9b76b638d65f37499d7041582b152f24ba703.jpg new file mode 100644 index 00000000..1004f177 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-226_3a14edbc8ebef7a7392a11d8bfc9b76b638d65f37499d7041582b152f24ba703.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-226_582df710b5a9b7aeb2bfcbd62b990a3a2a13844539013f445fcc2c0fd9869f9d.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-226_582df710b5a9b7aeb2bfcbd62b990a3a2a13844539013f445fcc2c0fd9869f9d.jpg new file mode 100644 index 00000000..8013dd56 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-226_582df710b5a9b7aeb2bfcbd62b990a3a2a13844539013f445fcc2c0fd9869f9d.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-226_5adebd49beebcf00462009619c385cc358e273e7de178552d2329f4356b5501b.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-226_5adebd49beebcf00462009619c385cc358e273e7de178552d2329f4356b5501b.jpg new file mode 100644 index 00000000..a15680df Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-226_5adebd49beebcf00462009619c385cc358e273e7de178552d2329f4356b5501b.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-226_633bda5ebb771d84c33f47f9fa8c7fff727a4e2bfd5b02eb6763ce2b41695456.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-226_633bda5ebb771d84c33f47f9fa8c7fff727a4e2bfd5b02eb6763ce2b41695456.jpg new file mode 100644 index 00000000..8a823790 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-226_633bda5ebb771d84c33f47f9fa8c7fff727a4e2bfd5b02eb6763ce2b41695456.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-226_dfbdea789d2e1c9f4bdbd43e148c094b69e86e03c635324c6bb7e7e3e1096c87.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-226_dfbdea789d2e1c9f4bdbd43e148c094b69e86e03c635324c6bb7e7e3e1096c87.jpg new file mode 100644 index 00000000..1958be63 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-226_dfbdea789d2e1c9f4bdbd43e148c094b69e86e03c635324c6bb7e7e3e1096c87.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-226_e7620479fb992b0c46f9b0ece6e298b1cfba44ec6083e310d28dc502e119f379.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-226_e7620479fb992b0c46f9b0ece6e298b1cfba44ec6083e310d28dc502e119f379.jpg new file mode 100644 index 00000000..8a72627e Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-226_e7620479fb992b0c46f9b0ece6e298b1cfba44ec6083e310d28dc502e119f379.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-226_ff49def1d14681d99bdf482e64aa83a0f91695fee01245334cd93f4db994ca91.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-226_ff49def1d14681d99bdf482e64aa83a0f91695fee01245334cd93f4db994ca91.jpg new file mode 100644 index 00000000..9b47b3c4 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-226_ff49def1d14681d99bdf482e64aa83a0f91695fee01245334cd93f4db994ca91.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-227_777be7ba0d1ba9c4bf05cc6667d2aad9a870eea4f8cc8c8844d2f98fb058febd.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-227_777be7ba0d1ba9c4bf05cc6667d2aad9a870eea4f8cc8c8844d2f98fb058febd.jpg new file mode 100644 index 00000000..a2d73c8a Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-227_777be7ba0d1ba9c4bf05cc6667d2aad9a870eea4f8cc8c8844d2f98fb058febd.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-227_84ed560e82e9c0f35106ef29bec36764ce231a20828abdf8bc2d28a8d4c54073.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-227_84ed560e82e9c0f35106ef29bec36764ce231a20828abdf8bc2d28a8d4c54073.jpg new file mode 100644 index 00000000..a7c0f27f Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-227_84ed560e82e9c0f35106ef29bec36764ce231a20828abdf8bc2d28a8d4c54073.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-227_f7f2ffdebc79856bf1c7bcc6ddcf9cd8dc3170f3bf43ead2eebbfbde1dbaa239.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-227_f7f2ffdebc79856bf1c7bcc6ddcf9cd8dc3170f3bf43ead2eebbfbde1dbaa239.jpg new file mode 100644 index 00000000..c20a396e Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-227_f7f2ffdebc79856bf1c7bcc6ddcf9cd8dc3170f3bf43ead2eebbfbde1dbaa239.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-228_07e84cb70d60d2b854966febc7b8be4293f2dbc4044c947bc7c43f32c4cdcfa4.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-228_07e84cb70d60d2b854966febc7b8be4293f2dbc4044c947bc7c43f32c4cdcfa4.jpg new file mode 100644 index 00000000..f1a2c640 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-228_07e84cb70d60d2b854966febc7b8be4293f2dbc4044c947bc7c43f32c4cdcfa4.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-228_1e2ac0cd1b1809ca36413c2511151f9b4394d326490f032b287404f608b40d8b.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-228_1e2ac0cd1b1809ca36413c2511151f9b4394d326490f032b287404f608b40d8b.jpg new file mode 100644 index 00000000..f717f4ad Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-228_1e2ac0cd1b1809ca36413c2511151f9b4394d326490f032b287404f608b40d8b.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-228_2713a3b7e25f23bdd3ceeb4d7913e2afa76b7b6f8b8a830c20edd44d7c4db4e2.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-228_2713a3b7e25f23bdd3ceeb4d7913e2afa76b7b6f8b8a830c20edd44d7c4db4e2.jpg new file mode 100644 index 00000000..d131498e Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-228_2713a3b7e25f23bdd3ceeb4d7913e2afa76b7b6f8b8a830c20edd44d7c4db4e2.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-228_46e02b6f424da44e314c3becb4548ba542de97358cb7486ee1448c77c77b9177.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-228_46e02b6f424da44e314c3becb4548ba542de97358cb7486ee1448c77c77b9177.jpg new file mode 100644 index 00000000..b3f3005f Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-228_46e02b6f424da44e314c3becb4548ba542de97358cb7486ee1448c77c77b9177.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-228_a1e1cd97b0bfcf2d90b11a6dd016274c4343f3c630e4504dbb1e15883fa90c7c.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-228_a1e1cd97b0bfcf2d90b11a6dd016274c4343f3c630e4504dbb1e15883fa90c7c.jpg new file mode 100644 index 00000000..5ed7df67 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-228_a1e1cd97b0bfcf2d90b11a6dd016274c4343f3c630e4504dbb1e15883fa90c7c.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-228_aec5f32ed5143fb105dd676f6df961374b9f1c2c87f84802fc422c7fc1b61acf.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-228_aec5f32ed5143fb105dd676f6df961374b9f1c2c87f84802fc422c7fc1b61acf.jpg new file mode 100644 index 00000000..347b44b0 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-228_aec5f32ed5143fb105dd676f6df961374b9f1c2c87f84802fc422c7fc1b61acf.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-228_c85a05b16dd83d5a2c58eff1cffc16794b290160ffc93be3125680e1ec8387e8.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-228_c85a05b16dd83d5a2c58eff1cffc16794b290160ffc93be3125680e1ec8387e8.jpg new file mode 100644 index 00000000..30d44c6d Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-228_c85a05b16dd83d5a2c58eff1cffc16794b290160ffc93be3125680e1ec8387e8.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-229_20eefb8d7a905d33fc3ad9a483c26d1493130ef27e81e8bb5fc3abf09f7dde4c.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-229_20eefb8d7a905d33fc3ad9a483c26d1493130ef27e81e8bb5fc3abf09f7dde4c.jpg new file mode 100644 index 00000000..083c80ce Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-229_20eefb8d7a905d33fc3ad9a483c26d1493130ef27e81e8bb5fc3abf09f7dde4c.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-229_36640847dfc171001b17e23a17ba234fa6e446f547bc1bc5a11cd17a4d18c82a.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-229_36640847dfc171001b17e23a17ba234fa6e446f547bc1bc5a11cd17a4d18c82a.jpg new file mode 100644 index 00000000..af5e6988 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-229_36640847dfc171001b17e23a17ba234fa6e446f547bc1bc5a11cd17a4d18c82a.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-229_88be4606006aed15e792b9b0ffb2c88f347502a13893886eac745ceff60db07a.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-229_88be4606006aed15e792b9b0ffb2c88f347502a13893886eac745ceff60db07a.jpg new file mode 100644 index 00000000..7758f628 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-229_88be4606006aed15e792b9b0ffb2c88f347502a13893886eac745ceff60db07a.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-229_fa5391dc977ff27b87b9e9231ef2529a73c6cac65775b45ba68ca62a953d3eb4.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-229_fa5391dc977ff27b87b9e9231ef2529a73c6cac65775b45ba68ca62a953d3eb4.jpg new file mode 100644 index 00000000..727fc779 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-229_fa5391dc977ff27b87b9e9231ef2529a73c6cac65775b45ba68ca62a953d3eb4.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-230_3444a2db834f6b580c523a79e97e1b80b9cfde9f3d3b0f7b2070873cbfa18f50.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-230_3444a2db834f6b580c523a79e97e1b80b9cfde9f3d3b0f7b2070873cbfa18f50.jpg new file mode 100644 index 00000000..41508b40 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-230_3444a2db834f6b580c523a79e97e1b80b9cfde9f3d3b0f7b2070873cbfa18f50.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-230_5969a6cddbc7884f5ecafb66cedb1d86da711737267497339ab73bf94ef9b153.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-230_5969a6cddbc7884f5ecafb66cedb1d86da711737267497339ab73bf94ef9b153.jpg new file mode 100644 index 00000000..ed28f3ab Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-230_5969a6cddbc7884f5ecafb66cedb1d86da711737267497339ab73bf94ef9b153.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-230_926109e042481cc7fa271686e8bcae0a07447fe9f699cf8d70b95cc18706a1b6.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-230_926109e042481cc7fa271686e8bcae0a07447fe9f699cf8d70b95cc18706a1b6.jpg new file mode 100644 index 00000000..1ac9bd59 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-230_926109e042481cc7fa271686e8bcae0a07447fe9f699cf8d70b95cc18706a1b6.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-230_93e316717cb7dbfafeddf4a26ebbeeeb8c68e355f199140220b817cd3a181272.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-230_93e316717cb7dbfafeddf4a26ebbeeeb8c68e355f199140220b817cd3a181272.jpg new file mode 100644 index 00000000..de9175f5 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-230_93e316717cb7dbfafeddf4a26ebbeeeb8c68e355f199140220b817cd3a181272.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-231_a0648648fd91b4dbb8ca8eb789dc62d520365b0f1cf2bb70c365c167fb31957e.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-231_a0648648fd91b4dbb8ca8eb789dc62d520365b0f1cf2bb70c365c167fb31957e.jpg new file mode 100644 index 00000000..83649c0b Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-231_a0648648fd91b4dbb8ca8eb789dc62d520365b0f1cf2bb70c365c167fb31957e.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-231_fb37979afda202c2a6af931de83d1e75b4445d641f3c7d41490d0357191a1eb5.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-231_fb37979afda202c2a6af931de83d1e75b4445d641f3c7d41490d0357191a1eb5.jpg new file mode 100644 index 00000000..f78b83b7 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-231_fb37979afda202c2a6af931de83d1e75b4445d641f3c7d41490d0357191a1eb5.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-232_2c96647c56e14b2cf26b23689bba993d9073980e8dd237ddb4988f7db0858fce.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-232_2c96647c56e14b2cf26b23689bba993d9073980e8dd237ddb4988f7db0858fce.jpg new file mode 100644 index 00000000..f0783cad Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-232_2c96647c56e14b2cf26b23689bba993d9073980e8dd237ddb4988f7db0858fce.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-232_5d318734b4c9b6725086ed9f8d3d8d35ad18f695a5f28cc7dc4660db83692739.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-232_5d318734b4c9b6725086ed9f8d3d8d35ad18f695a5f28cc7dc4660db83692739.jpg new file mode 100644 index 00000000..9157b4c1 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-232_5d318734b4c9b6725086ed9f8d3d8d35ad18f695a5f28cc7dc4660db83692739.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-232_8db278874ee22076baa3f9967ba0978f27958e99acfe72f1dfae12d562874bf1.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-232_8db278874ee22076baa3f9967ba0978f27958e99acfe72f1dfae12d562874bf1.jpg new file mode 100644 index 00000000..3410f9f0 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-232_8db278874ee22076baa3f9967ba0978f27958e99acfe72f1dfae12d562874bf1.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-232_f676c4a1c0181688c217af01df5091e586c93d17430e1f3ebd2ce56d27f1dd94.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-232_f676c4a1c0181688c217af01df5091e586c93d17430e1f3ebd2ce56d27f1dd94.jpg new file mode 100644 index 00000000..0b98df28 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-232_f676c4a1c0181688c217af01df5091e586c93d17430e1f3ebd2ce56d27f1dd94.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-233_06171aaa95c3dd4a1277d009bf0ecaa9862debba5b9b084fae4d1f1ce2b1e5e2.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-233_06171aaa95c3dd4a1277d009bf0ecaa9862debba5b9b084fae4d1f1ce2b1e5e2.jpg new file mode 100644 index 00000000..271a4b99 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-233_06171aaa95c3dd4a1277d009bf0ecaa9862debba5b9b084fae4d1f1ce2b1e5e2.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-233_2db98137c025cfbeda059fca9d807a6e8dbe66de7a948cf2b1d597c3c9c1d545.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-233_2db98137c025cfbeda059fca9d807a6e8dbe66de7a948cf2b1d597c3c9c1d545.jpg new file mode 100644 index 00000000..0510ae58 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-233_2db98137c025cfbeda059fca9d807a6e8dbe66de7a948cf2b1d597c3c9c1d545.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-233_2edb9d81f378d8c99f2fe5c849a1525f425f2180b2d991101711596ee0a4c31c.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-233_2edb9d81f378d8c99f2fe5c849a1525f425f2180b2d991101711596ee0a4c31c.jpg new file mode 100644 index 00000000..a0e004c1 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-233_2edb9d81f378d8c99f2fe5c849a1525f425f2180b2d991101711596ee0a4c31c.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-233_5be7ab683733ae53e184bc2d01003092b6343b08b31680597dcebb9084ad157c.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-233_5be7ab683733ae53e184bc2d01003092b6343b08b31680597dcebb9084ad157c.jpg new file mode 100644 index 00000000..41e52d17 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-233_5be7ab683733ae53e184bc2d01003092b6343b08b31680597dcebb9084ad157c.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-233_cbc859940b1ca71402a2d04189a0c970103ae9b8667073fe6c72089a282d26c1.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-233_cbc859940b1ca71402a2d04189a0c970103ae9b8667073fe6c72089a282d26c1.jpg new file mode 100644 index 00000000..a2c52efb Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-233_cbc859940b1ca71402a2d04189a0c970103ae9b8667073fe6c72089a282d26c1.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-233_eb97beddf18c03c7a5814c2a74c6b4766fa4a405c25d0d4cb8359faffe8f615e.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-233_eb97beddf18c03c7a5814c2a74c6b4766fa4a405c25d0d4cb8359faffe8f615e.jpg new file mode 100644 index 00000000..5b914cf3 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-233_eb97beddf18c03c7a5814c2a74c6b4766fa4a405c25d0d4cb8359faffe8f615e.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-234_0d7e602aadd37f60e6cd7bd3b47629498cf5c1ccf9cf7428406aeef0b07cfd9d.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-234_0d7e602aadd37f60e6cd7bd3b47629498cf5c1ccf9cf7428406aeef0b07cfd9d.jpg new file mode 100644 index 00000000..e6198f8e Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-234_0d7e602aadd37f60e6cd7bd3b47629498cf5c1ccf9cf7428406aeef0b07cfd9d.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-234_14c5892e7c930be362becba5c2a14ef4362c3cc3ae7e6feb7b0fec71cffc3a89.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-234_14c5892e7c930be362becba5c2a14ef4362c3cc3ae7e6feb7b0fec71cffc3a89.jpg new file mode 100644 index 00000000..91a05e15 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-234_14c5892e7c930be362becba5c2a14ef4362c3cc3ae7e6feb7b0fec71cffc3a89.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-234_17998e212720df3f9e9a6bc6245c9d16b579a94c76890b907ba0a8d4a7b25a6a.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-234_17998e212720df3f9e9a6bc6245c9d16b579a94c76890b907ba0a8d4a7b25a6a.jpg new file mode 100644 index 00000000..782328ff Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-234_17998e212720df3f9e9a6bc6245c9d16b579a94c76890b907ba0a8d4a7b25a6a.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-234_83e6a3198ef844aab377d1198d7156b0aae4798ae94d171ce3f9fa52ee71a758.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-234_83e6a3198ef844aab377d1198d7156b0aae4798ae94d171ce3f9fa52ee71a758.jpg new file mode 100644 index 00000000..33b58f6a Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-234_83e6a3198ef844aab377d1198d7156b0aae4798ae94d171ce3f9fa52ee71a758.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-235_3dac2c99faf614281a88c7ed292b254031091d47a2ced81ae71857ed9bc1547e.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-235_3dac2c99faf614281a88c7ed292b254031091d47a2ced81ae71857ed9bc1547e.jpg new file mode 100644 index 00000000..34533f0f Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-235_3dac2c99faf614281a88c7ed292b254031091d47a2ced81ae71857ed9bc1547e.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-235_5d1cf89ee2b64ffa0b3872c14e30b10e6f935a3753c540e876df4904ab771fcc.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-235_5d1cf89ee2b64ffa0b3872c14e30b10e6f935a3753c540e876df4904ab771fcc.jpg new file mode 100644 index 00000000..f23a06d4 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-235_5d1cf89ee2b64ffa0b3872c14e30b10e6f935a3753c540e876df4904ab771fcc.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-235_757ccffcf662a7c0c70bc579c4b917eabcf6ccf409819201f370d0981544535b.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-235_757ccffcf662a7c0c70bc579c4b917eabcf6ccf409819201f370d0981544535b.jpg new file mode 100644 index 00000000..441d2889 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-235_757ccffcf662a7c0c70bc579c4b917eabcf6ccf409819201f370d0981544535b.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-235_ec8f52c6fb97f38ff8acb97f1e9eddd3103a77021331210c0e8803e7621240f4.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-235_ec8f52c6fb97f38ff8acb97f1e9eddd3103a77021331210c0e8803e7621240f4.jpg new file mode 100644 index 00000000..d9f6d118 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-235_ec8f52c6fb97f38ff8acb97f1e9eddd3103a77021331210c0e8803e7621240f4.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-235_f1ae1f8d7f21beea8daaad1d7c0d578f53483bc584a941ef761ee092a34a85fc.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-235_f1ae1f8d7f21beea8daaad1d7c0d578f53483bc584a941ef761ee092a34a85fc.jpg new file mode 100644 index 00000000..6aba82d0 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-235_f1ae1f8d7f21beea8daaad1d7c0d578f53483bc584a941ef761ee092a34a85fc.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-236_3e176164eb024717349db4ee8fb9ada86923aac4148a1992b0d4b2eaadaf9f6e.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-236_3e176164eb024717349db4ee8fb9ada86923aac4148a1992b0d4b2eaadaf9f6e.jpg new file mode 100644 index 00000000..3ec2a227 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-236_3e176164eb024717349db4ee8fb9ada86923aac4148a1992b0d4b2eaadaf9f6e.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-236_86733af60ebeecb42bc803112ff855c762e69d2098a200e9b15b592516abab29.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-236_86733af60ebeecb42bc803112ff855c762e69d2098a200e9b15b592516abab29.jpg new file mode 100644 index 00000000..cff5beac Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-236_86733af60ebeecb42bc803112ff855c762e69d2098a200e9b15b592516abab29.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-236_96e6f34e07830c785e9ad0085d34499f323fe671d83b39bc64ed36dc4cda5e6f.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-236_96e6f34e07830c785e9ad0085d34499f323fe671d83b39bc64ed36dc4cda5e6f.jpg new file mode 100644 index 00000000..64e04c53 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-236_96e6f34e07830c785e9ad0085d34499f323fe671d83b39bc64ed36dc4cda5e6f.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-236_ad1a1c41e9ac73e5a96aab96cd700750fdc9de771c41fabb1e0c2f9c394ad4c9.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-236_ad1a1c41e9ac73e5a96aab96cd700750fdc9de771c41fabb1e0c2f9c394ad4c9.jpg new file mode 100644 index 00000000..0977956f Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-236_ad1a1c41e9ac73e5a96aab96cd700750fdc9de771c41fabb1e0c2f9c394ad4c9.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-236_cc412c02f39d62baadedd16129446e3c387eea0ac193c30e5cd1b4ffa59f9680.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-236_cc412c02f39d62baadedd16129446e3c387eea0ac193c30e5cd1b4ffa59f9680.jpg new file mode 100644 index 00000000..7373fb24 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-236_cc412c02f39d62baadedd16129446e3c387eea0ac193c30e5cd1b4ffa59f9680.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-236_cf07c3c0a033672d1b6cd8d20a8b44753595ed9739cc92acb40b3b3460da2bd7.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-236_cf07c3c0a033672d1b6cd8d20a8b44753595ed9739cc92acb40b3b3460da2bd7.jpg new file mode 100644 index 00000000..47d12f3e Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-236_cf07c3c0a033672d1b6cd8d20a8b44753595ed9739cc92acb40b3b3460da2bd7.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-237_0151bfe14797b5d421689ff8edf243564ef813f820cf1ca89de10cf4469a4dc6.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-237_0151bfe14797b5d421689ff8edf243564ef813f820cf1ca89de10cf4469a4dc6.jpg new file mode 100644 index 00000000..5dfa7ac2 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-237_0151bfe14797b5d421689ff8edf243564ef813f820cf1ca89de10cf4469a4dc6.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-237_127e55a86d9128c859f7dc548883e8af6fa94aa060de2e1f74181337e1ac9e05.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-237_127e55a86d9128c859f7dc548883e8af6fa94aa060de2e1f74181337e1ac9e05.jpg new file mode 100644 index 00000000..9b4ea7b8 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-237_127e55a86d9128c859f7dc548883e8af6fa94aa060de2e1f74181337e1ac9e05.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-237_1714ee63284ee4f5bf60b244130c792155e3a5c5cbddfd1b620f29403e1b9187.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-237_1714ee63284ee4f5bf60b244130c792155e3a5c5cbddfd1b620f29403e1b9187.jpg new file mode 100644 index 00000000..f1b0b706 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-237_1714ee63284ee4f5bf60b244130c792155e3a5c5cbddfd1b620f29403e1b9187.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-237_256b3ffd55cbdb26893c6cb0bfff63ddab26e27a92c7039f0dab41d22d547b6a.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-237_256b3ffd55cbdb26893c6cb0bfff63ddab26e27a92c7039f0dab41d22d547b6a.jpg new file mode 100644 index 00000000..fe806e3b Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-237_256b3ffd55cbdb26893c6cb0bfff63ddab26e27a92c7039f0dab41d22d547b6a.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-237_365d81b2034057ce4b9fcd79ec5533028a1e454eb3ff791c04f44afc976540cd.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-237_365d81b2034057ce4b9fcd79ec5533028a1e454eb3ff791c04f44afc976540cd.jpg new file mode 100644 index 00000000..32bc366e Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-237_365d81b2034057ce4b9fcd79ec5533028a1e454eb3ff791c04f44afc976540cd.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-237_41ae6f4e627f6623d48e8847aa186619b5f980bacc4b064eaab4f8e8838876b5.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-237_41ae6f4e627f6623d48e8847aa186619b5f980bacc4b064eaab4f8e8838876b5.jpg new file mode 100644 index 00000000..c267a755 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-237_41ae6f4e627f6623d48e8847aa186619b5f980bacc4b064eaab4f8e8838876b5.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-237_47a688a1af346e0ae6106570e7314916dec600dc172ac4091412dcfd0f6b42c1.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-237_47a688a1af346e0ae6106570e7314916dec600dc172ac4091412dcfd0f6b42c1.jpg new file mode 100644 index 00000000..f1153c83 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-237_47a688a1af346e0ae6106570e7314916dec600dc172ac4091412dcfd0f6b42c1.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-237_4f5c28435d1addb56394f957e9dedd6ab3e8429d3296b72a429235e0c5f9fcfc.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-237_4f5c28435d1addb56394f957e9dedd6ab3e8429d3296b72a429235e0c5f9fcfc.jpg new file mode 100644 index 00000000..c0381e9b Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-237_4f5c28435d1addb56394f957e9dedd6ab3e8429d3296b72a429235e0c5f9fcfc.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-237_de98e6344bf223c649a7a652154e933395b75ecbb9886f8a9bd541e32852ac2b.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-237_de98e6344bf223c649a7a652154e933395b75ecbb9886f8a9bd541e32852ac2b.jpg new file mode 100644 index 00000000..c2a31d88 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-237_de98e6344bf223c649a7a652154e933395b75ecbb9886f8a9bd541e32852ac2b.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-238_2929ace56a8914baf9678a1ec7ef29192c4a282f35fe1f0c4072b1eca0e66abb.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-238_2929ace56a8914baf9678a1ec7ef29192c4a282f35fe1f0c4072b1eca0e66abb.jpg new file mode 100644 index 00000000..8e28da6d Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-238_2929ace56a8914baf9678a1ec7ef29192c4a282f35fe1f0c4072b1eca0e66abb.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-238_34fa191477f19914442b0dfdc27dd080ffe9febc151375668652e03dfabdc0f4.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-238_34fa191477f19914442b0dfdc27dd080ffe9febc151375668652e03dfabdc0f4.jpg new file mode 100644 index 00000000..52f6bc0b Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-238_34fa191477f19914442b0dfdc27dd080ffe9febc151375668652e03dfabdc0f4.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-238_47131ef184d1ca8c9a5511d95533f9bf625bd0e2aec5dad6b6fc47817e8c9317.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-238_47131ef184d1ca8c9a5511d95533f9bf625bd0e2aec5dad6b6fc47817e8c9317.jpg new file mode 100644 index 00000000..4d45fca0 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-238_47131ef184d1ca8c9a5511d95533f9bf625bd0e2aec5dad6b6fc47817e8c9317.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-238_484b847a3422bf91697dd14f4aa86bbef765eddf9d1f71deb266ba3270de5078.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-238_484b847a3422bf91697dd14f4aa86bbef765eddf9d1f71deb266ba3270de5078.jpg new file mode 100644 index 00000000..0c27de21 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-238_484b847a3422bf91697dd14f4aa86bbef765eddf9d1f71deb266ba3270de5078.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-238_53437961aec76685655272518d0e95e9cd03c33708367e5d40a7699fa986b4f9.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-238_53437961aec76685655272518d0e95e9cd03c33708367e5d40a7699fa986b4f9.jpg new file mode 100644 index 00000000..bfd14a4b Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-238_53437961aec76685655272518d0e95e9cd03c33708367e5d40a7699fa986b4f9.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-238_5a3c98ef1e856868e94c3244ab3aff75ce5a9e04e2515587636d7b4824a4253d.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-238_5a3c98ef1e856868e94c3244ab3aff75ce5a9e04e2515587636d7b4824a4253d.jpg new file mode 100644 index 00000000..4b871c45 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-238_5a3c98ef1e856868e94c3244ab3aff75ce5a9e04e2515587636d7b4824a4253d.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-238_7066fa3a0c31e72c855497d955ea0b6b7a9bdbfba514c9c79bf4127df932bf0d.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-238_7066fa3a0c31e72c855497d955ea0b6b7a9bdbfba514c9c79bf4127df932bf0d.jpg new file mode 100644 index 00000000..670fb777 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-238_7066fa3a0c31e72c855497d955ea0b6b7a9bdbfba514c9c79bf4127df932bf0d.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-238_8ec6666a46f22231e69960d01428d5c1c4dfe2608f8a080578c17fe3ab1b5e68.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-238_8ec6666a46f22231e69960d01428d5c1c4dfe2608f8a080578c17fe3ab1b5e68.jpg new file mode 100644 index 00000000..5e3d5a29 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-238_8ec6666a46f22231e69960d01428d5c1c4dfe2608f8a080578c17fe3ab1b5e68.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-238_9ceac4bbcaeaee5170df7a69e90a249a911f07ba0c5eb189f2d1538a5597d810.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-238_9ceac4bbcaeaee5170df7a69e90a249a911f07ba0c5eb189f2d1538a5597d810.jpg new file mode 100644 index 00000000..6122b4cc Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-238_9ceac4bbcaeaee5170df7a69e90a249a911f07ba0c5eb189f2d1538a5597d810.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-238_b3351a34cf4172921e2ab97fdfb7afc149ba7273b1930868bbe66bd233ca8738.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-238_b3351a34cf4172921e2ab97fdfb7afc149ba7273b1930868bbe66bd233ca8738.jpg new file mode 100644 index 00000000..57dd781b Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-238_b3351a34cf4172921e2ab97fdfb7afc149ba7273b1930868bbe66bd233ca8738.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-238_ef2e80349a665605881c6be4d214287c0e7f687c5ab07711dfc9fe444e26d63a.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-238_ef2e80349a665605881c6be4d214287c0e7f687c5ab07711dfc9fe444e26d63a.jpg new file mode 100644 index 00000000..7a9b9912 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-238_ef2e80349a665605881c6be4d214287c0e7f687c5ab07711dfc9fe444e26d63a.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-239_721989fa7357b6b72ecaeaebddebb0128276efba9f7a4543b7d623242e9a92bc.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-239_721989fa7357b6b72ecaeaebddebb0128276efba9f7a4543b7d623242e9a92bc.jpg new file mode 100644 index 00000000..4f3b1705 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-239_721989fa7357b6b72ecaeaebddebb0128276efba9f7a4543b7d623242e9a92bc.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-239_730a9bf11f2ae712e9b4c59e40981d5c0fd6dc729e03d904b12cd4c05ad56849.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-239_730a9bf11f2ae712e9b4c59e40981d5c0fd6dc729e03d904b12cd4c05ad56849.jpg new file mode 100644 index 00000000..be66e1b7 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-239_730a9bf11f2ae712e9b4c59e40981d5c0fd6dc729e03d904b12cd4c05ad56849.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-239_93383093905136692433cac7ad565226d319cb2af9e70efe7dd19d1f8cb45416.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-239_93383093905136692433cac7ad565226d319cb2af9e70efe7dd19d1f8cb45416.jpg new file mode 100644 index 00000000..29726e80 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-239_93383093905136692433cac7ad565226d319cb2af9e70efe7dd19d1f8cb45416.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-239_b461bf8d11a76df1a61b264945dfdda9d1b1629f150bfde541e71b86e2ef2a05.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-239_b461bf8d11a76df1a61b264945dfdda9d1b1629f150bfde541e71b86e2ef2a05.jpg new file mode 100644 index 00000000..e78c502d Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-239_b461bf8d11a76df1a61b264945dfdda9d1b1629f150bfde541e71b86e2ef2a05.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-239_d6143e757865886375993d0df9edb4d89f3e1fdd44e0f0c8600522bebc0a18c5.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-239_d6143e757865886375993d0df9edb4d89f3e1fdd44e0f0c8600522bebc0a18c5.jpg new file mode 100644 index 00000000..894a9200 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-239_d6143e757865886375993d0df9edb4d89f3e1fdd44e0f0c8600522bebc0a18c5.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-239_e11bad904fc9ffc341b2a05314512e50f9f8d1866c4539caf25f7111227e21e8.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-239_e11bad904fc9ffc341b2a05314512e50f9f8d1866c4539caf25f7111227e21e8.jpg new file mode 100644 index 00000000..322ad21f Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-239_e11bad904fc9ffc341b2a05314512e50f9f8d1866c4539caf25f7111227e21e8.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-240_46051c1f6b2d71d9e830d6531a4751719bae05c4a2e5f05c93813669466d08dd.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-240_46051c1f6b2d71d9e830d6531a4751719bae05c4a2e5f05c93813669466d08dd.jpg new file mode 100644 index 00000000..7404a3aa Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-240_46051c1f6b2d71d9e830d6531a4751719bae05c4a2e5f05c93813669466d08dd.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-240_4e758ac5a2e2c22861c729dcf2d2428c575c6b32ca61a1f55c46b5cc074e18e6.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-240_4e758ac5a2e2c22861c729dcf2d2428c575c6b32ca61a1f55c46b5cc074e18e6.jpg new file mode 100644 index 00000000..91d290f9 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-240_4e758ac5a2e2c22861c729dcf2d2428c575c6b32ca61a1f55c46b5cc074e18e6.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-240_6f79cc3d474c5ea5ce957cdc4b12da1d8999e806729df6903fdf43ffd476ab22.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-240_6f79cc3d474c5ea5ce957cdc4b12da1d8999e806729df6903fdf43ffd476ab22.jpg new file mode 100644 index 00000000..0b37cafb Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-240_6f79cc3d474c5ea5ce957cdc4b12da1d8999e806729df6903fdf43ffd476ab22.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-240_81b4977a02537dcfbf5b6ae46b8c1b81eaf0c576cd5677d301709cea8f06e7aa.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-240_81b4977a02537dcfbf5b6ae46b8c1b81eaf0c576cd5677d301709cea8f06e7aa.jpg new file mode 100644 index 00000000..bd3f22d8 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-240_81b4977a02537dcfbf5b6ae46b8c1b81eaf0c576cd5677d301709cea8f06e7aa.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-240_adaa523e527c654f43568acd53867b6979536ea4aed0c49eb01c3feb8769016e.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-240_adaa523e527c654f43568acd53867b6979536ea4aed0c49eb01c3feb8769016e.jpg new file mode 100644 index 00000000..487f0844 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-240_adaa523e527c654f43568acd53867b6979536ea4aed0c49eb01c3feb8769016e.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-240_c1b9715149749904a7de1de4091eb2a2cd2c3eae00e0e838335680c6f084d028.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-240_c1b9715149749904a7de1de4091eb2a2cd2c3eae00e0e838335680c6f084d028.jpg new file mode 100644 index 00000000..1fea93e1 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-240_c1b9715149749904a7de1de4091eb2a2cd2c3eae00e0e838335680c6f084d028.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-240_f7e0aee10533a2580588519f62cb44d7ae10cac0730cda9ee8cbb2f473753139.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-240_f7e0aee10533a2580588519f62cb44d7ae10cac0730cda9ee8cbb2f473753139.jpg new file mode 100644 index 00000000..ebdc4658 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-240_f7e0aee10533a2580588519f62cb44d7ae10cac0730cda9ee8cbb2f473753139.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-240_f93b0f23621044a54df7af54d67cdecd949cc57a5f9ca51e2f207cd96274a712.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-240_f93b0f23621044a54df7af54d67cdecd949cc57a5f9ca51e2f207cd96274a712.jpg new file mode 100644 index 00000000..022167c5 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-240_f93b0f23621044a54df7af54d67cdecd949cc57a5f9ca51e2f207cd96274a712.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-241_0a34f1bf0a8514400d784192c04a85fd9a24a0898b8e528b2fbf2137c4a96671.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-241_0a34f1bf0a8514400d784192c04a85fd9a24a0898b8e528b2fbf2137c4a96671.jpg new file mode 100644 index 00000000..8ea50746 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-241_0a34f1bf0a8514400d784192c04a85fd9a24a0898b8e528b2fbf2137c4a96671.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-241_110157f0fa11685006d541ea60a00d3da5f6edc0319f8d537fbc3024b25fd2a7.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-241_110157f0fa11685006d541ea60a00d3da5f6edc0319f8d537fbc3024b25fd2a7.jpg new file mode 100644 index 00000000..89ef355d Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-241_110157f0fa11685006d541ea60a00d3da5f6edc0319f8d537fbc3024b25fd2a7.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-241_1cf6571e1cb5b94d7a7a2c6dcfc524fcc971e8662f94bfce6456c48a58386b57.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-241_1cf6571e1cb5b94d7a7a2c6dcfc524fcc971e8662f94bfce6456c48a58386b57.jpg new file mode 100644 index 00000000..4d2a90e6 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-241_1cf6571e1cb5b94d7a7a2c6dcfc524fcc971e8662f94bfce6456c48a58386b57.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-241_269ae5eeb414c18ad9dc344dee68d7bba6e75fa83beeab1c202a4d1b08b1c0aa.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-241_269ae5eeb414c18ad9dc344dee68d7bba6e75fa83beeab1c202a4d1b08b1c0aa.jpg new file mode 100644 index 00000000..ae9ac799 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-241_269ae5eeb414c18ad9dc344dee68d7bba6e75fa83beeab1c202a4d1b08b1c0aa.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-241_3da19f0bc1c8f021d14a4a83110930e0bf0770738d97792fc67abc460fd2c7e6.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-241_3da19f0bc1c8f021d14a4a83110930e0bf0770738d97792fc67abc460fd2c7e6.jpg new file mode 100644 index 00000000..7a6e4e52 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-241_3da19f0bc1c8f021d14a4a83110930e0bf0770738d97792fc67abc460fd2c7e6.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-241_7fcdde02f63e440db3cfb836fb89047aa9826bc26c986bc1f3ff3b920b2dd128.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-241_7fcdde02f63e440db3cfb836fb89047aa9826bc26c986bc1f3ff3b920b2dd128.jpg new file mode 100644 index 00000000..cdfb948b Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-241_7fcdde02f63e440db3cfb836fb89047aa9826bc26c986bc1f3ff3b920b2dd128.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-241_adec604f8bb2a1efe94803567d4df586043d803063ab7956ddcdd39b9df7576c.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-241_adec604f8bb2a1efe94803567d4df586043d803063ab7956ddcdd39b9df7576c.jpg new file mode 100644 index 00000000..116f8c20 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-241_adec604f8bb2a1efe94803567d4df586043d803063ab7956ddcdd39b9df7576c.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-241_e0b381e134869f8da8d766037c85b37ed78cfbdffe848c5f2d7410f751e4f2cd.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-241_e0b381e134869f8da8d766037c85b37ed78cfbdffe848c5f2d7410f751e4f2cd.jpg new file mode 100644 index 00000000..dfebd016 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-241_e0b381e134869f8da8d766037c85b37ed78cfbdffe848c5f2d7410f751e4f2cd.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-241_ee38928631b708bb6db12397d5c52410e8126e1bfeda4e79d8fcc82cc36100ae.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-241_ee38928631b708bb6db12397d5c52410e8126e1bfeda4e79d8fcc82cc36100ae.jpg new file mode 100644 index 00000000..f849425b Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-241_ee38928631b708bb6db12397d5c52410e8126e1bfeda4e79d8fcc82cc36100ae.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-242_0b136361c440c251ef0bbf104736118d44edd50e9b7363bd568953174eb70d19.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-242_0b136361c440c251ef0bbf104736118d44edd50e9b7363bd568953174eb70d19.jpg new file mode 100644 index 00000000..a8c35cd0 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-242_0b136361c440c251ef0bbf104736118d44edd50e9b7363bd568953174eb70d19.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-242_14ead239667494454b1008d735e5c2155b40fbc0cd7008f7e8155744ced6ed6d.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-242_14ead239667494454b1008d735e5c2155b40fbc0cd7008f7e8155744ced6ed6d.jpg new file mode 100644 index 00000000..bf862e8e Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-242_14ead239667494454b1008d735e5c2155b40fbc0cd7008f7e8155744ced6ed6d.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-242_61912e33be203c048b1573709022c81b9e81fa1484af048a4bcb5de265ba197c.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-242_61912e33be203c048b1573709022c81b9e81fa1484af048a4bcb5de265ba197c.jpg new file mode 100644 index 00000000..ba3cbf3a Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-242_61912e33be203c048b1573709022c81b9e81fa1484af048a4bcb5de265ba197c.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-242_96ea112d8d955c833b648cecc6bd290ca5e3aebfa9e0fbf1ce1969a9eff3f96d.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-242_96ea112d8d955c833b648cecc6bd290ca5e3aebfa9e0fbf1ce1969a9eff3f96d.jpg new file mode 100644 index 00000000..5d24b161 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-242_96ea112d8d955c833b648cecc6bd290ca5e3aebfa9e0fbf1ce1969a9eff3f96d.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-242_9ad5818f54817dd04952d68a0a5d181323d38602ea7fb3750efdcce2628ca515.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-242_9ad5818f54817dd04952d68a0a5d181323d38602ea7fb3750efdcce2628ca515.jpg new file mode 100644 index 00000000..cf57c41f Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-242_9ad5818f54817dd04952d68a0a5d181323d38602ea7fb3750efdcce2628ca515.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-242_9dfb4734ee7b81a84c18d6de8f64a7a06e1305d2084adef92c044d15cea576c8.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-242_9dfb4734ee7b81a84c18d6de8f64a7a06e1305d2084adef92c044d15cea576c8.jpg new file mode 100644 index 00000000..071552c0 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-242_9dfb4734ee7b81a84c18d6de8f64a7a06e1305d2084adef92c044d15cea576c8.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-242_ba31a0be908f80a4e4b68955611ad2ca3a17419b6bdb3ee6bde02934c7b2b62c.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-242_ba31a0be908f80a4e4b68955611ad2ca3a17419b6bdb3ee6bde02934c7b2b62c.jpg new file mode 100644 index 00000000..b0f1e6e8 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-242_ba31a0be908f80a4e4b68955611ad2ca3a17419b6bdb3ee6bde02934c7b2b62c.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-242_f042f7a046babbdad5f8e3244aa8b82b53b57852471ad71fadd568e6f85dd679.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-242_f042f7a046babbdad5f8e3244aa8b82b53b57852471ad71fadd568e6f85dd679.jpg new file mode 100644 index 00000000..4216e21c Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-242_f042f7a046babbdad5f8e3244aa8b82b53b57852471ad71fadd568e6f85dd679.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-243_0d1aad3f1e96df0196ccecd2925465ab70565b0a0683f015db3e3b065eb8cccf.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-243_0d1aad3f1e96df0196ccecd2925465ab70565b0a0683f015db3e3b065eb8cccf.jpg new file mode 100644 index 00000000..24183150 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-243_0d1aad3f1e96df0196ccecd2925465ab70565b0a0683f015db3e3b065eb8cccf.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-243_39abfc9587a8bb151ef91938b19babd7d6e344b58b3d318597ee198d053568ea.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-243_39abfc9587a8bb151ef91938b19babd7d6e344b58b3d318597ee198d053568ea.jpg new file mode 100644 index 00000000..9b891df4 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-243_39abfc9587a8bb151ef91938b19babd7d6e344b58b3d318597ee198d053568ea.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-243_3debf4d96caf3d912348958a012795c815e4ce973b3f85d79d644fa3da156b72.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-243_3debf4d96caf3d912348958a012795c815e4ce973b3f85d79d644fa3da156b72.jpg new file mode 100644 index 00000000..241b0df9 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-243_3debf4d96caf3d912348958a012795c815e4ce973b3f85d79d644fa3da156b72.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-243_8757dd83466b9f34e0dd96f3887924b3f7d6a104d1432f07cd6da6cbd89026bf.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-243_8757dd83466b9f34e0dd96f3887924b3f7d6a104d1432f07cd6da6cbd89026bf.jpg new file mode 100644 index 00000000..74cb0b04 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-243_8757dd83466b9f34e0dd96f3887924b3f7d6a104d1432f07cd6da6cbd89026bf.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-243_a0beedf63f6703496389af9ac3ad1c6ab293d41753f90debc58b0c67c3e7e9c7.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-243_a0beedf63f6703496389af9ac3ad1c6ab293d41753f90debc58b0c67c3e7e9c7.jpg new file mode 100644 index 00000000..3aed2073 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-243_a0beedf63f6703496389af9ac3ad1c6ab293d41753f90debc58b0c67c3e7e9c7.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-243_bac5ba7bbb5ddcabc17edd14a7d39f69de4690f143ec914cd0757dc521fd577e.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-243_bac5ba7bbb5ddcabc17edd14a7d39f69de4690f143ec914cd0757dc521fd577e.jpg new file mode 100644 index 00000000..6104b2cc Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-243_bac5ba7bbb5ddcabc17edd14a7d39f69de4690f143ec914cd0757dc521fd577e.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-243_c74d34eb141fd6ef3f4f2c7fa683812224206a46675a0fdd1420f117dcc36eb5.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-243_c74d34eb141fd6ef3f4f2c7fa683812224206a46675a0fdd1420f117dcc36eb5.jpg new file mode 100644 index 00000000..bb6fe361 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-243_c74d34eb141fd6ef3f4f2c7fa683812224206a46675a0fdd1420f117dcc36eb5.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-243_d2ad1d2574c7423e72c0f0905fba08c5fb06265a4f07ea9c0945ccd8ffd373fc.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-243_d2ad1d2574c7423e72c0f0905fba08c5fb06265a4f07ea9c0945ccd8ffd373fc.jpg new file mode 100644 index 00000000..61185553 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-243_d2ad1d2574c7423e72c0f0905fba08c5fb06265a4f07ea9c0945ccd8ffd373fc.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-244_1a2390a16092f3c97dd84755dfdaf0b2339571b9960fa7e0585e9a05b10115ec.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-244_1a2390a16092f3c97dd84755dfdaf0b2339571b9960fa7e0585e9a05b10115ec.jpg new file mode 100644 index 00000000..ba9b73cc Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-244_1a2390a16092f3c97dd84755dfdaf0b2339571b9960fa7e0585e9a05b10115ec.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-244_d9ac0dc694c1810d346e5ea3f2070d655aebbab5705bbffca95618f5b4dfed6e.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-244_d9ac0dc694c1810d346e5ea3f2070d655aebbab5705bbffca95618f5b4dfed6e.jpg new file mode 100644 index 00000000..2ca3f1c2 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-244_d9ac0dc694c1810d346e5ea3f2070d655aebbab5705bbffca95618f5b4dfed6e.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-244_ec1969495033fa9e8cb3ea91f0b678a7235a20d01d950b7b80d10802b21107a6.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-244_ec1969495033fa9e8cb3ea91f0b678a7235a20d01d950b7b80d10802b21107a6.jpg new file mode 100644 index 00000000..d8c67f0b Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-244_ec1969495033fa9e8cb3ea91f0b678a7235a20d01d950b7b80d10802b21107a6.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-244_f1f5d37a36ca2fc55478e934bf6e160ef1a07e613cbe1c37834ccf2d9d16d3be.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-244_f1f5d37a36ca2fc55478e934bf6e160ef1a07e613cbe1c37834ccf2d9d16d3be.jpg new file mode 100644 index 00000000..76b2da9d Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-244_f1f5d37a36ca2fc55478e934bf6e160ef1a07e613cbe1c37834ccf2d9d16d3be.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-245_14ec40d9756d0da546b025bacb09a57e1670b02f7d029ef39cfc3dbc3b173182.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-245_14ec40d9756d0da546b025bacb09a57e1670b02f7d029ef39cfc3dbc3b173182.jpg new file mode 100644 index 00000000..6d8c8a9c Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-245_14ec40d9756d0da546b025bacb09a57e1670b02f7d029ef39cfc3dbc3b173182.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-245_398a84418aab8bbf5379a46a62a88a6cc086272aec73620336a96370fc178070.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-245_398a84418aab8bbf5379a46a62a88a6cc086272aec73620336a96370fc178070.jpg new file mode 100644 index 00000000..2e763547 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-245_398a84418aab8bbf5379a46a62a88a6cc086272aec73620336a96370fc178070.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-245_3e0712169331359972afd9ae4c664d75bf7b8a3aeffa30e4c4c2b2293a22a44d.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-245_3e0712169331359972afd9ae4c664d75bf7b8a3aeffa30e4c4c2b2293a22a44d.jpg new file mode 100644 index 00000000..7cacb143 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-245_3e0712169331359972afd9ae4c664d75bf7b8a3aeffa30e4c4c2b2293a22a44d.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-245_593ce52b73faa3008445c626989f0f76d9328132d1d2139903bed5879557ae19.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-245_593ce52b73faa3008445c626989f0f76d9328132d1d2139903bed5879557ae19.jpg new file mode 100644 index 00000000..57d0b54b Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-245_593ce52b73faa3008445c626989f0f76d9328132d1d2139903bed5879557ae19.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-245_6a99c0d42a23514435118b647518b0001d6a9cd2d0ff94ad79011ef80e28b1d8.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-245_6a99c0d42a23514435118b647518b0001d6a9cd2d0ff94ad79011ef80e28b1d8.jpg new file mode 100644 index 00000000..a3c3ced2 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-245_6a99c0d42a23514435118b647518b0001d6a9cd2d0ff94ad79011ef80e28b1d8.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-245_6c9c91ad840a62fdb785e7387309340061e61d4e87b0460509dc4f40ccccd4d2.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-245_6c9c91ad840a62fdb785e7387309340061e61d4e87b0460509dc4f40ccccd4d2.jpg new file mode 100644 index 00000000..58de1fc8 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-245_6c9c91ad840a62fdb785e7387309340061e61d4e87b0460509dc4f40ccccd4d2.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-245_6d8d0dea154be1a3ef8fc681036a2adb8ad24a7798ae1586311128642557d04f.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-245_6d8d0dea154be1a3ef8fc681036a2adb8ad24a7798ae1586311128642557d04f.jpg new file mode 100644 index 00000000..1b91a657 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-245_6d8d0dea154be1a3ef8fc681036a2adb8ad24a7798ae1586311128642557d04f.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-246_01e32f317b9b6b2ba3bcc3cfbd55a19f60193c67102dd1ed41fd441141a01b82.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-246_01e32f317b9b6b2ba3bcc3cfbd55a19f60193c67102dd1ed41fd441141a01b82.jpg new file mode 100644 index 00000000..041880bb Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-246_01e32f317b9b6b2ba3bcc3cfbd55a19f60193c67102dd1ed41fd441141a01b82.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-246_10b4172b6fc7555133a3818faac56e5f019ccf55f76286be4301f8e4aeaf2cea.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-246_10b4172b6fc7555133a3818faac56e5f019ccf55f76286be4301f8e4aeaf2cea.jpg new file mode 100644 index 00000000..06707cac Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-246_10b4172b6fc7555133a3818faac56e5f019ccf55f76286be4301f8e4aeaf2cea.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-246_3d639102c6195171fee01e7b31a77ec401ffc63250dc6b1dda12382f1a5742a3.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-246_3d639102c6195171fee01e7b31a77ec401ffc63250dc6b1dda12382f1a5742a3.jpg new file mode 100644 index 00000000..25ecddba Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-246_3d639102c6195171fee01e7b31a77ec401ffc63250dc6b1dda12382f1a5742a3.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-246_63971f08f723b4316d7b68ecd9b7b55edabb7c88c6af21e1c13de4ce85f67440.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-246_63971f08f723b4316d7b68ecd9b7b55edabb7c88c6af21e1c13de4ce85f67440.jpg new file mode 100644 index 00000000..0af0a59a Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-246_63971f08f723b4316d7b68ecd9b7b55edabb7c88c6af21e1c13de4ce85f67440.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-246_fa897abb29009b16f93aac9936f5bd71da91f36ab7ed7c410704a183fe0e07fe.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-246_fa897abb29009b16f93aac9936f5bd71da91f36ab7ed7c410704a183fe0e07fe.jpg new file mode 100644 index 00000000..f30115dd Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-246_fa897abb29009b16f93aac9936f5bd71da91f36ab7ed7c410704a183fe0e07fe.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-247_70f1f32e35fa7973c290612732c7bf46a5ee8ef0377b73c3fd1aa0e0c2902e2e.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-247_70f1f32e35fa7973c290612732c7bf46a5ee8ef0377b73c3fd1aa0e0c2902e2e.jpg new file mode 100644 index 00000000..cbd8b4fa Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-247_70f1f32e35fa7973c290612732c7bf46a5ee8ef0377b73c3fd1aa0e0c2902e2e.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-249_3f9e32405df2d2b01fdfea8ada17cb5da5eac29edffda36e2334661a7e2471f0.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-249_3f9e32405df2d2b01fdfea8ada17cb5da5eac29edffda36e2334661a7e2471f0.jpg new file mode 100644 index 00000000..174c1199 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-249_3f9e32405df2d2b01fdfea8ada17cb5da5eac29edffda36e2334661a7e2471f0.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-249_d3c5bcf917dbf3b5a414ee804b2d748c6b7e8394f299e67ad0324d8e3b08a892.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-249_d3c5bcf917dbf3b5a414ee804b2d748c6b7e8394f299e67ad0324d8e3b08a892.jpg new file mode 100644 index 00000000..23f27c7e Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-249_d3c5bcf917dbf3b5a414ee804b2d748c6b7e8394f299e67ad0324d8e3b08a892.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-253_db480c75fce5f96e1dbfed47488c8c906a5eef22a224812170b7e76e7f8781cf.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-253_db480c75fce5f96e1dbfed47488c8c906a5eef22a224812170b7e76e7f8781cf.jpg new file mode 100644 index 00000000..6fb3e3ad Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-253_db480c75fce5f96e1dbfed47488c8c906a5eef22a224812170b7e76e7f8781cf.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-254_2e672c690088e3364d45cf17f4cdc448c6433ee4430c374ff2dacbbefa78b879.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-254_2e672c690088e3364d45cf17f4cdc448c6433ee4430c374ff2dacbbefa78b879.jpg new file mode 100644 index 00000000..8380c0d1 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-254_2e672c690088e3364d45cf17f4cdc448c6433ee4430c374ff2dacbbefa78b879.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-254_6fdf32bc60588c0294ed79c5603e839baca9bf90d054fa8a9af49fe0ac1ebc28.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-254_6fdf32bc60588c0294ed79c5603e839baca9bf90d054fa8a9af49fe0ac1ebc28.jpg new file mode 100644 index 00000000..0487f370 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-254_6fdf32bc60588c0294ed79c5603e839baca9bf90d054fa8a9af49fe0ac1ebc28.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-256_2572046d1ffa7d56e2e8e91b984a1d947a2754bf9adf8ec5944fcb30ec2bf7b8.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-256_2572046d1ffa7d56e2e8e91b984a1d947a2754bf9adf8ec5944fcb30ec2bf7b8.jpg new file mode 100644 index 00000000..f8b65e2a Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-256_2572046d1ffa7d56e2e8e91b984a1d947a2754bf9adf8ec5944fcb30ec2bf7b8.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-257_f98d44293cc5d88711333ea4595eeff1559472545a4ca148739aa2774327d039.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-257_f98d44293cc5d88711333ea4595eeff1559472545a4ca148739aa2774327d039.jpg new file mode 100644 index 00000000..1b20586c Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-257_f98d44293cc5d88711333ea4595eeff1559472545a4ca148739aa2774327d039.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-258_6149c8454fd607f897736cf6abc5897499499b951607d2363172abf36cf252b6.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-258_6149c8454fd607f897736cf6abc5897499499b951607d2363172abf36cf252b6.jpg new file mode 100644 index 00000000..89527c6a Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-258_6149c8454fd607f897736cf6abc5897499499b951607d2363172abf36cf252b6.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-258_7c446772b211117dabb50a77d44f16218ee1ad8f5a730ecb5fad035102ce8e55.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-258_7c446772b211117dabb50a77d44f16218ee1ad8f5a730ecb5fad035102ce8e55.jpg new file mode 100644 index 00000000..b8c18159 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-258_7c446772b211117dabb50a77d44f16218ee1ad8f5a730ecb5fad035102ce8e55.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-260_3115adacbb056a4c2f3c30be94f5c9db6bebf2cbbec0f7ed62db8d1087a57621.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-260_3115adacbb056a4c2f3c30be94f5c9db6bebf2cbbec0f7ed62db8d1087a57621.jpg new file mode 100644 index 00000000..11583d31 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-260_3115adacbb056a4c2f3c30be94f5c9db6bebf2cbbec0f7ed62db8d1087a57621.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-260_4eecc9641dcdbda4d93c6e11065a00a69393bceed6269e7334243db4925ee4d1.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-260_4eecc9641dcdbda4d93c6e11065a00a69393bceed6269e7334243db4925ee4d1.jpg new file mode 100644 index 00000000..c1b1deaa Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-260_4eecc9641dcdbda4d93c6e11065a00a69393bceed6269e7334243db4925ee4d1.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-261_4d973e0c0184d6194810b9ee905c78f3ace5b5cc15e29344ff0d61cb7f215798.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-261_4d973e0c0184d6194810b9ee905c78f3ace5b5cc15e29344ff0d61cb7f215798.jpg new file mode 100644 index 00000000..6817ebd9 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-261_4d973e0c0184d6194810b9ee905c78f3ace5b5cc15e29344ff0d61cb7f215798.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-261_6df8063e48ad99ead0dbb23c781c567da2d9b64534aa06296e01aac3cbbc0f19.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-261_6df8063e48ad99ead0dbb23c781c567da2d9b64534aa06296e01aac3cbbc0f19.jpg new file mode 100644 index 00000000..14b4f8de Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-261_6df8063e48ad99ead0dbb23c781c567da2d9b64534aa06296e01aac3cbbc0f19.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-261_cc3ee57c300e0947ee715773e0809cee784085859019c1dfb2a1ca6e698475d4.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-261_cc3ee57c300e0947ee715773e0809cee784085859019c1dfb2a1ca6e698475d4.jpg new file mode 100644 index 00000000..42c08146 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-261_cc3ee57c300e0947ee715773e0809cee784085859019c1dfb2a1ca6e698475d4.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-262_3527177768779bc7234199ebd27b665d568c9c1203da8f687ee43c28e87163a3.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-262_3527177768779bc7234199ebd27b665d568c9c1203da8f687ee43c28e87163a3.jpg new file mode 100644 index 00000000..72219f88 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-262_3527177768779bc7234199ebd27b665d568c9c1203da8f687ee43c28e87163a3.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-262_e2b93f428530a44b0ebf101d60bea166c5260d1c5e96cfc1ff465711e00a4358.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-262_e2b93f428530a44b0ebf101d60bea166c5260d1c5e96cfc1ff465711e00a4358.jpg new file mode 100644 index 00000000..829968ea Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-262_e2b93f428530a44b0ebf101d60bea166c5260d1c5e96cfc1ff465711e00a4358.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-263_19fb1def9e9e7b944da557f0e7c32d6881599b5e5bfac3dd4f199f94754f914f.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-263_19fb1def9e9e7b944da557f0e7c32d6881599b5e5bfac3dd4f199f94754f914f.jpg new file mode 100644 index 00000000..d779f5a6 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-263_19fb1def9e9e7b944da557f0e7c32d6881599b5e5bfac3dd4f199f94754f914f.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-263_d1ea0bb441bbc3b3e06105865aedf14aed92914a14528a874d28c9b1c4975921.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-263_d1ea0bb441bbc3b3e06105865aedf14aed92914a14528a874d28c9b1c4975921.jpg new file mode 100644 index 00000000..84e49b7a Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-263_d1ea0bb441bbc3b3e06105865aedf14aed92914a14528a874d28c9b1c4975921.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-265_ae2577081a0453232d314e3e76ebc33ae47b573593485ea9a762f417f2325e6e.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-265_ae2577081a0453232d314e3e76ebc33ae47b573593485ea9a762f417f2325e6e.jpg new file mode 100644 index 00000000..408d423b Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-265_ae2577081a0453232d314e3e76ebc33ae47b573593485ea9a762f417f2325e6e.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-266_213a5b4be103f0187ce95dc555c86763cc18abb123b585e278b400e88019a702.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-266_213a5b4be103f0187ce95dc555c86763cc18abb123b585e278b400e88019a702.jpg new file mode 100644 index 00000000..42b4fa8b Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-266_213a5b4be103f0187ce95dc555c86763cc18abb123b585e278b400e88019a702.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-267_1a8a20a03e7526711223c8610625af89629abdd8c32e590abac736d782a3d0dd.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-267_1a8a20a03e7526711223c8610625af89629abdd8c32e590abac736d782a3d0dd.jpg new file mode 100644 index 00000000..82e80efc Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-267_1a8a20a03e7526711223c8610625af89629abdd8c32e590abac736d782a3d0dd.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-267_c9b85d57d0a4ad29860fc3e0d0c3496eb4b2e725b511cb41eec8ad484cddc19d.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-267_c9b85d57d0a4ad29860fc3e0d0c3496eb4b2e725b511cb41eec8ad484cddc19d.jpg new file mode 100644 index 00000000..3122c74c Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-267_c9b85d57d0a4ad29860fc3e0d0c3496eb4b2e725b511cb41eec8ad484cddc19d.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-267_d6972c7a8d5040755c194ec44e380bf043744fc6c09fe95899db52b9fca72f2c.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-267_d6972c7a8d5040755c194ec44e380bf043744fc6c09fe95899db52b9fca72f2c.jpg new file mode 100644 index 00000000..2f77185d Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-267_d6972c7a8d5040755c194ec44e380bf043744fc6c09fe95899db52b9fca72f2c.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-267_df9de82578b1ef6ea73b43144a904a2f7b582970a78f07ac599abb10f68bbce5.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-267_df9de82578b1ef6ea73b43144a904a2f7b582970a78f07ac599abb10f68bbce5.jpg new file mode 100644 index 00000000..4ec83127 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-267_df9de82578b1ef6ea73b43144a904a2f7b582970a78f07ac599abb10f68bbce5.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-268_590079151921a2507123ba41fa63e7074174725f82bf2eeedca6e00b20f0540e.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-268_590079151921a2507123ba41fa63e7074174725f82bf2eeedca6e00b20f0540e.jpg new file mode 100644 index 00000000..a50dacb8 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-268_590079151921a2507123ba41fa63e7074174725f82bf2eeedca6e00b20f0540e.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-268_7e7c82c2bfbf29bded97364cb639c59a8d9dc85fed92d9f217be8c9accde866f.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-268_7e7c82c2bfbf29bded97364cb639c59a8d9dc85fed92d9f217be8c9accde866f.jpg new file mode 100644 index 00000000..5074c050 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-268_7e7c82c2bfbf29bded97364cb639c59a8d9dc85fed92d9f217be8c9accde866f.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-268_c35ac9f1197a5b9c3805b2c5f3b4d8af2209f7108088cbe5769ff8bcef56c54c.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-268_c35ac9f1197a5b9c3805b2c5f3b4d8af2209f7108088cbe5769ff8bcef56c54c.jpg new file mode 100644 index 00000000..b95db4e7 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-268_c35ac9f1197a5b9c3805b2c5f3b4d8af2209f7108088cbe5769ff8bcef56c54c.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-272_5949a753c352e813a047c22a1133b32ba93b52a239a0ff8f4187048b3de542a3.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-272_5949a753c352e813a047c22a1133b32ba93b52a239a0ff8f4187048b3de542a3.jpg new file mode 100644 index 00000000..edc0b538 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-272_5949a753c352e813a047c22a1133b32ba93b52a239a0ff8f4187048b3de542a3.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-272_d5e3706c62e3aa04f486ba4a7a967b3d5e5628b289d4d42a4b0811eca2c234c2.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-272_d5e3706c62e3aa04f486ba4a7a967b3d5e5628b289d4d42a4b0811eca2c234c2.jpg new file mode 100644 index 00000000..9e43e4fa Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-272_d5e3706c62e3aa04f486ba4a7a967b3d5e5628b289d4d42a4b0811eca2c234c2.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-272_f209ae3ddae5cfe9803a92d90c08f695f6a1d13acb0fa457b0e0a4cda02079e4.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-272_f209ae3ddae5cfe9803a92d90c08f695f6a1d13acb0fa457b0e0a4cda02079e4.jpg new file mode 100644 index 00000000..c12a12de Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-272_f209ae3ddae5cfe9803a92d90c08f695f6a1d13acb0fa457b0e0a4cda02079e4.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-273_3b2638e93ca693401baaa5436c9378c37c12a063be49af751c41b15ed9477243.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-273_3b2638e93ca693401baaa5436c9378c37c12a063be49af751c41b15ed9477243.jpg new file mode 100644 index 00000000..bd02587b Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-273_3b2638e93ca693401baaa5436c9378c37c12a063be49af751c41b15ed9477243.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-273_7b1aff624c5a8a846dc5831bd90b11f242091b8d6bd8418cb69e2644e2e0123d.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-273_7b1aff624c5a8a846dc5831bd90b11f242091b8d6bd8418cb69e2644e2e0123d.jpg new file mode 100644 index 00000000..118b8a83 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-273_7b1aff624c5a8a846dc5831bd90b11f242091b8d6bd8418cb69e2644e2e0123d.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-273_907240ec3c67cdaad19bb886e17751c07226b053ebaccf48056a32defe284128.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-273_907240ec3c67cdaad19bb886e17751c07226b053ebaccf48056a32defe284128.jpg new file mode 100644 index 00000000..1e7dfc88 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-273_907240ec3c67cdaad19bb886e17751c07226b053ebaccf48056a32defe284128.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-273_949697ecb66898432a5ec8607c8ad817468f384080b1769dcf89c8974402d2e8.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-273_949697ecb66898432a5ec8607c8ad817468f384080b1769dcf89c8974402d2e8.jpg new file mode 100644 index 00000000..6072b81d Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-273_949697ecb66898432a5ec8607c8ad817468f384080b1769dcf89c8974402d2e8.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-273_a143d68fb20704fb894dd76419381215a619f8943c650b54af42b09fdaf5d1e5.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-273_a143d68fb20704fb894dd76419381215a619f8943c650b54af42b09fdaf5d1e5.jpg new file mode 100644 index 00000000..a024a940 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-273_a143d68fb20704fb894dd76419381215a619f8943c650b54af42b09fdaf5d1e5.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-273_b2e0c3bf872562ea76364bd2a9b16ddfdf5d9c9e4daddaf390a5c1eb36413810.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-273_b2e0c3bf872562ea76364bd2a9b16ddfdf5d9c9e4daddaf390a5c1eb36413810.jpg new file mode 100644 index 00000000..133bcb58 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-273_b2e0c3bf872562ea76364bd2a9b16ddfdf5d9c9e4daddaf390a5c1eb36413810.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-274_39af2daf93374d3ac1b1550f4d07b82d159306b0400395a2d54c07847827635f.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-274_39af2daf93374d3ac1b1550f4d07b82d159306b0400395a2d54c07847827635f.jpg new file mode 100644 index 00000000..981cc5bb Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-274_39af2daf93374d3ac1b1550f4d07b82d159306b0400395a2d54c07847827635f.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-274_6b4b2340bbaf0114d7bee2defef827db84aa11d2f83cbce3a5ff61c801107c11.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-274_6b4b2340bbaf0114d7bee2defef827db84aa11d2f83cbce3a5ff61c801107c11.jpg new file mode 100644 index 00000000..ac556a09 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-274_6b4b2340bbaf0114d7bee2defef827db84aa11d2f83cbce3a5ff61c801107c11.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-275_1ce6caed0ef09524810af2a7044a34e981deee8fe9398edca14c9180ef74ee3d.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-275_1ce6caed0ef09524810af2a7044a34e981deee8fe9398edca14c9180ef74ee3d.jpg new file mode 100644 index 00000000..a290c60e Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-275_1ce6caed0ef09524810af2a7044a34e981deee8fe9398edca14c9180ef74ee3d.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-275_d5b5d6eee6767c228ab22cb5f3542c10b1fcdc5f89469c3cb4d42d382e899c18.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-275_d5b5d6eee6767c228ab22cb5f3542c10b1fcdc5f89469c3cb4d42d382e899c18.jpg new file mode 100644 index 00000000..dd5b4dfd Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-275_d5b5d6eee6767c228ab22cb5f3542c10b1fcdc5f89469c3cb4d42d382e899c18.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-276_3c1cd65038d00e90078076b655fe98057341973c73b7258d782449fa767b93ba.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-276_3c1cd65038d00e90078076b655fe98057341973c73b7258d782449fa767b93ba.jpg new file mode 100644 index 00000000..d35cfa49 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-276_3c1cd65038d00e90078076b655fe98057341973c73b7258d782449fa767b93ba.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-276_96aa823ed3046061a38a5a30e8139f78b5da2b106d32732cd7ace5586b15d21b.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-276_96aa823ed3046061a38a5a30e8139f78b5da2b106d32732cd7ace5586b15d21b.jpg new file mode 100644 index 00000000..d7a52a00 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-276_96aa823ed3046061a38a5a30e8139f78b5da2b106d32732cd7ace5586b15d21b.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-276_c7ab3d4cc5719cac4a4024396050df6854b04d52e2c04ab212045fa69aaa90ee.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-276_c7ab3d4cc5719cac4a4024396050df6854b04d52e2c04ab212045fa69aaa90ee.jpg new file mode 100644 index 00000000..09417673 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-276_c7ab3d4cc5719cac4a4024396050df6854b04d52e2c04ab212045fa69aaa90ee.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-276_eb9ac1d92edcc82f8da0d1707a9a3a6186438fd7c9b43c89f1c682fdb61b3b74.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-276_eb9ac1d92edcc82f8da0d1707a9a3a6186438fd7c9b43c89f1c682fdb61b3b74.jpg new file mode 100644 index 00000000..d0f288b5 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-276_eb9ac1d92edcc82f8da0d1707a9a3a6186438fd7c9b43c89f1c682fdb61b3b74.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-276_f154e0b80d8984ce4a6a095052c0ca6e7675c0b0db9905461ddcf00c4139a76f.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-276_f154e0b80d8984ce4a6a095052c0ca6e7675c0b0db9905461ddcf00c4139a76f.jpg new file mode 100644 index 00000000..f5a78eb7 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-276_f154e0b80d8984ce4a6a095052c0ca6e7675c0b0db9905461ddcf00c4139a76f.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-277_65547797836e77bb9e4da8221a6464c7c6b87c2f69202b1640f838d8e1f1ea59.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-277_65547797836e77bb9e4da8221a6464c7c6b87c2f69202b1640f838d8e1f1ea59.jpg new file mode 100644 index 00000000..8682e15f Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-277_65547797836e77bb9e4da8221a6464c7c6b87c2f69202b1640f838d8e1f1ea59.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-277_cf543ee197453de87f9c5afe6a2c2275d21bb8c4170217887a6acdcfc1886d21.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-277_cf543ee197453de87f9c5afe6a2c2275d21bb8c4170217887a6acdcfc1886d21.jpg new file mode 100644 index 00000000..f3b36ae0 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-277_cf543ee197453de87f9c5afe6a2c2275d21bb8c4170217887a6acdcfc1886d21.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-282_17eed4943b4a8d5527faaa3dbb2e793aa04bfd70b29f8e411c668388d1ec9704.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-282_17eed4943b4a8d5527faaa3dbb2e793aa04bfd70b29f8e411c668388d1ec9704.jpg new file mode 100644 index 00000000..68ade58a Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-282_17eed4943b4a8d5527faaa3dbb2e793aa04bfd70b29f8e411c668388d1ec9704.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-282_e5215dab3e9dd4b8bddcde700d214a36a48354589b93f8bb24f822d6d80209ab.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-282_e5215dab3e9dd4b8bddcde700d214a36a48354589b93f8bb24f822d6d80209ab.jpg new file mode 100644 index 00000000..e0d61d3d Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-282_e5215dab3e9dd4b8bddcde700d214a36a48354589b93f8bb24f822d6d80209ab.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-282_ed744834110f6339ae0ecc73988b9a7fefb76ffdf15d35adad7c5c0605dd71c6.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-282_ed744834110f6339ae0ecc73988b9a7fefb76ffdf15d35adad7c5c0605dd71c6.jpg new file mode 100644 index 00000000..911c39f0 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-282_ed744834110f6339ae0ecc73988b9a7fefb76ffdf15d35adad7c5c0605dd71c6.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-282_f21bdf66cc461cb920fe8906b29327254b9f530b6f789bd81029c7f4a5810bc4.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-282_f21bdf66cc461cb920fe8906b29327254b9f530b6f789bd81029c7f4a5810bc4.jpg new file mode 100644 index 00000000..2662a21d Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-282_f21bdf66cc461cb920fe8906b29327254b9f530b6f789bd81029c7f4a5810bc4.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-283_15cd530ffa59c8f7f0ed992353290b8277790c845ea40da245b2fe3a36941154.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-283_15cd530ffa59c8f7f0ed992353290b8277790c845ea40da245b2fe3a36941154.jpg new file mode 100644 index 00000000..26f84b95 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-283_15cd530ffa59c8f7f0ed992353290b8277790c845ea40da245b2fe3a36941154.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-283_58d6a60cad255e3e60a318febcd386d916221a0913a4e4ecf2f19093d9e164ab.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-283_58d6a60cad255e3e60a318febcd386d916221a0913a4e4ecf2f19093d9e164ab.jpg new file mode 100644 index 00000000..a53f7d8f Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-283_58d6a60cad255e3e60a318febcd386d916221a0913a4e4ecf2f19093d9e164ab.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-283_64544c69d1fa05f3096639e27aa62776ab04c1cdff87a68786886f8a42376114.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-283_64544c69d1fa05f3096639e27aa62776ab04c1cdff87a68786886f8a42376114.jpg new file mode 100644 index 00000000..5f8cfc54 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-283_64544c69d1fa05f3096639e27aa62776ab04c1cdff87a68786886f8a42376114.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-283_9987ec25bb00010f1e8d742a9d72f9c05b51035dc75083b4e518a208203d91ef.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-283_9987ec25bb00010f1e8d742a9d72f9c05b51035dc75083b4e518a208203d91ef.jpg new file mode 100644 index 00000000..886085bd Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-283_9987ec25bb00010f1e8d742a9d72f9c05b51035dc75083b4e518a208203d91ef.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-283_c9173dd8e9094c8e6427387a8088f6d02d7ae3cdf53880c70c9a6cb0849bdb97.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-283_c9173dd8e9094c8e6427387a8088f6d02d7ae3cdf53880c70c9a6cb0849bdb97.jpg new file mode 100644 index 00000000..e0b48481 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-283_c9173dd8e9094c8e6427387a8088f6d02d7ae3cdf53880c70c9a6cb0849bdb97.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-283_d1800d95cbfe0ce2b039df10dc3a97873ad2ec0e982336361c9067ace9d05d8f.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-283_d1800d95cbfe0ce2b039df10dc3a97873ad2ec0e982336361c9067ace9d05d8f.jpg new file mode 100644 index 00000000..1dd29513 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-283_d1800d95cbfe0ce2b039df10dc3a97873ad2ec0e982336361c9067ace9d05d8f.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-284_198bea51130ec98c661e5b17507142a37261fcd5cb7f2c9226435d2b93ab7902.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-284_198bea51130ec98c661e5b17507142a37261fcd5cb7f2c9226435d2b93ab7902.jpg new file mode 100644 index 00000000..59889482 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-284_198bea51130ec98c661e5b17507142a37261fcd5cb7f2c9226435d2b93ab7902.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-284_4ed77850972789a9f739a5548f8aca926b151f5dcfd51ec064f1e1a2abd95055.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-284_4ed77850972789a9f739a5548f8aca926b151f5dcfd51ec064f1e1a2abd95055.jpg new file mode 100644 index 00000000..57c4ea7b Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-284_4ed77850972789a9f739a5548f8aca926b151f5dcfd51ec064f1e1a2abd95055.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-284_5a38278e7217ffeaf70a50542c9e958b18ee1a143b1f9c8c51f6518295669f32.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-284_5a38278e7217ffeaf70a50542c9e958b18ee1a143b1f9c8c51f6518295669f32.jpg new file mode 100644 index 00000000..e46c3843 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-284_5a38278e7217ffeaf70a50542c9e958b18ee1a143b1f9c8c51f6518295669f32.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-284_6d23a40d59c259cda08486b6733228e429cff90fdc84baebc6bda63b00bcc24d.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-284_6d23a40d59c259cda08486b6733228e429cff90fdc84baebc6bda63b00bcc24d.jpg new file mode 100644 index 00000000..44134459 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-284_6d23a40d59c259cda08486b6733228e429cff90fdc84baebc6bda63b00bcc24d.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-284_7d00f2de4856cfe7e65156b328d9b68782b7b7ecc81f2d9aa5c86c720849cd5c.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-284_7d00f2de4856cfe7e65156b328d9b68782b7b7ecc81f2d9aa5c86c720849cd5c.jpg new file mode 100644 index 00000000..181ec5ee Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-284_7d00f2de4856cfe7e65156b328d9b68782b7b7ecc81f2d9aa5c86c720849cd5c.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-284_8fcde97c828ea51e69a04005b48691dff97db6a75f49fd33b45cce4cf6cf30b9.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-284_8fcde97c828ea51e69a04005b48691dff97db6a75f49fd33b45cce4cf6cf30b9.jpg new file mode 100644 index 00000000..7dec1edb Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-284_8fcde97c828ea51e69a04005b48691dff97db6a75f49fd33b45cce4cf6cf30b9.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-284_b8b40c78e5699d2e24a2879e565d4a732952f30840b2f9549b5ddf364bf8e71e.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-284_b8b40c78e5699d2e24a2879e565d4a732952f30840b2f9549b5ddf364bf8e71e.jpg new file mode 100644 index 00000000..2a075fa4 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-284_b8b40c78e5699d2e24a2879e565d4a732952f30840b2f9549b5ddf364bf8e71e.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-284_e619e24bc630acd95e0eed39908b9c4f9d6fcf9e968bcc343f32d3e4ad93b111.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-284_e619e24bc630acd95e0eed39908b9c4f9d6fcf9e968bcc343f32d3e4ad93b111.jpg new file mode 100644 index 00000000..0f3a9b8a Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-284_e619e24bc630acd95e0eed39908b9c4f9d6fcf9e968bcc343f32d3e4ad93b111.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-285_20346ceff0f06d1d833858884bf1c02cac04e1a684458be44d40dde1ee36f701.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-285_20346ceff0f06d1d833858884bf1c02cac04e1a684458be44d40dde1ee36f701.jpg new file mode 100644 index 00000000..d23f4f2c Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-285_20346ceff0f06d1d833858884bf1c02cac04e1a684458be44d40dde1ee36f701.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-285_49624b113dc31339592203adf204f82187776d2f87ca8abc07ff7bb7b4d20e6b.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-285_49624b113dc31339592203adf204f82187776d2f87ca8abc07ff7bb7b4d20e6b.jpg new file mode 100644 index 00000000..74a9abb2 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-285_49624b113dc31339592203adf204f82187776d2f87ca8abc07ff7bb7b4d20e6b.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-285_5715e03135c43f17905641bcd84fd459752a7e7ff1704b26dc3c744b7e649a55.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-285_5715e03135c43f17905641bcd84fd459752a7e7ff1704b26dc3c744b7e649a55.jpg new file mode 100644 index 00000000..e079facf Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-285_5715e03135c43f17905641bcd84fd459752a7e7ff1704b26dc3c744b7e649a55.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-285_7cbf4b02421994f75d31a8a0d0459058a0d7046d97a5fd01a3e12bf84f445896.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-285_7cbf4b02421994f75d31a8a0d0459058a0d7046d97a5fd01a3e12bf84f445896.jpg new file mode 100644 index 00000000..a692715d Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-285_7cbf4b02421994f75d31a8a0d0459058a0d7046d97a5fd01a3e12bf84f445896.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-285_abdce93c8e20f9d2d0df34fc2279dc6a2e167fc04881435b3557422872788616.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-285_abdce93c8e20f9d2d0df34fc2279dc6a2e167fc04881435b3557422872788616.jpg new file mode 100644 index 00000000..4da3898f Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-285_abdce93c8e20f9d2d0df34fc2279dc6a2e167fc04881435b3557422872788616.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-285_dcdfffffe8ec5294e6d84ccfc8b2c6c107591dd2d69d750b2ebcfbf1dff6845e.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-285_dcdfffffe8ec5294e6d84ccfc8b2c6c107591dd2d69d750b2ebcfbf1dff6845e.jpg new file mode 100644 index 00000000..534851ad Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-285_dcdfffffe8ec5294e6d84ccfc8b2c6c107591dd2d69d750b2ebcfbf1dff6845e.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-285_e630fa979e6080a2a899db9dee7ff52b872635bb12e9d22c390d09ef678cc6aa.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-285_e630fa979e6080a2a899db9dee7ff52b872635bb12e9d22c390d09ef678cc6aa.jpg new file mode 100644 index 00000000..1df9ab8c Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-285_e630fa979e6080a2a899db9dee7ff52b872635bb12e9d22c390d09ef678cc6aa.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-286_5acb2e91965840a93a26fa93f7b5d46c85971c503a404c4399b5e60d80dbc0de.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-286_5acb2e91965840a93a26fa93f7b5d46c85971c503a404c4399b5e60d80dbc0de.jpg new file mode 100644 index 00000000..dd8abe53 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-286_5acb2e91965840a93a26fa93f7b5d46c85971c503a404c4399b5e60d80dbc0de.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-286_8fce9b057749439d96e80ac1d2acf28593a7e24ca3fddca795ad63ab1077f277.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-286_8fce9b057749439d96e80ac1d2acf28593a7e24ca3fddca795ad63ab1077f277.jpg new file mode 100644 index 00000000..457fe280 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-286_8fce9b057749439d96e80ac1d2acf28593a7e24ca3fddca795ad63ab1077f277.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-286_a276b0f6dff4c711653f2aed5c9bba5980bce302f801f17627b247f14083a57d.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-286_a276b0f6dff4c711653f2aed5c9bba5980bce302f801f17627b247f14083a57d.jpg new file mode 100644 index 00000000..b31df370 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-286_a276b0f6dff4c711653f2aed5c9bba5980bce302f801f17627b247f14083a57d.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-286_cabd228d7cbd08ca7a0b47b906a7db440486e584763cfe25dd4f8fbfb44b7fea.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-286_cabd228d7cbd08ca7a0b47b906a7db440486e584763cfe25dd4f8fbfb44b7fea.jpg new file mode 100644 index 00000000..5acfc1a1 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-286_cabd228d7cbd08ca7a0b47b906a7db440486e584763cfe25dd4f8fbfb44b7fea.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-286_f25c9bbad6853b65deaebd6ab8d87e278d27c858f6576bf241e67bd035d7b235.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-286_f25c9bbad6853b65deaebd6ab8d87e278d27c858f6576bf241e67bd035d7b235.jpg new file mode 100644 index 00000000..9fdcfbe6 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-286_f25c9bbad6853b65deaebd6ab8d87e278d27c858f6576bf241e67bd035d7b235.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-287_010613394e2171fc6bde9aa65b74c32c1294afe13e4113fdbb3883325fb760dc.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-287_010613394e2171fc6bde9aa65b74c32c1294afe13e4113fdbb3883325fb760dc.jpg new file mode 100644 index 00000000..58910c25 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-287_010613394e2171fc6bde9aa65b74c32c1294afe13e4113fdbb3883325fb760dc.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-287_0d9dc4e1ed0676265b8599dda5b6b1bb01a8561c13ab6cbde95f67950edd394b.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-287_0d9dc4e1ed0676265b8599dda5b6b1bb01a8561c13ab6cbde95f67950edd394b.jpg new file mode 100644 index 00000000..69e6235f Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-287_0d9dc4e1ed0676265b8599dda5b6b1bb01a8561c13ab6cbde95f67950edd394b.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-287_1083419dad0257da32944b526346af92fe4a73c0690b88d0c0b88d17a8adc1ab.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-287_1083419dad0257da32944b526346af92fe4a73c0690b88d0c0b88d17a8adc1ab.jpg new file mode 100644 index 00000000..59d93c89 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-287_1083419dad0257da32944b526346af92fe4a73c0690b88d0c0b88d17a8adc1ab.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-287_17d531c73cd94230fce0627b5155fb366e5dd264dea13ca27989898ca899b16f.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-287_17d531c73cd94230fce0627b5155fb366e5dd264dea13ca27989898ca899b16f.jpg new file mode 100644 index 00000000..abd663c3 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-287_17d531c73cd94230fce0627b5155fb366e5dd264dea13ca27989898ca899b16f.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-287_31a30154b1537747a8ce8d356b09bb5895453dca846ff3fae198c7a9666dd78e.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-287_31a30154b1537747a8ce8d356b09bb5895453dca846ff3fae198c7a9666dd78e.jpg new file mode 100644 index 00000000..ea6f82ff Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-287_31a30154b1537747a8ce8d356b09bb5895453dca846ff3fae198c7a9666dd78e.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-287_73f23d260d2219d6d4f7abb76e216d923e135a092b9879f37b65407328757ffb.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-287_73f23d260d2219d6d4f7abb76e216d923e135a092b9879f37b65407328757ffb.jpg new file mode 100644 index 00000000..330aecd3 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-287_73f23d260d2219d6d4f7abb76e216d923e135a092b9879f37b65407328757ffb.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-287_a69a985e32138ebe541617f05338b9c3027d585214952b4bf79fe6ae6dc52cff.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-287_a69a985e32138ebe541617f05338b9c3027d585214952b4bf79fe6ae6dc52cff.jpg new file mode 100644 index 00000000..5235b284 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-287_a69a985e32138ebe541617f05338b9c3027d585214952b4bf79fe6ae6dc52cff.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-288_37d39492a91dbc9d3ea60b70dbbed406246f758fbc5b650fc94ebe636528b782.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-288_37d39492a91dbc9d3ea60b70dbbed406246f758fbc5b650fc94ebe636528b782.jpg new file mode 100644 index 00000000..53aa9ea9 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-288_37d39492a91dbc9d3ea60b70dbbed406246f758fbc5b650fc94ebe636528b782.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-288_960d6e9f9930310e41111ba6f7f8a39084cf633417e64c0dafac6b13bea53520.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-288_960d6e9f9930310e41111ba6f7f8a39084cf633417e64c0dafac6b13bea53520.jpg new file mode 100644 index 00000000..b80a78cd Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-288_960d6e9f9930310e41111ba6f7f8a39084cf633417e64c0dafac6b13bea53520.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-288_b2924c026650a75853092e81b01a40edb6d29c8dd37490574cfffe49382056e2.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-288_b2924c026650a75853092e81b01a40edb6d29c8dd37490574cfffe49382056e2.jpg new file mode 100644 index 00000000..4143cc14 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-288_b2924c026650a75853092e81b01a40edb6d29c8dd37490574cfffe49382056e2.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-288_ddc86cccd6c626cf529f28ee330bc6c4ec7a8359329cda8964a4afd477ac361e.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-288_ddc86cccd6c626cf529f28ee330bc6c4ec7a8359329cda8964a4afd477ac361e.jpg new file mode 100644 index 00000000..2195b40b Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-288_ddc86cccd6c626cf529f28ee330bc6c4ec7a8359329cda8964a4afd477ac361e.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-288_ef5cd78d4de83d64795b1a3e24ab1ab1c10352af4b42d88d80f2ac1fb09ca473.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-288_ef5cd78d4de83d64795b1a3e24ab1ab1c10352af4b42d88d80f2ac1fb09ca473.jpg new file mode 100644 index 00000000..c241fc65 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-288_ef5cd78d4de83d64795b1a3e24ab1ab1c10352af4b42d88d80f2ac1fb09ca473.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-288_fdc9702fb38aedf67507f6871f2f2b681271c531aec45cdaec2718b75caaee97.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-288_fdc9702fb38aedf67507f6871f2f2b681271c531aec45cdaec2718b75caaee97.jpg new file mode 100644 index 00000000..c4abbf40 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-288_fdc9702fb38aedf67507f6871f2f2b681271c531aec45cdaec2718b75caaee97.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-289_4490b7921a96e6cced7122e289275d278505b6d863ee87261a507f9ff783196b.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-289_4490b7921a96e6cced7122e289275d278505b6d863ee87261a507f9ff783196b.jpg new file mode 100644 index 00000000..4859786f Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-289_4490b7921a96e6cced7122e289275d278505b6d863ee87261a507f9ff783196b.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-289_61bd5338f00d0fa67f65f2a0a572e397f12b85f8c9a1397177f1348edf9c7c15.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-289_61bd5338f00d0fa67f65f2a0a572e397f12b85f8c9a1397177f1348edf9c7c15.jpg new file mode 100644 index 00000000..3208a7ff Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-289_61bd5338f00d0fa67f65f2a0a572e397f12b85f8c9a1397177f1348edf9c7c15.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-289_878aad813ddcbc6c8eb46e504014154102603c21fe8cd970aee0818ad033eafe.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-289_878aad813ddcbc6c8eb46e504014154102603c21fe8cd970aee0818ad033eafe.jpg new file mode 100644 index 00000000..8b6c3224 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-289_878aad813ddcbc6c8eb46e504014154102603c21fe8cd970aee0818ad033eafe.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-289_c5c6d87019aa337bae78eacda604d7d34b15f8dce3cc790102e79431821e7714.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-289_c5c6d87019aa337bae78eacda604d7d34b15f8dce3cc790102e79431821e7714.jpg new file mode 100644 index 00000000..d2b4f981 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-289_c5c6d87019aa337bae78eacda604d7d34b15f8dce3cc790102e79431821e7714.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-289_e8a9f94748c362611e6271a476634e8a460c77891f8bfababee5129f70be76f0.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-289_e8a9f94748c362611e6271a476634e8a460c77891f8bfababee5129f70be76f0.jpg new file mode 100644 index 00000000..6ebd1ee1 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-289_e8a9f94748c362611e6271a476634e8a460c77891f8bfababee5129f70be76f0.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-290_2dbd784e091445ede4a9534be261f4f3f702605f4c11ffa1c400a5c3cf8ab274.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-290_2dbd784e091445ede4a9534be261f4f3f702605f4c11ffa1c400a5c3cf8ab274.jpg new file mode 100644 index 00000000..11c7c0f1 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-290_2dbd784e091445ede4a9534be261f4f3f702605f4c11ffa1c400a5c3cf8ab274.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-290_2e1cb66e7f4bb57c679ab48844c3faab84c6c01d0af3f525648dd3d5db1b228d.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-290_2e1cb66e7f4bb57c679ab48844c3faab84c6c01d0af3f525648dd3d5db1b228d.jpg new file mode 100644 index 00000000..a59ff416 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-290_2e1cb66e7f4bb57c679ab48844c3faab84c6c01d0af3f525648dd3d5db1b228d.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-290_32820c9749eefb5e88072596d5f2b5ef823e5378d3ff811bd13cc357e6842692.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-290_32820c9749eefb5e88072596d5f2b5ef823e5378d3ff811bd13cc357e6842692.jpg new file mode 100644 index 00000000..5f0123ab Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-290_32820c9749eefb5e88072596d5f2b5ef823e5378d3ff811bd13cc357e6842692.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-290_3455ce3c312d5c0ca378d36cb52349f0451e91e1469724fe4e2a9f452559b53d.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-290_3455ce3c312d5c0ca378d36cb52349f0451e91e1469724fe4e2a9f452559b53d.jpg new file mode 100644 index 00000000..3c1588e9 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-290_3455ce3c312d5c0ca378d36cb52349f0451e91e1469724fe4e2a9f452559b53d.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-290_bc2a406ac422031c1abda585b709d52cec86e53c447c0e276ce0c7e37832eb92.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-290_bc2a406ac422031c1abda585b709d52cec86e53c447c0e276ce0c7e37832eb92.jpg new file mode 100644 index 00000000..e7a6cb9d Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-290_bc2a406ac422031c1abda585b709d52cec86e53c447c0e276ce0c7e37832eb92.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-290_e80e7c437e70389485c4bab2236d8b77ac9ecb079fe7c6f1832eddba5b852ae3.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-290_e80e7c437e70389485c4bab2236d8b77ac9ecb079fe7c6f1832eddba5b852ae3.jpg new file mode 100644 index 00000000..0c0c80b1 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-290_e80e7c437e70389485c4bab2236d8b77ac9ecb079fe7c6f1832eddba5b852ae3.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-290_f7b4f43f9ad726f0f019203021a2d70a3603fe6514a29d7cf2e3584f24b5f9ed.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-290_f7b4f43f9ad726f0f019203021a2d70a3603fe6514a29d7cf2e3584f24b5f9ed.jpg new file mode 100644 index 00000000..6b626364 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-290_f7b4f43f9ad726f0f019203021a2d70a3603fe6514a29d7cf2e3584f24b5f9ed.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-291_891c905528bf235f67a753ceec88809575cba85ddce8570ca02d832af01f1e18.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-291_891c905528bf235f67a753ceec88809575cba85ddce8570ca02d832af01f1e18.jpg new file mode 100644 index 00000000..b56bb2d7 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-291_891c905528bf235f67a753ceec88809575cba85ddce8570ca02d832af01f1e18.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-291_bbac8e2a81e56af9dcac37953af63fe5ad162eb55be401882ba3a06c6785050f.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-291_bbac8e2a81e56af9dcac37953af63fe5ad162eb55be401882ba3a06c6785050f.jpg new file mode 100644 index 00000000..8d840592 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-291_bbac8e2a81e56af9dcac37953af63fe5ad162eb55be401882ba3a06c6785050f.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-292_dd0bd1c5b3d69ce32bca506ff681d815b359be4ee2fc3475d6468efc5840cfb8.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-292_dd0bd1c5b3d69ce32bca506ff681d815b359be4ee2fc3475d6468efc5840cfb8.jpg new file mode 100644 index 00000000..838c0d72 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-292_dd0bd1c5b3d69ce32bca506ff681d815b359be4ee2fc3475d6468efc5840cfb8.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-293_6a958158b919dcde1ac8282fa5ef854ee700db022f1d8100b3af1ac5457aff5a.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-293_6a958158b919dcde1ac8282fa5ef854ee700db022f1d8100b3af1ac5457aff5a.jpg new file mode 100644 index 00000000..8cf5f109 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-293_6a958158b919dcde1ac8282fa5ef854ee700db022f1d8100b3af1ac5457aff5a.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-293_fd77ea0950d0fea322afa34499438560e4982a45e85c1ab3c00ac19b36cafd2c.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-293_fd77ea0950d0fea322afa34499438560e4982a45e85c1ab3c00ac19b36cafd2c.jpg new file mode 100644 index 00000000..02a7e78e Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-293_fd77ea0950d0fea322afa34499438560e4982a45e85c1ab3c00ac19b36cafd2c.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-295_297355334f1c5d0901301159c87fa9639b0a09a55cdd026d31912721b58afdae.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-295_297355334f1c5d0901301159c87fa9639b0a09a55cdd026d31912721b58afdae.jpg new file mode 100644 index 00000000..5c84b0af Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-295_297355334f1c5d0901301159c87fa9639b0a09a55cdd026d31912721b58afdae.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-295_7b83e81140eb4b8f921a1997fab65f8a213976be9d875d8b3d427cd976f927d8.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-295_7b83e81140eb4b8f921a1997fab65f8a213976be9d875d8b3d427cd976f927d8.jpg new file mode 100644 index 00000000..d2e96304 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-295_7b83e81140eb4b8f921a1997fab65f8a213976be9d875d8b3d427cd976f927d8.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-296_0a94ea11c86e43d2aa2da08c7b5fbaec6882a045af85cf42c6882a8e5f85fe1f.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-296_0a94ea11c86e43d2aa2da08c7b5fbaec6882a045af85cf42c6882a8e5f85fe1f.jpg new file mode 100644 index 00000000..3362cc99 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-296_0a94ea11c86e43d2aa2da08c7b5fbaec6882a045af85cf42c6882a8e5f85fe1f.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-296_71fe81ccda4b8f29c3943fca4e4f1940188400c065318cbe84685c06c7fb240a.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-296_71fe81ccda4b8f29c3943fca4e4f1940188400c065318cbe84685c06c7fb240a.jpg new file mode 100644 index 00000000..788a218b Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-296_71fe81ccda4b8f29c3943fca4e4f1940188400c065318cbe84685c06c7fb240a.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-296_ec40429731d143510c269a4e4569a066e5c159c9a61e0dc7cbebe327e7907702.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-296_ec40429731d143510c269a4e4569a066e5c159c9a61e0dc7cbebe327e7907702.jpg new file mode 100644 index 00000000..e2bff57b Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-296_ec40429731d143510c269a4e4569a066e5c159c9a61e0dc7cbebe327e7907702.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-299_3e31bd9cbc2d6dec2f26aff6654d3ead7270a794587c8f271ebb57eaec2a7927.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-299_3e31bd9cbc2d6dec2f26aff6654d3ead7270a794587c8f271ebb57eaec2a7927.jpg new file mode 100644 index 00000000..4e5e44c8 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-299_3e31bd9cbc2d6dec2f26aff6654d3ead7270a794587c8f271ebb57eaec2a7927.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-303_9738ec01837851c8998260a6da6a3fc4043f42621f53504096365d917ff0df17.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-303_9738ec01837851c8998260a6da6a3fc4043f42621f53504096365d917ff0df17.jpg new file mode 100644 index 00000000..a269eb92 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-303_9738ec01837851c8998260a6da6a3fc4043f42621f53504096365d917ff0df17.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-304_31aedfefb065a171f4e8528eee6f9024e92f6e3e455f83aac1d344b0af407120.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-304_31aedfefb065a171f4e8528eee6f9024e92f6e3e455f83aac1d344b0af407120.jpg new file mode 100644 index 00000000..9bb4f52e Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-304_31aedfefb065a171f4e8528eee6f9024e92f6e3e455f83aac1d344b0af407120.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-308_be08c3d52aa2bb7e67b916378b0f56192a669d5be8d7ed1d4c3514de49a1578d.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-308_be08c3d52aa2bb7e67b916378b0f56192a669d5be8d7ed1d4c3514de49a1578d.jpg new file mode 100644 index 00000000..b2730d36 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-308_be08c3d52aa2bb7e67b916378b0f56192a669d5be8d7ed1d4c3514de49a1578d.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-309_4adee727c7d6338e2319c295bfb9d92312cfde67ae7dd73fd1666cd931191bf2.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-309_4adee727c7d6338e2319c295bfb9d92312cfde67ae7dd73fd1666cd931191bf2.jpg new file mode 100644 index 00000000..c00cedc2 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-309_4adee727c7d6338e2319c295bfb9d92312cfde67ae7dd73fd1666cd931191bf2.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-309_faa6b1b4146a1a167a14e408d34250cb6c91daeaddfd230f0b16960bbf89c947.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-309_faa6b1b4146a1a167a14e408d34250cb6c91daeaddfd230f0b16960bbf89c947.jpg new file mode 100644 index 00000000..07702e05 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-309_faa6b1b4146a1a167a14e408d34250cb6c91daeaddfd230f0b16960bbf89c947.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-312_87284b6a0e2771ea43744971bdb2477b3b9991e1fd58d5170e8fd359fa193ac6.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-312_87284b6a0e2771ea43744971bdb2477b3b9991e1fd58d5170e8fd359fa193ac6.jpg new file mode 100644 index 00000000..cfe37e59 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-312_87284b6a0e2771ea43744971bdb2477b3b9991e1fd58d5170e8fd359fa193ac6.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-313_1b0145fbe7ff706c01da01ba6a6add2873907fbbda42818f64345546fa823ebb.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-313_1b0145fbe7ff706c01da01ba6a6add2873907fbbda42818f64345546fa823ebb.jpg new file mode 100644 index 00000000..7383204d Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-313_1b0145fbe7ff706c01da01ba6a6add2873907fbbda42818f64345546fa823ebb.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-314_59f884148dd0d0b3f7e699eac712dbec113631a33cefaf50ba8b1473af904a54.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-314_59f884148dd0d0b3f7e699eac712dbec113631a33cefaf50ba8b1473af904a54.jpg new file mode 100644 index 00000000..356974d2 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-314_59f884148dd0d0b3f7e699eac712dbec113631a33cefaf50ba8b1473af904a54.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-315_02b37f7b4de2f1fdeebdcc4930d17649b46a2d60ee51683acf932012aab39cb0.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-315_02b37f7b4de2f1fdeebdcc4930d17649b46a2d60ee51683acf932012aab39cb0.jpg new file mode 100644 index 00000000..7cf4c962 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-315_02b37f7b4de2f1fdeebdcc4930d17649b46a2d60ee51683acf932012aab39cb0.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-315_30d39ef1efbbbd9789453281ebc09c608b2b97865ee944f1b866688e1c3a6750.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-315_30d39ef1efbbbd9789453281ebc09c608b2b97865ee944f1b866688e1c3a6750.jpg new file mode 100644 index 00000000..72dd9e08 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-315_30d39ef1efbbbd9789453281ebc09c608b2b97865ee944f1b866688e1c3a6750.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-315_5a8cb273d0c515f4997e11a9233ce82db875f2c4ceaacd5cfc683882bbd6b5d7.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-315_5a8cb273d0c515f4997e11a9233ce82db875f2c4ceaacd5cfc683882bbd6b5d7.jpg new file mode 100644 index 00000000..03e35ddf Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-315_5a8cb273d0c515f4997e11a9233ce82db875f2c4ceaacd5cfc683882bbd6b5d7.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-315_d6c21c4e0ad83e9f6c015c913cc576931612f7888540f7e9102a4ba2c4125d8a.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-315_d6c21c4e0ad83e9f6c015c913cc576931612f7888540f7e9102a4ba2c4125d8a.jpg new file mode 100644 index 00000000..2af42a2e Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-315_d6c21c4e0ad83e9f6c015c913cc576931612f7888540f7e9102a4ba2c4125d8a.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-316_9ef581a3a53df1369c058d518686a45feb4b67332d3f3e460ae44f0f09882b4c.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-316_9ef581a3a53df1369c058d518686a45feb4b67332d3f3e460ae44f0f09882b4c.jpg new file mode 100644 index 00000000..30dd7211 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-316_9ef581a3a53df1369c058d518686a45feb4b67332d3f3e460ae44f0f09882b4c.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-316_d4216164d08dccb405f8dd971e27062ccea6c3a6975122da26e0d206464a8ad5.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-316_d4216164d08dccb405f8dd971e27062ccea6c3a6975122da26e0d206464a8ad5.jpg new file mode 100644 index 00000000..6d8e8b7d Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-316_d4216164d08dccb405f8dd971e27062ccea6c3a6975122da26e0d206464a8ad5.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-316_e01f2f825c09fb9022deff2043db762b47d12cb8fb8ebcb1f34805ff0cd250b9.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-316_e01f2f825c09fb9022deff2043db762b47d12cb8fb8ebcb1f34805ff0cd250b9.jpg new file mode 100644 index 00000000..7659829f Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-316_e01f2f825c09fb9022deff2043db762b47d12cb8fb8ebcb1f34805ff0cd250b9.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-317_39bd322860f9651a839c5b7599c3b9c8bbeaf7f692189bb357aca3e94e62d024.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-317_39bd322860f9651a839c5b7599c3b9c8bbeaf7f692189bb357aca3e94e62d024.jpg new file mode 100644 index 00000000..7085ac68 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-317_39bd322860f9651a839c5b7599c3b9c8bbeaf7f692189bb357aca3e94e62d024.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-317_955ac80eda66a1239c3676b81300712fc6f4a12ca9b75647eba7bf944c358e57.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-317_955ac80eda66a1239c3676b81300712fc6f4a12ca9b75647eba7bf944c358e57.jpg new file mode 100644 index 00000000..30e6430a Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-317_955ac80eda66a1239c3676b81300712fc6f4a12ca9b75647eba7bf944c358e57.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-317_9b370d453f2ac52967ab570ad83dcc0f085e8c578e350ba82ad0e0a77951ed85.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-317_9b370d453f2ac52967ab570ad83dcc0f085e8c578e350ba82ad0e0a77951ed85.jpg new file mode 100644 index 00000000..16aec52b Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-317_9b370d453f2ac52967ab570ad83dcc0f085e8c578e350ba82ad0e0a77951ed85.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-317_cf47ace282753853dccca35ed09d6e30658cf44f61e3a85949f4d519ff15208e.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-317_cf47ace282753853dccca35ed09d6e30658cf44f61e3a85949f4d519ff15208e.jpg new file mode 100644 index 00000000..d2d35c63 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-317_cf47ace282753853dccca35ed09d6e30658cf44f61e3a85949f4d519ff15208e.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-318_5da074351107be62c4c3e34cd5d3b06390d8e491c70cb78c552e40b92e99a729.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-318_5da074351107be62c4c3e34cd5d3b06390d8e491c70cb78c552e40b92e99a729.jpg new file mode 100644 index 00000000..7524690b Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-318_5da074351107be62c4c3e34cd5d3b06390d8e491c70cb78c552e40b92e99a729.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-318_784dc505f5ee1ff8b8ef71433609bceb063ae077f43f9c7adb3fd5346888a499.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-318_784dc505f5ee1ff8b8ef71433609bceb063ae077f43f9c7adb3fd5346888a499.jpg new file mode 100644 index 00000000..89d0a8d4 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-318_784dc505f5ee1ff8b8ef71433609bceb063ae077f43f9c7adb3fd5346888a499.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-318_8bbc5486c76b5be4b541d7567a4a4c13cf1f2ef5cf9e559dba62b271c83f4b96.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-318_8bbc5486c76b5be4b541d7567a4a4c13cf1f2ef5cf9e559dba62b271c83f4b96.jpg new file mode 100644 index 00000000..c7c39649 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-318_8bbc5486c76b5be4b541d7567a4a4c13cf1f2ef5cf9e559dba62b271c83f4b96.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-318_e0b9222ff149d078c1b483160c5f4606faae24e94053a8dc3dd168d3d85a7f8e.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-318_e0b9222ff149d078c1b483160c5f4606faae24e94053a8dc3dd168d3d85a7f8e.jpg new file mode 100644 index 00000000..8e623d34 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-318_e0b9222ff149d078c1b483160c5f4606faae24e94053a8dc3dd168d3d85a7f8e.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-318_ea9a847752c068e4af2288c89cd95ab6bc3bc3c0eaa125a3206988acb489c8b5.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-318_ea9a847752c068e4af2288c89cd95ab6bc3bc3c0eaa125a3206988acb489c8b5.jpg new file mode 100644 index 00000000..5b33eceb Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-318_ea9a847752c068e4af2288c89cd95ab6bc3bc3c0eaa125a3206988acb489c8b5.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-318_f57e742016482c81eab78301b9ba0811983b5a0337657f16ddf6762d3fe6c3d8.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-318_f57e742016482c81eab78301b9ba0811983b5a0337657f16ddf6762d3fe6c3d8.jpg new file mode 100644 index 00000000..80c19aeb Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-318_f57e742016482c81eab78301b9ba0811983b5a0337657f16ddf6762d3fe6c3d8.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-320_6577f7f1bb303e7ed4a86524a181efe727dbf1c5bf6f46cad43a40ed3d09509f.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-320_6577f7f1bb303e7ed4a86524a181efe727dbf1c5bf6f46cad43a40ed3d09509f.jpg new file mode 100644 index 00000000..8b2970ef Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-320_6577f7f1bb303e7ed4a86524a181efe727dbf1c5bf6f46cad43a40ed3d09509f.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-321_9aa9dc685fc07be69dfa6b9770fed2b9d64df5fa3709ad27edb45e6501614fc6.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-321_9aa9dc685fc07be69dfa6b9770fed2b9d64df5fa3709ad27edb45e6501614fc6.jpg new file mode 100644 index 00000000..c577ebe2 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-321_9aa9dc685fc07be69dfa6b9770fed2b9d64df5fa3709ad27edb45e6501614fc6.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-322_8039fbe3ea7f9baf1e40787c95fefe0aba203e06d87d0dcdbc83cf5a58e85704.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-322_8039fbe3ea7f9baf1e40787c95fefe0aba203e06d87d0dcdbc83cf5a58e85704.jpg new file mode 100644 index 00000000..520cf5a0 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-322_8039fbe3ea7f9baf1e40787c95fefe0aba203e06d87d0dcdbc83cf5a58e85704.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-323_770fb6cc0e20b57cd8456b712dabffe6de6698fcdd7e8b4b1b0e86f4cdf7f559.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-323_770fb6cc0e20b57cd8456b712dabffe6de6698fcdd7e8b4b1b0e86f4cdf7f559.jpg new file mode 100644 index 00000000..aa524550 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-323_770fb6cc0e20b57cd8456b712dabffe6de6698fcdd7e8b4b1b0e86f4cdf7f559.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-323_c616896949227cf527c47b3e75a30a48049ef37fc2fe2a7554defa12e83de6c5.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-323_c616896949227cf527c47b3e75a30a48049ef37fc2fe2a7554defa12e83de6c5.jpg new file mode 100644 index 00000000..fa29559e Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-323_c616896949227cf527c47b3e75a30a48049ef37fc2fe2a7554defa12e83de6c5.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-324_31942e6434ee77349c2e2c3abee02d214465daa05f913529c94ddfab5d10ea6e.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-324_31942e6434ee77349c2e2c3abee02d214465daa05f913529c94ddfab5d10ea6e.jpg new file mode 100644 index 00000000..f5470476 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-324_31942e6434ee77349c2e2c3abee02d214465daa05f913529c94ddfab5d10ea6e.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-324_640187fe8e37f21319feef2b83631fc3750a1863dddd900e94203d6b5f8c8be2.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-324_640187fe8e37f21319feef2b83631fc3750a1863dddd900e94203d6b5f8c8be2.jpg new file mode 100644 index 00000000..935e2935 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-324_640187fe8e37f21319feef2b83631fc3750a1863dddd900e94203d6b5f8c8be2.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-324_7795ed12f721505a20a3210d2deb4bfaee80163d8c9aefdb49104bdc54ead8f3.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-324_7795ed12f721505a20a3210d2deb4bfaee80163d8c9aefdb49104bdc54ead8f3.jpg new file mode 100644 index 00000000..e0f6a75f Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-324_7795ed12f721505a20a3210d2deb4bfaee80163d8c9aefdb49104bdc54ead8f3.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-324_8fa21c08236bf1acacf140d6af156ca27d5adcaf530028a9304d4eb039bfb9c7.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-324_8fa21c08236bf1acacf140d6af156ca27d5adcaf530028a9304d4eb039bfb9c7.jpg new file mode 100644 index 00000000..b6da72ca Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-324_8fa21c08236bf1acacf140d6af156ca27d5adcaf530028a9304d4eb039bfb9c7.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-324_98b7c492787113c1ed5c0f7ca49915bbaed134c1e5cb16234237fcf84282eac4.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-324_98b7c492787113c1ed5c0f7ca49915bbaed134c1e5cb16234237fcf84282eac4.jpg new file mode 100644 index 00000000..e8658a6a Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-324_98b7c492787113c1ed5c0f7ca49915bbaed134c1e5cb16234237fcf84282eac4.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-324_9e28ed058e22cb3fcf9987f31952de5244fc2d8227299efab9928cfe041aafb6.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-324_9e28ed058e22cb3fcf9987f31952de5244fc2d8227299efab9928cfe041aafb6.jpg new file mode 100644 index 00000000..65a77dc4 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-324_9e28ed058e22cb3fcf9987f31952de5244fc2d8227299efab9928cfe041aafb6.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-324_e7c4760cf37557b59e70a0ae735cd577802d310e919a6c68fd809db7452a079b.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-324_e7c4760cf37557b59e70a0ae735cd577802d310e919a6c68fd809db7452a079b.jpg new file mode 100644 index 00000000..375b5101 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-324_e7c4760cf37557b59e70a0ae735cd577802d310e919a6c68fd809db7452a079b.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-324_f1fc98c277f41be1b864bf2dace64e4f300b3bc83fc79d59a1b8fac92e93ab1a.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-324_f1fc98c277f41be1b864bf2dace64e4f300b3bc83fc79d59a1b8fac92e93ab1a.jpg new file mode 100644 index 00000000..689bbab2 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-324_f1fc98c277f41be1b864bf2dace64e4f300b3bc83fc79d59a1b8fac92e93ab1a.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-324_f64ecc69eeff059b3decad6fcace7e11e30a05bd86b64a64a691f6047d082e70.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-324_f64ecc69eeff059b3decad6fcace7e11e30a05bd86b64a64a691f6047d082e70.jpg new file mode 100644 index 00000000..6aec2d36 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-324_f64ecc69eeff059b3decad6fcace7e11e30a05bd86b64a64a691f6047d082e70.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-325_2fca247702e791287c06ed7f327f2b84c912902f3acf582ec3a618e8b8fba8e3.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-325_2fca247702e791287c06ed7f327f2b84c912902f3acf582ec3a618e8b8fba8e3.jpg new file mode 100644 index 00000000..65661f18 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-325_2fca247702e791287c06ed7f327f2b84c912902f3acf582ec3a618e8b8fba8e3.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-325_3f169859936abc7b5d5f400729aabaa2590f0f81585b7dc48b9a1baf8e6f25e1.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-325_3f169859936abc7b5d5f400729aabaa2590f0f81585b7dc48b9a1baf8e6f25e1.jpg new file mode 100644 index 00000000..1e4397a0 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-325_3f169859936abc7b5d5f400729aabaa2590f0f81585b7dc48b9a1baf8e6f25e1.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-326_7924df76bc76d6f893f1b4cbd3fe0960dd47c5527aa10f9b83aecac4a6d4c9be.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-326_7924df76bc76d6f893f1b4cbd3fe0960dd47c5527aa10f9b83aecac4a6d4c9be.jpg new file mode 100644 index 00000000..63893196 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-326_7924df76bc76d6f893f1b4cbd3fe0960dd47c5527aa10f9b83aecac4a6d4c9be.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-326_d1283a58adbdc45e2a4cc7406a2116047b3b45e899e9443db89b6598a5d7753c.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-326_d1283a58adbdc45e2a4cc7406a2116047b3b45e899e9443db89b6598a5d7753c.jpg new file mode 100644 index 00000000..d8b0a81a Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-326_d1283a58adbdc45e2a4cc7406a2116047b3b45e899e9443db89b6598a5d7753c.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-327_d5d97ed0479266601128485696b82d196a64e0e13edfeb460d3826e2ecdf1fc4.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-327_d5d97ed0479266601128485696b82d196a64e0e13edfeb460d3826e2ecdf1fc4.jpg new file mode 100644 index 00000000..fd044513 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-327_d5d97ed0479266601128485696b82d196a64e0e13edfeb460d3826e2ecdf1fc4.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-331_1b6b6bc36e0c896d03ad0db69f1280c45c890fc259155d0d983c6ce12627a939.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-331_1b6b6bc36e0c896d03ad0db69f1280c45c890fc259155d0d983c6ce12627a939.jpg new file mode 100644 index 00000000..c2efd795 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-331_1b6b6bc36e0c896d03ad0db69f1280c45c890fc259155d0d983c6ce12627a939.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-331_5ab8811172f280ecee14678e6f04a3d31197fb0689f0bf081dcf57133d23b979.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-331_5ab8811172f280ecee14678e6f04a3d31197fb0689f0bf081dcf57133d23b979.jpg new file mode 100644 index 00000000..33418960 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-331_5ab8811172f280ecee14678e6f04a3d31197fb0689f0bf081dcf57133d23b979.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-331_c38a9d08b3bfb455609f8cc23e9bac27ab1cdc5f650b4959d15e9a5238468b34.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-331_c38a9d08b3bfb455609f8cc23e9bac27ab1cdc5f650b4959d15e9a5238468b34.jpg new file mode 100644 index 00000000..16e99587 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-331_c38a9d08b3bfb455609f8cc23e9bac27ab1cdc5f650b4959d15e9a5238468b34.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-332_04be5fe0e75324761fcc60ec811bf63a340465fe11a4a833e5f083964433f750.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-332_04be5fe0e75324761fcc60ec811bf63a340465fe11a4a833e5f083964433f750.jpg new file mode 100644 index 00000000..ae593a6b Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-332_04be5fe0e75324761fcc60ec811bf63a340465fe11a4a833e5f083964433f750.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-332_2c0353ce0f6c9d9f18ee656dad04de56aa6921123ae1df5f18a33508b04f5e08.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-332_2c0353ce0f6c9d9f18ee656dad04de56aa6921123ae1df5f18a33508b04f5e08.jpg new file mode 100644 index 00000000..5e8302cd Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-332_2c0353ce0f6c9d9f18ee656dad04de56aa6921123ae1df5f18a33508b04f5e08.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-332_3f2bf506a0576e8659fb6bc13ed73e7b96428e84d8939a364a26ba9eecd75cdf.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-332_3f2bf506a0576e8659fb6bc13ed73e7b96428e84d8939a364a26ba9eecd75cdf.jpg new file mode 100644 index 00000000..c2ec51d7 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-332_3f2bf506a0576e8659fb6bc13ed73e7b96428e84d8939a364a26ba9eecd75cdf.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-332_474c1ef08f82c34c3511d819e3a086420faa5e6bb49149740ab5d4309f65eb5f.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-332_474c1ef08f82c34c3511d819e3a086420faa5e6bb49149740ab5d4309f65eb5f.jpg new file mode 100644 index 00000000..837ef5c3 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-332_474c1ef08f82c34c3511d819e3a086420faa5e6bb49149740ab5d4309f65eb5f.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-332_62acffae11cd7c9f75bb95254c9beaaf24f253f2f51bbc7d398f8aea615a868a.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-332_62acffae11cd7c9f75bb95254c9beaaf24f253f2f51bbc7d398f8aea615a868a.jpg new file mode 100644 index 00000000..6ed54acd Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-332_62acffae11cd7c9f75bb95254c9beaaf24f253f2f51bbc7d398f8aea615a868a.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-332_6d1f0439eade7d8f90759f20ac1d7b31c1eb991f2c3c5e23622c63168e3032dd.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-332_6d1f0439eade7d8f90759f20ac1d7b31c1eb991f2c3c5e23622c63168e3032dd.jpg new file mode 100644 index 00000000..c4aad89b Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-332_6d1f0439eade7d8f90759f20ac1d7b31c1eb991f2c3c5e23622c63168e3032dd.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-332_990cc312c04d754a81e1bac00a5013417234ab11169e990d0209103fd1a27945.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-332_990cc312c04d754a81e1bac00a5013417234ab11169e990d0209103fd1a27945.jpg new file mode 100644 index 00000000..c1aa45fe Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-332_990cc312c04d754a81e1bac00a5013417234ab11169e990d0209103fd1a27945.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-332_cf10b855944d2f4a970b661af2f21b32941bc9b5ee82a843e08e107757ab3765.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-332_cf10b855944d2f4a970b661af2f21b32941bc9b5ee82a843e08e107757ab3765.jpg new file mode 100644 index 00000000..e9836948 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-332_cf10b855944d2f4a970b661af2f21b32941bc9b5ee82a843e08e107757ab3765.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-333_332c5dc5bb961cf632a47d67ac9569de0d80c7256a38dec5ff8d9fbc48dac4ab.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-333_332c5dc5bb961cf632a47d67ac9569de0d80c7256a38dec5ff8d9fbc48dac4ab.jpg new file mode 100644 index 00000000..5ac49ed3 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-333_332c5dc5bb961cf632a47d67ac9569de0d80c7256a38dec5ff8d9fbc48dac4ab.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-333_48c2408f63c5d49e4f1127fc8fa55e87eb7005477b6734ac98e2859e69ed9126.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-333_48c2408f63c5d49e4f1127fc8fa55e87eb7005477b6734ac98e2859e69ed9126.jpg new file mode 100644 index 00000000..f35c4fbc Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-333_48c2408f63c5d49e4f1127fc8fa55e87eb7005477b6734ac98e2859e69ed9126.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-333_68a7353c11933abaf7333f170c2f0bcc89e248939869a63701f70e84c6d90660.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-333_68a7353c11933abaf7333f170c2f0bcc89e248939869a63701f70e84c6d90660.jpg new file mode 100644 index 00000000..a8ce2433 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-333_68a7353c11933abaf7333f170c2f0bcc89e248939869a63701f70e84c6d90660.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-333_a64f99f9a48f7a85f4c307cd238d5fc01755749e33a7d8df70f10c239dd5deef.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-333_a64f99f9a48f7a85f4c307cd238d5fc01755749e33a7d8df70f10c239dd5deef.jpg new file mode 100644 index 00000000..1fa87f40 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-333_a64f99f9a48f7a85f4c307cd238d5fc01755749e33a7d8df70f10c239dd5deef.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-333_bd48ad109ba28d563f3629c004b95bb319899ed81d5f332d3bebfc1cf3035557.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-333_bd48ad109ba28d563f3629c004b95bb319899ed81d5f332d3bebfc1cf3035557.jpg new file mode 100644 index 00000000..7126a95b Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-333_bd48ad109ba28d563f3629c004b95bb319899ed81d5f332d3bebfc1cf3035557.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-333_f1aae3b817e3264d807d2b5c0674c91002da54de88b0cf04b24dfb29345b589c.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-333_f1aae3b817e3264d807d2b5c0674c91002da54de88b0cf04b24dfb29345b589c.jpg new file mode 100644 index 00000000..47b23a88 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-333_f1aae3b817e3264d807d2b5c0674c91002da54de88b0cf04b24dfb29345b589c.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-333_fa546b2827ffa9d89ac7b3176b1665d1846b237a55918399ad4fafb001ba5b6e.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-333_fa546b2827ffa9d89ac7b3176b1665d1846b237a55918399ad4fafb001ba5b6e.jpg new file mode 100644 index 00000000..3a929b74 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-333_fa546b2827ffa9d89ac7b3176b1665d1846b237a55918399ad4fafb001ba5b6e.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-334_01bc5170b323d47eac05df013432388173be91aca920c366cc9ef05d6d5c659c.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-334_01bc5170b323d47eac05df013432388173be91aca920c366cc9ef05d6d5c659c.jpg new file mode 100644 index 00000000..6c8d3609 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-334_01bc5170b323d47eac05df013432388173be91aca920c366cc9ef05d6d5c659c.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-334_5f4e61c55b0647e8de16866e27a243bd610c9022ebd587800618733254ac847a.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-334_5f4e61c55b0647e8de16866e27a243bd610c9022ebd587800618733254ac847a.jpg new file mode 100644 index 00000000..dbb3c8a6 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-334_5f4e61c55b0647e8de16866e27a243bd610c9022ebd587800618733254ac847a.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-334_6829579a9db4c99069589dc0a83762e209333e74175237eed1579b0e8f8a8c42.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-334_6829579a9db4c99069589dc0a83762e209333e74175237eed1579b0e8f8a8c42.jpg new file mode 100644 index 00000000..3f8e7a8d Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-334_6829579a9db4c99069589dc0a83762e209333e74175237eed1579b0e8f8a8c42.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-334_9bf74529bf3e3352d68860e34f9e8bac030f148757efcf055c9b5a8bdd607642.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-334_9bf74529bf3e3352d68860e34f9e8bac030f148757efcf055c9b5a8bdd607642.jpg new file mode 100644 index 00000000..1c493b93 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-334_9bf74529bf3e3352d68860e34f9e8bac030f148757efcf055c9b5a8bdd607642.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-334_b5279899fc02758e93dce9a623019ecd1ccd392114eb55d32894c02b320395a9.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-334_b5279899fc02758e93dce9a623019ecd1ccd392114eb55d32894c02b320395a9.jpg new file mode 100644 index 00000000..e77035ac Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-334_b5279899fc02758e93dce9a623019ecd1ccd392114eb55d32894c02b320395a9.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-334_e9ba007126b9b73f332efc4794fbc754bf461d8f37fdaffda310edd9d9f69b22.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-334_e9ba007126b9b73f332efc4794fbc754bf461d8f37fdaffda310edd9d9f69b22.jpg new file mode 100644 index 00000000..62f9747f Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-334_e9ba007126b9b73f332efc4794fbc754bf461d8f37fdaffda310edd9d9f69b22.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-335_39ede3b19b128a65c2fe0a53d80c95e9a9822b6c32e65f613694071f466e0f89.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-335_39ede3b19b128a65c2fe0a53d80c95e9a9822b6c32e65f613694071f466e0f89.jpg new file mode 100644 index 00000000..b7e050c3 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-335_39ede3b19b128a65c2fe0a53d80c95e9a9822b6c32e65f613694071f466e0f89.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-335_540dfaaf24db73adaa7ce649468aba3590bc62cb906c38c7e2e369b1c13f6c9d.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-335_540dfaaf24db73adaa7ce649468aba3590bc62cb906c38c7e2e369b1c13f6c9d.jpg new file mode 100644 index 00000000..699e867e Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-335_540dfaaf24db73adaa7ce649468aba3590bc62cb906c38c7e2e369b1c13f6c9d.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-335_7f27580b96ad2027f79bf38000acab7b140c85c518ae19cd10c06298161ad2e4.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-335_7f27580b96ad2027f79bf38000acab7b140c85c518ae19cd10c06298161ad2e4.jpg new file mode 100644 index 00000000..b6a75493 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-335_7f27580b96ad2027f79bf38000acab7b140c85c518ae19cd10c06298161ad2e4.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-336_10d0e12f1708a289a17dbd2061b86581d12e2a61aa693adfac4a435e4fa13707.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-336_10d0e12f1708a289a17dbd2061b86581d12e2a61aa693adfac4a435e4fa13707.jpg new file mode 100644 index 00000000..b5b3144d Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-336_10d0e12f1708a289a17dbd2061b86581d12e2a61aa693adfac4a435e4fa13707.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-336_1a9a1cdd03e4148ef97e87459cf2d45a14f6150c75b3ec1220e0dd339752346d.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-336_1a9a1cdd03e4148ef97e87459cf2d45a14f6150c75b3ec1220e0dd339752346d.jpg new file mode 100644 index 00000000..3fd4513a Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-336_1a9a1cdd03e4148ef97e87459cf2d45a14f6150c75b3ec1220e0dd339752346d.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-336_8346b5f3819cf5d9873fd8fb056bc4cd0a1c0f6503d1a529f0cbb82e0796d1f5.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-336_8346b5f3819cf5d9873fd8fb056bc4cd0a1c0f6503d1a529f0cbb82e0796d1f5.jpg new file mode 100644 index 00000000..1afb1826 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-336_8346b5f3819cf5d9873fd8fb056bc4cd0a1c0f6503d1a529f0cbb82e0796d1f5.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-336_dd0db05d6e881cfce32a60356916c023eb92cb983ef3f21c26c00fa9a0d24e21.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-336_dd0db05d6e881cfce32a60356916c023eb92cb983ef3f21c26c00fa9a0d24e21.jpg new file mode 100644 index 00000000..21d32759 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-336_dd0db05d6e881cfce32a60356916c023eb92cb983ef3f21c26c00fa9a0d24e21.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-337_1c18d792a639494d1c9051705a3fc1fcbe38a6bd03ca0d5951dc15370d9a93ad.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-337_1c18d792a639494d1c9051705a3fc1fcbe38a6bd03ca0d5951dc15370d9a93ad.jpg new file mode 100644 index 00000000..bbead821 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-337_1c18d792a639494d1c9051705a3fc1fcbe38a6bd03ca0d5951dc15370d9a93ad.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-337_1c54e5f616cc4a82fd01a3dd332226e3080f43885a6a64f554ef893adb03762d.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-337_1c54e5f616cc4a82fd01a3dd332226e3080f43885a6a64f554ef893adb03762d.jpg new file mode 100644 index 00000000..68e26940 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-337_1c54e5f616cc4a82fd01a3dd332226e3080f43885a6a64f554ef893adb03762d.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-337_1fc8fe2148f5b8de4c5ffc9f1d00db15c7456adc4f8b3595bd6eb91ad06588d9.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-337_1fc8fe2148f5b8de4c5ffc9f1d00db15c7456adc4f8b3595bd6eb91ad06588d9.jpg new file mode 100644 index 00000000..ad43e075 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-337_1fc8fe2148f5b8de4c5ffc9f1d00db15c7456adc4f8b3595bd6eb91ad06588d9.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-337_6eca9d306b13d6532077d44a73c153760e3ecfb3127b66106875fc1b4d086b3a.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-337_6eca9d306b13d6532077d44a73c153760e3ecfb3127b66106875fc1b4d086b3a.jpg new file mode 100644 index 00000000..ec16d7c2 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-337_6eca9d306b13d6532077d44a73c153760e3ecfb3127b66106875fc1b4d086b3a.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-337_afc0d59b3a2d9ed1fd7cb999d3e084c7bb33eb191176f2a41f90f7b4953edb6c.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-337_afc0d59b3a2d9ed1fd7cb999d3e084c7bb33eb191176f2a41f90f7b4953edb6c.jpg new file mode 100644 index 00000000..87f37151 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-337_afc0d59b3a2d9ed1fd7cb999d3e084c7bb33eb191176f2a41f90f7b4953edb6c.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-337_f7fab1b759a85363114c4a68709d1594b9c6797b5425c673fb15fb35d42e0395.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-337_f7fab1b759a85363114c4a68709d1594b9c6797b5425c673fb15fb35d42e0395.jpg new file mode 100644 index 00000000..95720d00 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-337_f7fab1b759a85363114c4a68709d1594b9c6797b5425c673fb15fb35d42e0395.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-338_1ca223eeb7d5e17d8d39cdb5ef165f4a2580b4009da29100a13b536bbd528521.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-338_1ca223eeb7d5e17d8d39cdb5ef165f4a2580b4009da29100a13b536bbd528521.jpg new file mode 100644 index 00000000..037a6203 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-338_1ca223eeb7d5e17d8d39cdb5ef165f4a2580b4009da29100a13b536bbd528521.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-338_4474a4aef42b0f84dd11e40d6d204b7c705cf4daa76211e7f267beeea4a004bc.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-338_4474a4aef42b0f84dd11e40d6d204b7c705cf4daa76211e7f267beeea4a004bc.jpg new file mode 100644 index 00000000..d597b9db Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-338_4474a4aef42b0f84dd11e40d6d204b7c705cf4daa76211e7f267beeea4a004bc.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-338_4f123d953e6f0f62ebdc9b73a17ad69afff7b0d5495de99722dcf02d80222c17.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-338_4f123d953e6f0f62ebdc9b73a17ad69afff7b0d5495de99722dcf02d80222c17.jpg new file mode 100644 index 00000000..51cac214 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-338_4f123d953e6f0f62ebdc9b73a17ad69afff7b0d5495de99722dcf02d80222c17.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-338_5eb7ee76d21f1c593b78588b44ad90cb5b995147d9b24215c4da161ce93a7f28.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-338_5eb7ee76d21f1c593b78588b44ad90cb5b995147d9b24215c4da161ce93a7f28.jpg new file mode 100644 index 00000000..d6d39a2a Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-338_5eb7ee76d21f1c593b78588b44ad90cb5b995147d9b24215c4da161ce93a7f28.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-338_9c7df2504d2ab0c078c86ce8ddc818f6f8047a282ebcda0d62e9848cd529b197.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-338_9c7df2504d2ab0c078c86ce8ddc818f6f8047a282ebcda0d62e9848cd529b197.jpg new file mode 100644 index 00000000..49807fe6 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-338_9c7df2504d2ab0c078c86ce8ddc818f6f8047a282ebcda0d62e9848cd529b197.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-338_d47199d03161fd0a261f6bef563eaccc428d31615fc5049b167cf6ce3f6c237a.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-338_d47199d03161fd0a261f6bef563eaccc428d31615fc5049b167cf6ce3f6c237a.jpg new file mode 100644 index 00000000..ca204c25 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-338_d47199d03161fd0a261f6bef563eaccc428d31615fc5049b167cf6ce3f6c237a.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-338_e0b41aad7fc82d1de1e39ce7b52a5b460c8ecf2e0d4b9c991b8a99bda965a569.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-338_e0b41aad7fc82d1de1e39ce7b52a5b460c8ecf2e0d4b9c991b8a99bda965a569.jpg new file mode 100644 index 00000000..07e4ff18 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-338_e0b41aad7fc82d1de1e39ce7b52a5b460c8ecf2e0d4b9c991b8a99bda965a569.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-338_e957a209c9f408e8d2b0441ccdf044708895952d53e53b110639f9d0b1afb4da.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-338_e957a209c9f408e8d2b0441ccdf044708895952d53e53b110639f9d0b1afb4da.jpg new file mode 100644 index 00000000..207c24e7 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-338_e957a209c9f408e8d2b0441ccdf044708895952d53e53b110639f9d0b1afb4da.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-338_f613881aae546c553b7f5dc32efc94e0b7f731785fce0e24da1494ee315c4cee.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-338_f613881aae546c553b7f5dc32efc94e0b7f731785fce0e24da1494ee315c4cee.jpg new file mode 100644 index 00000000..a4101149 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-338_f613881aae546c553b7f5dc32efc94e0b7f731785fce0e24da1494ee315c4cee.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-339_528e998e81bee6e26f53178d3f757d72fef61b8afa1249ba863263a99a5b07b9.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-339_528e998e81bee6e26f53178d3f757d72fef61b8afa1249ba863263a99a5b07b9.jpg new file mode 100644 index 00000000..7d7f419c Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-339_528e998e81bee6e26f53178d3f757d72fef61b8afa1249ba863263a99a5b07b9.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-339_633b26a1417508a5f51b57d0ea84017c1c1224f2e88872ae9ff240f6f4281c09.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-339_633b26a1417508a5f51b57d0ea84017c1c1224f2e88872ae9ff240f6f4281c09.jpg new file mode 100644 index 00000000..48063d4c Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-339_633b26a1417508a5f51b57d0ea84017c1c1224f2e88872ae9ff240f6f4281c09.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-339_8821d31a51e4039c42fe8620b65a679a3ce9d50ac1578c58f51e324a1eb0ae2b.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-339_8821d31a51e4039c42fe8620b65a679a3ce9d50ac1578c58f51e324a1eb0ae2b.jpg new file mode 100644 index 00000000..81b1ed1e Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-339_8821d31a51e4039c42fe8620b65a679a3ce9d50ac1578c58f51e324a1eb0ae2b.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-339_e9aba0da6dee65beff5464ef5b8d2ed7168f7dd34291b3cb81b64b4ea3f53d8c.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-339_e9aba0da6dee65beff5464ef5b8d2ed7168f7dd34291b3cb81b64b4ea3f53d8c.jpg new file mode 100644 index 00000000..519b30b9 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-339_e9aba0da6dee65beff5464ef5b8d2ed7168f7dd34291b3cb81b64b4ea3f53d8c.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-339_f1e73150ae6b11d2b0081d83253b544c62b9b07322128e3efa3e1fe8c2b07a61.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-339_f1e73150ae6b11d2b0081d83253b544c62b9b07322128e3efa3e1fe8c2b07a61.jpg new file mode 100644 index 00000000..a9a3572b Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-339_f1e73150ae6b11d2b0081d83253b544c62b9b07322128e3efa3e1fe8c2b07a61.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-340_19a6b9645cebf299b47d24a2b5a9b66dd0569d49b73be884a71b2da51c813730.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-340_19a6b9645cebf299b47d24a2b5a9b66dd0569d49b73be884a71b2da51c813730.jpg new file mode 100644 index 00000000..c85492da Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-340_19a6b9645cebf299b47d24a2b5a9b66dd0569d49b73be884a71b2da51c813730.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-340_37b50e217b00bea02122a92f476eee1a6567c5a6b00d71ad5af5fa9d70dcafc5.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-340_37b50e217b00bea02122a92f476eee1a6567c5a6b00d71ad5af5fa9d70dcafc5.jpg new file mode 100644 index 00000000..f72e3fce Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-340_37b50e217b00bea02122a92f476eee1a6567c5a6b00d71ad5af5fa9d70dcafc5.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-340_4b0b6c6ff7d8002deb6f0fc40fc9520ea927aa0eb557f525f19b7e99d4a7b927.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-340_4b0b6c6ff7d8002deb6f0fc40fc9520ea927aa0eb557f525f19b7e99d4a7b927.jpg new file mode 100644 index 00000000..4e7b0325 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-340_4b0b6c6ff7d8002deb6f0fc40fc9520ea927aa0eb557f525f19b7e99d4a7b927.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-340_674ae974d207dd686f5f74c102bb27f90272c4067a25ccaef81785369755227c.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-340_674ae974d207dd686f5f74c102bb27f90272c4067a25ccaef81785369755227c.jpg new file mode 100644 index 00000000..df8f438c Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-340_674ae974d207dd686f5f74c102bb27f90272c4067a25ccaef81785369755227c.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-340_810c134335d5f86976f9e2138b49aa991f04caad08df925ff77b89ee74257289.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-340_810c134335d5f86976f9e2138b49aa991f04caad08df925ff77b89ee74257289.jpg new file mode 100644 index 00000000..1e7f5474 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-340_810c134335d5f86976f9e2138b49aa991f04caad08df925ff77b89ee74257289.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-340_8a6ebb872944bac8f8bb9387daa6e0d60d74cac6bdc7bb22e2ef867119de55c6.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-340_8a6ebb872944bac8f8bb9387daa6e0d60d74cac6bdc7bb22e2ef867119de55c6.jpg new file mode 100644 index 00000000..0151829d Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-340_8a6ebb872944bac8f8bb9387daa6e0d60d74cac6bdc7bb22e2ef867119de55c6.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-340_9dc34cd5194a324c31aea273c5ffd658be3c66545dfbc6b2229431ec9becd8f4.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-340_9dc34cd5194a324c31aea273c5ffd658be3c66545dfbc6b2229431ec9becd8f4.jpg new file mode 100644 index 00000000..d7badfe6 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-340_9dc34cd5194a324c31aea273c5ffd658be3c66545dfbc6b2229431ec9becd8f4.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-340_ec08ac7e1c621f5c739d0cbb740da560445ea22816516bf3a84f3be0b70b2346.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-340_ec08ac7e1c621f5c739d0cbb740da560445ea22816516bf3a84f3be0b70b2346.jpg new file mode 100644 index 00000000..c0d6582b Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-340_ec08ac7e1c621f5c739d0cbb740da560445ea22816516bf3a84f3be0b70b2346.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-341_092dd62e586fc93790452031e3d319d6150f90606de140b2ef4b4f73e53839ac.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-341_092dd62e586fc93790452031e3d319d6150f90606de140b2ef4b4f73e53839ac.jpg new file mode 100644 index 00000000..35d38dd6 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-341_092dd62e586fc93790452031e3d319d6150f90606de140b2ef4b4f73e53839ac.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-341_1db3b51bf8279dfd9fafbfa9e7231b96e2ad12a21353da480bc2442a43f6a6d4.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-341_1db3b51bf8279dfd9fafbfa9e7231b96e2ad12a21353da480bc2442a43f6a6d4.jpg new file mode 100644 index 00000000..406f316d Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-341_1db3b51bf8279dfd9fafbfa9e7231b96e2ad12a21353da480bc2442a43f6a6d4.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-341_44f93d2a0218593ff5e96d0ed33956c97812c3b2e5e4b2d10a20390b90ba6d07.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-341_44f93d2a0218593ff5e96d0ed33956c97812c3b2e5e4b2d10a20390b90ba6d07.jpg new file mode 100644 index 00000000..54b5755f Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-341_44f93d2a0218593ff5e96d0ed33956c97812c3b2e5e4b2d10a20390b90ba6d07.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-342_129442c8b5be6e374b8ea47c6bff19b053a98b376cfb0fd9b276ba404cdb0c43.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-342_129442c8b5be6e374b8ea47c6bff19b053a98b376cfb0fd9b276ba404cdb0c43.jpg new file mode 100644 index 00000000..1aa33d15 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-342_129442c8b5be6e374b8ea47c6bff19b053a98b376cfb0fd9b276ba404cdb0c43.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-343_5bd99d9564044dfd1a2729d74574247f41e35c758c3e6f473fcb9e2cfe1f9a70.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-343_5bd99d9564044dfd1a2729d74574247f41e35c758c3e6f473fcb9e2cfe1f9a70.jpg new file mode 100644 index 00000000..32bf49c1 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-343_5bd99d9564044dfd1a2729d74574247f41e35c758c3e6f473fcb9e2cfe1f9a70.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-346_1b2b673fb5cf18c73d176c13cb5581fd586b59cc51685e7a8d1b71787446c2cd.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-346_1b2b673fb5cf18c73d176c13cb5581fd586b59cc51685e7a8d1b71787446c2cd.jpg new file mode 100644 index 00000000..c57cdd2e Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-346_1b2b673fb5cf18c73d176c13cb5581fd586b59cc51685e7a8d1b71787446c2cd.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-346_92202ecf8994148364001119340eafd37ede66b7af77d6008b89c47347b7e615.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-346_92202ecf8994148364001119340eafd37ede66b7af77d6008b89c47347b7e615.jpg new file mode 100644 index 00000000..472f4278 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-346_92202ecf8994148364001119340eafd37ede66b7af77d6008b89c47347b7e615.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-357_44388d2b9273b6860372b0273fb3f6333303e4d33acbec0c8c5bcda22cc55164.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-357_44388d2b9273b6860372b0273fb3f6333303e4d33acbec0c8c5bcda22cc55164.jpg new file mode 100644 index 00000000..b5efa88d Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-357_44388d2b9273b6860372b0273fb3f6333303e4d33acbec0c8c5bcda22cc55164.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-358_645ab8ddf88be9537076f6e1702d87b898be2015b9067ee3985bab44743eca73.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-358_645ab8ddf88be9537076f6e1702d87b898be2015b9067ee3985bab44743eca73.jpg new file mode 100644 index 00000000..f8bdcde5 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-358_645ab8ddf88be9537076f6e1702d87b898be2015b9067ee3985bab44743eca73.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-358_fa58d107c20c213ac4a816f74c5bf531648eef61054abdf243faf4a8543d8aa0.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-358_fa58d107c20c213ac4a816f74c5bf531648eef61054abdf243faf4a8543d8aa0.jpg new file mode 100644 index 00000000..01b2491e Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-358_fa58d107c20c213ac4a816f74c5bf531648eef61054abdf243faf4a8543d8aa0.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-359_a28640cd410af76ff85758e9d9c2ee9a3b8a687c6fad77fd20db7cc907376fdd.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-359_a28640cd410af76ff85758e9d9c2ee9a3b8a687c6fad77fd20db7cc907376fdd.jpg new file mode 100644 index 00000000..a5041743 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-359_a28640cd410af76ff85758e9d9c2ee9a3b8a687c6fad77fd20db7cc907376fdd.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-360_f39271c68a18bd4b50dc64082d2d75c8db47b2eae4026920e44148685f0acc51.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-360_f39271c68a18bd4b50dc64082d2d75c8db47b2eae4026920e44148685f0acc51.jpg new file mode 100644 index 00000000..aadd5b5d Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-360_f39271c68a18bd4b50dc64082d2d75c8db47b2eae4026920e44148685f0acc51.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-364_76cf4479bdd944eb56178bb88c26c9c5ab9101b0210856e1ecadf6509456b9ea.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-364_76cf4479bdd944eb56178bb88c26c9c5ab9101b0210856e1ecadf6509456b9ea.jpg new file mode 100644 index 00000000..0922c689 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-364_76cf4479bdd944eb56178bb88c26c9c5ab9101b0210856e1ecadf6509456b9ea.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-366_4957bf2f81a024b454061e697cda2961788c966d555c1fc548ea6b98418d01e2.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-366_4957bf2f81a024b454061e697cda2961788c966d555c1fc548ea6b98418d01e2.jpg new file mode 100644 index 00000000..398caf8e Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-366_4957bf2f81a024b454061e697cda2961788c966d555c1fc548ea6b98418d01e2.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-367_7282858d1fec3c9bb621c16a1971c2eb4af44eb16bfaa5d614cb96f3b24826d7.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-367_7282858d1fec3c9bb621c16a1971c2eb4af44eb16bfaa5d614cb96f3b24826d7.jpg new file mode 100644 index 00000000..8bcd1b26 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-367_7282858d1fec3c9bb621c16a1971c2eb4af44eb16bfaa5d614cb96f3b24826d7.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-372_1a3fcf57afa9d87f293907bb7d0406530509a2ef9a7cb6b3162ff86d47f11c67.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-372_1a3fcf57afa9d87f293907bb7d0406530509a2ef9a7cb6b3162ff86d47f11c67.jpg new file mode 100644 index 00000000..af04ee36 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-372_1a3fcf57afa9d87f293907bb7d0406530509a2ef9a7cb6b3162ff86d47f11c67.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-372_6a02d0ad23d2e2aa665c24e901810ce9721a1d8f6748caad463fc8c5c3cc4e8f.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-372_6a02d0ad23d2e2aa665c24e901810ce9721a1d8f6748caad463fc8c5c3cc4e8f.jpg new file mode 100644 index 00000000..1dc9df1c Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-372_6a02d0ad23d2e2aa665c24e901810ce9721a1d8f6748caad463fc8c5c3cc4e8f.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-380_949931b2d11daf9bfc7dc0bcf8a47e8fa08c56f5adfa3422e63f364437764ab0.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-380_949931b2d11daf9bfc7dc0bcf8a47e8fa08c56f5adfa3422e63f364437764ab0.jpg new file mode 100644 index 00000000..81cf1744 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-380_949931b2d11daf9bfc7dc0bcf8a47e8fa08c56f5adfa3422e63f364437764ab0.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-381_b3c2f09d0e369b0e1b8120d8c03abe56c39443e5d0fc5930acdcbccfcc9c41ba.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-381_b3c2f09d0e369b0e1b8120d8c03abe56c39443e5d0fc5930acdcbccfcc9c41ba.jpg new file mode 100644 index 00000000..bd56eec9 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-381_b3c2f09d0e369b0e1b8120d8c03abe56c39443e5d0fc5930acdcbccfcc9c41ba.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-381_b700d1010a27f57feb7a76efafee3a8789cef5047e33dcdf6efdc9bf7ab49249.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-381_b700d1010a27f57feb7a76efafee3a8789cef5047e33dcdf6efdc9bf7ab49249.jpg new file mode 100644 index 00000000..972d8dfb Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-381_b700d1010a27f57feb7a76efafee3a8789cef5047e33dcdf6efdc9bf7ab49249.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-382_0621decd119f4bbbca726dc7dbb4527a89f3e1278dd046f811e28795732e2051.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-382_0621decd119f4bbbca726dc7dbb4527a89f3e1278dd046f811e28795732e2051.jpg new file mode 100644 index 00000000..cdef0376 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-382_0621decd119f4bbbca726dc7dbb4527a89f3e1278dd046f811e28795732e2051.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-382_71c9eba5498e223f7cf1f206b7258d729d665f0893a6f111526781f7df8772a3.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-382_71c9eba5498e223f7cf1f206b7258d729d665f0893a6f111526781f7df8772a3.jpg new file mode 100644 index 00000000..d6859c5c Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-382_71c9eba5498e223f7cf1f206b7258d729d665f0893a6f111526781f7df8772a3.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-382_73cfd3bcb53394c280cf719f78885a56207e394c6592af1168750d74661a3a24.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-382_73cfd3bcb53394c280cf719f78885a56207e394c6592af1168750d74661a3a24.jpg new file mode 100644 index 00000000..616296d5 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-382_73cfd3bcb53394c280cf719f78885a56207e394c6592af1168750d74661a3a24.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-382_c3db941fcc2a9223989fb6830dd8c8dd7b5f6481168692a49b7336dc94e9ba2d.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-382_c3db941fcc2a9223989fb6830dd8c8dd7b5f6481168692a49b7336dc94e9ba2d.jpg new file mode 100644 index 00000000..566037b5 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-382_c3db941fcc2a9223989fb6830dd8c8dd7b5f6481168692a49b7336dc94e9ba2d.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-383_100d1f93f3196d921f749d8e611af0b7beec302136fe8d170170f3d2a455fb4c.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-383_100d1f93f3196d921f749d8e611af0b7beec302136fe8d170170f3d2a455fb4c.jpg new file mode 100644 index 00000000..d6873c14 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-383_100d1f93f3196d921f749d8e611af0b7beec302136fe8d170170f3d2a455fb4c.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-388_499ebf63de94f10d0d8d7f0ddf6edeaa486964654815aa49fa968946662b8c91.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-388_499ebf63de94f10d0d8d7f0ddf6edeaa486964654815aa49fa968946662b8c91.jpg new file mode 100644 index 00000000..f61527f9 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-388_499ebf63de94f10d0d8d7f0ddf6edeaa486964654815aa49fa968946662b8c91.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-388_4d09be69acaa9b0093bf7de9ffcc7abfa6c28cce8ac41d95f2fd9d9b055583e6.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-388_4d09be69acaa9b0093bf7de9ffcc7abfa6c28cce8ac41d95f2fd9d9b055583e6.jpg new file mode 100644 index 00000000..cb4dd642 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-388_4d09be69acaa9b0093bf7de9ffcc7abfa6c28cce8ac41d95f2fd9d9b055583e6.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-388_74c1dad80c310b4dfd6b3a6740bb362a572588db25a938d6fe2ec7330817b577.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-388_74c1dad80c310b4dfd6b3a6740bb362a572588db25a938d6fe2ec7330817b577.jpg new file mode 100644 index 00000000..f4366784 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-388_74c1dad80c310b4dfd6b3a6740bb362a572588db25a938d6fe2ec7330817b577.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-389_11824050d11b765e9ee77792a4c9f27138fe60c587c0e6919ca34d44a625415c.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-389_11824050d11b765e9ee77792a4c9f27138fe60c587c0e6919ca34d44a625415c.jpg new file mode 100644 index 00000000..73e73eba Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-389_11824050d11b765e9ee77792a4c9f27138fe60c587c0e6919ca34d44a625415c.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-389_2a4f4ba5eeb425aefb5c700c585cf340c553182d48d16a3f03916093a38c25cb.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-389_2a4f4ba5eeb425aefb5c700c585cf340c553182d48d16a3f03916093a38c25cb.jpg new file mode 100644 index 00000000..3c8c3774 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-389_2a4f4ba5eeb425aefb5c700c585cf340c553182d48d16a3f03916093a38c25cb.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-389_4b89737f629b55c9458d5ae4f17d7d73eacfc68b395664d8dab5fafd901b0ef0.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-389_4b89737f629b55c9458d5ae4f17d7d73eacfc68b395664d8dab5fafd901b0ef0.jpg new file mode 100644 index 00000000..1c446db4 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-389_4b89737f629b55c9458d5ae4f17d7d73eacfc68b395664d8dab5fafd901b0ef0.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-389_507c8e79c8b85e91824e7ce33ef1ba4967bf182a3cde8b8fceba1deb152c36ad.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-389_507c8e79c8b85e91824e7ce33ef1ba4967bf182a3cde8b8fceba1deb152c36ad.jpg new file mode 100644 index 00000000..40df7ddc Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-389_507c8e79c8b85e91824e7ce33ef1ba4967bf182a3cde8b8fceba1deb152c36ad.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-389_980274d69c234e38ba04a6fac9cc0b4be38567d0acedf95b110aef0ac406cf25.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-389_980274d69c234e38ba04a6fac9cc0b4be38567d0acedf95b110aef0ac406cf25.jpg new file mode 100644 index 00000000..d25b69c5 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-389_980274d69c234e38ba04a6fac9cc0b4be38567d0acedf95b110aef0ac406cf25.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-389_b92f6cf058f3caaac64e51e8a834cd456cb3408b031e92e87d9d2f0725acc52c.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-389_b92f6cf058f3caaac64e51e8a834cd456cb3408b031e92e87d9d2f0725acc52c.jpg new file mode 100644 index 00000000..307bf5bd Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-389_b92f6cf058f3caaac64e51e8a834cd456cb3408b031e92e87d9d2f0725acc52c.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-389_bb851b1fe097d51a2508242886eb4374c31a215f0089739726a75fae30f8788d.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-389_bb851b1fe097d51a2508242886eb4374c31a215f0089739726a75fae30f8788d.jpg new file mode 100644 index 00000000..a8d7e769 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-389_bb851b1fe097d51a2508242886eb4374c31a215f0089739726a75fae30f8788d.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-390_09a8858d2e7b1f8016897297fabea5ac7f656982ba0109d9222828826791fb62.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-390_09a8858d2e7b1f8016897297fabea5ac7f656982ba0109d9222828826791fb62.jpg new file mode 100644 index 00000000..28f3bf61 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-390_09a8858d2e7b1f8016897297fabea5ac7f656982ba0109d9222828826791fb62.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-390_5bf942486a18d7ef48d98741e1ca7d9d352379d1d233f93bd7d94a8c626ee9af.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-390_5bf942486a18d7ef48d98741e1ca7d9d352379d1d233f93bd7d94a8c626ee9af.jpg new file mode 100644 index 00000000..b34cd6b5 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-390_5bf942486a18d7ef48d98741e1ca7d9d352379d1d233f93bd7d94a8c626ee9af.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-390_77e98989ddb2bad042a4bac46a1f4ea1bb4037a134b1f90cfe432ffb13af9414.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-390_77e98989ddb2bad042a4bac46a1f4ea1bb4037a134b1f90cfe432ffb13af9414.jpg new file mode 100644 index 00000000..b391ed98 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-390_77e98989ddb2bad042a4bac46a1f4ea1bb4037a134b1f90cfe432ffb13af9414.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-390_8484335044b7915b206279189b3a6c37c7fce9910c36c2b3d50628e58223ffed.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-390_8484335044b7915b206279189b3a6c37c7fce9910c36c2b3d50628e58223ffed.jpg new file mode 100644 index 00000000..eae3f783 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-390_8484335044b7915b206279189b3a6c37c7fce9910c36c2b3d50628e58223ffed.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-390_b949442ccbcad121d248c66c59b4b608dfd9d113f706a38d0f20205b8300b36e.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-390_b949442ccbcad121d248c66c59b4b608dfd9d113f706a38d0f20205b8300b36e.jpg new file mode 100644 index 00000000..c92003b8 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-390_b949442ccbcad121d248c66c59b4b608dfd9d113f706a38d0f20205b8300b36e.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-390_c042b1146136563f7426da9c4807ecd0ea8f6dbb552e16340cf0b831665e5ff4.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-390_c042b1146136563f7426da9c4807ecd0ea8f6dbb552e16340cf0b831665e5ff4.jpg new file mode 100644 index 00000000..f172a351 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-390_c042b1146136563f7426da9c4807ecd0ea8f6dbb552e16340cf0b831665e5ff4.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-390_e0d0826d8583e95364f6f87a2a7c69dc17e539f415830339bf1aaf39cd458cb9.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-390_e0d0826d8583e95364f6f87a2a7c69dc17e539f415830339bf1aaf39cd458cb9.jpg new file mode 100644 index 00000000..94cae792 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-390_e0d0826d8583e95364f6f87a2a7c69dc17e539f415830339bf1aaf39cd458cb9.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-390_fce0ecc75abaeec1583cb5e63cadc97b7b12987c9072a681f3762d5177a5df19.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-390_fce0ecc75abaeec1583cb5e63cadc97b7b12987c9072a681f3762d5177a5df19.jpg new file mode 100644 index 00000000..d07265f2 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-390_fce0ecc75abaeec1583cb5e63cadc97b7b12987c9072a681f3762d5177a5df19.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-391_cc536b2ad6d78ce4cf09f196936d7696b5cfeaf9da564d19df93da48d758e710.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-391_cc536b2ad6d78ce4cf09f196936d7696b5cfeaf9da564d19df93da48d758e710.jpg new file mode 100644 index 00000000..f1ac28c6 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-391_cc536b2ad6d78ce4cf09f196936d7696b5cfeaf9da564d19df93da48d758e710.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-392_177a8315c3c58c63434a51d07cf2dbc6001d0c96363bee674ee130cd8cff88b5.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-392_177a8315c3c58c63434a51d07cf2dbc6001d0c96363bee674ee130cd8cff88b5.jpg new file mode 100644 index 00000000..5d2a40c2 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-392_177a8315c3c58c63434a51d07cf2dbc6001d0c96363bee674ee130cd8cff88b5.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-392_3ce9570ffa950fb9defc931286b98dfda515cd566dcb88e2609e589b9ebbf6f5.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-392_3ce9570ffa950fb9defc931286b98dfda515cd566dcb88e2609e589b9ebbf6f5.jpg new file mode 100644 index 00000000..f5dc4826 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-392_3ce9570ffa950fb9defc931286b98dfda515cd566dcb88e2609e589b9ebbf6f5.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-392_4f516219496e18e4d76c40452a4019d9f9bd3307e30a7459097a33b6df147daf.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-392_4f516219496e18e4d76c40452a4019d9f9bd3307e30a7459097a33b6df147daf.jpg new file mode 100644 index 00000000..4156211d Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-392_4f516219496e18e4d76c40452a4019d9f9bd3307e30a7459097a33b6df147daf.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-392_8e916347537c16abcb8b1ac1f716ac6072bdd560fffe17523b02bddf25a09f99.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-392_8e916347537c16abcb8b1ac1f716ac6072bdd560fffe17523b02bddf25a09f99.jpg new file mode 100644 index 00000000..a87d8c65 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-392_8e916347537c16abcb8b1ac1f716ac6072bdd560fffe17523b02bddf25a09f99.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-392_bbc983bfb15846a24a612bb8e14bb09dc38d1bcb78db673b4934cd352d24a451.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-392_bbc983bfb15846a24a612bb8e14bb09dc38d1bcb78db673b4934cd352d24a451.jpg new file mode 100644 index 00000000..4c621292 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-392_bbc983bfb15846a24a612bb8e14bb09dc38d1bcb78db673b4934cd352d24a451.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-392_bd36fe291ca47949c81bff96d91822098dfc00763babbc2a762a595b4a4c9b52.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-392_bd36fe291ca47949c81bff96d91822098dfc00763babbc2a762a595b4a4c9b52.jpg new file mode 100644 index 00000000..e1b833b7 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-392_bd36fe291ca47949c81bff96d91822098dfc00763babbc2a762a595b4a4c9b52.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-392_e4512358f8e3d15e39639c13e5fe6f19603ad49591b93a0bf747292e7a94b03f.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-392_e4512358f8e3d15e39639c13e5fe6f19603ad49591b93a0bf747292e7a94b03f.jpg new file mode 100644 index 00000000..4d43b11e Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-392_e4512358f8e3d15e39639c13e5fe6f19603ad49591b93a0bf747292e7a94b03f.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-393_37344a4e7bb184c80faccc7e7ea04c26d2689a50dcf48d32a26ee05421ec4b1c.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-393_37344a4e7bb184c80faccc7e7ea04c26d2689a50dcf48d32a26ee05421ec4b1c.jpg new file mode 100644 index 00000000..065aad11 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-393_37344a4e7bb184c80faccc7e7ea04c26d2689a50dcf48d32a26ee05421ec4b1c.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-393_cf1633a78dd2a2fb70b11573d186998e6de386dc1528764ad16c61ef3c4b530b.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-393_cf1633a78dd2a2fb70b11573d186998e6de386dc1528764ad16c61ef3c4b530b.jpg new file mode 100644 index 00000000..cdf956c2 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-393_cf1633a78dd2a2fb70b11573d186998e6de386dc1528764ad16c61ef3c4b530b.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-393_ec184bb61d70133ef61d064620f9e581ac3a48729f5486a522fc9ed4e50dd834.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-393_ec184bb61d70133ef61d064620f9e581ac3a48729f5486a522fc9ed4e50dd834.jpg new file mode 100644 index 00000000..0afe92fa Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-393_ec184bb61d70133ef61d064620f9e581ac3a48729f5486a522fc9ed4e50dd834.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-394_c7bda09edbbc3660d016227a8b2e24557b0bf576c249c14c4211eedec03f853f.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-394_c7bda09edbbc3660d016227a8b2e24557b0bf576c249c14c4211eedec03f853f.jpg new file mode 100644 index 00000000..6164dba3 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-394_c7bda09edbbc3660d016227a8b2e24557b0bf576c249c14c4211eedec03f853f.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-395_6330c51d267796976a3169689b1119d90bbdd476f1fb933a8b06ed02b39430e2.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-395_6330c51d267796976a3169689b1119d90bbdd476f1fb933a8b06ed02b39430e2.jpg new file mode 100644 index 00000000..7f95343e Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-395_6330c51d267796976a3169689b1119d90bbdd476f1fb933a8b06ed02b39430e2.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-396_104f0935016921157976352f420faa99963a267da4dafe3f13f4823a435fa1f0.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-396_104f0935016921157976352f420faa99963a267da4dafe3f13f4823a435fa1f0.jpg new file mode 100644 index 00000000..f0c339cd Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-396_104f0935016921157976352f420faa99963a267da4dafe3f13f4823a435fa1f0.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-396_1fd0eee0afc25d347f02f2e4d74bebad0076139c51b4a94842c1eb2f90a5f85c.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-396_1fd0eee0afc25d347f02f2e4d74bebad0076139c51b4a94842c1eb2f90a5f85c.jpg new file mode 100644 index 00000000..f2e5c7f1 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-396_1fd0eee0afc25d347f02f2e4d74bebad0076139c51b4a94842c1eb2f90a5f85c.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-396_4baedfe7047300e840a28438adfb5e236d15d36039a76f70d1c5c1a0bbf79552.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-396_4baedfe7047300e840a28438adfb5e236d15d36039a76f70d1c5c1a0bbf79552.jpg new file mode 100644 index 00000000..c0dae65d Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-396_4baedfe7047300e840a28438adfb5e236d15d36039a76f70d1c5c1a0bbf79552.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-396_587189061e17b1d73017e0d7823139b3269d4ae856edbfdf430d4a13a97c6ee2.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-396_587189061e17b1d73017e0d7823139b3269d4ae856edbfdf430d4a13a97c6ee2.jpg new file mode 100644 index 00000000..22ad375f Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-396_587189061e17b1d73017e0d7823139b3269d4ae856edbfdf430d4a13a97c6ee2.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-396_736d99ed3c4dca3272a542c0303af2330f015c43d875d6cfdf8fd2590219a3a7.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-396_736d99ed3c4dca3272a542c0303af2330f015c43d875d6cfdf8fd2590219a3a7.jpg new file mode 100644 index 00000000..c744dadc Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-396_736d99ed3c4dca3272a542c0303af2330f015c43d875d6cfdf8fd2590219a3a7.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-396_97cdccaf63c16931483ba8a8d4eca37e9225b625db18676d01b4eaaf060eff90.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-396_97cdccaf63c16931483ba8a8d4eca37e9225b625db18676d01b4eaaf060eff90.jpg new file mode 100644 index 00000000..27b8db4c Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-396_97cdccaf63c16931483ba8a8d4eca37e9225b625db18676d01b4eaaf060eff90.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-396_b72b7e5914ffcc3b16c2db6bd5dc39ddd1beedfc9f3337f2430617f61716ad41.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-396_b72b7e5914ffcc3b16c2db6bd5dc39ddd1beedfc9f3337f2430617f61716ad41.jpg new file mode 100644 index 00000000..ab1447aa Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-396_b72b7e5914ffcc3b16c2db6bd5dc39ddd1beedfc9f3337f2430617f61716ad41.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-397_9fc0c93c6b404bddc555fe49e43b473eb5ecc351db663ed0119553e4d0b72f26.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-397_9fc0c93c6b404bddc555fe49e43b473eb5ecc351db663ed0119553e4d0b72f26.jpg new file mode 100644 index 00000000..8831376c Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-397_9fc0c93c6b404bddc555fe49e43b473eb5ecc351db663ed0119553e4d0b72f26.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-398_3e847f72941f85799026322d44c0f9c2747c46939599c8546534c44e22eb8aab.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-398_3e847f72941f85799026322d44c0f9c2747c46939599c8546534c44e22eb8aab.jpg new file mode 100644 index 00000000..61647ad1 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-398_3e847f72941f85799026322d44c0f9c2747c46939599c8546534c44e22eb8aab.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-398_579e93c7a940328535d4e271be78188e2a649d02815e28744c3140dd6b6398a6.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-398_579e93c7a940328535d4e271be78188e2a649d02815e28744c3140dd6b6398a6.jpg new file mode 100644 index 00000000..398d907f Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-398_579e93c7a940328535d4e271be78188e2a649d02815e28744c3140dd6b6398a6.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-398_60567e484946a0878cd753de5053bdc26fd74cddba18dd0f880c77321374e42b.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-398_60567e484946a0878cd753de5053bdc26fd74cddba18dd0f880c77321374e42b.jpg new file mode 100644 index 00000000..4c1e8f49 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-398_60567e484946a0878cd753de5053bdc26fd74cddba18dd0f880c77321374e42b.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-398_900d35928367a3e3a32f32c7c41632da96cdc6899c6060112fc8b8bbfa6b3367.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-398_900d35928367a3e3a32f32c7c41632da96cdc6899c6060112fc8b8bbfa6b3367.jpg new file mode 100644 index 00000000..9e7e03c5 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-398_900d35928367a3e3a32f32c7c41632da96cdc6899c6060112fc8b8bbfa6b3367.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-398_bb00ba2d876a806d87805bfbd63eacf82b1eb9c01aaf2ee0fcea4abb4d441320.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-398_bb00ba2d876a806d87805bfbd63eacf82b1eb9c01aaf2ee0fcea4abb4d441320.jpg new file mode 100644 index 00000000..8417bc7f Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-398_bb00ba2d876a806d87805bfbd63eacf82b1eb9c01aaf2ee0fcea4abb4d441320.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-398_dedf9f1fec601adccdd933b09e67b6f753fc60e4fd66d3b4f714564c5a29f2cd.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-398_dedf9f1fec601adccdd933b09e67b6f753fc60e4fd66d3b4f714564c5a29f2cd.jpg new file mode 100644 index 00000000..ae1e55a9 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-398_dedf9f1fec601adccdd933b09e67b6f753fc60e4fd66d3b4f714564c5a29f2cd.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-398_f1093f6cfc7d3b1b4e9baeeac58cede0ca5e62f228b0da531ec6497a4b432233.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-398_f1093f6cfc7d3b1b4e9baeeac58cede0ca5e62f228b0da531ec6497a4b432233.jpg new file mode 100644 index 00000000..9dafd447 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-398_f1093f6cfc7d3b1b4e9baeeac58cede0ca5e62f228b0da531ec6497a4b432233.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-399_05553bd872b916ba7f954b3d64145f89b164db044e6d9da976240d46b88501af.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-399_05553bd872b916ba7f954b3d64145f89b164db044e6d9da976240d46b88501af.jpg new file mode 100644 index 00000000..7908527b Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-399_05553bd872b916ba7f954b3d64145f89b164db044e6d9da976240d46b88501af.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-399_2daec7443b77e2e3722a380d3a9aa3d925e507243f72c669011a09dba8cc4ce1.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-399_2daec7443b77e2e3722a380d3a9aa3d925e507243f72c669011a09dba8cc4ce1.jpg new file mode 100644 index 00000000..d7a1f5dc Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-399_2daec7443b77e2e3722a380d3a9aa3d925e507243f72c669011a09dba8cc4ce1.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-399_30ac29df8a6f22bc77ea7c66f5f4a33029fd9e091eac429d48dc71e9edab7741.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-399_30ac29df8a6f22bc77ea7c66f5f4a33029fd9e091eac429d48dc71e9edab7741.jpg new file mode 100644 index 00000000..8982e6ec Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-399_30ac29df8a6f22bc77ea7c66f5f4a33029fd9e091eac429d48dc71e9edab7741.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-399_c00e5d0111644cb77db1fe3891496442afb10474f13d4bb74645ec4224eb8d80.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-399_c00e5d0111644cb77db1fe3891496442afb10474f13d4bb74645ec4224eb8d80.jpg new file mode 100644 index 00000000..bc13cc04 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-399_c00e5d0111644cb77db1fe3891496442afb10474f13d4bb74645ec4224eb8d80.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-399_c2ace47c1a10b4af8c71682f052b31a8f40deb27ad6430b83d99c83a409f06be.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-399_c2ace47c1a10b4af8c71682f052b31a8f40deb27ad6430b83d99c83a409f06be.jpg new file mode 100644 index 00000000..5c8820b1 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-399_c2ace47c1a10b4af8c71682f052b31a8f40deb27ad6430b83d99c83a409f06be.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-399_f8a1c85bb48fd50c6539fd110be8bed2a17e3ab71ced51f95af64fde3f914a46.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-399_f8a1c85bb48fd50c6539fd110be8bed2a17e3ab71ced51f95af64fde3f914a46.jpg new file mode 100644 index 00000000..0f5d5f8f Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-399_f8a1c85bb48fd50c6539fd110be8bed2a17e3ab71ced51f95af64fde3f914a46.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-400_07abc9885e6a82ab7ca3d98faffef5c7b0632da5266008864e2c605ec806a5d0.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-400_07abc9885e6a82ab7ca3d98faffef5c7b0632da5266008864e2c605ec806a5d0.jpg new file mode 100644 index 00000000..6fe45603 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-400_07abc9885e6a82ab7ca3d98faffef5c7b0632da5266008864e2c605ec806a5d0.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-400_424a5ff3ada12c070edc1174b0457d49f6f2b8f9218c3eca594464fabcb27220.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-400_424a5ff3ada12c070edc1174b0457d49f6f2b8f9218c3eca594464fabcb27220.jpg new file mode 100644 index 00000000..630032ae Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-400_424a5ff3ada12c070edc1174b0457d49f6f2b8f9218c3eca594464fabcb27220.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-400_5c55f913eefb640c0f1b476ae731fa57c8f26c7563cf584e81cd2b9ed4df6f6d.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-400_5c55f913eefb640c0f1b476ae731fa57c8f26c7563cf584e81cd2b9ed4df6f6d.jpg new file mode 100644 index 00000000..da63c5af Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-400_5c55f913eefb640c0f1b476ae731fa57c8f26c7563cf584e81cd2b9ed4df6f6d.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-400_bcf010c135ba8e05f7c489f54b96c51504fb5c4435ef1d8e9ccd0eb234503fa6.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-400_bcf010c135ba8e05f7c489f54b96c51504fb5c4435ef1d8e9ccd0eb234503fa6.jpg new file mode 100644 index 00000000..44f8ad37 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-400_bcf010c135ba8e05f7c489f54b96c51504fb5c4435ef1d8e9ccd0eb234503fa6.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-400_bd6ebccd41aff2b26335dad8cd1474eb69f45a4a891cb0f51846a43e7beafbc0.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-400_bd6ebccd41aff2b26335dad8cd1474eb69f45a4a891cb0f51846a43e7beafbc0.jpg new file mode 100644 index 00000000..d76706c0 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-400_bd6ebccd41aff2b26335dad8cd1474eb69f45a4a891cb0f51846a43e7beafbc0.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-400_d2c7c4631431054840aa1592f9f2d1cfc305933033b20a179c58384ce105e301.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-400_d2c7c4631431054840aa1592f9f2d1cfc305933033b20a179c58384ce105e301.jpg new file mode 100644 index 00000000..597fc304 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-400_d2c7c4631431054840aa1592f9f2d1cfc305933033b20a179c58384ce105e301.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-401_1dd51df32b6c2022d37db6334eebf55d63f44315079951d19cf06ff5396ec9c8.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-401_1dd51df32b6c2022d37db6334eebf55d63f44315079951d19cf06ff5396ec9c8.jpg new file mode 100644 index 00000000..ae5eca83 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-401_1dd51df32b6c2022d37db6334eebf55d63f44315079951d19cf06ff5396ec9c8.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-401_1fa7cb13db9e1da462a00db0e3c0e46c749c4a3bece28eac81bc82d44fdf890f.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-401_1fa7cb13db9e1da462a00db0e3c0e46c749c4a3bece28eac81bc82d44fdf890f.jpg new file mode 100644 index 00000000..6bc639a9 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-401_1fa7cb13db9e1da462a00db0e3c0e46c749c4a3bece28eac81bc82d44fdf890f.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-401_749dbbe3494ee8d37674a6eadfcf89dcc21d9c463a1729b251ab619eeeba0662.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-401_749dbbe3494ee8d37674a6eadfcf89dcc21d9c463a1729b251ab619eeeba0662.jpg new file mode 100644 index 00000000..422f16aa Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-401_749dbbe3494ee8d37674a6eadfcf89dcc21d9c463a1729b251ab619eeeba0662.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-401_d2dec1b1aeeb28c30d4a86bcb685af6d657def114bc1bd24b38a8ec18da01c60.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-401_d2dec1b1aeeb28c30d4a86bcb685af6d657def114bc1bd24b38a8ec18da01c60.jpg new file mode 100644 index 00000000..6de94a17 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-401_d2dec1b1aeeb28c30d4a86bcb685af6d657def114bc1bd24b38a8ec18da01c60.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-401_f3413591639f5b74023045abab43a7150c06739007f295fef0ea9738c5d395fd.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-401_f3413591639f5b74023045abab43a7150c06739007f295fef0ea9738c5d395fd.jpg new file mode 100644 index 00000000..91c74796 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-401_f3413591639f5b74023045abab43a7150c06739007f295fef0ea9738c5d395fd.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-402_4021a4150facacce6f3f47e4279a4b1c59e886589bc1949c029685c1fcdedde9.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-402_4021a4150facacce6f3f47e4279a4b1c59e886589bc1949c029685c1fcdedde9.jpg new file mode 100644 index 00000000..49f26b3f Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-402_4021a4150facacce6f3f47e4279a4b1c59e886589bc1949c029685c1fcdedde9.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-403_016a82b195acd55b89f16ccfae044bcc87d3595172e60184b80669e135f061e5.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-403_016a82b195acd55b89f16ccfae044bcc87d3595172e60184b80669e135f061e5.jpg new file mode 100644 index 00000000..90144a58 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-403_016a82b195acd55b89f16ccfae044bcc87d3595172e60184b80669e135f061e5.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-404_28d33b8ad96b2b2cda16fb04256cf5df08499c92d90878b39a7ada945df6a87b.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-404_28d33b8ad96b2b2cda16fb04256cf5df08499c92d90878b39a7ada945df6a87b.jpg new file mode 100644 index 00000000..693fe922 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-404_28d33b8ad96b2b2cda16fb04256cf5df08499c92d90878b39a7ada945df6a87b.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-404_2e75e7157851a965abebd2d90ffc38fa0c7a6e186dc35fdf08ecb1294cb29343.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-404_2e75e7157851a965abebd2d90ffc38fa0c7a6e186dc35fdf08ecb1294cb29343.jpg new file mode 100644 index 00000000..3b213d94 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-404_2e75e7157851a965abebd2d90ffc38fa0c7a6e186dc35fdf08ecb1294cb29343.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-413_c88d2a3ce79f8c6c5b924bf82ed1b30a501327c1281d8a9ad4db05ad04586e83.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-413_c88d2a3ce79f8c6c5b924bf82ed1b30a501327c1281d8a9ad4db05ad04586e83.jpg new file mode 100644 index 00000000..bb737580 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-413_c88d2a3ce79f8c6c5b924bf82ed1b30a501327c1281d8a9ad4db05ad04586e83.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-427_29460e91362a631c6e74197f3d1e34b20428cdcf1fd54582b26266dec80f1ea8.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-427_29460e91362a631c6e74197f3d1e34b20428cdcf1fd54582b26266dec80f1ea8.jpg new file mode 100644 index 00000000..07c6355f Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-427_29460e91362a631c6e74197f3d1e34b20428cdcf1fd54582b26266dec80f1ea8.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-429_01540f192aeb780208556db084304d00c88d9362be94a6b4ad6cfcda15e8d658.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-429_01540f192aeb780208556db084304d00c88d9362be94a6b4ad6cfcda15e8d658.jpg new file mode 100644 index 00000000..a6d566f4 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-429_01540f192aeb780208556db084304d00c88d9362be94a6b4ad6cfcda15e8d658.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-429_9eab36e055df7ae1e75062dfe8ce8527a79900a1040a8b707122cdcbade1cc02.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-429_9eab36e055df7ae1e75062dfe8ce8527a79900a1040a8b707122cdcbade1cc02.jpg new file mode 100644 index 00000000..f9b2c0f5 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-429_9eab36e055df7ae1e75062dfe8ce8527a79900a1040a8b707122cdcbade1cc02.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-431_8d54073203e1bb4957d16cc36ba48cda6337a735f346c47cf1fbe67934e3af73.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-431_8d54073203e1bb4957d16cc36ba48cda6337a735f346c47cf1fbe67934e3af73.jpg new file mode 100644 index 00000000..f1017334 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-431_8d54073203e1bb4957d16cc36ba48cda6337a735f346c47cf1fbe67934e3af73.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-431_eaa980c1f220dd59324168f134df3624983f2bb7e24a9d7bd12c5a78037df88e.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-431_eaa980c1f220dd59324168f134df3624983f2bb7e24a9d7bd12c5a78037df88e.jpg new file mode 100644 index 00000000..ac5a5003 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-431_eaa980c1f220dd59324168f134df3624983f2bb7e24a9d7bd12c5a78037df88e.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-432_b577deeefe5e1301ab205a51ab928b9a75a8ea94a5ffe40df56d3a93b899e519.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-432_b577deeefe5e1301ab205a51ab928b9a75a8ea94a5ffe40df56d3a93b899e519.jpg new file mode 100644 index 00000000..0ca084f6 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-432_b577deeefe5e1301ab205a51ab928b9a75a8ea94a5ffe40df56d3a93b899e519.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-433_97afa398ebad3d67bb1d4d5a8214769a8f028ae153e2bb3dc2fe78f8d3577fa0.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-433_97afa398ebad3d67bb1d4d5a8214769a8f028ae153e2bb3dc2fe78f8d3577fa0.jpg new file mode 100644 index 00000000..de284f9b Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-433_97afa398ebad3d67bb1d4d5a8214769a8f028ae153e2bb3dc2fe78f8d3577fa0.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-434_a25a23f53784a8ec9287724e7fd79a6699c0670ee71d3b945926545e35993013.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-434_a25a23f53784a8ec9287724e7fd79a6699c0670ee71d3b945926545e35993013.jpg new file mode 100644 index 00000000..851f261f Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-434_a25a23f53784a8ec9287724e7fd79a6699c0670ee71d3b945926545e35993013.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-434_af57c0d6db14c103667713fd1d93e8f6b5c182a5d826e61bfc600eaebefec384.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-434_af57c0d6db14c103667713fd1d93e8f6b5c182a5d826e61bfc600eaebefec384.jpg new file mode 100644 index 00000000..30db25f4 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-434_af57c0d6db14c103667713fd1d93e8f6b5c182a5d826e61bfc600eaebefec384.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-435_778d75e618c65082244b4f1874c1294d4f61f56584a4e9c8015be645d2846627.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-435_778d75e618c65082244b4f1874c1294d4f61f56584a4e9c8015be645d2846627.jpg new file mode 100644 index 00000000..d552831f Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-435_778d75e618c65082244b4f1874c1294d4f61f56584a4e9c8015be645d2846627.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-435_7b257f98fcb2bf5a43bf82e365e3277bc595034a866ad5f3c1fafd5b94aff362.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-435_7b257f98fcb2bf5a43bf82e365e3277bc595034a866ad5f3c1fafd5b94aff362.jpg new file mode 100644 index 00000000..2903adb2 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-435_7b257f98fcb2bf5a43bf82e365e3277bc595034a866ad5f3c1fafd5b94aff362.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-435_8219b20bb066d8d8ee9aa65fe234fb6e19e0d2b90f816281132343ada8f3c239.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-435_8219b20bb066d8d8ee9aa65fe234fb6e19e0d2b90f816281132343ada8f3c239.jpg new file mode 100644 index 00000000..7be8263f Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-435_8219b20bb066d8d8ee9aa65fe234fb6e19e0d2b90f816281132343ada8f3c239.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-435_b40a1fff68434703e13e7923d2425cdb7572778cd8a673e37fdf5130396ce04b.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-435_b40a1fff68434703e13e7923d2425cdb7572778cd8a673e37fdf5130396ce04b.jpg new file mode 100644 index 00000000..96060534 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-435_b40a1fff68434703e13e7923d2425cdb7572778cd8a673e37fdf5130396ce04b.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-435_e380905707e0194e9ec6de1ffaed7efd10bb977cd39e5f319cec26011f16659d.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-435_e380905707e0194e9ec6de1ffaed7efd10bb977cd39e5f319cec26011f16659d.jpg new file mode 100644 index 00000000..6064a979 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-435_e380905707e0194e9ec6de1ffaed7efd10bb977cd39e5f319cec26011f16659d.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-436_37b4d8839d34fb6e3530c0ec22622ca76c0e2fab4bf0283003f6d870cbed8e96.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-436_37b4d8839d34fb6e3530c0ec22622ca76c0e2fab4bf0283003f6d870cbed8e96.jpg new file mode 100644 index 00000000..57a04617 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-436_37b4d8839d34fb6e3530c0ec22622ca76c0e2fab4bf0283003f6d870cbed8e96.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-436_dd0dcb4a485c805dbcc5abb4ae30261a7c5ced37fd7449ac19d2415496347bbb.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-436_dd0dcb4a485c805dbcc5abb4ae30261a7c5ced37fd7449ac19d2415496347bbb.jpg new file mode 100644 index 00000000..abdf91dc Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-436_dd0dcb4a485c805dbcc5abb4ae30261a7c5ced37fd7449ac19d2415496347bbb.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-436_fa943596f9d1a41bc8de58d4364a9e4a3c9c0e91d228f25f8b87fe58795a0670.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-436_fa943596f9d1a41bc8de58d4364a9e4a3c9c0e91d228f25f8b87fe58795a0670.jpg new file mode 100644 index 00000000..fe16d926 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-436_fa943596f9d1a41bc8de58d4364a9e4a3c9c0e91d228f25f8b87fe58795a0670.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-436_fc87d9b5b8f5ca0cd3050285d6dea2e2c529eea9b837fad11002ebab5592763d.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-436_fc87d9b5b8f5ca0cd3050285d6dea2e2c529eea9b837fad11002ebab5592763d.jpg new file mode 100644 index 00000000..03f604b9 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-436_fc87d9b5b8f5ca0cd3050285d6dea2e2c529eea9b837fad11002ebab5592763d.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-437_f409e66836a7e9fa61df39579676198b76557ddf2f74785c0bf6a43f16fea0cf.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-437_f409e66836a7e9fa61df39579676198b76557ddf2f74785c0bf6a43f16fea0cf.jpg new file mode 100644 index 00000000..5d3435a7 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-437_f409e66836a7e9fa61df39579676198b76557ddf2f74785c0bf6a43f16fea0cf.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-438_6aa72dadd8aa2806bea974a97982cde45cadc66f50c6e0a6d1bdce8170f386b6.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-438_6aa72dadd8aa2806bea974a97982cde45cadc66f50c6e0a6d1bdce8170f386b6.jpg new file mode 100644 index 00000000..95489f77 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-438_6aa72dadd8aa2806bea974a97982cde45cadc66f50c6e0a6d1bdce8170f386b6.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-439_e79e0816d8afd26ac9d2521072c05e8557aa912b4222900bd1c5a98ce9d9a4b5.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-439_e79e0816d8afd26ac9d2521072c05e8557aa912b4222900bd1c5a98ce9d9a4b5.jpg new file mode 100644 index 00000000..40c65577 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-439_e79e0816d8afd26ac9d2521072c05e8557aa912b4222900bd1c5a98ce9d9a4b5.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-441_8f75f37bc4258965dd48ad992b15d6e07ae58b7e496007f37154a7493422bbbb.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-441_8f75f37bc4258965dd48ad992b15d6e07ae58b7e496007f37154a7493422bbbb.jpg new file mode 100644 index 00000000..720626b3 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-441_8f75f37bc4258965dd48ad992b15d6e07ae58b7e496007f37154a7493422bbbb.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-442_739ad0e3a1ffaf95de72787f33da5edd2f78a112ecb2a5344bb4f8381582a71d.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-442_739ad0e3a1ffaf95de72787f33da5edd2f78a112ecb2a5344bb4f8381582a71d.jpg new file mode 100644 index 00000000..97fca848 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-442_739ad0e3a1ffaf95de72787f33da5edd2f78a112ecb2a5344bb4f8381582a71d.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-442_8eb015e85ef291013d3ef2608c2e39eb1f9b9ae3fdc2c85f868df84b0cabbdae.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-442_8eb015e85ef291013d3ef2608c2e39eb1f9b9ae3fdc2c85f868df84b0cabbdae.jpg new file mode 100644 index 00000000..268b2350 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-442_8eb015e85ef291013d3ef2608c2e39eb1f9b9ae3fdc2c85f868df84b0cabbdae.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-442_ba0f54746fc42e9e71af9c34d7d95db0bc7d640e36ba8c3158f0b9dd9af2985d.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-442_ba0f54746fc42e9e71af9c34d7d95db0bc7d640e36ba8c3158f0b9dd9af2985d.jpg new file mode 100644 index 00000000..7688f43a Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-442_ba0f54746fc42e9e71af9c34d7d95db0bc7d640e36ba8c3158f0b9dd9af2985d.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-442_fb92d62b0eb77bffceee58985d7280738843f82db1f44a24c8be081b7b0093fa.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-442_fb92d62b0eb77bffceee58985d7280738843f82db1f44a24c8be081b7b0093fa.jpg new file mode 100644 index 00000000..1931d3bd Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-442_fb92d62b0eb77bffceee58985d7280738843f82db1f44a24c8be081b7b0093fa.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-443_031c4e3a010ea4529ef73f9d0dba8dcf6554915b218c4fed2de4a2a6b8ca4e3d.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-443_031c4e3a010ea4529ef73f9d0dba8dcf6554915b218c4fed2de4a2a6b8ca4e3d.jpg new file mode 100644 index 00000000..84c7a208 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-443_031c4e3a010ea4529ef73f9d0dba8dcf6554915b218c4fed2de4a2a6b8ca4e3d.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-443_51286827a2aeebb0dbaeb50fad9388d9ad98d016f71615a4c929ec11fa70410b.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-443_51286827a2aeebb0dbaeb50fad9388d9ad98d016f71615a4c929ec11fa70410b.jpg new file mode 100644 index 00000000..5b2d6e58 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-443_51286827a2aeebb0dbaeb50fad9388d9ad98d016f71615a4c929ec11fa70410b.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-443_5427a93a265c101d80084f8eaceb649182dd171d16a027939e55ead8f4553c22.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-443_5427a93a265c101d80084f8eaceb649182dd171d16a027939e55ead8f4553c22.jpg new file mode 100644 index 00000000..202268f1 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-443_5427a93a265c101d80084f8eaceb649182dd171d16a027939e55ead8f4553c22.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-443_990541b1cf0eedda4882d187f1dbe99ae3d59fe5fae809a44dee210e213e6fee.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-443_990541b1cf0eedda4882d187f1dbe99ae3d59fe5fae809a44dee210e213e6fee.jpg new file mode 100644 index 00000000..9098ad51 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-443_990541b1cf0eedda4882d187f1dbe99ae3d59fe5fae809a44dee210e213e6fee.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-443_d1e039e07406ff4ad12d684466238e417b452ce048c9bc31e5a1a41474e8fc04.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-443_d1e039e07406ff4ad12d684466238e417b452ce048c9bc31e5a1a41474e8fc04.jpg new file mode 100644 index 00000000..19a22308 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-443_d1e039e07406ff4ad12d684466238e417b452ce048c9bc31e5a1a41474e8fc04.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-444_98c66fd5746cbb024bda81df608fd6698a354e6c4cb6502fc872f3fe3409c07d.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-444_98c66fd5746cbb024bda81df608fd6698a354e6c4cb6502fc872f3fe3409c07d.jpg new file mode 100644 index 00000000..ab82ea63 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-444_98c66fd5746cbb024bda81df608fd6698a354e6c4cb6502fc872f3fe3409c07d.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-444_d753cd109ac617cfa81e090ae7067c7baf5062a1e18a9cc21064b01ad763dcf7.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-444_d753cd109ac617cfa81e090ae7067c7baf5062a1e18a9cc21064b01ad763dcf7.jpg new file mode 100644 index 00000000..d2abb612 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-444_d753cd109ac617cfa81e090ae7067c7baf5062a1e18a9cc21064b01ad763dcf7.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-445_5bc53b9ed7fd9f2900b2dc110d9eb11e68f996810fac1ecdc2028697c67aed3d.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-445_5bc53b9ed7fd9f2900b2dc110d9eb11e68f996810fac1ecdc2028697c67aed3d.jpg new file mode 100644 index 00000000..14e35d80 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-445_5bc53b9ed7fd9f2900b2dc110d9eb11e68f996810fac1ecdc2028697c67aed3d.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-445_d1611529f1f3b439f377dd55e3fc43d07957abca3e41befa99cdd7dade1c9b33.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-445_d1611529f1f3b439f377dd55e3fc43d07957abca3e41befa99cdd7dade1c9b33.jpg new file mode 100644 index 00000000..a797f0a8 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-445_d1611529f1f3b439f377dd55e3fc43d07957abca3e41befa99cdd7dade1c9b33.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-446_0e03a8892b0b392a9b767e2f1e80c60951e1345af7210eb052fb2ad71e405493.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-446_0e03a8892b0b392a9b767e2f1e80c60951e1345af7210eb052fb2ad71e405493.jpg new file mode 100644 index 00000000..eaf8c2ff Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-446_0e03a8892b0b392a9b767e2f1e80c60951e1345af7210eb052fb2ad71e405493.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-446_0f6d3d9417347496daf2baac4c0e6768c3b0f2519112d1139b319918fe20b847.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-446_0f6d3d9417347496daf2baac4c0e6768c3b0f2519112d1139b319918fe20b847.jpg new file mode 100644 index 00000000..452a9355 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-446_0f6d3d9417347496daf2baac4c0e6768c3b0f2519112d1139b319918fe20b847.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-446_73ac69057b6675d7707844c33ea3e53f989601f28d8d633f4ff7433df51a5ee7.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-446_73ac69057b6675d7707844c33ea3e53f989601f28d8d633f4ff7433df51a5ee7.jpg new file mode 100644 index 00000000..cb6c1274 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-446_73ac69057b6675d7707844c33ea3e53f989601f28d8d633f4ff7433df51a5ee7.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-446_b6ddf220df5891cdc3f8ba97a3fb19f7c3ed6bc45bb0583343aa66d56397b247.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-446_b6ddf220df5891cdc3f8ba97a3fb19f7c3ed6bc45bb0583343aa66d56397b247.jpg new file mode 100644 index 00000000..f76db8f0 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-446_b6ddf220df5891cdc3f8ba97a3fb19f7c3ed6bc45bb0583343aa66d56397b247.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-446_d0893c23873ef1e5855774d2de1529de9755c1606e22522273f93e7f1dbc389c.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-446_d0893c23873ef1e5855774d2de1529de9755c1606e22522273f93e7f1dbc389c.jpg new file mode 100644 index 00000000..4ce745e1 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-446_d0893c23873ef1e5855774d2de1529de9755c1606e22522273f93e7f1dbc389c.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-448_6edb9bb28f256d73be00da7b3916f4b9e5919629a886de9a403388fc194ec64c.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-448_6edb9bb28f256d73be00da7b3916f4b9e5919629a886de9a403388fc194ec64c.jpg new file mode 100644 index 00000000..2fed88b7 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-448_6edb9bb28f256d73be00da7b3916f4b9e5919629a886de9a403388fc194ec64c.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-448_7b83c53cadfa9d0aabfb731bd34cd15f7078b05e74ee1742535f76c841a587c0.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-448_7b83c53cadfa9d0aabfb731bd34cd15f7078b05e74ee1742535f76c841a587c0.jpg new file mode 100644 index 00000000..4c358dad Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-448_7b83c53cadfa9d0aabfb731bd34cd15f7078b05e74ee1742535f76c841a587c0.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-448_b37c753784b44147b038362413d309dfe84c50fb22f6d73cad1bdb9ba7eb6d9b.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-448_b37c753784b44147b038362413d309dfe84c50fb22f6d73cad1bdb9ba7eb6d9b.jpg new file mode 100644 index 00000000..62036ab8 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-448_b37c753784b44147b038362413d309dfe84c50fb22f6d73cad1bdb9ba7eb6d9b.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-448_b789f654a0a1753921fc63f9661e18d78f2c5d0244652165acdeaad4670a92d0.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-448_b789f654a0a1753921fc63f9661e18d78f2c5d0244652165acdeaad4670a92d0.jpg new file mode 100644 index 00000000..997c5c30 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-448_b789f654a0a1753921fc63f9661e18d78f2c5d0244652165acdeaad4670a92d0.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-449_0dc1da0404ab99889fd4cb242e3bd290652eb4eb06d655b24619ed4028ec826c.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-449_0dc1da0404ab99889fd4cb242e3bd290652eb4eb06d655b24619ed4028ec826c.jpg new file mode 100644 index 00000000..05cecb71 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-449_0dc1da0404ab99889fd4cb242e3bd290652eb4eb06d655b24619ed4028ec826c.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-449_59dc0486dbed19c1034edd4c357190a81ffe491d3d7eed2bb2c6376314c486c7.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-449_59dc0486dbed19c1034edd4c357190a81ffe491d3d7eed2bb2c6376314c486c7.jpg new file mode 100644 index 00000000..67dfe30c Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-449_59dc0486dbed19c1034edd4c357190a81ffe491d3d7eed2bb2c6376314c486c7.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-449_7bc8e46e93881007a582dd39fb40ec10ca5c97385286a323488ce6c41c6a34cd.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-449_7bc8e46e93881007a582dd39fb40ec10ca5c97385286a323488ce6c41c6a34cd.jpg new file mode 100644 index 00000000..f3fea23e Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-449_7bc8e46e93881007a582dd39fb40ec10ca5c97385286a323488ce6c41c6a34cd.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-449_d823e3093b0ad5c1b94c44769398f5f27f69e193c076554e2e8235560e923e6b.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-449_d823e3093b0ad5c1b94c44769398f5f27f69e193c076554e2e8235560e923e6b.jpg new file mode 100644 index 00000000..85ba68ea Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-449_d823e3093b0ad5c1b94c44769398f5f27f69e193c076554e2e8235560e923e6b.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-450_6090b7687404c9547c8f14823e298ed2446f2e3c7c096ff110080c9935f71d55.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-450_6090b7687404c9547c8f14823e298ed2446f2e3c7c096ff110080c9935f71d55.jpg new file mode 100644 index 00000000..4fc5f3dd Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-450_6090b7687404c9547c8f14823e298ed2446f2e3c7c096ff110080c9935f71d55.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-450_a037222c6b88a07135c1fb2788f14a75669bd4c8f01b3b853e6bc9e154d3b302.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-450_a037222c6b88a07135c1fb2788f14a75669bd4c8f01b3b853e6bc9e154d3b302.jpg new file mode 100644 index 00000000..caa96887 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-450_a037222c6b88a07135c1fb2788f14a75669bd4c8f01b3b853e6bc9e154d3b302.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-451_fc5e6ec025f5d2ad2cc58890a31055639904dac19ea27e926946f675364aa026.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-451_fc5e6ec025f5d2ad2cc58890a31055639904dac19ea27e926946f675364aa026.jpg new file mode 100644 index 00000000..7968f9d7 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-451_fc5e6ec025f5d2ad2cc58890a31055639904dac19ea27e926946f675364aa026.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-453_ee0bf3bb8dc2f23e67b0042cb0fbb44582aa8a794287cdba017e41d08cb91210.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-453_ee0bf3bb8dc2f23e67b0042cb0fbb44582aa8a794287cdba017e41d08cb91210.jpg new file mode 100644 index 00000000..5c27a9c9 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-453_ee0bf3bb8dc2f23e67b0042cb0fbb44582aa8a794287cdba017e41d08cb91210.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-466_0e4e073df9ba5d242b219151295ff47b26a9bb86b978c7d3aa561966752c07bb.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-466_0e4e073df9ba5d242b219151295ff47b26a9bb86b978c7d3aa561966752c07bb.jpg new file mode 100644 index 00000000..47e186cf Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-466_0e4e073df9ba5d242b219151295ff47b26a9bb86b978c7d3aa561966752c07bb.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-469_3c96e71c95f192b28d97fc8437730a8281962a19d55f0c4e561af530c6458b72.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-469_3c96e71c95f192b28d97fc8437730a8281962a19d55f0c4e561af530c6458b72.jpg new file mode 100644 index 00000000..cfb665e2 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-469_3c96e71c95f192b28d97fc8437730a8281962a19d55f0c4e561af530c6458b72.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-469_64cf4b39ff3cd68e891b7ea26bebea3f595bbc5b3ea7af5a76d4d95904ca3b9f.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-469_64cf4b39ff3cd68e891b7ea26bebea3f595bbc5b3ea7af5a76d4d95904ca3b9f.jpg new file mode 100644 index 00000000..a975ce5f Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-469_64cf4b39ff3cd68e891b7ea26bebea3f595bbc5b3ea7af5a76d4d95904ca3b9f.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-469_bc51ca27ea45df3b15bf84651b88a05ecaa35ae89a43f3eff4a464b7a4fae73e.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-469_bc51ca27ea45df3b15bf84651b88a05ecaa35ae89a43f3eff4a464b7a4fae73e.jpg new file mode 100644 index 00000000..1c870d64 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-469_bc51ca27ea45df3b15bf84651b88a05ecaa35ae89a43f3eff4a464b7a4fae73e.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-469_cc2150870f2e2312302b75eef24f278f27eb5ba3a92a83a251b45692cadd9d73.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-469_cc2150870f2e2312302b75eef24f278f27eb5ba3a92a83a251b45692cadd9d73.jpg new file mode 100644 index 00000000..37741189 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-469_cc2150870f2e2312302b75eef24f278f27eb5ba3a92a83a251b45692cadd9d73.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-469_f2765c9d7374a3b24e16811147e97fa597bc807d32bc32d6ebad163eed5682c3.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-469_f2765c9d7374a3b24e16811147e97fa597bc807d32bc32d6ebad163eed5682c3.jpg new file mode 100644 index 00000000..e21eec0c Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-469_f2765c9d7374a3b24e16811147e97fa597bc807d32bc32d6ebad163eed5682c3.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-470_1223531b36a8f36460cb4cf02a18883696b6a76784ae154818e3217a08179ebc.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-470_1223531b36a8f36460cb4cf02a18883696b6a76784ae154818e3217a08179ebc.jpg new file mode 100644 index 00000000..ec8e9ba2 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-470_1223531b36a8f36460cb4cf02a18883696b6a76784ae154818e3217a08179ebc.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-471_049925c12591b1ea9a55c65a8bc0937c394bb7029ed3470da9db2664248a9d16.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-471_049925c12591b1ea9a55c65a8bc0937c394bb7029ed3470da9db2664248a9d16.jpg new file mode 100644 index 00000000..45808fb8 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-471_049925c12591b1ea9a55c65a8bc0937c394bb7029ed3470da9db2664248a9d16.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-471_bfa00acbfeeac6bfe139752f34ac43b4b01b1c8f19da74699d3f161ba0a7052a.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-471_bfa00acbfeeac6bfe139752f34ac43b4b01b1c8f19da74699d3f161ba0a7052a.jpg new file mode 100644 index 00000000..ce6faa22 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-471_bfa00acbfeeac6bfe139752f34ac43b4b01b1c8f19da74699d3f161ba0a7052a.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-476_6695a0ad083c0fb279cbf2d320b7a40a02f8077f9d1daa7191702e82afec8862.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-476_6695a0ad083c0fb279cbf2d320b7a40a02f8077f9d1daa7191702e82afec8862.jpg new file mode 100644 index 00000000..b1a163f7 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-476_6695a0ad083c0fb279cbf2d320b7a40a02f8077f9d1daa7191702e82afec8862.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-477_24ccbf966f6788800bd555ee093cb817ff55260ce73040640affd90a9823ebaa.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-477_24ccbf966f6788800bd555ee093cb817ff55260ce73040640affd90a9823ebaa.jpg new file mode 100644 index 00000000..bc5600e2 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-477_24ccbf966f6788800bd555ee093cb817ff55260ce73040640affd90a9823ebaa.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-477_ecc108430ef084452617acc9554bcbf030803b9ac73b929c8032fe82e512a553.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-477_ecc108430ef084452617acc9554bcbf030803b9ac73b929c8032fe82e512a553.jpg new file mode 100644 index 00000000..456524ba Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-477_ecc108430ef084452617acc9554bcbf030803b9ac73b929c8032fe82e512a553.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-478_4f68ff1b8350888a800ba23a4ef7a7852d5542cb4f04234a23b68e4593be2f86.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-478_4f68ff1b8350888a800ba23a4ef7a7852d5542cb4f04234a23b68e4593be2f86.jpg new file mode 100644 index 00000000..e1dae437 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-478_4f68ff1b8350888a800ba23a4ef7a7852d5542cb4f04234a23b68e4593be2f86.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-479_601cb52d4c6ae34e0c2513913d0fcbb6ca4903875987bfa0b4ecce390ef7bfd9.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-479_601cb52d4c6ae34e0c2513913d0fcbb6ca4903875987bfa0b4ecce390ef7bfd9.jpg new file mode 100644 index 00000000..784fa2f0 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-479_601cb52d4c6ae34e0c2513913d0fcbb6ca4903875987bfa0b4ecce390ef7bfd9.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-479_a7b35188f680f06ada1158f14daec38a671c386f2aa4c239433c5baa634298e6.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-479_a7b35188f680f06ada1158f14daec38a671c386f2aa4c239433c5baa634298e6.jpg new file mode 100644 index 00000000..b611e31c Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-479_a7b35188f680f06ada1158f14daec38a671c386f2aa4c239433c5baa634298e6.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-480_20748838a4bdb070d61e733a520c8578fb521c027092f222f207c8478da9a4b7.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-480_20748838a4bdb070d61e733a520c8578fb521c027092f222f207c8478da9a4b7.jpg new file mode 100644 index 00000000..24e89027 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-480_20748838a4bdb070d61e733a520c8578fb521c027092f222f207c8478da9a4b7.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-481_b1e90d354184adf442e34304415c90c57e8b959408f2f548f54ecc6c32499476.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-481_b1e90d354184adf442e34304415c90c57e8b959408f2f548f54ecc6c32499476.jpg new file mode 100644 index 00000000..965d7704 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-481_b1e90d354184adf442e34304415c90c57e8b959408f2f548f54ecc6c32499476.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-482_55197535ee145ba3693bbaba9665fa74f63dc9b72f689d3e635062f6a27528b0.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-482_55197535ee145ba3693bbaba9665fa74f63dc9b72f689d3e635062f6a27528b0.jpg new file mode 100644 index 00000000..2f1872c1 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-482_55197535ee145ba3693bbaba9665fa74f63dc9b72f689d3e635062f6a27528b0.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-483_592a1e011f0f55eeaf859c278b66bcc503e535f1ebfbac36c9a5a996381ce692.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-483_592a1e011f0f55eeaf859c278b66bcc503e535f1ebfbac36c9a5a996381ce692.jpg new file mode 100644 index 00000000..3f4d3349 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-483_592a1e011f0f55eeaf859c278b66bcc503e535f1ebfbac36c9a5a996381ce692.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-484_10d3337788050e5c1797ee5e9c2144b31fbfca65af745f455e0894f6e0939bcf.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-484_10d3337788050e5c1797ee5e9c2144b31fbfca65af745f455e0894f6e0939bcf.jpg new file mode 100644 index 00000000..6a59937d Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-484_10d3337788050e5c1797ee5e9c2144b31fbfca65af745f455e0894f6e0939bcf.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-484_5a875958711d034c090db5b13d5f7be03eb5806a2d84e97f4ffe433bb8820c99.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-484_5a875958711d034c090db5b13d5f7be03eb5806a2d84e97f4ffe433bb8820c99.jpg new file mode 100644 index 00000000..26415071 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-484_5a875958711d034c090db5b13d5f7be03eb5806a2d84e97f4ffe433bb8820c99.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-484_71620ac5a4bfce4beec1921587d0a5a394d679aa4b78b00f0bd954349494b077.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-484_71620ac5a4bfce4beec1921587d0a5a394d679aa4b78b00f0bd954349494b077.jpg new file mode 100644 index 00000000..b7b6ad1f Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-484_71620ac5a4bfce4beec1921587d0a5a394d679aa4b78b00f0bd954349494b077.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-484_74f1342f816b695029ae01ca5d10b470d350b5631cd7dbf499daabefd0ffed89.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-484_74f1342f816b695029ae01ca5d10b470d350b5631cd7dbf499daabefd0ffed89.jpg new file mode 100644 index 00000000..7bcea613 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-484_74f1342f816b695029ae01ca5d10b470d350b5631cd7dbf499daabefd0ffed89.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-484_851ebc8b5bbbc9b2774b55921a2fe4064c658f0ef4a0cdd71f565f0aec2e0ef1.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-484_851ebc8b5bbbc9b2774b55921a2fe4064c658f0ef4a0cdd71f565f0aec2e0ef1.jpg new file mode 100644 index 00000000..729d7493 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-484_851ebc8b5bbbc9b2774b55921a2fe4064c658f0ef4a0cdd71f565f0aec2e0ef1.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-484_c6f29fbc3e530f355c0db96e2e0fcbc59516b70e07029f6f258b46b304c2447b.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-484_c6f29fbc3e530f355c0db96e2e0fcbc59516b70e07029f6f258b46b304c2447b.jpg new file mode 100644 index 00000000..42db4255 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-484_c6f29fbc3e530f355c0db96e2e0fcbc59516b70e07029f6f258b46b304c2447b.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-484_d068bc376c6b3c5cdf4e3de0058b6a7ba58e8873c486e5c81203ef5bbbe09d35.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-484_d068bc376c6b3c5cdf4e3de0058b6a7ba58e8873c486e5c81203ef5bbbe09d35.jpg new file mode 100644 index 00000000..3c8db08d Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-484_d068bc376c6b3c5cdf4e3de0058b6a7ba58e8873c486e5c81203ef5bbbe09d35.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-484_deceab8b35d48339d8d88a3bbfb76c326b437eb1cebb71e389dc2341960bcbf2.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-484_deceab8b35d48339d8d88a3bbfb76c326b437eb1cebb71e389dc2341960bcbf2.jpg new file mode 100644 index 00000000..14a50541 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-484_deceab8b35d48339d8d88a3bbfb76c326b437eb1cebb71e389dc2341960bcbf2.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-485_d148dc7626ec213e6dba034ae4bea07ef2100262204b0873c7dde104ff260ae3.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-485_d148dc7626ec213e6dba034ae4bea07ef2100262204b0873c7dde104ff260ae3.jpg new file mode 100644 index 00000000..eef85909 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-485_d148dc7626ec213e6dba034ae4bea07ef2100262204b0873c7dde104ff260ae3.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-486_19635f7795d9aede192e38a4eece7eb92701a0ce9103390b1be021eb42e099c9.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-486_19635f7795d9aede192e38a4eece7eb92701a0ce9103390b1be021eb42e099c9.jpg new file mode 100644 index 00000000..9b9fee93 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-486_19635f7795d9aede192e38a4eece7eb92701a0ce9103390b1be021eb42e099c9.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-487_003f06f2a43ed0a45f8d8b460f703ed76339b2f97ae0e05055ded29374ad0cc7.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-487_003f06f2a43ed0a45f8d8b460f703ed76339b2f97ae0e05055ded29374ad0cc7.jpg new file mode 100644 index 00000000..78014e57 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-487_003f06f2a43ed0a45f8d8b460f703ed76339b2f97ae0e05055ded29374ad0cc7.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-487_2326768d8294c219370fcc2bef129a9c59c983e67c869a75fc0de8aff44a0674.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-487_2326768d8294c219370fcc2bef129a9c59c983e67c869a75fc0de8aff44a0674.jpg new file mode 100644 index 00000000..60cb57e2 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-487_2326768d8294c219370fcc2bef129a9c59c983e67c869a75fc0de8aff44a0674.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-488_6345e44501381ef122005e669b3427933fa2b544420633ab76ad7a52d0ebef7e.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-488_6345e44501381ef122005e669b3427933fa2b544420633ab76ad7a52d0ebef7e.jpg new file mode 100644 index 00000000..d1e7b4d0 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-488_6345e44501381ef122005e669b3427933fa2b544420633ab76ad7a52d0ebef7e.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-488_e361e30b130b4b0921051fd7e92cb43aa705116f9957defade214558f683dff3.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-488_e361e30b130b4b0921051fd7e92cb43aa705116f9957defade214558f683dff3.jpg new file mode 100644 index 00000000..99bab9be Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-488_e361e30b130b4b0921051fd7e92cb43aa705116f9957defade214558f683dff3.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-489_33d4c5faf792059d7a101da1262a7ef707fbf12ed493f5f0a5c66ce1208f218d.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-489_33d4c5faf792059d7a101da1262a7ef707fbf12ed493f5f0a5c66ce1208f218d.jpg new file mode 100644 index 00000000..408f2efa Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-489_33d4c5faf792059d7a101da1262a7ef707fbf12ed493f5f0a5c66ce1208f218d.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-489_50875593bdd9c3a9e1aea7dbe77198f144a3967434436d1b1406f61f673331e7.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-489_50875593bdd9c3a9e1aea7dbe77198f144a3967434436d1b1406f61f673331e7.jpg new file mode 100644 index 00000000..a94f5986 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-489_50875593bdd9c3a9e1aea7dbe77198f144a3967434436d1b1406f61f673331e7.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-489_8d28be60b5da1df794b4b0771f25e4ed454daf29e58d87e9d10a65dedb7a6a1c.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-489_8d28be60b5da1df794b4b0771f25e4ed454daf29e58d87e9d10a65dedb7a6a1c.jpg new file mode 100644 index 00000000..a4252152 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-489_8d28be60b5da1df794b4b0771f25e4ed454daf29e58d87e9d10a65dedb7a6a1c.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-489_909d3f92f96bba6a21120f99f5e205698de595f8d5ddba3899969dbc3467f816.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-489_909d3f92f96bba6a21120f99f5e205698de595f8d5ddba3899969dbc3467f816.jpg new file mode 100644 index 00000000..1958d9dc Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-489_909d3f92f96bba6a21120f99f5e205698de595f8d5ddba3899969dbc3467f816.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-489_c6a92b77d59ca5ae901ae0640172c0bf25c6acca8a0dd7b8876f87587ec9c1f3.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-489_c6a92b77d59ca5ae901ae0640172c0bf25c6acca8a0dd7b8876f87587ec9c1f3.jpg new file mode 100644 index 00000000..242ab81c Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-489_c6a92b77d59ca5ae901ae0640172c0bf25c6acca8a0dd7b8876f87587ec9c1f3.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-489_eeaddfe97b435074858ef1db3520aa8821f81c5efd2601da37c0bd3032793c74.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-489_eeaddfe97b435074858ef1db3520aa8821f81c5efd2601da37c0bd3032793c74.jpg new file mode 100644 index 00000000..b4b6835d Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-489_eeaddfe97b435074858ef1db3520aa8821f81c5efd2601da37c0bd3032793c74.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-489_f96fac92c6a31780d039f5ec0b4c5bd5a0897ab6146ff9bd2b3cc5d6498014cd.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-489_f96fac92c6a31780d039f5ec0b4c5bd5a0897ab6146ff9bd2b3cc5d6498014cd.jpg new file mode 100644 index 00000000..00a5c5b3 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-489_f96fac92c6a31780d039f5ec0b4c5bd5a0897ab6146ff9bd2b3cc5d6498014cd.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-490_12c3427298f98ddc264437b4b8c5d1704b59bed0311132e9e7d9e51d283c71cb.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-490_12c3427298f98ddc264437b4b8c5d1704b59bed0311132e9e7d9e51d283c71cb.jpg new file mode 100644 index 00000000..c38bd685 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-490_12c3427298f98ddc264437b4b8c5d1704b59bed0311132e9e7d9e51d283c71cb.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-490_8aee6e3c2d4c88da034606e310ec7b5be09dd780ae1d18c6affb09b5d7f94c1e.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-490_8aee6e3c2d4c88da034606e310ec7b5be09dd780ae1d18c6affb09b5d7f94c1e.jpg new file mode 100644 index 00000000..a8393c9b Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-490_8aee6e3c2d4c88da034606e310ec7b5be09dd780ae1d18c6affb09b5d7f94c1e.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-490_d2768ceea2b34e9369331f1fff929a5e051154ff4f88847bf99880dbe211ab23.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-490_d2768ceea2b34e9369331f1fff929a5e051154ff4f88847bf99880dbe211ab23.jpg new file mode 100644 index 00000000..63ae91ef Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-490_d2768ceea2b34e9369331f1fff929a5e051154ff4f88847bf99880dbe211ab23.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-491_4f1d149968b65b41251fa0165bf4434708bb64e600334887efd3e740e92a508b.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-491_4f1d149968b65b41251fa0165bf4434708bb64e600334887efd3e740e92a508b.jpg new file mode 100644 index 00000000..2aa07925 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-491_4f1d149968b65b41251fa0165bf4434708bb64e600334887efd3e740e92a508b.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-491_7f8f4387e32a247b19dfacd0e2cd37c3cdfae6128505a1467b32b20ba9075a9b.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-491_7f8f4387e32a247b19dfacd0e2cd37c3cdfae6128505a1467b32b20ba9075a9b.jpg new file mode 100644 index 00000000..02d7c9fa Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-491_7f8f4387e32a247b19dfacd0e2cd37c3cdfae6128505a1467b32b20ba9075a9b.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-491_924003caf607603d5096c45d35a47b849c6a1dc5efe5b82c731d0211a520d925.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-491_924003caf607603d5096c45d35a47b849c6a1dc5efe5b82c731d0211a520d925.jpg new file mode 100644 index 00000000..e64f2cda Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-491_924003caf607603d5096c45d35a47b849c6a1dc5efe5b82c731d0211a520d925.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-491_c9141e3cab8fa5eb38e80646977d0d129a7d0d30752ae8395820b50632673f2f.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-491_c9141e3cab8fa5eb38e80646977d0d129a7d0d30752ae8395820b50632673f2f.jpg new file mode 100644 index 00000000..7b469ed7 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-491_c9141e3cab8fa5eb38e80646977d0d129a7d0d30752ae8395820b50632673f2f.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-491_f9120b4d6c380b61253d249f10ba8db7126b14d4967df3c5e10060057e793db5.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-491_f9120b4d6c380b61253d249f10ba8db7126b14d4967df3c5e10060057e793db5.jpg new file mode 100644 index 00000000..41acc804 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-491_f9120b4d6c380b61253d249f10ba8db7126b14d4967df3c5e10060057e793db5.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-492_7d4e158c96cc9420f985fa98b5b75f6abe9db6523b44cdb8a86f94f6aabe1772.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-492_7d4e158c96cc9420f985fa98b5b75f6abe9db6523b44cdb8a86f94f6aabe1772.jpg new file mode 100644 index 00000000..43d10d2b Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-492_7d4e158c96cc9420f985fa98b5b75f6abe9db6523b44cdb8a86f94f6aabe1772.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-493_0ee0aede32f28f7d499ad73eb62d154625fa41afe56b0ca3277641bf82d971ec.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-493_0ee0aede32f28f7d499ad73eb62d154625fa41afe56b0ca3277641bf82d971ec.jpg new file mode 100644 index 00000000..47395560 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-493_0ee0aede32f28f7d499ad73eb62d154625fa41afe56b0ca3277641bf82d971ec.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-493_3094c2583e7abab255422b44795464aa1904c998c7f9438819c6b4332aa0fab8.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-493_3094c2583e7abab255422b44795464aa1904c998c7f9438819c6b4332aa0fab8.jpg new file mode 100644 index 00000000..ae2794ca Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-493_3094c2583e7abab255422b44795464aa1904c998c7f9438819c6b4332aa0fab8.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-493_48d298f4acc01d368e24ca8604feafb1faf7d1f156001fae0877dc693ff4ce92.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-493_48d298f4acc01d368e24ca8604feafb1faf7d1f156001fae0877dc693ff4ce92.jpg new file mode 100644 index 00000000..7e308d9b Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-493_48d298f4acc01d368e24ca8604feafb1faf7d1f156001fae0877dc693ff4ce92.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-493_58ebd36489e2bd6f8302d9777a9e297fcee63c2590c03cc61af78b6c20325e83.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-493_58ebd36489e2bd6f8302d9777a9e297fcee63c2590c03cc61af78b6c20325e83.jpg new file mode 100644 index 00000000..dd444de0 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-493_58ebd36489e2bd6f8302d9777a9e297fcee63c2590c03cc61af78b6c20325e83.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-493_824cd89e24982497faaf4182c4fb9b1d535c46a79f6612411ed38808f96a81ed.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-493_824cd89e24982497faaf4182c4fb9b1d535c46a79f6612411ed38808f96a81ed.jpg new file mode 100644 index 00000000..472994e5 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-493_824cd89e24982497faaf4182c4fb9b1d535c46a79f6612411ed38808f96a81ed.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-493_88a2348e3f10f674275540837677f852673c5334f3babd81be0022f3c682477d.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-493_88a2348e3f10f674275540837677f852673c5334f3babd81be0022f3c682477d.jpg new file mode 100644 index 00000000..21c67951 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-493_88a2348e3f10f674275540837677f852673c5334f3babd81be0022f3c682477d.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-494_02588c174fc3c93ce1bb152838e5fddb743d8e420eba2c4ef3e23e3e8f8d0506.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-494_02588c174fc3c93ce1bb152838e5fddb743d8e420eba2c4ef3e23e3e8f8d0506.jpg new file mode 100644 index 00000000..3cf7c3e4 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-494_02588c174fc3c93ce1bb152838e5fddb743d8e420eba2c4ef3e23e3e8f8d0506.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-495_5b463522a8c1e893053fb6f83a127c38531d27effc16a1f6d068dc2d8e32bc2d.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-495_5b463522a8c1e893053fb6f83a127c38531d27effc16a1f6d068dc2d8e32bc2d.jpg new file mode 100644 index 00000000..87885b01 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-495_5b463522a8c1e893053fb6f83a127c38531d27effc16a1f6d068dc2d8e32bc2d.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-495_c1204912851b708520f39cb78f1a3f917be0745ceb57a80b9bb39084424813c6.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-495_c1204912851b708520f39cb78f1a3f917be0745ceb57a80b9bb39084424813c6.jpg new file mode 100644 index 00000000..33abd4bb Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-495_c1204912851b708520f39cb78f1a3f917be0745ceb57a80b9bb39084424813c6.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-495_d5e42f18cba26f5898bed6d0738e30c27101bdf377acd768e85075baf55ffcec.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-495_d5e42f18cba26f5898bed6d0738e30c27101bdf377acd768e85075baf55ffcec.jpg new file mode 100644 index 00000000..0de76660 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-495_d5e42f18cba26f5898bed6d0738e30c27101bdf377acd768e85075baf55ffcec.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-496_276fa9e248dc3845fa1310756a2cc61d8995dc66ee39845426d6e3c4df835fa3.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-496_276fa9e248dc3845fa1310756a2cc61d8995dc66ee39845426d6e3c4df835fa3.jpg new file mode 100644 index 00000000..db6fe690 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-496_276fa9e248dc3845fa1310756a2cc61d8995dc66ee39845426d6e3c4df835fa3.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-496_2c154586ae47e4f32a7f6a916da06d308b001e13b8fb3459d0b48bfb299394d6.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-496_2c154586ae47e4f32a7f6a916da06d308b001e13b8fb3459d0b48bfb299394d6.jpg new file mode 100644 index 00000000..3e7c3bba Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-496_2c154586ae47e4f32a7f6a916da06d308b001e13b8fb3459d0b48bfb299394d6.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-496_414724e906b3dcdb8e1ee21e29a13116e3165b0e5eaf176a77140d5a630a1975.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-496_414724e906b3dcdb8e1ee21e29a13116e3165b0e5eaf176a77140d5a630a1975.jpg new file mode 100644 index 00000000..f634309c Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-496_414724e906b3dcdb8e1ee21e29a13116e3165b0e5eaf176a77140d5a630a1975.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-496_6f922a44927c5e5c23ddc169f16076c8053be3f10dfad3d0b81eadad4218e583.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-496_6f922a44927c5e5c23ddc169f16076c8053be3f10dfad3d0b81eadad4218e583.jpg new file mode 100644 index 00000000..0426f301 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-496_6f922a44927c5e5c23ddc169f16076c8053be3f10dfad3d0b81eadad4218e583.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-496_aae34889a21312d31c2c4d6c73c8909e62b6bf9b9bade897b65aad2b947212a8.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-496_aae34889a21312d31c2c4d6c73c8909e62b6bf9b9bade897b65aad2b947212a8.jpg new file mode 100644 index 00000000..60147639 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-496_aae34889a21312d31c2c4d6c73c8909e62b6bf9b9bade897b65aad2b947212a8.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-496_f996bdda11acfe4f0e37a4dae848ebf7e869bd79350f92300ad50f39aa5453bb.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-496_f996bdda11acfe4f0e37a4dae848ebf7e869bd79350f92300ad50f39aa5453bb.jpg new file mode 100644 index 00000000..1c63966f Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-496_f996bdda11acfe4f0e37a4dae848ebf7e869bd79350f92300ad50f39aa5453bb.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-500_d1e07092b76e7e4aa5317eb07fd6dca744e4504942da55810ad5f5043ece45f1.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-500_d1e07092b76e7e4aa5317eb07fd6dca744e4504942da55810ad5f5043ece45f1.jpg new file mode 100644 index 00000000..92dbde5c Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-500_d1e07092b76e7e4aa5317eb07fd6dca744e4504942da55810ad5f5043ece45f1.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-501_0271b6aad4bcfc96c54d5e9c87b6deb4816a9e62cc9d0ff9abc642cd49af02b1.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-501_0271b6aad4bcfc96c54d5e9c87b6deb4816a9e62cc9d0ff9abc642cd49af02b1.jpg new file mode 100644 index 00000000..fae03407 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-501_0271b6aad4bcfc96c54d5e9c87b6deb4816a9e62cc9d0ff9abc642cd49af02b1.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-501_71d3a36d137a23204e2c2473c91e62f300f08a99420d04cb28f197f987537199.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-501_71d3a36d137a23204e2c2473c91e62f300f08a99420d04cb28f197f987537199.jpg new file mode 100644 index 00000000..47ffe77b Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-501_71d3a36d137a23204e2c2473c91e62f300f08a99420d04cb28f197f987537199.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-501_a4e53b0898430c20b8e248e120794bd374605f5f60564269cc8aa31383f76930.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-501_a4e53b0898430c20b8e248e120794bd374605f5f60564269cc8aa31383f76930.jpg new file mode 100644 index 00000000..3cb79248 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-501_a4e53b0898430c20b8e248e120794bd374605f5f60564269cc8aa31383f76930.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-501_bade829e0af56d4e7c0e0d848176e6684dc80f4cfec0091af024c0c1ce248124.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-501_bade829e0af56d4e7c0e0d848176e6684dc80f4cfec0091af024c0c1ce248124.jpg new file mode 100644 index 00000000..cdce056b Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-501_bade829e0af56d4e7c0e0d848176e6684dc80f4cfec0091af024c0c1ce248124.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-501_cafd24718877b0cad5cfe5de11a65c9536285939590ef719eecfb16b7b0b08d1.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-501_cafd24718877b0cad5cfe5de11a65c9536285939590ef719eecfb16b7b0b08d1.jpg new file mode 100644 index 00000000..3ea32e70 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-501_cafd24718877b0cad5cfe5de11a65c9536285939590ef719eecfb16b7b0b08d1.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-502_036b328b4ec7afb709e0093c77940fa2e325db85e0e24af7e49ff8faf6b8fad6.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-502_036b328b4ec7afb709e0093c77940fa2e325db85e0e24af7e49ff8faf6b8fad6.jpg new file mode 100644 index 00000000..f92a1a6f Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-502_036b328b4ec7afb709e0093c77940fa2e325db85e0e24af7e49ff8faf6b8fad6.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-502_0c7e79f9ef8e006364471de165af17c61745c9bfcb5c627956dbd8cd78e2d0b7.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-502_0c7e79f9ef8e006364471de165af17c61745c9bfcb5c627956dbd8cd78e2d0b7.jpg new file mode 100644 index 00000000..b49d0cfc Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-502_0c7e79f9ef8e006364471de165af17c61745c9bfcb5c627956dbd8cd78e2d0b7.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-502_2ef811480e9b2b81fb4b0bf516887cd87c62ed49e031e52cbc09f4e45dd8ca10.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-502_2ef811480e9b2b81fb4b0bf516887cd87c62ed49e031e52cbc09f4e45dd8ca10.jpg new file mode 100644 index 00000000..d53d8398 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-502_2ef811480e9b2b81fb4b0bf516887cd87c62ed49e031e52cbc09f4e45dd8ca10.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-502_40cfb4943124d37e1e8b33b1872400e9f4d53e77c62daec2974da9b6c320f9e1.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-502_40cfb4943124d37e1e8b33b1872400e9f4d53e77c62daec2974da9b6c320f9e1.jpg new file mode 100644 index 00000000..a176d8fa Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-502_40cfb4943124d37e1e8b33b1872400e9f4d53e77c62daec2974da9b6c320f9e1.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-502_523c4047f2163a68746b60b27f71dc6ab295b50a4324e505c47fa692aab87a50.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-502_523c4047f2163a68746b60b27f71dc6ab295b50a4324e505c47fa692aab87a50.jpg new file mode 100644 index 00000000..8bd08ff2 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-502_523c4047f2163a68746b60b27f71dc6ab295b50a4324e505c47fa692aab87a50.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-502_a4d218cbfce582f346287c08d39056f6e203c650080a2bd98ec7bc7445d6c00f.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-502_a4d218cbfce582f346287c08d39056f6e203c650080a2bd98ec7bc7445d6c00f.jpg new file mode 100644 index 00000000..f718c929 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-502_a4d218cbfce582f346287c08d39056f6e203c650080a2bd98ec7bc7445d6c00f.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-502_aab55d4cec05bd7ca3ffd021d920cf80629a4dfe516d669718718ec6e238c22d.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-502_aab55d4cec05bd7ca3ffd021d920cf80629a4dfe516d669718718ec6e238c22d.jpg new file mode 100644 index 00000000..9463cb79 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-502_aab55d4cec05bd7ca3ffd021d920cf80629a4dfe516d669718718ec6e238c22d.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-502_cf5c143cea6ba8e606fc87a322aed4a0a10f1be62a7a3aa2da4861e00dd39508.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-502_cf5c143cea6ba8e606fc87a322aed4a0a10f1be62a7a3aa2da4861e00dd39508.jpg new file mode 100644 index 00000000..93cb6f09 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-502_cf5c143cea6ba8e606fc87a322aed4a0a10f1be62a7a3aa2da4861e00dd39508.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-502_fce8e9c8af0623e6f753ab749090514987b110de145e832f599a7fcc9998cfa6.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-502_fce8e9c8af0623e6f753ab749090514987b110de145e832f599a7fcc9998cfa6.jpg new file mode 100644 index 00000000..027ac47c Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-502_fce8e9c8af0623e6f753ab749090514987b110de145e832f599a7fcc9998cfa6.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-504_2a3cb4fee0e20c242b8daffa75da0fd0c1fc5026ebebe8fd30e761b5a851f092.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-504_2a3cb4fee0e20c242b8daffa75da0fd0c1fc5026ebebe8fd30e761b5a851f092.jpg new file mode 100644 index 00000000..1932698b Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-504_2a3cb4fee0e20c242b8daffa75da0fd0c1fc5026ebebe8fd30e761b5a851f092.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-515_4d096649a32428bed3d7c5c72dd662ae64ec4db71b2575265055dceaae744574.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-515_4d096649a32428bed3d7c5c72dd662ae64ec4db71b2575265055dceaae744574.jpg new file mode 100644 index 00000000..c2fea259 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-515_4d096649a32428bed3d7c5c72dd662ae64ec4db71b2575265055dceaae744574.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-515_cd45b6825ab4aab35587dfff5a63fe1936f371146788a9578ce277d19ec8bc0c.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-515_cd45b6825ab4aab35587dfff5a63fe1936f371146788a9578ce277d19ec8bc0c.jpg new file mode 100644 index 00000000..da3c7fa5 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-515_cd45b6825ab4aab35587dfff5a63fe1936f371146788a9578ce277d19ec8bc0c.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-515_fc381d098ea396b849fdc22edb99ecf6686477091d01c3e533c9cbef76d19873.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-515_fc381d098ea396b849fdc22edb99ecf6686477091d01c3e533c9cbef76d19873.jpg new file mode 100644 index 00000000..43fafc91 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-515_fc381d098ea396b849fdc22edb99ecf6686477091d01c3e533c9cbef76d19873.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-516_64e9811d891a430b77e4aa0ae32fe4611ef9da0d65c1b8d27300acfb437d77c6.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-516_64e9811d891a430b77e4aa0ae32fe4611ef9da0d65c1b8d27300acfb437d77c6.jpg new file mode 100644 index 00000000..946235ad Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-516_64e9811d891a430b77e4aa0ae32fe4611ef9da0d65c1b8d27300acfb437d77c6.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-516_723712271ea4a4f66611cb75258fd1fcd705c57d230ad312c29ba69136a68883.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-516_723712271ea4a4f66611cb75258fd1fcd705c57d230ad312c29ba69136a68883.jpg new file mode 100644 index 00000000..edf5a548 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-516_723712271ea4a4f66611cb75258fd1fcd705c57d230ad312c29ba69136a68883.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-516_82ece0155fa01424b5b6f7c9a2ab605f0c5663a7da4cda62060c2e42a6a6a931.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-516_82ece0155fa01424b5b6f7c9a2ab605f0c5663a7da4cda62060c2e42a6a6a931.jpg new file mode 100644 index 00000000..09af231f Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-516_82ece0155fa01424b5b6f7c9a2ab605f0c5663a7da4cda62060c2e42a6a6a931.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-517_07f959ed92b0f0b051ede7ac3c0e910366e07d9651f8d5b761c8fdc976ce29d7.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-517_07f959ed92b0f0b051ede7ac3c0e910366e07d9651f8d5b761c8fdc976ce29d7.jpg new file mode 100644 index 00000000..decf69f8 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-517_07f959ed92b0f0b051ede7ac3c0e910366e07d9651f8d5b761c8fdc976ce29d7.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-517_7a918510729856b5fd416b17098e0defea56f33a030b0dd1d5c724dada863329.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-517_7a918510729856b5fd416b17098e0defea56f33a030b0dd1d5c724dada863329.jpg new file mode 100644 index 00000000..802c365f Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-517_7a918510729856b5fd416b17098e0defea56f33a030b0dd1d5c724dada863329.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-517_c8736af375b90eff6235adc387919a3c9d6bc3552dc17e6c04b2de5829d441d1.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-517_c8736af375b90eff6235adc387919a3c9d6bc3552dc17e6c04b2de5829d441d1.jpg new file mode 100644 index 00000000..4326f7a7 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-517_c8736af375b90eff6235adc387919a3c9d6bc3552dc17e6c04b2de5829d441d1.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-517_ccc6a86b9a7e8e761596de4087da02693704ebdff6e91287f17f0826540fc243.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-517_ccc6a86b9a7e8e761596de4087da02693704ebdff6e91287f17f0826540fc243.jpg new file mode 100644 index 00000000..7daed5f8 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-517_ccc6a86b9a7e8e761596de4087da02693704ebdff6e91287f17f0826540fc243.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-518_955fa91f75f37ccbebb705ef25915cf010f94e97fa008f3b04856027030823e6.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-518_955fa91f75f37ccbebb705ef25915cf010f94e97fa008f3b04856027030823e6.jpg new file mode 100644 index 00000000..76931221 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-518_955fa91f75f37ccbebb705ef25915cf010f94e97fa008f3b04856027030823e6.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-518_f65b8ce210e98c940f5e5fc6b59bc6e31b0583323b851af8df633a62815de5c3.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-518_f65b8ce210e98c940f5e5fc6b59bc6e31b0583323b851af8df633a62815de5c3.jpg new file mode 100644 index 00000000..28a02a59 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-518_f65b8ce210e98c940f5e5fc6b59bc6e31b0583323b851af8df633a62815de5c3.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-521_c2c2d3a17ec7d665da683e5edd1dd8234c832086a024bfaf67737ac8f848173d.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-521_c2c2d3a17ec7d665da683e5edd1dd8234c832086a024bfaf67737ac8f848173d.jpg new file mode 100644 index 00000000..33513834 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-521_c2c2d3a17ec7d665da683e5edd1dd8234c832086a024bfaf67737ac8f848173d.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-525_da0c545c27cc93f1d705f60223155b87ed6d5db9c71741b0a8e7c0c1943b64f5.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-525_da0c545c27cc93f1d705f60223155b87ed6d5db9c71741b0a8e7c0c1943b64f5.jpg new file mode 100644 index 00000000..707f691b Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-525_da0c545c27cc93f1d705f60223155b87ed6d5db9c71741b0a8e7c0c1943b64f5.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-527_d36229952997d60c74aacdc99a98ed0e65abde981b1b27d985612c4caa0e7975.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-527_d36229952997d60c74aacdc99a98ed0e65abde981b1b27d985612c4caa0e7975.jpg new file mode 100644 index 00000000..51fea624 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-527_d36229952997d60c74aacdc99a98ed0e65abde981b1b27d985612c4caa0e7975.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-527_d88eee7a785a1ae674196737d37c321c307aa55e8e73c366d6254c582d2a75d7.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-527_d88eee7a785a1ae674196737d37c321c307aa55e8e73c366d6254c582d2a75d7.jpg new file mode 100644 index 00000000..2b0c0b22 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-527_d88eee7a785a1ae674196737d37c321c307aa55e8e73c366d6254c582d2a75d7.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-531_0e8b033a23e60e8ee1954d59bbd71dcdf154f17a1044f44e523fc06eb71824f1.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-531_0e8b033a23e60e8ee1954d59bbd71dcdf154f17a1044f44e523fc06eb71824f1.jpg new file mode 100644 index 00000000..402d2695 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-531_0e8b033a23e60e8ee1954d59bbd71dcdf154f17a1044f44e523fc06eb71824f1.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-532_db8d714b7334d615d62a42da87b69772d654a30ad73dc6cbe6b15e03223e59e5.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-532_db8d714b7334d615d62a42da87b69772d654a30ad73dc6cbe6b15e03223e59e5.jpg new file mode 100644 index 00000000..2d456201 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-532_db8d714b7334d615d62a42da87b69772d654a30ad73dc6cbe6b15e03223e59e5.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-532_f82c0d77dbea18de15432a5a727f7a7f9604939329a884b2db94f9f999b20950.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-532_f82c0d77dbea18de15432a5a727f7a7f9604939329a884b2db94f9f999b20950.jpg new file mode 100644 index 00000000..e6d0e225 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-532_f82c0d77dbea18de15432a5a727f7a7f9604939329a884b2db94f9f999b20950.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-533_f4a67c13521d7d8b56b84aa1416026f0d92713341c88e20190318bc7949c75a1.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-533_f4a67c13521d7d8b56b84aa1416026f0d92713341c88e20190318bc7949c75a1.jpg new file mode 100644 index 00000000..07f232d4 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-533_f4a67c13521d7d8b56b84aa1416026f0d92713341c88e20190318bc7949c75a1.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-537_0625701cc012eea52e57340d9d51a79a355fa2b7275a3965e4f4602b2e591ef8.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-537_0625701cc012eea52e57340d9d51a79a355fa2b7275a3965e4f4602b2e591ef8.jpg new file mode 100644 index 00000000..01333e29 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-537_0625701cc012eea52e57340d9d51a79a355fa2b7275a3965e4f4602b2e591ef8.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-537_13237216444c84f65f9195596438898ecf31cf9528fd151d232693d6d025789e.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-537_13237216444c84f65f9195596438898ecf31cf9528fd151d232693d6d025789e.jpg new file mode 100644 index 00000000..2674ef60 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-537_13237216444c84f65f9195596438898ecf31cf9528fd151d232693d6d025789e.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-537_2fedecdb9fabf0f267d91faadc2fccb5745c0f25f84dea3ef72cd52c6d47e280.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-537_2fedecdb9fabf0f267d91faadc2fccb5745c0f25f84dea3ef72cd52c6d47e280.jpg new file mode 100644 index 00000000..17fe7e9b Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-537_2fedecdb9fabf0f267d91faadc2fccb5745c0f25f84dea3ef72cd52c6d47e280.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-537_a5369ccfe00ad9397c7b21d93a57c335eccafbb83b259d042e75f91ee5262f52.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-537_a5369ccfe00ad9397c7b21d93a57c335eccafbb83b259d042e75f91ee5262f52.jpg new file mode 100644 index 00000000..4af5b6a0 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-537_a5369ccfe00ad9397c7b21d93a57c335eccafbb83b259d042e75f91ee5262f52.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-537_e4989f884f80fc4e0d98bfe98b74f7c10deaeac987061fd77ba6d2c2798d2b56.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-537_e4989f884f80fc4e0d98bfe98b74f7c10deaeac987061fd77ba6d2c2798d2b56.jpg new file mode 100644 index 00000000..94f3f93b Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-537_e4989f884f80fc4e0d98bfe98b74f7c10deaeac987061fd77ba6d2c2798d2b56.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-539_6df032a0ce6750f3ca8830a372f9d68037ed5b6eee349e6ebf4a8db124aa9bfa.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-539_6df032a0ce6750f3ca8830a372f9d68037ed5b6eee349e6ebf4a8db124aa9bfa.jpg new file mode 100644 index 00000000..80784544 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-539_6df032a0ce6750f3ca8830a372f9d68037ed5b6eee349e6ebf4a8db124aa9bfa.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-540_00ef7126e7380bc71883987beac215f01550d130ab552fd8d8f608d19141d2d1.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-540_00ef7126e7380bc71883987beac215f01550d130ab552fd8d8f608d19141d2d1.jpg new file mode 100644 index 00000000..c534bbb1 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-540_00ef7126e7380bc71883987beac215f01550d130ab552fd8d8f608d19141d2d1.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-541_c359fa6888f7719232c5e4fdcf17cd0084b67c70445aa590a84523c8082bb646.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-541_c359fa6888f7719232c5e4fdcf17cd0084b67c70445aa590a84523c8082bb646.jpg new file mode 100644 index 00000000..d79bc26c Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-541_c359fa6888f7719232c5e4fdcf17cd0084b67c70445aa590a84523c8082bb646.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-541_cd50c7bba7097c2268ad16108b08d3fa98966b18e99b77d9d85e38d36f444c6f.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-541_cd50c7bba7097c2268ad16108b08d3fa98966b18e99b77d9d85e38d36f444c6f.jpg new file mode 100644 index 00000000..2fa0ab65 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-541_cd50c7bba7097c2268ad16108b08d3fa98966b18e99b77d9d85e38d36f444c6f.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-542_8170d780a9dc0ef300663ebf8cdc0e4f22af8879596f2eaca665e8e4521dcc65.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-542_8170d780a9dc0ef300663ebf8cdc0e4f22af8879596f2eaca665e8e4521dcc65.jpg new file mode 100644 index 00000000..425042f0 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-542_8170d780a9dc0ef300663ebf8cdc0e4f22af8879596f2eaca665e8e4521dcc65.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-542_a0690c43be26dd22fb5194451a3cc74fb85c061376703415555d1ff0002e5832.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-542_a0690c43be26dd22fb5194451a3cc74fb85c061376703415555d1ff0002e5832.jpg new file mode 100644 index 00000000..b4967266 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-542_a0690c43be26dd22fb5194451a3cc74fb85c061376703415555d1ff0002e5832.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-543_520348766c53759e27b78a7d34acebe2835b12d26fa3d0c6ab36d4260243890c.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-543_520348766c53759e27b78a7d34acebe2835b12d26fa3d0c6ab36d4260243890c.jpg new file mode 100644 index 00000000..fd96751e Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-543_520348766c53759e27b78a7d34acebe2835b12d26fa3d0c6ab36d4260243890c.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-547_34588f87816f3412c25fe6d57f3b4f3d84958b2dd35f4b930a2b069a286e9d17.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-547_34588f87816f3412c25fe6d57f3b4f3d84958b2dd35f4b930a2b069a286e9d17.jpg new file mode 100644 index 00000000..7a9da26d Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-547_34588f87816f3412c25fe6d57f3b4f3d84958b2dd35f4b930a2b069a286e9d17.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-548_00cc2795abd8c64e66bcf907fe65d4cca96480ecc3b73d9958c5c7f31843f41b.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-548_00cc2795abd8c64e66bcf907fe65d4cca96480ecc3b73d9958c5c7f31843f41b.jpg new file mode 100644 index 00000000..f97067b3 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-548_00cc2795abd8c64e66bcf907fe65d4cca96480ecc3b73d9958c5c7f31843f41b.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-548_306fc463b338c9cb12fa11a3c2b799a5f761ccd9d11aaedac8df20b0265fee40.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-548_306fc463b338c9cb12fa11a3c2b799a5f761ccd9d11aaedac8df20b0265fee40.jpg new file mode 100644 index 00000000..893186ee Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-548_306fc463b338c9cb12fa11a3c2b799a5f761ccd9d11aaedac8df20b0265fee40.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-548_433ce0c1d5ee1d3c0ae482c0c49d078bc265d79af6d1df4f7184cd50098b47d2.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-548_433ce0c1d5ee1d3c0ae482c0c49d078bc265d79af6d1df4f7184cd50098b47d2.jpg new file mode 100644 index 00000000..0489a1f6 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-548_433ce0c1d5ee1d3c0ae482c0c49d078bc265d79af6d1df4f7184cd50098b47d2.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-548_c7d964d472ecb876393b7424557333e5f2869c3e78a608a6299db464c9cb0ca8.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-548_c7d964d472ecb876393b7424557333e5f2869c3e78a608a6299db464c9cb0ca8.jpg new file mode 100644 index 00000000..1e9bfaa9 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-548_c7d964d472ecb876393b7424557333e5f2869c3e78a608a6299db464c9cb0ca8.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-548_c9cfda25cd855147b34c93ac914d8951c4dc2dbdf90ac89556616fdfbaf9e929.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-548_c9cfda25cd855147b34c93ac914d8951c4dc2dbdf90ac89556616fdfbaf9e929.jpg new file mode 100644 index 00000000..e13c37b1 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-548_c9cfda25cd855147b34c93ac914d8951c4dc2dbdf90ac89556616fdfbaf9e929.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-548_d13f9fa2a8659bd8b40340e3ccd78a4507a6636f9a6f9501d8b225985db943ab.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-548_d13f9fa2a8659bd8b40340e3ccd78a4507a6636f9a6f9501d8b225985db943ab.jpg new file mode 100644 index 00000000..7771d938 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-548_d13f9fa2a8659bd8b40340e3ccd78a4507a6636f9a6f9501d8b225985db943ab.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-549_1b7a6fc2949f4460c74ba1f43991f94456ede2c69e9e496937206cf846156766.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-549_1b7a6fc2949f4460c74ba1f43991f94456ede2c69e9e496937206cf846156766.jpg new file mode 100644 index 00000000..dc913bb9 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-549_1b7a6fc2949f4460c74ba1f43991f94456ede2c69e9e496937206cf846156766.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-549_28bbb2a4c7a9900ba6a9b5d7dbf66aa4f1b27154d6d1e7ff75c0010b6585ef60.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-549_28bbb2a4c7a9900ba6a9b5d7dbf66aa4f1b27154d6d1e7ff75c0010b6585ef60.jpg new file mode 100644 index 00000000..b2b58230 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-549_28bbb2a4c7a9900ba6a9b5d7dbf66aa4f1b27154d6d1e7ff75c0010b6585ef60.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-549_b49e74a25586ae7478336d3778177748674421876415363bfd2ebd6fa0d5d6c8.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-549_b49e74a25586ae7478336d3778177748674421876415363bfd2ebd6fa0d5d6c8.jpg new file mode 100644 index 00000000..2bd5dc7e Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-549_b49e74a25586ae7478336d3778177748674421876415363bfd2ebd6fa0d5d6c8.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-549_c479e2f4eb7f9f2b78d8fac9ab2edac8372bf9f885f3ea6395e35eb3ce542ce0.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-549_c479e2f4eb7f9f2b78d8fac9ab2edac8372bf9f885f3ea6395e35eb3ce542ce0.jpg new file mode 100644 index 00000000..83a38cf4 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-549_c479e2f4eb7f9f2b78d8fac9ab2edac8372bf9f885f3ea6395e35eb3ce542ce0.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-549_c5bb59f3f57fe8f4a17086b41cc2c77561aed5a55fc94f56eb92bdb6006dd044.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-549_c5bb59f3f57fe8f4a17086b41cc2c77561aed5a55fc94f56eb92bdb6006dd044.jpg new file mode 100644 index 00000000..f1984fcb Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-549_c5bb59f3f57fe8f4a17086b41cc2c77561aed5a55fc94f56eb92bdb6006dd044.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-549_f32c36c909f7d1620bceabb4097a6b829ca4f7f3a765d58a4dc3085766a18dbb.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-549_f32c36c909f7d1620bceabb4097a6b829ca4f7f3a765d58a4dc3085766a18dbb.jpg new file mode 100644 index 00000000..8ffd58d7 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-549_f32c36c909f7d1620bceabb4097a6b829ca4f7f3a765d58a4dc3085766a18dbb.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-550_fe7b55fadbbee6ea39b426099064f006a4ec1d21953b08922e77643eb09682a4.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-550_fe7b55fadbbee6ea39b426099064f006a4ec1d21953b08922e77643eb09682a4.jpg new file mode 100644 index 00000000..15977102 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-550_fe7b55fadbbee6ea39b426099064f006a4ec1d21953b08922e77643eb09682a4.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-551_a7115d66c45c567080c8e16b5343bfd6aa17c9e9a675196fa4c7e527edaac16f.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-551_a7115d66c45c567080c8e16b5343bfd6aa17c9e9a675196fa4c7e527edaac16f.jpg new file mode 100644 index 00000000..910aae0d Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-551_a7115d66c45c567080c8e16b5343bfd6aa17c9e9a675196fa4c7e527edaac16f.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-552_76d433fff3381f010ad1e6958d4127f01bc423f10eafc9faf669612057b6d6de.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-552_76d433fff3381f010ad1e6958d4127f01bc423f10eafc9faf669612057b6d6de.jpg new file mode 100644 index 00000000..8ed1d71b Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-552_76d433fff3381f010ad1e6958d4127f01bc423f10eafc9faf669612057b6d6de.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-553_b45eae24f564f55f6ced74263062099ae1f69303ee0cefe0d7ff1716a72b1525.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-553_b45eae24f564f55f6ced74263062099ae1f69303ee0cefe0d7ff1716a72b1525.jpg new file mode 100644 index 00000000..f2f5a14b Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-553_b45eae24f564f55f6ced74263062099ae1f69303ee0cefe0d7ff1716a72b1525.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-553_ce996f73216cb462a108d08f6609ddfd451e0962109718721c4fe27d51a37d12.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-553_ce996f73216cb462a108d08f6609ddfd451e0962109718721c4fe27d51a37d12.jpg new file mode 100644 index 00000000..74f6288f Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-553_ce996f73216cb462a108d08f6609ddfd451e0962109718721c4fe27d51a37d12.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-554_9f896744725d269051e3265d8252254fd2321452f2bb92502e68244a070c4255.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-554_9f896744725d269051e3265d8252254fd2321452f2bb92502e68244a070c4255.jpg new file mode 100644 index 00000000..3d856807 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-554_9f896744725d269051e3265d8252254fd2321452f2bb92502e68244a070c4255.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-555_11d2e58c57429b067dfd20f1c4cfd9181a26789f5c44d475c02f458d1b7796f3.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-555_11d2e58c57429b067dfd20f1c4cfd9181a26789f5c44d475c02f458d1b7796f3.jpg new file mode 100644 index 00000000..cab331c9 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-555_11d2e58c57429b067dfd20f1c4cfd9181a26789f5c44d475c02f458d1b7796f3.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-555_6b6bc185227ad058399c112b16f1f63e692a6ac68569f314b7ddf96aa06b938b.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-555_6b6bc185227ad058399c112b16f1f63e692a6ac68569f314b7ddf96aa06b938b.jpg new file mode 100644 index 00000000..86d21e36 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-555_6b6bc185227ad058399c112b16f1f63e692a6ac68569f314b7ddf96aa06b938b.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-555_99c1dd4d3fd635c00b9504e999ad98b2b59aade450f031c13985ab5da999b63d.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-555_99c1dd4d3fd635c00b9504e999ad98b2b59aade450f031c13985ab5da999b63d.jpg new file mode 100644 index 00000000..e1bfc0c7 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-555_99c1dd4d3fd635c00b9504e999ad98b2b59aade450f031c13985ab5da999b63d.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-556_111b99c27fdb0c636ffd902ac872e9f39d71b408463e99d503ef110090b58710.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-556_111b99c27fdb0c636ffd902ac872e9f39d71b408463e99d503ef110090b58710.jpg new file mode 100644 index 00000000..6512a0e5 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-556_111b99c27fdb0c636ffd902ac872e9f39d71b408463e99d503ef110090b58710.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-556_a93485fe37aa3e05414719864fcade11b1fd3be8dac7e65c5e8a1f84c5fa6d3c.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-556_a93485fe37aa3e05414719864fcade11b1fd3be8dac7e65c5e8a1f84c5fa6d3c.jpg new file mode 100644 index 00000000..fbc71b26 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-556_a93485fe37aa3e05414719864fcade11b1fd3be8dac7e65c5e8a1f84c5fa6d3c.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-556_b7265d0d2185dc16528627e40a53f830d93a2c9d317265543749b923dcd3c0a6.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-556_b7265d0d2185dc16528627e40a53f830d93a2c9d317265543749b923dcd3c0a6.jpg new file mode 100644 index 00000000..3b964646 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-556_b7265d0d2185dc16528627e40a53f830d93a2c9d317265543749b923dcd3c0a6.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-557_b0ecb0c9206e7b53e56625c3ab478faa2a57183e0c911b910ac436103deec882.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-557_b0ecb0c9206e7b53e56625c3ab478faa2a57183e0c911b910ac436103deec882.jpg new file mode 100644 index 00000000..5cbbc6a1 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-557_b0ecb0c9206e7b53e56625c3ab478faa2a57183e0c911b910ac436103deec882.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-558_7f645b998db6f9d172ce25b15d914bef976c5057825088b7d9a12498147da922.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-558_7f645b998db6f9d172ce25b15d914bef976c5057825088b7d9a12498147da922.jpg new file mode 100644 index 00000000..e9445b9a Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-558_7f645b998db6f9d172ce25b15d914bef976c5057825088b7d9a12498147da922.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-558_cb63912cd60792646db92df98d806505c919110c6c317226a2adbe8db643c088.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-558_cb63912cd60792646db92df98d806505c919110c6c317226a2adbe8db643c088.jpg new file mode 100644 index 00000000..3e9f203e Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-558_cb63912cd60792646db92df98d806505c919110c6c317226a2adbe8db643c088.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-559_0ae5fc244ebd565fc6ff295025531a548a94b00eaf3193e63803ca36d8872f82.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-559_0ae5fc244ebd565fc6ff295025531a548a94b00eaf3193e63803ca36d8872f82.jpg new file mode 100644 index 00000000..f3751246 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-559_0ae5fc244ebd565fc6ff295025531a548a94b00eaf3193e63803ca36d8872f82.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-560_67eda305c891dbdea8835054a18a93a47dd5849d852999b8ec2eb1ee20367eba.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-560_67eda305c891dbdea8835054a18a93a47dd5849d852999b8ec2eb1ee20367eba.jpg new file mode 100644 index 00000000..bd505b28 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-560_67eda305c891dbdea8835054a18a93a47dd5849d852999b8ec2eb1ee20367eba.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-560_b9677084c21d1aef80e1ab390754037d16032b59a6736f44349372649e2b6ffe.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-560_b9677084c21d1aef80e1ab390754037d16032b59a6736f44349372649e2b6ffe.jpg new file mode 100644 index 00000000..2fd5de0a Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-560_b9677084c21d1aef80e1ab390754037d16032b59a6736f44349372649e2b6ffe.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-561_d554060f311c9f320a7b4e64106ecf381c227343abce7cc1ccaa09dbb49bc7e5.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-561_d554060f311c9f320a7b4e64106ecf381c227343abce7cc1ccaa09dbb49bc7e5.jpg new file mode 100644 index 00000000..47986d72 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-561_d554060f311c9f320a7b4e64106ecf381c227343abce7cc1ccaa09dbb49bc7e5.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-562_3471da115ce952e3fda395fb24210bb65d1591675e00bdb6ebcb1e25a2c652d6.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-562_3471da115ce952e3fda395fb24210bb65d1591675e00bdb6ebcb1e25a2c652d6.jpg new file mode 100644 index 00000000..fb619fa2 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-562_3471da115ce952e3fda395fb24210bb65d1591675e00bdb6ebcb1e25a2c652d6.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-562_6879bb950e848849e9081af3cf1b7cae86290b429299afc7c222e0877c52536b.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-562_6879bb950e848849e9081af3cf1b7cae86290b429299afc7c222e0877c52536b.jpg new file mode 100644 index 00000000..62accd83 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-562_6879bb950e848849e9081af3cf1b7cae86290b429299afc7c222e0877c52536b.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-563_dc59c2aeaafb3b60a038cbb15edfd35d6ddac32f6895c0abf9457b1f9a24c388.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-563_dc59c2aeaafb3b60a038cbb15edfd35d6ddac32f6895c0abf9457b1f9a24c388.jpg new file mode 100644 index 00000000..a9df9837 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-563_dc59c2aeaafb3b60a038cbb15edfd35d6ddac32f6895c0abf9457b1f9a24c388.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-564_42b05c0a963e2f92db4c8b12fc22383a85aad417b8524be94c1d324fe4fd23d1.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-564_42b05c0a963e2f92db4c8b12fc22383a85aad417b8524be94c1d324fe4fd23d1.jpg new file mode 100644 index 00000000..b44d0049 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-564_42b05c0a963e2f92db4c8b12fc22383a85aad417b8524be94c1d324fe4fd23d1.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-565_0e69709f55f4e54bc2667f5f3264eb984c2c00f426ddeb4f30110947fc3cabd7.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-565_0e69709f55f4e54bc2667f5f3264eb984c2c00f426ddeb4f30110947fc3cabd7.jpg new file mode 100644 index 00000000..bf26d140 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-565_0e69709f55f4e54bc2667f5f3264eb984c2c00f426ddeb4f30110947fc3cabd7.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-565_484d3ad9260978feb6f169d74fee265537ee10c2e06b1e1d9f935f34355f7067.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-565_484d3ad9260978feb6f169d74fee265537ee10c2e06b1e1d9f935f34355f7067.jpg new file mode 100644 index 00000000..d59f4588 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-565_484d3ad9260978feb6f169d74fee265537ee10c2e06b1e1d9f935f34355f7067.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-565_74fa06746e27910d095d3bea968413960ad8a255adca4af2a0349d6248c1eef8.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-565_74fa06746e27910d095d3bea968413960ad8a255adca4af2a0349d6248c1eef8.jpg new file mode 100644 index 00000000..ae8c2231 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-565_74fa06746e27910d095d3bea968413960ad8a255adca4af2a0349d6248c1eef8.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-565_f0d0d24e80c7eeec64d6be12fc0b21c6a4610a45e211452cbc82476f825739c7.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-565_f0d0d24e80c7eeec64d6be12fc0b21c6a4610a45e211452cbc82476f825739c7.jpg new file mode 100644 index 00000000..c9a06d7b Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-565_f0d0d24e80c7eeec64d6be12fc0b21c6a4610a45e211452cbc82476f825739c7.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-566_7a61e27eee011d77314a23c5978a24e50f27758b7c5bdb0fc5f199341d33258f.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-566_7a61e27eee011d77314a23c5978a24e50f27758b7c5bdb0fc5f199341d33258f.jpg new file mode 100644 index 00000000..54c8cf09 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-566_7a61e27eee011d77314a23c5978a24e50f27758b7c5bdb0fc5f199341d33258f.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-567_feef4addd1927b71c4e8783119ffeb71dbe226612262e2fba0e2a33eb2bf759a.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-567_feef4addd1927b71c4e8783119ffeb71dbe226612262e2fba0e2a33eb2bf759a.jpg new file mode 100644 index 00000000..7e0fc537 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-567_feef4addd1927b71c4e8783119ffeb71dbe226612262e2fba0e2a33eb2bf759a.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-568_5fce3b12584dd4cbe59a461112f419363cd9f74faabf7da7a9a9fcd44a3ce52e.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-568_5fce3b12584dd4cbe59a461112f419363cd9f74faabf7da7a9a9fcd44a3ce52e.jpg new file mode 100644 index 00000000..6a3ace05 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-568_5fce3b12584dd4cbe59a461112f419363cd9f74faabf7da7a9a9fcd44a3ce52e.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-568_b7b1db326f0aca2ba9775c0f51c22988338cdd74f4cea02cc8d6d754f1488738.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-568_b7b1db326f0aca2ba9775c0f51c22988338cdd74f4cea02cc8d6d754f1488738.jpg new file mode 100644 index 00000000..368cacff Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-568_b7b1db326f0aca2ba9775c0f51c22988338cdd74f4cea02cc8d6d754f1488738.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-569_e8c4ac3bf83084c954ca1a314966032683a010273d3c2729a4eedc9ddee832ee.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-569_e8c4ac3bf83084c954ca1a314966032683a010273d3c2729a4eedc9ddee832ee.jpg new file mode 100644 index 00000000..ff701c5d Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-569_e8c4ac3bf83084c954ca1a314966032683a010273d3c2729a4eedc9ddee832ee.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-570_80df97c0cbe1fc446b8df16659fcf6950ad9befdc4d054a9e3e0f1ddeae09f9a.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-570_80df97c0cbe1fc446b8df16659fcf6950ad9befdc4d054a9e3e0f1ddeae09f9a.jpg new file mode 100644 index 00000000..8abc6220 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-570_80df97c0cbe1fc446b8df16659fcf6950ad9befdc4d054a9e3e0f1ddeae09f9a.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-571_29ae8c21c0ab8d69fb87ce26369d5cd392f9c31f9dc410d6560aa32cdeceddbc.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-571_29ae8c21c0ab8d69fb87ce26369d5cd392f9c31f9dc410d6560aa32cdeceddbc.jpg new file mode 100644 index 00000000..0979b4d6 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-571_29ae8c21c0ab8d69fb87ce26369d5cd392f9c31f9dc410d6560aa32cdeceddbc.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-571_2ea78168faf193560177105db30fd46e95a4096fcecb24992b2decd8fae4219a.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-571_2ea78168faf193560177105db30fd46e95a4096fcecb24992b2decd8fae4219a.jpg new file mode 100644 index 00000000..32953c45 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-571_2ea78168faf193560177105db30fd46e95a4096fcecb24992b2decd8fae4219a.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-571_755b2d40493db7574fd93d548560d1346df1205713b866c551b72d74c1c86bc2.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-571_755b2d40493db7574fd93d548560d1346df1205713b866c551b72d74c1c86bc2.jpg new file mode 100644 index 00000000..fa0dc7cf Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-571_755b2d40493db7574fd93d548560d1346df1205713b866c551b72d74c1c86bc2.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-571_8859ffd850b6e8326656fd9ae9304c02b3d0126b375782e07cb664fbfb193f97.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-571_8859ffd850b6e8326656fd9ae9304c02b3d0126b375782e07cb664fbfb193f97.jpg new file mode 100644 index 00000000..89f66d80 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-571_8859ffd850b6e8326656fd9ae9304c02b3d0126b375782e07cb664fbfb193f97.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-571_8c13d67d284de36537f4c76793893048533abed1ac544a67cd0e5d435d3be411.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-571_8c13d67d284de36537f4c76793893048533abed1ac544a67cd0e5d435d3be411.jpg new file mode 100644 index 00000000..31d7bad7 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-571_8c13d67d284de36537f4c76793893048533abed1ac544a67cd0e5d435d3be411.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-572_29ba8fb5c2fc7e7315ca303af924f51067bef0d5e90d8ede30479aac5bde735e.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-572_29ba8fb5c2fc7e7315ca303af924f51067bef0d5e90d8ede30479aac5bde735e.jpg new file mode 100644 index 00000000..2cc22568 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-572_29ba8fb5c2fc7e7315ca303af924f51067bef0d5e90d8ede30479aac5bde735e.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-572_2e5e4d0cedc6ca6cb8b16a5599832bf22507fefcac6e0c07eee735580823ddd7.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-572_2e5e4d0cedc6ca6cb8b16a5599832bf22507fefcac6e0c07eee735580823ddd7.jpg new file mode 100644 index 00000000..8f44b436 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-572_2e5e4d0cedc6ca6cb8b16a5599832bf22507fefcac6e0c07eee735580823ddd7.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-572_3c9dc8f3f85f2127043c9af9cd71eafe0ab288b94f76507cb4dbbc63288d8bec.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-572_3c9dc8f3f85f2127043c9af9cd71eafe0ab288b94f76507cb4dbbc63288d8bec.jpg new file mode 100644 index 00000000..a030c34c Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-572_3c9dc8f3f85f2127043c9af9cd71eafe0ab288b94f76507cb4dbbc63288d8bec.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-572_4b600b2aa769bdef7fb7acafd8796573abf4049f77cfd6bb773cbf037c66e1e6.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-572_4b600b2aa769bdef7fb7acafd8796573abf4049f77cfd6bb773cbf037c66e1e6.jpg new file mode 100644 index 00000000..1de3ee2d Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-572_4b600b2aa769bdef7fb7acafd8796573abf4049f77cfd6bb773cbf037c66e1e6.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-572_55d63ed66cf7f07734c9dde8cebfdc71d7b96a5d10d068095d346c692098176b.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-572_55d63ed66cf7f07734c9dde8cebfdc71d7b96a5d10d068095d346c692098176b.jpg new file mode 100644 index 00000000..1f657050 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-572_55d63ed66cf7f07734c9dde8cebfdc71d7b96a5d10d068095d346c692098176b.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-572_69d5475b4f162a254cc0b7a9a50833b851a5c332245c107132b9c912f22a9aaa.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-572_69d5475b4f162a254cc0b7a9a50833b851a5c332245c107132b9c912f22a9aaa.jpg new file mode 100644 index 00000000..b7a31ef6 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-572_69d5475b4f162a254cc0b7a9a50833b851a5c332245c107132b9c912f22a9aaa.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-572_71c1efc5d32304a7b7dfe2a8777680139601cbe775154bab0feaac07a753d03d.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-572_71c1efc5d32304a7b7dfe2a8777680139601cbe775154bab0feaac07a753d03d.jpg new file mode 100644 index 00000000..21913c1d Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-572_71c1efc5d32304a7b7dfe2a8777680139601cbe775154bab0feaac07a753d03d.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-572_98a8d02b9840b46b7ca4e1ea2e9b272d77ad247f6174bf96a128201ec5b0dfa7.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-572_98a8d02b9840b46b7ca4e1ea2e9b272d77ad247f6174bf96a128201ec5b0dfa7.jpg new file mode 100644 index 00000000..c31e4c56 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-572_98a8d02b9840b46b7ca4e1ea2e9b272d77ad247f6174bf96a128201ec5b0dfa7.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-572_abe14e198e82da18dbf618323718f006ab6b5ab4040f71148a4abb76c4aec0ea.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-572_abe14e198e82da18dbf618323718f006ab6b5ab4040f71148a4abb76c4aec0ea.jpg new file mode 100644 index 00000000..f35d0a42 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-572_abe14e198e82da18dbf618323718f006ab6b5ab4040f71148a4abb76c4aec0ea.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-572_d0397300a3103238ae0cd1b3c56f4c2a6d576695d5baf349129280cae89a7748.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-572_d0397300a3103238ae0cd1b3c56f4c2a6d576695d5baf349129280cae89a7748.jpg new file mode 100644 index 00000000..ff4dce06 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-572_d0397300a3103238ae0cd1b3c56f4c2a6d576695d5baf349129280cae89a7748.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-572_ed248c633f38ff90426341df81706b4cbe96ece6ab735d5e832a30c59b5df3b7.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-572_ed248c633f38ff90426341df81706b4cbe96ece6ab735d5e832a30c59b5df3b7.jpg new file mode 100644 index 00000000..f88b5de9 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-572_ed248c633f38ff90426341df81706b4cbe96ece6ab735d5e832a30c59b5df3b7.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-573_930d81891397a38d9c3631177eed4a26262e1a25327454dddbb7688bea957a31.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-573_930d81891397a38d9c3631177eed4a26262e1a25327454dddbb7688bea957a31.jpg new file mode 100644 index 00000000..42b276b2 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-573_930d81891397a38d9c3631177eed4a26262e1a25327454dddbb7688bea957a31.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-573_ab91d496aa027c39eaeef478fa4490ed948a725b710e45b31a6f4820b2665eea.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-573_ab91d496aa027c39eaeef478fa4490ed948a725b710e45b31a6f4820b2665eea.jpg new file mode 100644 index 00000000..2197cf35 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-573_ab91d496aa027c39eaeef478fa4490ed948a725b710e45b31a6f4820b2665eea.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-573_faa34db25ad9f41d11d0da230385fffbfa68bf7e2ed209d89dcebe357f8fa99d.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-573_faa34db25ad9f41d11d0da230385fffbfa68bf7e2ed209d89dcebe357f8fa99d.jpg new file mode 100644 index 00000000..9baeb485 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-573_faa34db25ad9f41d11d0da230385fffbfa68bf7e2ed209d89dcebe357f8fa99d.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-574_37b64ac9550c93cbd1d661687aa89c8b278a3dfc85f12d86a527520f9585d3a9.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-574_37b64ac9550c93cbd1d661687aa89c8b278a3dfc85f12d86a527520f9585d3a9.jpg new file mode 100644 index 00000000..ce13c0a4 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-574_37b64ac9550c93cbd1d661687aa89c8b278a3dfc85f12d86a527520f9585d3a9.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-574_6051533781a23e369f73a3e42a9554cc0c79a9f608b20065303be8c2f93fe6fd.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-574_6051533781a23e369f73a3e42a9554cc0c79a9f608b20065303be8c2f93fe6fd.jpg new file mode 100644 index 00000000..cd64fe83 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-574_6051533781a23e369f73a3e42a9554cc0c79a9f608b20065303be8c2f93fe6fd.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-574_8e89740fe2df400bb41a349294efd75dd88a3dbfaf895520f85e56defb81c472.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-574_8e89740fe2df400bb41a349294efd75dd88a3dbfaf895520f85e56defb81c472.jpg new file mode 100644 index 00000000..af8a0113 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-574_8e89740fe2df400bb41a349294efd75dd88a3dbfaf895520f85e56defb81c472.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-575_a567168409357fe537f4b6eb33977c0c3c1b1d3cd5b15dcca5397e3270c40560.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-575_a567168409357fe537f4b6eb33977c0c3c1b1d3cd5b15dcca5397e3270c40560.jpg new file mode 100644 index 00000000..8ea156a0 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-575_a567168409357fe537f4b6eb33977c0c3c1b1d3cd5b15dcca5397e3270c40560.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-576_22b3d09f279edd104b2a747f39eb3e3892485e4bb3ce945e68532fb32a0ed6d1.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-576_22b3d09f279edd104b2a747f39eb3e3892485e4bb3ce945e68532fb32a0ed6d1.jpg new file mode 100644 index 00000000..a98e2528 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-576_22b3d09f279edd104b2a747f39eb3e3892485e4bb3ce945e68532fb32a0ed6d1.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-576_34da9f03b73a011862d6cc75f789e57a4f97dcbe389d2c908340ed53d5cb3048.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-576_34da9f03b73a011862d6cc75f789e57a4f97dcbe389d2c908340ed53d5cb3048.jpg new file mode 100644 index 00000000..fff09ca7 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-576_34da9f03b73a011862d6cc75f789e57a4f97dcbe389d2c908340ed53d5cb3048.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-577_ae65fd0d29f9fa6854945918f9f8a6255faae7f6e54493d9b63076f1be89e9f1.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-577_ae65fd0d29f9fa6854945918f9f8a6255faae7f6e54493d9b63076f1be89e9f1.jpg new file mode 100644 index 00000000..6886a66b Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-577_ae65fd0d29f9fa6854945918f9f8a6255faae7f6e54493d9b63076f1be89e9f1.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-577_f6e94e16f7d8f5cb0e1ee6272a0a5ec15c0af1eed74026385aa7cebcedcd0b62.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-577_f6e94e16f7d8f5cb0e1ee6272a0a5ec15c0af1eed74026385aa7cebcedcd0b62.jpg new file mode 100644 index 00000000..7f96a859 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-577_f6e94e16f7d8f5cb0e1ee6272a0a5ec15c0af1eed74026385aa7cebcedcd0b62.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-578_7f92922e7785131826590ce1f26064aff66021d24d45d2a23cbbad768a72000f.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-578_7f92922e7785131826590ce1f26064aff66021d24d45d2a23cbbad768a72000f.jpg new file mode 100644 index 00000000..686e3529 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-578_7f92922e7785131826590ce1f26064aff66021d24d45d2a23cbbad768a72000f.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-579_8253154f6c40d23ba5645deb459c536dd671c97d43f002e780ffb333f3f8ee47.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-579_8253154f6c40d23ba5645deb459c536dd671c97d43f002e780ffb333f3f8ee47.jpg new file mode 100644 index 00000000..611ec9d3 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-579_8253154f6c40d23ba5645deb459c536dd671c97d43f002e780ffb333f3f8ee47.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-579_ec9313858a07c0021b819a5e339ab127ae5794dcda0ce5eaece26e06e919409e.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-579_ec9313858a07c0021b819a5e339ab127ae5794dcda0ce5eaece26e06e919409e.jpg new file mode 100644 index 00000000..40f518e3 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-579_ec9313858a07c0021b819a5e339ab127ae5794dcda0ce5eaece26e06e919409e.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-580_39d295a6002721a5dd651b6619e65c816c52523d27f363c40ade0abfa23ce933.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-580_39d295a6002721a5dd651b6619e65c816c52523d27f363c40ade0abfa23ce933.jpg new file mode 100644 index 00000000..033dd94a Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-580_39d295a6002721a5dd651b6619e65c816c52523d27f363c40ade0abfa23ce933.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-581_a584f43fc1c84fce914892a8fab32520ca4b30efdf3c7744fb5857e97922dffc.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-581_a584f43fc1c84fce914892a8fab32520ca4b30efdf3c7744fb5857e97922dffc.jpg new file mode 100644 index 00000000..0c1fc587 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-581_a584f43fc1c84fce914892a8fab32520ca4b30efdf3c7744fb5857e97922dffc.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-582_ac53f94e02a806e4d14b89a707bd13e7e3aef0506895926441c55269347bd2aa.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-582_ac53f94e02a806e4d14b89a707bd13e7e3aef0506895926441c55269347bd2aa.jpg new file mode 100644 index 00000000..19bc3ce2 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-582_ac53f94e02a806e4d14b89a707bd13e7e3aef0506895926441c55269347bd2aa.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-583_ef861ea72758c80721788ed4664c523921cb529e4964d544434d829d200e4847.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-583_ef861ea72758c80721788ed4664c523921cb529e4964d544434d829d200e4847.jpg new file mode 100644 index 00000000..887043d1 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-583_ef861ea72758c80721788ed4664c523921cb529e4964d544434d829d200e4847.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-584_6750e09b3a99457116e26e0c78798a0c9e19a159767b004ae40b1db3be573373.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-584_6750e09b3a99457116e26e0c78798a0c9e19a159767b004ae40b1db3be573373.jpg new file mode 100644 index 00000000..93095aab Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-584_6750e09b3a99457116e26e0c78798a0c9e19a159767b004ae40b1db3be573373.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-584_88036c74c59ccb11db5c3879b2505a4a2dd3be7cd0e1121655cbe6ff3bcdb4a4.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-584_88036c74c59ccb11db5c3879b2505a4a2dd3be7cd0e1121655cbe6ff3bcdb4a4.jpg new file mode 100644 index 00000000..47aa986c Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-584_88036c74c59ccb11db5c3879b2505a4a2dd3be7cd0e1121655cbe6ff3bcdb4a4.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-585_070b6d06f44a6535840b1068d77e05813e2f986acf7613faf14ce08d1e8ba905.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-585_070b6d06f44a6535840b1068d77e05813e2f986acf7613faf14ce08d1e8ba905.jpg new file mode 100644 index 00000000..e7559574 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-585_070b6d06f44a6535840b1068d77e05813e2f986acf7613faf14ce08d1e8ba905.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-585_5879ea82aac6a414e808326798a682fd4d6bba66e0076b2f8e95a94b332e546e.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-585_5879ea82aac6a414e808326798a682fd4d6bba66e0076b2f8e95a94b332e546e.jpg new file mode 100644 index 00000000..dc9d55e4 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-585_5879ea82aac6a414e808326798a682fd4d6bba66e0076b2f8e95a94b332e546e.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-586_ec51171cd08ac16944a325fc290966c7753b539b0e65b628ddb8302dcb6e7f89.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-586_ec51171cd08ac16944a325fc290966c7753b539b0e65b628ddb8302dcb6e7f89.jpg new file mode 100644 index 00000000..03eb7477 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-586_ec51171cd08ac16944a325fc290966c7753b539b0e65b628ddb8302dcb6e7f89.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-587_3b9710d08b47ca85d94433664156ef4b55cba0c912fdfa9f0b65c3012645b02c.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-587_3b9710d08b47ca85d94433664156ef4b55cba0c912fdfa9f0b65c3012645b02c.jpg new file mode 100644 index 00000000..ba71b2ab Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-587_3b9710d08b47ca85d94433664156ef4b55cba0c912fdfa9f0b65c3012645b02c.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-587_4160156c7c0dc9e8f94a8a68783a11454b297c64ce7bb3897d5b540ccae7c281.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-587_4160156c7c0dc9e8f94a8a68783a11454b297c64ce7bb3897d5b540ccae7c281.jpg new file mode 100644 index 00000000..e5b23034 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-587_4160156c7c0dc9e8f94a8a68783a11454b297c64ce7bb3897d5b540ccae7c281.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-588_b0b45acf1efa300bf5405d82ad665f59fadcda9b784e91e45ddc210701d44145.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-588_b0b45acf1efa300bf5405d82ad665f59fadcda9b784e91e45ddc210701d44145.jpg new file mode 100644 index 00000000..fa83d39a Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-588_b0b45acf1efa300bf5405d82ad665f59fadcda9b784e91e45ddc210701d44145.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-588_b9f943e98b5b6c42e13e0f4b19660f65b114d1c5e9707abc9df0897bde59ad51.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-588_b9f943e98b5b6c42e13e0f4b19660f65b114d1c5e9707abc9df0897bde59ad51.jpg new file mode 100644 index 00000000..d66e7d5c Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-588_b9f943e98b5b6c42e13e0f4b19660f65b114d1c5e9707abc9df0897bde59ad51.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-589_74a74c7c887a7e2affe452e3be25276bc0ae82e67dbbedcfd89c4386848756b1.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-589_74a74c7c887a7e2affe452e3be25276bc0ae82e67dbbedcfd89c4386848756b1.jpg new file mode 100644 index 00000000..5769e24e Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-589_74a74c7c887a7e2affe452e3be25276bc0ae82e67dbbedcfd89c4386848756b1.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-589_b4638cbb20429fea4dd99c588a1f561db59360909d6bf23c2891eba9fc1fb199.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-589_b4638cbb20429fea4dd99c588a1f561db59360909d6bf23c2891eba9fc1fb199.jpg new file mode 100644 index 00000000..a189423e Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-589_b4638cbb20429fea4dd99c588a1f561db59360909d6bf23c2891eba9fc1fb199.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-590_1d7ad2c5df3fff9c9f96fee3432510dc134d8765cbfbaa058b706ea3e9f8a3cb.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-590_1d7ad2c5df3fff9c9f96fee3432510dc134d8765cbfbaa058b706ea3e9f8a3cb.jpg new file mode 100644 index 00000000..0019a39b Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-590_1d7ad2c5df3fff9c9f96fee3432510dc134d8765cbfbaa058b706ea3e9f8a3cb.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-590_87f8db47b83736438b5e29404593620fb86da4a6342d5849c4b1346094d80282.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-590_87f8db47b83736438b5e29404593620fb86da4a6342d5849c4b1346094d80282.jpg new file mode 100644 index 00000000..914f6174 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-590_87f8db47b83736438b5e29404593620fb86da4a6342d5849c4b1346094d80282.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-591_7b934b6f60deb89421100f46bea2f9db299be8a8bd4da3ddc5e354b14ea71935.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-591_7b934b6f60deb89421100f46bea2f9db299be8a8bd4da3ddc5e354b14ea71935.jpg new file mode 100644 index 00000000..e0b3d283 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-591_7b934b6f60deb89421100f46bea2f9db299be8a8bd4da3ddc5e354b14ea71935.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-591_c495d4856ab01ae5980d77153341d21b6d6495d750590be483c1159db87fe98a.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-591_c495d4856ab01ae5980d77153341d21b6d6495d750590be483c1159db87fe98a.jpg new file mode 100644 index 00000000..3a1abe35 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-591_c495d4856ab01ae5980d77153341d21b6d6495d750590be483c1159db87fe98a.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-591_d5e0c3ff2ae49b71f17b01dd19264faee2370acaa3f3a7974cecff3bb951aac1.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-591_d5e0c3ff2ae49b71f17b01dd19264faee2370acaa3f3a7974cecff3bb951aac1.jpg new file mode 100644 index 00000000..e35523ed Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-591_d5e0c3ff2ae49b71f17b01dd19264faee2370acaa3f3a7974cecff3bb951aac1.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-591_f4a8b15f364c3f69e280e57f1cd5052510a394a003933d36d5efc4d4c17783e4.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-591_f4a8b15f364c3f69e280e57f1cd5052510a394a003933d36d5efc4d4c17783e4.jpg new file mode 100644 index 00000000..b598e899 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-591_f4a8b15f364c3f69e280e57f1cd5052510a394a003933d36d5efc4d4c17783e4.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-591_ff49985916681f294bf13d92d18cacc1c8f6c63842fa01948be2c1e9d5d671ec.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-591_ff49985916681f294bf13d92d18cacc1c8f6c63842fa01948be2c1e9d5d671ec.jpg new file mode 100644 index 00000000..69c6fb01 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-591_ff49985916681f294bf13d92d18cacc1c8f6c63842fa01948be2c1e9d5d671ec.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-592_205a5a304ef9aea1d38929b99b5bfe283aede75fc8dbc65be421c805299023c9.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-592_205a5a304ef9aea1d38929b99b5bfe283aede75fc8dbc65be421c805299023c9.jpg new file mode 100644 index 00000000..620e55b9 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-592_205a5a304ef9aea1d38929b99b5bfe283aede75fc8dbc65be421c805299023c9.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-592_a9909858fcb12b424c47c7c7c50e633e2b77b97e89405fc6664703ecce929ace.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-592_a9909858fcb12b424c47c7c7c50e633e2b77b97e89405fc6664703ecce929ace.jpg new file mode 100644 index 00000000..702dbd36 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-592_a9909858fcb12b424c47c7c7c50e633e2b77b97e89405fc6664703ecce929ace.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-592_c3582b571250276f4c2c1788167a4fae4c6c254e5887a061c62ed0f3e305c800.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-592_c3582b571250276f4c2c1788167a4fae4c6c254e5887a061c62ed0f3e305c800.jpg new file mode 100644 index 00000000..2b3ea4dd Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-592_c3582b571250276f4c2c1788167a4fae4c6c254e5887a061c62ed0f3e305c800.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-593_1f35a8395a172bfeb53c471d43b45c78ecf14d9d8f8fb29919bfc856b1551a47.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-593_1f35a8395a172bfeb53c471d43b45c78ecf14d9d8f8fb29919bfc856b1551a47.jpg new file mode 100644 index 00000000..e5a08912 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-593_1f35a8395a172bfeb53c471d43b45c78ecf14d9d8f8fb29919bfc856b1551a47.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-593_f8d9bd0ac891363c288629d7d887b98c7ee9a7f31c3308b724a0139f5bc79e3e.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-593_f8d9bd0ac891363c288629d7d887b98c7ee9a7f31c3308b724a0139f5bc79e3e.jpg new file mode 100644 index 00000000..008a0471 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-593_f8d9bd0ac891363c288629d7d887b98c7ee9a7f31c3308b724a0139f5bc79e3e.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-594_b901ed7754f2a183b2089ed70489ef8dc76ec78ebfe71e3fbb4dd308e07a4eb9.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-594_b901ed7754f2a183b2089ed70489ef8dc76ec78ebfe71e3fbb4dd308e07a4eb9.jpg new file mode 100644 index 00000000..e61fb0d0 Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-594_b901ed7754f2a183b2089ed70489ef8dc76ec78ebfe71e3fbb4dd308e07a4eb9.jpg differ diff --git a/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-594_cd2444fdf3b78d338f931707d2bcb7b3a58f46c2113a5c4648ceb600e6808db8.jpg b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-594_cd2444fdf3b78d338f931707d2bcb7b3a58f46c2113a5c4648ceb600e6808db8.jpg new file mode 100644 index 00000000..424e2fcf Binary files /dev/null and b/.raw/FiniteElementsinPlasticityTheoryandPractice/images/page-594_cd2444fdf3b78d338f931707d2bcb7b3a58f46c2113a5c4648ceb600e6808db8.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_001.md b/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_001.md new file mode 100644 index 00000000..3cfee33c --- /dev/null +++ b/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_001.md @@ -0,0 +1,226 @@ + + +MIDAS Family Program은 + +주식회사 마이다스아이티에서 개발한 + +구조해석 및 설계, 지반 및 터널해석, 가시설전용 해석 및 설계용 소프트웨어 패키지입니다. + +MIDAS Family Program과 관련 책자는 + +컴퓨터 프로그램 보호법과 저작권법에 의하여 보호를 받고 있습니다. + +프로그램이나 관련 자료에 대한 문의는 아래 연락처를 참조 바랍니다. + +![](images/page-001_9d9d07a2fdc50e84d4b5539b3fde05f70467736df5c1373d697a90e0aaa40602.jpg) + +# OOO + +MIDAS Information Technology Co.,Ltd. + +경기도 성남시 분당구 삼평동 633 판교세븐벤처밸리 마이다스아이티동 + +Phone : 031-789-2000 + +Fax : 031-789-2100 + +E-mail : midas@midasit.com + +http : //www.midasuser.com + +Modeling, Integrated Design & Analysis Software + +본 사용자 지침서의 작성에 인용된 상표(trademark) 및 등록상표(registered trademark)는 + +다음과 같습니다. + +ADINA is a trademark of ADINA R&D, Inc. + +AutoCAD is a registered trademark of Autodesk, Inc. + +SAP2000 is registered trademark of Computer and Structure, Inc. + +Excel is a trademark of Microsoft Corporation. + +IBM is a registered trademarks of International Business Machines Corporation. + +Intel 386, 486, and Pentium are trademark of Intel Corporation. + +MIDAS is a trademark of MIDAS Information Technology Corporation. + +MSC/NASTRAN is a registered trademark of the National Aeronautics and Space Administration(NASA). + +NISAⅡis a trademark of Engineering Mechanics Research Corporation. + +ScreenCam is a trademark of Lotus Development Corporation. + +Sentinel is a trademark of Rainbow Technologies, Inc. + +Windows is a trademark of Microsoft Corporation. + +Internet Explorer is a trademark of Microsoft Corporation. + + + +MIDAS Family Program은 개발단계에서 수천종의 예제 문제를 통하여 이론치 그리고 타 S/W와의 비교검증을 마친바 있으며 최신의 이론을 내장하여 우수한 해석결과를 산출합니다. + +그리고 1989년 개발 이후 관공서를 포함해 국내외 5,000여 프로젝트에 적용하여 정확성과 효용성이 입증되었습니다. + +MIDAS Family Program은 사단법인 한국전산구조공학회와 사단법인 한국터널공학회의 엄격한 검증과정을거친 프로그램입니다. + +그러나 방대한 양의 이론과 설계지식이 집적되는 구조해석 및 설계 프로그램의 특성상, MIDAS FamilyProgram을 사용함으로써 발생될 수 있는 어떠한 이익과 손실에 대해서도 MIDAS Family Program의 개발후원자와 개발자 그리고, 검증참여기관에게는 권리와 책임이 없습니다. + +따라서 프로그램을 사용하기 전에 사용자지침서에 대한 충분한 이해과정이 필요하며 프로그램의 수행결과에대해서도 사용자의 검증이 반드시 필요합니다. + +# DISCLAIMER + +Developers and sponsors assume no responsibility for the use of MIDAS Family Program (midas Civil, midas FEA, midas UMD, midas Abutment, midas Pier, midas Deck, midas GTS, SoilWorks, GeoXD, midas Gen, midas ADS, midas Modeler, midas Drawing, midas Desiign+, midas SDS, midas eGen, midas NFX, Nastran FX ; hereinafter referred to as “MIDAS package”) or for the accuracy or validity of any results obtain from the MIDAS package. + +Developers and sponsors shall not be liable for loss of profit, loss of business, or financial loss which may be caused directly or indirectly by the MIDAS package, when used for any purpose or use, due to any defect or deficiency therein. + + + +MIDAS Family Program은 포스코그룹의 창사 이래 엔지니어링과 건설분야에서 축적한 구조설계 및 설계기술을 집적하여 지반, 터널 및 가시설분야로 영역을 확대하여 국내의 여러 교수님들과 기술자 여러분들의 도움으로 만들어진 것입니다. + +본 프로그램의 개발과 사용자지침서의 작성에 도움을 주신 학계 교수님, 그리고 관련 분야의 기술자여러분께감사드립니다. + +특히, 프로그램 개발 전반에 걸쳐 조언과 알고리즘을 제공해 주신 서울산업대 정완진 교수, VSL Korea의 이만섭 사장, COWI Korea의 박찬민 사장, 박정호 실장, 박인교 부장, Parsons Brinckerhoff의 이승우 박사, 울산대학교 차수원 교수님께 깊이 감사 드립니다. + +그리고 본 프로그램의 개발을 위해 지원을 아끼지 않으신 대한토목학회, 한국강구조공학회, 한국전산구조공학회의 여러분께도 본 지면을 빌어 감사의 말씀을 올립니다. + +폐사는 본 프로그램이 우리나라 구조해석 및 설계 분야의 기술 신장과 대외 기술 경쟁력 확보에 다소나마 기여할 수 있기를 바랍니다. + +주식회사 마이다스아이티 + + + +# 머리말 + +midas Civil은 토목구조물의 구조해석과 설계를 빠른 시간내에 완성할 수 있도록 개발된 “토목전용 구조해석및 최적설계 프로그램”이며, Civil은 “Civil structure analysis/design”을 의미합니다. + +# midas Civil과 MIDAS Family Program에 대하여 + +midas Civil은 1989년부터 개발되기 시작한 MIDAS Family Program 중 하나입니다. MIDAS FamilyProgram은 구조물 해석 및 설계에 수반되는 단위설계 업무의 전 과정을 자동화하기 위한 목적으로 개발된package software로서 다음과 같이 구성되어 있습니다. + +
토목분야midas Civil토목분야 범용 구조해석 및 최적설계 시스템
midas FEA건설분야 비선형 해석 및 상세해석 시스템
midas UMD토목분야 단위 부재 설계 프로그램
midas Abutment교대설계(계산서, 도면, 수량) 자동화 시스템
midas Pier교각설계(계산서, 도면, 수량) 자동화 시스템
midas Deck콘크리트 바닥판 설계(계산서, 도면, 수량) 자동화 시스템
지반분야midas GTS지반 및 터널구조물 전용 해석 시스템
SoilWorks지반분야 최적 설계용 토탈 솔루션
GeoXD가시설 구조설계 및 도면생성 시스템
건축분야midas Gen건축분야 범용 구조해석 및 최적설계 시스템
midas ADS전단벽식 아파트 전용 구조해석 및 설계 시스템
midas Modeler3D 구조해석모델 자동 생성 프로그램
midas Drawing구조도면 및 물량산출 자동 생성 프로그램
midas Design+부재설계 및 구조도면 자동생성 프로그램
midas SDS바닥판/기초판 구조해석 및 설계 시스템
midas eGen저층 건축물 전용 구조설계 프로그램
기계분야midas NFX최적설계용 다분야 통합해석 솔루션
Nastran FX기계분야 토탈 구조해석 시스템
+ + + +midas Civil은 포스코 그룹의 실무 및 연구 기술진을 중심으로 관련 학계의 교수 여러분과 업계의 실무자 여러분들의 조력으로 개발된 프로그램이며 윈도우즈기반의 객체 지향적 특성과 기존 Gen의 장점을 최대한 반영하여 Visual C++로 개발 되었기 때문에 빠르고 쉽게 익혀서 실무에 적용할 수 있습니다. + +특히, 정교하게 설계된 GUI(Graphic User Interface)기능과 도화처리(Graphic display)기능을 이용하여 구조모델의 형상과정을 입력 단계 별로 확인할 수 있으며, 출력결과를 바로 문서화할 수 있도록 개발되었습니다. + +또한 midas Civil은 개발 과정에서 수천종의 검증용 문제를 통하여 모든 기능에 대해 이론치 및 타 범용 프로그램과의 비교 검토를 마쳤으며, 이미 다양한 실무 프로젝트에 적용되어 신뢰성과 효용성에 대한 검증이 이루어진 바 있습니다. + +이들 중에서 대표적인 검증 예제를 발췌하여 작성한 Verification을 홈페이지(www.midasuser.com)에 등재하였습니다. 해석 결과의 정확도를 결정하는 유한요소의 알고리즘 측면에서도 최신의 이론을 적용하였기 때문에 타 유사 프로그램에 비해 우수한 결과를 산출합니다. + +# 끝으로 + +midas Civil은 국내의 수많은 기술자들과 학계에 계신 교수님들의 노력과 협조로 잉태된 것입니다. + +이제부터 midas Civil을 사용하실 여러분의 성공적인 성과를 기대하며, 사용상 불편한 점이나 개선사항이 있으면 연락 바랍니다. + +끝으로 midas Civil을 개발하는 동안 참여해 주신 여러분께 감사 드리며, 특히 헌신적인 희생을 감내하여 주신 개발자와 가족 여러분께 지면을 빌어 감사의 말을 전합니다. + + + +midas Civil의 사용자지침서는 다음과 같이 2권의 책자와 Online manual로 구성되어 있습니다. + +제 1권, Getting Started + +프로그램의 개요와 프로그램 사용 전에 알아야 할 사항 + +제 2권, Analysis Reference + +수치해석모델과 요소 및 해석기능에 대한 해설 + +Online Manual + +프로그램에 내장되어 있으며 각 기능에 대한 자세한 사용법과 각 입력항목에 대한 설명 + +midas Civil의 특성과 기능을 효과적으로 이해하고 습득하기 위해서는 다음과 같은 순서에 따라 지침서에 포함된 내용을 먼저 이해한 후에 프로그램을 사용하는 것이 바람직합니다. + +우선 midas Civil의 구조해석 기능에 대한 해설내용을 포함하고 있는 제 2권을 보시기 바랍니다. + +midas Gen이나 타 프로그램에 대한 사용경험이 있는 경우에는 이 과정을 생략하고, 프로그램 사용 중에 참고가 필요할 때 원하는 부분을 찾아보시면 됩니다. + +제 2권에 기술된 내용들은 midas Civil로 유한요소해석을 수행하는데 필요한 기초적 고려사항과 설계과정에서 기본적으로 숙지해야 하는 내용들을 포함하고 있습니다. 실제로 구조해석 이론과 사용 프로그램에 대한이해가 부족한 상태에서 구조해석을 수행할 경우, 오류가 포함될 확률이 90%를 상회하는 것으로 외국 저널등에 심각하게 보고되고 있습니다. + +제 1권의 “설치하기”부분을 보시고 안내된 절차에 따라 midas Civil을 설치하십시오. + +그리고 midas Civil을 사용하는데 필요한 기본개념을 포함하고 있는 제 1권의 나머지 부분들을 읽어 보시기바랍니다. + +제 1권에는 모델링을 위한 “작업환경 설정하기”, “데이터 입력하기”, “모델화면 다루기”, “선택기능과 활성화/비활성화기능” 등과 같이 midas Civil의 효과적인 운용을 위해 반드시 숙지해야 할 GUI 환경에 대한 사용법과“모델링하기”, “해석하기”, “해석결과 분석하기” 등 실제 해석 작업에 필수적으로 수반되는 기능들에 대한 설명이 포함되어 있습니다. + + + +참고로 각 기능에 대한 자세한 사용법과 각 입력항목에 대한 설명은 midas Civil의 help메뉴에 내장된Online manual의 “midas Civil의 기능” 부분에 기술되어 있습니다. + +MIDAS IT의 홈페이지(www.midasuser.com)에는 midas Civil의 주요 해석기능에 대하여 다양한 예제를 통하여 이론치 또는 타 구조해석 프로그램의 결과치와 비교, 검증한 내용을 담은 verification examples가 등재되어 있습니다. 검증예제들은 대부분 대학의 학과과정에서 접할 수 있는 간단한 문제들로 구성되어 있기 때문에 구조해석에 입문하는 초보 사용자들이 구조해석에 관한 개념을 파악하고 이해하는데 자료로 활용될 수 있습니다. + +MIDAS IT의 홈페이지(www.midasuser.com)에는 이외에도 “프로그램 기술자료”, “교육센터>온라인학습”, “교육센터>세미나 다시보기”, “질문과 답변” 그리고 “전문가 칼럼” 등의 코너를 운용하고 있습니다. midas Civil의사용법은 물론 실무를 수행하는데 필요한 구조기술을 전파하고 기타 유용한 자료를 실시간으로 제공하여, 항상 사용자 여러분 옆에 존재하는 살아있는 매뉴얼의 역할을 수행하고자 최선을 다하고 있습니다. + + + + + +# CONTENTS + +# Part 1 midas Civil의 수치해석 모델 + +Chapter 1. 수치해석 모델 ···· 001 + +Chapter 2. 좌표계와 절점··· 003 + +Chapter 3. 유한요소의 종류 ····· 005 + +3-1 트러스요소 (Truss Element) / 006 +3-2 인장력 전담요소 (Tension-only Element) / 010 +3-3 케이블 요소(Cable Element) / 012 +3-4 압축력 전담요소 (Compression-only Element) / 016 +3-5 보요소 (Beam Element) / 018 +3-6 평면응력요소 (Plane Stress Element) / 022 +3-7 평면변형요소 (2D Plane Strain Element) / 031 +3-8 축대칭요소 (2D Axisymmetric Element) / 041 +3-9 판요소 (Plate Element) / 048 +3-10 입체요소 (Solid Element) / 058 + +Chapter 4. 요소 입력시 주요 고려사항····· 072 + +4-1 트러스요소, 인장력 전담요소, 압축력 전담요소 / 075 +4-2 보요소 / 078 +4-3 평면응력요소 / 081 +4-4 평면변형요소 / 083 +4-5 축대칭요소 / 084 +4-6 판요소 / 085 +4-7 입체요소 / 087 +4-8 직교이방성재질 입력시 주요 고려사항 / 088 + + + +# CONTENTS + +# Chapter 5. 요소의 강성 데이터 ·· 089 + +5-1 단면적 (Area : Cross Sectional Area) / 091 +5-2 유효전단면적 (Asy, Asz : Effective Shear Area) / 092 +5-3 비틀림강성 (Ixx: Torsional Resistance) / 094 +5-4 단면2차모멘트 (Iyy, Izz: Area Moment of Inertia) / 101 +5-5 단면상승모멘트 (Iyz: Area Product Moment of Inertia) / 103 +5-6 단면1차모멘트 (Qy, Qz: First Moment of Area) / 106 +5-7 전단계수 (Qyb, Qzb: Shear Factors of Shear Stress due to Bending) / 107 +5-8 합성단면의 강성계산 / 108 + +# Chapter 6. 경계조건 109 + +6-1 경계조건 / 109 +6-2 자유도 구속조건 / 110 +6-3 탄성경계요소 / 113 +6-4 범용경계요소(General Spring Supports) / 116 +6-5 Distributed Spring(Winkler Spring) / 118 +6-6 탄성연결요소 / 121 +6-7 범용연결요소 (General Link) / 122 +6-8 요소의 단부해제조건 / 130 +6-9 강성역 / 132 +6-10 주절점과 종속절점(강체연결기능) / 145 +6-11 지지점의 강제변위 / 154 diff --git a/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_002.md b/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_002.md new file mode 100644 index 00000000..01182d48 --- /dev/null +++ b/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_002.md @@ -0,0 +1,227 @@ + + +# CONTENTS + +# Part 2 midas Civil의 구조해석 기능 + +Chapter 1. 구조해석 기능 ···· 157 + +Chapter 2. 정적해석 159 + +Chapter 3. 자유진동 해석··· 160 + +3-1 고유벡터 해석 / 160 + +3-2 Ritz벡터 해석 / 166 + +Chapter 4. 감쇠의 고려 171 + +4-1 감쇠의 개요 / 171 + +4-2 비례감쇠 / 175 + +4-3 Rayleigh 감쇠 / 177 + +4-4 변형율 에너지에 기초한 모드 감쇠 / 181 + +4-5 모드별 감쇠 / 185 + +4-6 요소별 Rayleigh 감쇠 / 186 + +4-7 감쇠행렬의 구성 / 187 + +4-8 범용연결요소의 선형감쇠의 고려 / 188 + +Chapter 5. 응답스펙트럼 해석 ···· 189 + + + +# CONTENTS + +# Chapter 6. 시간이력해석···· 193 + +6-1 모드 중첩법 / 194 +6-2 직접적분법 / 196 +6-3 다중지점 지진입력 하중에 대한 해석 / 200 + +# Chapter 7. 좌굴해석 203 + +# Chapter 8. 비선형해석 208 + +8-1 개요 / 208 +8-2 기하비선형 해석 / 210 +8-3 P-Delta / 216 +8-4 경계비선형 해석 / 221 +8-5 경계비선형 시간이력해석 (Boundary Nonlinear Time History Analysis)/ 224 +8-6 재료비선형 해석 (Material Nonlinear Analysis) / 254 +8-7 정적증분해석 (Pushover 해석) / 275 + +# Chapter 9. 비선형 시간이력해석 338 + +9-1 개요 / 338 +9-2 비탄성 요소 / 347 +9-3 비선형 이력 모델의 개요 / 355 +9-4 일축-힌지 이력모델(Hysteresis Model for Uni-axial Hinge) / 358 +9-5 다축-힌지 이력모델(Hysteresis Model for Multi-axial Hinge) / 398 +9-6 파이버 모델 / 406 + + + +# CONTENTS + +# Chapter 10. 시공단계해석··· 431 + +10-1 개요 / 431 +10-2 시간의존적 재질 / 433 +10-3 시공단계의 정의 및 구성 / 444 +10-4 비선형 시공단계 해석 / 449 +10-5 현수교 평형상태 해석 / 452 + +# Chapter 11. 수화열해석 ·· 458 + +11-1 열전달해석 (Heat Transfer Analysis) / 458 +11-2 열응력해석(Thermal Stress Analysis) / 463 +11-3 수화열 해석과정 / 467 + +# Chapter 12. PSC 해석 470 + +12-1 프리스트레스트 콘크리트의 해석 / 470 +12-2 프리스트레스의 손실 / 472 +12-3 프리스트레스 하중 / 479 + +# Chapter 13. 이동하중해석··· 481 + +13-1 차선과 차선면 / 484 +13-2 차량이동하중 / 493 +13-3 차량하중의 재하조건 / 508 + + + +# CONTENTS + +Chapter 14. 구조물의 지점침하를 자동 고려한 해석······ 518 + +Chapter 15. 강합성단면의 합성전∙후 해석 ·· 519 + +Chapter 16. 최적화기법을 사용한 미지하중의 해 ········ 520 + +Chapter 17. 임의형상 기둥의 부재설계 ··· 525 + +17-1 확대모멘트 계산 / 525 + +17-2 기둥부재 설계 / 529 + +17-3 임의 단면에 대한 3차원 축력-모멘트 상관도 분석 / 533 + +17-4 임의 단면에 대한 기둥의 전단설계 / 537 + +Chapter 18. Wave Load 하중생성 539 + +18-1 개요 / 539 + +18-2 파랑이론 / 543 + +18-3 파랑이론의 적용한계/ 550 + +18-4 Flow Chart / 551 + + + +# Part 1 midas Civil의 수치해석 모델 + + + +# Chapter 1. 수치해석 모델 + +구조물의 구조해석 모델은 절점과 유한요소 그리고 경계조건 데이터로 구성됩니다.절점은 구조부재의 위치를 지정하는데 사용되고, 유한요소는 구조부재를 수치해석적데이터로 입력하는데 사용되며, 경계조건은 해석대상의 구조모델과 인접 구조체와의연결상태를 고려하는 데 사용됩니다. + +구조해석이란, 구조물의 거동을 분석하기 위해 수치해석 모델을 이용하여 예견되는가상적 상황에 대한 이론적 모의실험을 수행하는 것입니다. + +그러므로 성공적인 해석작업을 위해서는 구조물의 구조적 성질과 외부환경적 조건에대한 정확한 묘사가 전제되어야 합니다. 여기서 외부환경적 요인인 하중조건은 적용법규 또는 확률론적 접근방법에 의해 결정된 결과를 따르면 되지만, 구조물의 구조적성질에 대해서는 수치해석 모델을 구성하고 있는 유한요소의 종류 및 모델링기법에따라 해석결과가 크게 달라질 수 있습니다. + +따라서 실제 구조물의 거동에 영향을 미치는 강성성분에 대해 충분히 파악한 후, 해당 강성을 실제구조물과 가능한 한 근접하게 반영될 수 있도록 유한요소를 선택하여야 합니다. + +그러나 실제의 구조물은 일반적으로 복잡한 형상과 여러 가지 물성치를 가진 재료로구성되기 때문에, 구조물이 지니고 있는 모든 강성성분과 질량성분을 정확하게 수치해석 모델에 반영한다는 것은 매우 어려운 작업이며 비경제적일 수 있습니다. + +따라서 해석하고자 하는 목적을 벗어나지 않는 한도 내에서 수치해석 모델을 단순화하거나 조정할 필요가 있습니다. + +예를 들어 교량주형을 모델링할 때 판형요소(평면응력요소 또는 판요소)를 사용하는것보다 선요소(트러스 요소, 보요소 등)를 사용하는 것이 소요시간측면이나 설계적용측면에서 보다 효과적입니다. + + + +유한요소(Fnite Element)란 구조물을 구성하는 각 구조요소의 구조적 특성을 수학적인 방법으로 이상화한 것이기 때문에 해당 구조요소의 특성을 모든 경우에 대해 완벽하게 묘사하기는 어렵습니다. + +따라서 전술한 바와 같이 사용하고자 하는 유한요소의 특성을 충분히 파악하고, 각유한요소간의 접합시 나타나는 거동을 정확하게 분석하여 설계에 적용하는 것이 바람직합니다. + + + +# Chapter 2. 좌표계와 절점 + +midas Civil에서는 다음과 같은 좌표계를 사용하고 있습니다. + +전체좌표계 (Global Coordinate System) + +요소좌표계 (Element Coordinate System) + +절점좌표계 (Node Local Coordinate System) + +전체좌표계는 오른손법칙을 따르는 X, Y, Z축의 직교좌표계(Conventional CartesianCoordinate System)를 사용하며, 대문자로 “X, Y, Z”축으로 표현합니다. 절점데이터,절점과 관련하여 입력되는 대부분의 데이터, 절점변위 그리고 반력 등이 본 좌표계를따르게 됩니다. + +전체좌표계는 구조해석을 수행하고자 하는 구조물의 기하학적 위치를 입력하는데 사용되며, 이때의 기준점(Reference Point)은 프로그램 내부에서 X=0, Y=0, Z=0인 위치에 자동으로 설정됩니다. midas Civil에서는 프로그램 화면의 수직방향이 전체좌표계Z축방향으로 구성되어 있기 때문에 구조물의 수직방향(중력가속도 작용방향의 반대방향)을 전체좌표계 Z방향과 평행하도록 입력하는 것이 편리합니다. + +요소좌표계는 오른손법칙을 따르는 x, y, z축의 직교좌표계를 사용하며, 소문자 “x, y,z”로 축을 표현합니다. + +요소내력, 응력 등과 요소와 관련되어 입력되는 대부분의 데이터가 이 좌표계를 따릅니다. + +절점좌표계(Node local Coordinate System)는 절점에 전체좌표계와 일치하지 않는 임의의 방향으로 구속조건, 경계스프링 또는 강제변위 등의 경계조건을 입력하거나 임의의 방향으로 반력을 계산하여 출력하고자 할 경우에 사용됩니다. + +절점좌표계는 오른손법칙을 따르는 x, y, z축의 직교좌표계를 사용하며, “x, y, z”축으로 표현합니다. + + + +![](images/page-019_f3768e4767a79825ff3cecb12ded7391e8d1297e37b8824b213c1fb5b15afc7d.jpg) + +
+text_image + +Z +a node(Xi, Yi, Zi) +Y +Zi +Reference point +(origin)of the Global +Coordinate System +Xi +Yi +X +
+ +그림 1.2.1 전체좌표계와 절점좌표 + + + +# Chapter 3. 유한요소의 종류 + +midas Civil에서 사용 가능한 유한요소의 종류(Element Library)는 다음과 같습니다. + +트러스요소 (Truss Element) + +인장력전담요소 (Tension-only Element, Hook 기능 포함) + +케이블요소 (Cable Element) + +압축력전담요소 (Compression-only Element, Gap 기능 포함) + +보요소/변단면요소 (Beam Element/Tapered Beam Element) + +평면응력요소 (Plane Stress Element) + +평면변형요소 (2D Plane Strain Element) + +축대칭요소 (2D Axisymmetric Element) + +판요소 (Plate Element) + +입체요소 (Solid Element) + +유한요소의 입력은 유한요소종류와 재료적 성질, 강성데이터 그리고 위치, 모양, 크기를 입력하기 위한 연결절점번호를 입력함으로써 이루어 집니다. diff --git a/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_003.md b/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_003.md new file mode 100644 index 00000000..102b02c5 --- /dev/null +++ b/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_003.md @@ -0,0 +1,391 @@ + + +# 3-1 트러스요소 (Truss Element) + +# 3-1-1 일반사항 + +이 요소는 2개의 절점에 의해 정의되는 "Uniaxial Tension-Compression 3D Line Element"로서, 일반적으로 공간트러스(Space Truss) 또는 대각부재(Diagonal Brace) 등을 모델링 하는데 사용되며 요소 축방향의 힘만 전달할 수 있습니다. + +# 3-1-2 요소자유도 및 요소좌표계 + +요소자유도는 요소좌표계 x방향의 변위자유도만 갖습니다. + +요소좌표계는 모든 요소의 부재력 또는 응력의 출력 기준이 되고, 특히 보요소의 전단강성과 힘강성의 입력방향을 정하는 기준이 되기 때문에 이 개념을 정확히 이해하는 것이 중요합니다. + +트러스나 인장력/압축력 전담요소와 같이 축방향 강성만 가지는 요소의 경우는 요소 좌표계 x축만 의미를 가지며 y, z축은 의미를 가지지 않지만, 그래픽 화면상에 부재 단면의 배치방향을 지정하는데 필요합니다. + +midas Civil은 사용자 편의를 위해 요소좌표계 y, z축의 방향을 지정하는데 Beta Angle(β) 개념을 사용합니다. + +모든 선요소의 요소좌표계 x축은 N1 절점에서 N2 절점으로 진행하는 방향이 됩니다. (그림 1.3.2 참조) 선요소의 요소좌표계 x축이 전체좌표계 Z축과 평행하면 Beta Angle은 전체좌표계 X축과 요소좌표계 z축이 이루는 각도가 됩니다. 각도의 부호는 요소좌표계 x축을 회전축으로 한 오른손법칙을 따릅니다. 그리고, 요소좌표계 x축이 전체좌표계 Z축과 평행하지 않으면 Beta Angle은 전체좌표계 Z축과 요소좌표계 x-z 평면이 이루는 수직각도가 됩니다. + +midas Civil에서 선요소란 트러스, 인장력전담, 압축력전담, 보, 변단면요소와 같이 선형인 요소를 통칭하며, 면요소(또는 판형요소)는 평면응력요소, 판요소, 평면변형요소, 축대칭 요소 등을 의미한다. + + + +![](images/page-022_7e88388a9d95fb73fccfee28bfe20b9338dc4867bb987c3a6bfe899406e254a2.jpg) + +
+text_image + +Y' +z +β +X' +X' +Y +Z' +Z +X +GCS +Y' +X' +Z' +y +
+ +g through node N1 and parallel with the global X-axis g through node N1 and parallel with the global Y-axis through node N1 and parallel with the global Z-axis + +(a) 수직부재인 경우 (요소좌표계 x축이 전체좌표계 Z축과 평행할 경우) +![](images/page-022_2f7d4eb903cf2dc9bbaf473441bcd6e85566ac9e4399a0d26621e2d3be0d4102.jpg) + +
+text_image + +z +β +y +z +Y' +x +X' +Z' +Y +GCS +
+ +![](images/page-022_b85c414a1f543c7a534c456716cb78be860d7d2b62bbcb27255c76c7071ec75f.jpg) + +
+text_image + +Z' +z +x +X' +y +Y' +β +
+ +(b) 수평 또는 대각부재인 경우 (요소좌표계 x축이 전체좌표계 Z축과 평행하지 않을 경우) +그림 1.3.1 Beta Angle의 개념도 + + + +# 3-1-3 요소관련 기능 + +
Create Elements요소의 입력
Material재료적 성질 입력
Section단면성질 입력
Pretension Loads프리텐션하중 입력
+ +# 3-1-4 요소내력 출력내용 + +요소내력의 출력치에 대한 부호규약은 그림 1.3.2와 같고, 화살표 방향이 양(+)의 방향을 의미합니다. + +![](images/page-023_8a05876f23d0fed75b81c84c4efcedd60b125c49c736482b5047d4c4647cf879.jpg) + +
+text_image + +※ 화살표 방향이 양(+)의 방향을 의미한다. +ECS x-axis +Axial Force +ECS z-axis +N1 +ECS y-axis +N2 +Axial Force +
+ +그림 1.3.2 트러스요소의 요소좌표계 및 요소내력(또는 응력) 출력치의 부호규약 + + + +TRUSS ELEMENT FORCES DEFAULT PR INTOUT + +Unit System：kN，m + +
ELEMMATSECLCFORCE-IFORCE-J
2016sLCB1-22.81527-22.77868
sLCB2-20.20131-20.16994
sLCB3-27.16766-27.13630
2116sLCB1-42.00574-41.93256
sLCB2-36.21717-36.15445
sLCB3-49.17972-49.11699
2216sLCB1-43.48228-43.37251
sLCB2-36.49496-36.40087
sLCB3-50.08010-49.98602
+ +TRUSSELEMENT STRESSES DEFAULTPRINTOUT + +Unit System :N，mm + +
ELEMMATSECLCSTRESS-ISTRESS-J
2016sLCB1-16.8006-16.7737
sLCB2-14.8758-14.8527
sLCB3-20.0056-19.9825
2116sLCB1-30.9321-30.8782
sLCB2-26.6695-26.6233
sLCB3-36.2148-36.1686
2216sLCB1-32.0194-31.9385
sLCB2-26.8740-26.8048
sLCB3-36.8778-36.8086
+ +그림 1.3.3 트러스요소의 요소내력 및 요소응력 출력 예 + + + +# 3-2 인장력 전담요소 (Tension-only Element) + +# 3-2-1 일반사항 + +이 요소는 2개의 절점에 의해 정의되는 “Tension-only 3D Line Element” 로서, 일반적으로 Wind Brace나 Hook Element 등을 모델링 하는데 사용되며, 요소 축방향의 인장력만 전달할 수 있습니다. + +인장력전담요소에서 입력할 수 있는 요소의 종류는 다음과 같습니다. + +# Truss + +인장력전담요소로 인장력만을 받을 수 있는 트러스요소를 정의하는데 사용됩니다. + +# Hook + +인장력전담요소로 일정한 초기간격(Hook Distance)을 가지며, 이 간격만큼 변위가 발생한 후에 요소의 강성이 발현됩니다. + +![](images/page-025_7e48e29670efde6c12578cacda61e8de4fbbed5e426a8883044c75b85f54ab13.jpg) + +
+text_image + +N1 +N2 +X +if hook distance = 0 +
+ +(a) Truss Type + +![](images/page-025_21e250cad96a512afd51f94043c6df26bbfe6aa0bde2869773b97fc974b92cbe.jpg) + +
+text_image + +N1 +N2 +X +if hook distance > 0 +
+ +(b) Hook Type +그림 1.3.4 인장력 전담요소의 형태에 따른 개념도 + + + +# 3-2-2 요소자유도 및 요소좌표계 + +“트러스요소”와 동일한 요소자유도를 가지며, 요소좌표계도 동일한 체계를 따릅니다. + +# 3-2-3 요소관련 기능 + +Main Control Data 인장력전담요소의 반복해석시 사용되는 수렴조건 입력 + +Material 재료적 성질 입력 + +Section 단면성질 입력 + +Pretension Loads 프리텐션하중 입력 + +# 3-2-4 요소내력 출력 내용 + +“트러스요소”와 동일한 부호체계를 따릅니다. + + + +# 3-3 케이블 요소(Cable Element) + +# 3-3-1 일반사항 + +이 요소는 2개의 절점에 의해 정의되는 “Tension-only 3D Line Element”로서, 요소 축방향의 인장력만 전달할 수 있으며, 부재의 장력에 따라 강성이 변화하는 케이블의 특성을 고려하는데 사용됩니다. + +케이블요소는 선형해석시 등가 트러스요소로, 기하비선형 해석시 탄성현수선 요소로 자동 전환되어 해석에 적용됩니다. + +![](images/page-027_4fce5a675ce95e8631372522df83a20ee1c79495a001ac4075633fd3bb56bf60.jpg) + +
+flowchart + +```mermaid +graph TD + N1 --> N2 + N2 --> X + style N1 fill:#f9f,stroke:#333 + style N2 fill:#f9f,stroke:#333 + style pretension fill:#ccf,stroke:#333 +``` +
+ +그림 1.3.5 케이블 요소의 개념도 + +# 3-3-2 등가 트러스요소 + +등가 트러스요소의 강성은 일반 탄성강성과 처짐(Sag)에 의한 강성으로 구성되며 장력의 변화에 따른 강성은 아래와 같은 식으로 산정합니다. + +$$ +K _ {c o m b} = \frac {1}{1 / K _ {s a g} + 1 / K _ {e l a s t i c}} +$$ + +$$ +K _ {c o m b} = \frac {E A}{L \left[ 1 + \frac {\left(w L _ {H}\right) ^ {2} E A}{1 2 T ^ {3}} \right]} +$$ + +$$ +K _ {\text { elastic }} = \frac {E A}{L}, \quad K _ {\text { sag }} = \frac {1 2 T ^ {3}}{w ^ {2} L L _ {H} ^ {2}} +$$ + +여기서 + +E:탄성계수 + +A: 단면적 + +L: 길이 + +w:단위길이당 자중 + +T:장력 + +LH:수평 길이 + + + +# 3-3-3 탄성현수선요소 (Elastic Catenary Cable Element) + +midas Civil에서 기하비선형 해석에 적용하는 케이블요소에 대한 접선강성은 다음과 같은 방법으로 계산합니다. + +그림 1.3.6과 같이 두 점을 갖는 케이블 요소에 i점에서의 변위 $\Delta_{1}$ , $\Delta_{2}$ , $\Delta_{3}$ 와 j점에서의 변위 $\Delta_{4}$ , $\Delta_{5}$ , $\Delta_{6}$ 가 발생하여 절점력이 $F_{1}^{0}, F_{2}^{0}, F_{3}^{0}, F_{4}^{0}, F_{5}^{0}, F_{6}^{0}$ 에서 $F_{1}, F_{2}, F_{3}, F_{4}, F_{5}, F_{6}$ 으로 변환되었을 때 절점력과 변위의 평형관계는 다음과 같습니다. + +$$ +F _ {4} = - F _ {1} +$$ + +$$ +F _ {5} = - F _ {2} +$$ + +$$ +F _ {6} = - F _ {3} - \omega_ {0} L _ {0} \text {(단,} \omega_ {0} = \omega \text {로 가정 가능)} +$$ + +$$ +l _ {x} = l _ {x} ^ {0} - \Delta_ {I} + \Delta_ {4} = f (F _ {1}, F _ {2}, F _ {3}) +$$ + +$$ +l _ {y} = l _ {y} ^ {0} - \Delta_ {2} + \Delta_ {5} = g (F _ {1}, F _ {2}, F _ {3}) +$$ + +$$ +l _ {z} = l _ {z} ^ {0} - \Delta_ {3} + \Delta_ {6} = h (F _ {1}, F _ {2}, F _ {3}) +$$ + +![](images/page-028_c735a9aec2feac9d6a1748cf92494d61c816f00715c7c801a3967f51b913ba0c.jpg) + +
+text_image + +F⁰₂ +F⁰₃ +i +{Δ₁, Δ₂, Δ₃} +F⁰₁ +x⁰ +y⁰ +F⁰₃ +F₂ +i′ +F₁ +x +y +F₃ +F⁰₅ +F⁰₄ +{Δ₄, Δ₅, Δ₆} +F⁰₆ +j +j′ +F₄ +F₅ +F₆ +F₆ +w₀, A₀, L₀ +w, A, L +z⁰ +z +
+ +그림 1.3.6 탄성현수선요소(케이블)의 접선강성 개념도 + + + +케이블의 전체좌표계 방향별 길이에 대한 미분식은 아래와 같고, 변위와 하중에 대한 관계를 정리하면 유연도 행렬([F])을 구할 수 있습니다. 유연도 행렬의 역행렬을 계산하여 케이블의 접선강성([K])을 구합니다. 케이블의 강성은 한번에 구해지는 것이 아니라 평형 상태에 도달할 때까지 반복적인 해석을 통하여 구할 수 있습니다. + +$$ +d l _ {x} = \frac {\partial f}{\partial F _ {1}} d F _ {1} + \frac {\partial f}{\partial F _ {2}} d F _ {2} + \frac {\partial f}{\partial F _ {3}} d F _ {3} +$$ + +$$ +d l _ {y} = \frac {\partial g}{\partial F _ {1}} d F _ {1} + \frac {\partial g}{\partial F _ {2}} d F _ {2} + \frac {\partial g}{\partial F _ {3}} d F _ {3} +$$ + +$$ +d l _ {z} = \frac {\partial h}{\partial F _ {1}} d F _ {1} + \frac {\partial h}{\partial F _ {2}} d F _ {2} + \frac {\partial h}{\partial F _ {3}} d F _ {3} +$$ + +$$ +\left\{ \begin{array}{l} d l _ {x} \\ d l _ {y} \\ d l _ {z} \end{array} \right\} = [ F ] \left\{ \begin{array}{l} d F _ {1} \\ d F _ {2} \\ d F _ {3} \end{array} \right\}, \quad \left(\left[ F \right] = \left[ \begin{array}{c c c} \frac {\partial f}{\partial F _ {1}} & \frac {\partial f}{\partial F _ {2}} & \frac {\partial f}{\partial F _ {3}} \\ \frac {\partial g}{\partial F _ {1}} & \frac {\partial g}{\partial F _ {2}} & \frac {\partial g}{\partial F _ {3}} \\ \frac {\partial h}{\partial F _ {1}} & \frac {\partial h}{\partial F _ {2}} & \frac {\partial h}{\partial F _ {3}} \end{array} \right] = \left[ \begin{array}{c c c} f _ {1 1} & f _ {1 2} & f _ {1 3} \\ f _ {2 1} & f _ {2 2} & f _ {2 3} \\ f _ {3 1} & f _ {3 2} & f _ {3 3} \end{array} \right]\right) +$$ + +$$ +\left\{ \begin{array}{l} d F _ {1} \\ d F _ {2} \\ d F _ {3} \end{array} \right\} = [ K ] \left\{ \begin{array}{l} d l _ {x} \\ d l _ {y} \\ d l _ {z} \end{array} \right\}, \quad (K = F ^ {- 1}) +$$ + +유연도 행렬의 각 원소들은 아래 식과 같이 구성됩니다. + +$$ +f _ {1 1} = \frac {\partial f}{\partial F _ {1}} = - \frac {L _ {0}}{E A _ {0}} - \frac {1}{w} \left[ \ln \left\{F _ {3} + w L _ {0} + B \right\} - \ln \left\{F _ {3} + A \right\} \right] +$$ + +$$ +- \frac {F _ {1} ^ {2}}{w} \left[ \frac {1}{B ^ {2} + \left(F _ {3} + w L _ {0}\right) B} - \frac {1}{A ^ {2} + F _ {3} A} \right] +$$ + +$$ +f _ {1 2} = \frac {\partial f}{\partial F _ {2}} = - \frac {F _ {1} F _ {2}}{w} \left[ \frac {1}{B ^ {2} + \left(F _ {3} + w L _ {0}\right) B} - \frac {1}{A ^ {2} + F _ {3} A} \right] +$$ + +$$ +f _ {1 3} = \frac {\partial f}{\partial F _ {3}} = - \frac {F _ {1}}{w} \left[ \frac {F _ {3} + w L _ {0} + B}{B ^ {2} + \left(F _ {3} + w L _ {0}\right) B} - \frac {F _ {3} + A}{A ^ {2} + F _ {3} A} \right] +$$ + + + +$$ +\begin{array}{l} f _ {2 1} = \frac {\partial g}{\partial F _ {1}} = f _ {1 2} \\ f _ {2 2} = \frac {\partial g}{\partial F _ {2}} = - \frac {L _ {0}}{E A _ {0}} - \frac {1}{w} \left[ \ln \left\{F _ {3} + w L _ {0} + B \right\} - \ln \left\{F _ {3} + A \right\} \right] \\ - \frac {F _ {2} ^ {2}}{w} \left[ \frac {1}{B ^ {2} + \left(F _ {3} + w L _ {0}\right) B} - \frac {1}{A ^ {2} + F _ {3} A} \right] \\ \end{array} +$$ + +$$ +f _ {2 3} = \frac {\partial g}{\partial F _ {3}} = \frac {F _ {2}}{F _ {1}} f _ {1 3} +$$ + +$$ +f _ {3 1} = \frac {\partial h}{\partial F _ {1}} = - \frac {F _ {1}}{w} \left[ \frac {1}{B} - \frac {1}{A} \right] +$$ + +$$ +f _ {3 2} = \frac {\partial h}{\partial F _ {2}} = \frac {F _ {2}}{F _ {1}} f _ {3 1} +$$ + +$$ +f _ {3 3} = \frac {\partial h}{\partial F _ {3}} = - \frac {L _ {0}}{E A _ {0}} - \frac {1}{w} \left[ \frac {F _ {3} + w L _ {0}}{B} - \frac {F _ {3}}{A} \right] +$$ + +$$ +A = \left(F _ {1} ^ {2} + F _ {2} ^ {2} + F _ {3} ^ {2}\right) ^ {1 / 2}, \quad B = \left(F _ {1} ^ {2} + F _ {2} ^ {2} + \left(F _ {3} + w L _ {0}\right) ^ {2}\right) ^ {1 / 2}) +$$ + +$$ +\left\{d F \right\} = K _ {T} \left\{d \Delta \right\} +$$ + +$$ +\left( \right.\text {단}, K _ {T} = \left[\begin{array}{c c c c c c}\frac {\partial F _ {1}}{\partial \Delta_ {1}}&\frac {\partial F _ {1}}{\partial \Delta_ {2}}&\frac {\partial F _ {1}}{\partial \Delta_ {3}}&\frac {\partial F _ {1}}{\partial \Delta_ {4}}&\frac {\partial F _ {1}}{\partial \Delta_ {5}}&\frac {\partial F _ {1}}{\partial \Delta_ {6}}\\\frac {\partial F _ {2}}{\partial \Delta_ {1}}&\frac {\partial F _ {2}}{\partial \Delta_ {2}}&\frac {\partial F _ {2}}{\partial \Delta_ {3}}&\frac {\partial F _ {2}}{\partial \Delta_ {4}}&\frac {\partial F _ {2}}{\partial \Delta_ {5}}&\frac {\partial F _ {2}}{\partial \Delta_ {6}}\\\frac {\partial F _ {3}}{\partial \Delta_ {1}}&\frac {\partial F _ {3}}{\partial \Delta_ {2}}&\frac {\partial F _ {3}}{\partial \Delta_ {3}}&\frac {\partial F _ {3}}{\partial \Delta_ {4}}&\frac {\partial F _ {3}}{\partial \Delta_ {5}}&\frac {\partial F _ {3}}{\partial \Delta_ {6}}\\- \frac {\partial F _ {1}}{\partial \Delta_ {1}}&- \frac {\partial F _ {1}}{\partial \Delta_ {2}}&- \frac {\partial F _ {1}}{\partial \Delta_ {3}}&- \frac {\partial F _ {1}}{\partial \Delta_ {4}}&- \frac {\partial F _ {1}}{\partial \Delta_ {5}}&- \frac {\partial F _ {1}}{\partial \Delta_ {6}}\\- \frac {\partial F _ {1}}{\partial \Delta_ {1}}&- \frac {\partial F _ {2}}{\partial \Delta_ {2}}&- \frac {\partial F _ {2}}{\partial \Delta_ {3}}&- \frac {\partial F _ {2}}{\partial \Delta_ {4}}&- \frac {\partial F _ {2}}{\partial \Delta_ {5}}&- \frac {\partial F _ {2}}{\partial \Delta_ {6}}\\- \frac {\partial F _ {1}}{\partial \Delta_ {1}}&- \frac {\partial F _ {3}}{\partial \Delta_ {2}}&- \frac {\partial F _ {3}}{\partial \Delta_ {3}}&- \frac {\partial F _ {3}}{\partial \Delta_ {4}}&- \frac {\partial F _ {3}}{\partial \Delta_ {5}}&- \frac {\partial F _ {3}}{\partial \Delta_ {6}}\end{array}\right] = \left[\begin{array}{c c}F _ {i i}&F _ {i j}\\- F _ {i i}&- F _ {i j}\end{array}\right]) +$$ diff --git a/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_004.md b/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_004.md new file mode 100644 index 00000000..ac4cab3c --- /dev/null +++ b/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_004.md @@ -0,0 +1,307 @@ + + +# 3-4 압축력 전담요소 (Compression-only Element) + +# 3-4-1 일반사항 + +이 요소는 2개의 절점에 의해 정의되는 “Compression-only 3D Line Element”로서, 접촉문제나 지반경계조건 등을 모델링 하는데 사용되며, 요소 축방향의 압축력만 전달할 수 있습니다. 압축력전담요소에서 입력할 수 있는 요소의 종류는 다음과 같습니다. + +# Truss + +압축력전담요소로 압축력만을 받을 수 있는 트러스요소를 정의하는데 사용됩니다. + +# Gap + +압축력전담요소로 일정한 초기간격(Gap Distance)을 가지며, 그 간격만큼 변위가 발생한 후에 요소의 강성이 발현 됩니다. + +# 3-4-2 요소자유도 및 요소좌표계 + +“트러스요소”와 동일한 요소자유도를 가지며, 요소좌표계도 동일한 체계를 따릅니다. + +# 3-4-3 요소관련 기능 + +Main Control Data 인장력전담요소의 반복해석시 사용되는 수렴조건 입력 + +Material 재료적 성질 입력 + +Section 단면성질 입력 + +Pretension Loads 프리텐션하중 입력 + + + +# 3-4-4 요소내력 출력내용 + +“트러스요소”와 동일한 부호체계를 따릅니다. + +![](images/page-032_a8ee8c223860ca09f573aa76835448d38d48ff1644b2735ec0b22c55141896f7.jpg) + +
+text_image + +N1 +N2 +X +if gap distance = 0 +
+ +(a) Truss Type + +![](images/page-032_0ded6619645f01c9fc4fd949a8ab2629c3b0153d068d7409ddee58220ab70a7e.jpg) + +
+text_image + +N1 +N2 +X +if gap distance > 0 +
+ +(b)Gap Type +그림 1.3.7 압축력 전담요소의 형태에 따른 개념도 + + + +# 3-5 보요소 (Beam Element) + +# 3-5-1 일반사항 + +이 요소는 2개의 절점에 의해 정의되는 보요소(Prismatic/Non-prismatic 3D Beam Element)로서, 인장 및 압축, 전단, 굽힘, 비틀림 등의 거동에 대한 강성을 갖도록 정식화(Timoshenko Beam Theory) 되어 있습니다. + +보요소의 단면이 전체길이에 걸쳐 균일한 경우(Prismatic Beam Element)에는 한개의 단면을 Section 대화상자에서 입력하고, 비균일단면(Non-prismatic Beam Element)을 가진 경우에는 양단의 단면 두개를 각각 입력하게 됩니다. + +midas Civil에서 비균일단면을 가진 보요소의 단면성질 중 단면적과 유효전단면적, 비틀림강성에 대해서는 요소좌표계 x축을 따라 한쪽 단부에서 반대쪽 단부까지 1차적으로 변화(Linear Variation)하는 것으로 가정하고, 강축 또는 약축방향에 대한 단면2차모멘트에 대해서는 사용자의 선택에 따라 1차, 2차 또는 3차(Linear, Parabolic or Cubic Variation) 함수 형태의 변화를 고려할 수 있습니다. + +# 3-5-2 요소자유도 및 요소좌표계 + +요소자유도는 요소좌표계 또는 전체좌표계에 관계없이 절점당 세 가지의 이동변위 (Translation) 성분과 세 가지의 회전변위(Rotation) 성분을 가지게 됩니다. + +요소좌표계는 트러스요소와 동일한 체계를 따릅니다. + + + +3-5-3 요소관련 기능 + +
Create Elements요소 입력
Material재료적 성질 입력
Section단면성질 입력
Beam End Release양 절점의 접속상태 (단부해제, 강접 또는 힌지접합 등) 지정
Beam End Offsets양단의 강체이격거리(rigid end offset distance)입력
Element Beam Loads보하중의 입력 (보요소의 중간에 작용하는 집중 또는 분포하중)
Line Beam Loads재하범위를 지정하여 보하중을 입력
Assign Floor Loads바닥판하중을 보하중 형태로 치환하여 입력
Prestress Beam Loads프리스트레스하중 입력
Temperature Gradient온도구배 입력
Beam Section Temperature비선형 온도 분포 하중 입력
Tendon Prestress Loads텐던을 사용한 프리스트레스 하중 입력
+ + + +# 3-5-4 요소내력 출력내용 + +요소내력의 출력치에 대한 부호규약은 그림 1.3.8과 같고, 화살표 방향이 양(+)의 방향을 의미합니다. + +부재응력의 부호규약은 요소내력과 동일합니다. + +단, 휨모멘트에 의한 응력의 경우는 인장일 때 ‘+’ 그리고 압축일 때 ‘-’부호를가집니다. + +※ 내력 출력치의 부호는 화살표 방향이 양(+)의 부호를 가진다. + +![](images/page-035_c35d20dde99c1208a95764cdca9b21fc1fb005e546cf3a1d1b27cdc33dbd79ab.jpg) + +
+text_image + +ECS x-axis +Axial Force +Torque +Momenty +Sheary +Shearz +Momentz +N2 +3/4pt +1/2pt +1/4pt +N1 +ECS y-axis +Sheary +Momenty +Torque +Axial Force +Momentz +Shearz +
+ +그림 1.3.8 보요소의 요소좌표계 및 요소내력(또는 응력) 출력치의 부호규약 + + + +
BEAM ELEMENT FORCES & MOMENTS DEFAULT PRINTOUT Unit System : kN , m
ELEMMATSECLCPTAXIALSHEAR-ySHEAR-zTORSIONMOMENT-yMOMENT-z
111sLCB11-279.924284.87200-12.409710.00000.00000.0000
1/4-279.244334.87200-12.409710.00009.3073-3.6540
CNT-278.564384.87200-12.409710.000018.6146-7.3080
3/4-277.884434.87200-12.409710.000027.9219-10.9620
J-277.204474.87200-12.409710.000037.2291-14.6160
sLCB21-504.104889.77588-22.485810.00000.00000.0000
1/4-503.522079.77588-22.485810.000016.8644-7.3319
CNT-502.939259.77588-22.485810.000033.7287-14.6638
3/4-502.356439.77588-22.485810.000050.5931-21.9957
J-501.773629.77588-22.485810.000067.4574-29.3276
sLCB31-515.264899.78124-23.077980.00000.00000.0000
1/4-514.682089.78124-23.077980.000017.3085-7.3359
CNT-514.099269.78124-23.077980.000034.6170-14.6719
3/4-513.516449.78124-23.077980.000051.9254-22.0078
J-512.933639.78124-23.077980.000069.2339-29.3437
+ +
BEAM ELEMENT STRESSES DEFAULT PRINTOUT Unit System : N , mm
ELEMMATSECLCPTAXIALSHEAR-ySHEAR-z(+y)-BENDING-(-y)(+z)-BENDING-(-z)
111sLCB1I-33.27681.4000-4.20840.00000.00000.00000.0000
1/4-33.19591.4000-4.208421.0000-21.0000-7.85427.8542
CNT-33.11511.4000-4.208442.0000-42.0000-15.708515.7085
3/4-33.03431.4000-4.208463.0000-63.0000-23.562723.5627
J-32.95351.4000-4.208484.0000-84.0000-31.417031.4170
sLCB2I-59.92692.8092-7.62540.00000.00000.00000.0000
1/4-59.85762.8092-7.625442.1374-42.1374-14.231514.2315
CNT-59.78832.8092-7.625484.2748-84.2748-28.463028.4630
3/4-59.71902.8092-7.6254126.4122-126.4122-42.694642.6946
J-59.64972.8092-7.6254168.5496-168.5496-56.926156.9261
sLCB3I-61.25362.8107-7.82630.00000.00000.00000.0000
1/4-61.18432.8107-7.826342.1605-42.1605-14.606314.6063
CNT-61.11502.8107-7.826384.3210-84.3210-29.212629.2126
3/4-61.04572.8107-7.8263126.4816-126.4816-43.818943.8189
J-60.97642.8107-7.8263168.6421-168.6421-58.425358.4253
+ +그림 1.3.9 보요소의 요소내력 및 요소응력 출력 예 + + + +# 3-6 평면응력요소 (Plane Stress Element) + +# 3-6-1 일반사항 + +이 요소는 동일평면상에 위치한 3개 또는 4개의 절점에 의해 정의되는 평면응력요소 (3D Plane Stress Element)로서, 평면방향으로만 하중을 받을 수 있고 두께가 요소면의 전체에 걸쳐 균일한 박판(Membrane)의 모델링에 사용됩니다. + +midas Civil에서 이 요소는 비적합모드를 가진 등매개 평면응력이론(Isoparametric Plane Stress Formulation with Incompatible Modes)을 사용하여 정식화 되었습니다. + +따라서 이 요소의 두께방향 응력성분은 존재하지 않으며 두께방향의 변형율은 Poisson Effects에 의해 존재하는 것으로 가정합니다. + +# 3-6-2 요소자유도 및 요소좌표계 + +요소자유도는 요소좌표계를 기준으로 x, y방향의 변위자유도만을 가지게 됩니다. + +요소좌표계는 오른손법칙에 준한 x, y, z축의 직교좌표계를 따르며, 요소좌표계의 방향은 그림 1.3.10과 같이 설정됩니다. + +사각형요소의 경우는 연결절점의 입력순서대로 오른손법칙에 따라 회전할 때(N1→N2→N3→N4) 요소중심에서 요소면의 수직방향으로 엄지손가락 방향이 요소좌표계 z 축이 됩니다. 그리고 요소좌표계 x축 방향은 N1과 N4를 잇는 선분의 중심에서 N2와 N3을 잇는 선분의 중심까지 직선으로 연결할 때 그 직선의 진행방향이 되며, 요소평면상에서 오른손 좌표계를 기준으로 x축과 수직을 이루는 축이 요소좌표계 y축이 됩니다. + +삼각형요소의 경우는 면의 중심점에서 N1부터 N2로 진행하는 방향이 요소좌표계의 x방향이 되고, 나머지 y, z축 방향은 사각형요소의 경우와 동일합니다. + + + +![](images/page-038_b2bc54418bb878e4838704f3bda76895ef8113aec8728a07b106493ee8f99c15.jpg) + +
+text_image + +ECS z-axis (normal to the element surface) +Node numbering order for creating the element (N1→N2→N3→N4) +ECS y-axis (perpendicular to ECS x-axis in the element plane) +Center of Element +N1 +N2 +N3 +N4 +ECS x-axis +(N1 to N2 direction) +
+ +(a) 사각형요소의 요소좌표계 + +![](images/page-038_8e1cea396a0724f919ac55eeb47499529be69bae47c06e6580ba2f6f1cec367d.jpg) + +
+text_image + +ECS z-axis (normal to the element surface) +Node numbering order for creating the element (N1→N2→N3) +N3 +ECS y-axis (perpendicular to ECS x-axis in the element plane) +N1 +Center of Element +ECS x-axis +(N1 to N2 direction) +N2 +
+ +(b) 삼각형요소의 요소좌표계 +그림 1.3.10 평면응력요소의 배치 및 요소좌표계 + + + +3-6-3 요소관련 기능 + +
Create Elements요소 입력
Material재료적 성질 입력
Thickness요소두께의 입력
Pressure Loads요소의 변에 수직방향으로 압력하중 입력평면응력요소의 압력하중은 그림 1.3.11과 같이 각 변에 수직방향으로 입력됩니다.
+ +![](images/page-039_37d3cb9d141600a144d408bdc9d9469e6a4903834b81e84aceaa20f47cf28041.jpg) + +
+flowchart + +```mermaid +graph TD + P1 --> N4 + N4 --> P1 + N4 --> P2 + P2 --> P1 + P1 --> N4 + N4 --> P2 + P2 --> P1 + P1 --> N4 + N4 --> P2 + P2 --> P1 + P1 --> N4 + N4 --> P2 + P2 --> P1 + P1 --> N4 + N4 --> P2 + P2 --> P1 + P1 --> N4 + N4 --> P2 + P2 --> P1 + + subgraph Edge No.1 + N1 --> Edge No.1 + N2 --> Edge No.1 + N3 --> Edge No.2 + N4 --> Edge No.3 + N5 --> Edge No.3 + N6 --> Edge No.4 + end + + subgraph Edge No.2 + N1 --> Edge No.1 + N2 --> Edge No.1 + N3 --> Edge No.2 + N4 --> Edge No.3 + end + + subgraph Edge No.3 + N1 --> Edge No.1 + N2 --> Edge No.1 + N3 --> Edge No.2 + end + + subgraph Edge No.4 + N1 --> Edge No.1 + N2 --> Edge No.1 + N3 --> Edge No.2 + end + + style Edge No.1 fill:#f9f,stroke:#333 + style Edge No.2 fill:#f9f,stroke:#333 + style Edge No.3 fill:#f9f,stroke:#333 + style Edge No.4 fill:#f9f,stroke:#333 +``` +
+ +그림 1.3.11 평면응력요소의 압력하중 + + + +# 3-6-4 적분점 + +# 3절점 삼각형 요소 + +이 요소는 1 Point Gauss 적분을 이용하므로, 적분에 적용되는 자연좌표계에서 적분점 좌표는 (1/3, 1/3) 입니다. + +![](images/page-040_8b9f7566b6e4166ec090d218b41f8083d74ec4ff925cb5e400bb569d635985e9.jpg) + +
+text_image + +N3 +η +P = (1/3, 1/3) +N1 +N2 +ξ +y +x +
+ +그림 1.3.12 3절점 평면응력요소의 적분점위치 + +이 요소의 기하학적 형상함수는 $N_{1}=1-\xi-\eta$ , $N_{2}=\xi$ , $N_{3}=\eta$ 이므로 요소내 특정 위치에서의 좌표값은 형상함수를 이용하여 다음 식과 같이 구할 수 있습니다. + +$$ +x _ {p} = \sum_ {i = 1} ^ {N} N _ {i} x _ {i}, y _ {p} = \sum_ {i} ^ {N} N _ {i} y _ {i} +$$ + +이 요소의 적분점 좌표인 $\xi=1/3$ , $\eta=1/3$ 을 형상함수에 대입하면 전체좌표계에서 적분점의 좌표를 구할 수 있습니다. + +$$ +x _ {p} = \sum_ {i = 1} ^ {3} N _ {i} x _ {i} = \left(1 - \frac {1}{3} - \frac {1}{3}\right) x _ {1} + \frac {1}{3} x _ {2} + \frac {1}{3} x _ {3} = \frac {1}{3} \left(x _ {1} + x _ {2} + x _ {3}\right) +$$ + +$$ +y _ {p} = \frac {1}{3} \left(y _ {1} + y _ {2} + y _ {3}\right) +$$ diff --git a/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_005.md b/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_005.md new file mode 100644 index 00000000..ed720106 --- /dev/null +++ b/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_005.md @@ -0,0 +1,407 @@ + + +# 4절점 사각형 요소 + +이 요소는 4 Point Gauss 적분을 이용하므로, 적분에 적용되는 자연좌표계에서 적분점 좌표 $P_{i}$ 는 다음 그림과 같습니다. + +![](images/page-041_a61117c569826167a7a9129a01ca2d3646e4de0d8bb10c0d675c166fd4e815ab.jpg) + +
+text_image + +N4 +η +N3 +P4 +P3 +η = 1/√3 +ξ +P1 +P2 +η = -1/√3 +N1 +ξ = -1/√3 +ξ = 1/√3 +N2 +y +x +P1 = (-1/√3, -1/√3) +P2 = (1/√3, -1/√3) +P3 = (1/√3, 1/√3) +P4 = (-1/√3, 1/√3) +
+ +그림 1.3.13 4절점 평면응력요소의 적분점위치 + +이 요소의 기하학적 형상함수는 다음 식과 같습니다. + +$$ +N _ {1} = \frac {1}{4} (1 - \xi) (1 - \eta), N _ {2} = \frac {1}{4} (1 + \xi) (1 - \eta), N _ {3} = \frac {1}{4} (1 + \xi) (1 + \eta), N _ {4} = \frac {1}{4} (1 - \xi) (1 + \eta) +$$ + +이 요소의 적분점 좌표인 $P_{i}$ 를 형상함수에 대입하면 전체좌표계에서 적분점의 좌표를 구할 수 있습니다. 예를 들어 첫 번째 적분점 좌표 $P_{1}$ 에 대한 전체좌표계에서 x 좌표를 구하면 다음과 같습니다. + +$$ +x _ {p 1} = \sum_ {i = 1} ^ {4} N _ {i} x _ {i} = \frac {1}{6} \left[ (2 + \sqrt {3}) x _ {1} + x _ {2} + (2 - \sqrt {3}) x _ {3} + x _ {4} \right] +$$ + +같은 방법으로 각 적분점에 대해 전체좌표계에서 좌표를 구하면 다음과 같습니다. + +$$ +x _ {p} = \frac {1}{6} \left[ \begin{array}{c c c c} 2 + \sqrt {3} & 1 & 2 - \sqrt {3} & 1 \\ & 2 + \sqrt {3} & 1 & 2 - \sqrt {3} \\ & & 2 + \sqrt {3} & 1 \\ \text {symmetry} & & & 2 + \sqrt {3} \end{array} \right] \left\{ \begin{array}{c} x _ {1} \\ x _ {2} \\ x _ {3} \\ x _ {4} \end{array} \right\} +$$ + + + +$$ +y _ {p} = \frac {1}{6} \left[ \begin{array}{c c c c} 2 + \sqrt {3} & 1 & 2 - \sqrt {3} & 1 \\ & 2 + \sqrt {3} & 1 & 2 - \sqrt {3} \\ & & 2 + \sqrt {3} & 1 \\ \text {symmetry} & & & 2 + \sqrt {3} \end{array} \right] \left\{ \begin{array}{c} y _ {1} \\ y _ {2} \\ y _ {3} \\ y _ {4} \end{array} \right\} +$$ + +# 3-6-5 응력계산법(Extrapolation) + +3절점 삼각형 요소의 경우 1 Point Gauss 적분을 하므로 모든 절점에 대해 적분점에서 계산된 응력을 동일하게 적용합니다. + +![](images/page-042_6bbeb739eb3670ec438016c941677e44692009a9f295cfc5b414d909c797565e.jpg) + +
+text_image + +ξ = -1 +η = 1 +ξ = 1 +η = 1 +t = 1 +η = -1 +ξ = -1 +s = -1 +s = 1 +ξ = 1 +η +t = -1 +η = -1 +ξ = 1 +
+ +그림 1.3.14 4절점 평면응력요소에 대한 적분점에서 응력에 대한 외삽법 + +4절점 사각형 요소의 경우 각 적분점은 요소좌표계의 좌표절점과 다음과 같은 관계를 갖습니다. + +$$ +s = \xi \sqrt {3}, t = \eta \sqrt {3} +$$ + +요소 내부의 특정 위치에서 응력은 형상함수를 이용하여 구할 수 있습니다. + +$$ +\sigma_ {N} = \sum N _ {i} \sigma_ {i} \quad i = 1, 2, 3, 4 +$$ + + + +예를 들어 절점 1에서 응력을 계산하면 다음과 같습니다. + +$$ +\begin{array}{l} \sigma_ {N 1} = \sum_ {i = 1} ^ {4} N _ {i} \sigma_ {i} = \frac {1}{4} \left[ (1 + \sqrt {3}) (1 + \sqrt {3}) \sigma_ {1} + (1 - \sqrt {3}) (1 + \sqrt {3}) \sigma_ {2} \right. \\ \left. + (1 - \sqrt {3}) (1 - \sqrt {3}) \sigma_ {3} + (1 + \sqrt {3}) (1 - \sqrt {3}) \sigma_ {4} \right] \\ = \frac {1}{4} \left[ (4 + 2 \sqrt {3}) \sigma_ {1} - 2 \sigma_ {2} + (4 - 2 \sqrt {3}) \sigma_ {3} - 2 \sigma_ {4} \right] \\ \end{array} +$$ + +같은 방법으로 각 절점에서 응력을 구하면 다음과 같습니다. + +$$ +\left\{ \begin{array}{l} \sigma_ {N 1} \\ \sigma_ {N 2} \\ \sigma_ {N 3} \\ \sigma_ {N 4} \end{array} \right\} = \frac {1}{2} \left[ \begin{array}{c c c c} 2 + \sqrt {3} & - 1 & 2 - \sqrt {3} & - 1 \\ & 2 + \sqrt {3} & - 1 & 2 - \sqrt {3} \\ & & 2 + \sqrt {3} & - 1 \\ & & & 2 + \sqrt {3} \end{array} \right] \left\{ \begin{array}{l} \sigma_ {1} \\ \sigma_ {2} \\ \sigma_ {3} \\ \sigma_ {4} \end{array} \right\} +$$ + +# 3-6-6 요소내력 출력내용 + +평면응력요소의 요소내력 및 응력은 다음과 같이 출력되며 부호와 방향은 요소좌표계 또는 전체좌표계를 따릅니다. 여기서는 요소좌표계를 기준으로 설명합니다. + +■ 연결절점에서의 요소내력 출력 +■연결절점과 요소중심에서 요소응력 출력 + +연결절점에서의 요소내력은 절점에서 산출된 각 성분별 변위와 해당요소 강성성분을 곱한 값으로 출력됩니다. 연결절점과 요소중심에서의 응력은 요소내의 적분점(Gauss Point)에서 연산된 응력을 이용하여 외삽법(Extrapolation)에 의해 산출됩니다. + +▪ 요소내력의 출력 + +요소내력의 출력치에 대한 부호규약은 그림 1.3.15와 같고, 화살표방향이 양 (+)의 방향을 의미합니다. + + + +#  요소응력의 출력 + +요소응력의 출력치에 대한 부호규약은 그림 1.3.16과 같고, 화살표방향이 양(+)의 방향을 의미합니다. + +![](images/page-044_736eff59cc4118578cbf5fd3bb3f6115e273ae7af4ede4fcf08536c6375272ea.jpg) + +
+text_image + +Fx4 +Fy4 +N4 +Fx1 +N1 +Center of +Element +z +y +x +N3 +Fy3 +Fx3 +N2 +Fx2 +Fy2 +
+ +(a) 사각형요소의 절점내력 + +![](images/page-044_73c197e349ce94121cb4cce185d80b5fe54c383c3495e0bc17d7b2be68480173.jpg) + +
+text_image + +F_{x1} +F_{y1} +N1 +Center of Element +F_{x2} +F_{y2} +N2 +F_{y2} +N3 +F_{y3} +F_{x3} +F_{y3} +(b) 상각현 요소의 적절내력 +
+ +(b) 삼각형요소의 절점내력 +그림 1.3.15 평면응력요소의 연결절점에서의 내력출력치 부호규약 + + + +※ 요소응력의 출력은 요소좌표계를 따르며 화살표 방향이 양(+)의 방향을 의미한다. + +![](images/page-045_30a17ccecf0303f57917e5e25c16f13b931afde9a216c418123d917f2d918947.jpg) + +
+text_image + +σy +τxy +y +x +τxy +σx +τxy +τxy +σx +σy +
+ +(a) 축응력 및 전단응력 성분 + +![](images/page-045_a06cfa2fae9bf4ea92460c53116c09fa1952a574d5d613281f748a5120cd9f12.jpg) + +
+text_image + +σ₂ +y +1 +2 +θ +x +σ₁ +σ₁ +σ₂ +
+ +(b) 주응력 성분 + +$\sigma _ { x }$ : Axial stress in the ECS x - direction + +$\sigma _ { x }$ : Axial stress in the ECS y - direction + +$\tau _ { x y }$ : Shear stress in the ECS x - y plane + +$\sigma _ { I }$ : Maximum principal stress $= \displaystyle \frac { \sigma _ { x } + \sigma _ { y } } { 2 } + \sqrt { \left( \displaystyle \frac { \sigma _ { x } - \sigma _ { y } } { 2 } \right) ^ { 2 } + \tau _ { x y } ^ { 2 } }$ + +$\sigma _ { 2 }$ : Minimum principal stress $= \displaystyle \frac { \sigma _ { x } + \sigma _ { y } } { 2 } - \sqrt { \left( \displaystyle \frac { \sigma _ { x } - \sigma _ { y } } { 2 } \right) ^ { 2 } + \tau _ { x y } ^ { 2 } }$ 2xy+τ + +$\tau _ { x y }$ : Maximum shear stress $= \sqrt { \left( \frac { \sigma _ { x } - \sigma _ { y } } { 2 } \right) ^ { 2 } + \tau _ { x y } ^ { 2 } }$ 2 xy+τ + +:θ Angle between the x - axis and the principal axis,1 + +$\sigma _ { e f f }$ : von - Mises Stress $= \sqrt { ( { \sigma } _ { I } ^ { 2 } - { \sigma } _ { I } { \sigma } _ { 2 } + { \sigma } _ { 2 } ^ { 2 } ) }$ + +그림 1.3.16 평면응력요소의 응력출력위치 및 출력치의 부호규약 + + + +# 3-7 평면변형요소 (2D Plane Strain Element) + +# 3-7-1 일반사항 + +이 요소는 댐(Dam) 또는 터널(Tunnel) 등과 같이 일정한 단면을 유지하면서 길이가 긴 구조물의 해석에 사용될 수 있으며, 등매개 평면변형이론(Isoparametric Plane Strain Formulation with Incompatible Modes)을 근거로 개발되었습니다. + +이 요소는 다른 종류의 요소와 훈용할 수 없으며 요소의 특성상 선형정적해석에만적용 가능합니다. + +midas Civil에서는 요소가 X-Z 평면상에 위치하도록 입력되며 요소의 두께는 그림 1.3.17과 같이 1.0(단위 폭)으로 자동 고려됩니다. + +이 요소는 평면변형적 특성을 근거로 하기 때문에 두께방향 변형율성분은 존재하지 않으며, 두께방향의 응력성분은 Poisson Effects에 의해 존재하는 것으로 가정합니다. + +![](images/page-046_cf4a14e7aabdfaee985c5fa2763909fa29bf489c90968c7abdc67268ffa1683c.jpg) + +
+text_image + +Z +Y +1.0(Unit thickness) +X +Plane strain elements +
+ +그림 1.3.17 2차원 평면변형요소의 두께 + + + +# 3-7-2 요소자유도 및 요소좌표계 + +midas Civil에서 평면변형요소의 요소좌표계는 프로그램 내부에서 요소강성행렬을계산하거나, 후처리 모드(Post-processing Mode)에서 사용자가 요소좌표계를 기준으로 응력성분을 도화처리할 때 사용됩니다. + +요소자유도는 전체좌표계를 기준으로 X, Z방향의 변위자유도만을 가지게 됩니다. + +요소좌표계는 오른손법칙에 준한 x, y, z축의 직교좌표계를 따르며, 요소좌표계의 방향은 그림 1.3.18과 같이 설정됩니다. + +사각형요소의 경우는 연결절점의 입력순서대로 오른손법칙에 따라 회전할 때(N1→N2→N3→N4) 요소중심에서 요소면의 수직방향으로 엄지손가락 방향이 요소좌표계 z축이 됩니다. 그리고 요소좌표계 x축 방향은 N1과 N4를 잇는 선분의 중심에서 N2와N3을 잇는 선분의 중심까지 직선으로 연결할 때 그 직선의 진행방향이 되며, 요소평면상에서 오른손 좌표계를 기준으로 x축과 수직을 이루는 축이 요소좌표계 y축이 됩니다. + +삼각형요소의 경우는 면의 중심점에서 N1부터 N2로 진행하는 방향이 요소좌표계의x방향이 되고, 나머지 y, z축 방향은 사각형요소의 경우와 동일합니다. + + + +![](images/page-048_6bdf5048281a6eff23fa9ff9f0cfcb22a3d2b146e48e5d608734273440cf1ba3.jpg) + +
+text_image + +ESC y-axis (perpendicular to ESC x-axis in the element plane) +F +Fx3 +N3 +Node numbering order for creating the element +(N1→N2→N3→N4) +ECS x-axis +(N1 to N2 direction) +Center of Element +ECS z-axis (normal to the element surface, out of the paper) +Fx1 +N1 +Fz1 +N2 +Fx2 +Fz2 +Z +GCS +X +
+ +(a) 사각형요소 + +![](images/page-048_9139140b1fa77709f27ed5dfa924bc88005dd2cddea9473a402430896cccd21d.jpg) + +
+text_image + +ECS y-axis (perpendicular ECS x- +axis in the element plane +Node numbering order +for creating the element +(N1→N2→N3) +ECS z-axis (normal to the element +surface, out of the paper) +Center of Element +ECS x-axis +(N1 to N2 +direction) +Z +X +
+ +(b) 삼각형요소 +그림 1.3.18 평면변형요소의 배치 및 요소좌표계, 절점내력 + + + +# 3-7-3 요소관련 기능 + +
Create Elements요소 입력
Material재료적 성질 입력
Pressure Loads요소의 변에 수직방향으로 압력하중 입력
+ +평면변형요소의 압력하중은 그림 1.3.19와 같이 각 변에 수직방향으로 입력되며, 압력하중의 작용면적은 그림 1.3.17과 같이 단위폭(1.0)만큼 자동 고려됩니다. + +![](images/page-049_ed8a9df00858c8baba840ac944a0b8a0c98bc9168de7df748e6c57eaf3526953.jpg) + +
+text_image + +P1 +P2 +N4 +edge number 3 +N3 +P2 +edge number 4 +edge number 2 +N1 +N2 +P1 +P2 +Z +GCS +X +P1 +P2 +
+ +그림 1.3.19 평면변형요소의 압력하중 + + + +# 3-7-4 적분점 + +# 3절점 삼각형 요소 + +이 요소는 1 Point Gauss 적분을 이용하므로, 적분에 적용되는 자연좌표계에서 적분점 좌표는 (1/3, 1/3) 입니다. + +![](images/page-050_83bcd0a03bd1dd4095165951d7a890764d963be836702813b7c356d6d680ff7e.jpg) + +
+text_image + +N3 +η +P = (1/3, 1/3) +N1 +ξ +N2 +z +x +
+ +그림 1.3.20 3절점 평면변형률요소의 적분점위치 + +이 요소의 기하학적 형상함수는 $N_{1}=1-\xi-\eta$ , $N_{2}=\xi$ , $N_{3}=\eta$ 이므로 요소내 특정 위치에서의 좌표값은 형상함수를 이용하여 다음 식과 같이 구할 수 있습니다. + +$$ +x _ {p} = \sum_ {i = 1} ^ {N} N _ {i} x _ {i}, z _ {p} = \sum_ {i} ^ {N} N _ {i} z _ {i} +$$ + +이 요소의 적분점 좌표인 $\xi=1/3$ , $\eta=1/3$ 을 형상함수에 대입하면 전체좌표계에서 적분점의 좌표를 구할 수 있습니다. + +$$ +x _ {p} = \sum_ {i = 1} ^ {3} N _ {i} x _ {i} = \left(1 - \frac {1}{3} - \frac {1}{3}\right) x _ {1} + \frac {1}{3} x _ {2} + \frac {1}{3} x _ {3} = \frac {1}{3} \left(x _ {1} + x _ {2} + x _ {3}\right) +$$ + +$$ +z _ {p} = \frac {1}{3} \left(z _ {1} + z _ {2} + z _ {3}\right) +$$ diff --git a/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_006.md b/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_006.md new file mode 100644 index 00000000..33e95a93 --- /dev/null +++ b/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_006.md @@ -0,0 +1,397 @@ + + +# 4절점 사각형 요소 + +이 요소는 4 Point Gauss 적분을 이용하며, 적분에 적용되는 자연좌표계에서 적분점 좌표 $P_{i}$ 는 다음 그림과 같습니다. + +![](images/page-051_efd051425f58cc453461716bb0c49f9aa32eeb8a3e62d709a19615bfc6aa892f.jpg) + +
+text_image + +N4 +η +N3 +η = 1/√3 +P4 +P3 +ξ +P1 +P2 +N1 +ξ = -1/√3 +ξ = 1/√3 +N2 +η = -1/√3 +Z +x +
+ +$$ +P _ {1} = \left(- \frac {1}{\sqrt {3}}, - \frac {1}{\sqrt {3}}\right) +$$ + +$$ +P _ {2} = \left(\frac {1}{\sqrt {3}}, - \frac {1}{\sqrt {3}}\right) +$$ + +$$ +P _ {3} = \left(\frac {1}{\sqrt {3}}, \frac {1}{\sqrt {3}}\right) +$$ + +$$ +P _ {4} = \left(- \frac {1}{\sqrt {3}}, \frac {1}{\sqrt {3}}\right) +$$ + +그림 1.3.21 4절점 평면변형률요소의 적분점위치 + +이 요소의 기하학적 형상함수는 다음 식과 같습니다. + +$$ +N _ {1} = \frac {1}{4} (1 - \xi) (1 - \eta), N _ {2} = \frac {1}{4} (1 + \xi) (1 - \eta), N _ {3} = \frac {1}{4} (1 + \xi) (1 + \eta), N _ {4} = \frac {1}{4} (1 - \xi) (1 + \eta) +$$ + +이 요소의 적분점 좌표인 $P_{i}$ 를 형상함수에 대입하면 전체좌표계에서 적분점의 좌표를 구할 수 있습니다. 예를 들어 첫 번째 적분점 좌표 $P_{1}$ 에 대한 전체좌표계에서 x 좌표를 구하면 다음과 같습니다. + +$$ +x _ {p 1} = \sum_ {i = 1} ^ {4} N _ {i} x _ {i} = \frac {1}{6} \left[ (2 + \sqrt {3}) x _ {1} + x _ {2} + (2 - \sqrt {3}) x _ {3} + x _ {4} \right] +$$ + +같은 방법으로 각 적분점에 대한 전체좌표계에서 좌표를 구하면 다음과 같습니다. + +$$ +x _ {p} = \frac {1}{6} \left[ \begin{array}{c c c c} 2 + \sqrt {3} & 1 & 2 - \sqrt {3} & 1 \\ & 2 + \sqrt {3} & 1 & 2 - \sqrt {3} \\ & & 2 + \sqrt {3} & 1 \\ \text { symmetry } & & & 2 + \sqrt {3} \end{array} \right] \left\{ \begin{array}{l} x _ {1} \\ x _ {2} \\ x _ {3} \\ x _ {4} \end{array} \right\} +$$ + + + +# 3-7-5 응력계산법(Extrapolation) + +3절점 삼각형 요소의 경우 1 Point Gauss 적분을 하므로 모든 절점에 대해 적분점에서 계산된 응력을 동일하게 적용합니다. + +![](images/page-052_28527dad12f58192e1d7172232a6f64fbbcb88250ad844821abda428a24432ea.jpg) + +
+text_image + +ξ = -1 +η = 1 +ξ = 1 +η = 1 +t = 1 +η = -1 +ξ = -1 +s = -1 +s = 1 +ξ = 1 +η +ξ = -1 +η +ξ = 1 +t = -1 +η = -1 +ξ = 1 +
+ +그림 1.3.22 4절점 평면응력요소에 대한 적분점에서 응력에 대한 외삽법 + +4절점 사각형 요소의 경우 각 적분점은 요소좌표계의 좌표절점과 다음과 같은 관계를 갖습니다. + +$$ +s = \xi \sqrt {3}, t = \eta \sqrt {3} +$$ + +요소 내부의 특정 위치에서 응력은 형상함수를 이용하여 구할 수 있습니다. + +$$ +\sigma_ {N} = \sum_ {i = 1} ^ {4} N _ {i} \sigma_ {i} +$$ + +예를 들어 절점 1에서 응력에 대해 형상함수에 $\xi,\eta$ 대신 앞의 s, t 를 대입하여 정리하면 다음과 같습니다. + +$$ +\begin{array}{l} \sigma_ {N 1} = \sum_ {i = 1} ^ {4} N _ {i} \sigma_ {i} = \frac {1}{4} \left[ (1 + \sqrt {3}) (1 + \sqrt {3}) \sigma_ {1} + (1 - \sqrt {3}) (1 + \sqrt {3}) \sigma_ {2} \right. \\ \left. + (1 - \sqrt {3}) (1 - \sqrt {3}) \sigma_ {3} + (1 + \sqrt {3}) (1 - \sqrt {3}) \sigma_ {4} \right] \\ = \frac {1}{4} \left[ (4 + 2 \sqrt {3}) \sigma_ {1} - 2 \sigma_ {2} + (4 - 2 \sqrt {3}) \sigma_ {3} - 2 \sigma_ {4} \right] \\ \end{array} +$$ + + + +같은 방법으로 각 절점에서 응력을 구하면 다음과 같습니다. + +$$ +\left\{ \begin{array}{l} \sigma_ {N 1} \\ \sigma_ {N 2} \\ \sigma_ {N 3} \\ \sigma_ {N 4} \end{array} \right\} = \frac {1}{2} \left[ \begin{array}{c c c c} 2 + \sqrt {3} & - 1 & 2 - \sqrt {3} & - 1 \\ & 2 + \sqrt {3} & - 1 & 2 - \sqrt {3} \\ & & 2 + \sqrt {3} & - 1 \\ & & & 2 + \sqrt {3} \end{array} \right] \left\{ \begin{array}{l} \sigma_ {1} \\ \sigma_ {2} \\ \sigma_ {3} \\ \sigma_ {4} \end{array} \right\} +$$ + +# 3-7-6 요소내력 출력내용 + +평면변형요소의 요소내력 및 응력은 다음과 같이 출력되며 부호와 방향은 요소좌표계 또는 전체좌표계를 따릅니다. 그림 1.3.23은 요소좌표계의 축방향 또는 주응력방향의 단위 segment에서 발생되는 응력의 부호규약을 설명한 것입니다. + +■ 연결절점에서의 요소내력 출력 +■연결절점과 요소중심에서 요소응력 출력 + +연결절점에서의 요소내력은 절점에서 산출된 각 성분별 변위와 해당요소 강성성분을 곱한 값으로 출력됩니다. + +연결절점과 요소중심에서의 응력은 요소내의 적분점(Gauss Point)에서 연산된 응력을 이용하여 외삽법(Extrapolation)에 의해 산출됩니다. + +▪ 요소내력의 출력 + +요소내력의 출력치에 대한 부호규약은 그림 1.3.18과 같고, 화살표방향이 양 (+)의 방향을 의미합니다. + +■ 요소응력의 출력 + +요소응력의 출력치에 대한 부호규약은 그림 1.3.23과 같고, 화살표방향이 양 (+)의 방향을 의미합니다. + + + +※ 요소응력의 출력은 요소좌표계를 따르며 화살표 방향이 양(+)의 방향을 의미한다. + +![](images/page-054_21f2c5a33262bd26f0dcfb4cc9f24ddf62c8df604e1314880879fadaccdf8f46.jpg) + +
+text_image + +y +σyy +σyx +σzz +σxx +σxy +σzz +σxy +σxx +x +z +σyx +σyy +
+ +(a) 축응력 및 전단응력 성분 + +![](images/page-054_130eb60266cc4dbed9cd42d3302b11b560f80a7321bad533bcb1e2acf8af2aa8.jpg) + +
+text_image + +y +3 +2 +σ₃ +σ₂ +σ₁ +θ +z +x +σ₂ +σ₃ +
+ +(b) 주응력 성분 + +$\sigma_{xx}$ : Axial stress in the ECS x - direction + +$\sigma_{yy}$ : Axial stress in the ECS y - direction + +$\sigma_{zz}$ : Axial stress in the ECS z - direction + +$\sigma_{xy} = \sigma_{yx}$ : Shear stress in the ECS x - y plane + +$\sigma_{1}, \sigma_{2}, \sigma_{3}$ : Principal stresses in the directions of the principal axes, 1, 2 and 3 + +where, $\sigma^3 - I_1\sigma^2 - I_2\sigma - I_3 = 0$ + +$$ +I _ {l} = \sigma_ {x x} + \sigma_ {y y} + \sigma_ {z z} +$$ + +$$ +I _ {2} = - \left| \begin{array}{c c} \sigma_ {x x} & \sigma_ {x y} \\ \sigma_ {x y} & \sigma_ {y y} \end{array} \right| - \left| \begin{array}{c c} \sigma_ {x x} & \sigma_ {x z} \\ \sigma_ {x z} & \sigma_ {z z} \end{array} \right| - \left| \begin{array}{c c} \sigma_ {y y} & \sigma_ {y z} \\ \sigma_ {y z} & \sigma_ {z z} \end{array} \right| +$$ + +$$ +I _ {3} = \left| \begin{array}{c c c} \sigma_ {x x} & \sigma_ {x y} & \sigma_ {x z} \\ \sigma_ {x y} & \sigma_ {y y} & \sigma_ {y z} \\ \sigma_ {x z} & \sigma_ {y z} & \sigma_ {z z} \end{array} \right|, \sigma_ {x z} = \sigma_ {z y} = 0 +$$ + +$\theta$ : Angle between the x-axis and the principal axis, 1 in the ECS x-y plane + +$\tau_{max}$ : Maximum shear stress = max $\left[\frac{\left|\sigma_{1}-\sigma_{2}\right|}{2},\frac{\left|\sigma_{2}-\sigma_{3}\right|}{2},\frac{\left|\sigma_{3}-\sigma_{1}\right|}{2}\right]$ + +$\sigma_{eff}:$ von - Mises Stress $= \sqrt{\frac{1}{2}\left[\left(\sigma_{1} - \sigma_{2}\right)^{2} + \left(\sigma_{2} - \sigma_{3}\right)^{2} + \left(\sigma_{3} - \sigma_{1}\right)^{2}\right]}$ + +$\sigma_{oct}:$ Octahedral Normal Stress $= \frac{1}{3}\big(\sigma_{1} + \sigma_{2} + \sigma_{3}\big)$ + +$\tau_{oct}:$ Octahedral Shear Stress $= \sqrt{\frac{1}{9}\left[\left(\sigma_{1} - \sigma_{2}\right)^{2} + \left(\sigma_{2} - \sigma_{3}\right)^{2} + \left(\sigma_{3} - \sigma_{1}\right)^{2}\right]}$ + +그림 1.3.23 평면변형요소의 요소내력 및 요소응력 출력 예 + + + +PLANE STRAIN ELEMENT FORCES(GLOBAL）DEFAULT OUTPUT +Unit System：kN，m + +
ELEMMATLCNODEFXFZ
11LCOMB110.00000-792.25733
2-368.40881-769.85496
13-434.10321734.84737
12-69.04498789.21492
LCOMB210.00000-1028.47676
2-352.46066-978.32967
13-433.19898947.23949
12-85.897361021.51694
+ +PLANE STRAIN ELEMENT STRESSES(GLOBAL）DEFAULT OUTPUT +Unit System:N，mm + +
ELEMMATLCNODESig-XXSig-YYSig-ZZSig-XZ
11LCOMB1Cent82.62870.0000-1.9933-3.7578
182.30880.0000-2.0162-3.4446
282.99370.0000-2.0140-3.4873
1382.94400.0000-1.9716-4.0557
1282.25640.0000-1.9716-4.0429
NODESig-P1Sig-P2Sig-P3MAX-SHRSig-EFFSig-Oct
Cent82.79520.0000-2.159942.477583.896039.5489
182.44920.0000-2.156742.303083.548539.3851
283.13650.0000-2.156842.646784.235639.7091
1383.13730.0000-2.164842.651184.240639.7114
1282.45010.0000-2.165242.307683.553739.3876
LCNODESig-XXSig-YYSig-ZZSig-XZ
LCOMB2Cent82.62870.0000-1.9933-3.7578
182.30880.0000-2.0162-3.4446
282.99370.0000-2.0140-3.4873
1382.94400.0000-1.9716-4.0557
1282.25640.0000-1.9716-4.0429
NODESig-P1Sig-P2Sig-P3MAX-SHRSig-EFFSig-Oct
Cent82.79520.0000-2.159943.477583.896039.5489
182.44920.0000-2.156742.303083.548539.3851
283.13650.0000-1.971642.646784.235639.7091
1383.13730.0000-2.164842.651184.240639.7114
1282.45010.0100-2.165242.307683.553739.3876
+ +그림 1.3.24 평면변형요소의 요소내력 및 요소응력 출력 예 + + + +# 3-8 축대칭요소 (2D Axisymmetric Element) + +# 3-8-1 일반사항 + +이 요소는 형상, 재질, 하중조건 등이 임의 축에 대해 회전대칭 조건을 만족하는 구조체(Pipe, Cylindrical Vessel Body 또는 Head 등)의 해석에 사용될 수 있으며 등매개변수 정식화이론(Isoparametric Formulation)을 근거로 개발되었습니다. + +이 요소는 다른 종류의 요소와 혼용할 수 없으며 요소의 특성상 선형정적해석에만적용 가능합니다. + +축대칭 요소는 3차원 축대칭 모델을 축대칭적 특성을 고려하여 2차원 요소로 이상화한 것입니다. midas Civil에서는 전체좌표계 Z축이 회전대칭을 위한 기준축이 되고, 전체좌표계 X-Z 평면의 Z축의 오른쪽 평면에 위치하도록 입력되어야 합니다. 이 경우 반경방향은 전체좌표계 X축 방향이 되며, 모든 절점의 X방향 좌표는 양(X≥0)의 값을 가지도록 모델링 되어야 합니다. + +요소의 두께는 그림 1.3.25와 같이 1.0 Radian(단위폭)으로 자동 고려됩니다. + +이 요소는 구조물의 축대칭적 특성을 근거로 하기 때문에 원주방향에 대한 변위, 전단변형률( $\gamma_{XY}, \gamma_{YZ}$ ) 그리고 원주방향 전단응력( $\tau_{XY}, \tau_{YZ}$ )은 모두 존재하지 않습니다. + + + +![](images/page-057_ac00094b8247f94aea221fdcc1c875c38b8e6de06b02a03eaa243d98618c4fde.jpg) + +
+text_image + +Z (axis of rotation) +1.0 radian (unit width) +N4 +N3 an axisymmetric element +N1 +N2 +X (radial direction) +
+ +그림 1.3.25 축대칭요소의 단위 폭 + +# 3-8-2 요소자유도 및 요소좌표계 + +midas Civil에서 축대칭요소의 요소좌표계는 프로그램 내부에서 요소강성행렬을 계산하거나, 후처리 모드(Post-processing Mode)에서 사용자가 요소좌표계를 기준으로 응력성분을 도화처리할 때 사용됩니다. + +# 요소자유도는 전체좌표계를 기준으로 X, Z방향의 변위자유도만을 가지게 됩니다. + +요소좌표계는 오른손법칙에 준한 x, y, z축의 직교좌표계를 따르며, 요소좌표계의 방향은 그림 1.3.26과 같이 설정됩니다. + +사각형요소의 경우는 연결절점의 입력순서대로 오른손법칙에 따라 회전할 때(N1→N2→N3→N4) 요소중심에서 요소면의 수직방향으로 엄지손가락 방향이 요소좌표계 z축이 됩니다. 그리고 요소좌표계 x축 방향은 N1과 N4를 잇는 선분의 중심에서 N2와N3을 잇는 선분의 중심까지 직선으로 연결할 때 그 직선의 진행방향이 되며, 요소평면상에서 오른손좌표계를 기준으로 x축과 수직을 이루는 축이 요소좌표계 y축이 됩니다. + + + +삼각형요소의 경우는 면의 중심점에서 N1부터 N2로 진행하는 방향이 요소좌표계의x방향이 되고, 나머지 y, z축 방향은 사각형요소의 경우와 동일합니다. + +※ 요소내력의 출력은 전체좌표계를 따르며 화살표 방향이 양(+)의 방향을 의미한다. + +![](images/page-058_d024cc4d1fea99300a41342e44a1525b2306327d5ef2f80ba0d91e308dc2ab77.jpg) + +
+text_image + +ECS y-axis (perpendicular to ECS x-axis +in the element plane) +Fz3 +Fx3 +N3 +Node numbering order for +creating the element +(N1→N2→N3→N4) +ECS x-axis +(N1 to N2 direction) +Center of Element +ECS z-axis (normal to the element +surface, out of the paper) +Fx1 +N1 +N2 +Fx2 +Fx3 +Fz1 +Fz2 +(a) 사각형 요소 +Z +GCS +X +Fz3 +ECS y-axis (perpendicular to ECS x- +axis) +Fx3 +N3 +Node numbering order for creating +the element (N1→N2→N3) +ECS z-axis (normal to the element +surface, +Center of Element +ECS x-axis +(N1→N2 direction) +Fx1 +N1 +Fz1 +N2 +Fx2 +Fz2 +(b) 삼각형 요소 +GCS +X +Z +GCS +
+ +그림 1.3.26 축대칭요소의 배치 및 요소좌표계, 절점내력 + + + +# 3-8-3 요소관련 기능 + +
Create Elements요소 입력
Material재료적 성질 입력
Pressure Loads요소의 변에 수직방향으로 압력하중 입력
+ +축대칭요소의 압력하중은 그림 1.3.27과 같이 각 변에 수직방향으로 입력되며, 압력하중의 작용면적은 그림 1.3.25와 같이 1.0 Radian의 폭만큼 자동 고려됩니다. + +![](images/page-059_7aac4086e8a712f361d222ca4bd04e4eed8dce4efe92ea1179526ec9abf13b61.jpg) + +
+text_image + +P1 +P2 +P1 +N4 +edge number 3 +N3 +edge number 4 +edge number 2 +N1 +N2 +P2 +P1 +P2 +Z +GCS +X +edge number 1 +
+ +그림 1.3.27 축대칭요소의 압력하중 + + + +# 3-8-4 요소내력 출력내용 + +축대칭요소의 요소내력 및 응력은 다음과 같이 출력되며 부호와 방향은 요소좌표계또는 전체좌표계를 따릅니다. 그림 1.3.28은 요소좌표계의 축방향 또는 주응력방향의단위 Segment에서 발생되는 응력의 부호규약을 설명한 것입니다. + + 연결절점에서의 요소내력 출력 + 연결절점과 요소중심에서 요소응력 출력 + +연결절점에서의 요소내력은 절점에서 산출된 각 성분별 변위와 해당요소 강성성분을곱한 값으로 출력됩니다. + +연결절점과 요소중심에서의 응력은 요소내의 적분점(Gauss Point)에서 연산된 응력을이용하여 외삽법(Extrapolation)에 의해 산출됩니다. + + 요소내력의 출력 +요소내력의 출력치에 대한 부호규약은 그림 1.3.26과 같고, 화살표방향이 양(+)의 방향을 의미합니다. + + 요소응력의 출력 + +요소응력의 출력치에 대한 부호규약은 그림 1.3.28과 같고, 화살표방향이 양(+)의 방향을 의미합니다. diff --git a/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_007.md b/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_007.md new file mode 100644 index 00000000..12271bed --- /dev/null +++ b/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_007.md @@ -0,0 +1,503 @@ + + +※ 요소응력의 출력은 요소좌표계를 따르며 화살표 방향이 양(+)의 방향을 의미한다. + +![](images/page-061_959d29ff7b510100b80dca7073975416186c76bfb2b783c86ce6d7bb07ef6750.jpg) + +
+text_image + +y +σyy +σyx +σzz +σxx +σxy +σzz +σxy +σxx +x +z +σyy +
+ +(a) 축응력 및 전단응력 성분 + +![](images/page-061_f85e4616056bf3a935a07e678fc010e30269c09fff661cf588b446488671ad72.jpg) + +
+text_image + +y +3 +2 +σ₃ +σ₂ +σ₁ +θ +z +x +σ₂ +σ₃ +
+ +(b) 주응력 성분 + +$\sigma_{xx}$ : Axial stress in the ECS x - direction + +$\sigma_{yy}$ : Axial stress in the ECS y - direction + +$\sigma_{zz}$ : Axial stress in the ECS z - direction + +$\sigma_{xy} = \sigma_{yx}$ : Shear stress in the ECS x - y plane + +$\sigma_{1}, \sigma_{2}, \sigma_{3}:$ Principal stresses in the directions of the principal axes, 1, 2 and 3 + +where, $\sigma^3 - I_1\sigma^2 - I_2\sigma - I_3 = 0$ + +$$ +I _ {l} = \sigma_ {x x} + \sigma_ {y y} + \sigma_ {z z} +$$ + +$$ +I _ {2} = - \left| \begin{array}{c c} \sigma_ {x x} & \sigma_ {x y} \\ \sigma_ {x y} & \sigma_ {y y} \end{array} \right| - \left| \begin{array}{c c} \sigma_ {x x} & \sigma_ {x z} \\ \sigma_ {x z} & \sigma_ {z z} \end{array} \right| - \left| \begin{array}{c c} \sigma_ {y y} & \sigma_ {y z} \\ \sigma_ {y z} & \sigma_ {z z} \end{array} \right| +$$ + +$$ +I _ {3} = \left| \begin{array}{c c c} \sigma_ {x x} & \sigma_ {x y} & \sigma_ {x z} \\ \sigma_ {x y} & \sigma_ {y y} & \sigma_ {y z} \\ \sigma_ {x z} & \sigma_ {y z} & \sigma_ {z z} \end{array} \right|, \quad \sigma_ {y z} = \sigma_ {z x} = 0 +$$ + +θ : Angle between the x - axis and the principal axis, 1 in the ECS x - y plane + +$\tau_{max}:\text{Maximum shear stress} = \max \left[\frac{\left|\sigma_1 - \sigma_2\right|}{2},\frac{\left|\sigma_2 - \sigma_3\right|}{2},\frac{\left|\sigma_3 - \sigma_1\right|}{2}\right]$ + +$\sigma_{eff}:$ von - Mises Stress $= \sqrt{\frac{1}{2}\left[\left(\sigma_{1} - \sigma_{2}\right)^{2} + \left(\sigma_{2} - \sigma_{3}\right)^{2} + \left(\sigma_{3} - \sigma_{1}\right)^{2}\right]}$ + +$\sigma_{oct}:$ Octahedral Normal Stress $= \frac{1}{3}\big(\sigma_{1} + \sigma_{2} + \sigma_{3}\big)$ + +$\tau_{oct}:$ Octahedral Shear Stress $= \sqrt{\frac{1}{9}\left[\left(\sigma_{1} - \sigma_{2}\right)^{2} + \left(\sigma_{2} - \sigma_{3}\right)^{2} + \left(\sigma_{3} - \sigma_{1}\right)^{2}\right]}$ + +그림 1.3.28 축대칭요소의 응력출력치의 부호규약 + + + +AXISYMMETRIC ELEMENT FORCES(GLOBAL）DEFAULTOUTPUT +Unit System：kN，m + +
ELEMMATLCNODEFXFZ
11LCOMB110.000000.00000
2-42.269700.00000
13-42.269700.00000
120.000000.00000
LCOMB210.00000-36.76077
2-42.26970-38.83923
13-42.269700.00000
120.000000.00000
+ +AXISYMMETRIC ELEMENT STRESSES(GLOBAL） DEFAULTOUTPUT +Unit System：N，mm + +
ELEMMATLCNODESig-XXSig-YYSig-ZZSig-XZ
11LCOMB1Cent-87.6928154.42280.00000.0000
1-87.6928166.14400.00000.0000
2-87.6928143.06790.00000.0000
13-87.6928143.06790.00000.0000
12-87.6928166.14400.00000.0000
NODESig-P1Sig-P2Sig-P3MAX-SHRSig-EFFSig-Oct
Cent154.42280.0000-87.6928121.0578212.3162100.0868
1166.14400.0000-87.6928126.9184223.3013105.2652
2143.06790.0000-87.6928115.3803201.753595.1075
13143.06790.0000-87.6928115.3803201.753595.1075
12166.14400.0000-87.6928126.9184223.3013105.2652
LCNODESig-XXSig-YYSig-ZZSig-XZ
LCOMB2Cent-87.6928154.42280.00000.0000
1-87.6928166.14400.00000.0000
2-87.6928143.05790.00000.0000
13-87.6928143.06790.00000.0000
12-87.6928166.14400.00000.000
NODESig-P1Sig-P2Sig-P3MAX-SHRSig-EFFSig-Oct
Cent154.42280.0000-87.6928121.0578212.3162100.0868
1166.14400.0000-87.6928126.9184223.3013
2143.06790.0000-87.6928115.3803201.7535
13143.06790.0000-87.6928115.3803201.7535
12166.14400.0000-87.6928126.9184223.3013
+ +그림 1.3.29 축대칭요소의 요소내력 및 요소응력 출력 예 + + + +# 3-9 판요소 (Plate Element) + +# 3-9-1 일반사항 + +이 요소는 동일평면상에 위치한 3개 또는 4개의 절점에 의해 정의되는 판요소(3D Plate Element)로서 평면인장/압축거동, 평면전단거동, 두께방향의 휩거동, 두께방향의 전단거동을 고려할 수 있습니다. + +midas Civil에 사용된 판요소의 면외강성은 DKT, DKQ(Discrete Kirchhoff Element)와 DKMT, DKMQ(Discrete Kirchhoff-Mindlin Element)의 두가지 종류로 구분됩니다. DKT, DKQ인 경우에는 얇은 판 이론(Kirchhoff Plate Theory)에 의해 개발된 것이고, DKMT, DKMQ요소는 두꺼운 판 이론(Mindlin-Reissner Plate Theory)에 의해 개발되었으나, 적절한 전단변형율장을 가정함으로서 얇은 요소부터 두꺼운 판요소까지 우수한 성능을 나타내고 있습니다. 판요소의 면내강성은 3각형인 경우는 LST(Linear Strain Triangle) 이론이 사용하고 있고 4각형인 경우에는 면내 회전자유도에 대한 강성의 고려여부에 따라 2가지 방법을 사용하고 있습니다. 회전자유도에 대한 강성을 고려하지 않는 경우에는 비적합모드를 포함하는 등매개 평면응력이론(Isoparametric Plane Stress Formulation with Incompatible Modes)을 사용합니다. 회전자유도에 대한 강성을 고려하는 경우에는 Allman의 2차 변위장 가정을 도입하여 면내 회전자유도를 변위에 반영하는 방법을 사용하고 있습니다. 이때 사각형 판요소의 경우에는 모든 절점이 동일평면에 존재하지 않는 경우에 발생하는 해석상의 문제를 해결하기 위해 warping항을 고려할 수 있도록 하였으며, DKQ는 Taylor& Simon에 의해 개발된 알고리즘을 사용하여 새롭게 보완되었습니다. + +판요소 두께의 입력은 면내강성(In-plane Stiffness)을 계산하기 위한 것과 면외강성(Out-of-Plane Stiffness)을 계산하기 위한 것으로 구분하여 입력할 수 있습니다. 일반적으로 자중이나 질량의 계산에는 면내강성의 계산을 위한 두께가 사용되며, 면외강성의 계산을 위한 두께만 입력되는 경우에는 면외방향 두께가 사용됩니다. + +# 3-9-2 요소자유도 및 요소좌표계 + +요소자유도는 요소좌표계 x, y, z 축 방향의 이동변위자유도와 x, y 축에 대한 회전변위자유도를 가집니다. + + + +요소좌표계는 오른손법칙에 준한 x, y, z축의 직교좌표계를 따르며 요소좌표계의 방향은 평면응력요소의 경우와 같이 설정됩니다. (그림 1.3.30 참조) + +3-9-3 요소관련 기능 + +
Create Elements요소 입력
Material재료적 성질 입력
Thickness요소두께의 입력
Pressure Loads요소의 면에 수직한 방향 또는 변에 압력하중 입력
Temperature Gradient온도구배 입력
+ +![](images/page-064_8ff7d6bcd8b34f4a8bdb451b53ea42cf9fb6e3f827a73a54450d9b8eb55b5361.jpg) + +
+text_image + +ECS z-axis (normal to the element +Node numbering +order for creating the +element +ECS y-axis (perpendicular to the ECS +x-axis in the element plane) +Center +of +N1 +N2 +N3 +N4 +
+ +(a) 사긱형요소의 요소좌표계 +![](images/page-064_7bf972e23dcc7987be94a4cab0b872c1832cd8edf60cf849edbfa3a24a4a7259.jpg) + +
+text_image + +ECS z-axis (normal to the element surface, out of the +Node numbering order +for creating the element +(N1→N2→N3) +ECS y-axis (perpendicular to +ECS x-axis in the element +Center of Element +N1 +N2 +ECS x-axis +(N1→N2 direction) +
+ +(b) 삼각형요소의 요소좌표계 +그림 1.3.30 판요소의 배치 및 요소좌표계 + + + +# 3-9-4 적분점 + +# 3절점 삼각형 요소 + +이 요소는 1 Point Gauss 적분을 이용하므로, 적분에 적용되는 자연좌표계에서 적분점 좌표는 (1/3, 1/3) 입니다. + +![](images/page-065_ea90809b6f3031b3601684c160c5c7f97cfcd33cb30a49ab886426b631698b8d.jpg) + +
+text_image + +N3 +η +P = (1/3, 1/3) +N1 +N2 +ξ +y +x +
+ +그림 1.3.31 3절점 평면응력요소의 적분점위치 + +이 요소의 기하학적 형상함수는 $N_{1}=1-\xi-\eta$ , $N_{2}=\xi$ , $N_{3}=\eta$ 이므로 요소내 특정 위치에서의 좌표값은 형상함수를 이용하여 다음 식과 같이 구할 수 있습니다. + +$$ +x _ {p} = \sum_ {i = 1} ^ {N} N _ {i} x _ {i}, y _ {p} = \sum_ {i} ^ {N} N _ {i} y _ {i} +$$ + +이 요소의 적분점 좌표인 $\xi=1/3$ , $\eta=1/3$ 을 형상함수에 대입하면 전체좌표계에서 적분점의 좌표를 구할 수 있습니다. + +$$ +x _ {p} = \sum_ {i = 1} ^ {3} N _ {i} x _ {i} = \left(1 - \frac {1}{3} - \frac {1}{3}\right) x _ {1} + \frac {1}{3} x _ {2} + \frac {1}{3} x _ {3} = \frac {1}{3} \left(x _ {1} + x _ {2} + x _ {3}\right) +$$ + +$$ +y _ {p} = \frac {1}{3} \left(y _ {1} + y _ {2} + y _ {3}\right) +$$ + + + +# 4절점 사각형 요소 + +이 요소는 4 Point Gauss 적분을 이용하므로, 적분에 적용되는 자연좌표계에서 적분점 좌표 $P_{i}$ 는 다음 그림과 같습니다. + +![](images/page-066_c75b84c91f51782630690eb0406117207ce47de6f4a3e3e11e118f2c4a57d912.jpg) + +
+text_image + +N4 +η +N3 +P4 +P3 +η = 1/√3 +ξ +P1 +P2 +η = -1/√3 +N1 +ξ = -1/√3 +ξ = 1/√3 +N2 +y +x +P1 = (-1/√3, -1/√3) +P2 = (1/√3, -1/√3) +P3 = (1/√3, 1/√3) +P4 = (-1/√3, 1/√3) +
+ +그림 1.3.32 4절점 평면응력요소의 적분점위치 + +이 요소의 기하학적 형상함수는 다음 식과 같습니다. + +$$ +N _ {1} = \frac {1}{4} (1 - \xi) (1 - \eta), N _ {2} = \frac {1}{4} (1 + \xi) (1 - \eta), N _ {3} = \frac {1}{4} (1 + \xi) (1 + \eta), N _ {4} = \frac {1}{4} (1 - \xi) (1 + \eta) +$$ + +이 요소의 적분점 좌표인 $P_{i}$ 를 형상함수에 대입하면 전체좌표계에서 적분점의 좌표를 구할 수 있습니다. 예를 들어 첫 번째 적분점 좌표 $P_{1}$ 에 대한 전체좌표계에서 x 좌표를 구하면 다음과 같습니다. + +$$ +x _ {p 1} = \sum_ {i = 1} ^ {4} N _ {i} x _ {i} = \frac {1}{6} \left[ (2 + \sqrt {3}) x _ {1} + x _ {2} + (2 - \sqrt {3}) x _ {3} + x _ {4} \right] +$$ + +같은 방법으로 각 적분점에 대해 전체좌표계에서 좌표를 구하면 다음과 같습니다. + +$$ +x _ {p} = \frac {1}{6} \left[ \begin{array}{c c c c} 2 + \sqrt {3} & 1 & 2 - \sqrt {3} & 1 \\ & 2 + \sqrt {3} & 1 & 2 - \sqrt {3} \\ & & 2 + \sqrt {3} & 1 \\ \text {symmetry} & & & 2 + \sqrt {3} \end{array} \right] \left\{ \begin{array}{c} x _ {1} \\ x _ {2} \\ x _ {3} \\ x _ {4} \end{array} \right\} +$$ + +$$ +y _ {p} = \frac {1}{6} \left[ \begin{array}{c c c c} 2 + \sqrt {3} & 1 & 2 - \sqrt {3} & 1 \\ & 2 + \sqrt {3} & 1 & 2 - \sqrt {3} \\ & & 2 + \sqrt {3} & 1 \\ \text {symmetry} & & & 2 + \sqrt {3} \end{array} \right] \left\{ \begin{array}{c} y _ {1} \\ y _ {2} \\ y _ {3} \\ y _ {4} \end{array} \right\} +$$ + + + +# 3-9-5 응력계산법(Extrapolation) + +3절점 삼각형 요소의 경우 1 Point Gauss 적분을 하므로 모든 절점에 대해 적분점에서 계산된 응력을 동일하게 적용합니다. + +![](images/page-067_55721316182253d4b27cab30740630ab9fa5906c12c8d6b2c3b76ea932d3b866.jpg) + +
+text_image + +ξ = -1 +η = 1 +ξ = 1 +η = 1 +t = 1 +η = -1 +ξ = -1 +s = -1 +s = 1 +ξ = 1 +η +ξ = -1 +η +ξ = 1 +t = -1 +η = -1 +ξ = 1 +
+ +그림 1.3.33 4절점 평면응력요소에 대한 적분점에서 응력에 대한 외삽법 + +4절점 사각형 요소의 경우 각 적분점은 요소좌표계의 좌표절점과 다음과 같은 관계를 갖습니다. + +$$ +s = \xi \sqrt {3}, t = \eta \sqrt {3} +$$ + +요소 내부의 특정 위치에서 응력은 형상함수를 이용하여 구할 수 있습니다. + +$$ +\sigma_ {N} = \sum N _ {i} \sigma_ {i} \quad i = 1, 2, 3, 4 +$$ + +예를 들어 절점 1에서 응력을 계산하면 다음과 같습니다. + +$$ +\begin{array}{l} \sigma_ {N 1} = \sum_ {i = 1} ^ {4} N _ {i} \sigma_ {i} = \frac {1}{4} \left[ (1 + \sqrt {3}) (1 + \sqrt {3}) \sigma_ {1} + (1 - \sqrt {3}) (1 + \sqrt {3}) \sigma_ {2} \right. \\ \left. + (1 - \sqrt {3}) (1 - \sqrt {3}) \sigma_ {3} + (1 + \sqrt {3}) (1 - \sqrt {3}) \sigma_ {4} \right] \\ = \frac {1}{4} \left[ (4 + 2 \sqrt {3}) \sigma_ {1} - 2 \sigma_ {2} + (4 - 2 \sqrt {3}) \sigma_ {3} - 2 \sigma_ {4} \right] \\ \end{array} +$$ + + + +같은 방법으로 각 절점에서 응력을 구하면 다음과 같습니다. + +$$ +\left\{ \begin{array}{l} \sigma_ {N 1} \\ \sigma_ {N 2} \\ \sigma_ {N 3} \\ \sigma_ {N 4} \end{array} \right\} = \frac {1}{2} \left[ \begin{array}{c c c c} 2 + \sqrt {3} & - 1 & 2 - \sqrt {3} & - 1 \\ & 2 + \sqrt {3} & - 1 & 2 - \sqrt {3} \\ & & 2 + \sqrt {3} & - 1 \\ & & & 2 + \sqrt {3} \end{array} \right] \left\{ \begin{array}{l} \sigma_ {1} \\ \sigma_ {2} \\ \sigma_ {3} \\ \sigma_ {4} \end{array} \right\} +$$ + +# 3-9-6 요소내력 출력내용 + +판요소의 요소내력 및 응력은 다음과 같이 출력되며 부호와 방향은 요소좌표계 또는 전체좌표계를 따릅니다. 여기서는 요소좌표계를 기준으로 설명합니다. + +■ 연결절점에서의 요소내력 출력 +■연결절점과 요소중심에서의 단위길이당 요소내력 출력 +▪ 연결절점과 요소중심에서 판의 상단면과 하단면에 대한 응력출력 + +연결절점에서의 요소내력은 절점에서 산출된 각 성분별 변위와 해당요소 강성성분을 곱한 값으로 출력됩니다. + +연결절점과 요소중심에서의 단위길이당 요소내력은 면내평면거동과 면외굽힘거동을 분리하여 해당점에서 계산된 응력을 두께방향으로 적분하여 산출합니다. + +단위길이당 요소내력은 철근콘크리트 부재의 설계시 효과적으로 사용될 수 있습니다. + +연결절점과 요소중심에서의 응력은 요소내의 적분점(Gauss Point)에서 연산된 응력을 이용하여 외삽법(Extrapolation)에 의해 산출됩니다. + +■ 요소내력의 출력 +요소내력의 출력치에 대한 부호규약은 그림 1.3.34와 같고, 화살표방향이 양 (+)의 방향을 의미합니다. + +■ 단위길이당 요소내력의 출력 + + + +연결절점과 요소중심에서의 단위길이당 요소내력의 출력치에 대한 부호규약은 그림 1.3.35와 같고, 화살표방향이 양(+)의 방향을 의미합니다. + +#  요소응력의 출력 + +연결절점과 요소중심에서의 요소응력은 그림 1.3.36 (a)에서와 같이 상단면(Top Surface)과 하단면(Bottom Surface)에 대해서 출력되고 부호규약은1.3.36(b)와 같습니다. + +※ 요소내력의 출력은 요소좌표계를 따르며 화살표 방향이 양(+)의 방향을 의미한다. + +![](images/page-069_22bce5bf7e9d21dacc63f9d96de20a2a4bdb81a37ae48de359bb7c9ca3f16cbd.jpg) + +
+text_image + +Fz4 +Fx4 +Mx4 +My4 +Fy4 +N4 +Fz3 +Mx3 +Fy3 +Fx1 +Mx1 +N1 +y +z +Fz2 +My2 +Mx2 +Fy2 +N2 +Fx2 +Fy2 +Mx3 +Fx3 +
+ +(a) 사각형요소의 절점내력 +![](images/page-069_354aa5f2c6fce4e21728d22302d7f858c3ffe823113b13bb696b039517c93e19.jpg) + +
+text_image + +Fz3 +Mx3 +My3 +Fy3 +Fx3 +Fx1 +Mx1 +N3 +y +Fz2 +Mx2 +Fx2 +My2 +Fy2 +N2 +Fy1 +My1 +N1 +
+ +(b) 삼각형요소의 절점내력 +그림 1.3.34 판요소의 각 연결절점에서의 내력출력치에 대한 부호규약 + + + +※ 요소내력의 출력은 요소좌표계를 따르며 화살표 방향이 양(+)의 방향을 의미한다. + +![](images/page-070_bfb3d42420478afe6a0dd0e8ef1f6c4b90e7d64dd3796cd8fac6fae213943f4b.jpg) + +
+text_image + +N4 +y +z +Center point +N3 +N1 +x +N2 +
+ +![](images/page-070_c4162fd1f4ed38ad247be9999ce8d425686b68d7e93aa0b98e54aed3207761c4.jpg) + +
+text_image + +N1 +N2 +N3 +Center point +x +y +z +
+ + : Out put locations of element forces per unit length + +(a) 내력 출력위치 +![](images/page-070_a74937ca1e3f67d48b18db11de0184af444a33632696fcb5d74783c87deaeb2a.jpg) + +
+text_image + +y +Vyy Fyy +Fxy +Fxy +Fxx → x +Vxx +
+ +![](images/page-070_5eaf37e05681119836280307e6e7b9c93327da0a3718c81baa2658997a2e586e.jpg) + +
+text_image + +y +Fyy +Fxx +Fxy +Fmin +Fmax +x +Angle of +principal axis +
+ +(b) 출력위치에서의 면내평면거동에 의한 단위길이당의 힘 + +![](images/page-070_2386f02c195cdd4f6d327d86b5314f8f602f914a059bf492c72d2808d681d5fc.jpg) + +
+text_image + +Mxy +Myy +Mxy → x +Mxx +
+ +![](images/page-070_df836a996980e0bd904219ef23bd82b8821e550e2de81e101a0846130399aa7d.jpg) + +
+text_image + +y +Mxy +Mxy +Mxy +Mxx +Mmin +Mmax +Angle of +principal axis +x +
+ +(c) 출력위치에서의 면외굽힘거동에 의한 단위길이당의 모멘트 +그림 1.3.35 판요소의 단위길이당 요소내력의 출력위치 및 부호규약 diff --git a/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_008.md b/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_008.md new file mode 100644 index 00000000..6b732846 --- /dev/null +++ b/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_008.md @@ -0,0 +1,404 @@ + + +※ 요소응력의 출력은 요소좌표계를 따르며 화살표 방향이 양(+)의 방향을 의미한다. + +![](images/page-071_5ba47eb18611096aa2bda5a62f709d9917132bf2e11fa582e741b6882fb6c743.jpg) + +
+text_image + +Center of Element +
+ + : Output locations of the element stresses (at each connecting node and the center at top/bottom surfaces) + +(a) 응력출력 위치 +![](images/page-071_c9393d2349116d25129025c55a36400deab7b59225369a68a64a378f12a65b56.jpg) + +
+text_image + +σy +τxy +σx ← +τxy +τxy ← +σx +τxy +σy +τxy +x +y +
+ +![](images/page-071_36d8d6e3871984d2986f87e082f1ee3c05264cd0a9fb0427f43f3cf64ac0baa2.jpg) + +
+text_image + +σ₂ +y +1 +θ +x +2 +σ₁ +σ₂ +
+ +$\sigma _ { x }$ : Axial stress in the ECS x - direction + +$\sigma _ { x }$ : Axial stress in the ECS y - direction + +$\tau _ { x y }$ : Shear stress in the ECS x - y plane + +$$ +\sigma_ {l}: \text { Maximum principal stress } = \frac {\sigma_ {x} + \sigma_ {y}}{2} + \sqrt {\left(\frac {\sigma_ {x} - \sigma_ {y}}{2}\right) ^ {2} + \tau_ {x y} ^ {2}} +$$ + +$$ +\sigma_ {2}: \text { Minimum principal stress } = \frac {\sigma_ {x} + \sigma_ {y}}{2} - \sqrt {\left(\frac {\sigma_ {x} - \sigma_ {y}}{2}\right) ^ {2} + \tau_ {x y} ^ {2}} +$$ + +$$ +\tau_ {x y}: \text { Maximum shear stress } = \sqrt {\left(\frac {\sigma_ {x} - \sigma_ {y}}{2}\right) ^ {2} + \tau_ {x y} ^ {2}} +$$ + +:θ Angle between the x - axis and the principal axis,1 + +$$ +\sigma_ {e f f}: \text { von - Mises Stress } = \sqrt {\left(\sigma_ {1} ^ {2} - \sigma_ {1} \sigma_ {2} + \sigma_ {2} ^ {2}\right)} +$$ + +(b) 응력 출력치의 부호규약 + +그림 1.3.36 판요소의 응력출력위치 및 출력치의 부호규약 + + + +
PLATE ELEMENT FORCES (LOCAL) DEFAULT PRINTOUT Unit System : lbf , in
ELEMMATSECLCNODEFxFyFzMxMyMz
111LCOMB116.66-0.131.66-0.0-1.1-0.0
10-7.510.120.25-0.00.6-0.0
110.610.570.210.80.5-0.0
20.25-0.56-0.330.8-0.8-0.0
PLATE ELEMENT FORCES (LOCAL, UNIT LENGTH) PRINTOUT Unit System : lbf , in
ELEMMATSECLCNODEFxxFyyFxyFmaxFminANGLE
111LCOMB1Cent-4.40.00.40.0-4.584.41
1-18.3-0.70.5-0.6-18.388.44
10-18.30.70.40.7-18.388.80
119.50.70.59.50.73.11
29.5-0.70.49.5-0.72.24
NODEMxxMyyMxyMmaxMminANGLE
Cent-1.00.40.20.5-1.080.73
1-1.7-0.00.2-0.0-1.784.30
10-0.80.00.30.1-0.970.89
11-0.60.80.30.9-0.679.31
2-0.80.90.10.9-0.885.99
NODEVxxVyy
Cent-0.3-0.5
1-0.5-0.4
10-0.5-0.5
11-0.1-0.5
2-0.1-0.4
+ +
PLATE ELEMENT STRESSES(LOCAL) DEFAULT PRINTOUT Unit System : lbf , in
ELEMSECLCNODESig-xxSig-yySig-xySig-MAXSig-MINANGLESig-EFF
11LCOMB1Cent T3.5e+003-1.5e+003-847.42503.6e+003-1.7e+003-9.354.7e+003
B-3.7e+0031.5e+003869.32371.7e+003-3.8e+00380.814.9e+003
1 T5.8e+00352.3091-613.66265.9e+003-12.2579-6.015.9e+003
B-6.7e+003-85.1531637.6313-24.5580-6.8e+00384.576.8e+003
10 T2.5e+003-43.3576-1.2e+0033.0e+003-511.7862-21.373.3e+003
B-3.5e+00377.52791.2e+003455.6133-3.8e+00372.744.1e+003
11 T2.4e+003-3.0e+003-990.57962.5e+003-3.2e+003-10.174.9e+003
B-1.9e+0033.0e+0031.0e+0033.2e+003-2.1e+00378.764.6e+003
2 T3.1e+003-3.2e+003-418.55793.1e+003-3.3e+003-3.765.5e+003
B-2.6e+0033.2e+003438.42123.2e+003-2.7e+00385.725.1e+003
+ +그림 1.3.37 판요소의 요소내력 및 요소응력 출력 예 + + + +# 3-10 입체요소 (Solid Element) + +# 3-10-1 일반사항 + +이 요소는 임의 3차원 공간상에 위치한 4개, 6개 또는 8개의 절점으로 정의되는 3차원 입체요소(3D Solid Element)로서, Solid Structure 또는 Thick Shell 등의 모델링에 주로 사용됩니다. + +이 요소는 삼각뿔(Tetrahedron) 또는 삼각기둥(Wedge), 육면체(Hexahedron) 등의 입체 형상을 가질 수 있고, 절점당 3방향의 이동변위 자유도를 가집니다. + +midas Civil에서 이 요소는 비적합모드를 가진 등매개 정식화이론(Isoparametric Formulation with Incompatible Modes)을 사용하여 정식화 되었습니다. + +# 3-10-2 요소자유도, 요소좌표계, 요소의 종류 + +midas Civil에서 3차원 입체요소의 요소좌표계는 프로그램 내부에서 요소강성행렬을 계산하거나, 후처리 모드(Post-processing Mode)에서 사용자가 요소좌표계를 기준으로 응력성분을 도화처리할 때 사용됩니다. + +# 요소자유도는 전체좌표계를 기준으로 X, Y, Z 방향의 이동변위자유도를 가집니다. + +요소좌표계는 오른손 법칙에 준한 x, y, z축의 직교좌표계를 따르며, 원점은 요소의 중심이고, 좌표축의 방향은 면번호 1번의 형상과 동일한 판요소의 요소좌표축 방향과 같습니다. + +요소의 종류는 그림 1.3.38에서와 같이 요소 형상에 따라 8절점요소, 6절점요소 그리고 4절점요소 세 가지가 있으며, 각 요소 종류별로 절점번호의 부여 순서는 N1부터 마지막 번호까지 해당 위치의 절점번호를 순차적으로 입력합니다. + + + +![](images/page-074_e7fda3dac02264bfb1245f110a87ecacf8902b998d9b0c4b79543ea2e78fdba4.jpg) + +
+text_image + +N8 +Plane no. 2 +N5 +Plane no. 6 +N7 +N4 +Plane no. 5 +N6 +Plane no. 4 +N3 +N1 +Plane no. 3 +N2 +Plane no. 1 +
+ +(a) 8절점요소 (Hexahedron) + +![](images/page-074_3ea7f382d8f9991dd757262458f92126a3e125f480cc01f293de909ea913a90c.jpg) + +
+text_image + +N6 +N4 +Plane no. 2 +(triangular plane defined +by nodes N4, N5 and N6) +N5 +Plane no. 4 +Plane no. 5 +Plane no. 3 +N3 +N1 +N2 +Plane no. 1 +(triangular plane defined by nodes N1, N2 and N3) +
+ +![](images/page-074_0998dce0600aa7f3c8677ae785ccd4168df12641341c07397cf510a2eabddf77.jpg) + +
+text_image + +(b) 6절점요소 (Wedge) +N4 +Plane no. 4 +N3 +N1 +Plane no. 2 +Plane no. 3 +N2 +Plane no. 1 +
+ +(c) 4절점요소 (Tetrahedron) +그림 1.3.38 3차원 입체요소의 형상별요소 종류 및 절점번호 부여 순서 + + + +3-10-3 요소관련 명령어 + +
Create Elements요소 입력
Material재료적 성질 입력
Pressure Loads요소의 면에 수직방향으로 압력하중 입력
+ +요소하중은 압력형태로 그림 1.3.39와 같이 요소의 각 면에 입력됩니다 + +※ 화살표 방향이 양(+)의 방향을 의미한다. + +![](images/page-075_76e5d33f6c788ed8aa8486c12b52d1da7cfa72e6cc46aa88b0705bcd13afba15.jpg) + +
+text_image + +Pressure loads acting on the plane no. 2 +Pressure loads acting on the plane no. 6 +Pressure loads acting on the plane no. 5 +Pressure loads acting on the plane no. 4 +Pressure loads acting on the plane no. 3 +Pressure loads acting on the plane no. 1 +
+ +그림 1.3.39 3차원 입체요소의 면에 작용하는 압력하중 + + + +# 3-10-4 적분점 + +# 4절점 사면체 요소 + +이 요소는 1 Point Gauss 적분을 이용하며, 적분에 적용되는 자연좌표계에서 적분점 좌표는 (1/4, 1/4, 1/4) 입니다. + +![](images/page-076_c590a2e012be12fbb985ae731f1be08d19c445b1e4f29da53a782b8165888601.jpg) + +
+text_image + +N1 = 1 - ξ - η - ζ +N2 = ξ +N3 = η +N4 = ζ +P +N1 +N2 +ξ +z +y +x +P = (1/4, 1/4, 1/4) +
+ +그림 1.3.40 4절점 4면체 Solid요소의 적분점 + +적분점 좌표 P 에 대한 전체좌표계에서 좌표를 구하면 다음과 같습니다. + +$$ +x _ {p} = \sum_ {i = 1} ^ {4} N _ {i} x _ {i} = \left(1 - \frac {1}{4} - \frac {1}{4} - \frac {1}{4}\right) x _ {1} + \frac {1}{4} x _ {2} + \frac {1}{4} x _ {3} + \frac {1}{4} x _ {4} = \frac {1}{4} \left(x _ {1} + x _ {2} + x _ {3} + x _ {4}\right) +$$ + +$$ +y _ {p} = \sum_ {i = 1} ^ {4} N _ {i} y _ {i} = \frac {1}{4} \left(y _ {1} + y _ {2} + y _ {3} + y _ {4}\right) +$$ + + + +# 6절점 5면체 요소 + +이 요소는 6 Point Gauss 적분을 이용하며, 적분에 적용되는 자연좌표계에서 적분점 좌표 $P_{i}$ 는 다음 그림과 같습니다. + +![](images/page-077_5b09c7622fa899df8d32f89180ba2ffa9e35ad6f133a36ceadf8f1919e0b3480.jpg) + +
+text_image + +ζ +N4 +ζ = 1/√3 +N5 +P4 +x +P5 x +P6 +x +N6 +z +x +y +P1 x +P2 x +N1 +P3 +x +N2 +ζ = -1/√3 +
+ +$$ +P _ {1} = \left(\frac {1}{6}, \frac {1}{6}, - \frac {1}{\sqrt {3}}\right) +$$ + +$$ +P _ {2} = \left(\frac {2}{3}, \frac {1}{6}, - \frac {1}{\sqrt {3}}\right) +$$ + +$$ +P _ {3} = \left(\frac {1}{6}, \frac {2}{3}, - \frac {1}{\sqrt {3}}\right) +$$ + +$$ +P _ {4} = \left(\frac {1}{6}, \frac {1}{6}, \frac {1}{\sqrt {3}}\right) +$$ + +$$ +P _ {5} = \left(\frac {2}{3}, \frac {1}{6}, \frac {1}{\sqrt {3}}\right) +$$ + +$$ +P _ {6} = \left(\frac {1}{6}, \frac {2}{3}, \frac {1}{\sqrt {3}}\right) +$$ + +그림 1.3.41 6절점 5면체 Solid요소의 적분점 + +이 요소의 기하학적 형상함수는 다음 식과 같습니다. + +$$ +\lambda = 1 - \xi - \eta +$$ + +$$ +N _ {1} = \frac {\lambda}{2} (1 - \zeta) \quad N _ {2} = \frac {\xi}{2} (1 - \zeta) \quad N _ {3} = \frac {\eta}{2} (1 - \zeta) +$$ + +$$ +N _ {4} = \frac {\lambda}{2} (1 + \zeta) \quad N _ {5} = \frac {\xi}{2} (1 + \zeta) \quad N _ {6} = \frac {\eta}{2} (1 + \zeta) +$$ + +적분점 좌표 $P_{1}$ 에 대한 전체좌표계에서 좌표를 구하면 다음과 같습니다. + +$$ +x _ {p 1} = \sum_ {i = 1} ^ {6} N _ {i} x _ {i} = \frac {1}{3 6} \left[ (1 2 + 4 \sqrt {3}) x _ {1} + (3 + \sqrt {3}) x _ {2} + (3 + \sqrt {3}) x _ {3} \right. +$$ + +$$ +\left. + (1 2 - 4 \sqrt {3}) x _ {4} + (3 - \sqrt {3}) x _ {5} + (3 - \sqrt {3}) x _ {6} \right] +$$ + + + +같은 방법으로 각 적분점에 대한 전체좌표계에서 좌표를 구하면 다음과 같습니다. + +$$ +x _ {p} = \frac {1}{3 6} \left[ \begin{array}{c c c c c c} 1 2 + 4 \sqrt {3} & 3 + \sqrt {3} & 3 + \sqrt {3} & 1 2 - 4 \sqrt {3} & 3 - \sqrt {3} & 3 - \sqrt {3} \\ & 1 2 + 4 \sqrt {3} & 3 + \sqrt {3} & 3 - \sqrt {3} & 1 2 - 4 \sqrt {3} & 3 - \sqrt {3} \\ & & 1 2 + 4 \sqrt {3} & 3 - \sqrt {3} & 3 - \sqrt {3} & 1 2 - 4 \sqrt {3} \\ & & & 1 2 + 4 \sqrt {3} & 3 + \sqrt {3} & 3 + \sqrt {3} \\ & & & & 1 2 + 4 \sqrt {3} & 3 + \sqrt {3} \\ & & & & & 1 2 + 4 \sqrt {3} \end{array} \right] \left\{ \begin{array}{l} x _ {1} \\ x _ {2} \\ x _ {3} \\ x _ {4} \\ x _ {5} \\ x _ {6} \end{array} \right\} +$$ + +# 8절점 6면체 요소 + +이 요소는 8 Point Gauss 적분을 이용하며, 적분에 적용되는 자연좌표계에서 적분점 좌표 $P_{i}$ 는 그림과 같습니다. + +![](images/page-078_6868f2d4499f81d4f850ce71765c566384ff2523664c1b90b28b26ff441772ec.jpg) + +
+text_image + +3D geometric diagram with labeled points and coordinate axes, including mathematical expressions for P and ζ. +
+ +그림 1.3.42 8절점 6면체 Solid요소의 적분점 + + + +이 요소의 기하학적 형상함수는 다음 식과 같습니다. + +$$ +\begin{array}{l} N _ {1} = \frac {1}{8} (1 - \xi) (1 - \eta) (1 - \zeta) \quad N _ {2} = \frac {1}{8} (1 + \xi) (1 - \eta) (1 - \zeta) \\ N _ {3} = \frac {1}{8} (1 + \xi) (1 + \eta) (1 - \zeta) \quad N _ {4} = \frac {1}{8} (1 - \xi) (1 + \eta) (1 - \zeta) \\ N _ {5} = \frac {1}{8} (1 - \xi) (1 - \eta) (1 + \zeta) \quad N _ {6} = \frac {1}{8} (1 + \xi) (1 - \eta) (1 + \zeta) \\ N _ {7} = \frac {1}{8} (1 + \xi) (1 + \eta) (1 + \zeta) \quad N _ {8} = \frac {1}{8} (1 - \xi) (1 + \eta) (1 + \zeta) \\ \end{array} +$$ + +적분점 좌표 $P_{1}$ 에 대한 전체좌표계에서 좌표를 구하면 다음과 같습니다. + +$$ +\begin{array}{l} x _ {p 1} = \sum_ {i = 1} ^ {8} N _ {i} x _ {i} = \frac {1}{3 6} \left[ (9 + 5 \sqrt {3}) x _ {1} + (3 + \sqrt {3}) x _ {2} + (3 - \sqrt {3}) x _ {3} + (3 + \sqrt {3}) x _ {4} \right. \\ \left. + \left(3 + \sqrt {3}\right) x _ {5} + \left(3 - \sqrt {3}\right) x _ {6} + \left(9 - 5 \sqrt {3}\right) x _ {7} + \left(3 - \sqrt {3}\right) x _ {8} \right] \\ \end{array} +$$ + +같은 방법으로 각 적분점에 대한 전체좌표계에서 좌표를 구하면 다음과 같습니다. + +$$ +x _ {p} = \frac {1}{3 6} \left[ \begin{array}{c c c c c c c c} 9 + 5 \sqrt {3} & 3 + \sqrt {3} & 3 - \sqrt {3} & 3 + \sqrt {3} & 3 + \sqrt {3} & 3 - \sqrt {3} & 9 - 5 \sqrt {3} & 3 - \sqrt {3} \\ & 9 + 5 \sqrt {3} & 3 + \sqrt {3} & 3 + \sqrt {3} & 3 - \sqrt {3} & 3 + \sqrt {3} & 3 - \sqrt {3} & 9 - 5 \sqrt {3} \\ & & 9 + 5 \sqrt {3} & 3 + \sqrt {3} & 9 - 5 \sqrt {3} & 3 - \sqrt {3} & 3 + \sqrt {3} & 3 - \sqrt {3} \\ & & & 9 + 5 \sqrt {3} & 3 - \sqrt {3} & 9 - 5 \sqrt {3} & 3 - \sqrt {3} & 3 + \sqrt {3} \\ & & & & 9 + 5 \sqrt {3} & 3 + \sqrt {3} & 3 - \sqrt {3} & 3 + \sqrt {3} \\ & & & & & 9 + 5 \sqrt {3} & 3 + \sqrt {3} & 3 + \sqrt {3} \\ & & & & & & 9 + 5 \sqrt {3} & 3 + \sqrt {3} \\ & & & & & & & 9 + 5 \sqrt {3} \end{array} \right] \left\{ \begin{array}{l} x _ {1} \\ x _ {2} \\ x _ {3} \\ x _ {4} \\ x _ {5} \\ x _ {6} \\ x _ {7} \\ x _ {8} \end{array} \right\} +$$ + + + +# 3-10-5 응력계산법(Extrapolation) + +4절점 4면체 요소의 경우 1 point gauss 적분을 하므로 모든 절점에 대해 적분점에서 계산된 응력을 동일하게 적용합니다. + +![](images/page-080_9b4fd65f06f79a03486e98bbc243fbd5c8b74e845387e8770f4b0cfa393591d3.jpg) + +
+text_image + +ξ = -\frac{1}{\sqrt{3}} +u +P1 +P2 +P3 +s +t +η +
+ +그림 1.3.43 6절점 5면체요소에 대한 적분점에서 응력에 대한 외삽법 + +6절점 5면체 요소의 경우 각 적분점은 요소좌표계의 좌표절점과 다음 식과 같은 관계를 갖습니다. + +$$ +\lambda = 2 \left(1 - \xi - \eta - \frac {1}{6}\right), s = 2 \left(\xi - \frac {1}{6}\right), t = 2 \left(\eta - \frac {1}{6}\right), u = \zeta \sqrt {3} +$$ + +요소 내부의 특정 위치에서의 응력은 형상함수를 이용하여 구할 수 있습니다. + +$$ +\sigma_ {N} = \sum_ {i = 1} ^ {6} N _ {i} \sigma_ {i} +$$ + +예를 들어 절점 1에서 응력을 계산하면 다음과 같습니다. + +$$ +\sigma_ {N 1} = \sum_ {i = 1} ^ {6} N _ {i} \sigma_ {i} = \frac {1}{6} \left[ 5 (1 + \sqrt {3}) \sigma_ {1} - (1 + \sqrt {3}) \sigma_ {2} - (1 + \sqrt {3}) \sigma_ {3} \right. +$$ + +$$ +\left. + 5 (1 - \sqrt {3}) \sigma_ {4} - (1 - \sqrt {3}) \sigma_ {5} - (1 - \sqrt {3}) \sigma_ {6} \right] +$$ diff --git a/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_009.md b/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_009.md new file mode 100644 index 00000000..fb3e6705 --- /dev/null +++ b/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_009.md @@ -0,0 +1,236 @@ + + +같은 방법으로 각 절점에서 응력을 구하면 다음과 같습니다. + +$$ +\left\{ \begin{array}{l} \sigma_ {N 1} \\ \sigma_ {N 2} \\ \sigma_ {N 3} \\ \sigma_ {N 4} \\ \sigma_ {N 5} \\ \sigma_ {N 6} \end{array} \right\} = \frac {1}{6} \left[ \begin{array}{c c c c c c} 5 + 5 \sqrt {3} & - 1 - \sqrt {3} & - 1 - \sqrt {3} & 5 - 5 \sqrt {3} & - 1 - \sqrt {3} & - 1 + \sqrt {3} \\ & 5 + 5 \sqrt {3} & - 1 - \sqrt {3} & - 1 + \sqrt {3} & 5 - 5 \sqrt {3} & - 1 + \sqrt {3} \\ & & 5 + 5 \sqrt {3} & - 1 - \sqrt {3} & - 1 - \sqrt {3} & 5 - 5 \sqrt {3} \\ & & & 5 + 5 \sqrt {3} & - 1 - \sqrt {3} & - 1 - \sqrt {3} \\ & & & & 5 + 5 \sqrt {3} & - 1 - \sqrt {3} \\ & & & & & 5 + 5 \sqrt {3} \end{array} \right] \left\{ \begin{array}{l} \sigma_ {1} \\ \sigma_ {2} \\ \sigma_ {3} \\ \sigma_ {4} \\ \sigma_ {5} \\ \sigma_ {6} \end{array} \right\} +$$ + +8절점 6면체 요소의 경우 각 적분점은 요소좌표계의 좌표절점과 다음과 같은 관계를 갖습니다. + +![](images/page-081_c197644b6d4d4018f1a66ed9e2cc795bfe31929c1a39b222da01caa9e0d1ea21.jpg) + +
+text_image + +ξ = -1 +η = 1 +ξ = -1 +η = 1 +t = 1 +ξ = -1/√3 +η = -1 +ξ = -1 +s = -1 +s = 1 +ξ = 1 +η = -1 +η = -1 +ξ = 1 +s = 1 +ξ = 1 +
+ +그림 1.3.44 8절점 6면체요소에 대한 적분점에서 응력에 대한 외삽법 + +$$ +s = \xi \sqrt {3}, t = \eta \sqrt {3}, u = \zeta \sqrt {3} +$$ + + + +예를 들어 절점 1에서의 응력을 형상함수를 이용하여 구하면 다음과 같습니다. + +$$ +\begin{array}{l} \sigma_ {N 1} = \sum_ {i = 1} ^ {8} N _ {i} \sigma_ {i} = \frac {1}{8} \left[ (1 + \sqrt {3}) (1 + \sqrt {3}) (1 + \sqrt {3}) \sigma_ {1} + (1 - \sqrt {3}) (1 + \sqrt {3}) (1 + \sqrt {3}) \sigma_ {2} \right. \\ + (1 - \sqrt {3}) (1 - \sqrt {3}) (1 + \sqrt {3}) \sigma_ {3} + (1 + \sqrt {3}) (1 - \sqrt {3}) (1 + \sqrt {3}) \sigma_ {4} \\ + (1 + \sqrt {3}) (1 + \sqrt {3}) (1 - \sqrt {3}) \sigma_ {5} + (1 - \sqrt {3}) (1 + \sqrt {3}) (1 - \sqrt {3}) \sigma_ {6} \\ \left. + (1 - \sqrt {3}) (1 - \sqrt {3}) (1 - \sqrt {3}) \sigma_ {7} + (1 + \sqrt {3}) (1 - \sqrt {3}) (1 - \sqrt {3}) \sigma_ {8} \right] \\ = \frac {1}{8} \left[ (1 0 + 6 \sqrt {3}) \sigma_ {1} + (- 2 - 2 \sqrt {3}) \sigma_ {2} + (- 2 + 2 \sqrt {3}) \sigma_ {3} + (- 2 - 2 \sqrt {3}) \sigma_ {4} \right. \\ \left. + \left(- 2 - 2 \sqrt {3}\right) \sigma_ {5} + \left(- 2 + 2 \sqrt {3}\right) \sigma_ {6} + \left(1 0 - 6 \sqrt {3}\right) \sigma_ {7} + \left(- 2 + 2 \sqrt {3}\right) \sigma_ {8} \right] \\ \end{array} +$$ + +같은 방법으로 각 적분점에 대한 전체좌표계에서 좌표를 구하면 다음과 같습니다. + +$$ +\left\{ \begin{array}{l} \sigma_ {N 1} \\ \sigma_ {N 2} \\ \sigma_ {N 3} \\ \sigma_ {N 4} \\ \sigma_ {N 5} \\ \sigma_ {N 6} \\ \sigma_ {N 7} \\ \sigma_ {N 8} \end{array} \right\} = \frac {1}{4} \left[ \begin{array}{c c c c c c c c} 5 + 3 \sqrt {3} & - 1 - \sqrt {3} & - 1 + \sqrt {3} & - 1 - \sqrt {3} & - 1 - \sqrt {3} & - 1 + \sqrt {3} & 5 - 3 \sqrt {3} & - 1 + \sqrt {3} \\ & 5 + 3 \sqrt {3} & - 1 - \sqrt {3} & - 1 + \sqrt {3} & - 1 + \sqrt {3} & - 1 - \sqrt {3} & - 1 + \sqrt {3} & 5 - 3 \sqrt {3} \\ & & 5 + 3 \sqrt {3} & - 1 - \sqrt {3} & 5 - 3 \sqrt {3} & - 1 + \sqrt {3} & - 1 - \sqrt {3} & - 1 + \sqrt {3} \\ & & & 5 + 3 \sqrt {3} & - 1 + \sqrt {3} & 5 - 3 \sqrt {3} & - 1 + \sqrt {3} & - 1 - \sqrt {3} \\ & & & & 5 + 3 \sqrt {3} & - 1 - \sqrt {3} & - 1 + \sqrt {3} & - 1 - \sqrt {3} \\ & & & & & 5 + 3 \sqrt {3} & - 1 - \sqrt {3} & - 1 + \sqrt {3} \\ & & & & & & 5 + 3 \sqrt {3} & - 1 - \sqrt {3} \\ & & & & & & & 5 + 3 \sqrt {3} \end{array} \right] \left\{ \begin{array}{l} \sigma_ {1} \\ \sigma_ {2} \\ \sigma_ {3} \\ \sigma_ {4} \\ \sigma_ {5} \\ \sigma_ {6} \\ \sigma_ {7} \\ \sigma_ {8} \end{array} \right\} +$$ + + + +# 3-10-6 요소내력 출력내용 + +3차원 입체요소의 요소내력 및 응력은 다음과 같이 출력되며, 부호와 방향은 요소좌표계 또는 전체좌표계를 따릅니다. + + 연결절점에서의 요소내력 출력 + 연결절점과 요소중심에서 3차원 응력성분 출력 + +연결절점에서의 요소내력은 절점에서 산출된 각 성분별 변위와 해당 요소의 강성성분을 곱한 값으로 출력됩니다. + +연결절점과 요소중심에서의 응력은 요소내의 적분점(Gauss Point)에서 계산된 응력을 이용하여 외삽법(Extrapolation)에 의해 산출됩니다. + + 요소내력의 출력 +요소내력의 출력치에 대한 부호규약은 그림 1.3.45와 같고, 화살표방향이 양(+)의 방향을 의미합니다. + + 요소응력의 출력 + +요소응력의 출력치에 대한 부호규약은 그림 1.3.46과 같고, 화살표 방향이양(+)의 방향을 의미합니다. + + + +※ 요소내력의 출력은 전체좌표계를 따르며 화살표 방향이 양(+)의 방향을 의미한다. + +![](images/page-084_94e239d668d740ca9131dd24fed42501ef79fcc5fd3e58a05f9b3592f8c1405a.jpg) + +
+text_image + +Diagram showing force vectors (F) acting on a 3D grid with labeled axes X, Y, Z and force components FX1 to FX8 +
+ +그림 1.3.45 3차원 입체요소의 연결절점에 대한 내력출력치 및 부호규약 + + + +※ 요소응력의 출력은 요소좌표계를 따르며 화살표 방향이 양(+)의 방향을 의미한다. + +![](images/page-085_e4b898f67e38c2edeff237a0b401a70f7a69c9aace1f87bd0506843c71c05582.jpg) + +
+text_image + +σ_zz +σ_zx +σ_zy +σ_xz +σ_yz +σ_xx +σ_xy +σ_yx +σ_yy +σ_zz +
+ +(a) 축응력 및 전단응력 성분 + +![](images/page-085_94914a67bd38e93f81f65019b193aa51e53163f4adc353bce36dc284eb7550da.jpg) + +
+text_image + +σ₃ +σ₂ +σ₁ +
+ +(b) 주응력 성분 + +$\sigma_{xx}$ : Axial stress in the ECS x - direction + +$\sigma_{yy}$ : Axial stress in the ECS y - direction + +$\sigma_{zz}$ : Axial stress in the ECS z - direction + +$\sigma_{xz} = \sigma_{zx}$ : Shear stress in the ECS x - z direction + +$\sigma_{xy} = \sigma_{yx}$ : Shear stress in the ECS x - y direction + +$\sigma_{yz} = \sigma_{zy}$ : Shear stress in the ECS y - z direction + +$\sigma_{1}, \sigma_{2}, \sigma_{3}$ : Principal stresses in the directions of the principal axes, 1, 2 and 3 + +where, $\sigma^3 - I_1\sigma^2 - I_2\sigma - I_3 = 0$ + +$$ +I _ {l} = \sigma_ {x x} + \sigma_ {y y} + \sigma_ {z z} +$$ + +$$ +I _ {2} = - \left| \begin{array}{c c} \sigma_ {x x} & \sigma_ {x y} \\ \sigma_ {x y} & \sigma_ {y y} \end{array} \right| - \left| \begin{array}{c c} \sigma_ {x x} & \sigma_ {x z} \\ \sigma_ {x z} & \sigma_ {z z} \end{array} \right| - \left| \begin{array}{c c} \sigma_ {y y} & \sigma_ {y z} \\ \sigma_ {y z} & \sigma_ {z z} \end{array} \right| +$$ + +$$ +I _ {3} = \left| \begin{array}{c c c} \sigma_ {x x} & \sigma_ {x y} & \sigma_ {x z} \\ \sigma_ {x y} & \sigma_ {y y} & \sigma_ {y z} \\ \sigma_ {x z} & \sigma_ {y z} & \sigma_ {z z} \end{array} \right| +$$ + +$\tau_{max}$ : Maximum shear stress = max $\left[\frac{\left|\sigma_{1}-\sigma_{2}\right|}{2},\frac{\left|\sigma_{2}-\sigma_{3}\right|}{2},\frac{\left|\sigma_{3}-\sigma_{1}\right|}{2}\right]$ + +$\sigma_{eff}:$ von - Mises Stress $= \sqrt{\frac{1}{2}\left[\left(\sigma_{1} - \sigma_{2}\right)^{2} + \left(\sigma_{2} - \sigma_{3}\right)^{2} + \left(\sigma_{3} - \sigma_{1}\right)^{2}\right]}$ + +$\sigma_{oct}:$ Octahedral Normal Stress $= \frac{1}{3}\big(\sigma_{1} + \sigma_{2} + \sigma_{3}\big)$ + +$\tau_{oct}:$ Octahedral Shear Stress $= \sqrt{\frac{1}{9}\left[\left(\sigma_{1} - \sigma_{2}\right)^{2} + \left(\sigma_{2} - \sigma_{3}\right)^{2} + \left(\sigma_{3} - \sigma_{1}\right)^{2}\right]}$ + +그림 1.3.46 3차원 입체요소의 연결절점에 대한 응력출력치 및 부호규약 + + + +
SOLID ELEMENT FORCES(GLOBAL) DEFAULT OUTPUT Unit System : kN , m
ELEMMATLCNODEFXFYFZ
11LCOMB11-2.45166-0.28377-0.28377
52.45166-0.07604-0.07604
62.451660.07604-0.07604
2-2.451660.28377-0.28377
3-2.45166-0.283770.28377
72.45166-0.076040.07604
82.451660.076040.07604
4-2.451660.283770.28377
LCOMB21-49.03325-2.45166-6.30239
544.129922.45166-0.98543
6-44.129922.451660.98543
249.03325-2.451666.30239
3-49.03325-2.451666.30239
744.129922.451660.98543
8-44.129922.45166-0.98543
449.03325-2.45166-6.30239
+ +
SOLID ELEMENT STRESSES(GLOBAL) DEFAULT OUTPUT Unit System : N , mm
ELEMMATLCNODESig-XXSig-YYSig-ZZSig-XYSig-YZSig-XZ
11LCOMB1Cent0.03920.00290.00290.00000.00000.0000
10.03920.00790.00790.00000.00000.0000
50.0392-0.0021-0.00210.00000.00000.0000
60.0392-0.0021-0.00210.00000.00000.0000
20.03920.00790.00790.00000.00000.0000
30.03920.00790.00790.00000.00000.0000
70.0392-0.0021-0.00210.00000.00000.0000
80.0392-0.0021-0.00210.00000.00000.0000
40.03920.00790.00790.00000.00000.0000
NODESig-P1Sig-P2Sig-P3MAX-SHRSig-EFFSig-Oct
Cent0.03920.00290.00290.01820.03630.0171
10.03920.00000.00000.01960.03920.0185
50.0392-0.0021-0.00210.02070.04130.0195
60.03920.00000.00000.01960.03920.0185
20.03920.00790.00790.01570.03140.0148
30.03920.00790.00790.01570.03140.0148
70.0392-0.0021-0.00210.02070.04130.0195
80.0392-0.0021-0.00210.02070.04130.0195
40.03920.00790.00790.01570.03140.0148
LCNODESig-XXSig-YYSig-ZZSig-XYSig-YZSig-XZ
LCOMB2Cent0.00000.00000.00000.03920.00000.0000
12.27860.04260.38760.03920.08500.0850
52.1933-0.0426-0.03780.0392-0.08500.0850
6-2.19330.04260.03780.0392-0.0850-0.0850
2-2.2786-0.0426-0.38760.03920.0850-0.0850
32.27860.04260.38760.0392-0.0850-0.0850
72.1933-0.0426-0.03780.03920.0850-0.0850
8-2.19330.04260.03780.03920.08500.0850
4-2.2786-0.0426-0.38760.0392-0.08500.0850
NODESig-P1Sig-P2Sig-P3MAX-SHRSig-EFFSig-Oct
Cent0.03920.0000-0.03920.03920.06790.0320
12.28320.40300.02271.13032.09640.9883
52.19710.0443-0.12861.16282.24421.0579
60.1286-0.0443-2.19711.16282.24421.0579
2-0.0227-0.4030-2.28321.13032.09640.9883
32.28320.40300.02271.13032.09640.9883
72.19710.0443-0.12861.16282.24421.0579
80.1286-0.0443-2.19711.16282.24421.0579
4-0.0227-0.4030-2.28321.13032.09640.9883
+ +그림 1.3.47 3차원 입체요소의 요소내력 및 요소응력 출력 예 + + + +# Chapter 4. 요소 입력시 주요 고려사항 + +“midas Civil의 수치해석 모델”의 서론에서 언급한 바와 같이 구조해석이란 구조물의 구조적 거동을 파악하기 위해 수치해석 모델을 이용하여 예견되는 가상적 상황에대한 이론적 모의실험을 수행하는 것이기 때문에 실제상황과 얼마나 근접한 모델을입력하는가가 해석작업의 성패를 좌우하는 요인이 됩니다. + +따라서 해당구조물의 구조적 거동을 가장 근접하게 반영할 수 있는 유한요소의 선택과 입력과정이 모델링 전반에 걸쳐 가장 중요한 사항이라 할 수 있습니다. + +유한요소의 선택과 모델링의 범위는 해석의 목적에 따라 달라지게 되는데, 가령 해석후 설계작업까지 수행할 경우에는 설계시 필요한 변위, 부재내력, 응력 등의 설계변수를 구할 수 있도록 절점과 요소분할이 이뤄져야 합니다. 요소내력 및 응력의 출력치도 추가로 변환할 필요없이 설계에 그대로 사용할 수 있도록 요소를 선택하는 것이 효과적입니다. 그리고 해석목적이 변위를 구하는 것인지, 요소내력을 구하는 것인지 또는 고유치해석 수행에 있는지 등에 따라 요소의 선택과 모델링의 범위, 요소분할의 정도가 달라집니다. 변위를 구하거나 고유치해석만을 수행할 경우에는 모델을단순화하는 것이 효과적이지만 요소내력을 구할 경우에는 일반적으로 요소를 세분화하는 것이 바람직합니다. + +구조물의 전체거동을 파악하기 위한 고유치해석 문제인 경우에는 국부모드(LocalMode)의 발생을 억제하기 위하여 모델을 단순화하는 것이 좋습니다. 특히 토목구조물의 예비설계단계(Preliminary Design Phase)에서는 상세모델보다는 구조물의 전체강성을 이론적 등가강성을 가진 보요소로 치환하는 것이 효과적입니다. + + + +수치해석 모델을 입력하는데 주요한 고려사항은 다음과 같습니다. + +절점의 위치를 선정할 때의 주요 고려사항은 구조물의 기하학적 형상, 재료, 단면의종류, 하중상태 등으로 절점이 필요한 위치는 다음과 같습니다. + + 해석결과가 필요한 위치 + 하중의 입력이 필요한 위치 + 강성(단면 또는 두께)이 변하는 위치 또는 구획선의 경계 + 재질이 변하는 위치 또는 구획선의 경계 + 개구부 주위와 같이 응력의 변화가 심한 위치 또는 구획선의 경계 + 구조물의 경계부분 + 구조형상이 변하는 위치 또는 구획선의 경계 + +선요소(트러스요소, 보요소 ... 등)의 경우는 요소의 크기에 영향을 받지 않지만, 판형요소(평면응력요소, 평면변형요소, 축대칭요소, 판요소) 또는 입체요소의 경우는 요소의 크기, 형상, 분포에 따라 결과치에 큰 영향을 받게 됩니다. 판형요소 또는 입체요소의 크기와 분포방법을 결정할 때, 응력의 변화가 심한 곳과 정확한 해가 요구되는부분에 대해서는 세분화하고, 예상되는 응력의 분포형태를 고려하여 응력등고선(Contour)을 따라 분할하는 것이 바람직합니다. + +일반적으로 요소의 세분화가 필요한 부위는 다음과 같습니다. + + 기하학적 불연속부위 또는 개구부 주위 + 하중의 변화가 심한 부위, 특히 상대적으로 큰 집중하중이 작용하는 위치의인접부위 + 강성 또는 물성치가 변하는 부위 + 불규칙한 경계부위 + 응력집중이 예상되는 부위 + 정밀한 요소내력 또는 응력결과가 필요한 부위 + + + +요소의 크기 및 형상을 결정할 때 기본적으로 고려해야 하는 사항은 다음과 같습니다. + + 요소의 크기와 모양은 가능한 한 일정하게 유지해야 합니다. + 요소크기의 변화가 필요한 부위에 대해서는 로그분포(LogarithmicConfiguration)를 가지도록 합니다. + 인접요소간의 요소크기 차이가 1/2이하가 되도록 합니다. + 응력을 구할 경우에는 4절점요소(면요소) 또는 8절점요소(입체요소)를 사용하고, 요소형상비는 1:1일 경우가 최적의 조건이며 불가피할 경우에는 1:4를 넘지않도록 합니다. 그리고 강성전달 목적이거나 변위를 구할 경우에는 1:10을넘지 않도록 하는 것이 좋습니다. + 이상적인 모서리각도는 요소의 면이 사각형일 때는 90°이고 삼각형일 때는60°입니다. + 불가피한 경우라도 사각면의 모서리각도는 가능한 한 45°이하 또는 135°이상이 되지 않도록 하며, 삼각면일 경우는 내부각이 30°이하나 150°이상이 되지않도록 합니다. + 사각형일 경우 모서리절점은 가능한 한 동일평면상에 존재하는 것이 좋으며높이차가 장변길이에 대해 1/100을 넘지 않도록 합니다. + + + +다음은 요소종류별 주요용도와 사용상 고려해야 할 사항에 대해 서술합니다. + +# 4-1 트러스요소, 인장력 전담요소, 압축력 전담요소 + +이 요소들은 공간트러스, 케이블, 대각부재 등과 같이 부재의 축방향으로만 힘을 받는 부재나 접촉면의 모델링에 주로 사용됩니다. + +예를 들어 트러스요소는 축방향으로 압축 및 인장을 받을 수 있는 트러스구조의 모델에 사용될 수 있으며, 인장력전담요소는 Sagging을 무시할 수 있는 케이블 또는 대각부재 중 세장비가 커서 압축력을 거의 전달할 수 없는 Wind Bracing 등과 같은 부재에 사용될 수 있습니다. + +그리고 압축력전담요소는 구조체간의 접촉면이나 인장을 받을 수 없는 지반경계조건 등을 반영하는데 응용될 수 있습니다. 프리스트레스를 받는 경우에는 Pretension Loads를 이용할 수 있습니다. + +이 요소들은 회전강성이 없어서 양단의 연결절점에서 회전변위에 대한 자유도를 가지지 못하기 때문에 이 요소들 또는 기타 회전자유도가 없는 요소끼리 접하는 절점에서는 해석과정에서 특이성 오류(Singular Error)가 발생됩니다. midas Civil에서는 이러한 경우 해당절점의 회전자유도를 자동구속시킴으로써 특이성 오류의 발생을 방지하고 있습니다. + +그러나 이 요소들이 회전방향강성을 가진 보요소와 연결될 때는 별도로 특이성 오류를 방지하기 위한 조치가 필요 없습니다. + +그림 1.4.1과 같이 트러스요소끼리 연결할 경우에는 불안정구조체가 되지 않도록 주의해야 합니다. + +그림 1.4.1(a)의 경우는 평면방향으로 하중이 가해질 때 하중을 지지하여 전달할 수 있는 회전강성이 없기 때문에 불안정구조체가 되며 그림 1.4.1(b), (c)의 경우도 마찬 가지로 Y-Z 평면에 대해서는 안정적이지만 하중작용방향인 X-Z 평면방향의 거동에 diff --git a/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_010.md b/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_010.md new file mode 100644 index 00000000..a4d30408 --- /dev/null +++ b/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_010.md @@ -0,0 +1,237 @@ + + +대해서는 불안정구조체가 됩니다. + +압축력전담요소나 인장력전담요소를 사용할 경우에는 하중의 크기에 따라 요소의 강성이 발현되지 않을 수 있으므로(예: 인장력전담요소가 압축력을 받는 경우) 주의해야 합니다. + +![](images/page-091_99f83215705d171ef7b2c0935d618980a2fd3ac45575455d727f9da2e99bfea6.jpg) +(c) Y-Z 평면에 X방향으로하중이 작용하는 경우 +그림 1.4.1 트러스요소(인장력 또는 압축력전담요소)로 형성된 대표적 불안정구조체의 예 + +인장력 전담요소와 압축력 전담요소는 비선형탄성(Nonlinear Elastic)요소로서 다음 그림과 같은 거동을 한다고 가정합니다. + +![](images/page-091_36b6d14a395c8149378eb6b31e31e5f0515cd0f4bb15656ad5e918db360dc48a.jpg) + +
+text_image + +P +Tension +Limit +u +
+ +(a) 인장력 전담요소 + +![](images/page-091_1043621a4bc00fa01b3d34f1cbc487849e9a22bddd14dd8cbf6ba9c6a5ecc82c.jpg) + +
+text_image + +P +u +Compression +Limit +
+ +(b) 압축력 전담요소 +그림 1.4.2 인장력 전담요소와 압축력 전담요소의 비선형 거동 + + + +위의 그림 1.4.2(a)에서와 같이 인장력 전담요소는 인장력만 받을 수 있는 요소로서,인장력이 Tension Limit보다 작을 경우 일반적인 트러스와 같이 거동하지만 인장력이Tension Limit보다 크거나 압축을 받을 경우에는 외력에 저항하지 못하는 특징을 가집니다. 반면, 그림 1.4.2(b)에서와 같이 압축력 전담요소는 압축력만 받을 수 있는 요소로서, 압축력이 Compression Limit보다 작을 경우 일반적인 트러스와 같이 거동하지만 압축력이 Compression Limit보다 크거나 인장을 받을 경우에는 외력에 저항하지못하는 특징을 가집니다. + +압축력 전담요소가 인장을 받거나 압축력이 Compression Limit을 초과하는 경우 요소의 강성은 0이 되며 이는 수치에러를 유발하는 주요한 요인이 됩니다. 이러한 이유로인장력 전담요소와 압축력 전담요소를 사용할 경우에는 특별한 주의를 요합니다. 다음 그림은 압축력 전담요소를 사용하는 경우 발생되는 문제를 예시한 것입니다. + +![](images/page-092_c4185d227a64bdc5423a425a4aef7c4c022f7955eb4c3c5131868d343ed5733e.jpg) + +
+flowchart + +```mermaid +graph LR + A["Grid with input and output arrows"] --> B["Grid with input and output arrows"] + B --> C["Output with input and output arrows"] + style A fill:#f9f,stroke:#333 + style C fill:#f9f,stroke:#333 +``` +
+ +(a) 압축력 전담요소에 인장력이 작용하여 구조물이 분리되는 경우 + +![](images/page-092_18b5622a939b3548c0f8651270eb808a0d0b9b26b8644e8c914caa295cb90363.jpg) + +
+text_image + +Diagram showing two grid-based configurations with arrows indicating direction, possibly representing a mathematical or logical operation. +
+ +(b) 압축력 전담요소에 인장력이 작용하여도 구조물이 분리되지 않는 경우 +그림 1.4.3 구조물의 분리 여부에 대한 해석의 안정성 + +위의 그림 1.4.3(a)의 경우 왼쪽 구조물은 압축력 전담요소를 통해 하나의 구조물로연결된 것으로 보입니다. 그러나, 압축력 전담요소에 인장이 발생하도록 하중을 재하할 경우 오른쪽 그림과 같이 두 개의 구조물로 분리됨을 알 수 있습니다. 이때 분리된 각각의 구조물이 용인되는(Admissible) 조건을 가진다면 문제가 발생하지 않지만그림과 같이 용인되지 못하는 조건을 가진 경우에는 수치에러가 발생합니다. 따라서이러한 경우 분리되는 두 구조물을 사전에 예상하여 경계조건을 적절히 추가하여 용 + + + +인되는 조건을 만족시켜주어야 합니다. 그림 1.4.3(b)의 경우는 압축력 전담요소에 인장이 발생하더라도 구조물이 두 개의 구조물로 분리되지 않기 때문에 압축력 전담요소의 비선형거동이 적용된 안정적인 해석이 가능합니다. 이와 같은 문제는 압축력 전담요소에 Compression Limit보다 큰 압축력이 발생하거나 인장력 전담요소에 압축력 또는 Tension Limit보다 큰 인장력이 발생하는 경우에도 동일하게 적용됩니다. + +앞에서 거론된 바와 같이 인장력 전담요소나 압축력 전담요소는 비선형 탄성거동을 하는 요소로써 근본적으로 반복해석을 수행하는 비선형 해석이 필요합니다. 비선형 해석은 근사해석으로써 이전단계 해석 결과와 현재 단계 해석 결과의 비율로서 수렴비를 산정하며 이를 특정 기준값과 비교하여 수렴여부를 판단합니다. 수렴비를 산정하기 위해 사용되는 해석 결과로는 증분 변위, 증분 하중, 증분 에너지가 있으며 midas Civil 에서는 증분 변위값을 사용하여 수렴비를 산정하고 이 값이 기준값인 Tolerance값보다 작을 경우 수렴이 된 것으로 판단합니다. 통상적으로 비선형 해석에서 Tolerance는 1/100\~1/1000의 값을 사용하는 것이 일반적이며 midas Civil 은 Default로 1/1000값을 사용하고 있습니다. + +인장력 전담요소 및 압축력 전담요소는 일반선형해석, 선형시공단계해석 및 비선형시공단계해석에서 사용가능하며, Civil 2009 부터는 재료비선형해석 및 기하비선형해석에서도 사용가능합니다. 그러나, 좌굴해석, P-Delta 해석, Pushover 해석, 수화열해석에서는 사용할 수 없습니다. 특히, 고유치해석, 응답스펙트럼해석, 시간이력해석, 이동하중해석, 지점침하해석에서는 일반 트러스로 변환하여 해석을 수행합니다. + +# 4-2 보요소 + +단면의 치수에 비해 길이가 긴 균일단면의 골조부재나 변단면부재(Tapered Member)의 모델링에 주로 사용됩니다. 그리고 보요소는 절점당 6개의 자유도를 가지기 때문에 자유도가 서로 다른 요소끼리 연결될 때 하중전달용 요소로도 사용될 수 있습니다. + +보요소에 재하할 수 있는 하중의 종류는 골조부재에 작용하는 중간 집중하중, 분포하중, 온도구배하중 등이며 프리스트레스하중을 고려할 수 있습니다. + + + +보요소는 인장, 압축, 전단, 굽힘, 비틀림 등의 강성을 가지기 때문에 절점당 6개의자유도를 가질 수 있습니다. 보요소에서 전단변형을 무시하고자 할 경우에는 단면성질을 입력할 때 전단면적을 입력하지 않습니다. + +보요소의 정식화에는 Timoshenko Beam Theory(중립축에 수직한 단면은 변형 후에평면을 유지하지만 중립축에 수직일 필요는 없다.)가 사용되었으며 보의 전단변형을고려할 수 있습니다. 길이에 대한 단면의 폭 또는 높이비가 대략 1/5 보다 커질 경우에는 전단변형에 의한 영향이 커지게 되므로 판형요소를 사용하여 조밀한 요소망이형성되도록 모델링하는 것이 바람직합니다. + +보요소의 단면성질중 비틀림강성(Torsional Resistance)은 단면의 극관성모멘트(PolarMoment of Inertia)와는 다르며(원형 또는 원통형 단면의 경우는 동일) 실험적 방법에의해 결정되기 때문에 비틀림변형의 영향이 클 경우에는 주의해야 합니다. + +보요소(또는 트러스요소)는 선요소(Line Element)로 이상화되어 있기 때문에 단면방향의 크기가 없는 것으로 가정되며, 단면의 성질이 양절점간을 연결하는 중립축에 집중되어 있는 것으로 간주되기 때문에 부재간의 Panel Zone(기둥과 보부재의 접합부위)에 의한 효과나 중립축의 불일치에 따른 영향을 고려하지 않습니다. 따라서 PanelZone에 의한 효과나 중립축의 불일치에 따른 효과를 고려할 경우에는 강성역(BeamEnd Offset) 기능을 이용하거나 기하학적 구속조건을 사용해야 합니다. + +부재의 단면이 비균일단면(Non-prismatic Section)일 경우에는 Tapered Section을 사용하고, 굽은보를 모델링에 반영할 경우에는 가능한 한 여러 개로 분할한 요소를 사용하는 것이 바람직합니다. + +보요소로 모델링할 부재의 양단부가 핀접합(Pin Connection) 또는 슬롯홀(Slot Hole)등에 의해 연결될 경우에는 단부자유도해제조건(Beam End Release)을 이용하여 모델에 반영해야 합니다. + +이때 한 절점의 임의 자유도에 대해 중복하여 단부자유도해제조건을 부여할 경우에는 해당 자유도의 강성이 없어져서 특이성 오류가 발생될 수 있기 때문에 주의해야하며, 불가피한 경우에는 해당 자유도에 미소량의 스프링요소(또는 탄성경계요소)를추가하여야 합니다. + + + +여러개의 보요소가 한 절점에 핀접합으로 만날 경우 특이성 오류를 피하기위해 한 개 요소의 끝단은단부해제조건을 부여하지않고 나머지요소의 단부에대해서는 단부자유도해제조건(Beam End Release)를 부여한다. + +![](images/page-095_b067f8a628c78c08921434fcfa393db3a538c01cc1ac3c4d59d3f054ab276074.jpg) + +![](images/page-095_7d804cf2be5eccf844340311ff4e6f24f2cd147c79972aebe2addd8ee350158c.jpg) + +
+text_image + +Rotational d.o.f. released +Rigid connection +
+ +(c) 여러 개의 보요소가 한 절점에 핀접합으로 연결된 경우 + +![](images/page-095_d8e67f34e0db551f8cda61cb5274109b97dff09c382743776402d9f2d1b42182.jpg) + +
+text_image + +Rigid connection +Beam element +Rigid beam element +for connectivity +All rotational degrees of +freedom and vertical +displacement degree of +freedom released +beam +beam +wall +Plane stress or plate elements +
+ +(d) 절점자유도가 서로 다른 요소끼리 연결된 경우 +그림 1.4.4 단부자유도해제조건(Beam End Release)의 적용 예 + + + +그리고 절점자유도가 서로 다른 요소끼리 접하는 경우에는 강체 보요소(Rigid Beam Element)를 사용하면 효과적입니다. 일반적으로 강체 보요소의 강성은 연산오류를 감안하여 인접한 요소의 탄성계수에 비해 대략 $10^{5} \sim 10^{8}$ 배 정도의 값을 사용하는 것이 타당합니다. + +그림 1.4.4(d)에서 벽체와 보부재가 연결될 경우에 벽체를 평면응력요소 또는 판요소로 모델링하고 보부재를 보요소로 입력하면 평면응력요소(또는 판요소, 입체요소)는 면의 수직방향에 대한 회전강성을 가지지 않기 때문에 보요소를 연결하더라도 보요소와의 회전방향 자유도에 대한 연결성이 확보되지 않고 핀접합한 것과 같은 결과가 됩니다. 이때 연결성의 확보를 위해 강체 보요소를 사용하게 되는데, 강체 보요소의 단부접합조건은 보요소와 연결되는 단부에 대해서는 별도의 해제조건을 부여하지 않고, 반대편 단부에 대해서는 회전자유도와 축방향 변위자유도를 해제하는 방법을 사용합니다. + +# 4-3 평면응력요소 + +인장 또는 압축을 받는 막구조나 평면방향으로만 하중을 전달할 수 있는 구조물의 부재에 사용될 수 있습니다. + +평면응력요소는 각 변에 대해 수직방향으로 압력하중을 받을 수 있습니다. + +평면응력요소는 사각형 또는 삼각형모양을 가지며 평면내의 인장, 압축, 전단강성만을 가집니다. + +사각형요소(4절점 요소)는 요소의 특성상 변위 및 응력에 대해 근접한 결과를 산출하지만, 삼각형요소(3절점 요소)의 경우 변위는 비교적 정확하나 응력은 정확성이 떨어지는 경향이 있습니다. 따라서 정밀한 해석결과가 필요한 부위에서는 삼각형요소의 사용을 피해야 합니다. + +체눈의 크기를 변화시키고자 하는 경우 사각형요소간의 연결을 위해 삼각형요소가 주로 사용됩니다. (그림 1.4.5 참조) + + + +평면응력요소는 회전강성이 없어서 연결절점에서 회전변위에 대한 자유도가 없기 때문에 회전자유도가 없는 요소끼리 접하는 절점에서는 해석과정에서 특이성오류가 발생됩니다. midas Civil에서는 이러한 경우 해당절점의 회전자유도를 자동구속시킴으로써 특이성오류의 발생을 방지하고 있습니다. + +그리고 회전강성을 가진 보요소나 판요소 등과 연결될 때는 강체구속조건(주절점, 종속절점기능)을 이용하거나 강체 보요소 등을 이용하여 요소간의 연결성을 유지시키도록 하여야 합니다. + +요소의 적정 형상비(Aspect Ratio)는 요소의 종류, 기하학적 형상, 구조형태 등에 따라 다릅니다. 그러나 일반적으로는 요소형상비를 가능한 한 1.0에 가깝도록 하고 사각형요소의 경우는 네 모서리각이 90°에 근접하도록 하는 것이 바람직합니다. 이와같은 조건으로 모델링하기 어려울 경우에는 응력의 변화가 심한 부분이나 엄밀해가요구되는 부위만이라도 정사각형에 가깝도록 유지하는 것이 좋습니다. + +또한, 요소의 크기는 상대적으로 작을수록 수렴성이 우수합니다. + +![](images/page-097_3775f151192e3b180ecf5ce645eb249e46d66bfad17e8d18f6a7675773907448.jpg) + +
+text_image + +(a) Crack +(b) 유한요소모델 +Triangular elements are used +for connecting the +quadrilateral +elements +
+ +그림 1.4.5 Crack 모델에서 삼각형/사각형요소를 사용한 예 + + + +# 4-4 평면변형요소 + +평면변형요소는 댐(Dam) 또는 터널(Tunnel)등과 같이 일정한 단면을 유지하면서 길이가 긴 구조물의 해석에 사용될 수 있으며 다른 종류의 요소들과는 혼용될 수 없습니다. + +평면변형요소는 각 변에 대해 수직방향으로 압력하중을 받을 수 있습니다. + +이 요소는 평면변형적 특성을 근거로 하고 있기 때문에 선형정적해석에만 사용할 수 있고 두께방향의 변형율은 존재하지 않으며, 두께방향의 응력성분은 Poisson Effect에 의해 존재하는 것으로 가정합니다. + +평면변형요소는 사각형 또는 삼각형모양을 가질 수 있으며 평면내의 인장, 압축, 전단강성과 두께방향의 인장, 압축강성을 가집니다. + +평면변형요소는 평면응력요소와 마찬가지로 삼각형요소보다는 사각형요소를 사용하는 것이 바람직하며, 요소형상비는 1.0에 가깝도록 하는 것이 좋습니다. + + + +# 4-5 축대칭요소 + +축대칭요소는 형상, 재질, 하중조건 등이 임의 축에 대해 회전대칭조건을 만족하는 구조체(Pipe, Vessel, Tank, Bin 등)의 해석에 사용될 수 있으며, 다른 종류의 요소들과는 흔용될 수 없습니다. + +축대칭요소는 각 변에 대해 수직방향으로 압력하중을 받을 수 있습니다. + +이 요소는 구조물의 축대칭적 특성을 근거로 하고 있기 때문에 선형정적해석에만 사용할 수 있고 원주방향에 대한 변위, 전단변형률, 전단응력은 “0”으로 가정합니다. + +축대칭요소는 평면응력요소와 마찬가지로 삼각형요소보다는 사각형요소를 사용하는 것이 바람직하며, 요소형상비는 1.0에 가깝도록 하는 것이 좋습니다. + + + +# 4-6 판요소 + +평면방향 거동과 면외 hover거동을 일으킬 수 있는 압력용기, 토류벽, 교량의 상판, 구조물의 바닥 및 기초판 등의 모델에 사용할 수 있습니다. + +판요소는 전체좌표계 또는 요소좌표계를 기준으로 임의 방향에 대해 면상에 압력하 중을 받을 수 있습니다. + +판요소는 사각형 또는 삼각형 모양을 가지며 평면내의 압축, 인장, 전단강성과 두께 방향의 휈강성, 전단강성을 가집니다. + +midas Civil에 사용된 판요소의 면외강성은 DKT, DKQ(Discrete Kirchhoff Element)와 DKMT, DKMQ(Discrete Kirchhoff-Mindlin Element)의 두가지 종류로 구분됩니다. DKT, DKQ인 경우에는 얇은 판 이론(Kirchhoff Plate Theory)에 의해 개발된 것이고, DKMT, DKMQ요소는 두꺼운 판 이론(Mindlin-Reissner Plate Theory)에 의해 개발되었으나 적절한 전단변형률장을 가정함으로서 얇은 요소부터 두꺼운 판요소까지 우수한 성능을 나타내고 있는 요소입니다. 판요소의 면내강성은 3각형인 경우는 LST(Linear Strain Triangle)이론을 사용하였고 4각형인 경우에는 비적합모드를 포함하는 등매개 평면응력이론(Isoparametric Plane Stress Formulation with Incompatible Modes)을 사용하여 정식화하였습니다. + +판요소 두께의 입력은 면내강성(Inplane Stiffness)을 계산하기 위한 것과 면외강성(Out of Plane Stiffness)을 계산하기 위한 것으로 구분하여 입력할 수 있습니다. 일반적으로 자중이나 질량의 계산에는 면내강성의 계산을 위한 두께가 사용되지만 면외강성의 계산을 위한 두께만 입력되는 경우에는 면외방향 두께를 사용합니다. + +![](images/page-100_f18f7374967857b56ec5f25de1594f01eceeafd3c32aa17bd82b8bd27769b7c7.jpg) + +
+text_image + +angle between adjacent elements +node +plate element +
+ +그림 1.4.6 구형 또는 원통형 모델에 사용된 판요소의 예 diff --git a/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_011.md b/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_011.md new file mode 100644 index 00000000..09240db6 --- /dev/null +++ b/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_011.md @@ -0,0 +1,208 @@ + + +판요소도 평면응력요소와 마찬가지로 가능한 한 4절점요소를 사용하는 것이 바람직합니다. 그리고 판요소로 곡면구조(곡률을 가진 판)를 모델링할 때는 인접한 요소간의 각도가 10°를 넘지 않도록 해야 하며, 엄밀해가 요구되는 부위에서는 2\~3°를넘지 않도록 하는 것이 바람직합니다. + +응력의 변화가 심한 부분이나 엄밀해가 요구되는 부위에 대해서는 정사각형에 가까운 4절점요소로 세분화하는 것이 바람직합니다. + +이론적으로는 판의 전체적인 거동이 면외 휨이 지배할 경우에는 얇은 판(Thin Plate)을 사용하고, 면외 전단변형의 영향도 고려해야할 경우에는 두꺼운 판(Thick Plate)을사용하는 것이 적합하지만, 위에서 언급했듯이 midas Civil의 Thick Plate는 적절한 전단변형률장을 가정했기 때문에 대부분의 경우에 Thick Plate를 사용해도 우수한 성능을 나타냅니다. 다만 구분해서 사용할 때 그 판단이 어려울 경우, 아주 간단하게 모델의 평면상의 가장 긴 쪽 지간 길이와 두께의 비율이 10일 때를 기준으로 두께가얇은 경우에는 Thin을, 두꺼운 경우에는 Thick을 사용하는 방법도 있습니다. + + + +# 4-7 입체요소 + +3차원 입체구조물의 모델링에 사용되며, 삼각뿔, 삼각기둥, 육면체 등의 입체모양을 가집니다. + +압력하중은 요소의 각 면에 수직방향이나 전체좌표계 X, Y, Z 방향으로 입력이 가능합니다. + +육면체요소(8절점 요소)는 요소의 특성상, 변위 및 응력에 대해 근접한 결과를 산출하지만, 삼각뿔요소(4절점 요소) 또는 삼각기둥요소(6절점 요소)의 경우에 변위는 비교적 정확하나 응력은 정확성이 떨어지기 때문에 정밀한 해석결과가 필요한 부위에서는 사용을 피해야 합니다. + +체눈의 크기를 변화시키고자 하는 경우 육면체 요소간의 연결을 위해 주로 사용됩니다. + +입체요소는 회전강성이 없어서 연결절점에서 회전변위에 대한 자유도가 없기 때문에 기타 회전자유도가 없는 요소끼리 접하는 절점에서는 해석과정에서 특이성오류가 발생됩니다. midas Civil에서는 이러한 경우 해당절점의 회전자유도를 자동구속시킴으로써 특이성오류의 발생을 방지하고 있습니다. + +그리고 회전강성을 가진 보요소나 판요소 등과 연결될 때는 강체구속조건(주절점, 종속절점기능)을 이용하거나 강체 보요소 등을 이용하여 요소간의 연결성을 유지시키도록 하여야 합니다. + +요소의 적정 형상비(Aspect Ratio)는 요소의 종류, 기하학적 형상, 구조형태 등에 따라 다릅니다. 그러나 일반적으로는 요소형상비를 가능한 한 1.0에 가깝도록 하고, 육면체요소의 경우는 8개의 모서리각이 90°에 근접하도록 하는 것이 바람직합니다. 이러한 조건으로 모델링하기 어려울 경우에는 응력의 변화가 심한 부분이나 엄밀해가 요구되는 부위만이라도 정육면체에 가깝도록 유지하는 것이 좋습니다. + +또한, 요소의 크기는 상대적으로 작을수록 수렴성이 우수합니다. + + + +# 4-8 직교이방성재질 입력시 주요 고려사항 + +물체의 물리적 성질이 방향에 따라 달라지지 않는 경우를 등방성(Isotropic) 이라 하고, 방향에 따라 성질이 달라지는 경우를 이방성(Anisotropic) 이라고 합니다. 이방성 중 서로 직교하는 세 면에 관해서 대칭인 성질을 가질때 직교이방성(Orthotropic) 이라고 합니다. 예를 들어, 섬유보강 플라스틱(FRP)과 같은 복합 재료는 직교이방성 재질입니다. + +직교이방성 재질은 서로 직교하는 세 방향에 대한 탄성계수와 선열팽창계수, 그리고 직교하는 세 면에 대한 전단탄성계수와 포와송비를 물성치로 가집니다. 이러한 물성치는 해당 재질에 대한 실험값 또는 제조사에서 제공하는 값을 사용합니다. + +직교이방성 재질 사용시 주의점은 다음과 같습니다. + +- 재질특성은 요소의 Local 축을 기준으로 반영이 됩니다. 따라서, 요소의 Local 축과 직교이방성 재질의 방향과의 관계를 고려하여 모델링하는 것이 필요합니다. +▪ 요소의 종류에 따라 적용되는 탄성계수의 성분은 다음과 같습니다. + +1차원 요소의 경우(트러스, 보): Local-x + +2차원 요소의 경우(판, 평면) : Local-x, Local-y + +3차원 요소의 경우(입체) : Local-x, Local-y, Local-z + +\- 이방성 재질에 입력되는 탄성계수와 프아송비는 아래와 같은 조건을 만족하여야 합니다. + +$$ +\frac {\nu_ {x y}}{E _ {x}} = \frac {\nu_ {y x}}{E _ {y}}, \quad \frac {\nu_ {x z}}{E _ {x}} = \frac {\nu_ {z x}}{E _ {z}}, \quad \frac {\nu_ {y z}}{E _ {y}} = \frac {\nu_ {z y}}{E _ {z}} +$$ + +직교 이방성 재질은 다음과 같은 경우에 주로 사용합니다. + +▪ 철근 배근으로 요소의 Local-x와 Local-y의 강성이 다른 벽체 +▪ 보강판 등으로 요소의 Local-x와 Local-y의 강성이 달라진 바닥판 + + + +# Chapter 5. 요소의 강성 데이터 + +요소의 강성을 계산하는데는 재질데이터와 단면(또는 두께)데이터가 사용됩니다. + +재질데이터는 Properties탭>Material그룹>Material Properties 기능을 통해서 입력되며, 단면데이터는 Properties탭>Section그룹>Section Properties 또는 Thickness 기능을 통해서 입력됩니다. + +요소종류별 요소강성데이터를 위해 필요한 명령어는 표 1.5.1과 같습니다. + +
요소구분재질단면 / 두께비고
트러스요소MaterialSection트러스요소의 경우는 해석을 하기 위해서 단면적(Cross Sectional Area)만 필요하지만 설계작업 또는 부재형상을 화면상에서 표현하기 위해 단면 형상을 입력해야 한다.
인장력전담요소MaterialSection
압축력전담요소MaterialSection
보요소MaterialSection보요소가 SRC(철골철근콘크리트)기둥으로 사용될 경우 강재와 콘크리트가 공존하는데에 따른 등가강성의 계산은 프로그램 내부에서 자동적으로 이루어진다.
평면응력요소MaterialThickness
판요소MaterialThickness
평면변형요소Material-평면변형요소와 축대칭요소의 경우는 각각 단위폭(1.0)과 단위각도(1.0 rad)의 두께가 프로그램 내부에서 자동적으로 주어지기 때문에 별도로 단면데이터를 입력할 필요가 없다.
축대칭요소Material-
입체요소Material-입체요소는 요소를 구성하는 모서리 절점들에 의해 요소의 크기가 결정되기 때문에 별도로 단면데이터를 입력할 필요가 없다.
+ +표 1.5.1 요소종류별 요소강성데이터 명령어 + + + +선요소의 단면성질에 대한 정의 및 계산방법은 다음과 같습니다. + +선요소(트러스요소, 보요소... 등)의 단면성질을 직접 계산하여 입력할 때는 각각의단면성질이 구조적 거동에 미치는 영향을 충분히 이해하고 입력해야 합니다. + +또한, 부재의 부식이나 마모 등으로 부재단면의 감소요인이 있을 경우에는 이를 반영하여 단면성질을 계산하여야 합니다. + +midas Civil에서는 아래의 3가지 방법을 이용하여 단면성질을 입력할 수 있도록 되어 있습니다. + +1. 단면의 주요치수만 입력하여 midas Civil 내부에서 자동연산하는 방법 +2. 모든 단면성질을 사용자가 직접 계산하여 입력하는 방법 +3. KS, JIS, AISC 규준에 등록된 단면의 경우, 데이터베이스에서 공칭명을 선택하여 입력하는 방법 + +일반단면(Prismatic Section), 비균일 단면(Tapered Section), 합성단면(CombinedSection) 그리고 SRC단면인 경우에는 단면특성을 각각 고유의 단면번호를 사용하여입력할 수 있으나 시공단면(Construction Section)인 경우에는 미리 입력된 2개의단면특성을 사용하여 입력하게 됩니다. 시공단면은 철골과 콘크리트의 합성형태로 구성되어 구조물의 시공단계(큰크리트의 타설 및 양생)에 따라 다른 단면특성을 갖는경우에 사용됩니다. + +다음은 midas Civil 내부에서 단면성질을 계산하는데 사용된 방법과 각 단면성질을계산할 때 고려해야 하는 일반적인 사항을 서술합니다. + + + +# 5-1 단면적 (Area : Cross Sectional Area) + +단면적(Cross Sectional Area)은 부재가 인장 또는 압축력(Axial Force)을 받는 경우 이에 저항하는 강성(Axial Stiffness)을 계산하거나 부재에 발생한 응력을 계산하는데 사용되며 계산방법은 그림 1.5.1과 같습니다. + +midas Civil 내부에서 단면적을 계산하거나 데이터베이스로부터 입력되는 경우에는 접합부의 볼트접합구멍 또는 리벳접합구멍 등에 의한 단면적의 감소요인은 고려하지 않으므로 필요시 전술한 “단면성질 입력방법 2” 를 사용하여 사용자의 판단에 따라 조정된 단면적을 입력해야 합니다. + +![](images/page-106_ce2ce8eecb3ac84056764bc6d89c2cb6fe04b3015922608eb351bd4894fee2dd.jpg) + +
+text_image + +300 +15 +A1 +A2 +10 +600 +A3 +320 +12 +
+ +$$ +\begin{array}{l} \text { Area } = \int d A = A 1 + A 2 + A 3 \\ = (3 0 0 \times 1 5) + (5 7 3 \times 1 0) + (3 2 0 \times 1 2) \\ = 1 4 0 7 0 \\ \end{array} +$$ + +그림 1.5.1 단면적의 계산 예 + + + +# 5-2 유효전단면적 ( $A_{sy}$ , $A_{sz}$ : Effective Shear Area) + +전단력에 대한 유효전단면적(Effective Shear Area)은 부재단면의 요소좌표계 y축 또는 z축 방향으로 작용하는 전단력(Shear Force)에 저항하는 강성(Shear Stiffness)의 계산에 필요합니다. + +만약 유효전단면적을 입력하지 않았을 경우 해당 방향의 전단변형이 무시됩니다. + +midas Civil내부에서 단면성질을 계산하거나 데이터베이스로부터 입력되는 경우에는 해당 전단강성성분이 자동고려되며 계산방법은 표 1.5.2와 같습니다. + +$A_{sy}$ : 요소좌표계 y축 방향으로 작용하는 전단력에 저항하는 유효전단면적 + +$A_{sz}$ : 요소좌표계 z축 방향으로 작용하는 전단력에 저항하는 유효전단면적 + + + +
Section ShapeEffective Shear AreaSection ShapeEffective Shear Area
1. Angle $A_{sy} = \frac{5}{6} B \times t_f$ $A_{sz} = \frac{5}{6} H \times t_w$ 2. Channel $A_{sy} = \frac{5}{6} (2 \times B \times t_f)$ $A_{sz} = H \times t_w$
3. I-Section $A_{sy} = \frac{5}{6} (2 \times B \times t_f)$ $A_{sz} = H \times t_w$ 4. Tee $A_{sy} = \frac{5}{6} (B \times t_f)$ $A_{sz} = H \times t_w$
5. Thin Walled Tube $A_{sy} = 2 \times B \times t_f$ $A_{sz} = 2 \times H \times t_w$ 6. Thin Walled Pipe $A_{sy} = \pi \times r \times t_w$ $A_{sz} = \pi \times r \times t_w$
7. Solid Round Bar $A_{sy} = 0.9 \pi r^2$ $A_{sz} = 0.9 \pi r^2$ 8. Solid Rectangular Bar $A_{sy} = \frac{5}{6} BH$ $A_{sz} = \frac{5}{6} BH$
+ +표 1.5.2 단면형상별 유효전단면적 + + + +# 5-3 비틀림강성 ( $I_{xx}$ : Torsional Resistance) + +비틀림강성은 비틀림모멘트에 저항하는 강성으로 식 (1)과 같이 표현됩니다. + +$$ +I _ {x x} = \frac {T}{\theta} \tag {1} +$$ + +여기서 $I_{xx}$ : 비틀림강성 (Torsional Resistance) + +T : 비틀림모멘트 (Torsional Moment or Torque) + +θ : 비틀림각도 (Angle of Twist) + +비틀림강성은 상기 식에서와 같이 비틀림에 저항하는 강성이며, 비틀림에 의한 전단 응력을 결정하는 극관성 단면 2차 모멘트(Polar Moment of Inertia)와는 다릅니다. (단, 원형단면 또는 두께가 두꺼운 원통단면의 경우는 비틀림모멘트와 극관성 단면 2차 모멘트가 일치합니다.) + +그리고 단면의 형태가 개방형단면(Open Section)인지 또는 밀폐형단면(Closed Section)인지에 따라 비틀림강성의 계산방법이 다르고, 단면의 두께가 얇은지 또는 두꺼운지에 따라서도 계산방법이 다르기 때문에 모든 종류의 단면에 공통적으로 적용할 수 있는 일반식은 없습니다. + +개방형단면의 비틀림강도 계산은 개방형단면을 여러 개의 직사각형 단면으로 분할하여 식 (2)를 이용하여 계산하고, 그 계산 결과치를 합산함으로써 근사적으로 구할 수 있습니다. + +$$ +I _ {x x} = \sum i _ {x x} +$$ + +$$ +i _ {x x} = a b ^ {3} \left[ \frac {1 6}{3} - 3. 3 6 \frac {b}{a} \left(1 - \frac {b ^ {4}}{1 2 a ^ {4}}\right) \right] \text { 단, } a \geq b \tag {2} +$$ + +여기서 $i_{xx}$ : 분할단면(직사각형)의 비틀림강성 + +2a : 분할단면의 긴 변의 길이 + +2b : 분할단면의 짧은 변의 길이 + + + +그리고 얇은 튜브형태의 밀폐형 단면에 대한 비틀림강성의 계산식은 식 (3)과 같습니다. (그림 1.5.2 참조) + +$$ +I _ {x x} = \frac {4 A ^ {2}}{\int d _ {s} / t} \tag {3} +$$ + +여기서 A : 튜브의 단면적 + +$d _ { S }$ + +t ： + +또한 교량의 박스형 단면과 같이 두꺼운 튜브형태의 밀폐형 단면에 대한 비틀림강성은 상기의 식(1)과 (3)을 합산함으로써 구할 수 있습니다. + +![](images/page-110_278d58a31f638767250a99728cb7597939cb709051f22860f984ce157234c2a3.jpg) + +
+text_image + +d_{s1} +t_s +
+ +Torsional resistance : xxI $I _ { _ { x x } } = \frac { 4 A ^ { 2 } } { \displaystyle \int d _ { s } / t _ { s } }$ + +Shear stress at a given point : T  $\tau _ { T } = \frac { T } { 2 A t _ { s } }$ 2 sAt + +$t _ { s }$ : Thickness of tube at a given point + +그림 1.5.2 얇은 튜브형 밀폐단면의 비틀림강성 및 전단응력 diff --git a/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_012.md b/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_012.md new file mode 100644 index 00000000..9e20b070 --- /dev/null +++ b/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_012.md @@ -0,0 +1,293 @@ + + +
Section ShapeTorsional ResistanceSection ShapeTorsional Resistance
1. Solid Round Bar $I_{xx} = \frac{1}{2} \pi r^{2}$ 2. Solid Square Bar $I_{xx} = 2.25a^{4}$
3. Solid Rectangular Bar $I_{xx} = ab^{3} \left[ \frac{16}{3} - 3.36 \frac{b}{a} \left( I - \frac{b^{4}}{12a^{4}} \right) \right]$ (where, $a \geq b$ )
+ +표 1.5.3 Solid Section의 비틀림강성 + +
Section ShapeTorsional ResistanceSection ShapeTorsional Resistance
1. Rectangular Tube (Box) $I_{xx} = \frac{2(b \times h)^2}{\left(\frac{b}{t_f} + \frac{h}{t_w}\right)}$ 2. Circular Tube(Pipe)[IMAGE] $I_{xx} = \frac{1}{2} \pi \left[ \left( \frac{D_o}{2} \right)^4 - \left( \frac{D_i}{2} \right)^4 \right]$
+ +표 1.5.4 두께가 얇은 폐쇄형 단면의 비틀림 강성 + + + +
Section ShapeTorsional Resistance
1. Angle $I_{xx} = I_1 + I_2 + \alpha D^4$ $I_1 = ab^3 \left[ \frac{1}{3} - 0.21 \frac{b}{a} \left( 1 - \frac{b^4}{12a^4} \right) \right]$ $I_2 = cd^3 \left[ \frac{1}{3} - 0.105 \frac{d}{c} \left( 1 - \frac{d^4}{192c^4} \right) \right]$ $\alpha = \frac{d}{b} \left( 0.07 + 0.076 \frac{r}{b} \right)$ $D = 2 \left[ d + b + 3r - \sqrt{2(2r + b)(2r + d)} \right]$ (where, $b < 2(d + r)$ )
2. TeeIF $b < d$ : $t = b$ , $t_1 = d$ IF $b > d$ : $t = d$ , $t_1 = b$ $I_{xx} = I_1 + I_2 + \alpha D^4$ $I_1 = ab^3 \left[ \frac{1}{3} - 0.21 \frac{b}{a} \left( 1 - \frac{b^4}{12a^4} \right) \right]$ $I_2 = cd^3 \left[ \frac{1}{3} -0.105 \frac{d}{c} \left( 1 - \frac{d^4}{192c^4} \right) \right]$ $\alpha = \frac{t}{t_1} \left( 0.15 + 0.10 \frac{r}{b} \right)$ $D = \frac{(b + r)^2 + rd + \frac{d^2}{4}}{(2r + b)}$ (where, $d < 2(b + r)$ )
3. ChainSum of torsional stiffnesses of 2 angles
4. I-Section $I_{xx} = 2I_1 + I_2 + 2\alpha D^4$ $I_1 = ab^3 \left[ \frac{1}{3} - 0.21 \frac{b}{a} \left( 1 - \frac{b^4}{12a^4} \right) \right]$ $I_2 = \frac{1}{3} cd^3$ $\alpha = \frac{t}{t_1} \left( 0.15 + 0.10 \frac{r}{b} \right)$ $D = \frac{(b + r)^2 + rd + \frac{d^2}{4}}{(2r + b)}$ (where, $d < 2(b + r)$ )
+ +표 1.5.5 두께가 두꺼운 개방형 단면의 비틀림강성 + + + +
Section ShapeTorsional Resistance
1. Angle $I_{xx} = \frac{1}{3} \left( h \times t_{w}^{3} + b \times t_{f}^{3} \right)$
2. Channel $I_{xx} = \frac{1}{3} \left( h \times t_{w}^{3} + 2 \times b \times t_{f}^{3} \right)$
3. I-Section $I_{xx} = \frac{1}{3} \left( h \times t_{w}^{3} + 2 \times b \times t_{f}^{3} \right)$
4. Tee $I_{xx} = \frac{1}{3} \left( h \times t_{w}^{3} + b \times t_{f}^{3} \right)$
5. I-Section $I_{xx} = \frac{1}{3} \left( h \times t_{w}^{3} + b_{1} \times t_{f1}^{3} + b_{2} \times t_{f2}^{3} \right)$
+ +표 1.5.6 두께가 얇은 개방형 단면의 비틀림강성 + + + +2개 이상의 형강을 조합하여 하나의 단면으로 만들 경우, 조합하는 형태에 따라 폐쇄형 단면과 개방형 단면이 동시에 생길 수 있습니다. 이 경우 비틀림강성의 계산은 폐쇄형 단면 부분과 개방형 단면 부분으로 나누어 각각 계산한 다음 그 값을 더하는 방법을 사용합니다. + +예를 들면, 이종 H형 단면(Double H-section)의 경우 1.5.3(a)와 같이 단면의 중앙에는 폐쇄형 단면이 형성되고, 외곽 플랜지들은 개방형 단면이 됩니다. + +-폐쇄형 단면 부분(빗금친 부분)의 비틀림강성 + +$$ +I _ {C} = \frac {2 (b _ {1} \times h _ {1}) ^ {2}}{\left(\frac {b _ {1}}{t _ {f}} + \frac {h _ {1}}{t _ {w}}\right)} \tag {4} +$$ + +-개방형 단면 부분(돌출된 플랜지 부분)의 비틀림강성 + +$$ +I _ {o} = 2 \left[ \frac {1}{3} \left(2 b - b _ {1} - t _ {w}\right) \times t _ {w} ^ {3} \right] \tag {5} +$$ + +\- 전체단면에 대한 비틀림강성 + +$$ +I _ {x x} = I _ {C} + I _ {O} \tag {6} +$$ + +H형 단면을 2개의 Flat Bar로 보강할 경우에는 그림 1.5.3(b)와 같이 폐쇄형 단면이 2개 이상 생길 수 있으며 이 때의 단면 비틀림강성은 다음과 같이 계산합니다. + +플랜지 끝단부의 개방형 단면에 대한 비틀림강성이 전체단면의 비틀림강성에 비해 무시할 정도로 작은 값일 경우, H형 단면의 상 하 플랜지와 보강재로 사용된 2개의 Flat Bar에 의해 형성되는 최외곽의 폐쇄 단면에 대하여 비틀림강성을 계산하면 다음과 같습니다. + +$$ +I _ {x x} = \frac {2 (b _ {1} \times h _ {1}) ^ {2}}{\left(\frac {b _ {1}}{t _ {f}} + \frac {h _ {1}}{t _ {w}}\right)} \tag {7} +$$ + + + +그리고 전체단면을 구성하는 요소중에서 개방형 단면의 비틀림강성이 무시할 수 없을 정도로 큰 값일 경우 개방형 단면에 대한 비틀림강성을 계산하여 더합니다. + +![](images/page-115_ee2e8069d47db4fb988472e3428ec01da7b12bf8d55068f9c4183ab452b3b655.jpg) + +
+text_image + +z +b +t_f +h_l +y +t_w +h +b_l +
+ +(a) 폐쇄형과 개방형 단면이 함께 존재하는 경우 + +![](images/page-115_dd2cc8459ea4fbabd7fa2447abc1c912fd7401b590982843121d9932a8eae77f.jpg) + +
+text_image + +z +h₁ +tₛ +tᵥ +y +h +b₁ +b +
+ +(b) 폐쇄형 단면이 2개 이상 존재하는 경우 +그림 1.5.3 두개 이상의 형강을 조합한 단면의 비틀림강성 + + + +# 5-4 단면2차모멘트 ( $I_{yy}$ , $I_{zz}$ : Area Moment of Inertia) + +단면2차모멘트(Area Moment of Inertia)는 힘모멘트(Bending Moment)에 저항하는 강성(Flexural Stiffness)을 계산하는데 사용되며, 해당 단면의 도심축에서 다음의 식에 따라 계산됩니다.- 요소좌표계 y축에 대한 단면2차모멘트 + +$$ +I _ {y y} = \int z ^ {2} d A \tag {8} +$$ + +\- 요소좌표계 z축에 대한 단면2차모멘트 + +$$ +I _ {z z} = \int y ^ {2} d A \tag {9} +$$ + +![](images/page-116_5b4208ca92138cab56f6c44cb51d2e142d388793fcc01c987d9caa042b53315b.jpg) + +
+text_image + +8 +z +③ +② +Centroid +y +10 +3 +17 +Neutral axis +z' +2 +4 +y' +① +y +Reference point +for the centroid position +calculation +y +10 +
+
123Total
$b$ 1028-
$h$ 4103-
$A_{i}$ 40202484
$\overline{z}_{i}$ 2915.5-
$Q_{yi}$ 8018037263.2
$\overline{y}_{i}$ 555-
$Q_{zi}$ 200100120420
+ +$A_{i}$ : area + +$\overline{z}_{i}$ : distance from the reference point to the centroid of the section element in the $z'$ -axis direction + +$\overline{y}_{i}$ : distance from the reference point to the centroid of the section element in the $y'$ -axis direction + +$Q_{yi}$ : first moment of area relative to the reference point in the $y'$ -axis direction + +$Q_{zi}$ : first moment of area relative to the reference point in the $z'$ -axis direction + + + +\- 중립축 위치 계산 ( $\bar{Z}$ , $\bar{Y}$ ) + +$$ +\overline {{{Y}}} = \frac {\int \overline {{{z}}} d A}{A r e a} = \frac {Q _ {y}}{A r e a} = \frac {6 3 2}{8 4} = 7. 5 2 3 8 +$$ + +$$ +\overline {{{Z}}} = \frac {\int \overline {{{y}}} d A}{A r e a} = \frac {Q _ {z}}{A r e a} = \frac {4 2 0}{8 4} = 5. 0 0 0 0 +$$ + +\- 단면 2차 모멘트 계산 (Iyy, Izz) +
Section element $A_i$ $\overline{Z} - \overline{z}_i$ $I_{y1}$ $I_{y2}$ $I_{yy}$ $\overline{Y} - \overline{y}_i$ $I_{z1}$ $I_{z2}$ $I_{zz}$
1405.53281224.553.31277.800333.3333.3
2201.467243.1166.7209.8006.76.7
3247.97621526.918.01544.900128.0128.0
Total2794.5238.03032.50468.0468.0
$$ +I _ {y 1} = A _ {i} \times (\overline {{{Z}}} - \overline {{{z}}} _ {i}) ^ {2}, \quad I _ {y 2} = \frac {b h ^ {3}}{1 2}, \quad I _ {y y} = I _ {y 1} + I _ {y 2} +$$ + +$$ +I _ {z 1} = A _ {i} \times (\overline {{{Y}}} - \overline {{{y}}} _ {i}) ^ {2}, \quad I _ {z 2} = \frac {h b ^ {3}}{1 2}, \quad I _ {z z} = I _ {z 1} + I _ {z 2} +$$ + + + +# 5-5 단면상승모멘트 ( $I_{yz}$ : Area Product Moment of Inertia) + +단면상승모멘트(Area Product Moment of Inertia)는 비대칭단면의 응력성분을 계산하는데 사용되며 그 정의는 식 (10)과 같습니다. + +$$ +I _ {y z} = \int y \cdot z d A \tag {10} +$$ + +H, Pipe, Box, Channel, Tee형 단면과 같이 요소좌표계 y, z축 어느 1개의 축에 대해서 대칭인 경우에는 $I_{yz}=0$ 이 되며, Angle형 단면과 같이 어느 1개 축에 대해서도 대칭이 아닌 경우에는 $I_{yz} \neq 0$ 이므로 응력성분 계산시 고려하여야 합니다. + +Angle형 단면의 단면상승모멘트의 계산방법은 아래와 같습니다. + +![](images/page-118_7f8cdce052cea00faf202e5564f654d8bacf30745781a0f30db79442bce05b52.jpg) + +
+text_image + +z +B +① +tf +centroid +H +② +y +Z +tw +Y +
+ +$$ +\begin{array}{l} I _ {y z} = \sum A _ {i} \times e _ {y i} \times e _ {z j} \\ = (B \times t _ {f}) \times (B / 2 - \bar {Y}) \times \{(H - t _ {f} / 2) - \bar {Z} \} \\ + \left\{\left(H - t _ {f}\right) \times t _ {w} \right\} \times \left\{t _ {w} / 2 - \bar {Y}\right) \times \left\{\left(H - t _ {f} / 2\right) - \bar {Z} \right\} \\ \end{array} +$$
Section Element $A_i$ $e_{yi}$ $e_{zi}$
1 $B \times t_f$ $B/2-\overline{Y}$ $(H-t_f/2)-\overline{Z}$
2 $(H-t_f) \times t_w$ $t_w/2-\overline{Y}$ $(H-t_f/2)-\overline{Z}$
+ + + +![](images/page-119_0ca1147aff6b561a0aa1e637f52bdb2cf844c501614d7688987c0b381f75e0d7.jpg) + +
+text_image + +z +m +φ +x +y +φ +n +z +x +y +M_y +V_y +M +M_z +V_z +V +
+ +그림 1.5.4 비대칭형 단면에서의 휈응력 분포도 + +중립축(Neutral Axis)은 휨모멘트에 의한 부재내 휨응력이 '0(Zero)' 이 되는 위치를 통과하는 축을 말하며, 그림 1.5.4의 우측 그림에서와 같이 n-축이 중립축이 됩니다. m-축은 n-축에 대하여 수직을 이루는 축입니다. + +중립축에서는 휨모멘트에 의한 휨응력이 '0' 이므로 식 (11)로부터 중립축 방향을 구할 수 있습니다. + +$$ +\left(M _ {y} \times I _ {z z} + M _ {z} \times I _ {y z}\right) \times z - \left(M _ {z} \times I _ {y y} + M _ {y} \times I _ {y z}\right) = 0 +$$ + +$$ +\tan \phi = \frac {y}{z} = \frac {M _ {y} \times I _ {z z} + M _ {z} \times I _ {y z}}{M _ {z} \times I _ {y y} + M _ {y} \times I _ {y z}} \tag {11} +$$ + +힘모멘트에 의한 단면의 힜응력을 계산하는데 적용되는 일반식은 식 (12)와 같습니다. + +$$ +f _ {b} = \frac {M _ {y} - M _ {z} \left(I _ {y z} / I _ {z z}\right)}{I _ {y y} - \left(I _ {y z} ^ {2} / I _ {z z}\right)} \cdot z + \frac {M _ {z} - M _ {y} \left(I _ {y z} / I _ {y y}\right)}{I _ {z z} - \left(I _ {y z} ^ {2} / I _ {y y}\right)} \cdot y \tag {12} +$$ + + + +만일 H형 단면일 경우에는 $I_{yz}=0$ 이 되므로, + +$$ +f _ {b} = \frac {M _ {y}}{I _ {y y}} \cdot z + \frac {M _ {z}}{I _ {z z}} \cdot y = f _ {b y} + f _ {b x} \tag {13} +$$ + +여기서, $l_{yy}$ : 요소좌표계 y축에 대한 단면2차모멘트 + +$I_{zz}$ : 요소좌표계 z축에 대한 단면2차모멘트 + +$I_{yz}$ : 단면상승모멘트 + +y : 요소단면의 중립축으로부터 힘응력을 계산하고자 하는 위치까지의 요소좌 표계 y축 방향의 거리 + +z : 요소단면의 중립축으로부터 휈응력을 계산하고자 하는 위치까지의 요소좌 표계 z축 방향의 거리 + +$M_{y}$ : 요소좌표계 y축에 대한 힘모멘트 + +$M_{z}$ : 요소좌표계 z축에 대한 힘모멘트 + +요소좌표계 y축 및 z축 방향으로 작용하는 전단력에 대한 전단응력을 계산하는데 적용되는 일반식은 식 (14), (15)와 같습니다. + +$$ +\tau_ {y} = \frac {V _ {y}}{b _ {z} \times \left(I _ {y y} \cdot I _ {z z} - I _ {y z} ^ {2}\right)} \times \left(I _ {y y} \cdot Q _ {z} - I _ {y z} \cdot Q _ {y}\right) = \left(\frac {I _ {y y} \cdot Q _ {z} - I _ {y z} \cdot Q _ {y}}{I _ {y y} \cdot I _ {z z} - I _ {y z} ^ {2}}\right) \times \left(\frac {V _ {y}}{b _ {z}}\right) \tag {14} +$$ + +$$ +\tau_ {z} = \frac {V _ {z}}{b _ {y} \times \left(I _ {y y} \cdot I _ {z z} - I _ {y z} ^ {2}\right)} \times \left(I _ {z z} \cdot Q _ {y} - I _ {y z} \cdot Q _ {z}\right) = \left(\frac {I _ {z z} \cdot Q _ {y} - I _ {y z} \cdot Q _ {z}}{I _ {y y} \cdot I _ {z z} - I _ {y z} ^ {2}}\right) \times \left(\frac {V _ {z}}{b _ {y}}\right) \tag {15} +$$ + +여기서,Vy : 요소좌표계 y축 방향으로 작용하는 전단력 + +Vz : 요소좌표계 z축 방향으로 작용하는 전단력 + +Qy : 요소좌표계 y축에 대한 단면 1차모멘트 + +Qz : 요소좌표계 z축에 대한 단면 1차모멘트 + +by : 전단응력을 계산하는 위치에서의 요소좌표계 z축과 직각을 이루는 단면의 두께 + +bz : 전단응력을 계산하는 위치에서의 요소좌표계 y축과 직각을 이루는 단면의 두께 diff --git a/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_013.md b/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_013.md new file mode 100644 index 00000000..720936d4 --- /dev/null +++ b/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_013.md @@ -0,0 +1,294 @@ + + +# 5-6 단면1차모멘트 ( $Q_{y}$ , $Q_{z}$ : First Moment of Area) + +단면 1차모멘트(First Moment of Area)는 단면의 임의 위치에서의 전단응력을 계산하는데 사용되며 아래와 같이 계산합니다. + +$$ +Q _ {y} = \int z d A \tag {16} +$$ + +$$ +Q _ {z} = \int y d A \tag {17} +$$ + +단면이 y, z 양축 중에서 어느 한 축에 대하여 대칭일 경우, 임의 위치에서의 전단응력은 다음과 같이 계산합니다. + +$$ +\tau_ {y} = \frac {V _ {y} \cdot Q _ {z}}{I _ {z z} \cdot b _ {z}} \tag {18} +$$ + +$$ +\tau_ {z} = \frac {V _ {z} \cdot Q _ {y}}{I _ {y y} \cdot b _ {y}} \tag {19} +$$ + +여기서, $V_{y}$ : 요소좌표계 y축 방향으로 작용하는 전단력 + +$V_{z}$ : 요소좌표계 z축 방향으로 작용하는 전단력 + +$I_{yy}$ : 요소좌표계 y축에 대한 단면2차모멘트 + +$I_{zz}$ : 요소좌표계 z축에 대한 단면2차모멘트 + +$b_{y}$ : 전단응력을 계산하고자 하는 위치에서의 요소좌표계 z축과 직각을 이루는 단면의 두께 + +$b_{z}$ : 전단응력을 계산하고자 하는 위치에서의 요소좌표계 y축과 직각을 이루는 단면의 두께 + + + +# 5-7 전단계수 ( $Q_{yb}$ , $Q_{zb}$ : Shear Factors of Shear Stress due to Bending) + +전단계수는 힘모멘트에 의한 전단응력을 계산하는데 사용되며 부재단면중 전단응력을 계산하고자 하는 위치에서의 단면1차모멘트를 동일한 위치의 단면두께로 나눈 값입니다. + +$$ +\tau_ {y} = \frac {V _ {y} \cdot Q _ {z}}{I _ {z z} \cdot b _ {z}} = \frac {V _ {y}}{I _ {z z}} \left(\frac {Q _ {z}}{b _ {z}}\right) = \frac {V _ {y}}{I _ {z z}} Q _ {z b}, Q _ {z b} = \frac {Q _ {z}}{b _ {z}} \tag {20} +$$ + +$$ +\tau_ {z} = \frac {V _ {z} \cdot Q _ {y}}{I _ {y y} \cdot b _ {y}} = \frac {V _ {z}}{I _ {y y}} \left(\frac {Q _ {y}}{b _ {y}}\right) = \frac {V _ {z}}{I _ {y y}} Q _ {y b}, Q _ {y b} = \frac {Q _ {y}}{b _ {y}} \tag {21} +$$ + +![](images/page-122_f129298830a1b2bed20bee1d1da5d671d76fa266ad6f56904d73a2c652150cd1.jpg) + +
+text_image + +point of shear stress calculation +τz = VzQy / Iyyby = Vz / IyyQyb +Qy = ∫zdA = (B×tf)×z̄ +by = tw +Qyb = {(B·tf)×z̄} / tw +
+ +그림 1.5.5 전단계수의 계산 예 + + + +# 5-8 합성단면의 강성계산 + +midas Civil에서 철골-철근콘크리트 합성부재의 강성은 콘크리트 단면(철근의 단면은 콘크리트단면에 포함됨)과 철골단면이 구조적으로 완전 합성된 것으로 가정하여 등가환산단면성질(Equivalent Sectional Properties) 형태로 고려됩니다. + +등가환산 단면성질의 계산에서 강재의 탄성계수(Es)와 콘크리트의 탄성계수(Ec)는 철골-철근콘크리트규준(SSRC79(Structural Stability Research Council, 1979, USA))에 명기된 수치를 사용하되, Ec값은 Eurocode 4에 따라 20% 감소한 값을 사용합니다. + +\- 등가환산 단면적 + +$$ +\text { Area } _ {e q} = A _ {s t l} + \frac {0 . 8 E _ {c}}{E _ {s}} A _ {c o n} = A _ {s t l} + 0. 8 \frac {A _ {c o n}}{R E N} +$$ + +\- 등가환산 유효전단면적 + +$$ +A s _ {e q} = A s _ {s t l} + \frac {0 . 8 E _ {c}}{E _ {s}} A s _ {c o n} = A s _ {s t l} + \frac {A s _ {c o n}}{R E N} +$$ + +\- 등가환산 단면2차모멘트 + +$$ +I _ {e q} = I _ {s t l} + \frac {0 . 8 E _ {c}}{E _ {s}} I _ {c o n} = I _ {s t l} + 0. 8 \frac {I _ {c o n}}{R E N} +$$ + +여기서, $A_{stl}$ : 철골의 단면적 + +$A_{con}$ : 콘크리트의 단면적 + +$As_{stl}$ : 철골의 유효전단면적 + +$As_{con}$ : 콘크리트의 유효전단면적 + +$I_{stl}$ : 철골의 단면2차모멘트 + +$I_{con}$ : 콘크리트의 단면2차모멘트 + +REN : 콘크리트의 탄성계수(Ec)에 대한 철골의 탄성계수(Es)의 비 (Es/Ec) + + + +# Chapter 6. 경계조건 + +# 6-1 경계조건 + +midas Civil에서 경계조건은 다음과 같이 절점 경계조건과 요소의 경계조건으로 구분할 수 있습니다. + +# - 절점 경계조건 + +자유도 구속 (Support) + +탄성 경계요소 (Spring Support) + +탄성연결요소 (Elastic Link) + +비선형연결요소 (General Link) + +# ▪ 요소의 경계조건 + +요소의 단부해제 조건 (Beam End Release, Plate End Release) + +강성역 (Beam End Offsets 참조) + +강체연결기능 (Rigid Link 참조) + + + +# 6-2 자유도 구속조건 + +자유도 구속(Constraint) 기능은 임의 절점의 변위를 구속시키거나 자유도가 부족한 요소(트러스, 평면응력, 판요소 등)끼리 접합될 때 해당 자유도성분을 구속하는데 사용됩니다. + +자유도 구속조건은 임의 절점에 전체좌표계(Global Coordinate System) 또는 절점좌표계(Node Local Coordinate System)를 기준으로 6개 자유도에 대해 입력됩니다. + +예를 들어 그림 1.6.1과 같은 평면골조모델에 자유도 구속조건을 부여하는 방법은 다음과 같습니다. + +이 모델은 전체좌표계 X-Z 평면내에서만 거동이 허용되는 2차원 모델이기 때문에 Boundary 탑>Supports그룹>Supports 기능으로 모든 절점에 대해 전체좌표계 Y방향 변위자유도와 X방향 및 Z방향에 대한 회전자유도를 구속하여야 합니다. + +![](images/page-125_b20948b2204987833ab1d3fe09e4d32a6fca5cc5fca6ab0c29ec4cfe552f3476.jpg) + +
+text_image + +N2 +N4 +N6 +N1 +N3 +z +angle of +inclination +y +x +N5 +NCS +z +Y GCS X +•: fixed support condition +○: pinned support condition +▲: roller support condition +
+ +그림 1.6.1 자유도 구속조건이 고려된 평면골조모델 + +그리고 고정지지조건인 N1 절점에 대해서는 Supports 기능으로 전체좌표계 X, Z방향 변위자유도와 Y방향에 대한 회전자유도를 추가로 구속합니다. + + + +핀접합이면서 로울러 지지조건인 N3에 대해서는 Z방향 변위자유도를 추가로 구속합니다. + +절점좌표계에 대해 로울러 지지조건이 부여된 N5 절점에 대해서는 전체좌표계 X축에 대해 경사각만큼 회전한 절점좌표계를 설정한 다음, Supports기능으로 전체좌표계 Z축 방향 변위자유도를 구속합니다. 절점좌표계가 선언되어 있는 절점에 입력되는 구속조건은 절점좌표계를 따라 구속을 수행하게 됩니다. + +절점변위를 구속하는 기능은 변위를 무시할 수 있는 지지조건(Supports) 등에 주로이용되며 임의 절점에 대해 구속조건이 주어지면 해당절점에 대한 반력이 발생합니다. + +절점에서의 반력은 전체좌표계를 기준으로 출력되며, 절점좌표계가 부여된 경우에는 절점좌표계 기준으로 반력을 출력할 수 있습니다. + +그림 1.6.2는 Supports기능을 부족한 자유도의 구속조건에 사용한 예입니다. + +그림 1.6.2(a)의 경우는 트러스요소가 축방향의 변위자유도만 가지기 때문에 연결절점에서의 X방향 변위와 모든 회전방향 변위성분은 구속되었습니다. + +그림 1.6.2(b)는 상　하부 플랜지를 보요소로 대신한 예로써 보요소가 절점당 6개의 자유도를 가지기 때문에 보요소와 연결되는 절점에서는 별도의 구속조건이 필요없고, 평면응력요소끼리 만나는 절점에 대해서는 평면응력요소가 면내의 거동에대한 자유도만 가지기 때문에 면외변위성분인 Y방향 변위자유도와 모든 회전자유도를 구속해야 합니다. + + + +![](images/page-127_c1a79c56d71725a1a7f41994a0e689efb5e4296790a0d193e88b9161dd480e32.jpg) + +
+text_image + +connecting node +(DX, RX, RY and RZ are constrained) +supports (all degrees of +freedom are constrained) +
+ +(a) 트러스요소끼리 접합된 경우 + +![](images/page-127_7ed146049321d04f92e82e2c9c680a88e525d2ab565c6828a9cfc1f9c9aa173f.jpg) + +
+text_image + +supports (all degrees of freedom are constrained) +top flange (beam element) +Z +web (plane stress element) +Y +bottom flange (beam element) +X +in-plane vertical load +●: nodes with +○: DY, RX, R +DX: displaced +DY: displaced +DZ: displaced +RX: rotation a +RY: rotation a +
+ +(b) H형 외팔보를 상/하부 플랜지를 보요소로 모델링하고 웨브를 평면응력요소로 모델링한 경우 +그림 1.6.2 자유도 구속조건의 사용 예 + + + +# 6-3 탄성경계요소 + +탄성경계요소는 모델의 경계부분에 위치한 인접구조 또는 지반 등의 강성을 고려할 때나 자유도가 부족한 요소(트러스, 평면응력, 판요소 등)가 상호 접합될 경우 접합절점에서 발생할 수 있는 특이성 오류(Singular Error)를 방지하기위해 주로 사용됩니다. + +탄성경계요소는 임의 절점당 전체좌표계 기준의 6개 자유도(선방향 3개성분, 회전방향 3개성분)에 대해 모두 입력이 가능하며 선방향 탄성성분은 단위길이당 힘의 단위로 입력되고 회전방향 탄성성분은 단위각도(Radian)당 모멘트단위로 입력됩니다. + +선방향 탄성경계요소는 해석대상 구조물 하부의 기둥이나 Pile 또는 지반강성을 반영하는데 유용하게 사용됩니다. 지반을 모델링할 때는 지반반력계수(Modulus of Subgrade Reaction)에 해당절점의 유효면적(Tributary Area)을 곱한 값이 사용됩니다. 이때 토질의 특성이 압축에는 유효한 반면, 인장력에 대해서는 저항할 수 없기 때문에 주의해야 합니다. + +midas Civil에서는 토질과 접하는 면의 경계조건을 쉽게 모델링할 수 있도록 Surface Spring Supports 기능을 내장하고 있습니다. Boundary탭>Spring Supports그룹>Surface Spring Supports 기능에서 절점스프링을 선택하고 단위면적당 지반반력계수를 입력하면, 절점이 차지하고 있는 유효면적과 지반반력계수의 곱으로 강성을 계산하여 절점스프링 형태로 경계조건을 고려합니다. 또한 압축력만을 부담할 수 있는 지반의 특성을 고려한 해석을 수행하고자 할 경우, 탄성연결요소(압축전담)를 선택하고 지반반력계수를 입력하여 압축력만을 받을 수 있는 탄성연결요소로 경계조건을 구성하게 됩니다. + +표 1.6.1는 실무설계시 일반적으로 접할 수 있는 토질의 종류에 대한 지반반력계 수를 정리한 것으로 표의 상한치와 하한치의 값을 사용하여 각각 해석을 수행한 다음, 불리한 값을 사용하여 설계에 적용합니다. + +대상구조물과 접하는 기둥이나 Pile의 축방향 강성성분을 고려할 경우에 탄성경계 요소의 강성은 EA/H로 계산될 수 있고, 여기서 E는 지지부재의 탄성계수, A는 유 + + + +효단면적, H는 유효길이입니다. + +![](images/page-129_63e5dcb87354b8d5cb7946544e55b8d8cfd34738b3e7b8f828b959e60d9c1192.jpg) + +
+text_image + +SRY +SY +SX +SZ +SRX +SRZ +Z +Y +X +
+ +(a) Point Spring Support 기능을 사용한 경계조건의 입력 + +![](images/page-129_12b23b002d2e15e2e5a2e989beafc0c0d4c820f778ed887b4eaf416ca421ea59.jpg) + +
+text_image + +Diagram of a structural frame with springs labeled K and a rotation arrow indicating upward motion +
+ +(b) Surface Spring Supports 기능을 사용한 경계조건의 입력 +그림 1.6.3 탄성경계요소의 입력 예 + +회전방향 탄성성분은 해석대상 구조물의 인접경계부위의 회전강성을 반영하는데주로 사용되며, 인접경계부위가 기둥일 경우에는 αEI/H의 값으로 결정됩니다. 여기서 α는 기둥의 연결상태에 따라 결정되는 회전강성성분계수이고, I는 유효단면2 + + + +"Foundation Analysis and Design" by Joseph E. Bowles 4th Edition 참조 + +차모멘트, 그리고 H는 기둥의 유효길이입니다. + +절점에 사용되는 경계스프링은 일반적으로 각 자유도 방향별로 입력되는데, 정밀한 해석을 하고자 할 경우 다른 자유도와 연관(Coupled)되는 강성까지 고려해야합니다. 즉 이동변위가 발생할 때 동시에 발생되는 회전변위 등을 고려하기 위해서는 적절히 연관된 강성을 고려한 스프링의 입력이 필요합니다. 예를 들어 구조물의 기초에 사용되는 파일(Pile)을 경계스프링으로 모델링하고자 할 경우에 각 방향별 강성이외에 연관된 강성을 추가로 입력하여 보다 정밀한 해석을 수행할 수있습니다. + +절점에 입력되는 경계스프링은 일반적으로 전체좌표계를 따르지만 절점에 절점좌표계가 도입된 경우 절점좌표계를 따릅니다. + +해석단계에서 강성행렬의 조합후에 특정자유도에 대한 강성성분이 없을 경우 발생할 수 있는 특이성 오류(Singular Error)를 피하기 위해 회전탄성성분을 입력할 경우에는 사용단위계에 따라 차이가 있지만 0.0001 \~ 0.001의 값이 주로 사용됩니다. + +midas Civil에서는 이와 같은 특이성오류를 방지하기 위해 해석결과에 거의 영향이없을 정도의 강성을 자동으로 부여하는 기능을 내장하고 있습니다. + +
토질종류지반 반력계수( $tonf/m^{3}$ )
연약 점토1200 ~ 2400
중간정도 점토2400 ~ 4800
굳은 점토4800 ~ 11200
느슨한 모래480 ~ 1600
중간정도 다져진 모래960 ~ 8000
실트질 중간정도 다져진 모래2400 ~ 4800
점토질 자갈4800 ~ 9600
점토질 중간정도 다져진 모래3200 ~ 8000
다져진 모래6400 ~ 13000
잘다져진 모래8000 ~ 19000
실트질 자갈8000 ~ 19000
+ +표 1.6.1 토질종류별 대표적 지반 반력계수 diff --git a/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_014.md b/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_014.md new file mode 100644 index 00000000..8ffd839c --- /dev/null +++ b/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_014.md @@ -0,0 +1,274 @@ + + +On-line Manual의 "Boundary탭>Spring Supports그룹>General Spring Supports" 참조 + +# 6-4 범용경계요소(General Spring Supports) + +탄성경계요소는 모델의 경계부분에 위치한 인접구조 또는 지반 등의 강성을 고려할 때나 자유도가 부족한 요소(트러스, 평면응력, 판요소 등)가 상호 접합될 경우 접합절점에서 발생할 수 있는 특이성 오류(Singular Error)를 방지하기위해 주로 사용됩니다. + +탄성경계요소는 임의 절점당 전체좌표계 기준의 6개 자유도(선방향 3개성분, 회전방향 3개성분)에 대해 모두 입력이 가능하며 선방향 탄성성분은 단위길이당 힘의 단위로 입력되고 회전방향 탄성성분은 단위각도(Radian)당 모멘트단위로 입력됩니다. + +선방향 탄성경계요소는 해석대상 구조물 하부의 기둥이나 Pile 또는 지반강성을 반영하는데 유용하게 사용됩니다. 지반을 모델링할 때는 지반반력계수(Modulus of Subgrade Reaction)에 해당절점의 유효면적(Tributary Area)을 곱한 값이 사용됩니다. 이때 토질의 특성이 압축에는 유효한 반면, 인장력에 대해서는 저항할 수 없기 때문에 주의해야 합니다. + +탄성경계요소는 질량 및 감쇠의 설정이 가능합니다. 질량은 고유치해석, 응답스펙트럼해석, 시간이력해석에 적용됩니다. 감쇠는 응답스펙트럼해석과 시간이력해석에 적용됩니다. + +탄성경계요소에 입력되는 감쇠는 해석의 종류에 따라 다음과 같이 고려됩니다. + +1. 정적해석 및 고유치해석에서는 감쇠는 무시됩니다. +2. 응답스펙트럼해석에서는 구조물의 감쇠설정을 Strain Energy Proportional로 선택한 경우만 모드감쇠비를 통해 해석에 반영됩니다. +3. 모드중첩법에 기초한 시간이력해석을 수행하는 경우에는 구조물의 감쇠설정을 Strain Energy Proportional로 선택한 경우만 모드감쇠비를 통해 해석에 반영됩니다. +4. 직접적분법에 의한 시간이력해석을 수행하는 경우에는 구조물의 감쇠설정을 Mass & Stiffness Proportional 혹은 Element Mass & Stiffness + + + +Proportional로 설정한 경우, 요소감쇠행렬을 통해서 해석에 반영됩니다. 만약탄성경계요소에 요소강성 또는 요소질량에 비례하는 감쇠가 지정된 경우에는다음과 같이 탄성경계요소의 속성에서 지정된 감쇠와 합하여 해석을 수행합니다. + +$$ +M \ddot {u} + C \dot {u} + C _ {e f f} \dot {u} + K _ {S} u = p +$$ + +여기서 M : 질량행렬 + +C : 감쇠행렬 + +$C _ { e f f }$ Cef : 탄성경계요소의 감쇠 + +$K _ { s }$ : 탄성 요소의 강성행렬 + +$u , \dot { u } , \ddot { u }$ : 절점에 대한 변위, 속도 및 가속도 응답 + +p : 절점에 대한 동적하중 + +또한, Strain Energy Proportional을 선택한 경우는 탄성경계요소의 감쇠를반영한 모드감쇠비를 이용하여, 전체 구조물의 감쇠행렬을 구성합니다. + +이상에서 설명한 탄성경계요소의 감쇠적용규칙을 정리하면 다음과 같습니다. + + Response Spectrum + +: Strain Energy Proportional로 선택한 경우만 모드감쇠비를 통해 반영 + + Modal Analysis에 의한 시간이력해석 + +: Strain Energy Proportional로 선택한 경우만 모드감쇠비를 통해 반영 + + 직접적분법에 의한 시간이력해석 + +: Mass & Stiffness Proportional 혹은 Element Mass & Stiffness Proportional로설정한 경우, 요소감쇠행렬을 통해서 해석에 반영, Strain Energy Proportional인 경우는 모드감쇠비를 통해 해석에 반영 + + + +# 6-5 Distributed Spring(Winkler Spring) + +지반과 구조물의 상호작용을 고려하는 가장 쉬운 방법은 지반의 강성을 여러 개의 절점스프링으로 치환하여 해석하는 것입니다. 특히 탄성 지반 위의 기초를 모델링하는데 가장 널리 사용되는 스프링은 원클러(Winkler)모델입니다. + +midas Civil에서는 Boundary 탑>Spring Supports그룹>Surface Spring Supports 기능에서 분포스프링을 선택하면 원클러 스프링모델을 사용할 수 있습니다. + +이 모델의 기본 가정은 기초가 강성이 있는 구조물과 탄성지반으로 구성되어 있고, 지반이 개별 스프링들로 표현되므로 요소들간에 상호작용 없이 독립적으로 작용한다는 것입니다. + +이와 같은 원클러 스프링 경계조건은 보나 판 혹은 솔리드의 면에 동일한 방식으로 부여할 수 있습니다. 보의 경우에는 유효폭을 가진 길이에, 판이나 솔리드의 경우에는 면에 분포하고 있다는 가정으로 강성을 계산합니다. + +여기서는 보를 기준으로 원클러 스프링의 정식화 과정을 소개합니다. + +가상일의 원리(Principle of Virtual Work)에 따라 내적 가상 변형율 에너지(Internal Virtual Strain Energy)는 다음과 같이 표현할 수 있습니다. + +$$ +\delta U = \int [ \delta \varepsilon ] ^ {T} [ \sigma ] d V = \int [ \delta v ] k _ {v} [ v ] d V +$$ + +여기서 $\delta U$ : 내적 가상변형율 에너지 (Internal Virtual Strain Energy) + +$\delta\varepsilon$ : 가상 변형율(Virtual Strain) + +σ : 응력 + +V : 부피 + +δv : 가상 변위(Virtual Displacement) + +$k_{v}$ : 연직지반반력계수(Modulus of Vertical Subgrade Reaction) + + + +![](images/page-134_b188ff5277de47f404aa5dd0b5902c84bc830d470f0b5293a384fa166a69a767.jpg) + +
+text_image + +L +x +θ₁ +q₁ +y +kᵥ·v +θ₂ +q₂ +
+ +그림 1.6.4 Beam on elastic foundation element + +내적 가상일과 외적 가상일(External Virtual Work)이 동일하므로 다음과 같은 수식 전개를 할 수 있습니다. + +$$ +\int [ \delta q ] ^ {T} [ f ] ^ {T} k _ {v} [ f ] [ q ] d V = [ \delta q ] ^ {T} \cdot [ P ] +$$ + +$$ +k _ {v} \int [ f ] ^ {T} [ f ] d V \cdot [ q ] = [ P ] +$$ + +$$ +[ K _ {r} ] \cdot [ q ] = [ P ] +$$ + +$$ +[ K _ {r} ] = k _ {v} \int [ f ] ^ {T} [ f ] d V +$$ + +여기서, P : 보요소에 가해지는 외부 하중 + +f : 변형 형상 함수 + +q : 절점 변위 + +$K_{r}$ : 탄성지반위의 보요소를 위한 지반강성행렬 + +요소의 길이가 L 인 탄성지반위의 보에 대한 원클러 스프링 강성은 다음과 같고, 이 스프링강성과 요소강성이 더해져서 최종 요소강성 행렬을 구성합니다. + +$$ +\left[ K _ {r} \right] = \frac {k _ {v} L}{4 2 0} \left[ \begin{array}{c c c c} 1 5 6 & 2 2 L & 5 4 & - 1 3 L \\ 2 2 L & 4 L ^ {2} & 1 3 L & - 3 L ^ {2} \\ 5 4 & 1 3 L & 1 5 6 & - 2 2 L \\ - 1 3 L & - 3 L ^ {2} & - 2 2 L & 4 L ^ {2} \end{array} \right] +$$ + + + +보요소의 내부변위에 따라 스프링의 힘이 작용하기 때문에 보부재력을 보정할 필요가 있습니다. 이 힘을 구하기 위한 보요소의 임의의 위치 x 에서의 내부변위는 절점변위 q 와 θ를 기준으로 다음과 같이 계산합니다. + +$$ +v = y (x) = \frac {1}{L ^ {3}} \left[ \begin{array}{l} \left(2 x ^ {3} - 3 L x ^ {2} + L ^ {3}\right) q _ {1} + \left(- L x ^ {3} + 2 L ^ {2} x ^ {2} - L ^ {3} x\right) \theta_ {1} \\ + \left(- 2 x ^ {3} + 3 L x ^ {2}\right) q _ {2} + \left(- L x ^ {3} + L ^ {2} x ^ {2}\right) \theta_ {2} \end{array} \right] +$$ + +원클러 스프링에 의한 전단력과 모멘트는 보요소의 내부변위와 지반력계수를 이용하여 다음식과 같이 구합니다. 보의 부재력에 이와 같은 원클러 스프링의 내력을 더하여 최종 보 부재력이 구해집니다. + +$$ +\begin{array}{l} V (x ^ {\prime}) = k _ {v} \int_ {0} ^ {x ^ {\prime}} y (x) d x \\ = \frac {k _ {v}}{L ^ {3}} \left[ \begin{array}{l} \left(\frac {1}{2} x ^ {4} - L x ^ {3} + L ^ {3} x\right) q _ {1} + \left(- \frac {L}{4} x ^ {4} + \frac {2 L ^ {2}}{3} x ^ {3} - \frac {L ^ {3}}{2} x ^ {2}\right) \theta_ {1} \\ + \left(- \frac {1}{2} x ^ {4} + L x ^ {3}\right) q _ {2} + \left(- \frac {L}{4} x ^ {4} + \frac {L ^ {2}}{3} x ^ {3}\right) \theta_ {2} \end{array} \right] \\ \end{array} +$$ + +$$ +\begin{array}{l} M (x ^ {\prime}) = k _ {v} \int_ {0} ^ {x ^ {\prime}} y (x) (x ^ {\prime} - x) d x = k _ {v} x ^ {\prime} \int_ {0} ^ {x ^ {\prime}} y (x) d x - k _ {v} \int_ {0} ^ {x ^ {\prime}} y (x) \cdot x d x \\ = \frac {k _ {v}}{L ^ {3}} \left[ \begin{array}{l} \left(\frac {1}{2} x ^ {5} - L x ^ {4} + L ^ {3} x ^ {2}\right) q _ {1} + \left(- \frac {L}{4} x ^ {5} + \frac {2 L ^ {2}}{3} x ^ {4} - \frac {L ^ {3}}{2} x ^ {3}\right) \theta_ {1} \\ + \left(- \frac {1}{2} x ^ {5} + L x ^ {4}\right) q _ {2} + \left(- \frac {L}{4} x ^ {5} + \frac {L ^ {2}}{3} x ^ {4}\right) \theta_ {2} \end{array} \right] \\ - \frac {k _ {v}}{L ^ {3}} \left[ \begin{array}{l} \left(\frac {2}{5} x ^ {5} - \frac {3 L}{4} x ^ {4} + \frac {L ^ {3}}{2} x ^ {2}\right) q _ {1} + \left(- \frac {L}{5} x ^ {5} + \frac {2 L ^ {2}}{4} x ^ {4} - \frac {L ^ {3}}{3} x ^ {3}\right) \theta_ {1} \\ + \left(- \frac {2}{5} x ^ {5} + \frac {3 L}{4} x ^ {4}\right) q _ {2} + \left(- \frac {L}{5} x ^ {5} + \frac {L ^ {2}}{4} x ^ {4}\right) \theta_ {2} \end{array} \right] \\ \end{array} +$$ + + + +# 6-6 탄성연결요소 + +탄성연결요소는 두개의 절점을 사용자가 입력한 강성으로 연결하여 요소처럼 거동할 수 있도록 하는 기능입니다. 두개의 절점은 트러스나 보요소를 사용하여 연결할 수도 있지만, 사용자가 원하는 크기와 방향의 강성을 만들기에는 적절하지 못합니다. + +탄성연결요소의 입력은 각각 3방향의 이동 및 회전 강성으로 구성되어 있으며 방향은 요소좌표계를 따릅니다. + +탄성연결요소의 강성 크기는, 선방향은 단위길이당 힘으로 회전방향은 단위각도 (Radian)당 모멘트로 입력되고 요소좌표계의 방향은 그림 1.6.5과 같습니다. + +탄성연결요소에는 인정전담이나 압축전담특성을 부여할 수 있는데 이러한 경우에는 요소좌표계 x방향으로만 강성을 입력할 수 있습니다. + +탄성연결요소의 사용 예로는 교량구조물의 상부와 하부교각부를 연결해주는 탄성받침 등이 있고, 탄성연결요소에 압축전담특성을 부여하여 지반 경계조건으로 사용할 수 있습니다. 또한 강체 연결기능을 선택하면 Rigid Link와 같이 두 절점을 강체 연결할 수도 있습니다. + +![](images/page-136_63ab3629a0640a714b5dfa9f0b4e377cf4a7c82657f6690b6cf62a5f9323bebc.jpg) + +
+text_image + +z +Ref. +x +y +N1 +N2 +
+ +그림 1.6.5 2절점을 연결하는 탄성연결요소의 요소좌표계 + + + +# 6-7 범용연결요소 (General Link) + +![](images/page-137_02b73c9d9eef2214adfc26a2bfea4b6adc3824f2b6518ccfd4939bcf8a8fe869.jpg) + +
+text_image + +L +Lyi = cyiL Lzi = cziL +Lyj = cyjL Lzj = czjL +단, cyi + cyj = czi + czj = 1.0 +jointi +kx +Lyj +ky +Lzi +Lzj +kz +jointj +kθx +kθy +kθz +local coordinate axis +y +x +z +
+ +그림 1.6.6 범용연결요소의 구성 + +범용연결요소는 제진장치, 면진장치, 압축 또는 인장 전담요소, 소성힌지, 지반스프링 등을 모델링하는데 사용되는 요소입니다. 범용연결요소는 그림 1.6.6과 같이 2개의 절점을 연결하는 6개의 스프링으로 구성됩니다. 요소좌표계는 트러스 요소와 동일한 체계를 따르며, General Link에서 다양한 요소축 설정이 가능합니다. + +범용연결요소의 요소자유도는 요소좌표계 또는 전체좌표계에 관계없이 절점당 각각 세가지의 이동변위(Translation) 성분과 회전변위(Rotation) 성분을 가지게 되며, 6개의 변위 성분은 1개의 재축방향 변형, 2개의 전단 변형, 1개의 비틀림 변형 및 2개의 힜 변형 성분으로 구분됩니다. 6개의 변위 성분은 각각 독립된 6개의 스프링으로 표현되며, 이 가운데 일부의 스프링만을 선택하여 속성을 부여할 수 있습니다. + + + +범용연결요소의 속성은 General Link Properties에서 정의되며, 해석에 적용하는 방식에 따라서 크게 Element Type과 Force Type으로 구분됩니다. + +Element Type 범용연결요소에는 Spring, Linear Dashpot 및 Spring and LinearDashpot의 세가지 Property Type이 제공됩니다. Spring은 6개 성분 별로 선형탄성인 Stiffness만을 가지며 Linear Dashpot은 6개 성분 별로 선형점성인 Damping만을갖습니다. Spring and Linear Dashpot은 Spring과 Linear Dashpot이 병렬로 연결된형태입니다. + +Element Type 범용연결요소는 기본적으로 Linear Properties만을 갖는 선형요소로해석됩니다. Spring은 직접적분법에 의한 비선형시간이력해석 시에 Inelastic HingeProperties를 부여하여 비선형 요소로 사용가능하며, 비선형 해석과정에서 요소 강성행렬을 갱신함으로써 요소의 비선형 거동을 직접적으로 반영합니다. 이는 주로구조물에 부분적으로 발생하는 소성힌지나 지반의 비선형성을 모델링하기 위해 사용됩니다. + +Force Type 범용연결요소는 제진장치로 이용되는 Visco-elastic Damper, HystereticSystem, 면진장치로 이용되는 Lead Rubber Bearing Isolator, Friction PendulumSystem Isolator, 압축전담요소인 Gap 및 인장전담요소인 Hook 등의 Property Type이 제공되며, 비선형 시간이력해석인 경계비선형해석에 사용됩니다. Force Type 범용연결요소의 각각의 성분은 Linear Properties로서 Effective Stiffness 및 EffectiveDamping을 가지며, 사용자가 선택한 성분에 대해서 Nonlinear Properties를 입력할수 있습니다. + +Force Type 범용연결요소는 정적해석, 응답스펙트럼해석에서는 Effective Stiffness를갖는 선형요소로서 해석되며 Effective Damping는 무시됩니다. 선형시간이력해석에서는 Effective Stiffness에 기초한 선형요소로서 해석됩니다. 비선형시간이력해석에서는 Effective Stiffness가 가상의 선형강성 역할을 하며, 요소 강성행렬을 갱신하지않고, 비선형 속성에 따라 계산된 부재력을 외부 하중으로 치환해 줌으로써 간접적으로 비선형성을 고려합니다. + +범용연결요소의 Linear Properties로서 입력되는 Damping(Element Type) 또는Effective Damping(Force Type)은 해석의 종류에 따라 다음과 같이 고려됩니다. + + + +1. 정적해석에서는 Damping 또는 Effective Damping은 무시됩니다. +2. 응답스펙트럼해석에서는 구조물의 감쇠설정을 Strain Energy Proportional로 선택한 경우만 모드감쇠비를 통해 해석에 반영되며, 요소절점력 계산시Damping 또는 Effective Damping은 무시됩니다. +3. 모드중첩법에 기초한 선형 및 비선형 해석을 수행하는 경우에는 구조물의감쇠설정을 Strain Energy Proportional로 선택한 경우만 모드감쇠비를 통해 해석에 반영되며, 요소절점력 계산시에도 Damping 또는 EffectiveDamping을 반영합니다. +4. 직접적분법에 의한 선형 및 비선형 해석을 수행하는 경우에는 구조물의 감쇠설정을 Mass & Stiffness Proportional 혹은 Element Mass &Stiffness Proportional로 설정한 경우, 요소감쇠행렬을 통해서 해석에 반영됩니다. 만약 범용연결요소에 요소강성 또는 요소질량에 비례하는 감쇠가 지정된 경우에는 다음과 같이 범용연결요소의 속성에서 지정된 감쇠 또는 유효감쇠와 합하여 해석을 수행합니다. + +$$ +M \ddot {u} + C \dot {u} + C _ {e f f} \dot {u} + K _ {s} u = p +$$ + +여기서 M : 질량행렬 + +C : 감쇠행렬 + +$C _ { e f f }$ Cef : Damping 또는 Effective Damping + +$K _ { s }$ : 탄성 요소의 강성행렬 + +u u u , ,  : 절점에 대한 변위, 속도 및 가속도 응답 + +p : 절점에 대한 동적하중 + +또한, Strain Energy Proportional을 선택한 경우는 범용연결요소의 Damping또는 Effective Damping를 반영한 모드감쇠비를 이용하여, 전체 구조물의감쇠행렬을 구성합니다. 단, 요소절점력 계산시는 Damping 또는 EffectiveDamping을 반영하지 않습니다. + + + +이상에서 설명한 범용연결요소의 적용규칙을 정리하면 표1.6.2과 같습니다. + +
Application TypeElement Type General LinkForce Type General Link
Property TypeSpringLinear DashpotSpring and Linear DashpotSpring and DampingEffective StiffnessEffective Damping
Linear PropertiesStiffnessDampingStiffnessDampingStiffnessDamping
Static Analysis탄성-탄성-탄성-
Response Spectrum탄성 $\text{선형}^{1)}$ 탄성 $\text{선형}^{1)}$ 탄성 $\text{선형}^{1)}$
Linear Time HistoryModal Analysis탄성 $\text{선형}^{1)}$ 탄성 $\text{선형}^{1)}$ 탄성 $\text{선형}^{1)}$
Direct Integration탄성 $\text{선형}^{2)}$ 탄성 $\text{선형}^{2)}$ 탄성 $\text{선형}^{2)}$
Nonlinear Time HistoryModal Analysis탄성 $\text{선형}^{1)}$ 탄성 $\text{선형}^{1)}$ 탄성(가상) $\text{선형}^{1)}$
Direct Integration탄성 $\text{선형}^{2)}$ 탄성 $\text{선형}^{2)}$ 탄성(가상) $\text{선형}^{2)}$
+ +1) Strain Energy Proportional로 선택한 경우만 모드감쇠비를 통해 해석에 반영 +2) Mass & Stiffness Proportional 혹은 Element Mass & Stiffness Proportional로 설정한 경우, 요소감쇠행렬을 통해서 해석에 반영, Strain Energy Proportional인 경우는 모드감쇠비를 통해 해석에 반영 + +# 표 1.6.2 범용연결요소의 적용규칙 + +범용연결요소의 Linear Properties로서 입력되는 Damping(Element Type) 또는Effective Damping(Force Type)은 다음과 같이 강성비례형으로 계산하는 것이 일반적입니다. diff --git a/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_015.md b/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_015.md new file mode 100644 index 00000000..c389dfe7 --- /dev/null +++ b/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_015.md @@ -0,0 +1,361 @@ + + +$$ +C _ {e f f} = \frac {2 \xi}{\omega} K +$$ + +여기서 $C _ { e f f }$ : Damping 또는 Effective Damping + +K : 범용연결요소의 강성 + +： + +： + +![](images/page-141_b9f308dcc7db8a450d5486bc5d0587b221c62bf6c520148d3ff4fe9704c2654f.jpg) + +
+text_image + +F = K_s u +u +
+ +(a) Damping 또는 Effective Damping을 고려하지 않는 경우의 요소절점력과 변형 + +![](images/page-141_71a4663434cbbd0f76766263975eb682b04388304b5b66ec6d8d618c6e021f3d.jpg) + +
+text_image + +F = C_eff \dot{u} + K_s u +u +
+ +(b) Damping 또는 Effective Damping을 고려한 경우의 요소절점력과 변형 +그림 1.6.7 범용연결요소의 요소절점력과 변형관계 + + + +
Damping MethodDamping 또는 Effective Damping이 입력되지 않은 경우의 요소력Damping 또는 Effective Damping이 입력된 경우의 요소력
Static Analysis- $F = K_{S}u$ $F = K_{S}u$
Response spectrumModal $F = K_{S}u$ $F = K_{S}u$
Mass & Stiffness Proportional
Strain Energy Proportional
Modal AnalysisModal $F = K_{S}u$ $F = K_{S}u$
Mass & Stiffness Proportional $F = K_{S}u$
Strain Energy Proportional $F = C_{eff} \dot{u} + K_{S}u$
Direct IntegrationModal $F = K_{S}u$ $F = K_{S}u$
Mass & Stiffness Proportional $F = C_{eff} \dot{u} + K_{S}u$
Strain Energy Proportional $F = K_{S}u$
Element Mass & Stiffness Propor. $F = C_{eff} \dot{u} + K_{S}u$
+ +표 1.6.3 Damping 또는 Effective Damping에 의한 범용연결요소의 요소절점력 계산방법 + +요소내력의 출력치는 절점당 1개의 축방향력, 2개의 전단력, 1개의 비틀림 모멘트,2개의 휨 모멘트로 구성되며 부호규약은 보 요소와 동일합니다. 요소절점력과 변형의 관계는 범용연결요소의 Damping 또는 Effective Damping의 고려 유무에 따라그림 1.6.7과 같이 계산되며, 요소절점력의 계산방법은 표1.6.2를 기준으로 해석종류 및 감쇠방법에 따라 표1.6.3과 같이 표현됩니다. 단, 요소질량 또는 요소강성비례감쇠에 의한 절점력은 무시됩니다. + +범용연결요소의 자중은 Self Weight의 Total Weight에 입력합니다. Total Weight에 입력된 값은 정적하중으로서 Load탭>Load Type그룹>Static Loads>StructureLoads/Masses그룹>Self Weight에서 지정한 하중에 부가적인 절점하중으로 작용하며, 절점질량으로 변환됩니다. 또한, 질량을 별도로 정의할 경우는 Use Mass를 선택하고 Total Mass를 직접 입력하면, Total Weight를 절점질량으로 변환한 값 대신에 Total Mass로 입력된 값이 사용됩니다. 이와 같이 설정된 질량은 고유치해석 및동적해석에 반영됩니다. 단, Structure탭>Type그룹>Structure Type Conversion of + + + +Structure Self-weight into Masses에서 Do not Covert를 선택하면, Total Weight를 절점질량으로 변환한 값과 Total Mass로 직접 입력된 질량은 양쪽 모두 고유치해석및 동적해석에 반영되지 않으므로, 주의할 필요가 있습니다. + +2개의 전단 스프링은 부재내의 위치를 별도로 입력할 수 있습니다. 입력형식은 첫번째 절점으로부터의 거리를 전체 부재길이로 나눈 비율인 Shear Spring Location으로 입력합니다. Shear Spring Location은 해석에서 다음과 같이 반영됩니다. + +![](images/page-143_8c61fffdcd95606255ffbc0fc84e912bdc4a3ffd7e5caa008c16892f3b5f6c7f.jpg) + +
+text_image + +y +Momenty +(-) +Sheary +(-) +Axial +(-) +Momentz +(-) +Shearz +(-) +Momentx +(-) +x +L +Lzi +Lzj +i +j +Lyi +Lyj +y +Momenty +(+) +Sheary +(+) +Axial +(+) +Momentz +(+) +Shearz +(+) +Momentx +(+) +z +location of shear spring +in direction 3 +location of shear spring +in direction 2 +
+ +Lyi , Lyj : i, j-node에서 y-axis 방향 전단 스프링까지의 거리 + +Lzi , Lzj : i, j-node에서 z-axis 방향 전단 스프링까지의 거리 + +그림 1.6.8 범용연결요소의 요소좌표계 및 전단 스프링 위치 + +#  Shear Spring Location을 지정하지 않은 경우 + +전단스프링 위치를 지정하지 않으면, 독립적인 스프링 6개를 서로 다른 요소로 입력한 것과 동일하게 처리됩니다. 이 경우, 전단력은 모멘트의 미분이라는 일반적인 전단력-모멘트 관계가 성립하지 않습니다. 따라서, 전단력이 작용해도 양단의 모멘트는 동일합니다. + +#  Shear Spring Location을 지정한 경우 + +전단 스프링의 위치를 입력하면, 그림1.6.9에 나타낸 것과 같이 휨 스프링과 전단 스프링의 위치는 동일하게 간주되며, 전단력 작용시 단부에서 서로 다른 휨모멘트를 가집니다. 휨 변형은 스프링에서만 발생하며, 절점과휨 스프링 사이는 강체로 거동합니다. 모멘트는 스프링의 위치에 따라서변하므로 스프링 위치는 회전 변형에 영향을 줍니다. 따라서 부재에 하중이 재하되지 않는다면 전단력은 부재전체에 걸쳐서 동일하지만, 양단의 모멘트 차이는 전단력과 입력된 Shear Spring Location의 곱으로 표현됩니다. + + + +![](images/page-144_dd3e667ce4102e14c5f8bec6967b99700bd7b4b878d627b7745d31bd5da26f0a.jpg) + +
+text_image + +y +L_yjθ_zj +θ_zj +k_θz +k_y +v +L_yiθ_zi +θ_zi +v_i +i +j +x +z +L_yi +L_yj +L_y +θ_z = θ_zj - θ_zi +v_j +
+ +Shear force diagram + +Bending moment diagram +![](images/page-144_1028014255611eb677af5fddbea499b1da5599d59893c42e2a73141a558ef4f2.jpg) + +
+text_image + +q_yi +q_yj +q_yi +q_yj +V +m_zi +m_zj +M +
+ +$$ +\begin{array}{l} v = \left(v _ {j} - L _ {y j} \theta_ {z j}\right) - \left(v _ {i} + L _ {y i} \theta_ {z i}\right) \\ = \left(v _ {j} - v _ {i}\right) - \left(L _ {y j} \theta_ {z j} + L _ {y i} \theta_ {z i}\right) \\ \end{array} +$$ + +$$ +q _ {y} = k _ {y} v +$$ + +$$ += k _ {y} \left\{\left(v _ {j} - v _ {i}\right) - \left(L _ {y j} \theta_ {z j} + L _ {y i} \theta_ {z i}\right) \right\} +$$ + +$$ += q _ {y j} = - q _ {y i} +$$ + +$$ +\begin{array}{l} m _ {z} = k _ {\theta z} \theta_ {z} \\ = k _ {\theta z} \left(\theta_ {z j} - \theta_ {z i}\right) \quad \left| q _ {y i} = - \frac {z _ {j}}{L} \right. \\ = m _ {z j} - L _ {y j} q _ {z j} \quad \left| \begin{array}{c} m _ {z j} - m _ {z i} \end{array} \right| \\ = - (m _ {z i} + L _ {y i} q _ {z i}) \\ \end{array} +$$ + +$$ +m _ {z i} = - m _ {z} - L _ {y i} q _ {y i} +$$ + +$$ +m _ {z j} = m _ {z} + L _ {y j} q _ {y j} +$$ + +그림 1.6.9 전단스프링의 위치를 입력한 경우의 전단력과 모멘트 관계 + + + +# 6-8 요소의 단부해제조건 + +일반적으로 요소와 요소가 만나게 되면 각각의 요소가 갖고 있는 자유도에 대해 각 요소의 강성으로 서로 연결이 이루어지게 되는데, 이러한 연결을 해제하고자할 경우에 요소의 단부해제조건을 도입하게 됩니다. 단부해제조건의 입력이 가능한 요소는 보요소와 판요소이고 각 요소의 단부해제조건의 입력방법과 기능은 다음과 같습니다. + +보요소의 단부해제조건은 요소를 구성하는 두 절점의 모든 자유도에 대하여 입력이 가능하고, Partial Fixity를 고려하는 계수를 입력하여 연결된 요소의 전체강성 중 일부만 고려하여 해석할 수도 있습니다. 보요소의 두 절점에 회전방향에 대한 단부해제조건을 입력하면 구조적으로 트러스요소와 같은 거동을 하게 됩니다. + +판요소의 단부해제조건은 요소를 구성하는 3\~4개의 절점에 대해 평면의 수직방향에 대한 회전자유도를 제외한 모든 자유도에 대해 입력이 가능합니다. 판요소를 구성하는 모든 절점에 면외방향의 힘에 대한 단부해제조건을 입력하면 구조적으로 평면응력요소와 같은 거동을 하게 됩니다. + +단부해제가 수행되는 방향은 요소좌표계를 따르므로, 전체좌표계에 대한 강성의 연결해제를 입력할 경우에는 요소좌표계와의 관계에 주의해야 합니다. 또한 요소의 단부해제에 따른 강성의 변화가 구조해석을 수행하는 과정에서 특이성오류를 발생시킬수 있으므로 전체구조물에 대한 충분한 이해를 필요로 합니다. + +그림 1.6.10에서는 보요소와 판요소의 단부해제조건을 적절히 사용하여 교량의 교각과 상판의 연결부의 경계조건을 모델링하는 방법을 보여주고 있습니다. + + + +![](images/page-146_b2e8d6f29539c39c64dd98411d196407c13c4b4f68530d5a5a63fa4611a65229.jpg) + +
+natural_image + +Pure structural diagram of a T-shaped beam with supports and a triangular load, no text or symbols present +
+ +(a) 교량의 교각과 상판의 연결부 + +![](images/page-146_6836ce8f86636835d324e2e7db65870787bea965055158e36c21bc3297bedd22.jpg) + +
+flowchart + +```mermaid +graph TD + 1 --> 4 + 4 --> 2 + 4 --> 3 + 2 --> 3 + 3 --> 4 + 4 -->|①| 4 + 4 -->|②| 3 + 4 -->|③| 1 + 1 -->|Z| 2 + 2 -->|Z| 3 +``` +
+ +Element 1 – Node 4 end release of Fx & My Element 2 – Node 4 end release of $M _ { y }$ + +(b) 보요소를 사용하여 모델링한 경우 +![](images/page-146_a66f5099279f685c57fbce2362dbb30c290d4e3f18fea530782dd4827971b3e1.jpg) + +
+text_image + +Z +2 +X +4 +Z +X +1 +① +② +6 +3 +5 +③ +
+ +Element 1 – Node 3 & 4 end release of Fx & My Element 2 – Node 3 & 4 end release of $M _ { y }$ +(c) 판요소를 사용하여 모델링한 경우 +그림 1.6.10 보요소와 판요소의 단부해제조건을 사용한 모델링 + + + +# 6-9 강성역 + +토목, 건축구조물에서 골조부재로 구성되는 구조체는 요소종립축간의 교차점 사이의 거리를 해당요소의 길이로 간주하여 해석을 수행하기 때문에 실제의 경우보다 다소 큰 변위가 계산되고, 단부 및 중앙부의 모멘트 또한 크게 계산됩니다. midas Civil에서는 이와 같이 단부에 형성되는 편심 및 기둥과 보의 접합부에 형성되는 Panel Zone의 효과를 고려하기 위해 2가지 방법이 사용됩니다. (그림 1.6.11 참조) + +1. 기둥부재와 보부재가 만나는 모든 panel Zone에 대해 강성역을 자동고려하도록 하는 방법 +2. 보요소의 양단에 강성역을 직접 입력하는 방법 + +midas Civil은 보요소(또는 변단면요소)에 대해서만 강성역을 고려합니다. + +# 6-9-1 Panel Zone의 강성을 자동 고려하도록 하는 방법 + +Panel Zone에서 휨변형과 전단변형 등이 발생되지 않는다고 가정하면 골조부재의 휨변형과 전단변형에 대한 유효강성길이는 아래와 같이 표현할 수 있습니다. + +$$ +L 1 = L - (R i + R j) +$$ + +여기서, L은 부재의 양 중립축 교차점(양단절점) 사이의 길이이고, Ri와 Rj는 양단의 강성역(Rigid End Offset Length)입니다. 여기서 요소의 길이를 상기의 L1으로만 고려하게 되면, 미소하지만 접합부위에서 발생되는 변형(Rigid Zone Deformation)을 무시하는데 따른 오차가 발생하게 됩니다. + +midas Civil은 이와 같은 오차를 사용자가 보정할 수 있도록 강성역 보정계수 (Offset Factor)를 사용하고 있습니다. + +$$ +L 1 = L - Z F (R i + R j) +$$ + +여기서, ZF는 강성역 보정계수를 의미합니다. 강성역 보정계수는 0부터 1.0까지의 값으로 입력되며, 접합부의 기하학적 형상과 보강재의 사용여부에 따라 달라지기 + + + +때문에 사용자가 주의해서 적절한 값을 입력해야 합니다. + +강성역 보정계수는 축방향변형(Axial deformation)과 비틀림변형(Torsional + +Deformation)에 대해서는 영향을 미치지 않으며, 이들 변형을 계산할 때는 요소의전체길이(L)가 사용됩니다. + +![](images/page-148_3c43a0d5261a5fa082de0eb652d2b90e19815ec399764eab97a520d1a7fcfc3b.jpg) +(a) 기둥과 보의 연결부에 형성된 강성역 + +![](images/page-148_67004aed6c7c9945833f873592ddbab876e2ae063026542083ebbc29aa55846e.jpg) + +
+text_image + +eccentricity +
+ +(b) 기둥이 편심접합되는 경우 + +![](images/page-148_76c9c08ace7d145120af9935a58c2c5ab0aa0736c808b4910d6736f575602c1d.jpg) + +
+text_image + +eccentricity in +the Y-direction +eccentricity in +the Z-direction +Z +Y +X +eccentricity in +the X-direction +
+ +(c) 기둥과 보가 편심접합되는 경우 +그림 1.6.11 부재가 편심상태로 접합되는 경우의 이격거리 + + + +midas Civil에서 Boundary탭>Etc.그룹>Panel Zone Effects 기능을 이용하면 전체좌표계 Z축이 중력 반대방향으로 자동 설정되고, 강성역에 대해 이격거리도 자동 고려됩니다. + +강성역은 거더부재와 기둥부재가 접합되는 부위에 대해서만 고려됩니다. + +여기서, 기둥부재란 Z축에 평행하게 위치한 보요소를 의미하며, 거더부재란 전체좌표계 X-Y 평면에 평행한 평면상에 위치한 보요소를 의미합니다. + +Panel Zone Effects 기능을 사용하여 강단이력거리을 자동으로 고려하고자 하는 경우에 부재력 출력위치 “Output Position”에서 “Offset Position”을 선택하면, 요소강성과 자중 및 분포하중의 고려방법, 그리고 부재력의 출력위치가 강단이력거리 보정계수에 의해 조정된 이격위치에 따라 변하게 됩니다. 그리고 “ PanelZone”을 선택하면 요소강성의 계산에 사용되는 요소의 길이만 강단이력거리 보정계수에 따라 조정됩니다. 자중 및 분포하중의 고려방법, 부재력의 출력위치를 결정하는 이격 위치는 Panel Zone의 경계위치(보의 경우는 기둥의 면과 보의 끝단부가 접하는 위치, 기둥의 경우는 보의 상하부 면과 기둥이 만나는 위치)로 고정됩니다. + +참고로, Panel Zone Effects 기능에서 부재력 출력위치로 “Offset Position”을 선택하고 강단이력거리 보정계수를 1.0으로 하면, “Panel Zone”을 선택하고 강단이력거리 보정계수를 1.0으로 하는 것과 동일한 조건이 됩니다. 그리고 부재력 출력위치로 “Offset Position”을 선택하고 강단이력거리 보정계수를 0.0으로 하면 강단이력거리을 고려하지 않은 경우와 같은 조건이 됩니다. + + + +Panel Zone Effects 기능을 사용하여 강단이력거리를 자동으로 고려하고자 하는 경우에는 부재력 출력위치의 선택에 따라 자중 및 분포하중의 고려방법이나 부재력의 출력위치가 결정되기 때문에 다음 사항에 유의하여야 합니다. + +#  요소강성의 계산 + +요소의 강성을 계산할 때, 축방향강성과 비틀림강성에 대해서는 양 절점사이의 길이가 사용되고, 전단강성과 휨강성을 계산할 때는 부재력 출력위치의 선택에 관계없이 강단이력거리 보정계수가 고려된 길이(L1 = L -ZF (Ri + Rj))가 사용됩니다. (그림 1.6.12 참조) + +#  분포하중의 계산 + +부재력 출력위치를 “Panel Zone”으로 하면, 강단이격위치와 절점 사이의구간에 재하되는 분포하중은 해당 절점상에 전단력으로만 고려되며, 나머지 구간에 재하된 분포하중은 그림 1.6.13에서와 같이 전단력과 모멘트로치환되어 고려됩니다. 부재력 출력위치를 “Offset Position”으로 하는 경우에는 강성역 보정계수가 고려된 위치(강단이격 조정위치)를 사용하여 계산합니다. + +#  자중의 고려길이 + +기둥부재의 자중은 강성역을 고려하지 않은 양 절점 사이의 길이에 대해고려됩니다. 거더부재의 경우, 부재력 출력위치가 “Panel Zone”일때는 양절점 사이의 길이에서 양단의 강성역을 제외한 길이(L1 = L - (Ri+ Rj))가자중의 계산에 사용되고, 부재력 출력위치가 “Offset Position”일때는 강성역 보정계수에 의해 조정된 길이를 뺀 길이(L1 = L - ZF (Ri+ Rj))가 사용됩니다. 그리고 이와 같이 결정된 자중은 전술한 분포하중 계산법에 따라 전단력과 모멘트로 치환되어 해석에 고려됩니다. + +#  부재력의 출력위치 + +기둥 및 거더부재의 부재력은 부재력 출력위치가 “Panel Zone”이면,Panel Zone의 끝단부와 Panel Zone 사이의 구간을 4등분한 위치에서 출력됩니다. 부재력 출력위치가 “Offset Position”이면, 강단이격 조정위치가주어진 거더부재의 경우는 양 절점 사이의 길이에서 강단이격 조정길이를뺀 구간을 4등분한 위치에서 출력됩니다. 참고로 부재력 출력위치가“Panel Zone”이면, 부재력 출력위치가 “Offset Position”이면서 강성역 보정계수가 1.0인 경우와 부재력의 출력위치가 동일합니다. diff --git a/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_016.md b/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_016.md new file mode 100644 index 00000000..4fb7c19a --- /dev/null +++ b/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_016.md @@ -0,0 +1,410 @@ + + +#  단부자유도 해제조건(Beam End Release)이 고려되었을 때의 강성역 + +기둥 및 거더부재의 어느 한쪽 또는 양쪽 연결점이 핀접합에 의해 자유도해제조건이 부여되었을 때 해당 연결점에 대해서도 강성역을 고려합니다. + +#  기둥부재의 강성역 고려방법 + +기둥부재의 강성역은 기둥의 상단과 하단부에서 각각 계산하게 됩니다.(그림 1.6.12 참조) + +기둥부재와 거더부재의 연결부에서 기둥부재의 강단이력거리는 연결되는거더부재의 춤(Depth)과 방향에 의해 결정되며 그림 1.6.15에서와 같이 기둥부재와 보부재가 연결될 경우, 기둥부재의 강단이력거리는 요소좌표계 y축과 z축방향 각각에 대해 산정됩니다. + +기둥부재에 여러 방향으로부터 거더부재가 접합될 경우, 각 방향별 강성역의 산정방법은 다음과 같습니다. (그림 1.6.16 참조) + +$$ +R C _ {y} = B D \times \cos^ {2} \theta \quad R C _ {z} = B D \times \sin^ {2} \theta +$$ + +RCy : 기둥부재 상부의 요소좌표계 y축 방향에 대한 강성역 + +RC : 기둥부재 상부의 요소좌표계 z축 방향에 대한 강성역 + +BD:(Depth) + +θ : 기둥부재의 요소좌표계 z축과 거더부재가 이루는 각도 + +기둥부재의 각 방향별 강성역은 기둥부재에 연결된 거더부재들에 대해 각방향별 강성역을 구한 다음, 그 중 가장 큰 값으로 결정합니다. + + + +column centerline axis (parallel with the z-axis) +![](images/page-152_03ccfd3acf6c5ff18bd07ce1a8e34336751cb0650f85cb26102ec1b718dedf5a.jpg) + +
+text_image + +Panel Zone +rigid end offset distance +of column (A) +Panel Zone +column length(L) +when the centerline of beam section +coincides with the story level +
+ +column centerline axis (parallel with the z-axis) +![](images/page-152_a5ec3d0899e0b54059425fd6d4dc591a84d4963d18c2f4a129b5c77be1a1263f.jpg) + +
+text_image + +Panel Zone +rigid end offset +distance of +column (A) +Panel Zone +when the top of beam is flush with +the story level +
+ +(a) 기둥부재의 강성역 +![](images/page-152_ab1003d0518f6dafa11396afdb47f5467749ef89db47f3d94345d76af2b2d1fb.jpg) + +
+text_image + +column centerline axis +column centerline axis +column member +beam member +column member +Panel Zone +Panel Zone +B +clear length of beam +A +length between nodes (L) +
+ +(b) 보부재의 강성역 + +
Offset Factoreffective length for stiffness calculation
1.00L-1.00×(A+B)
0.75L-0.75×(A+B)
0.50L-0.50×(A+B)
0.25L-0.25×(A+B)
0.00L-0.00×(A+B)
+ +Offset Factor: rigid end offset factor entered in “Panel Zone Effects” +(c) 강성고려길이 (기둥의 경우 B=0) +그림 1.6.12 Panel Zone Effects 기능을 사용하여 강단이격거리를 고려할 때 +요소의 휨/전단강성의 계산에 사용되는 길이 + + + +![](images/page-153_026038e388d25ff1bbd03d2caa5dc7948831774b155da7a691b73586193ced9a.jpg) + +
+text_image + +rigid end offset location at i-th node +rigid end offset location at j-th node +distributed load on beam element +i-end +L₁ +L₁ (length for shear/bending stiffness calculation) +L +j-end +L +zone in which load is converted into shear force only at i-th node +zone in which load is converted into both shear and moment +zone in which load is converted into shear force only at j-end +V₃ +V₁ +V₂ +V₄ +M₁ +M₂ +locations for member force output(▼) +" " " " " +
+ +Li = 1.0  Ri : “Panel Zone”is selected for the locations of member force output + +Li=ZF  Ri : “Offset Position”is selected for the locations of member force output + +Lj = 1.0  Rj : “Panel Zone”is selected for the locations of member force output + +Lj=ZF  Rj : “Offset Position”is selected for the locations of member force output + +Ri : rigid end offset distance at i-th node + +Rj : rigid end offset distance at j-th node + +ZF : rigid end Offset Factor + +V1, V2 : shear forces due to distributed load between the offset ends + +M1, M2 : moments due to distributed load between the offset ends + +V3, V4 : shear forces due to distributed load between the offset ends and the nodal points + +(a) 보부재 + +그림 1.6.13 Panel Zone Effects 기능을 사용하여 강성역을 고려할 때 + +분포하중의 고려방법 및 부재력 출력위치 + + + +![](images/page-154_20e81430706f91b91c6fcef0675fd0227c85ed646fd12196ef46704269b5b049.jpg) + +
+text_image + +top node +L_R +zone in which load +is converted into +shear force only +at the top node +rigid end +offset location +distributed load on column element +L_1 +L +L1 zone in which distributed load is +bottom node +V_3 +V_2 +M_2 +locations for member force output( ) +V_1 +M_1 +
+ +LR=1.0R “Panel Zone” is selected for the location of member force output + +LR=ZFR “Offset Position”is selected for the locations of member force output Where R is the rigid end offset factor + +V1, V2 shear forces due to distributed load between the offset end and the bottomnode + +M1, M2 :moments due to distributed load between the offset end and the bottom node + +V3 : shear force due to distributed load between the offset end and the top node + +(b) 기둥부재 + +그림 1.6.14 Panel Zone Effects 기능을 사용하여 강성역을 고려할 때 + +분포하중의 고려방법 및 부재력 출력위치 + + + +![](images/page-155_33db6eb0665835fe26364c500186f6861fd99e12fc0eca939453af48b85fd89d.jpg) + +
+text_image + +ECS y axis of column +column member +ECS z - axis +of column +beam member 2 +column centerline axis +(parallel with the GCS Z-axis) +beam member 1 +Y +Z +X +
+ +(a) 평면도 + +![](images/page-155_bfcb91747c53d9dd697ce35bcb2889058f053e3f6319a587d7bbfca8ba4d65b8.jpg) + +
+text_image + +column centerline axis +beam member 2 +Story(Floor)Lev +rigid end offset distance +at the top of the column +for bending about the +ECS z - axis +beam member 1 +rigid end offset distance +at the top of the column +for bending about the +ECS y - axis +Z +Y +X +
+ +(b) 정면도 +그림 1.6.15 Panel Zone Effect 기능을 사용할 경우, 기둥 부재의 강성역 + + + +![](images/page-156_7dd7c38ec042d6e1981a16e0348867ed2317bb520696d20e3e7cd5d87e2a6ea0.jpg) + +
+text_image + +beam member 3 +column centerline axis +column member +column member +Y +Z +X +Y +Z +X +column centerline axis +beam member 2 +beam member 1 +ECS z - axis of the column +θ +ECS y - axis of the column +
+ +beam member1: BD = 250 $\theta = 0^{\circ}$ $RC_{z} = 250 \times \sin^{2}0^{\circ} = 0.0$ $RC_{y} = \cos^{2}0^{\circ} = 250.0$ + +beam member2: BD = 200 $\theta = 40^{\circ}$ $RC_{z} = 200 \times \sin^{2} 40^{\circ} = 82.6$ $RC_{y} = 200 \times \cos^{2} 0^{\circ} = 117.4$ + +beam member3: BD = 150 $\theta = 90^{\circ}$ $RC_{z} = 150 \times \sin^{2} 90^{\circ} = 150$ $RC_{y} = 150 \times \cos^{2} 90^{\circ} = 0.0$ + +rigid end offset distance of the column: $RC_y = \mathrm{MAX}(250.0, 117.4, 0.0) = 250.0$ + +$$ +R C _ {z} = \mathrm{MAX} (0. 0, 8 2. 6, 1 5 0. 0) = 1 5 0. 0 +$$ + +where, BD : beam depth + +RCz : rigid end offset distance for bending about the minor axis + +RCy : rigid end offset distance for bending about the major axis + +그림 1.6.16 Panel Zone Effects 기능을 사용할 경우, 기둥부재의 강성역 산정 예 + + + +#  거더부재의 강성역 고려방법 + +거더부재의 강성역은 거더부재 양 끝단에 대한 기둥부재의 높이(Depth)과폭(Width)에 의해 결정되며 산정식은 다음과 같다. + +\- 각 방향별 강단이격거리에 의한 이격거리 산정식 (그림 1.6.17 참조) + +$$ +R B = \frac {\text { Depth } \times \cos^ {2} \theta}{2} + \frac {\text { Width } \times \sin^ {2} \theta}{2} +$$ + +Depth : 기둥부재의 요소좌표계 z축 방향의 단면길이 + +Width : 기둥부재의 요소좌표계 y축 방향의 단면길이 + +![](images/page-157_79d573c4f700b4b2d3777dd349e4840d00d67e61399807486a672bc87428837b.jpg) + +
+text_image + +beam member 3 +rigid end offset distance +for beam member 2 +beam member 2 +column width +θ +ECS z – axis +of the column +beam member 1 +rigid end offset distance +for beam member 3 +ECS x – axis +of the column +column depth +rigid end offset distance +for beam member 1 +ECS y – axis +of the column +Y +Z +X +
+ +그림 1.6.17 Panel Zone Effects 기능을 사용할 경우, 거더부재의 강단이격거리 + + + +![](images/page-158_52ee3171262615e29efd499abc7cd1f70f7c34e75b017ad544b8b72c089ba380.jpg) + +
+text_image + +beam member +ECS z - axis +of column +θ +ECS x - axis +of beam +
+ +column centerline at i– th node +column centerline at j– th node + +150, 100depth of column section = width of column section= , for $\theta = 4 0 ^ { \circ }$ + +150 0 100 0 2 2cos sin    rigid end offset distance at i-th node = $: \frac { 1 5 0 \times { c o s } ^ { 2 } 0 ^ { \circ } } { 2 } + \frac { 1 0 0 \times { s i n } ^ { 2 } 0 ^ { \circ } } { 2 } = 7 5 . 0$ + +150 40 1002cos  rigid end offset distance at j-th node = 40 2 sin  ${ \mathsf { n o d e } } = { \frac { 1 5 0 \times c o s ^ { 2 } 4 0 ^ { \circ } } { 2 } } + { \frac { 1 0 0 \times s i n ^ { 2 } 4 0 ^ { \circ } } { 2 } } = 6 4 . 7$ + +그림 1.6.18 Panel Zone Effects 기능을 사용할 경우, 거더부재의 강성역 산정 예 + + + +# 6-9-2 Beam End Offsets 기능을 이용하여 보요소의 양단에 강성역을직접 입력하는 방법 + +Beam End Offsets에서는 다음의 2가지 방법으로 보요소의 양단에 강성역 거리를직접 입력하게 됩니다. + +1. 양절점에서의 이격거리(Offset Length)를 전체좌표계 기준으로 X, Y, Z축방향의 성분거리로 입력 +2. 양절점에서의 이격거리를 요소좌표계 x축 방향의 거리로 입력 + +첫째 방법은 접합부의 방향별 편심거리를 입력합니다. 요소강성을 계산하거나 분포하중 또는 자중을 계산할 때 고려되는 거리는 이격된 양절점 사이의 전길이가고려됩니다. 그리고 부재력의 출력위치 또는 단부자유도해제조건에 대해서도 이격된 위치를 기준으로 조정됩니다. (그림 1.6.11 (b), (c) 참조) + +둘째 방법은 축방향의 편심거리를 입력합니다. 이 방법은 요소강성의 계산과 부재력의 출력위치 또는 단부자유도해제조건에 대해서는 Panel Zone Effects 기능에서“Panel Zone" 을 선택하고 강성역 보정계수를 1.0을 입력한 경우와 같은 효과를가지나, 분포하중에 대해서는 조정된 거리 대신 양절점 사이의 전체길이를 사용합니다. + + + +# 6-10 주절점과 종속절점(강체연결기능) + +강체연결기능(Boundary탭>Link그룹>Rigid Link)은 구조물의 기하학적 상대거동을 상호 구속하는 기능입니다. + +기하학적 상대거동의 구속은 임의 절점의 자유도에 한 개 또는 그 이상의 절점의 자유도를 종속시킴으로써 이루어지며 여기서 임의 절점을 주절점(Master Node)이라 하고 자유도가 종속되는 절점을 종속절점(slave node)이라 합니다. +강체연결기능에는 다음과 같이 네 가지 종류가 있습니다. + +Rigid Body Connection + +Rigid Plane Connection + +Rigid Translation Connection + +Rigid Rotation Connection + +Rigid Body Connection은 주절점과 종속절점들이 3차원 강체로 연결된 것처럼 상호거동이 구속되는 방법으로, 각 절점간의 거리가 일정하게 유지되며 상호구속방정식은 다음과 같습니다. + +$$ +U _ {X s} = U _ {X m} + R _ {Y m} \Delta Z - R _ {Z m} \Delta Y +$$ + +$$ +U _ {Y s} = U _ {Y m} + R _ {Z m} \Delta X - R _ {X m} \Delta Z +$$ + +$$ +U _ {Z s} = U _ {Z m} + R _ {X m} \Delta Y - R _ {Y m} \Delta X +$$ + +$$ +R _ {X s} = R _ {X m} +$$ + +$$ +R _ {Y s} = R _ {Y m} +$$ + +$$ +R _ {Z s} = R _ {Z m} +$$ + +여기서, $\Delta X = X_{m} - X_{s}$ , $\Delta Y = Y_{m} - Y_{s}$ , $\Delta Z = Z_{m} - Z_{s}$ + +상기 식에서 첨자 m과 s는 각각 주절점과 종속절점을 의미하며, $U_{X}$ , $U_{Y}$ , $U_{Z}$ 는 각각 전체좌표계 기준의 X방향변위, Y방향변위, Z방향변위 성분을, 그리고 $R_{X}$ , $R_{Y}$ , $R_{Z}$ 는 각각 전체좌표계 기준의 X방향에 대한 회전변위, Y방향에 대한 회전변위, Z방향에 대한 회전변위 성분을 의미합니다. 그리고 $X_{m}$ , $Y_{m}$ , $Z_{m}$ 는 주절점의 좌표를 diff --git a/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_017.md b/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_017.md new file mode 100644 index 00000000..271bd444 --- /dev/null +++ b/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_017.md @@ -0,0 +1,354 @@ + + +Xs, Ys, Zs는 종속절점의 좌표를 각각 의미합니다. 이 기능은 강성이 타 구조부재보다 훨씬 커서 변형효과를 무시할 수 있는 부재의 모델링이나 Stiffened Plate에서 Plate와 Stiffener를 상호 연결하는데 활용될 수 있습니다. + +Rigid Plane Connection은 주절점과 종속절점들이 X-Y평면 또는 Y-Z평면, 또는Z-X평면들과 평행한 평면상에서 평면강체로 연결된 것처럼 상호거동이 구속되는방법으로, 평면상에 투영된 각 절점간의 거리가 일정하게 유지되며 상호구속 방정식은 다음과 같습니다. + +\- X-Y 평면거동에 대해 Rigid Plane Connection을 부여할 경우 + +$$ +U _ {X s} = U _ {X m} - R _ {Z m} \Delta Y +$$ + +$$ +U _ {Y s} = U _ {Y m} + R _ {Z m} \Delta X +$$ + +$$ +R _ {Z s} = R _ {Z m} +$$ + +\- Y-Z 평면거동에 대해 Rigid Plane Connection을 부여할 경우 + +$$ +U _ {Y s} = U _ {Y m} - R _ {X m} \varDelta Z +$$ + +$$ +U _ {Z s} = U _ {Z m} + R _ {X m} \varDelta Y +$$ + +$$ +R _ {X s} = R _ {X m} +$$ + +\- Z-X 평면거동에 대해 Rigid Plane Connection을 부여할 경우 + +$$ +U _ {Z s} = U _ {Z m} - R _ {Y m} \Delta X +$$ + +$$ +U _ {X s} = U _ {X m} + R _ {Y m} \Delta Z +$$ + +$$ +R _ {Y s} = R _ {Y m} +$$ + +이 기능은 평면내 상대거동이 무시될 수 있는 바닥판의 모델링에 주로 활용됩니다. + + + +Rigid Translation Connection은 주절점과 종속절점들의 X축, Y축 또는 Z축방향 거동을 상호 구속시키는 방법으로 상호구속 방정식은 다음과 같습니다. + +\- X축방향 거동에 대해 상호구속할 경우 + +$$ +U _ {X s} = U _ {X m} +$$ + +\- Y축방향 거동에 대해 상호구속할 경우 + +$$ +U _ {Y s} = U _ {Y m} +$$ + +\- Z축방향 거동에 대해 상호구속할 경우 + +$$ +U _ {Z s} = U _ {Z m} +$$ + +Rigid Rotation Connection은 주절점과 종속절점들의 X축, Y축 또는 Z축에 대한 회전거동을 상호 구속시키는 방법으로 상호구속방정식은 다음과 같습니다. + +\- X축에 대한 회전거동을 상호구속할 경우 + +$$ +R _ {X s} = R _ {X m} +$$ + +\- Y축에 대한 회전거동을 상호구속할 경우 + +$$ +R _ {Y s} = R _ {Y m} +$$ + +\- Z축에 대한 회전거동을 상호구속할 경우 + +$$ +R _ {Z s} = R _ {Z m} +$$ + +다음은 강체연결기능에 대한 이해를 돕기 위해 Rigid Plane Connection 기능을 구조물 바닥판의 모델에 적용한 예를 개념적으로 서술한 것입니다. + +일반적으로 구조물이 횡력을 받을 때 바닥판 내의 모든 위치에서의 횡방향 상대변위는 다른 구조부재(기둥, 벽, 대각부재)의 상대변위에 비해 거의 무시할 만큼 작습니다. 이와 같은 바닥판의 강막작용(Rigid Diaphragm Action)은 바닥판 내의 모 + + + +든 면내거동을 상호 구속함으로써 고려될 수 있습니다. 이때 면내거동은 바닥판의면내 이동변위 2개의 성분과 면의 수직방향에 대한 회전변위성분입니다. + +![](images/page-163_867b19d6abf207efd8d88334c63a6f13ae641fb1ea5304fa9346c7b273ea3318.jpg) +그림 1.6.19 바닥판이 있는 일반구조물이 횡력을 받는 경우 + +그림 1.6.19에서 구조물에 횡력이 가해질 때 바닥판의 면내강성이 수직기둥부재의횡방향강성에 비해 무한대로 클 경우, 바닥판의 면내변형은 구조적으로 무시될 수있고, 따라서 δ1과 δ2는 거의 같은 값을 가지게 됩니다. + + + +![](images/page-164_3b6e55ae98acab5f03f45e634f8df6881c6d104b477f4656a61c0cc13755926a.jpg) + +
+text_image + +floor diaphragm +torsional moment +1 +2 +3 +4 +Z +Y +X +
+ +![](images/page-164_38b5ebd933852092a4ee50e5e5c3b84173fab177ae7f5aca497b9839e47be3d3.jpg) + +
+text_image + +Y +φ4 +4 +3 +φ3 +φ +φ1 +1 +2 +X +φ2 +
+ +그림 1.6.20 바닥판이 있는 단층구조물이 수직축에 대해 비틀림모멘트를 받는 경우 + +그림 1.6.20의 예에서 바닥판이 있는 단층구조물이 비틀림모멘트를 받을 때, 바닥판의 면내강성이 수직기둥부재의 강성에 비해 무한히 클 경우, 바닥판 전체가 만큼 회전하게 되고 가 됩니다. 따라서 4개의 자유도를 1개의자유도로 축약시킬 수 있습니다. + +그림 1.6.21는 강막작용을 고려하여 절점당 6개씩의 자유도(64), 총 24개의 자유도를 15개의 자유도로 축약하는 과정을 나타낸 그림입니다. + + + +![](images/page-165_f286257f178c84356b8eabcc0c243dd5898d1570e8b300a340f1fe6fc1266517.jpg) + +
+text_image + +U_X U_Y U_Z R_X R_Y R_Z +U_X U_Y U_Z R_X R_Y R_Z +U_X U_Y U_Z R_X R_Y R_Z +U_X U_Y U_Z R_X R_Y R_Z +X +
+ +![](images/page-165_5634678b0c56eb4ac41ab1279797b10f68f1799e5ec158d6baded4738e8c2ff1.jpg) + +
+text_image + +UZRXRY +4 +UZRXRY +3 +Z +UZRXRY +1 +2 +UZRXRY +Y +X +
+ +${ \sf U } _ { \sf X }$ : displacement degree of freedom in the X-direction at the corresponding node + +$\mathsf { U } _ { \mathsf { Y } } :$ displacement degree of freedom in the Y-direction at the corresponding node + +$\mathsf { U } _ { Z } :$ displacement degree of freedom in the Z-direction at the corresponding node + +$\mathsf { R } _ { \mathsf { X } }$ : rotational degree of freedom about the X-axis at the corresponding node + +$\mathsf { R } _ { \mathsf { Y } }$ : rotational degree of freedom about the Y-axis at the corresponding node + +$\mathsf { R } _ { Z }$ : rotational degree of freedom about the Z-axis at the corresponding node + +그림 1.6.21 면내 무한강성을 가진 바닥판의 자유도 축약 개념도 + + + +![](images/page-166_7c0db9b7bc40a0a4fabaca7270ddd38df7195e9fbffc5eba7a68b2b6ecd26de9.jpg) + +
+text_image + +Y +U_Ym +R_m +U_Xm +ΔY +ΔX +U_Xs +U_Ys +X +R_s +
+ +$\mathsf { U } _ { \mathsf { X } \mathsf { m } }$ : X-direction displacement of master node + +$\mathsf { U } _ { \mathsf { Y m } }$ : Y-direction displacement of master node + +$\mathsf { R } _ { Z \mathsf { m } }$ : rotation about Z-axis at master node + +$\mathsf { U } _ { \mathsf { X } \mathsf { s } }$ : X-direction displacement of slave node + +$\mathsf { U } _ { \mathsf { Y s } }$ : X-direction displacement of slave node + +$\mathsf { R } _ { \mathsf { Z } \mathsf { s } }$ : rotation about Z-axis at slave node + +그림 1.6.22 무한강성을 가진 바닥판이 횡력에 의해 변위가 발생하였을 경우 + +그림 1.6.22에서 무한강성을 가진 바닥판에 횡력에 의한 평면방향변위 및 회전변위가 동시에 발생하였을 경우 바닥판내의 임의 점에서의 변위는 아래와 같이 계산됩니다. + +$$ +U _ {X s} = U _ {X m} - R _ {Z m} \Delta Y +$$ + +$$ +U _ {Y s} = U _ {Y m} + R _ {Z m} \Delta X +$$ + +$$ +R _ {Z s} = R _ {Z m} +$$ + + + +기구학적 구속기능을 이용하여 자유도를 축약시키게 되면 해석 소요시간을 단축시키는데도 상당히 효과적입니다. 가령 구조물의 구조해석시 바닥판을 판요소(또는평면응력요소) 등으로 모델링 한다면 층당 수많은 절점이 필요합니다. 이 경우 횡방향 자유도만 고려해도 절점갯수×3 만큼의 자유도수가 늘어나기 때문에 몇 개 바닥만 모델링 하더라도 구조해석 프로그램의 해석능력을 초과하거나, 해석이 가능하더라도 상당한 시간이 소요됩니다. 일반적으로 해를 구하는데 소요되는 시간은자유도수의 세제곱에 비례하기 때문에 해의 정확도를 크게 떨어뜨리지 않는 한도내에서 자유도수를 줄이는 것이 효과적입니다. + +그림 1.6.23은 Rigid Body Connection과 Rigid Plane Connection 기능을 이용한 예를 나타낸 것입니다. + +그림 1.6.23(a)는 사각튜브의 구조적 거동을 정밀해석하기 위해 정밀검토가 필요한부분에 대해서는 판요소로 세분화하고 나머지 부분은 보요소를 사각튜브의 중립축선상에 모델링하여 두 모델사이를 Rigid Body Connection 기능을 이용하여 강체연결시킨 예입니다. + +그림 1.6.23(b)는 2차원 평면상에 있는 두 개의 기둥이 편심되어 만나는 경우에 절점에서의 편심효과를 고려하기 위해 Rigid Plane Connection 기능을 이용한 예입니다. 이와 같이 임의 평면내에 강체연결기능을 사용하고자 할 때에는 반드시 평면내의 두 개의 선변위성분과 수직방향에 대한 회전변위성분에 대해 기구학적 구속조건을 부여해야 합니다. 마찬가지로 그림 1.6.23(a)와 같이 모든 방향성분에 대해강체연결을 할 경우에는 6개 자유도 전부에 대해 기구학적 구속조건을 부여해야합니다. + +기구학적 구속조건을 동적해석모델에 고려하고자 할 경우에는 주절점의 위치가 종속절점에 입력된 모든 질량성분(자중을 질량으로 환산하여 고려할 경우에는 자중에 의한 질량성분도 포함)의 질량중심(Mass Center)과 일치하도록 입력되어야 합니다. + + + +![](images/page-168_2f41c05451cb42adfef4b1498ab171bea4546fe682834fc06a048b1a6d181a65.jpg) + +
+text_image + +rectangular tube modeled with plate elements +Rigid Link +rectangular tube modeled as a beam element +master node +Z +Y +X +○ : slave nodes (12 nodes) +* all 6 degrees of freedom of +the slave nodes are linked +to the master node. +
+ +(a)한 개의 튜브를 보요소와 판요소로 부위별로 모델링하여 상호연결한 경우 (Rigid Body Connection) +![](images/page-168_cf1e1364063655360c43af06448523913a394e3f0ba0eac34b8b3bcabfbd5a37.jpg) + +
+text_image + +P +P +slave node +master node +eccentricity +eccentricity +* all slave node's d.o.f in the X-Z plane are linked to the master node (translational displacement d.o.f in the X and Z-directions and rotational d.o.f about the Y-axis +
+ +(b) 두개의 기둥이 편심되어 만나는 경우 (Rigid Plane Connection) +그림 1.6.23 강체연결기능의 사용 예 + + + +# 6-11 지지점의 강제변위 + +지지점의 강제변위(Specified Displacements)는 구속되어 있는 자유도의 변위량을 알고 있을 때 그 변위 조건하에서의 구조적 거동을 분석하는데 사용됩니다. + +일반적으로 실무문제에 있어서 이 기능이 효과적으로 사용되는 경우는 다음과 같습니다. + +■ 기존의 구조물에 변형이 발생하여 정밀안전진단이 요구될 경우 +- 특정부위의 거동에 대해 상세모델을 이용하여 정밀분석하고자 할 경우, 구조물의 전체모델에 대한 해석을 수행하여 해당부위의 변위값을 상세모델의 경계조건으로 이용할 경우 +- 기존의 구조물에 지점침하가 발생하여 이를 고려한 해석을 수행하고자 할 경우 +▪ 교량구조물의 지점침하를 고려한 해석을 수행할 경우 + +midas Civil에서 지지점의 강제변위는 하중조건별로 입력이 가능합니다. 또한 구속되지 않은 자유도에 강제변위를 입력하면 프로그램 내부에서 자동으로 해당 자유도에 구속조건을 도입하고 강제변위를 적용하게 됩니다. 만일 강제변위를 부여하고자 하는 자유도에 구속을 도입하지 않은 해석결과를 원한다면 별도의 모델을 만들어 해석을 수행하여야 합니다. + +강제변위를 입력할 때는 미소한 차이에도 구조적 거동이 민감하게 변하기 때문에 정확한 값을 사용하여야 하며, 가능한 한 6개 자유도에 대해 모두 고려하는 것이 바람직합니다. 변형된 구조물의 안전성을 평가할 때와 같이 회전변위를 측정하기가 어려울 경우에는 이동변위만으로도 근사적 해석이 가능하지만 이 경우에는 해석 후 해당부위의 변형형상이 구조물의 실제 변형형상과 유사한지 검토하여야 합니다. + +특정부위의 정밀해석을 위해 전체모델의 해석결과로부터 도출된 변위량을 사용할 경우에는 정밀해석 모델의 경계면에 위치한 절점에는 반드시 6개 자유도 모두에 대해 이동변위 및 회전변위성분을 입력해야 하며, 정밀해석 모델 내에 존재하는 모든 하중조건에 대해서도 추가로 고려하여야 합니다. + + + +강제변위는 일반적으로 전체좌표계를 따라 입력되지만 절점에 절점좌표계가 도입되는 경우에는 절점좌표계를 따라 입력됩니다. + +![](images/page-170_41b8e46ff046435c4b8537d2056fa9d6fb995ccfc025d6c30f501334a656b3d2.jpg) + +
+text_image + +Rigid +are a +section +displa +obtain +the end +the m +each s +
+ +links (master and slave nodes) assigned to the boundary ons, and the specified acements, the displacements ned from the initial analysis for ntire structure, are assigned to master node at the centroid of section. + +(a) 전체 모델과 접합부 상세도 +![](images/page-170_a2e5ec25f807545e0d9108db67d8d289be6574df82e403a7a17a8542d4255806.jpg) + +
+text_image + +connection for a detail analysis +boundary section +boundary section +column member +boundary section +beam (girder) member +beam (girder) member +boundary section +• : node +○ : boundaries for the detail model +(displacements of the total +analysis at this node are +assigned to the detail model +as specified displacements +
+ +(b) 접합부에 대한 상세유한요소 해석모델 +그림 1.6.24 강제변위 기능을 이용한 접합부 정밀해석 예 diff --git a/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_018.md b/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_018.md new file mode 100644 index 00000000..320d284f --- /dev/null +++ b/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_018.md @@ -0,0 +1,231 @@ + + +그림 1.6.24은 구조물의 모서리 접합부에 대한 정밀해석을 수행한 예로써 그 절차는 다음과 같습니다. + +1. 그림 1.6.24(a)의 전체모델에 대한 해석을 수행한 후 정밀해석을 요하는 접합부와 경계부분에서의 변위를 발췌합니다. +2. 경계부분 4개소에서 발췌된 24개(각 절점당 6개 성분)의 변위성분을 그림1.6.24(a)의 모델에 입력합니다. 이때 상세모델의 경계부분에 주절점(Master Node)과 종속절점(Slave Nodes) 관계를 지정하여 전체모델 중하나의 절점으로부터 추출한 변위성분을 상세모델의 전 경계면에 영향을미칠 수 있도록 강체연결기능을 이용하면 편리합니다. 그리고 강체연결기능을 이용하는데 따른 오차를 줄이기 위해 경계면은 가능한 정밀분석 대상부위에서 먼거리에 위치해야 합니다. +3. 전체모델에 고려된 하중조건 중에서 상세해석 모델의 범위 내에 포함되는하중조건을 추가로 입력하고 해석을 수행합니다. + + + + + +# Part 2 midas Civil의 구조해석 기능 + + + +# Chapter 1. 구조해석 기능 + +하중조건하에서 구조물의 실제거동은 엄밀히 재료적으로 비선형성을 가지게 되지만구성부재의 내력이 설계규준에서 정하고 있는 허용범위 내에 있는 경우에는 거의 선형적 거동에 근접하므로 설계목적의 구조해석에서 재료적 비선형성은 일반적으로 고려하지 않습니다. + +midas Civil은 선형해석을 근간으로 하고 있으며 인장 또는 압축력전담요소, P-Delta해석 등의 기하학적 비선형을 고려할 수 있습니다. midas Civil의 구조해석기능은 기본적인 선형해석과 비선형해석으로 구성되어 있고, 실무에서 필요로 하는 다양한 해석기능들을 포함하고 있습니다. + +midas Civil의 구조해석기능의 구체적인 내용들은 다음과 같습니다. + + 정적해석 (Static Analysis) +선형 정적해석 (Linear Static Analysis) +열응력 해석 (Thermal Stress Analysis) +재료비선형 해석 (Material Nonlinear Analysis) +기하비선형 해석 (Geometric Nonlinear Analysis) +대변형 해석 (Large Displacement Analysis) +P-Delta 해석 (P-Delta Analysis) + + 좌굴해석 (Buckling Analysis) + 정적증분해석 (Pushover Analysis) + 수화열 해석 (Heat of Hydration Analysis) + 시공단계별 해석 (Construction Stage Analysis) + 이동하중 해석 (Moving Load Analysis)영향선 해석 (Influence Line Analysis)영향면 해석 (Influence Surface Analysis) + + 동적해석 (Dynamic Analysis)자유진동해석 (Free Vibration Analysis)고유벡터해석 (Eigen Vector Analysis)Ritz벡터 해석 (Ritz Vector Analysis) + + + +응답스펙트럼 해석 (Response Spectrum Analysis) + +시간이력해석 (Time History Analysis) + +경계비선형 시간이력해석 (Boundary Nonlinear Time History Analysis) + +비탄성 시간이력해석 (Inelastic Time History Analysis) + + 구조물의 지점침하를 고려한 해석 (Analysis of Structures subjected to SupportSettlement) + 강합성단면의 합성 전후 단면성질을 고려한 해석 (Composite SectionAnalysis) + 최적화 기법을 사용한 미지하중 계산기능 (Calculation of Unknown Loads byOptimizing Techniques) + +midas Civil은 상기의 각종 하중조건에 대한 해석이 동시에 수행되도록 고안되었습니다. (단, 응답스펙트럼과 시간이력해석은 동시수행 불가) + + + +# Chapter 2. 정적해석 + +midas Civil의 선형정적해석(Linear Static Analysis)에 사용된 기본방정식은 다음과같습니다. + +$$ +[ K ] \{U \} = \{P \} +$$ + +여기서, [ ] K : 구조물의 전체강성행렬 (Stiffness Matrix) + +{ } U : 모든 자유도의 변위벡터 (Displacement Vector) + +{ } P : 작용된 하중벡터 (Load Vector) + +midas Civil은 정적 단위하중 조건과 하중조합 수에 제한이 없습니다. + + + +# Chapter 3. 자유진동 해석 + +# 3-1 고유벡터 해석 + +구조물의 동적 특성을 나타내는 지표인 고유진동수와 모드 형상을 계산하는 방법으로서 midas Civil에서는 고유벡터 해석과 Ritz벡터 해석의 두 가지 방법을 채택하고 있습니다. 두 방법 모두 구조물의 고유치 문제의 특성방정식을 구성하고 그 해를 구하는 방법이지만 후자의 해석 결과를 이용하는 것이 응답스펙트럼해석이나 시간이력해석에서 보다 높은 효율성을 갖는 것으로 알려져 있습니다. 다음은 고유벡터 해석에 관한 설명이며 Ritz벡터 해석에 관해서는 다음 절에서 설명합니다. + +midas Civil에서 비감쇠 자유진동(Undamped Free Vibration) 조건하의 모드형상 (Mode Shape)과 고유주기(Natural Periods)를 구하기 위해 사용된 특성방정식은 다음과 같습니다. + +$$ +\left[ K \right] \left\{\Phi_ {n} \right\} = \omega_ {n} ^ {2} \left[ M \right] \left\{\Phi_ {n} \right\} +$$ + +여기서 [K] : 구조물의 강성행렬 (Stiffness Matrix) + +[M] : 구조물의 질량행렬 (Mass Matrix) + +$\omega_{n}^{2}$ : n번째 모드의 고유치 (Eigenvalue) + +$\{\Phi_{n}\}$ : n번째 모드의 모드형상 (Mode Vector) + +고유치해석은 구조물 고유의 동적특성을 분석하는데 사용되며 자유진동해석(Free Vibration Analysis) 이라고도 합니다. + +고유치해석을 통해 구해지는 구조물의 주요한 동적특성은 고유모드(또는 모드형상), 고유주기(또는 고유진동수), 그리고 모드기여계수(Modal Participation Factor) 등이며 이들은 구조물의 질량과 강성에 의해 결정됩니다. + +고유모드(Vibration Modes)는 구조물이 자유진동(또는 변형) 할 수 있는 일종의 고 + + + +유형상이며, 주어진 모양으로 변형시키기 위해 소요되는 에너지(또는 힘)가 제일적은 것부터 순차적으로 1차 모드형상(또는 기본진동형상), 2차 모드형상, …, n차모드형상이라고 합니다. 그림 2.3.1은 외팔보의 진동모드를 저차부터(적은 에너지로 변형시킬 수 있는 모양부터) 순차적으로 나타낸 것입니다. + +고유주기는 고유모드와 일대일 대응되는 고유한 값으로 구조물이 자유진동상태에서 해당 모드형상으로 1회 진동하는데 소요되는 시간을 의미합니다. + +참고로 단일자유도계에서 고유주기를 구하는 방법은 다음과 같습니다. 단일자유도계의 운동방정식에서 하중과 감쇠항을 0으로 가정하여 자유진동 방정식을 만들면식 (1)과 같은 선형 2차 미분방정식이 됩니다. + +$$ +m \ddot {u} + c \dot {u} + k u = p (t) \tag {1} +$$ + +$$ +m \ddot {u} + k u = 0 +$$ + +여기서 u가 진동에 의한 변위이기 때문에 이를 단순히 u Acos  ωt (여기서 A는초기 변위치와 관련한 상수)라고 가정하면 위 식은 식 (2)와 같습니다. + +$$ +\left(- m \omega^ {2} + k\right) A \cos \omega t = 0 \tag {2} +$$ + +상기의 등식이 항상 만족하기 위해서는 좌변의 괄호내의 값이 0 이 되어야 하므로고유치는 식 (3)과 같은 형태로 구해지게 됩니다. + +$$ +\omega^ {2} = \frac {k}{m}, \omega = \sqrt {\frac {k}{m}}, f = \frac {\omega}{2 \pi}, T = \frac {1}{f} \tag {3} +$$ + +여기서, 2  을 고유치(Eigenvalue)라고 하고,  를 회전고유진동수(RotationalNatural Frequency), f 를 고유진동수(Natural Frequency), T 를 고유주기(NaturalPeriod)라 합니다. + + + +![](images/page-179_15872902395d4cfd5615a3fc0a52c186fa261347b2705543fae6b12fb1ffe4b0.jpg) + +
+natural_image + +Simple curved line diagram with no text or symbols +
+ +1st mode + +![](images/page-179_2d014fd59d897fe80ae479239d5947d9c63614482f0ec03cbc0ea816585f9e5b.jpg) + +
+natural_image + +Pure curved line diagram without any text, numbers, or symbols +
+ +2nd mode + +![](images/page-179_b557e1cf26f49b3375eee1afcda7e78d6f99fdd2bba8c1b98288532ac0ac5810.jpg) + +
+natural_image + +Pure abstract curved line drawing without any text, numbers, or symbols +
+ +3rd mode + +(a) 고유모드형상 +![](images/page-179_cd8281d69551d7908f9e6d8ab351a3ee3eba86ccca0d3c1b1f58ae735187c173.jpg) + +
+text_image + +sec +
+ +λ1=1.87510407 +T1=1.78702sec + +![](images/page-179_f50253c21439439dcafa7eb0fd094fae446f8643dce08958a3fff68ec9d17bdc.jpg) + +
+text_image + +sec +
+ +λ= 4.69409113 +T2=0.28515sec + +![](images/page-179_54d52560112aa8d2d11bcc67193a571d73594ad47db1673b68329660f3b18fc5.jpg) + +
+text_image + +sec +
+ +λ3 = 7.85475744 +T3=0.10184sec + +(b) 고유주기 + +그림 2.3.1 균일단면을 가진 외팔보의 고유모드형상 및 고유주기 + + + +그리고 모드기여계수는 해당 모드의 영향을 총 모드에 대한 비율로 나타낸 것으로식 (4)와 같이 표현됩니다. + +$$ +\tau_ {m} = \frac {\sum M _ {i} \varphi_ {i m}}{\sum M _ {i} \varphi_ {i m} ^ {2}} \tag {4} +$$ + +여기서 $\tau _ { m }$ : 모드기여계수(Modal Participation Factor) + +m : 임의의 모드차수 (Mode Number) + +$M _ { \scriptscriptstyle i }$ : 임의의 i 위치의 질량 (Mass) + +$\varphi _ { _ { i m } }$ ：i m ≌ (Mode Shape) + +일반내진설계기준에서는 해석에 포함되는 모드별 유효질량(Effective Modal Mass)의합이 전체 질량의 90% 이상을 확보하도록 요구하고 있습니다. 이는 해석결과에영향을 주는 대부분의 주요모드를 포함하도록 하기 위한 것입니다. + +$$ +M _ {m} = \frac {\left[ \sum \varphi_ {i m} M _ {i} \right] ^ {2}}{\sum \varphi_ {i m} ^ {2} M _ {i}} \tag {5} +$$ + +여기서 $M _ { { \scriptscriptstyle m } } \Subset \Sigma \subseteq \varXi$ 유효질량(Effective Modal Mass)입니다. + +임의 질량의 자유도가 구속되어 있을 경우 총 질량에는 반영되지만 해당 자유도의모드벡터가 억제되어 질량성분이 유효질량에는 포함되지 않습니다. 그러므로 모드별 유효질량을 계산하여 전체질량에 대한 비를 평가하고자 할 경우에는 질량이 입력된 성분의 자유도가 구속되지 않도록 해야 합니다. + +특히 건축구조물에서 지하구조물의 횡변위가 구속된 경우, 해당층의 횡질량성분은입력할 필요가 없습니다. + +구조물의 동적거동을 제대로 분석하기 위해서는 고유치를 결정하는 질량과 강성을정확하게 반영하는 것이 가장 기본이 되는 작업입니다. 여기서 강성은 구조부재를유한요소로 모델링 함으로써 거의 모든 강성성분을 비교적 근접하게 반영할 수 있으나, 질량은 구조부재 자체의 질량이 전체 질량에 비해 적기 때문에 바닥 슬래브등 모델에 포함되지 않은 재료에 대한 질량성분을 정확하게 파악하여 입력하는 것 diff --git a/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_019.md b/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_019.md new file mode 100644 index 00000000..23183e22 --- /dev/null +++ b/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_019.md @@ -0,0 +1,292 @@ + + +이 매우 중요합니다. + +질량성분은 절점당 6개의 자유도성분에 따라 일반적으로 이동질량성분(Translational Masses) 3개와 회전질량성분(Rotational Mass Moment of Inertia) 3개로입력됩니다. 여기서 회전질량성분은 회전질량관성에 기인한 것으로 내진설계에서는 지진이 선방향 지진가속도로 가해지기 때문에 동적응답에 직접적인 영향을 미치지는 않으나, 구조물이 비정형일 때(질량중심과 강성중심이 일치하지 않을 때)는모드형상을 일부 변형시킴으로써 동적응답에 간접적인 영향을 미치게 됩니다. 질량성분은 다음과 같이 계산됩니다. (표 2.3.1 참조) + +\- 이동질량성분 + +$$ +\int d m +$$ + +\- 회전질량관성모멘트 + +$$ +\int r ^ {2} d m +$$ + +여기서, r은 전체 무게중심에서 해당 미소질량성분 중심까지의 거리 + +질량의 입력단위계는 중량을 중력가속도로 나눈 단위([중량(시간2 /길이)])와 같으며회전질량관성모멘트의 단위계는 질량에 길이단위의 제곱을 곱한 단위([중량(시간2 /길이)길이2 ])와 같습니다. 예를 들면 MKS 또는 English 단위계를 사용할 경우에는중량에 중력가속도를 나눈 값을 질량으로 입력해야 하며, SI 단위계를 사용할 경우에는 MKS 단위계에서 사용되는 중량치를 그대로 질량으로 입력합니다. 그리고 탄성계수나 하중을 입력할 때는 MKS 단위계에서 사용되는 값에 중력가속도를 곱하여 입력하면 됩니다. + +midas Civil에서는 해석작업의 효율성을 고려하여 집중질량(Lumped Mass)을 사용하고 있습니다. 질량데이터는 Main Menu의 Load탭>Load Type그룹>Static Loads>Structure Loads/Masses그룹>Nodal Masses 또는 Loads to Masses 기능을 이용하여 입력됩니다. + +midas Civil에서 사용된 고유치 해석방법은 Subspace Iteration Method와 대형 구조물의 해석에 적합한 Lanczos Method가 있습니다. + + + +
ShapeTranslation MassRotational Mass Moment of Inertial
Rectangular Shape $M = \rho bd$ $I_m = \rho \left( \frac{bd^3}{12} + \frac{db^3}{12} \right)$ $= \frac{M}{12} \left( b^2 + d^2 \right)$
Triangular Shape $M = \rho \times \text{area of triangle}$ $I_m = \rho \left( I_x + I_y \right)$
Circular Shap $M = \rho \left( \frac{\pi d^2}{4} \right)$ $I_m = \rho \left( \frac{\pi d^4}{32} \right)$
General Shape $M = \rho \times \int dA$ $I_m = \rho \left( I_x + I_y \right)$
Linear Shape $\rho_L = \text{mass per unit length}$ $M = \rho_L \times L$ $I_m = \rho_L \left( \frac{L^3}{12} \right)$
Eccentric Masseccentric mass : mM = mrotational mass moment of inertia about its mass center : $I_o$ $I_m = I_o + mr^2$
+ +표 2.3.1 질량데이터의 산정방법 + + + +# 3-2 Ritz벡터 해석 + +Ritz 벡터 해석은 구조물의 동적 특성을 나타내는 고유진동수와 모드형상을 구하는 방법으로써 이렇게 구해진 고유진동수와 모드형상은 구조물의 동적 특성을 표현하는데 있어서 고유벡터 해석에 의한 것보다 효율적인 것으로 알려져 있습니다. 이 방법은 다자유도 구조물의 모드 형상을 가정하여 단자유도 구조물로 치환한 뒤고유진동수를 구하는 Rayleigh-Ritz 방법을 확장한 것입니다. + +먼저 n 자유도의 구조물 운동방정식에서 변위 벡터가 다음과 같이 p개의 Ritz Vector의 조합으로 표현된다고 가정합니다. 이 때 p는 n보다 작거나 같습니다. + +$$ +M \ddot {u} (t) + C \dot {u} (t) + K u (t) = p (t) \tag {6} +$$ + +$$ +u (t) = \sum_ {i = 1} ^ {p} \psi_ {i} z _ {i} (t) = \Psi z (t) \tag {7} +$$ + +여기서 M : 구조물의 질량행렬 + +C : 구조물의 감쇠행렬 + +K : 구조물의 강성행렬 + +u(t) : n자유도 구조물의 변위 벡터 + +$z(t)$ : 일반화 좌표(Generalized Coordinate) 벡터 + +$p(t)$ : 동적 하중 벡터 + +$\psi_{i}$ : i번째 Ritz 벡터 + +$z_{i}(t)$ : i 번째 일반화 좌표 + +$\Psi=\left[\psi_{I}\cdots\psi_{i}\cdots\psi_{p}\right]^{T}$ : Ritz 벡터 행렬 + +위 가정에 의해서 n자유도의 운동방정식은 다음과 같이 p자유도의 운동방정식으로 축소됩니다. + +$$ +\tilde {M} \ddot {z} (t) + \tilde {C} \dot {z} (t) + \tilde {K} z (t) = \tilde {p} (t) \tag {8} +$$ + +여기서, $\tilde{M}=\Psi^{T}M\Psi$ : 축소된 운동방정식의 질량행렬 + +$\tilde{C}=\Psi^{T}C\Psi$ : 축소된 운동방정식의 감쇠행렬 + + + +$$ +\tilde {K} = \Psi^ {T} K \Psi \quad : \text { 축소된 운동방정식의 질량행렬 } +$$ + +$$ +\tilde {p} (t) = \Psi^ {T} p (t) \quad : \text { 축소된 운동방정식의 동적하중 벡터 } +$$ + +축소된 운동방정식에 대해서 다음과 같은 고유치 문제를 구성하고 해석을 수행합니다. + +$$ +\tilde {K} \tilde {\varphi} _ {i} = \tilde {\omega} _ {i} ^ {2} \tilde {M} \tilde {\varphi} _ {i} \tag {9} +$$ + +여기서, $\tilde{\varphi}_{i}$ : 축소된 운동방정식의 모드 형상 + +$\tilde{\omega}_{i}$ : 축소된 운동방정식의 고유진동수 + +위 고유치 문제의 해를 이용하면 고전적 감쇠행렬을 가정할 때 축소된 운동방정식을 다음과 같이 각 모드의 단자유도 운동방정식으로 분리할 수 있습니다. + +$$ +\ddot {q} _ {i} (t) + 2 \xi_ {i} \tilde {\omega} _ {i} \dot {q} _ {i} (t) + \tilde {\omega} _ {i} ^ {2} q (t) = \frac {\Psi^ {T} \tilde {p} _ {i} (t)}{\Psi^ {T} M \Psi} \tag {10} +$$ + +$$ +z (t) = \sum_ {i = 1} ^ {p} \tilde {\varphi} _ {i} q _ {i} (t) \tag {11} +$$ + +여기서, $q_{i}(t)$ : i번째 모드 좌표 + +$\xi_{i}$ : i번째 모드 감쇠비 + +축소된 운동방정식의 고유치 해석해 $\tilde{\omega}_{i}$ 는 원래 운동방정식의 고유진동수에 대한 근사해를 의미합니다. + +$$ +\omega_ {i} = \tilde {\omega} _ {i} \tag {12} +$$ + +여기서, $\omega_{i}$ : i번째 모드형상의 근사해 + +구조물의 모드형상은 운동방정식의 변위벡터와 모드좌표 사이의 사상(寫像; Mapping) 관계를 정의해주는 벡터입니다. 따라서 Ritz 벡터 해석에 의한 근사적 모드 형상은 원래 운동방정식의 변위 벡터 $u(t)$ 와 모드좌표 $q_{i}(t)$ 사이의 관계식에 의해 정의되며 그 식은 다음과 같습니다. + + + +$$ +u (t) = \Psi z (t) = \sum_ {i = 1} ^ {p} \left[ \Psi \tilde {\varphi} _ {i} \right] q _ {i} (t) \tag {13} +$$ + +따라서 i 번째 모드형상의 근사해는 다음과 같이 정의됩니다. + +$$ +\varphi_ {i} = \Psi \tilde {\varphi} _ {i} \tag {14} +$$ + +여기서 φi : i번째 모드형상의 근사해 + +Ritz 벡터 해석에 의한 근사적 모드형상 벡터는 고유치해석에 의한 것과 마찬가지로 원래의 질량 및 강성행렬에 대하여 직교성(Orthogonality)을 갖습니다. + +Ritz 벡터 해석에 의한 고유진동수와 모드형상의 근사해는 일반 고유치해석의 해와 마찬가지로 모드기여계수(Modal Participation Factor)와 모드별 유효질량(Effective Modal Mass)의 계산에 사용됩니다. + +Ritz 벡터 해석 결과를 기초로 모드중첩법에 의한 시간이력해석을 수행하는 경우에는 상기 모드 운동방정식 (10)을 사용합니다. + +구조물의 변형형상을 가정하는 Ritz 벡터는 일반적으로 구조물에 가해지는 하중에대한 변위를 반복적으로 계산하여 생성하게 됩니다. + +먼저 사용자가 초기하중벡터(Initial Load Vector)를 선정합니다. 여기서 기본가정은동적하중이 시간에 따라 변화하지만 각 자유도별 분포는 사용자가 지정한 초기하중벡터를 따른다는 것입니다. 다음으로는 선정된 초기하중벡터에 대해서 일차적으로 정적 해석을 수행하여 첫번째 Ritz 벡터를 구합니다. + +$$ +K \psi^ {(1)} = r ^ {(1)} +$$ + +$$ +\psi^ {(1)} = K ^ {- 1} r ^ {(1)} +$$ + +여기서, K : 구조물의 강성행렬 + + + +(1) ψ : 첫번째 Ritz 벡터 + +(1) r : 사용자 지정 초기하중벡터 + +이렇게 해서 얻어진 첫번째 Ritz 벡터를 구조물의 변위로 가정합니다. 그러나 위의정적해석은 구조물의 동적 응답에 의해서 발생하는 관성력의 영향을 무시하고 있습니다. 따라서 추가적인 반복계산을 통해 관성력에 의한 변위를 계산하게 됩니다.먼저 구조물의 가속도 분포는 앞서 계산된 변위벡터, 즉 첫번째 Ritz 벡터를 따른다고 가정합니다. 따라서 가속도에 의해 발생하는 관성력은 질량벡터를 곱해서 계산되며, 이 관성력이 구조물에 추가적인 변위를 발생시키는 하중으로 작용한다고가정하여 다시 정적해석을 수행합니다. + +$$ +K \psi^ {(2)} = M \psi^ {(1)} +$$ + +$$ +\psi^ {(2)} = K ^ {- 1} M \psi^ {(1)} +$$ + +여기서, M : 구조물의 질량행렬 + +( 2 ) ψ : 두번째 Ritz 벡터 + +이렇게 해서 얻어진 두번째 Ritz 벡터 역시 정적평형(Static Equilibrium)만을 나타내는 위 식에서 고려되지 못한 가속도 분포를 나타낸다고 가정하여 상기 과정을 반복하면서 사용자가 지정한 개수만큼의 Ritz 벡터를 계산합니다. + +사용자는 복수의 초기하중벡터를 지정할 수 있으며 각각에 대해서 생성할 Ritz 벡터의 개수를 개별적으로 지정할 수 있습니다. 단, 생성할 Ritz 벡터의 전체 개수는구조물 운동방정식에 존재하는 실제 모드 개수를 넘을 수 없습니다. 또한 반복과정에서 이미 생성된 Ritz 벡터에 대해서 선형종속(Linearly Dependent)인 Ritz 벡터가 계산되면 이를 삭제하게 됩니다. 이 때문에 더 이상 선형독립(LinearlyIndependent)인 Ritz벡터가 계산될 수 없으면 반복과정을 종료하게 되고, 이는 사용자가 지정한 초기하중벡터만으로는 지정한 개수만큼의 모드를 구할 수 없음을의미합니다. + +midas Civil에서 사용자가 선정할 수 있는 초기하중벡터는 전체좌표계 X, Y 및 Z + + + +방향 지반가속도에 의한 관성력, 사용자가 입력한 모든 정적하중 조건(Static LoadCase), 비선형 연결요소의 부재력 벡터(Nonlinear Link Force Vector)입니다. + +전체좌표계 X, Y 및 Z 방향 지반가속도에 의한 관성력은 주로 해당 방향의 지반가속도에 발생하는 변위에 관련된 Ritz 벡터를 구하기 위해 사용됩니다. + +사용자 입력 정적하중 조건은 특정한 분포를 갖는 동적하중에 대한 Ritz 벡터를구하고자 할 때에 사용합니다. 통상적인 정적하중 조건(고정하중, 적재하중, 풍하중등)을 사용하거나 Ritz 벡터 생성을 위한 목적으로 인위적인 정적하중 조건을 만들어 사용할 수 있습니다. + +비선형 연결요소의 부재력 벡터는 각 비선형 연결요소에서 발생하는 부재력의 구조물에 대한 영향을 반영하는 Ritz 벡터를 생성하기 위한 것입니다. 요소가 가진6개의 변위 자유도 가운데 사용자가 체크한 것에 대해서만 개별적으로 단위 힘을갖는 초기하중벡터를 구성하여 Ritz 벡터 생성에 이용합니다. (그러나 비선형 연결요소를 포함한 구조물의 해석에 있어서 반드시 이를 이용해야 하는 것은 아니며,사용자의 판단에 따라 주어진 해석조건 하의 구조물 변형형상을 충분히 반영할 수있는 초기하중벡터를 선정합니다.) + +고유치해석과 비교할 때에 Ritz 벡터 해석의 장점은 다음과 같습니다. + +Ritz 벡터는 적은 수의 모드를 계산하더라도 실제 하중에 대한 정적해석 해에 기초하기 때문에 그 안에 고차모드의 영향이 자동적으로 반영되어 있습니다. 예를들어서 Ritz 벡터 해석에 의해 구해진 1차 모드의 모드형상과 고유치해석에 의해구해진 1차 모드의 모드형상의 차이는 전자에 반영된 고차모드의 성분을 의미한다고 할 수 있습니다. 또한 구조물에 작용하는 하중에 의해 가진 되는 모드형상만이구해지므로 불필요한 모드의 계산이 배제됩니다. + +이상과 같은 원리로 Ritz 벡터 해석은 정확한 해석 결과를 얻기 위해 필요한 모드의 개수를 줄여줍니다. 예를 들면, 특정한 모드별 유효질량의 합계를 확보하기 위해 필요로 하는 모드의 개수는 일반적으로 Ritz 벡터 해석의 경우가 고유치해석의경우보다 적은 것으로 알려져 있습니다. + + + +# Chapter 4. 감쇠의 고려 + +# 4-1 감쇠의 개요 + +동적해석에서 구조물의 감쇠는 크게 다음과 같이 분류할 수 있습니다. + +▪ 비례 감쇠 + +질량비례형 + +강성비례형 + +Rayleigh 형 + +Caughey 형 + +■ 비비례 감쇠 + +에너지 비례형 + +■ 요소별 감쇠 + +점성감쇠 (Voigt 형태, Maxwell 형태) + +이력형감쇠 + +마찰감쇠 + +내부마찰 감쇠(재료감쇠) + +외부마찰 감쇠 + +미끌림 마찰감쇠 + +일산감쇠 + +동적해석에서 구조물의 감쇠는 운동 방정식을 구성하는 감쇠행렬을 강성과 질량의 비율로 표현할 수 있는지의 여부에 따라서, 크게 비례 감쇠(Proportional Damping)와 비비례 감쇠(Non-proportional Damping)로 분류할 수 있습니다. + +비비례 감쇠는 구조물의 부분별로 서로 다른 감쇠특성을 갖는 재질을 사용하거나 + + + +부가적인 감쇠장치가 도입된 경우에 구조물을 구성하는 각각의 감쇠기구를 별개로평가하여 각 부분의 감쇠모델로부터 감쇠행렬을 구성하는 방법입니다. 그러나, 실제 구조물에 있어서 감쇠문제는 매우 복잡하여 세부적인 감쇠 매카니즘을 파악하기 곤란한 경우가 많습니다. + +따라서, 진동해석에서는 고유진동해석으로부터 얻어진 주요한 모드성분의 감쇠 성질을 적절히 표현하여 얻어진 비례감쇠로 감쇠행렬을 가정하는 경우가 일반적입니다. 비례감쇠는 ‘고전적감쇠(Classical Damping)’라고도 불리우며, 고유모드행렬을감쇠행렬 좌우에 곱하였을 때 대각성분만을 갖는 행렬을 구할 수 있으므로 모드별로 감쇠비를 분리할 수 있습니다. + +한편, 비비례 감쇠 특성을 고려하는 경우, 모드별로 감쇠비를 분리할 수 없으므로구조물의 고유치 해석을 통해 얻어지는 모드 형상에 기초하여 변형율 에너지 개념을 도입하여 모드별 감쇠비를 구할 수 있습니다. + +midas Civil에서 감쇠방법은 응답 스펙트럼 해석에서는 Response Spectrum LoadCases에서 설정하며, 시간이력 해석에서는 Time History Load Cases에서 설정합니다. 동적해석의 해석방법에 따라서, 설정가능한 감쇠방법은 다음과 같습니다. + + 응답 스펙트럼 해석 및 모드중첩법에 의한 시간이력해석의 감쇠설정 +Modal +Mass & Stiffness Proportional (질량비례형, 강성비례형, Rayleigh형 감쇠) +Strain Energy Proportional + + 직접 적분법에 의한 시간이력해석의 감쇠설정 + +Modal + +Mass & Stiffness Proportional (질량비례형, 강성비례형, Rayleigh형 감쇠) + +Strain Energy Proportional + +Element Mass & Stiffness Proportional (Rayleigh형 감쇠) + + + +또한, 범용연결요소에 선형점성 감쇠인 Damping 혹은 Effective Damping을 설정하여 구조물에 부가적으로 설치되는 선형감쇠기(Kelvin Model)의 모델링도 가능합니다. 단, 범용연결요소의 선형 점성감쇠는 응답 스펙트럼 해석 및 모드중첩법에 의한 해석인 경우, 감쇠방법을 Strain Energy Proportional로 선택한 경우만 모드별 감쇠비에 반영되어 간접적으로 적용됩니다. 직접적분법에 의한 시간이력해석인 경우,감쇠방법을 Mass & Stiffness Proportional 혹은 Element Mass & StiffnessProportional로 설정한 경우에 요소감쇠행렬을 통해서 해석에 직접적으로 반영되며,Strain Energy Proportional인 경우는 모드별 감쇠비에 반영되어 간접적으로 적용됩니다. + +다음은 모드중첩법과 직접적분법에서의 감쇠의 고려방법에 대해 설명합니다. 구조물의 운동방정식은 다음과 같이 구성됩니다. + +$$ +M \ddot {u} (t) + C \dot {u} (t) + K u (t) = p (t) \tag {1} +$$ + +여기서, M : 질량행렬 + +C : 감쇠행렬 + +K : 강성행렬 + +u t( ) , u t  ( ) , u t ( ) : 절점의 변위, 속도, 가속도 + +p ( )t : 동적하중 + +응답 스펙트럼 해석 및 모드중첩법에 의한 진동해석의 개념은 식 (1)을 모드의 직교성을 이용하여 모드 분해하여 식(2)와 같이 모드 분해된 각 모드의 운동 방정식의 해를 중첩하여 해석합니다. 따라서, 고유치해석이 필수적으로 선행되어야 합니다. + +$$ +\ddot {q} _ {i} (t) + 2 \xi_ {i} \omega_ {i} \dot {q} _ {i} (t) + \omega_ {i} ^ {2} q (t) = \frac {\phi_ {i} ^ {T} p (t)}{\phi_ {i} ^ {T} M \phi_ {i}} \tag {2} +$$ + +여기서, i : i-번째 모드의 모드형상 벡터 + +i : i -번째 모드의 감쇠비 diff --git a/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_020.md b/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_020.md new file mode 100644 index 00000000..151309e0 --- /dev/null +++ b/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_020.md @@ -0,0 +1,355 @@ + + +：ーー + +( ) q t , ( ) q t  , ( ) q t  : i -번째 모드의 일반화 변위, 속도, 가속도 + +따라서, 응답 스펙트럼 해석 및 모드중첩법에 의한 진동해석에서는 감쇠의 방법에관계없이 모드 감쇠비 i 에 의해 감쇠가 고려됨을 알 수 있습니다. + +직접적분법에 의한 시간이력해석은 수치적분법에 의해 2계의 운동 방정식 (1)의해를 직접 구합니다. 따라서, 운동 방정식 구성시에 감쇠행렬을 작성할 필요가 있습니다. + +이하에서는 각 해석방법과 감쇠방법에 대한 감쇠비  와 감쇠행렬 C 의 작성방법에대해 설명합니다. + + + +# 4-2 비례감쇠 + +질량 비례형 감쇠는 공기 저항 등에 의한 외부 점성 감쇠를 표현한 것으로, 감쇠 행렬이 질량에 비례한다고 가정하고 있습니다. 한편 강성 비례형 감쇠는 일산감쇠 효과(진동에너지의 지반에의 방출 효과)가 직접 표현되기 어려우므로 그 효과를 강성에 비례한다고 가정하기 때문에 고차 모드의 감쇠를 과대평가할 우려가 있습니다. + +비례 감쇠 매트릭스 C 의 일반형태는, Caughey에 의해 다음과 같이 정의됩니다. + +$$ +C = M \cdot \sum_ {j = 0} ^ {N - 1} a _ {j} \left(M ^ {- 1} K\right) ^ {j} \tag {3} +$$ + +여기서, j, N: 절점의 자유도(모드 차수) + +식(1)에서 $M^{-1}K$ 는 이하와 같이 비감쇠계의 자유진동식에서 구할 수 있습니다. + +$$ +M \{\ddot {y} \} + K \{y \} = 0 \tag {4} +$$ + +$$ +\{y \} = \{u \} e ^ {i a x} \tag {5} +$$ + +로 가정하고, 이것을 식 (4)에 대입하면 + +$$ +\left(- \omega^ {2} M + K\right) \{u \} = \{0 \} \tag {6} +$$ + +로 되어 식 (6)로부터 $M^{-1}K = \omega^{2}$ 로 됩니다. 여기서, $\omega^{2}$ 는 모드 수만큼 존재하기 때문에 모드 차수를 고려하여 $\omega_{s}^{2}$ 로 표기합니다. + +식 (4)\~(6)에서 구한 $M^{-l}K$ 을 식 (3)에 대입하고, 좌측에 $\{u_{s}\}^{T}$ 을 곱하고 우측에 $\{u_{s}\}$ 을 곱하면 식 (3)은 다음과 같이 표현됩니다. + +$$ +\left\{u _ {s} \right\} ^ {T} C \left\{u _ {s} \right\} = C _ {s} = \sum_ {j = 0} ^ {N - 1} a _ {j} \cdot \omega_ {s} ^ {2 j} \cdot \left\{u _ {s} \right\} ^ {T} M \left\{u _ {s} \right\} = \sum_ {j = 0} ^ {N - 1} a _ {j} \cdot \omega_ {s} ^ {2 j} \cdot M _ {s} \tag {7} +$$ + +또한 s차 모드의 감쇠 정수, $\xi_{s}$ 는 다음과 같이 표현할 수 있습니다. + + + +$$ +C _ {s} = 2 \xi_ {s} \cdot \omega_ {s} \cdot M _ {s} \tag {8} +$$ + +식(7), (8)에서 N개의 고유 모드에 대한 감쇠정수 $\xi_{s}$ 는 다음과 같이 결정됩니다. + +$$ +\begin{array}{l} \xi_ {s} = \frac {C _ {s}}{2 \omega_ {s} \cdot M _ {s}} = \frac {1}{2 \omega_ {s}} \sum a _ {j} \cdot \omega_ {s} ^ {2 j} \\ = \frac {1}{2} \left(\frac {a _ {0}}{\omega_ {s}} + a _ {1} \cdot \omega_ {s} + a _ {2} \cdot \omega_ {s} ^ {3} + \dots + a _ {N - 1} \cdot \omega_ {s} ^ {2 N - 3}\right), \quad s = 1 - N \tag {9} \\ \end{array} +$$ + +질량 비례형, 강성 비례형태의 감쇠 정수 및 감쇠 행렬은 각각 다음과 같이 표현됩니다. + +$$ +\xi_ {s} = \frac {a _ {0}}{2 \omega_ {s}}, \quad C = a _ {0} M = 2 \xi_ {s} \omega_ {s} M: \text {질량 비례형} \tag {10} +$$ + +$$ +\xi_ {s} = \frac {a _ {I} \cdot \omega_ {s}}{2}, \quad C = a _ {I} K = \frac {2 \xi_ {s}}{\omega_ {s}} K: \text { 강성 비례형 } \tag {11} +$$ + +질량비례형 혹은 강성비례형 감쇠는 Response Spectrum Load Cases 혹은 Time History Load Cases에서 Damping Method를 Mass & Stiffness Proportional로 선택하고, 질량비례형은 Mass Proportional, 강성비례형은 Stiffness Proportional을 선택하여 설정합니다. 구체적인 설정방법은 Rayleigh감쇠에서 다루고 있습니다. + +![](images/page-193_e940d939326294f20793b6a424858aab84bf2d58a0acd12f1e0c0dbc98a66ba2.jpg) + +
+line +| Natural frequencies ωs | ξs | +| --------------------- | ------ | +| ω₁ | High | +| ω₂ | Medium | +| ω₃ | Low | +| ω₄ | Very Low | +
+ +(a) Mass Proportional Damping + +![](images/page-193_2c4d5cb2bc8fc6eb3f06186979b8a0661d41459b80943acc25c54fce083682b2.jpg) + +
+line + +| Natural frequencies ωs | Stiffness Proportional | +| --------------------- | --------------------- | +| ω₁ | ξs | +| ω₂ | ξs = (a₁·ωs)/2 | +| ω₃ | ξs | +| ω₄ | ξs | +
+ +(b) Stiffness Proportional Damping +그림 2.4.1 모드별 감쇠율 + + + +# 4-3 Rayleigh 감쇠 + +Rayleigh형 감쇠는 강성 비례형 감쇠에서의 고차 모드의 감쇠 정수를 수정한 것으로, 그림 2.4.2(b)에 나타낸 것과 같이 감쇠행렬을 구조물의 질량행렬과 강성행렬의 선형합으로서 구성합니다. i차모드의 감쇠정수 $\xi_{i}$ 와 고유진동수 $\omega_{i}$ 및 j차 모드의 감쇠정수 $\xi_{j}$ 와 고유진동수 $\omega_{j}$ 가 주어졌을 때 Rayleigh형 감쇠의 감쇠행렬은 다음과 같이 표현됩니다. 단, i, j차 모드는 구조물의 주요한 2개의 모드를 의미합니다. + +$$ +C = a _ {0} M + a _ {I} K \tag {12} +$$ + +$$ +\xi_ {s} = \frac {1}{2} \left(\frac {a _ {0}}{\omega_ {s}} + a _ {1} \cdot \omega_ {s}\right) \tag {13} +$$ + +여기서, + +$$ +a _ {0} = \frac {2 \cdot \omega_ {i} \cdot \omega_ {j} \left(\xi_ {i} \cdot \omega_ {j} - \xi_ {j} \cdot \omega_ {i}\right)}{\left(\omega_ {j} ^ {2} - \omega_ {i} ^ {2}\right)} \tag {14} +$$ + +$$ +a _ {I} = \frac {2 \left(\xi_ {j} \cdot \omega_ {j} - \xi_ {i} \cdot \omega_ {i}\right)}{\left(\omega_ {j} ^ {2} - \omega_ {i} ^ {2}\right)} \tag {15} +$$ + +![](images/page-194_a219847656a5a6585c27bde7a84d18b7c646ac177356c3e7cabb541ba8a953a2.jpg) + +
+line +| Natural frequencies ωs | Mass Proportional (C = a₀M) | Stiffness Proportional (C = a₁K) | +| --------------------- | --------------------------- | ------------------------------- | +| ω₁ | ~0 | ~0 | +| ω₂ | ~-0.5 | ~0.5 | +| ω₃ | ~-1.5 | ~1.5 | +| ω₄ | ~-2.5 | ~2.5 | +
+ +(a) Mass Proportional Damping과 Stiffness Proportional Damping + +![](images/page-194_e43e326fa83e2f1a98ce2333dada1e75874b32c16a6929c2316371a223589564.jpg) + +
+line + +| Natural frequencies ωs | ξs (solid line) | ξs (dashed line) | +| ---------------------- | --------------- | ---------------- | +| ωi | High | Low | +| ωj | Low | High | +
+ +(b) Rayleigh Damping +그림 2.4.2 모드별 감쇠율과 고유진동수와의 관계 + + + +$a_{0}$ 과 $a_{1}$ 은 Response Spectrum Load Cases 혹은 Time History Load Cases에서 다음과 같은 방법으로 설정할 수 있습니다. + +# 1. Direct Specification + +사용자가 $a_{0}$ , $a_{1}$ 값을 직접 입력합니다. + +# 2. Calculate from Modal Damping + +고유치해석을 통해 얻어진 진동수 혹은 고유주기, 그리고 i, j차 모드의 감쇠비를 사용자가 입력하면, 식 (14), (15)를 이용하여 $a_{0}$ , $a_{1}$ 값을 자동계산합니다. + +예를 들어, i, j차 진동수와 모드의 감쇠비가 각각 $f_{i}=1.0Hz$ , $f_{j}=1.25Hz$ $\xi_{i}=0.05$ , $\xi_{j}=0.05$ 라 할 때, $a_{0}$ , $a_{1}$ 값을 구하면 다음과 같습니다. + +■ 고유진동수 + +$$ +\omega_ {1} = \frac {2 \pi}{1 . 0} = 6. 2 8, \omega_ {2} = \frac {2 \pi}{0 . 8} = 7. 8 5 +$$ + +■ 식(14), (15)를 이용한 $a_{0}$ , $a_{1}$ 의 수계산 + +$$ +a _ {0} = \frac {2 \cdot 6 . 2 8 \cdot 7 . 8 5 (0 . 0 5 \cdot 7 . 8 5 - 0 . 0 5 \cdot 6 . 2 8)}{7 . 8 5 ^ {2} - 6 . 2 8 ^ {2}} = 0. 3 4 9 +$$ + +$$ +a _ {I} = \frac {2 (0 . 0 5 \cdot 7 . 8 5 - 0 . 0 5 \cdot 6 . 2 8)}{7 . 8 5 ^ {2} - 6 . 2 8 ^ {2}} = 0. 0 0 7 +$$ + +\- midas Civil에서의 $a_{0}, a_{1}$ 의 자동계산 + +![](images/page-195_780c73f23f94b0a1dfb3927c1d866643e67646670f3bde5edcec6eb51e768544.jpg) + +
+text_image + +Damping +Damping Method : Mass & Stiffness Proportional +Mass and Stiffness Coefficients +Damping Type : Mass +Proportional Stiffness +Proportional +Direct Specification : 0,3492 0,0052 +Calculate from Modal Damping : 0,34906584444 0,00707355314 +Coefficients Calculation +Frequency [Hz] : 1 1,25 +Period [sec] : 1,0 0,8 +Damping Ratio : 0,05 0,05 +
+ + + +Rayleigh형 감쇠는 응답 스펙트럼해석, 모드중첩법 및 직접적분법에 의한 시간이력해석에서 사용가능하며, Response Spectrum Load Cases 혹은 Time History Load Cases에서 Damping Method를 Mass & Stiffness Proportional로 선택하고, Mass Proportional과 Stiffness Proportional을 모두 선택하여 설정합니다. 해석방법에 따른 감쇠의 고려방법은 다음과 같습니다. + +# 4-3-1 응답 스펙트럼 해석 및 모드중첩법에서의 Rayleigh 감쇠의 고려 + +응답 스펙트럼 해석 및 모드중첩법에 의한 진동해석은 구조물의 운동방정식을고유치해석시에 설정한 모드의 수만큼 분해하여 각 모드의 운동 방정식을 중첩하여 해석합니다. 따라서, Rayleigh형 감쇠를 사용할 경우, 주요한 2개의 모드만으로 결정된 $a_{0}$ , $a_{1}$ 값을 식(13)에 대입하여 사용하는 모드의 수만큼의 감쇠비를 구할 필요가 있습니다. + +midas Civil에서 주요한 2개의 모드만으로 결정된 $a_{0}$ , $a_{1}$ 값을 이용하여 모드별 감쇠비를 구하는 방법은 이하와 같습니다. + +예를 들어, $a_{0}=0.35$ , $a_{1}=0.005$ 이고, 3차 모드까지를 고려할 경우의 모드별 감쇠 비 $\xi_{s}$ 를 구하면 다음과 같습니다. 단, 고유진동수는 $\omega_{1}=4.59215$ , $\omega_{2}=9.81814$ , $\omega_{3}=14.57793$ 라 가정합니다. + +■ 1, 2, 3차 모드의 감쇠비 계산 + +$$ +\xi_ {s} = \frac {1}{2} \left(\frac {a _ {0}}{\omega_ {s}} + a _ {1} \cdot \omega_ {s}\right) +$$ + +$$ +\xi_ {1} = \frac {1}{2} \left(\frac {1}{4 . 5 9 2 1 5} 0. 3 5 + 0. 0 0 5 \cdot 4. 5 9 2 1 5\right) = 0. 0 4 9 5 9 +$$ + +$$ +\xi_ {2} = \frac {1}{2} \left(\frac {1}{9 . 8 1 8 1 4} 0. 3 5 + 0. 0 0 5 \cdot 9. 8 1 8 1 4\right) = 0. 0 4 2 3 7 +$$ + +$$ +\xi_ {3} = \frac {1}{2} \left(\frac {1}{1 4 . 5 7 7 9 3} 0. 3 5 + 0. 0 0 5 \cdot 1 4. 5 7 7 9 3\right) = 0. 0 4 8 4 5 +$$ + + + + 1, 2, 3차 모드의 감쇠비 계산 + +RAYLEIGH DAMPING COEFFICIENT, TIME LOADCASE = 1 + +MASS COEFFICIENT. : 0.35000 STIFFNESS COEFFICIENT. : 0.00500 + +MODE FREQUENCY DAMPING RATIO NO. [RAD/SEC] + +1 4.59215E+00 4.95889E-02 +2 9.81814E+00 4.23695E-02 +3 1.45779E+01 4.84493E-02 + +단, 위와 같이 구한 모드별 감쇠비가  s >1 인 경우는  s  0.9999 , $\xi _ { s } < 0$ 인 경우는=0.0 + +# 4-3-2 직접적분법에서의 Rayleigh감쇠의 고려 + +직접적분법에서의 Rayleigh형 감쇠는 주요한 2개의 모드만으로 결정된 $a _ { 0 } \ , a _ { 1 }$ 값을이용하여 $C = a _ { \theta } M + a _ { I } K$ 와 같이 감쇠행렬을 작성하여 운동 방정식을 구성하고,수치적분법에 의해 해를 구합니다. + +직접적분법을 이용한 비선형시간이력해석에서 구조물이 탄성한계를 넘어 소성영역으로 들어갈 경우, 감쇠행렬 $C = a _ { \theta } M + a _ { I } K$ 에서 강성 K 를 초기상태의 강성을 그대로 사용하면 감쇠력이 과대하게 평가될 가능성이 있습니다. + +midas Civil에서는 부재가 항복하여 강성이 갱신되면 갱신된 강성을 감쇠행렬 구성시에 반영하는 기능을 제공합니다. 감쇠행렬 구성시 강성의 갱신은 Rayleigh형 감쇠에 근거하여 감쇠행렬을 구성하는 Mass & Stiffness Proportional과 Element Mass& Stiffness Proportional 감쇠인 경우만 적용됩니다. + +설정방법은 Time History Load Cases에서 감쇠방법을 Mass & Stiffness Proportional혹은 Element Mass & Stiffness Proportional을 선택하고, Damping Matrix Update를Yes로 선택하면 감쇠행렬 구성시 갱신된 강성을 사용하고, No로 선택하면 부재의상태에 관계없이 초기강성으로 감쇠행렬을 구성합니다. + + + +# 4-4 변형율 에너지에 기초한 모드 감쇠 + +# 4-4-1 변형율 에너지에 기초한 모드 감쇠의 개요 + +실제 구조물은 재료에 따라서 감쇠특성이 상이하며 국부적으로 감쇠장치를 설치하기도 합니다. midas Civil에서는 Element Mass & Stiffness Proportional을 이용하여 요소별로 다른 감쇠 특성을 지정할 수 있습니다. 그러나 요소별로 감쇠 특성을 각각 고려할 경우, 감쇠 행렬은 대부분 비고전적 감쇠가 되어서 모드 분리가 되지 않습니다. 따라서 응답스펙트럼해석 및 모드중첩법을 이용한 해석에서는 요소별로 서로 다른 감쇠 특성을 반영하기 위해서 변형율에너지의 개념에 기초해 모드별 감쇠비를 산정합니다. + +midas Civil에서는 변형율에너지에 기초한 모드감쇠는 응답스펙트럼해석 및 모드 중첩법, 직접 적분법에 의한 시간이력해석에 사용 가능합니다. 변형율에너지에 기초한 모드감쇠는 Response Spectrum Load Cases 혹은 Time History Load Cases에서 Damping Method를 Strain Energy Proportional로 선택하여 설정합니다. 단, 직접 적분법에 의한 시간이력해석에 변형율에너지에 기초한 모드감쇠를 고려할 경우, 감쇠행렬은 Full Matrix형태가 되므로, 모드 중첩법에 비해 계산시간이 과도하게 늘어나는 문제가 발생할 수 있습니다. + +점성감쇠를 갖는 단자유도 진동계의 감쇠비는 조화운동(Harmonic Motion)에서 소산되는 에너지(Dissipated Energy)와 구조물에 저장되는 변형율에너지(Strain Energy) 사이의 비율로 정의할 수 있으며 다음 식과 같습니다. + +$$ +\xi = \frac {E _ {D}}{4 \pi E _ {S}} \tag {16} +$$ + +여기서 + +$E_{D}$ : 소산에너지 + +$E_{S}$ : 변형율에너지 + + + +![](images/page-199_4d7b1b4024249e8941faffb84a028b68543046daa071ee50b2c785f4e5f38fc0.jpg) + +
+text_image + +F = F_D + F_S +E_D = 2\pi \cdot h \cdot KA^2 +: Dissipated Energy +F_S = K_S u +u +A +E_S = \frac{1}{2} KA^2 : Strain Energy +F_D = C u +
+ +그림 2.4.3 소산에너지와 변형율에너지 + +다자유도 구조물에 있어서, 특정 모드의 동적거동은 해당되는 고유진동수를 갖는 단자유도 진동계의 동적거동으로 파악될 수 있습니다. 이 때 특정 요소를 대상으로 소산에너지와 변형율에너지를 계산하는데 있어서 두 가지 가정을 사용합니다. 먼저 구조물의 변형은 모드형상에 비례한다고 가정합니다. i-번째 모드만이 해당 고유진동수로 조화진동을 하는 구조물에서 요소 절점의 변위 및 속도 벡터는 다음과 같이 쓸 수 있습니다. + +$$ +\begin{array}{l} u _ {i, n} = \phi_ {i, n} \sin \left(\omega_ {i} t + \theta_ {i}\right) \\ \dot {u} _ {i, n} = \omega_ {i} \phi_ {i, n} \cos \left(\omega_ {i} t + \theta_ {i}\right) \tag {17} \\ \end{array} +$$ + +여기서 $u_{i,n}$ : i-번째 모드의 진동에 의한 n-번째 요소 절점 변위 + +$\dot{u}_{i,n}$ : i -번째 모드의 진동에 의한 n-번째 요소 절점 속도 + +$\phi_{i,n}$ : n-번째 요소의 자유도에 해당되는 i-번째 모드의 형상 + +$\omega_{i}$ : i-번째 모드의 고유진동수 + +$\theta_{i}$ : i-번째 모드의 위상각(Phase Angle) + + + +두 번째로, 요소의 감쇠는 요소강성에 비례하는 점성감쇠로 가정합니다. + +$$ +C _ {n} = \frac {2 h _ {n}}{\omega_ {i}} K _ {n} \tag {18} +$$ + +여기서 $C_{n}$ : n-번째 요소의 감쇠행렬 + +$K_{n}$ : n-번째 요소의 강성행렬 + +$h_{n}$ : n-번째 요소의 감쇠비 + +위 식의 가정에 의해, 요소의 소산에너지와 변형율 에너지는 다음과 같이 나타낼 수 있습니다. + +$$ +\begin{array}{l} E _ {D} (i, n) = \pi u _ {i, n} ^ {T} C _ {n} \dot {u} _ {i, n} = 2 \pi h _ {n} \phi_ {i, n} ^ {T} K _ {n} \phi_ {i, n} \\ E _ {S} (i, n) = \frac {1}{2} u _ {i, n} ^ {T} K _ {n} u _ {i, n} = \frac {1}{2} \phi_ {i, n} ^ {T} K _ {n} \phi_ {i, n} \tag {19} \\ \end{array} +$$ + +여기서 $E_{D}(i,n)$ : i-번째 모드의 진동에 의한 n-번째 요소의 소산에너지 + +ES (i, n) : i-번째 모드의 진동에 의한 n-번째 요소의 변형율에너지 + +전체구조물의 i-번째 모드 감쇠비는 모든 요소의 i-번째 모드에 해당되는 에너지의 합으로 계산할 수 있으며 다음과 같습니다. + +$$ +\xi_ {i} = \frac {\sum_ {n = 1} ^ {N} E _ {D} (i , n)}{4 \pi \cdot \sum_ {n = 1} ^ {N} E _ {S} (i , n)} = \frac {\sum_ {n = 1} ^ {N} h _ {n} \phi_ {n , i} ^ {T} K _ {n} \phi_ {n , i}}{\sum_ {n = 1} ^ {N} \phi_ {n , i} ^ {T} K _ {n} \phi_ {n , i}} \tag {20} +$$ + +# 4-4-2 변형율 에너지에 기초한 모드 감쇠의 설정 및 계산 + +midas Civil에서 변형율 에너지에 기초한 모드 감쇠의 설정은, 우선, Group에서 서로 다른 감쇠 특성을 지정할 요소와 경계를 그룹으로 지정합니다. Group Damping의 Damping Ratio for Specified Elements and Boundaries내의 Strain Energy Proportional Damping에서 요소그룹 및 경계그룹별로 Damping Ratio를 지정합니다. 요소그룹 및 경계그룹에 포함되지 않는 부분은 Default Values for Unspecified Elements and Boundaries의 Strain Energy Proportional Damping에서 Damping Ratio diff --git a/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_021.md b/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_021.md new file mode 100644 index 00000000..12c65dd2 --- /dev/null +++ b/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_021.md @@ -0,0 +1,321 @@ + + +를 지정합니다. + +고유치해석을 수행하면, 이상의 과정으로 설정한 요소그룹 및 경계그룹별Damping Ratio를 이용하여 변형율 에너지에 기초한 각 모드별 감쇠비를 계산하며,Modal Damping Ratio based on Group Damping의 Modal Damping Ratio를 통하여계산된 결과를 확인할 수 있습니다. 단, Group Damping에서 Calculate Only WhenUsed를 선택하면, 시간이력해석에서 감쇠방법을 Strain Energy Proportional으로 지정했을 경우만, 모드별 감쇠비를 계산하므로 주의할 필요가 있습니다. + +![](images/page-201_bf2134dfa5165f70e89cb7b0f1ceed2b06aea3a19ac75dac56c6f6795a90d414.jpg) + +
+text_image + +Group Damping : Strain Energy Proportional +Default Values for Unspecified Elements and Boundaries +Damping Ratio : 0,05 (0,00 ~ 1,00) +Damping Ratio for Specified Elements and Boundaries +Type: Material Structure Boundary +Name of Material / Group : Cross beam +Use Material Data Direct Define +Damping Ratio : 0,05 (0,00 ~ 1,00) +Name Type Damping Ratio +SM490 Material 0,02 +SM400 Material 0,02 +C27 Material 0,05 +Cross beam Structure 0,05 +Strt Group 2 Structure 0,05 +Add +Modify +Delete +Calculate Only When Used +Damping Ratio Select Option OK Cancel +
+ +![](images/page-201_5c39fe2b8e1b30e6dd3261e19aebe047c7ff6dec633c43f9dc901e4d7c9d3524.jpg) + +
+scatter + +Modal Damping Ratio based on Group Damping +| Mode No. | Frequency(Hz) | Period (sec) | M.P.M. X (%) | M.P.M. Y (%) | M.P.M. Z (%) | Modal Damping Ratio | +|---|---|---|---|---|---|---| +| 1 | 1.516833 | 0.059269 | 0.421441 | 62.89271 | 0.000006 | 0.038782 | +| 2 | 1.947404 | 0.513504 | 63.51830 | 2.460253 | 0.088640 | 0.033136 | +| 3 | 2.131171 | 0.465225 | 0.096991 | 0.006675 | 0.001452 | 0.023911 | +| 4 | 2.489797 | 0.401639 | 16.04919 | 2.228820 | 0.790667 | 0.026394 | +| 5 | 2.652879 | 0.376949 | 0.421931 | 3.149565 | 1.388275 | 0.026245 | +| 6 | 3.192462 | 0.313238 | 13.47970 | 0.213074 | 0.103476 | 0.026844 | +| 7 | 3.720826 | 0.268758 | 0.064054 | 0.016115 | 58.53938 | 0.021697 | +Calculation of Mass and Stiffness Coefficients +Damping Option | Mass Proportional | Stiffness Proportional +Mode No. | Frequency(Hz) | Damping Ratio | Calc. | C=Alpha+M + Beta+K +Mode 1 | Mode 1 | Mode 2 | Alpha | Beta | Alpha 0.0 | Alpha 0.0 | +|---|---|---|---|---|---|---| +| Mode 1 | 1.51683 | | 0.0387819 | | | | +| Mode 2 | 1.51683 | | 0.0387819 | | | | +
+ +(a) 요소그룹 및 경계그룹별 Damping Ratio의 설정 (b) 변형율 에너지에 기초하여 계산된 모드별감쇠 + +그림 2.4.4 Strain Energy Damping의 설정 및 모드별 감쇠 + +응답스펙트럼해석 및 모드중첩법에 의한 해석에서는 구조물의 운동 방정식을 모드별로 분해하여, 각 모드의 운동 방정식에 변형율에너지에 기초하여 구한 모드별감쇠비  s 을 적용하여 해를 구합니다. + +직접적분법에 의한 시간이력해석에서는 변형율에너지에 기초하여 구한 모드별 감쇠비  s 와 고유진동수 i , 모드행렬 등을 이용하여 전체 구조물의 감쇠행렬을 작성하여 운동 방정식을 구성합니다. 감쇠행렬의 작성방법은 별도로 다루도록 합니다. + + + +# 4-5 모드별 감쇠 + +모드별 감쇠는 각 모드별로 사용자가 직접 감쇠비를 정의하고 정의된 모드별 감쇠비에 따라서 모드별 응답을 계산합니다. 모드별 감쇠는 응답 스펙트럼해석 및 모드중첩법, 직접적분법에 의한 시간이력해석에서 사용가능합니다. 단, 직접적분법에 의한 시간이력해석에 모드별 감쇠를 고려할 경우, 감쇠행렬은 비대칭 행렬이 되므로 모드 중첩법에 비해 계산시간이 과도하게 늘어나는 문제가 발생할 수 있습니다. + +모드별 감쇠의 설정은 Response Spectrum Load Cases 혹은 Time History Load Cases에서 Damping Method를 Modal로 선택하여, Modal Damping Overrides에서 모드별로 감쇠비를 입력합니다. 모드별로 감쇠비를 입력하지 않는 모드의 감쇠비는 Damping Ratio for All Modes로 입력합니다. + +응답스펙트럼해석과 모드중첩법에 의한 해석에서는 구조물의 운동 방정식을 모드별로 분해하여, 각 모드의 운동 방정식에 사용자가 직접 입력한 모드별 감쇠비 $\xi_{s}$ 을 적용하여 해를 구합니다. + +직접적분법에 의한 시간이력해석에서는 입력된 모드별 감쇠비 $\xi_{s}$ 와 고유진동수 $\omega_{s}$ 그리고 모드행렬 등을 이용하여 전체 구조물의 감쇠행렬을 작성하여 운동 방정식을 구성합니다. 감쇠행렬의 작성은 별도로 다루도록 합니다. + + + +# 4-6 요소별 Rayleigh 감쇠 + +요소별 Rayleigh 감쇠는 구조물을 구성하는 특정한 부재 또는 경계부분에 요소별로 다른 감쇠를 적용할 수 있는 기능으로, 구조물에서 감쇠가 서로 다른 재료가 흔재하거나 제진 및 면진장치가 설치되어 있는 경우 등에 사용합니다. + +요소별로 감쇠 특성을 각각 고려할 경우, 감쇠 행렬은 대부분 비비례 감쇠가 되어서 모드 분리가 되지 않습니다. 따라서, 요소별 Rayleigh 감쇠는 감쇠행렬을 직접 작성하는 직접적분법에 의한 시간이력해석시에만 적용가능하며, Time History Load Cases에서 Damping Method를 Element Mass & Stiffness Proportional로 선택하여 설정합니다. + +응답스펙트럼해석 및 모드중첩법을 이용한 해석에서 요소별로 서로 다른 감쇠 특성을 반영하기 위해서는 Group Damping에서 요소그룹 및 경계조건 그룹별로 지정된 감쇠비를 설정하여, 고유치해석을 통해 변형율에너지의 개념에 기초한 모드별 감쇠비를 산정하여 해석을 수행할 필요가 있습니다. + +midas Civil에서 요소별 Rayleigh 감쇠의 설정은, 우선, Group에서 서로 다른 감쇠 특성을 지정할 요소와 경계를 그룹으로 지정합니다. Group Damping의 Damping Ratio for Specified Elements and Boundaries내의 Element Mass & Stiffness Proportional Damping에서 요소그룹 및 경계그룹별로 Mass Coefficient(α)와 Stiffness Coefficient(β)를 지정합니다. 요소그룹 및 경계그룹에 포함되지 않은 부분은 Default Values for Unspecified Elements and Boundaries의 Element Mass & Stiffness Proportional Damping에서 지정합니다. + +요소별 Rayleigh 감쇠는 요소별로 입력된 $\alpha,\beta$ 값을 이용하여 $C=\alpha M+\beta K$ 와 같이 요소의 감쇠행렬을 작성하여 운동 방정식을 구성합니다. 요소별 Rayleigh 감쇠는 Rayleigh감쇠에 기초하므로, 부재 n의 $\alpha_{n},\beta_{n}$ 은 Rayleigh감쇠와 동일하게 계산합니다. + +단, 현재의 midas Civil에서는 Mass Coefficient(α)는 지원되지 않으므로, 요소별 강성비례형 감쇠로 취급됩니다. + + + +# 4-7 감쇠행렬의 구성 + +직접적분법에 의한 시간이력해석에서 감쇠방법을 Modal 혹은 Strain Energy Proportional로 선택한 경우, 감쇠행렬은 Full Matrix 형태가 되며, 입력된 모드별 감쇠비 $\xi_{s}$ 와 고유진동수 $\omega_{i}$ 그리고 모드행렬 등을 이용하여 전체 구조물의 감쇠행렬 작성할 필요가 있습니다. + +전체 구조물의 감쇠행렬은 다음 식으로 구성됩니다. + +$$ +\mathbf {C} = \mathbf {M} \boldsymbol {\Phi} \left[ \begin{array}{c c c} \ddots & & \\ & 2 \xi_ {i} \omega_ {i} & \\ & & \ddots \end{array} \right] \boldsymbol {\Phi} ^ {T} \mathbf {M} +$$ + +여기서 C: 전체구조물의 감쇠행렬 + +M: 전체구조물의 질량 행렬 + +$\xi_{i}$ 전체구조물의 i-번째 모드 감쇠비 + +Φ: 모드 형상 + +$\Phi=\left\{\Phi_{1}\quad\Phi_{2}\quad\ldots\quad\Phi_{i}\quad\ldots\quad\Phi_{nf}\right\}\quad nf:$ 사용되는 모드 수 + + + +# 4-8 범용연결요소의 선형감쇠의 고려 + +범용연결요소는 제진장치, 면진장치, 압축 또는 인장 전담요소, 소성힌지, 지반스프링 등을 모델링 하는데 사용되는 요소로서 2개의 절점을 연결하는 6개의 스프링으로 구성됩니다. 범용연결요소는 선형점성감쇠를 설정하여 구조물에 부가적으로 설치되는 감쇠기를 모델링 할 수 있습니다. + +범용연결요소의 선형점성감쇠는 Element Type인 경우, Linear Dashpot과 Spring and Linear Dashpot를 선택하여 Linear Properties의 Damping을 통해 설정되며, Force Type인 경우 Linear Properties의 Effective Damping을 통해 설정됩니다. + +범용연결요소의 선형점성감쇠에 관한 상세한 사항은 범용연결요소에서 다루기로 합니다. 여기서는 변형율 에너지에 기초한 모드 감쇠를 고려할 경우 범용연결요소의 선형점성감쇠를 고려하여 모드별 감쇠비를 구하는 방법에 관해서 설명합니다. + +범용연결요소의 선형점성감쇠 Damping 혹은 Effective Damping는 다음과 같이 입력된다고 가정합니다. + +$$ +C _ {e f f} = \frac {2 \xi_ {e f f}}{\omega_ {e f f}} K _ {e f f} +$$ + +여기서 $C_{eff}$ : Damping 또는 Effective Damping + +$K_{eff}$ : 범용연결요소의 강성 + +$\xi_{eff}$ : 범용연결요소의 감쇠비 + +$\omega_{eff}$ : 범용연결요소의 고유진동수 + +위 식의 가정에 의해, 범용연결요소의 변형율 에너지의 계산시에 선형점성감쇠를 반영하면 i-번째 모드 감쇠비는 다음과 같이 표현됩니다. + +$$ +\xi_ {i} = \frac {\sum_ {n = 1} ^ {N} E _ {D} (i , n)}{4 \pi \cdot \sum_ {n = 1} ^ {N} E _ {S} (i , n)} = \frac {\sum_ {n = 1} ^ {N} \left(h _ {n} \phi_ {n , i} ^ {T} K _ {n} \phi_ {n , i} + 0 . 5 \omega_ {i} \phi_ {n , i} ^ {T} C _ {\text { eff }} \phi_ {n , i}\right)}{\sum_ {n = 1} ^ {N} \phi_ {n , i} ^ {T} K _ {n} \phi_ {n , i}} +$$ + +위 식으로 계산된 모드별 감쇠비는 응답 스펙트럼해석 및 모드중첩법, 직접적분법에 의한 시간이력해석에서 동일하게 적용됩니다. + + + +# Chapter 5. 응답스펙트럼 해석 + +midas Civil의 응답스펙트럼해석(Response Spectrum Analysis)에서 지반운동이 가해지는 구조물의 동적 평형방정식은 다음과 같습니다. + +$$ +[ M ] \ddot {u} (t) + [ C ] \dot {u} (t) + [ K ] u (t) = - [ M ] w _ {g} (t) +$$ + +여기서 [M] : 질량행렬 (Mass Matrix) + +[C] : 감쇠행렬 (Damping Matrix) + +[K] : 강성행렬 (Stiffness Matrix) + +wg $\mathsf { w } _ { \mathfrak { g } }$ : 지반가속도 + +이고, u(t)와 u t ( ) , u t ( ) 은 각각 상대변위, 속도, 가속도를 의미합니다. + +응답스펙트럼해석법은 다자유도시스템을 단일자유도시스템의 복합체로 가정하여미리 수치적분 과정을 통해 계산된 임의 주기(또는 진동수) 영역내의 최대응답치에대한 스펙트럼(가속도, 속도, 변위 등)을 이용하여 조합 해석하는 방법으로 설계용스펙트럼을 이용한 내진설계에 주로 활용됩니다. + +응답스펙트럼해석법에서는 임의 모드에서의 최대응답치를 각 모드별로 구한 다음,적정한 조합방법을 이용하여 조합함으로써 최대응답치를 예견하게 됩니다. 예를들어 내진 해석시 임의 모드의 임의 자유도에 대한 변위와 관성력은 식 (1)과 같이 계산됩니다. + +$$ +d _ {x m} = \Gamma_ {m} \varphi_ {x m} S _ {d m}, F _ {x m} = \Gamma_ {m} \varphi_ {x m} S _ {a m} W _ {x} \tag {1} +$$ + +여기서 + +$\Gamma _ { \mathsf { m } }$ : m차 모드에서의 모드기여계수 + + mx : 임의 x위치에서의 m차 모드벡터 + +$\mathsf { S } _ { \mathsf { d m } }$ : m차 주기에서의 Normalized Spectral Displacement + +$\mathsf { S } _ { \mathsf { a m } }$ : m차 주기에서의 Normalized Spectral Acceleration + +${ \sf W } _ { \sf x }$ : 임의 x 위치에서의 질량 + +구조해석 프로그램에서 임의 주기치에 대한 Spectral Data가 입력되면 해석된 고유 + + + +midas Civil에서는 모드별 조합과정에서 삭제된 부호를 재생하여 응답스펙트럼해석결과에 반영할 수 있다. + +주기에 해당하는 Spectral Value를 찾기 위해 일반적으로 선형보간법을 사용하기 때문에 Spectral Curve의 변화가 많은 부위에 대해서는 가능한 세분화된 데이터를 사용하는 것이 바람직합니다. (그림 2.5.1 참조) 그리고 Spectral Data의 주기범위는 반드시 고유치해석시 산출된 최소·최대 주기범위를 포함할 수 있도록 입력되어야 합니다. 그리고 midas Civil에서는 내진해석시 사용되는 Spectral Data를 규준에 따른 동적계수, 지반계수, 지역계수, 중요도계수, 반응수정계수 등의 입력으로 쉽게 생성할 수 있어 편리합니다. 반응수정계수 값은 부재 설계시 적용되는 것이 보다 일반적인 방법입니다. + +midas Civil은 전체좌표계 X-Y 평면의 임의의 방향과 Z방향에 대한 응답스펙트럼해석이 가능하며, 모드별 해석결과의 조합(Modal Combination)은 사용자의 선택에 따라 CQC(Complete Quadratic Combination)방법과 SRSS(Square Root of the Sum of the Squares)방법 등을 사용할 수 있습니다. + +각 모드별 응답을 조합하는 방법은 다음과 같습니다. + +\- SRSS (Square Root of the Sum of the Squares + +$$ +R _ {\max} = \left[ R _ {1} ^ {2} + R _ {2} ^ {2} + \dots + R _ {n} ^ {2} \right] ^ {1 / 2} \tag {2} +$$ + +\- ABS (ABsolute Sum) + +$$ +R _ {\max} = \left| R _ {1} \right| + \left| R _ {2} \right| + \dots + \left| R _ {n} \right| \tag {3} +$$ + +– CQC (Complete Quadratic Combination) + +$$ +R _ {\max} = \left[ \sum_ {i = 1} ^ {N} \sum_ {j = 1} ^ {N} R _ {i} \rho_ {i j} R _ {j} \right] ^ {1 / 2} \tag {4} +$$ + +여기서 $\rho_{ij}=\frac{8\xi^{2}(1+r)r^{3/2}}{(1-r^{2})^{2}+4\xi^{2}r(1+r)^{2}}$ ， $r=\frac{\omega_{j}}{\omega_{i}}$ + +$R_{max}$ : 최대응답치 + +$R_{i}$ : 임의 i차 모드에서의 최대응답치 + +r :i번째 모드에 대한 j번째 모드의 고유진동수 비율 + +ξ : 감쇠비 (Damping Ratio) + + + +상기 식 (4)에서 i = j 이면, 감쇠비(ξ)에 관계없이 $\rho_{ij} = 1$ 이 되고, 감쇠비(ξ)가 0인 경우 CQC와 SRSS의 결과가 동일한 값을 가집니다. + +위 방법 중에서 ABS가 가장 큰 조합치를 산출합니다. SRSS는 고유진동수들이 근접한 값을 가질 경우, 조합결과가 과대 또는 과소평가 되는 경향이 있기 때문에 종래에는 SRSS가 주로 사용되었으나 최근에는 모드간 확률적인 상관도를 고려할 수 있도록 고안된 CQC방법의 사용이 늘고 있습니다. + +예를 들어, 감쇠비가 0.05이고 3개의 자유도를 가진 구조물의 고유진동수와 각 모드별 변위가 다음과 같이 계산되었을 경우, SRSS와 CQC의 적용결과를 비교하면 다음과 같습니다. + +-고유진동수 + +$$ +\omega_ {1} = 0. 4 6, \omega_ {2} = 0. 5 2, \omega_ {3} = 1. 4 2 +$$ + +\- 모드별 최대변위 : $D_{ij}$ (j 고유차수에 대한 i 자유도의 변위성분) + +$$ +D _ {i j} = \left| \begin{array}{c c c} 0. 0 3 6 & 0. 0 1 2 & 0. 0 1 9 \\ - 0. 0 1 2 & 0. 0 4 4 & - 0. 0 0 5 \\ 0. 0 4 9 & 0. 0 0 2 & - 0. 0 1 7 \end{array} \right| +$$ + +\- 각 자유도에 대한 응답치를 SRSS에 의해 구하면 + +$$ +R _ {\max} = \left[ R _ {1} ^ {2} + R _ {2} ^ {2} + R _ {3} ^ {2} \right] ^ {1 / 2} = \left\{0. 0 4 2 \quad 0. 0 4 6 \quad 0. 0 5 2 \right\} +$$ + +그리고 CQC를 사용하여 구하면 + +$$ +\rho_ {1 2} = \rho_ {2 1} = 0. 3 9 8 5 +$$ + +$$ +\rho_ {1 3} = \rho_ {3 1} = 0. 0 0 6 1 +$$ + +$$ +\rho_ {2 3} = \rho_ {3 2} = 0. 0 0 8 0 +$$ + +$$ +R _ {\max} = \left[ R _ {1} ^ {2} + R _ {2} ^ {2} + R _ {3} ^ {2} + 2 \rho_ {1 2} R _ {1} R _ {2} + 2 \rho_ {1 3} R _ {1} R _ {3} + 2 \rho_ {2 3} R _ {2} R _ {3} \right] ^ {1 / 2} +$$ + +$$ += \left\{ \begin{array}{c c c} 0. 0 4 6 & 0. 0 4 1 & 0. 0 5 3 \end{array} \right\} +$$ + + + +상기의 두 가지 결과를 비교해보면 SRSS 방법을 사용할 경우가 CQC에 비해 첫번째 자유도성분에 대해서는 과소평가되고, 두번째 자유도성분에 대해서는 과대평가 되었습니다. 따라서 고유진동수들이 상대적으로 근접한 값을 가질 때 SRSS 방법은 과소 또는 과대평가된 결과를 산출함을 알 수 있습니다. + +![](images/page-209_636e8fde130c06228e09dbe29264b97ff7e0f13044df1d8d2b2bc3843c0a5a80.jpg) + +
+line + +| Period (Sec) | Special Data | +| ------------ | ------------ | +| T1 | S1 | +| T2 | S2 | +| T3 | S3 | +| T4 | S4 | +| T5 | S5 | +| T6 | S6 | +| T7 | S7 | +| Tn-1 | Sn-1 | +| Tn | Sn | +
+ +그림 2.5.1 스펙트럼곡선 및 임의 주기에 대한 스펙트럼데이터의 참조 방법 + + + +# Chapter 6. 시간이력해석 + +midas Civil의 시간이력해석(Time History Analysis)에 사용된 동적평형방정식은 다음과 같습니다. + +$$ +[ M ] \ddot {u} (t) + [ C ] \dot {u} (t) + [ K ] u (t) = p (t) +$$ + +여기서 [M] : 질량행렬 (Mass Matrix) + +[C] : 감쇠행렬 (Damping Matrix) + +[K] : 강성행렬 (Stiffness Matrix) + +p(t) : 동적하중 + +이고, u(t)와 u t ( ) , u t ( ) 는 각각 변위, 속도, 가속도를 의미합니다. + +시간이력해석은 구조물에 동적하중이 작용할 경우에 동적 평형방정식의 해를 구하는 것으로, 구조물의 동적특성과 가해지는 하중을 사용하여 임의의 시간에 대한구조물 거동(변위, 부재력 등)을 계산하게 됩니다. midas Civil에서는 시간이력해석을 위해 모드중첩법(Modal Superposition Method)과 직접적분법(Direct Integration)을사용하고 있습니다. + +다음은 모드중첩법과 직접적분법에 대한 개략적인 개념과 데이터의 입력시 주의해야 할 사항을 서술합니다. diff --git a/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_022.md b/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_022.md new file mode 100644 index 00000000..c9c704eb --- /dev/null +++ b/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_022.md @@ -0,0 +1,336 @@ + + +# 6-1 모드 중첩법 + +구조물의 변위를 서로 직교성을 갖는 변위형상의 선형조합 형태로 구하는 방법으로 다음 식과 같이 표현됩니다. 이 방법에서는 감쇠행렬이 질량행렬과 강성행렬의 선형조합으로 이루어질 수 있다고 가정합니다. + +$$ +[ C ] = \alpha [ M ] + \beta [ K ] \tag {1} +$$ + +$$ +\Phi^ {T} M \Phi \dot {q} (t) + \Phi^ {T} C \Phi \dot {q} (t) + \Phi^ {T} K \Phi q (t) = \Phi^ {T} F (t) \tag {2} +$$ + +$$ +m _ {i} \ddot {q} _ {i} (t) + c _ {i} \dot {q} _ {i} (t) + k _ {i} q _ {i} (t) = P _ {i} (t) \quad (i = 1, 2, 3, \dots , m) \tag {3} +$$ + +$$ +u (t) = \sum_ {i = j} ^ {m} \Phi_ {i} q _ {i} (t) \tag {4} +$$ + +$$ +q _ {i} (t) = e ^ {- \xi_ {i} \omega_ {i} t} \left[ q _ {i} (0) \cos \omega_ {D i} t + \frac {\xi_ {i} \omega_ {i} q _ {i} (0) + q _ {i} (0)}{\omega_ {D i}} \sin \omega_ {D i} t \right] \tag {5} +$$ + +$$ ++ \frac {1}{m _ {i} \omega_ {D i}} \int_ {0} ^ {t} P _ {i} (\tau) e ^ {- \xi_ {i} \omega_ {i} (t - \tau)} \sin \omega_ {D i} (t - \tau) d \tau +$$ + +$$ +\text { 여기서 } \quad \omega_ {D i} = \omega_ {i} \sqrt {1 - \xi_ {i} ^ {2}} +$$ + +α, β : Rayleigh 계수 + +ξi : i 번째 모드의 감쇠비 + +ωi : i 번째 모드의 고유주기 + +$\Phi_{i}$ : i 번째 모드 형상 + +$q_{i}(t)$ : i 번째 모드에 의한 단자유도 방정식의 해 + +시간이력해석에서 구조물의 변위는 식 (4)와 같이 모드 형상과 단일자유도계 방정식의 해와의 곱으로 결정되고, 변위의 정확성은 사용하는 모드 수에 영향을 받습니다. 이 방법은 구조해석 프로그램에서 가장 많이 사용되는 것으로 대형구조물의 선형 동적해석에 매우 효과적인 방법입니다. 그러나 비선형 동적해석이나 특별한 감쇠장치가 포함되어 감쇠를 강성과 질량의 선형 조합으로 가정할 수 없을 경우 사용할 수 없는 단점이 있습니다. + +모드중첩법을 이용할 경우 요구되는 데이터와 입력시 주의사항은 다음과 같습니다. + + + + 전체 해석시간(또는 해석횟수) : 해석하고자 하는 시간이나 해석횟수 + 해석시간 간격 : 해석에 사용되는 시간간격으로, 해석의 정확성에 상당 한영향을 미칠 수 있으며, 시간간격의 크기는 구조물의 고차모드의 주기나하중의 주기와 밀접한 관계를 갖습니다. 해석시간 간격은 식 (5)의 적분항에 직접적인 영향을 주게 되어 부적절한 값이 입력되는 경우 부정확한 결과를 나타낼 수 있습니다. 일반적으로 고려하고자 하는 최고차모드 주기의1/10 정도의 시간간격이 타당합니다. 또한, 해석시간 간격은 입력된 하중의 시간간격보다는 작아야 합니다. + +$$ +\Delta t = \frac {T _ {p}}{1 0} +$$ + +Tp = 고려하고자 하는 최고차모드 주기 + + 모드별 감쇠비(또는 Rayleigh 계수) : 구조물의 감쇠를 결정하기 위해 필요로 하는 값으로 전체 구조물의 감쇠비나 각 모드별 감쇠비 + 동적하중 : 구조물의 절점이나 기초부에 직접 가해지는 동적하중으로 시간의 함수로 표시되며, 전체 하중 변화를 충분히 나타낼 수 있어야 합니다.입력되지 않은 시간에서의 하중 값은 선형보간하여 사용합니다. + + + +# 6-2 직접적분법 + +직접적분법은 동적평형방정식을 시간에 따라 점진적으로 적분하여 해를 구하는 방법입니다. 평형방정식의 형태 변화 없이 시간단계마다 적분을 사용하여 해를 구하게 되고 사용방법에 따라 다양한 적분방법이 사용될 수도 있습니다. midas Civil 에서는 수렴성이 좋은 Newmark 방법을 사용하여 직접적분법을 수행하고 있습니다. 기본적인 가정과 적분방법은 아래와 같습니다. + +[기본가정] + +$$ +{ } ^ { t + \Delta t } \dot { u } = { } ^ { t } \dot { u } + [ ( 1 - \delta ) { } ^ { t } \ddot { u } + \delta { } ^ { t + \Delta t } \ddot { u } ] \Delta t \tag {6} +$$ + +$$ +u ^ {t + \Delta t} = ^ {t} u + ^ {t} \dot {u} \Delta t + \left[ \left(\frac {1}{2} - \alpha\right) ^ {t} \ddot {u} + \alpha^ {t + \Delta t} \ddot {u} \right] \Delta t ^ {2} \tag {7} +$$ + +식 (7)에서 $t+\Delta t$ $\ddot{u}$ 를 구하고 이 값을 식 (6)에 대입하여 $t+\Delta t$ $\dot{u}$ 를 계산하면 식 (8)과 같이 이전 단계의 변위, 속도, 가속도와 현재의 변위로 표현할 수 있습니다. + +$$ +{ } ^ { t + \Delta t } \ddot { u } = f \left( \begin{array} { c c c c c c } { } ^ { t + \Delta t } u & , & ^ { t } u & , & ^ { t } \dot { u } & , & ^ { t } \ddot { u } \end{array} \right) +$$ + +$$ +{ } ^ { t + \Delta t } \dot { u } = f \left( { } ^ { t + \Delta t } u , { } ^ { t } u , { } ^ { t } \dot { u } , { } ^ { t } \ddot { u } \right) \tag {8} +$$ + +식 (8)의 값들을 식 (9)와 같은 동적 평형방정식에 대입하면 이전 단계의 변위, 속도, 가속도와 현단계 변위에 대한 식으로 나타낼 수 있고, 식 (11)과 같은 식으로 현단계의 변위를 계산할 수 있습니다. 현단계의 변위를 구하면 이 값과 이전 단계의 값들을 사용하여 식 (12)과 같이 현단계의 가속도와 속도를 계산할 수 있습니다. 감쇠는 식 (13)과 같이 강성과 질량을 사용하여 비례식으로 계산합니다. + +$$ +[ M ] ^ {t + \Delta t} \ddot {u} + [ C ] ^ {t + \Delta t} \dot {u} + [ K ] ^ {t + \Delta t} u = ^ {t + \Delta t} p \tag {9} +$$ + +$$ +\left[ [ K ] + a _ {0} [ M ] + a _ {1} [ C ] \right] ^ {t + \Delta t} u = ^ {t + \Delta t} p + [ M ] \left(a _ {0} ^ {t} u + a _ {2} ^ {t} \dot {u} + a _ {3} ^ {t} \ddot {u}\right) \tag {10} +$$ + +$$ ++ [ C ] (a _ {1} ^ {t} u + a _ {4} ^ {t} \dot {u} + a _ {5} ^ {t} \ddot {u}) +$$ + + + +$$ +[ \hat {K} ] ^ {t + \Delta t} u = ^ {t + \Delta t} \hat {p} \tag {11} +$$ + +$$ +[ \hat {K} ] = [ K ] + a _ {0} [ M ] + a _ {1} [ C ] +$$ + +$$ +{ } ^ { t + \Delta t } \hat { p } = { } ^ { t + \Delta t } p + [ M ] ( a _ { 0 } { } ^ { t } u + a _ { 2 } { } ^ { t } \dot { u } + a _ { 3 } { } ^ { t } \ddot { u } ) + [ C ] ( a _ { 1 } { } ^ { t } u + a _ { 4 } { } ^ { t } \dot { u } + a _ { 5 } { } ^ { t } \ddot { u } ) +$$ + +$$ +{ } ^ { t + \Delta t } \ddot { u } = a _ { 0 } \left( { } ^ { t + \Delta t } u - { } ^ { t } u \right) - a _ { 2 } { } ^ { t } \dot { u } - a _ { 3 } { } ^ { t } \ddot { u } \quad { } ^ { t + \Delta t } \dot { u } = { } ^ { t } \dot { u } + a _ { 6 } { } ^ { t } \ddot { u } + a _ { 7 } { } ^ { t + \Delta t } \ddot { u } \tag {12} +$$ + +$$ +\text { 여기서 } a _ {0} = \frac {l}{\alpha \Delta t ^ {2}} \quad a _ {1} = \frac {\delta}{\alpha \Delta t} \quad a _ {2} = \frac {l}{\alpha \Delta t} \quad a _ {3} = \frac {l}{2 \alpha} - l +$$ + +$$ +a _ {4} = \frac {\delta}{\alpha} - 1 \quad a _ {5} = \frac {\Delta t}{2} \left(\frac {\delta}{\alpha} - 2\right) \quad a _ {6} = \Delta t (1 - \delta) \quad a _ {7} = \delta \Delta t +$$ + +α, δ : Newmark 적분 변수 (α = 0.5, δ = 0.25 인 경우에는 항상 안정) + +Δt : 적분 시간간격 + +$$ +[ C ] = a [ K ] + b [ M ] \tag {13} +$$ + +여기서 a, b : 감쇠계산을 위한 질량과 강성의 비례상수 + +강성이나 감쇠의 비선형성을 고려한 해석을 위해서는 대부분의 경우 직접적분방법을 사용해야 합니다. 직접적분법의 경우는 모든 시간단계에 대하여 해석을 수행하기 때문에 시간단계의 수에 비례하여 해석시간이 소요됩니다. 직접적분방법의 사용시에 요구되는 데이터와 입력시 주의사항은 다음과 같습니다. + +▪ 전체 해석시간(또는 해석횟수): 해석하고자 하는 시간이나 해석횟수 +- 해석시간 간격 : 해석에 사용되는 시간간격으로, 해석의 정확성에 상당 한 영향을 미칠 수 있으며, 시간간격의 크기는 구조물의 고차모드의 주기나 하중의 주기와 밀접한 관계를 갖습니다. 해석시간 간격은 식 (10)의 적분 항에 직접적인 영향을 주게 되어 부적절한 값이 입력되는 경우 부정확한 결과를 나타낼 수 있습니다. 일반적으로 고려하고자 하는 최고차모드 주기 + + + +의 1/10 정도의 시간간격이 타당합니다. 또한, 해석시간 간격은 입력된 하중의 시간간격보다는 작아야 합니다. + +$$ +\Delta t = \frac {T _ {p}}{1 0} +$$ + +Tp = 고려하고자 하는 최고차모드 주기 + + 해석시간 간격의 수에 따라 해석 소요시간이 비례하여 증가하기 때문에필요이상으로 작게 하지 않는 것이 적합합니다. + 강성과 질량을 사용한 감쇠의 정의 : 강성과 질량의 비례식으로 감쇠를 정의합니다. + 시간적분방법 : Newmark 방법의 적용 시 필요한 적분변수를 입력합니다.Constant Acceleration 인 경우에는 모든 조건하에서 발산하지 않고 안정적으로 수렴한 값을 계산하지만 Linear Acceleration 인 경우에는 조건에 따라서 수렴하지 않을 수도 있습니다. 가능한 Constant Acceleration에 해당하는 적분변수를 사용하는 것이 타당합니다. + 동적하중 : 구조물의 절점이나 기초부에 직접 가해지는 동적하중으로 시간의 함수로 표시되며, 전체 하중 변화를 충분히 나타낼 수 있어야 합니다.입력되지 않은 시간에서의 하중 값은 선형보간하여 사용합니다. + +다음은 사용자의 이해를 돕기 위해 구조물의 동적해석에 필요한 기초적인 사항을서술한 것입니다. + +그림 2.6.1은 단일자유도 구조물의 운동을 이상화한 것입니다. 단일자유도계에 작용하는 힘들에 대한 평형방정식은 다음과 같습니다. + +$$ +f _ {I} (t) + f _ {D} (t) + f _ {E} (t) = f (t) \tag {14} +$$ + +( ) f t (관성력)는 구조물의 운동속도가 변화하는데 대해 저항하려는 관성효과를 힘으로 나타낸 것으로, 크기는 mu t ( ) 가 되며 작용방향은 가속도의 반대방향입니다.( ) Ef t (탄성력)는 구조물에 변형이 발생하면 구조계가 이에 저항하여 원위치로 복귀하려는 성질에 따른 탄성복원력으로, 그 크기는 ku t( ) 이며 작용방향은 변위와 + + + +반대방향입니다. ( ) Df t (감쇠력)는 구조물에 추가의 외력을 가하지 않을 경우, 내부마찰 등으로 인한 운동에너지의 소멸로 운동의 진폭이 점점 작아지는 현상을 고려하기 위한 구조계 내부의 가상의 힘이며, 그 크기는 cu t ( ) 이고, 작용방향은 운동속도와 반대방향입니다. + +![](images/page-216_82e614e33e9d0f3985b022eb427682e3c4e8fcb60acd5c1d5e05399ccded40b7.jpg) + +
+text_image + +(a) 모델 +(b) 평형 상태도 +
+ +그림 2.6.1 단일자유도 구조물 운동계 + +위의 각 힘들을 정리하면 다음과 같습니다. + +$$ +f _ {I} = m \ddot {u} (t) +$$ + +$$ +f _ {D} = c \dot {u} (t) \tag {15} +$$ + +$$ +f _ {E} = k u (t) +$$ + +위에서 m은 질량, c는 감쇠계수, k는 탄성계수입니다. 그림 2.6.1(b)의 힘의 평형관계로부터 변위에 대한 단일자유도 구조물의 운동방정식은 다음과 같습니다. + +$$ +m \ddot {u} + c \dot {u} + k u = f (t) \tag {16} +$$ + +위 식에서 f t( ) 0  으로 두면 자유진동에 대한 방정식이 되고, 여기에 c  0 인조건을 추가하면 비감쇠 자유진동방정식이 됩니다. 그리고 f ( )t 를 임의 시간에 대한 가진력(또는 가진변위, 속도, 가속도 등)으로 두면 강제진동 해석문제가 되며,모드중첩법(Mode Superposition Method) 또는 직접적분법(Direct Integration Method)을 이용하여 해를 구할 수 있습니다. + + + +# 6-3 다중지점 지진입력 하중에 대한 해석 + +장대교량과 같이 지지점이 공간적으로 멀리 떨어져 있는 경우에 구조물에 입력되는 지진하중에 시간지연효과가 발생하게 됩니다. 이러한 경우에는 각 지지점 별로 해당 지지점에 맞는 지진하중을 입력하여 해석하는 것이 좀더 나은 해를 구할 수 있습니다. 다중지점의 지진하중을 받는 구조물의 운동방정식은 식 (27)과 같습니다. 다중지점의 지진하중 입력을 고려한 해석시에는 모든 변위, 속도, 가속도값이 절대값에 대하여 계산됩니다. 다중지점 지진하중을 고려한 해석 결과는 여러 개의 지지점에서 지진하중이 입력되기 때문에 기준점을 정할 수 없어 절대값에 대하여 결과를 출력합니다. 절대값으로 결과를 출력하면 변위, 속도, 가속도 결과값이 지반거동을 포함하게 됩니다. 부재력이나 반력의 경우에는 절점간의 상대변위를 사용하여 계산하므로 절대값이나 상대값 출력에 영향을 받지 않습니다. + +$$ +\left[ \begin{array}{l l} M _ {s s} & M _ {s g} \\ M _ {g s} & M _ {g g} \end{array} \right] \left\{ \begin{array}{l} \ddot {u} _ {s} ^ {t} (t) \\ \ddot {u} _ {g} ^ {t} (t) \end{array} \right\} + \left[ \begin{array}{l l} C _ {s s} & C _ {s g} \\ C _ {g s} & C _ {g g} \end{array} \right] \left\{ \begin{array}{l} \dot {u} _ {s} ^ {t} (t) \\ \dot {u} _ {g} ^ {t} (t) \end{array} \right\} + \left[ \begin{array}{l l} K _ {s s} & K _ {s g} \\ K _ {g s} & K _ {g g} \end{array} \right] \left\{ \begin{array}{l} u _ {s} ^ {t} (t) \\ u _ {g} ^ {t} (t) \end{array} \right\} = \left\{ \begin{array}{l} 0 \\ P _ {g} (t) \end{array} \right\} \tag {17} +$$ + +여기에서 s 와 g 는 각각 상부구조물과 지점부를 의미합니다. t 는 절대변위를 의미합니다. + +지지점의 운동 $\ddot{u}_{g}^{t}(t)$ , $\dot{u}_{g}^{t}(t)$ , $u_{g}^{t}(t)$ 은 지지점별로 입력을 해야합니다. 절대변위를 지반변위에 의한 변위( $u_{s}^{s}(t)$ )와 이를 제외한 동적변위( $u_{s}(t)$ )로 구분하면 식(2)와 같습니다. + +$$ +\left\{ \begin{array}{l} u _ {s} ^ {t} (t) \\ u _ {g} ^ {t} (t) \end{array} \right\} = \left\{ \begin{array}{l} u _ {s} ^ {s} (t) \\ u _ {g} (t) \end{array} \right\} + \left\{ \begin{array}{c} u _ {s} (t) \\ 0 \end{array} \right\} \tag {18} +$$ + +$u_{g}(t)$ 는 지지점 변위이며 $u_{s}^{s}(t)$ 를 정적으로 구조물에 작용하였을 때 발생하는 상부구조물의 변위로서 유사정적변위(Quasi-static Displacement) 라고 하며 식 (19)와 같은 관계를 갖습니다. $P_{g}^{s}(t)$ 은 지지점의 변위( $u_{g}(t)$ )를 구조물에 가하기 위해 필요한 유사정적 지점하중(Quasi-static Support Force)입니다. + + + +$$ +\left[ \begin{array}{l l} K _ {s s} & K _ {s g} \\ K _ {g s} & K _ {g g} \end{array} \right] \left\{ \begin{array}{l} u _ {s} ^ {s} (t) \\ u _ {g} (t) \end{array} \right\} = \left\{ \begin{array}{c} 0 \\ P _ {g} ^ {s} (t) \end{array} \right\} \tag {19} +$$ + +식 (17)의 첫번째 식을 전개하고 식 (18)를 적용하여 정리하면 식 (20)와 같습니다. + +$$ +M _ {s s} \left(\ddot {u} _ {s} ^ {s} (t) + \ddot {u} _ {s} (t)\right) + M _ {s g} \ddot {u} _ {g} (t) + C _ {s s} \left(\dot {u} _ {s} ^ {s} (t) + \dot {u} _ {s} (t)\right) + C _ {s g} \dot {u} _ {g} (t) \tag {20} +$$ + +$$ ++ K _ {s s} \left(u _ {s} ^ {s} (t) + u _ {s} (t)\right) + K _ {s g} u _ {g} (t) = 0 +$$ + +$$ +M _ {s s} \ddot {u} _ {s} (t) + C _ {s s} \dot {u} _ {s} (t) + K _ {s s} u _ {s} (t) = P _ {e f f} (t) +$$ + +$$ +P _ {e f f} (t) = - M _ {s s} \ddot {u} _ {s} ^ {s} (t) - M _ {s g} \ddot {u} _ {g} (t) - C _ {s s} \dot {u} _ {s} ^ {s} (t) \tag {21} +$$ + +$$ +- C _ {s g} \dot {u} _ {g} (t) - K _ {s s} u _ {s} ^ {s} (t) - K _ {s g} u _ {g} (t) +$$ + +식 (19)의 첫번째 항을 정리하면 식 (6)과 같습니다. + +$$ +K _ {s s} u _ {s} ^ {s} (t) + K _ {s g} u _ {g} (t) = 0 +$$ + +$$ +u _ {s} ^ {s} (t) = I _ {f} u _ {g} (t) \tag {22} +$$ + +$$ +I _ {f} = - K _ {s s} ^ {- 1} K _ {s g} +$$ + +$I_{f}$ 는 영향행렬이라 하며 지지점의 변위에 의한 구조물의 변위를 나타냅니다. 식 (22)을 식 (21)에 대입하여 정리하면 다음과 같습니다. + +$$ +P _ {e f f} (t) = - \left(M _ {s s} I _ {f} + M _ {s g}\right) \ddot {u} _ {g} (t) - \left(C _ {s s} I _ {f} + C _ {s g}\right) \dot {u} _ {g} (t) \tag {23} +$$ + +감쇠행렬이 강성비례감쇠 행렬이라고 가정하면 식 (24)로 나타낼 수 있습니다. + +$$ +\left(C _ {s s} I _ {f} + C _ {s g}\right) \dot {u} _ {g} (t) = \beta \left(K _ {s s} I _ {f} + K _ {s g}\right) \dot {u} _ {g} (t) = 0 \tag {24} +$$ + +$$ +P _ {e f f} (t) = - \left(M _ {s s} I _ {f} + M _ {s g}\right) \ddot {u} _ {g} (t) \tag {25} +$$ + + + +질량이 집중질량이라고 가정하면 식 (21)에서의 운동방정식과 유효지진하중은 식 (26)과 같습니다. + +$$ +\begin{array}{l} \begin{array}{l} M _ {s s} \ddot {u} _ {s} (t) + C _ {s s} \dot {u} _ {s} (t) + K _ {s s} u _ {s} (t) = P _ {\text {eff}} (t) \\ \text {P} _ {s s} (t) = M _ {s s} \ddot {u} _ {s} (t) \end{array} \tag {26} \\ P _ {e f f} (t) = - M _ {s s} I _ {f} \ddot {u} _ {g} (t) \\ \end{array} +$$ + +참고로 균등 지진하중이 입력되는 경우의 운동방정식은 다음과 같습니다. + +$$ +\begin{array}{l} \begin{array}{l} M \ddot {u} (t) + C \dot {u} (t) + K u (t) = P _ {\text {eff}} (t) \\ P _ {\text {eff}} (t) = M (1) \ddot {u} (t) \end{array} \tag {27} \\ P _ {e f f} (t) = - M \left\{1 \right\} \ddot {u} _ {g} (t) \\ \end{array} +$$ + +최종적인 절대값 기준의 변위, 속도, 가속도는 식 (28)과 같이 계산됩니다. + +$$ +\left\{ \begin{array}{l} u _ {s} ^ {t} (t) \\ \dot {u} _ {s} ^ {t} (t) \\ \ddot {u} _ {s} ^ {t} (t) \end{array} \right\} = \left\{ \begin{array}{l} u _ {s} ^ {s} (t) \\ \dot {u} _ {s} ^ {s} (t) \\ \ddot {u} _ {s} ^ {s} (t) \end{array} \right\} + \left\{ \begin{array}{l} u _ {s} (t) \\ \dot {u} _ {s} (t) \\ \ddot {u} _ {s} (t) \end{array} \right\} \tag {28} +$$ + +다중지점 지진입력에 대한 시간이력해석은 선형시간이력 해석에만 반영되고 있습니다. + + + +# Chapter 7. 좌굴해석 + +midas Civil의 선형좌굴해석기능(Linear Buckling Analysis)은 트러스, 보요소 판요소또는 솔리드요소로 구성된 구조물의 임계하중계수(Critical Load Factor)와 그에 해당되는 좌굴모드형상(Buckling Mode Shape)을 해석하는데 사용됩니다. 일정한 변형상태에서구조물의 정적 평형방정식은 다음과 같습니다. + +$$ +[ K ] \{U \} + [ K _ {G} ] \{U \} = \{P \} \tag {1} +$$ + +여기서 [ ] K : 구조물의 탄성강성행렬 + +[ ] K G : 구조물의 기하강성행렬 + +U : 구조물의 전체변위 + +P : 구조물에 작용하는 하중 + +구조물의 기하강성행렬은 각 요소의 기하강성행렬을 합하여 계산되고, 각 요소별기하강성 행렬은 다음과 같이 구해집니다. 여기에서 기하강성행렬은 구조물이 변형된 상태에서 강성이 변화된 상태를 나타내며 작용하는 하중과 직접적인 관계를갖습니다. 가령 임의의 부재에 압축력이 작용하면 강성이 감소하는 경향을 가지고,반면에 인장력이 작용하면 강성이 증가하는 경향을 가집니다. + +$$ +[ K _ {G} ] = \sum [ k _ {G} ] +$$ + +$$ +[ k _ {G} ] = F [ \overline {{k}} _ {G} ] +$$ + +여기서 [ ] Gk : 각 부재별 표준 기하강성행렬 + +F : 부재력 (축력 - 트러스, 보요소의 경우) diff --git a/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_023.md b/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_023.md new file mode 100644 index 00000000..a1cd79ee --- /dev/null +++ b/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_023.md @@ -0,0 +1,295 @@ + + +-트러스 부재의 표준 기하강성행렬 + +$$ +\left[ k _ {G} \right] = \left[ \begin{array}{c c c c c c} 0 & & & & & \\ 0 & \frac {1}{L} & & & \text {symm.} \\ 0 & 0 & \frac {1}{L} & & \\ 0 & 0 & 0 & 0 & & \\ 0 & - \frac {1}{L} & 0 & 0 & \frac {1}{L} & \\ 0 & 0 & - \frac {1}{L} & 0 & 0 & \frac {1}{L} \end{array} \right] +$$ + +-보부재의 표준 기하강성행렬 + +$$ +[ \overline {{k}} _ {G} ] = [ \overline {{k}} _ {G 1} ] + [ \overline {{k}} _ {G 2} ] +$$ + +여기서, $[\overline{k}_{G1}]$ : 축방향 부재력에 의한 기하강성행렬 + +$[\bar{k}_{G2}]$ : 전단력과 모멘트에 의한 기하강성행렬 + +$$ +\left[ \overline {{k}} _ {G} \right] = \left[ \begin{array}{c c c c c c c c c c c c} 0 & & & & & & & & & & & \\ 0 & \frac {6}{5 L} & & & & & & & & & & \\ 0 & 0 & \frac {6}{5 L} & & & & & & & & & \\ 0 & 0 & 0 & 0 & & & & & & & & \\ 0 & 0 & - \frac {1}{1 0} & 0 & \frac {2 L}{1 5} & & & & & & & \\ 0 & \frac {1}{1 0} & 0 & 0 & 0 & \frac {2 L}{1 5} & & & & & & \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & & & & & \\ 0 & - \frac {6}{5 L} & 0 & 0 & 0 & - \frac {1}{1 0} & 0 & \frac {6}{5 L} & & & & \\ 0 & 0 & - \frac {6}{5 L} & 0 & \frac {1}{1 0} & 0 & 0 & 0 & \frac {6}{5 L} & & & \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & & \\ 0 & 0 & - \frac {1}{1 0} & 0 & - \frac {L}{3 0} & 0 & 0 & 0 & \frac {1}{1 0} & 0 & \frac {2 L}{1 5} & \\ 0 & \frac {1}{1 0} & 0 & 0 & 0 & - \frac {L}{3 0} & 0 & - \frac {1}{1 0} & 0 & 0 & 0 & \frac {2 L}{1 5} \end{array} \right] +$$ + +$$ +[ \overline {{k}} _ {G 2} ] = \int_ {L} [ G ] ^ {T} [ S ] [ G ] d L +$$ + +여기서, [G] : 변형률과 변위관계 행렬 + +[S] : 힜 모멘트, 비틀림 모멘트와 전단력으로 구성된 행렬 + + + +-판요소 및 솔리드요소의 기하강성행렬 + +$$ +\left[ k _ {G} \right] = \int_ {v} [ G ] ^ {T} \left[ \begin{array}{c c c} s & 0 & 0 \\ 0 & s & 0 \\ 0 & 0 & s \end{array} \right] [ G ] d V +$$ + +여기서 [G] : 변형률과 변위관계 행렬 + +$$ +[ S ] = \left[ \begin{array}{c c c} \sigma_ {x x} & \sigma_ {x y} & \sigma_ {z x} \\ \sigma_ {x y} & \sigma_ {y y} & \tau_ {y z} \\ \sigma_ {z x} & \sigma_ {y z} & \sigma_ {z z} \end{array} \right]: \text { 요소의 응력 행렬 } +$$ + +기하강성행렬을 하중계수와 입력된 하중을 받는 구조물의 기하강성행렬의 곱으로 나타내면 식 (2)와 같습니다. + +$$ +[ K _ {G} ] = \alpha [ \overline {{{K}}} _ {G} ] \tag {2} +$$ + +여기서 α: 임계하중계수 + +$[K_{G}]$ : 좌굴해석을 위해 입력된 하중을 받고 있는 구조물의 기하강성행렬 + +$$ +[ K + \lambda K _ {G} ] \{u \} = \{p \} \tag {3} +$$ + +$$ +[ K _ {e q} ] = [ K + \lambda K _ {G} ] +$$ + +위 식과 같은 평형방정식에서 구조물이 불안정한 상태가 되려면 특이해를 가져야 합니다. 즉 등가강성행렬의 행렬식이 영(Zero)이 되는 경우에 좌굴이 발생하게 됩니다. + +$$ +\left| \left[ K _ {e q} \right] \right| < 0 \left(\lambda > \lambda_ {c r}\right): \text { 불안정한 평형상태 } +$$ + +$$ +\left| \left[ K _ {e q} \right] \right| = 0 \left(\lambda = \lambda_ {c r}\right): \text { 불안정한 상태 } +$$ + +$$ +\left| \left[ K _ {e q} \right] \right| > 0 \left(\lambda < \lambda_ {c r}\right): \text {안정한 상태} +$$ + +그러므로, 식 (3)에서 좌굴해석을 위한 문제는 식 (4)와 같은 고유치 문제로 귀결됩니다. + + + +$$ +\left| K + \lambda_ {I} \left[ K _ {G} \right] \right| = 0, \quad \lambda_ {i}: \text { 고유치(임계하중계수) } \tag {4} +$$ + +이 문제는 "고유치 해석"에서와 같은 방법으로 풀 수 있습니다. + +고유치 해석을 통해 얻어지는 값은 고유치값과 고유모드가 있는데, 고유치값은 임계하중계수가 되고 고유모드는 임계하중에 해당하는 좌굴형상이 됩니다. 임계하중은 초기하중으로 주어진 값과 임계하중계수를 곱한 값으로 구해집니다. 임계하중과 좌굴형상은 입력된 구조물에 임계하중이 작용할 경우, 좌굴모드와 같은 형상으로 구조물의 좌굴이 발생하게 된다는 것을 의미합니다. + +예를 들어 초기하중이 10만큼 작용하는 구조물에 좌굴해석을 수행하여 임계하중계수 5를 얻었다면, 이 구조물은 50의 하중이 작용할 때 좌굴이 발생하게 됩니다.그러나, 구조물의 좌굴은 대부분 기하적으로나 재료적으로 대변형이나 비선형상태에서 발생하기 때문에 실제 문제에 있어서 적용은 제한적이라고 할 수 있습니다. + + + +midas Civil에서의 선형좌굴해석은 트러스, 보요소, 판요소, 솔리드요소로 제한되며,해석과정은 다음과 같은 2단계의 해석과정이 수반되고, 해석의 순서도는 그림2.7.1과 같습니다. + +1. 사용자가 입력하는 하중조건하에서 선형정적해석을 수행하는 과정으로, 해석된 구조 부재의 부재력 또는 응력을 적용하여 해당부재의 기하강성행렬(Geometric Stiffness Matrix)을 구성합니다. +2. 위에서 계산된 기하강성행렬과 탄성강성행렬을 사용하여 고유치 문제를 계산합니다. + +이 과정에서 얻어지는 고유치값은 임계하중계수가 되고 고유모드는 좌굴형상이 됩니다. + +![](images/page-224_76b3fe55d303642b897540bd843f94b79f2595102b0f1979c20a00b781f92180.jpg) + +
+flowchart + +```mermaid +graph TD + A["구조물 해석모델 입력"] --> B["전체 강성행렬 및 좌굴해석을 위한 하중행렬 구성"] + B --> C["정적해석 수행 및 요소별 기하강성행렬 구성"] + C --> D["전체 기하강성행렬 구성"] + D --> E["전체강성과 기하강성을 사용한 고유치 해석"] +``` +
+ +그림 2.7.1 좌굴해석 개념도 + + + +# Chapter 8. 비선형 해석 + +# 8-1 개요 + +구조물의 선형-탄성 거동을 해석하는 경우에는 변위와 하중이 서로 비례관계에 있다는 가정을 전제로 합니다. 이러한 가정은 재하 되는 하중에 대하여 재료의 응력-변형율 관계가 선형이고, 하중이 구조물의 강성에 비해 상대적으로 작아서 발생하는 변위가 미소하여 기하학적 형태가 변하지 않는 경우에 적용 가능합니다. + +일반적인 설계조건에서는 대부분의 구조물에서 선형가정을 전제로 해석을 수행하게 되지만 대변형이 발생하는 경우나 응력이 탄성범위를 초과하는 경우에는 반드시 구조물의 비선형 해석을 수행하여야 합니다. + +비선형 해석은 아래와 같이 크게 세 가지로 구분할 수 있습니다. + +첫째, 구조물에 상대적으로 큰 하중이 재하 되어 응력이 커지면 응력-변형율 관계가 비선형으로 변하여 비선형 거동을 하게 되는데 이를 재료 비선형이라고 합니다. 아래 그림 2.8.1과 같은 응력-변형율의 관계로 표현되며 하중재하 방법과 재료에 따라서 다양한 형태의 응력-변형율 관계가 발생합니다. + +![](images/page-225_ede9f93d9ce1ee89d847875c3917aeae4d247db3c6f8c85bb2cb75e15e020f89.jpg) + +
+line + +| ε | σ | +| --- | ---- | +| 0 | 0 | +| ε | σy | +| >ε | σy | +
+ +그림 2.8.1 재료 비선형 해석에 사용되는 응력-변형율 관계 + +둘째, 구조물에 상대적으로 큰 변형이 발생하고 기하학적 형태가 변하여 변위-변형 + + + +율 관계가 비선형이 되는 경우에 미소변형해석에서 무시하였던 변위-변형율 관계의 고차항을 포함하여 해석을 하게 되는데 이를 기하학적 비선형이라고 합니다.기하비선형성은 재료의 선형 상태에서도 발생이 가능하고 변형이 크게 발생할 수있는 구조물의 경우 설계를 위한 해석에서도 사용됩니다. 기하비선형은 재질과 관계없이 구조물의 형상에 따라 발생하게 되는데 하중에 따른 변위가 크게 발생하여구조물의 좌표가 변화하거나 모멘트와 같은 부가 하중이 발생할 경우에는 반드시고려해야 합니다. (그림 2.8.2 참조) + +셋째, 하중에 의한 구조물의 변형에 따라 경계조건이 변화하는 구조물에서 발생하는 하중-변위의 비선형 관계를 경계비선형이라 합니다. 지반과 접하는 구조물의 압축전담 경계조건 등에 대한 문제들이 경계비선형 문제에 해당됩니다. + +midas Civil에서의 비선형 해석 기능은 비선형 요소(인장/압축 전담 요소)를 사용한경계비선형 해석기능과 구조물에 상대적으로 큰 변위가 발생하는 경우에 필요로하는 기하비선형 해석기능을 포함하고 있습니다. 또한 재료의 비탄성을 고려하는정적재료비선형 및 비탄성 시간이력해석 기능을 포함하고 있습니다. + +![](images/page-226_fe24cd77280aa650b49c56fe0aab57295264205b510fc9039dfdb92a0e3f5ea1.jpg) + +
+text_image + +V +
+ +(a) 구조물의 대변형에 따른 강성의 변화 +![](images/page-226_f8b4133aa719f4ea2e1e8e413200c13df3e2dbb56b753d9c56a70427767a5d1a.jpg) + +
+text_image + +V +H +
+ +(b) 변형에 따른 부가 하중의 발생 +그림 2.8.2 기하비선형 해석이 요구되는 구조계 + + + +# 8-2 기하비선형 해석 + +선형해석에서 사용하는 미소변형( $\varepsilon_{ij}$ )은 회전이 작다는 가정하에서 다음과 같이 정의됩니다. + +$$ +\varepsilon_ {i j} = \frac {1}{2} (u _ {i, j} + u _ {j. i}) +$$ + +u는 변위이며 $(u_{i,j}, u_{j,i})$ 는 최초 좌표에 대한 미분을 나타냅니다. 그림 2.8.3과 같이 대변형이 발생하는 경우에 더 이상 Small Strain으로는 구조물의 변형을 정확히 표현할 수 없게 됩니다. 대변형은 다음 식과 같이 회전성분과 회전이 아닌 성분으로 분리할 수 있습니다. F는 변위텐서(Deformation Tensor), R은 회전변위 텐서 (Rotation Tensor) 그리고 U는 변형텐서 (Stretch Tensor)를 나타냅니다. 실제 구조물에 발생하는 변형은 U에 따라 결정됩니다. + +$$ +F = R U, \quad \varepsilon = f (U) +$$ + +![](images/page-227_9226db8367d5a93a33aa3186b84f5ff53e3713017f5313915d5d45d8a4aafd14.jpg) + +
+text_image + +Z +Initial +Configuration +Final +Configuration +ê₀₃ +ê₀₂ +ê₀₁ +ê₁ᵗ +ê₁ᵗ +e₃ +e₂ +e₁ +X +Y +
+ +그림 2.8.3 대변형에 의한 기하학적 비선형성 + +위 식의 전체 변형에서 회전성분을 제거하여야 정확한 변형률(Strain)의 계산이 가능하므로 회전량이 큰 경우 처음부터 정확한 변형률-변위 관계를 알 수는 없습니다. 즉 선형해석에서 계산된 변위에 따라 변형률이 변하게 되므로 기하학적 비선형이 도입됩니다. + + + +midas Civil의 기하비선형 해석은 Co-rotational 방법을 사용하는데 이 방법은 변형되는 요소에 부착되어 요소의 회전에 따라 움직이는 Co-rotational 좌표계에서Strain을 사용하여 기하학적 비선형성을 고려하는 방법입니다. Co-rotational 좌표계에서의 변형률-변위 관계는 행렬식 ˆ ˆ  Buˆ 와 같이 표현할 수 있으며 선형해석에서와 같은 변형률-변위 관계 행렬을 사용할 수 있습니다. 즉 기하학적 비선형이 도입되어도 선형해석에 사용된 요소의 안정성 및 수렴성이 유지된다는 것을 의미하므로 우수한 선형요소의 특성이 유지된다는 장점을 가지고 있습니다. + +Co-rotational 좌표계에서의 변위 uˆ 는 관계식 ^ $\boldsymbol { u } = f ( e , e _ { 1 } , e _ { 2 } , \hat { e } _ { 3 } , \hat { e } _ { 1 } , \hat { e } _ { 2 } , e _ { 3 } )$ 에 의하여계산되며 미소변위 uˆ 는 선형화하여 u T uˆ   으로 표현할 수 있습니다. Co-rotational 좌표계에서의 선형 탄성문제의 경우 동시회전 좌표계에서의 요소내력$\hat { p } ^ { i n t }$ 는 다음 식으로 구할 수 있습니다. + +$$ +\hat {p} ^ {i n t} = \int_ {d v} B ^ {T} \hat {\sigma} d V _ {0} +$$ + +여기서 ˆ 는 동시회전 좌표계에서 표현된 응력이고, 위 식에 대해 변분을 취하면다음 식을 얻을 수 있습니다. + +$$ +\delta \hat {p} ^ {i n t} = (K + \hat {K} _ {\sigma}) \delta \hat {u} +$$ + +위 식에서 $\hat { K } _ { \sigma } \ \in \ \mathrm { ~ \mathfrak { z } ~ }$ (nitial Stress StiffnessMatrix)이고, 내력과 외력의 평형관계를( $\boldsymbol { p } ^ { e x t } - \boldsymbol { p } ^ { i n t } = 0$ ) 사용하면 다음 식과 같은비선형 평형방정식을 구성할 수 있습니다. + +$$ +(K + K _ {\sigma}) u = p ^ {e x t} +$$ + +비선형 평형방정식의 해를 구하는 방법으로는 Newton-Raphson 방법과 Arc-length방법이 있습니다. 일반적인 해석인 경우에는 하중제어 방법인 Newton-Raphson 방법을 사용하고 Snap-through나 Snap-back과 같은 문제에 대하여는 변위제어 방법인 Arc-length 방법을 사용하면 적절한 해석을 할 수 있습니다. + + + +midas Civil의 기하비선형 해석에 사용할 수 있는 요소에는 트러스, 보, 판요소가있으며 기타의 요소가 같이 사용될 경우에는 강성만 고려되고 기하비선형성은 고려되지 않습니다. + +# 8-2-1 Newton-Raphson 반복법 + +외력이 작용하는 구조물의 기하비선형 해석에서는 기하강성이 변위의 함수 형태로나타나고 변위가 다시 기하강성의 영향을 받기 때문에 반복해석이 필요합니다.Newton-Raphson 방법은 일반적으로 많이 사용되는 방법으로 그림 2.8.4와 같이주어진 외력과 평형조건을 이루는 변위를 계산하게 됩니다. 하중-변위의 평형방정식에서 주어진 하중에 대해 평형이 만족되도록 반복 계산할 때마다 강성행렬을 재구성하고, 이를 이용하여 근사해를 반복적으로 수정하여 허용오차의 범위내의 근사해를 구합니다. + +$$ +(K + K _ {\sigma}) u = p, \quad K _ {T} u = p +$$ + +$$ +K _ {T} = K + K _ {\sigma}, \quad K _ {\sigma} = f (u) +$$ + +$$ +K _ {T} (u _ {m - 1}) (u _ {m - 1} + \Delta u _ {m}) = R _ {m} +$$ + +![](images/page-229_7b4e956b6adb911170146d44052f851e1af3d549e398b75fbaa075f54c97f6ef.jpg) + +
+line +| Displacement | Force (K_T(um)) | Force (K_T(um-1)) | +| ------------ | --------------- | ----------------- | +| Um-1 | Rm | Rm-1 | +| Um | Rm-1 | Rm-1 | +
+ +그림 2.8.4 Newton-Raphson Method + + + +Taylor 전개식 $( y ( x _ { n } + h ) = y ( x _ { n } ) + y ^ { \prime } ( x _ { n } ) h )$ 에 의해 위 식의 좌변을 전개하면, 다음과 같습니다. + +$$ +K _ {T} (u _ {m - 1}) (u _ {m - 1} + \Delta u _ {m}) = K _ {T} (u _ {m - 1}) u _ {m - 1} + \frac {d R}{d u _ {m - 1}} \Delta u _ {m} +$$ + +$\frac { d R } { d u _ { m - 1 } } = K _ { T } ( u _ { m - 1 } ) R _ { m } - F _ { m - 1 } = R ^ { R }$ 1 1( ) RT m m mK u R F R    의 관계를 위 식에 대입하여 정리하면, 다음과 같1mdu 습니다. + +$$ +K _ {T} \left(u _ {m - 1}\right) \Delta u _ {m} = R _ {m} + R _ {m - 1} = R ^ {R} \quad \left(R ^ {R}: \text { Residual Force }\right) +$$ + +해석의 과정은 그림 2.8.4에서 묘사됩니다. $\varDelta u _ { { \scriptscriptstyle m } }$ 이 계산되면, 변위를$\textit { u } _ { m } = u _ { m - 1 } + \varDelta u _ { m }$ 의 식에 의해 보정합니다. 다시 반복 과정을 적용하기 위해 새로운 접선 강성 $K _ { \scriptscriptstyle T } ( u _ { { m } } )$ 과, 불균형 하중 $R _ { m + 1 } - R _ { _ m } \ \stackrel { \circ } { \equiv }$ 계산하고 이에 의해 보정된변위 $u _ { m + 1 } \equiv$ 구합니다. + +이상의 반복과정 중에서 한 스텝에서의 변위, 에너지, 혹은 하중의 증분량이 수렴한계 내에 들어올 때까지 반복하여 해를 계산합니다. + +# 8-2-2 Arc-length 반복법 + +일반의 반복 과정에서는 하중-변위 곡선이 거의 수평인 경우, 변위 증분의 계산값이 매우 커질 수 있습니다. 즉, 하중 증분을 고정적으로 두면, 변위는 매우 큰 값을 갖습니다. Arc-length 방법을 사용하면 이 문제를 해결할 수 있으며, 변위 제어법을 사용하는 경우와 같이 Snap-through 거동(그림 2.8.5(a) 참조)을 해석할 수 있습니다. 또한, Arc-length 방법은 변위 제어법으로 해석할 수 없는 Snap-back 거동도 해석할 수 있습니다. (그림 2.8.5(b) 참조) + +Arc-length 방법은 증분 변위의 Norm을 미리 정의된 값으로 구속합니다. 이 증분의 크기는 반복과정 내에서는 고정적으로 적용되지만, 증분의 시작시에는 고정되어 있지 않습니다. 증분의 크기를 결정하기 위해서 다음과 같은 과정을 따릅니다.(그림 2.8.5(c) 참고) diff --git a/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_024.md b/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_024.md new file mode 100644 index 00000000..b4a6b376 --- /dev/null +++ b/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_024.md @@ -0,0 +1,362 @@ + + +![](images/page-231_8ec34551462efde1af4f7b6a098b69910e8f140808bd090073606bc89a6a3048.jpg) + +
+line + +| Displacement | Force | +| ------------ | ----- | +| 0 | 0 | +| Peak | ~0.8 | +| Low | ~0.3 | +| High | ~1.0 | +
+ +(a) Snap-through + +![](images/page-231_5f41bfea0b700bdc2fb3b5c14084822c25d807702f63ac03f386d339d8106d57.jpg) + +
+line + +| Displacement | Force | +| ------------ | ----- | +| 0 | 0 | +| 0.5 | 0.5 | +| 1.0 | 1.0 | +| 1.5 | 1.5 | +| 2.0 | 2.0 | +| 2.5 | 2.5 | +| 3.0 | 3.0 | +| 3.5 | 3.5 | +| 4.0 | 4.0 | +| 4.5 | 4.5 | +| 5.0 | 5.0 | +| 5.5 | 5.5 | +| 6.0 | 6.0 | +| 6.5 | 6.5 | +| 7.0 | 7.0 | +| 7.5 | 7.5 | +| 8.0 | 8.0 | +| 8.5 | 8.5 | +| 9.0 | 9.0 | +| 9.5 | 9.5 | +| 10.0 | 10.0 | +
+ +(b) Snap-back +![](images/page-231_5f4ec4bff10e9c8bbb30b94a4bfc7395e728f97d14b701f06362a7289022069c.jpg) + +
+line + +| Displacement R | Force R | Label | +| -------------- | ------- | ------------ | +| 0 | 0 | (u1, λ1f) | +| 0 | 0 | (u2, λ2f) | +| 0 | 0 | Δl | +| 0 | 0 | Δλ1f | +| 0 | 0 | Δλ2f | +| 0 | 0 | λαf | +| 0 | 0 | um-1 | +| 0 | 0 | Δu1 | +| 0 | 0 | Δu2 | +| 0 | 0 | Δu1-1 | +| 0 | 0 | Δu2 | +
+ +(c) Arc-length Method의 개념도 +그림 2.8.5 Arc-length Method + +증분 시작시의 외력 벡터를 $R_{m}$ 으로, 외력 벡터의 증분을 $\Delta\lambda_{i}f$ 로 정의합니다. 하중 계수 $\Delta\lambda_{i}$ 는 단위 하중 f에 곱해지고 매 반복단계마다 변하게 됩니다. + +$$ +K _ {T} (u _ {i - 1}) \delta u _ {i} = \Delta R _ {i} ^ {R} +$$ + +$$ +\delta u _ {i} = K _ {T} (u _ {i - 1}) ^ {- 1} (f _ {\text { int }} (u _ {m - 1}) - f _ {\text { int }} (u _ {i})) +$$ + +위 방정식의 해는 다음과 같이 두 부분으로 나눌 수 있고, 증분 변위는 다음과 같이 구할 수 있습니다. + +$$ +\delta u _ {i} ^ {I} = K _ {T} (u _ {i - 1}) ^ {- 1} (f _ {\text { int }} (u _ {m - 1}) - f _ {\text { int }} (u _ {i})), \quad \delta u _ {i} ^ {I I} = K _ {T} (u _ {i - 1}) ^ {- 1} f +$$ + +$$ +\delta u _ {i} = \delta u _ {i} ^ {I} + \Delta \lambda_ {i} \delta u _ {i} ^ {I I} +$$ + + + +midas Civil에서는 하중 계수 $\Delta\lambda_{i}$ 를 구면경로법(Spherical Path)을 사용하여 구하고 이 방법의 구속 조건식은 다음과 같습니다. + +$$ +\Delta u _ {i} ^ {T} \Delta u _ {i} = \Delta l ^ {2} +$$ + +$\Delta l$ 은 구속하고자 하는 변위의 길이이고, $\Delta u_{i} = \Delta u_{i-1} + \delta u_{i}$ 의 식을 위 식에 대입하면 하중 계수 $\Delta \lambda_{i}$ 는 다음과 같이 계산됩니다. + +$$ +a _ {1} \Delta \lambda^ {2} + a _ {2} \Delta \lambda + a _ {3} = 0 +$$ + +$$ +\Delta \lambda_ {i} = \frac {- a _ {2} + \sqrt {a _ {2} ^ {2} - 4 a _ {1} a _ {3}}}{2 a _ {1}} +$$ + +여기서 + +$$ +\begin{array}{l} a _ {1} = \left(\delta u _ {i} ^ {I I}\right) ^ {T} \delta u _ {i} ^ {I I} \\ a _ {2} = 2 \left(\delta u _ {i} ^ {I}\right) ^ {T} \delta u _ {i} ^ {I I} + 2 \left(\Delta u _ {i - 1}\right) ^ {T} \delta u _ {i} ^ {I I} \\ a _ {3} = 2 \left(\Delta u _ {i - 1}\right) ^ {T} \delta u _ {i} ^ {I} + \left(\delta u _ {i} ^ {I}\right) ^ {T} \delta u _ {i} ^ {I} + \left(\Delta u _ {i - 1}\right) ^ {T} \Delta u _ {i - 1} - \Delta l ^ {2} \\ \end{array} +$$ + +일반적으로 위 식의 해는 2개 이지만, 복소수 해의 경우에는 구면경로법의 선형 동등해를 사용합니다. 두 실수해 중에서 어느 것을 사용할지 결정하기 위해서, 이전 반복과정과 현재 과정 사이의 변위 증분 벡터간의 각도 $\theta$ 를 다음 식으로 계산하여 판단합니다. + +$$ +\cos \theta = \frac {\left(\Delta u _ {i - 1}\right) ^ {T} \delta u _ {i}}{\left\| \Delta u _ {i - 1} \right\| \left\| \delta u _ {i} \right\|} +$$ + +하나의 해가 음의 값이고 다른 해가 양의 값이면 양의 값을 선택하고, 두 해가 모두 예각이면 선형 해답 $\Delta\lambda_{i} = -a_{3}/a_{2}$ 에 가까운 해를 사용합니다. + + + +# 8-3 P-Delta + +midas Civil의 P-Delta 해석 기능은 보요소가 횡력과 축력을 동시에 받을 때 2차적인 구조적 거동을 고려하기 위한 것으로 기하학적 비선형성(Geometric Nonlinearity)의 일종입니다. + +ACI318 Code나 AISC-LRFD Code에서는 실제적인 부재내력을 설계에 반영하기 위해 P-Delta 효과를 고려한 구조해석을 요구하고 있습니다. + +![](images/page-233_ccae5dfb0ace4919c6961c4bc4b95082c53ccfeab1732d6994bc8264c630db48.jpg) + +
+flowchart + +```mermaid +graph TD + A["해석모델 입력"] --> B["강성행렬 구성"] + B --> C["초기 정적해석수행"] + C --> D["기하강성행렬 구성"] + D --> E["수정된 강성행렬 구성"] + E --> F["정적해석 수행"] + F --> G{수렴어부확인} + G -->|정적해석| H["해석결과출력"] + G -->|동적해석| I["고유치해석"] +``` +
+ +그림 2.8.6 P-Delta 해석 수행개념도 + + + +midas Civil의 P-Delta 해석 기능은 좌굴문제(Buckling Problem)를 수치해석적인 방법으로 해를 구할 때 사용되는 개념을 응용한 것으로써, 먼저 주어진 하중조건에대해 정적해석을 수행한 다음, 각 요소에 발생한 응력을 이용하여 기하강성행렬(Geometric Stiffness Matrix)을 만들고 수정된 강성행렬을 사용하여 주어진 조건을만족할 때까지 해석을 반복 수행하게 됩니다. + +그림 2.8.6과 같이 동적해석에 P-Delta 효과를 고려할 경우에도 기하강성행렬의 구성을 위해 정적하중조건의 입력이 필요합니다. + +midas Civil에 사용된 P-Delta 해석의 개념은 그림 2.8.7과 같습니다. + +외력에 의해 횡방향으로 모멘트와 전단력을 받는 기둥부재가 축력에 의해 인장 또는 압축을 추가로 받을 때, 인장력은 기둥부재가 모멘트와 전단력에 대해 저항하게 하는 반면, 압축력은 모멘트와 전단력에 대해 약해지게 합니다. + +즉, 인장력은 횡방향 거동에 대한 기둥부재의 강성을 증가시키고, 압축력은 그 강성을 감소시키는 효과를 가집니다. + +만약 압축력에 의한 응력이 아주 커져서 횡방향거동에 대한 강성감소치가 해당 부재의 횡방향강성과 같아지면 그 부재에 좌굴이 발생하게 되는데 그 때의 압축하중을 임계좌굴하중(Critical Buckling Load)이라 합니다. 이 효과를 축력과 횡력을 받는기둥부재에 대해 예를 들면 다음과 같습니다. + +그림 2.8.7(a)에서 기둥부재가 인장력과 횡력을 동시에 받는 경우, P-Delta 효과를고려하지 않을 때(횡변형과 수직하중에 의한 2차 변형효과를 고려하지 않을 때)모멘트는 기둥부재 끝단의 M=0에서부터 하단의 M=VL까지 일정한 비율로 증가합니다. 그러나 실제의 경우는 횡력때문에 Δ만큼의 횡변위가 발생하고 이 횡변위 Δ와 인장력 P에 의해 P·Δ만큼의 모멘트가 감소하게 됩니다. 따라서 기둥부재의 횡방향 강성이 증가한 것과 같은 효과를 가지게 됩니다. + +반대로 압축력과 횡력을 동시에 받는 경우는 P·Δ 만큼의 모멘트가 증가하게 되고,그 결과 기둥부재의 횡방향 강성이 감소한 것과 같은 영향을 나타내게 됩니다. + + + +따라서 횡방향 변위는 횡력과 축력의 변수가 됩니다. 이를 수식으로 표현하면 다음과 같습니다. + +$$ +\Delta = V / K, \quad K = K _ {o} + K _ {G} +$$ + +여기서 KO는 기둥부재 고유의 횡방향 강성을 의미하고, KG는 축력에 따른 강성증감효과를 나타낸 것입니다. 트러스, 보, 판요소의 기하강성행렬 구성은 “Chapter 7.좌굴해석"을 참조하시기 바랍니다. + +![](images/page-235_7a642b1652e9367e6389c0222aa94797b5f0a175635cef64b35c93aa079dc0b5.jpg) + +
+text_image + +P +V +L +Before deflection +After deflection +P +Δ +P +V +y +My +My = Vy - Px +body +P-Delta effect +Px +Vy +P-Delta effect +PΔ +M = VL +
+ +(a) 기둥부재에 인장력과 횡력이 동시에 작용하는 경우 + +![](images/page-235_b3dd0e557589996e501c1de6a27877c0e35e4a6b5a060d8d3f84d10edfdef811.jpg) + +
+text_image + +Before deflection +After deflection +P +V +L +P +x +P +y +My +My = Vy + Px +Free body diagram +P-Delta effect +P-Delta effect +M = VL +
+ +(b) 기둥부재에 압축력과 횡력이 동시에 작용하는 경우 +그림 2.8.7 P-Delta 효과를 고려한 기둥부재의 거동 + + + +P-Delta 해석을 단계별로 정리하면 다음과 같습니다. + +-1단계 해석 + +$$ +\Delta_ {I} = V / K _ {0} +$$ + +-2단계 해석 + +$$ +\Delta_ {2} = f (P, \Delta_ {1}), \quad \Delta = \Delta_ {1} + \Delta_ {2} +$$ + +-3단계 해석 + +$$ +\Delta_ {3} = f (P, \Delta_ {2}), \quad \Delta = \Delta_ {1} + \Delta_ {2} + \Delta_ {3} +$$ + +-4단계 해석 + +$$ +\Delta_ {4} = f (P, \Delta_ {3}), \quad \Delta = \Delta_ {1} + \Delta_ {2} + \Delta_ {3} + \Delta_ {4} +$$ + +\- n단계 해석 + +$$ +\Delta_ {n} = f \left(P, \Delta_ {n - 1}\right), \quad \Delta = \Delta_ {1} + \Delta_ {2} + \Delta_ {3} + \dots + \Delta_ {n} +$$ + +: + +midas Civil의 내부에서 수행되는 P-Delta 해석 과정을 각 단계별로 설명하면 다음과 같습니다. + +1단계 해석을 통하여 횡력에 의한 $\Delta_{7}$ 을 계산한 다음, 축력에 따른 기하강성행렬을 구하고 초기의 강성행렬에 기하강성행렬을 더하여 새로운 강성행렬을 구성합니다. 새로 구성된 강성행렬을 이용하여 P-Delta 효과를 고려한 $\Delta_{2}$ 를 계산하고 수렴조건의 만족 여부를 검토합니다. 수렴조건은 P-Delta Analysis Control 에서 주어진 최대 반복수행 횟수 및 허용변위차(Displacement Tolerance)에 대한 검토를 의미합니다. 수렴조건을 만족할 경우에는 반복수행 과정을 종료하게 되고 만족하지 못할 경우에는 동일한 절차에 따라 수렴조건을 만족할 때까지 상기의 과정을 반복 수행하게 됩니다. + + + +midas Civil의 P-Delta 해석에 사용된 정적 평형방정식을 정리하면 다음과 같습니다. + +$$ +[ K ] \{u \} + [ K _ {G} ] \{u \} = \{P \} +$$ + +여기서 [ ] K : 변형전 모델의 강성행렬(Stiffness Matrix) + +$[ K _ { G } ]$ : 매 반복 과정에서 새로 구성되는 부재력과 응력에 따른 기하강성행렬(Geometric Stiffness Matrix) + +{ } P : 정적하중벡터 + +{ }u : 변위벡터 + +midas Civil의 P-Delta해석 기능은 다음의 가정하에 수행됩니다. + + P-Delta 효과를 고려하기 위한 기하강성행렬은 트러스요소, 보요소, 벽요소에 대해서만 구성이 가능합니다. + 보요소의 횡변위(굽힘 및 전단 변형)는 축력에 의한 Large-Stress Effect에대해서만 고려됩니다. + P-Delta 해석은 탄성영역에서 유효합니다. + +일반적으로 P-Delta 효과를 고려한 해석은 소요시간이 길기 때문에 구조설계의 완료단계에서 적용하는 것이 바람직합니다. + + + +# 8-4 경계비선형 해석 + +# 8-4-1 비선형요소를 사용한 해석 + +비선형 요소(인장/압축 전담요소)를 사용한 경계비선형 해석에서는 구조계 전체를 선형으로 가정하고 일부 비선형요소에 대해서만 비선형 거동 특성을 고려 하는 방법을 사용하고 있으며 내용을 정리하면 다음과 같습니다. + +비선형요소를 사용한 정적해석에서 사용할 수 있는 비선형요소에는 인장전담 트러스요소, Hook 요소, Cable 요소, 압축전담 트러스요소, Gap 요소, Elastic Link의 인장 또는 압축전담조건 등이 있습니다. + +비선형요소를 사용한 구조계의 정적평형 방정식을 정리하면 식 (1)과 같습니다. + +$$ +\left[ K + K _ {N} \right] \{U \} = \{P \} \tag {1} +$$ + +여기서 K : 선형구조물의 강성 + +$K_{N}$ : 비선형요소의 강성 + +식 (1)과 같은 비선형강성을 포함하는 평형방정식의 해를 구하는 방법으로는 변위나 부재력 조건에 따른 비선형요소의 강성을 재구성하고 반복해석을 통하여 평형방정식이 수렴하도록 하는 방법을 사용하고 있습니다. + + + +# 8-4-2 비선형요소의 강성( ${ \cal K } _ { N } )$ + +midas Civil에서 사용되는 비선형요소의 강성은 해석결과로부터 구해지는 변위와부재력에 의해 결정됩니다. Truss, Hook, Gap 형태의 비선형강성은 양단부의 변위와 Hook나 Gap의 간격에 따라 부재의 비선형 강성이 결정되고, Cable 형태의 요소는 해석결과로 발생하는 부재력을 통해 비선형 강성이 결정됩니다 + +Truss, Hook, Gap 형태의 인장 및 압축전담요소의 비선형강성은 식 (2)와 같이 결정되고, cable의 비선형강성은 부재에 작용하는 인장력의 변화에 따른 강성의 변화를 고려하고자 식 (3)과 같이 산정된 유효강성을 통해 비선형 강성이 결정됩니다. + +$$ +K _ {N} = f (D - d) \tag {2} +$$ + +여기서 D : 초기상태의 간격(Hook나 Gap의 간격) + +d : 해석결과 발생하는 부재의 길이 변화량 + +$$ +K _ {e f f} = \frac {1}{1 / K _ {s a g} + 1 / K _ {e l a s t i c}} = \frac {E A}{L (1 + \frac {\omega^ {2} L ^ {2} E A}{1 2 T ^ {3}})} \tag {3} +$$ + +여기서 sagK $K _ { _ { s a g } } = \frac { 1 2 T ^ { 3 } } { \omega ^ { 2 } L ^ { 3 } } , K _ { e l a s t i c } = \frac { E A } { L }$ 2 3 , elastic K + +Cabl + +L : 중력방향에 대한 수평길이 + +T : Cable의 인장력 + + + +비선형요소의 경계조건에 따른 비선형성을 고려하는 해석에서는 구조물 재질의 비선형성 등을 고려하지 않기 때문에 다음과 같은 몇 가지 적용상의 제약이 있습니다. + +1. 구조물의 재료 비선형성은 고려하지 않습니다. +2. 비선형요소만으로 구성된 구조형태는 하중에 따라 불안정성이 발생할 수있기 때문에 비선형요소만으로 구성된 절점의 사용은 제한됩니다. +3. 하중에 따라 발생한 변위와 부재력에 의해 요소의 강성이 변화하기 때문에하중조건결과들의 선형조합은 사용할 수 없습니다. +4. 비선형요소를 사용한 구조물의 동적해석 시에는 선형상태의 강성을 사용하여 해석을 수행합니다 + +비선형요소를 사용한 해석과정은 다음과 같습니다. + +1. 구조물의 선형강성과 비선형요소의 선형상태의 강성을 사용하여 구조물의전체강성행렬과 하중벡터를 구성합니다. +2. 전체강성과 하중벡터를 사용하여 정적해석을 수행하고 변위와 부재력을 구합니다. +3. 구조물의 전체강성을 재구성합니다. +4. 강성을 변화시켜 해석을 수행하는 경우에는 구해진 변위와 부재력을 사용하여 비선형요소의 강성을 계산하고 전체구조물의 강성을 재구성합니다. +5. 2와 3 과정을 반복하여 해석결과가 수렴조건을 만족할 때까지 수행합니다. diff --git a/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_025.md b/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_025.md new file mode 100644 index 00000000..5fd5d879 --- /dev/null +++ b/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_025.md @@ -0,0 +1,262 @@ + + +# 8-5 경계비선형 시간이력해석 (Boundary Nonlinear Time History Analysis) + +# 8-5-1 해석 모델 구성 + +해석 대상 구조물은 비선형연결요소를 포함하는 구조물로서 비선형연결요소를 제외한 나머지 부재들은 모두 선형 탄성인 것으로 가정합니다. 비선형연결요소는 구조물의 두 절점 사이를 연결하거나 구조물과 지지점을 연결할 수 있습니다. + +하나의 비선형연결요소는 총 6개의 스프링 (1개의 부재 축방향 스프링, 2개의 전단 스프링, 1개의 비틀림 변형 스프링, 2개의 휨 변형 스프링)으로 구성되며, 이 가운데 일부의 스프링만을 선택하여 이용할 수 있습니다. + +각각의 스프링은 기본적으로 선형 속성을 가지며 사용자의 선택에 의해 비선형 속성을 가지도록 할 수 있습니다. 선형 속성만을 가지는 스프링은 실제 물리적으로 선형 탄성인 스프링입니다. (이하 선형 스프링으로 표기합니다.) 비선형 속성을 가지는 스프링은 실제 물리적으로 비선형성을 갖는 스프링을 표현하며 이 스프링이 갖는 선형 속성은 해석 알고리줌 상의 필요에 의해서만 요구되는 속성입니다. (이하 비선형 스프링으로 표기합니다.) + +선형 속성은 요소를 구성하는 각 스프링의 유효강성으로서 비선형연결요소의 요소 +강성행렬과 전체 구조물의 강성행렬을 구성하는데 사용됩니다. 비선형연결요소를 +포함한 구조물의 선형 및 비선형 정적해석과 선형 동적해석에서는 이 유효강성만 +으로 해석을 수행합니다. + +비선형 속성은 비선형 스프링의 동적특성을 나타내는 파라미터로서 midas Civil 에서는 6가지 타입(Visco-elastic Damper, Gap, Hook, Hysteretic System, Lead Rubber Bearing Isolator, Friction Pendulum System Isolator)의 비선형 속성이 제공됩니다. + + + +# 8-5-2 경계비선형 시간이력해석의 개요 + +경계비선형 시간이력해석은 구조물의 일부분에 비선형성이 국한된 경우에 적용할수 있는 비선형 시간이력해석 방법으로, 면진 및 제진장치가 설치된 구조물의 비선형 거동 특성을 파악하기 위한 해석기능입니다. 면진 및 제진장치는 설계하중에대한 구조물의 소성변형을 방지하거나 최소화 시켜주므로 구조부재들은 탄성거동하고, 구조물의 비선형 거동은 주로 면진 및 제진장치에서 발생한다고 볼 수 있습니다. midas Civil의 경계비선형 해석에서는 이러한 점에 근거하여 면진 및 제진장치를 비선형 경계요소로서, 비선형연결요소(Force Type General Link)를 통해 모델링 되고, 나머지 부분은 선형 탄성 구조물로 가정됩니다. 편의상 전자를 비선형계,후자를 선형계라고 칭합니다. 경계비선형 시간이력 해석은 비선형계에서 발생하는부재력을 선형계에 가해지는 외부 동적하중으로 치환하는 방법으로 해석을 수행합니다. + +![](images/page-242_dc767c909d274f76b797dde06a5d47e3d1934c1f7e832b8782d4f3810fe306c2.jpg) + +
+flowchart + +```mermaid +graph TD + A["p"] --> B["●"] + B --> C["●"] + C --> D["●"] + D --> E["f_N"] + E --> F["×"] + F --> G["→"] + G --> H["p"] + H --> I["●"] + I --> J["●"] + J --> K["●"] + K --> L["f_N"] + L --> M["×"] + M --> N["f_L"] + N --> O["→"] + O --> P["f_L"] + P --> Q["→"] +``` +
+ +fN : Nonlinear spring & force +fL : Effective Stiffness & force + +그림 2.8.8 경계비선형 해석의 해석방법 + +midas Civil의 경계비선형 해석은 비선형 모드 중첩법(Nonlinear Modal Time HistoryAnalysis)과 비선형 직접적분법(Nonlinear Direct Integration)에 의해 수행됩니다. 경계비선형 해석에 사용되는 면진 및 제진 장치는, 각 요소특성이 상미분방정식으로표현되며, 상미분방정식의 수치해법인 Runge-Kutta Method에 의해 계산됩니다. 구해진 결과는 유효하중으로 반영하여, 미지수를 구하는 방법으로 계산을 수행합니다. + + + +비선형 모드 중첩법은 면진 및 제진장치를 제외한 모든 구조부재가 탄성영역에 있다는 전제하에 수행되며, 고유치 해석이 반드시 선행되어야 합니다. 또한, 선형 시간이력 해석과 같이 중첩의 원리에 의해 해석하기 때문에 모든 시간증분마다 전체구조물에 대한 평형방정식을 풀게 되는 직접적분법에 비해 해석속도가 빠르다는장점이 있습니다. + +한편, 구조부재의 비탄성거동까지를 해석할 경우에는 구조부재도 비탄성 요소로보고 구조물 전체를 비선형 해석할 필요가 있습니다. 비선형 직접적분법은 비선형경계요소뿐만 아니라, 구조부재에 비선형 거동까지 고려 할 수 있습니다. 또한, 구조부재의 비탄성 거동은 비선형 이력모델에 의해서 산정됩니다. + +# 8-5-3 모드 중첩법에 의한 경계비선형 시간이력해석 + +# 모드 중첩법에 의한 경계비선형 시간이력해석의 개요 + +Force Type 범용연결요소가 포함된 구조물의 운동방정식은 다음과 같이 구성됩니다. + +$$ +M \ddot {u} (t) + C \dot {u} (t) + \left(K _ {S} + K _ {N}\right) u (t) = B _ {P} p (t) + B _ {N} \left(f _ {L} (t) - f _ {N} (t)\right) +$$ + +여기서 M : 질량행렬 + +$C$ : 감쇠행렬 + +$K _ { s }$ : Force Type 범용연결요소를 제외한 나머지 탄성강성 + +$K _ { \scriptscriptstyle { N } }$ : Force Type 범용연결요소의 유효강성 + +$B _ { { } _ { P } }$ , $B _ { \scriptscriptstyle { N } }$ : 변환행렬 + +u t( ) , u t  ( ) , u t ( ) : 절점의 변위, 속도, 가속도 + +$p ( t )$ : 동적하중 + +$f _ { L } ( t )$ : Force Type 범용연결요소에 포함된 비선형 성분의 유효강성에의한 내력 + +$f _ { _ { N } } ( t )$ : Force Type 범용연결요소에 포함된 비선형 성분의 실제 내력 + + + +우변의 ( ) Lf t 는 좌변의 KN 에 의해 발생하는 절점력 가운데 Force Type 범용연결요소의 비선형 성분에 해당되는 것과 상쇄되며 ( ) Nf t 만이 동적 거동에 영향을 주게 됩니다. 유효강성행렬 KN 을 사용하는 이유는 Force Type 범용연결요소의 연결위치에 따라서 원래 구조물의 강성행렬 KS 만으로는 불안정 구조물이 될 수 있기때문입니다. + +질량행렬과 강성행렬에 대한 모드 형상(Mode Shape)과 고유진동수(NaturalFrequency)는 고유치 해석(Eigenvalue Analysis), 또는 Ritz벡터 해석을 통해 계산할수 있습니다. 감쇠는 모드감쇠비를 통해 고려됩니다. + +모드의 직교성을 이용하여 위의 운동방정식은 다음과 같이 모드좌표계(ModalCoordinate)의 운동방정식으로 변환됩니다. + +$$ +\ddot {q} _ {i} (t) + 2 \xi_ {i} \omega_ {i} \dot {q} _ {i} (t) + \omega_ {i} ^ {2} q (t) = \frac {\phi_ {i} ^ {T} B _ {P} p (t)}{\phi_ {i} ^ {T} M \phi_ {i}} + \frac {\phi_ {i} ^ {T} B _ {N} f _ {L} (t)}{\phi_ {i} ^ {T} M \phi_ {i}} + \frac {\phi_ {i} ^ {T} B _ {N} f _ {N} (t)}{\phi_ {i} ^ {T} M \phi_ {i}} +$$ + +여기서  : i-번째 모드의 모드형상 벡터 + + : i-번째 모드의 감쇠비 + +@：- + +( ) q t , ( ) q t  , ( ) q t  : i-번째 모드의 일반화 변위, 속도, 가속도 + +우변의 ( ) f t 와 ( ) f t 는 해당 Force Type 범용연결요소의 요소좌표계에서의 실제변형 및 변형의 변화율에 의해 결정됩니다. 그러나 요소의 실제 변형은 특정 모드가 아닌 전체 모드의 성분을 모두 포함하고 있습니다. 따라서 상기의 모드좌표계운동방정식은 모드별로 독립적이라고 할 수 없습니다. 모드해석의 장점을 이용하기 위해서 각 해석시간 단계에서의 ( ) Nf t 와 ( ) Lf t 를 가정하게 되면 독립적인 모드좌표계 운동방정식이 됩니다. + +먼저 이전 단계의 해석결과로부터 현재 단계에서의 모드 일반화 변위와 속도를 가정하고 이를 기초로 현재단계에서의 ( ) f t 와 ( ) f t 를 계산합니다. 이로부터 현재단계에서의 모드 일반화 변위 및 속도를 조합해서 Force Type 범용연결요소의 변형 및 변형 변화율을 계산합니다. ( ) f t 와 ( ) f t 의 계산과 그에 따르는 모드 일반 + + + +화 변위 및 속도의 계산 과정을 다음 수렴오차가 허용치 이내로 들어올 때까지 계속 반복합니다. + +$$ +\varepsilon_ {q} = \max _ {i} \left\{\frac {q _ {i} ^ {(j + 1)} (n \Delta t) - q _ {i} ^ {(j)} (n \Delta t)}{q _ {i} ^ {(j + 1)} (n \Delta t)} \right\} +$$ + +$$ +\varepsilon_ {\dot {q}} = \max _ {i} \left\{\frac {\dot {q} _ {i} ^ {(j + 1)} (n \Delta t) - \dot {q} _ {i} ^ {(j)} (n \Delta t)}{\dot {q} _ {i} ^ {(j + 1)} (n \Delta t)} \right\} +$$ + +$$ +\varepsilon_ {f _ {M}} = \max _ {i} \left\{\frac {f _ {M , i} ^ {(j + 1)} (n \Delta t) - f _ {M , i} ^ {(j)} (n \Delta t)}{f _ {M , i} ^ {(j + 1)} (n \Delta t)} \right\} +$$ + +여기서 $f_{M,i}^{(j)}(n\Delta t)=\frac{\phi_{i}^{T}B_{N}f_{N}^{(j)}(n\Delta t)}{\phi_{i}^{T}M\phi_{i}}$ + +Δt : 타임스텝 크기 + +n : 타임스텝 + +j : 반복계산 스텝 + +i : 모드 차수 + +이상의 과정을 각 해석시간 단계별로 반복하며 최대반복회수 및 수렴 허용오차는 Time History Load Cases에서 사용자가 직접 입력합니다. 만약 수렴하지 않으면 자동적으로 해석시간 간격 $\Delta t$ 를 세분하여 다시 해석하게 됩니다. + +Force Type 범용연결요소의 비선형 특성은 미분방정식으로 표현되며 각 반복과정에서 비선형 성분에 해당되는 내력을 산정하기 위해서는 이 미분방정식의 수치해석 해가 필요하게 됩니다. midas Civil에서는 수치해석 방법으로서 Runge-Kutta Method를 사용하고 있으며 이 방법은 미분방정식의 수치해석에 가장 널리 사용되는 방법으로서 빠른 해석 속도와 정확성을 가집니다. + +# 고유치 해석시 유의점 + +비선형 모드 중첩법에 의한 경계비선형 시간이력 해석은 모드해석을 기반으로 하기 때문에 구조물의 응답을 표현하기에 충분한 수의 모드를 사용할 필요가 있습니다. 특히 Force Type 범용연결요소의 변형을 표현하기에 충분한 수의 모드를 필요로 합니다. + + + +대표적인 경우로 마찰진자형 면진장치의 지진응답해석을 들 수 있습니다. 마찰진자형 면진장치는 요소의 축방향 성분 내력이 전단방향 성분의 거동을 결정하는 중요한 인자입니다. 따라서 일반적인 지진응답 해석과는 달리 연직방향 모드의 중요성이 매우 크며, 연직방향 모드질량의 합계가 전체질량에 가깝도록 충분한 모드수를 확보할 필요가 있습니다. + +고유치해석 방법을 사용할 경우 이와 같은 목표를 달성하기 위해서는 매우 많은수의 모드를 필요로 할 수 있으므로 해석 시간이 길어질 수 있습니다. Ritz 벡터해석을 사용하면 각 자유도에 대한 동적하중의 분포를 고려한 모드형상과 고유진동수를 구해주며, 적은 수의 모드로 고차모드의 영향까지 포함시킬 수 있습니다. + +예를 들면 마찰진자형 면진장치의 경우에 Ritz벡터 해석을 위한 입력 대화상자에서 구조물의 Z방향(중력방향) 가속도나 구조물의 자중에 관계된 정적하중의 단위하중조건을 선택하면 주로 구조물의 연직방향 거동에 관계되는 고유진동수와 모드형상을 구할 수 있습니다. 많은 경우에 있어서 Ritz 벡터해석을 이용하는 것이 고유치해석을 이용하는 것 보다 적은 수의 모드로 더 정확한 해석결과를 산출해 주는 것으로 알려져 있습니다. + +# 정적하중과 동적하중의 조합 + +비선형 모드중첩법에 의한 시간이력 해석은 선형 시간이력 해석과는 달리 중첩의원리가 적용될 수 없습니다. 즉, 정적하중에 대한 해석결과와 동적하중에 대한 해석 결과를 단순히 합하여 두 하중이 동시에 작용한 결과로 사용할 수 없습니다.따라서 정적하중과 동적하중의 영향을 동시에 고려하기 위해서는 정적하중을 동적하중 형태로 입력하여 경계비선형 시간이력해석을 수행해야 합니다. + +이를 위해 midas Civil에서는 Time Varying Static Loads 기능을 통해 정적하중을 동적하중 형태로 입력할 수 있도록 지원합니다. + +먼저, Time Forcing Function에 Time Function Data Type이 Normal인 Ramp 함수를입력합니다. 다음으로 Time Varying Static Load에서 연직방향 정적하중의 StaticLoad 및 이미 지정된 Function Name을 입력합니다. Ramp 함수의 형상은 지반가 + + + +속도의 도달시간(Arrival Time) 이전에 정적하중의 재하가 완료되고 이로 인한 진동이 충분히 감쇠될 수 있도록 지정합니다. 이와 관련하여 정적하중 재하로 인한 진동의 감쇠에 소요되는 시간을 줄이기 위해 Time History Load Cases에서 해석 초기에 사용자가 지정한 시간동안 99%의 감쇠율을 적용하는 옵션을 선택할 수 있습니다. 또한 지반가속도가 작용하는 동안에도 정적하중이 계속 지속되도록 합니다. + +![](images/page-247_95e012dc83ab6236600e56600f4a94e3b35385712598b8b28653c0a44342dc0a.jpg) +Dynamic Load (t0 + +직접 적분법에 의한 경계비선형 시간이력해석에서 Force Type 범용연결요소가 포함된 구조물의 운동방정식은 다음과 같이 구성됩니다. + +$$ +M \ddot {u} (t) + C \dot {u} (t) + K _ {N} u (t) = p (t) + \left(f _ {L} (t) - f _ {N} (t)\right) +$$ + +여기서 M : 질량행렬 + +C : 감쇠행렬 + +$K_{N}$ : Force Type 범용연결요소의 유효강성 + +$u(t)$ , $\dot{u}(t)$ , $\ddot{u}(t)$ : 절점의 변위, 속도, 가속도 + +$p(t)$ : 동적하중 + +$f_{L}(t)$ : Force Type 범용연결요소의 유효강성에 의한 내력 + +$f_{N}(t)$ : Force Type 범용연결요소의 실제 내력 + +Newmark-β 법에 의해, 증분변위 $\delta u$ 에 관한 평형방정식은 다음과 같이 표현됩니다. + +$$ +K _ {E f f} \cdot \delta u = \Delta p _ {E f f} +$$ + +여기서 $K_{Eff}$ : 유효강성행렬 + +$$ +K _ {E f f} = \frac {1}{\beta (\Delta t) ^ {2}} M + \frac {1}{\beta \Delta t} C + K _ {N} +$$ + +$\Delta p_{Eff}$ : 각 반복해석 단계에서의 유효하중벡터 + +$$ +\Delta p _ {E f f} = p (t) - \left\{M \ddot {u} (t) + C \dot {u} (t) + \left(f _ {L} (t) - f _ {N} (t)\right) \right\} +$$ + +$\delta u$ : 각 반복해석 단계에서의 변위증분벡터 + +β : Newmark-β 법 관련 파라미터 + + + +# 8-5-5 유효강성 + +경계비선형 시간이력 해석은 전체 구조물을 선형계와 비선형계로 나누며, 비선형계에서 발생하는 비선형 부재력을 선형계에 가해지는 외부 동적하중으로 치환하여해석합니다. 여기서 비선형계를 구성하는 Force Type 범용연결요소의 위치에 따라선형계 만으로는 불안정 구조가 될 수 있으므로 유효강성을 사용하여 안정한 구조물로 만든 뒤 시간이력해석을 수행합니다. + +만약 Force Type 범용연결요소를 제거할 경우에 구조물이 불안정해진다면, 고유진동수와 모드형상이 실제의 비선형 거동과 유사하도록 적절한 유효강성을 입력할필요가 있습니다. 이 경우에 적절한 유효강성은 일반적으로 0보다 크고 비선형 특성상의 초기강성 보다 작거나 같은 값을 사용합니다. 초기강성은 뒤에서 설명할요소의 종류별 동적특성에서 Viscoelastic Damper 의 bk , Gap, Hook 및 HystereticSystem의 k , Lead Rubber Bearing Isolator와 Friction Pendulum System의 $k _ { y }$ , $k _ { b }$ 가 이에 해당됩니다. + +비선형 거동 이전의 응답을 계산하기 위한 선형정적해석 또는 선형동적해석을 수행하려면 초기강성을 유효강성으로 입력합니다. 근사적인 선형동적해석을 수행하려면 비선형 연결요소가 비선형 해석시와 유사한 거동을 하도록, 다음 그림과 같이 예상되는 최대 변형을 기준으로 적절한 할선강성(Secant Stiffness)을 유효강성으로 입력합니다. 만약 해석결과가 수렴하지 않는 경우에 유효강성 값을 조정하면수렴하는 결과를 얻을 수도 있습니다. + +![](images/page-249_b1aa300340258038b51ad2c844b8ad28395a1f2ab7409a978c41ee8d00fe669f.jpg) + +
+text_image + +Force +F+ +k_eff +Δ- +Δ+ +Displacement +F- +
+ +그림 2.8.10 경계비선형 요소의 유효강성 + + + +# 8-5-6 Force Type 범용연결요소의 동적특성 + +midas Civil의 경계 비선형 시간이력 해석기능에서 제공되는 Force Type 범용연결요소는 점탄성감쇠기(Viscoelastic Damper), 갭(Gap), 후크(Hook), 이력거동 시스템(Hysteretic System), 납삽입 고무베어링형 면진장치(Lead Rubber Bearing Isolator),마찰진자형 면진장치(Friction Pendulum System Isolator)의 6가지이며 각각의 동적특성은 다음과 같습니다. + +# 8-5-7 점탄성감쇠기 (Visco-Elastic Damper) + +점탄성 감쇠기는 변형에 비례해서 힘이 발생하는 탄성, 그리고 변형의 속도에 비례해서 힘이 발생하는 점성을 동시에 갖습니다. 점탄성 감쇠기는 구조물의 감쇠능력을 증대시켜 지진, 바람 등에 의해 발생하는 동적 응답을 감소시켜 구조물의 안전성과 사용성을 확보하기 위한 목적으로 사용됩니다. + +탄성체는 그림 2.8.11(a)에 나타낸 것과 같이 힘을 가해면 가한 힘에 비례하여 늘어나고, 가해진 힘을 제거하면 완전히 원래의 형태로 돌아오는 물체를 말하며, 고무 또는 Spring이 여기에 해당합니다. 점성체는 힘을 가하면 점점 변형하여, 힘이없어져도 변형된 형태를 유지하는 성질을 가집니다. 그림 2.8.11(b)는 점성체(Dashpot) 모델과 그 특성을 나타내며, 점토 등이 여기에 해당합니다. 점탄성체는탄성과 점성 양쪽의 성질을 갖는 물체로서, 그림 2.8.11(c),(d)와 같이 힘을 가하면변형이 점점 증가하고, 어느 순간 힘을 빼면, 순간적으로 변형이 줄어드는 특징을가집니다. 점탄성 댐퍼는 점탄성체의 이러한 성질을 이용한 감쇠장치입니다. diff --git a/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_026.md b/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_026.md new file mode 100644 index 00000000..64f97312 --- /dev/null +++ b/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_026.md @@ -0,0 +1,444 @@ + + +![](images/page-251_dd1ae40099276684a20e38d7aae5761ef9c099ac59d1573f90bc020c4400a360.jpg) + +
+text_image + +k_d +f ← → f +deformation +Loading Unloading Time +
+ +(a) Elastic Object + +![](images/page-251_2f3fa6a97a36bc365f71ded5832a6ba31fbbc89a648ef9bacda5103c5e039dec.jpg) + +
+text_image + +f ← ●─→ f + c_d + defarmonation + Loading → Unloading → Time +
+ +(b) Visco Object + +![](images/page-251_530f5fce94cd5f7d3af18005b962fb7b0b82d828522c51055299256fa1897854.jpg) + +
+chemical + +Electrical circuit diagram with springs and capacitors labeled with f and k_d +
+ +![](images/page-251_1bc2442505343c9fe4957d915a928e2f5172591da37a31474633a1e84afd4dcd.jpg) + +![](images/page-251_4a283880d5f3458e3e00a0bea1639c05174bfd455779cfdaeda43a63b40f31d6.jpg) + +
+line + +| Time | Deformation | +|------|-------------| +| Loading | 0 | +| Unloading | 0 | +
+ +![](images/page-251_ea2735813773a91adc37f35ee56755d6db975b5a070fcb35a1fa10792a18b46b.jpg) + +
+line + +| Time | deformation | +| ---------- | ----------- | +| Loading | 0 | +| Unloading | 1 | +| Unloading | 0 | +
+ +(c) Viscoelastic Object(Maxwell Model) +![](images/page-251_2f828a627ecb70decceb3136b32384b4afb9c93408a83b6fac8fdcff9da1cfac.jpg) + +
+text_image + +c_d +f ← ●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─ ●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●○─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─○─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●─●○ +deformation +Loading +Unloading +Time +
+ +(d) Viscoelastic Object(Kelvin Model) +그림 2.8.11 Elastic Object와 Visco Object + + + +midas Civil의 점탄성 감쇠기(Viscoelastic Damper) 요소는 점탄성 감쇠기의 특성을가진 6개의 독립적인 스프링으로 구성됩니다. + +점탄성 감쇠기의 대표적인 수학적 모델은 선형스프링과 점성감쇠가 직렬로 연결된Maxwell 모델과, 병렬로 연결된 Kelvin 모델이 있으며, midas Civil의 점탄성 감쇠기는 Maxwell Model, Kelvin Model, 그리고 Kelvin Model에 스프링이 연결된 DamperBrace Assembly Model로 구분됩니다. + +점탄성 감쇠기의 Damper Brace Assembly Model의 힘-변형 관계 기본식은 다음과같습니다. + +$$ +f = k _ {d} d _ {d} + c _ {d} \operatorname{sign} \left(\dot {d} _ {d}\right) \left| \dot {d} _ {d} \right| ^ {s} = k _ {b} d _ {b} +$$ + +$$ +d = d _ {d} + d _ {b} +$$ + +여기서 f : 점탄성감쇠기의 요소내력 + +$k _ { d }$ : 점탄성감쇠기 강성 + +$k _ { b }$ : 연결 부재 강성 + +$c _ { d }$ : 점탄성감쇠기 감쇠 상수 + +s : 점탄성감쇠기의 비선형 감쇠 특성을 정의하는 지수(Exponent) + +$d$ : 요소의 두 절점 사이의 변형 + +$d _ { d }$ : 점탄성감쇠기의 변형 + +$d _ { b }$ : 연결부재의 변형 + +위 식에서 볼 수 있는 바와 같이 점성감쇠는 변형의 변화율에 비례하는 선형 점성감쇠( s  1.0 ) 뿐만 아니라 변형 변화율의 지수승에 비례하는 비선형 점성감쇠( 0.0 1.0  s )로 모델링 할 수 있습니다. 비선형 감쇠 특성 지수 s 는 0.35\~1.00 범위의 값이 일반적으로 사용됩니다. midas Civil에서는 s 값을 0.20\~1.00로 제한하고있습니다. + +위 식에서 $c _ { d } s i g n \big ( \dot { d } _ { d } \big ) \big | \dot { d } _ { d } \big | ^ { s }$ 항은 감쇠력을 나타내므로, 힘의 단위를 가지게 됩니다.그러나, 비선형 감쇠 특성 지수가 s  1.0 인 경우, 다음 식과 같이 감쇠력, $f _ { d } ^ { D } \ \stackrel { \circ } { = }$ + + + +$N \cdot \left( m / s e c \right) ^ { s - 1 }$ 와 같은 단위위가 되는 문제 가 발생하게 됩됩니다. + +$$ +\underbrace {c _ {d}} _ {\frac {N}{m / s e c}} \text { sign } \left(\dot {d} _ {d}\right) \underbrace {\left| \dot {d} _ {d} \right| ^ {s}} _ {(m / s e c) ^ {s}} = \underbrace {f _ {d} ^ {D}} _ {N \cdot (m / s e c) ^ {s - 1}} +$$ + +따라서, mid as Civil에서는 $\dot { d } _ { d }$ 가 무차원량량이 되도록, Reeference Velocitty, $\nu _ { 0 } \equivq$ 채용합니다. $\nu _ { 0 } \equivq$ 채용하면, 감쇠력과 점탄탄성 감쇠기의 힘 -변형 관계는는 다음과 같이표현됩니다. + +$$ +f = k _ {d} d _ {d} + c _ {d} \operatorname{sign} \left(\dot {d} _ {d}\right) \left| \frac {\dot {d} _ {d}}{v _ {0}} \right| ^ {c e x p} = k _ {b} d _ {b} +$$ + +여기서, 감쇠쇠 상수 $c _ { d }$ 의 단위는 본래, $N / m / s e c$ 와 같같지만, 속도항이이 $\nu _ { 0 }$ 로 일반화되므로 힘힘의 단위인 N , tonf 등의 단위를 갖게 됩니다. 따라서서, ReferenceVelocity, $\nu _ { 0 }$ 는 일반적으로로 1.0값을 입력 합니다. 단, mi das Civil이 제공공하는 단위계의 변환시에에는 변환된 길 이단위에 따라 $\nu _ { 0 }$ 값이 자동 으로 변환되므 로, 주의할 필요가 있습니니다. + +위 식에 기초초하여, 점탄성 감쇠기의 비선선형 물성치의 단단위는 다음과 같습니다. +
비선형 속성단 위
점탄성 감쇠기의 요소내력, $f$ $N$ , $tonf$
점탄성감쇠기의 변형의 변화율, $\dot{d}_{d}$ $m/sec$ , $cm/sec$
Reference Velocity, $v_{0}$ $m/sec$ , $cm/sec$
$\dot{d}_{d}/v_{0}$ 무차원량
점탄성감쇠기의 감쇠상수, $c_{d}$ $N$ , $tonf$
+ +표표 2.8.1 점탄성 감쇠쇠기의 비선형 속성 단위 + + + +midas Civil에서 단위계 변환시에 점탄성 댐퍼의 비선형 물성치는 다음과 같이 변환됩니다. + +# 1. 초기 설정(단위계는 kN, m로 설정) + +점탄성 댐퍼의 비선형속성을 다음과 같이 설정합니다. + +-Nonlinear Properties + +
Damper Stiffness (kd): 1000kN/m
Damping (Cd): 1kN
Reference Velocity (V0): 1m/sec
Damping Exponent (s): 1
Bracing Stiffness (kb): 1000kN/m
+ +# 2. 단위계를 N, cm 로 변환 + + Reference Velocity, $\nu _ { 0 } \equivq$ 고려하지 않는 경우 + +$$ +k _ {d} = 1 0, 0 0 0 N / c m +$$ + +$$ +C _ {d} = \frac {1 \cdot 1 0 0 0}{1 0 0} = 1 0 N \cdot \sec / c m +$$ + +$$ +s = 1 +$$ + +$$ +k _ {b} = 1 0, 0 0 0 N / c m +$$ + + 경우Reference Velocity, $\nu _ { 0 }$ 를 고려하는 경우 + +$$ +C _ {d} = 1 k N = 1, 0 0 0 N +$$ + +$$ +v _ {0} = 1 m / \sec = 1 0 0 c m / \sec +$$ + +Nonlinear Properties +
Damper Stiffness (kd):10000N/cm
Damping (Cd):1000N
Reference Velocity (V0):100cm/sec
Damping Exponent (s):1
Bracing Stiffness (kb):10000N/cm
+ + + +![](images/page-255_834bf081bd764073f4e4d6daf1a55ec7178c4c65a3c12985c6d1de39d298a995.jpg) + +
+text_image + +d_d +f ← ●⋯⋯●—[c_d]—[k_b]—●⋯⋯●→ f +N1 +d_b +N2 +
+ +(a) Maxwell Model + +![](images/page-255_40ef40430f7ee2f8f3de306f990a38a6f7d835815b1b760133c0394263ed3feb.jpg) + +
+text_image + +d_d +f ←●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● +N1 +k_d +N2 +f +
+ +(b) Kelvin Model + +![](images/page-255_c01b1935ce30007f2a52bec194a6aa33842dc0c14e6f36848431120f6012d6c4.jpg) + +
+text_image + +d_d +c_d +f ←●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● +N1 +k_d +k_b +N2 +f +
+ +(c) Damper Brace Assembly Model +그림 2.8.12 점탄성 감쇠기 + +# Maxwell Model + +Maxwell Model은 그림 2.8.12(a)에 나타낸 것과 같이, 선형스프링과 점성감쇠가 직렬로 연결된 모델로서 Fluid Viscoelastic Device의 해석에 사용됩니다. + +Maxwell Model의 힘-변형 관계식은 다음과 같습니다. + +$$ +f = c _ {d} \operatorname{sign} \left(\dot {d} _ {d}\right) \left| \frac {\dot {d}}{v _ {0}} \right| ^ {s} = k _ {b} d _ {b} +$$ + + + +위 식은 상미분방정식의 초기치 문제가 되므로, Runge-Kutta Method를 사용하여,미지수인 점탄성감쇠기의 변형 $d _ { d }$ 를 구합니다. + +# Kelvin(Voigt) Model + +Kelvin Model은 그림 2.8.12(b)에 나타낸 것과 같이, 선형스프링과 점성감쇠가 병렬로 연결된 모델로서 Solid Viscoelastic Device의 해석에 사용됩니다. + +Kelvin Model의 힘-변형 관계식은 다음과 같으며, 우변의 모든 항이 기지값이므로,식을 직접 풀어 점탄성 감쇠기에 작용하는 힘을 구할 수 있습니다. + +$$ +f = k _ {d} d + c _ {d} \operatorname{sign} (\dot {d}) \left| \frac {\dot {d}}{v _ {0}} \right| ^ {s} +$$ + +# Damper Brace Assembly Model + +Damper Brace Assembly Model은 Kelvin Model에 스프링이 연결된 모델로서, 그림2.8.12(c)와 같이 제진용 가새 해석시에 사용됩니다. + +Damper Brace Assembly Model의 힘-변형 관계식은 다음과 같습니다. 아래 식은 상미분방정식의 초기치 문제가 되므로, Runge-Kutta Method를 사용하여 미지수인 점탄성감쇠기의 변형 dd 를 구합니다. + +$$ +f = k _ {d} d _ {d} + c _ {d} \operatorname{sign} \left(\dot {d} _ {d}\right) \left| \frac {\dot {d}}{v _ {0}} \right| ^ {s} = k _ {b} d _ {b} +$$ + +Maxwell Model과 Damper Brace Asssembly Model은 앞서 언급한 바와 같이, 미지수인 점탄성감쇠기의 변형 $d _ { d }$ 을 구하기 위해서 미분방정식의 수치해법인 Runge-Kutta Method을 사용합니다. midas Civil에서는 비선형 감쇠 특성 지수 s 가 1인 선형 점성감쇠( s  1.0 )를 갖는 경우, 해석의 효율성을 높이기 위해서, Runge-KuttaMethod 대신에, 다음과 같은 근사식을 통하여 미지수인 점탄성감쇠기의 변형 $d _ { d }$ 를 구하는 방법을 사용합니다. + + + +$$ +d _ {d} (t + \Delta t) = \frac {k _ {b} d (t + \Delta t) + \frac {c _ {d}}{v _ {0}} \frac {1}{\Delta t} d _ {d} (t)}{k _ {d} + k _ {b} + \frac {c _ {d}}{v _ {0}} \frac {1}{\Delta t}}; (s = 1. 0) +$$ + +단, Maxwell Model : $k _ { d } = 0 . 0$ + +# 점탄성 감쇠기의 정적 비선형 해석 + +정적 비선형해석시에는 댐퍼의 변형 변화율을 $\dot { d } _ { d } = 0 . 0 \subseteq$ 로 두고, 점탄성 감쇠기의 유효강성 k 을 구하여 계산합니다. + +$$ +f = k \cdot d +$$ + +여기서, Maxwell Model : k  0.0 + +Kelvin Model : $k = k _ { d }$ + +Damper Brace Assembly Model : k  $k = \frac { k _ { b } \cdot k _ { d } } { k _ { b } + k _ { d } }$ + +# 8-5-8 갭 (Gap) + +갭은 다른 경계요소와 마찬가지로 6개의 성분으로 구성되며, 요소좌표계에서 6개자유도 별로 N1절점에 대한 N2절점의 상대변위가 초기간격보다 큰 절대값의 음수가 되면 해당 성분의 강성이 발현됩니다. 축방향 성분만을 사용하는 경우에는 압축력전담요소가 되며 접촉문제 등을 모델링 하는데 사용될 수 있습니다. + +![](images/page-257_b2e54033e5e20170042b43cf0f7fccdbce623e106de5b9644ab5fef00b8b4041.jpg) + +
+text_image + +o +d +f ←●─| |─●─●─●─→ f +N1 k N2 +
+ +그림 2.8.13 갭(Gap) + + + +6개의 성분은 독립적으로 거동하며 다음과 같은 힘-변형 관계식을 갖습니다. + +$$ +f = \left\{ \begin{array}{l l} k (d + o) & \text { if } d + o < 0 \\ 0 & \text { otherwise } \end{array} \right. +$$ + +여기서 k : 강성 + +o : 초기간격 + +d : 변형 + +# 8-5-9 후크 (Hook) + +후크는 다른 경계요소와 마찬가지로 6개의 성분으로 구성되며, 요소좌표계에서 6개 자유도 별로 N1절점에 대한 N2절점의 상대변위가 초기간격보다 큰 절대값의양수가 되면 해당 성분의 강성이 발현됩니다. 축방향 성분만을 사용하면 인장력전담요소가 되며, Wind Brace나 Hook Element 등을 모델링 하는데 사용할 수 있습니다. + +![](images/page-258_593e63df3d6b0bdc7309f29f9d64188ea58115a3d0ff4b72f02a1abd354c40ba.jpg) + +
+text_image + +o +d +f ← ● → f +N1 +k +N2 +
+ +그림 2.8.14 후크(Hook) + +6개의 성분은 독립적으로 거동하며 다음과 같은 힘-변형 관계식을 갖습니다. + +$$ +f = \left\{ \begin{array}{l l} k (d - o) & \text { if } d - o > 0 \\ 0 & \text { otherwise } \end{array} \right. +$$ + +여기서 k : 강성 + +o : 초기간격 + +d : 변형 + + + +# 8-5-10 이력거동시스템 (Hysteretic System) + +이력거동시스템은 1축 소성(Uniaxial Plasticity)의 특성을 가진 6개의 독립적인 성분으로 구성됩니다. 이력거동 시스템은 이력거동을 통한 에너지 소산장치(EnergyDissipation Device)를 모델링 하는데 사용되며 대표적인 것으로는 금속항복형 감쇠기(Metallic Yield Damper)가 있습니다. 금속항복형 감쇠기는 주구조물(PrimaryStructure)보다 상대적으로 큰 강성을 가지면서 낮은 항복강도를 갖도록 제작되어주변 부재보다 먼저 소성변형을 일으킴으로써 주구조물을 보호할 목적으로 사용됩니다. 강재 댐퍼 등과 비교하여 많은 반복에 대하여 안전한 성능을 발휘할 수 있고, 진폭 및 진동수에 의존하지 않기 때문에 일정한 마찰력을 얻을 수 있습니다. + +![](images/page-259_56ce5ea0c6a7154ffaf286030323634f5eb989133172ab5f1d39acd92978bda1.jpg) + +
+text_image + +f ←●─N1 +Fy | r·k +k +d +N2 +f +
+ +그림 2.8.15 이력거동 시스템 + +이력거동시스템의 성분 별 힘-변형 관계식은 Park, Wen and Ang(1986)에 의해 제안된 다음 식에 의해서 표현됩니다. + +$$ +f = r \cdot k \cdot d + (1 - r) \cdot F _ {y} \cdot z +$$ + +여기서 k : 초기 강성 + +Fy : 항복 강도 + +r : 항복 후 강성 저하율 + +d : 두 절점 사이의 변형 + +z : 이력거동 내부변수 + + + +z 는 이력거동을 나타내는 내부변수로서, Wen(1976)에 의해 제안된 다음의 미분방정식에 의해 정의됩니다. + +$$ +\dot {z} = \frac {k}{F _ {y}} \left[ 1 - | z | ^ {s} \left\{\alpha \operatorname{sgn} (\dot {d} z) + \beta \right\} \right] \dot {d} +$$ + +여기서 + +$\alpha , \beta \qquad : 0 | \Xi | \Xi \Lambda \vdash 1 \underline { { \circ } } | \bigcirc | \bar { \Xi } \Lambda \vdash \Xi \exists \Xi \mid \Xi \vdash \big \forall \bar { \Xi } \Lambda \vdash \Xi \mid , \ | \alpha | + | \beta | = 1 . 0$ + +s : 항복점의 전이영역(Transition Region)의 크기를 결정하는 상수 + +d : 두 절점 사이의 변형 변화율 + +α와 β는 항복후의 거동을 결정하는 상수로서 α+β > 0인 경우에는 SofteningSystem, α+β < 0 인 경우에는 Hardening System을 모델링 할 수 있습니다. 이력거동에 의한 에너지 소산량은 이력곡선에 의한 폐곡선의 면적이 커질수록 증가하며Softening System의 경우에 (β-α)가 작은 값을 가질수록 증가합니다. α와 β의 변화에 따른 이력거동의 변화 예는 그림 2.8.16과 같습니다. + +s는 탄성변형과 소성변형사이의 전이구간, 즉 항복 발생 구간의 형상을 결정짓는상수로서 큰 값을 가질수록 항복점이 뚜렷해 지고 이상적인 Bi-linear Elasto-PlasticSystem에 가까워집니다. s 는 30이하의 값을 사용하는 것이 일반적이며, midasCivil에서는 s 값을 1.0\~50.0으로 제한하고 있습니다. s에 따른 전이구간의 변화 예는 그림 2.8.17과 같습니다. + +![](images/page-260_5e9e715589697d6d78d7231b90d5999189faa9b58cb1bf290d3afe9f4ac17766.jpg) + +
+line +| d | f (Line 1) | f (Line 2) | f (Line 3) | +|------|------------|------------|------------| +| -2.0 | -1.0 | -1.0 | -1.0 | +| -1.5 | -0.5 | -0.5 | -0.5 | +| -1.0 | 0.0 | 0.0 | 0.0 | +| -0.5 | 0.5 | 0.5 | 0.5 | +| 0.0 | 0.75 | 0.75 | 0.75 | +| 0.5 | 0.875 | 0.875 | 0.875 | +| 1.0 | 0.9375 | 0.9375 | 0.9375 | +| 1.5 | 0.96875 | 0.96875 | 0.96875 | +| 2.0 | 1.0 | 1.0 | 1.0 | +
+ +$\mathrm { ( a ) } ~ \mathrm { a } = 0 . 9 , ~ \beta = 0 . 1$ + +![](images/page-260_5799dab032d9fa71878fb1c364b96fe78f843d76a0f3efa96153cd3c6f6cb195.jpg) + +
+line + +| d | f (Line 1) | f (Line 2) | f (Line 3) | f (Line 4) | f (Line 5) | +|------|------------|------------|------------|------------|------------| +| -2.0 | -1.0 | -1.0 | -1.0 | -1.0 | -1.0 | +| -1.5 | -0.75 | -0.75 | -0.75 | -0.75 | -0.75 | +| -1.0 | -0.5 | -0.5 | -0.5 | -0.5 | -0.5 | +| -0.5 | -0.25 | -0.25 | -0.25 | -0.25 | -0.25 | +| 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | +| 0.5 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | +| 1.0 | 0.5 | 0.5 | 0.5 | 0.5 | 0.5 | +| 1.5 | 0.75 | 0.75 | 0.75 | 0.75 | 0.75 | +| 2.0 | 1.0 | 1.0 | 1.0 | 1.0 | 1.0 | +
+ +(b) α = 0.1, β = 0.9 diff --git a/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_027.md b/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_027.md new file mode 100644 index 00000000..7503fd9a --- /dev/null +++ b/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_027.md @@ -0,0 +1,353 @@ + + +![](images/page-261_0ce0dcec42247e74da6db7a2771c805d264c7f7af465d9a5d1a6507b6d77d86a.jpg) + +
+line + +| d | f (Line 1) | f (Line 2) | f (Line 3) | f (Line 4) | f (Line 5) | +|------|------------|------------|------------|------------|------------| +| -2.0 | -3.0 | -3.0 | -3.0 | -3.0 | -3.0 | +| -1.5 | -2.0 | -1.5 | -1.0 | -0.5 | 0.0 | +| -1.0 | -1.0 | -0.5 | 0.0 | 0.5 | 1.0 | +| -0.5 | 0.0 | 0.5 | 1.0 | 1.5 | 2.0 | +| 0.0 | 1.0 | 1.5 | 2.0 | 2.5 | 3.0 | +| 0.5 | 1.5 | 2.0 | 2.5 | 3.0 | 3.5 | +| 1.0 | 2.0 | 2.5 | 3.0 | 3.5 | 4.0 | +| 1.5 | 2.5 | 3.0 | 3.5 | 4.0 | 4.5 | +| 2.0 | 3.0 | 3.5 | 4.0 | 4.5 | 5.0 | +
+ +(c) α = 0.5, β = -0.5 + +![](images/page-261_adb97aa5e811fac84b067d0b4496d222e31d2ae92c2245a76daa58d931a8781f.jpg) + +
+line + +| d | f (Line 1) | f (Line 2) | f (Line 3) | f (Line 4) | f (Line 5) | +|------|------------|------------|------------|------------|------------| +| -2.0 | -5.0 | -5.0 | -5.0 | -5.0 | -5.0 | +| -1.5 | -3.0 | -2.5 | -2.0 | -1.5 | -1.0 | +| -1.0 | -1.0 | -0.5 | 0.0 | 0.5 | 1.0 | +| -0.5 | 0.5 | 1.0 | 1.5 | 2.0 | 2.5 | +| 0.0 | 1.0 | 1.5 | 2.0 | 2.5 | 3.0 | +| 0.5 | 1.5 | 2.0 | 2.5 | 3.0 | 3.5 | +| 1.0 | 2.0 | 2.5 | 3.0 | 3.5 | 4.0 | +| 1.5 | 2.5 | 3.0 | 3.5 | 4.0 | 4.5 | +| 2.0 | 3.0 | 3.5 | 4.0 | 4.5 | 5.0 | +
+ +(d) α = 0.25, β = -0.75 + +그림 2.8.16 이력거동에 의한 변위-내력 관계 (r = 0, k = Fy = s = 1.0) +![](images/page-261_7d1042ff6872468f38a1a1a10bbba1fb75588a2454fc7eca3307a8073af87aaa.jpg) + +
+text_image + +f +s=∞ +Fy +s=2 +s=1 +r·k +k +k +d +
+ +![](images/page-261_64ac4d67b0fc6c8cf891914c068947e7aa431d52c511bb926e4543281aa0f510.jpg) + +
+line + +| d | s = 1.0 | s = 2.0 | s = 10.0 | s = 100.0 | +| ---- | ------- | ------- | -------- | --------- | +| 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | +| 0.2 | 0.2 | 0.2 | 0.2 | 0.2 | +| 0.4 | 0.4 | 0.4 | 0.4 | 0.4 | +| 0.6 | 0.6 | 0.6 | 0.6 | 0.6 | +| 0.8 | 0.8 | 0.8 | 0.8 | 0.8 | +| 1.0 | 1.0 | 1.0 | 1.0 | 1.0 | +| 1.2 | 1.0 | 1.0 | 1.0 | 1.0 | +| 1.4 | 1.0 | 1.0 | 1.0 | 1.0 | +| 1.6 | 1.0 | 1.0 | 1.0 | 1.0 | +| 1.8 | 1.0 | 1.0 | 1.0 | 1.0 | +| 2.0 | 1.0 | 1.0 | 1.0 | 1.0 | +
+ +그림 2.8.17 탄성변형과 소성변형 사이의 전이구간 (항복구간) + + + +# 8-5-11 납삽입 고무베어링형 면진장치 (Lead Rubber Bearing Type Isolator) + +면진장치는 지반의 진동으로부터 구조물을 보호하기 위해 진동의 전달을 차단하는장치로서 교량의 교각과 상판 사이, 혹은 건축물의 지상구조와 기초 사이에 설치합니다. 납삽입 고무베어링형 면진장치는 납의 낮은 항복 후 강성에 의해 구조물의 고유진동수를 지반 진동의 주요 진동수 성분과 격리시키고 이력거동에 의해 면진장치 내에서 진동에너지를 소산시키는 작용을 합니다. + +납삽입 고무베어링형은 2개의 전단 성분에 대해서는 상호 연관된 2축 소성(BiaxialPlasticity)의 특성을 가지며, 나머지 축력, 비틀림과 2개의 휨 성분에 대해서는 상호 독립된 선형탄성 스프링의 특성을 갖습니다. + +![](images/page-262_dbfc06e459bd66133e6f527e3869129bfae511726160e0980593c97ae0c53b07.jpg) + +
+text_image + +Elastic Axial Spring +kₓ +Nonlinear Shear Spring +(Hysteretic System) +d₃ +f₃ +f₂ +d₂ +x +y +z +Fᵧ +Kᵧ +K₀ +dᵧ +
+ +그림 2.8.18 납삽입면진장치의 스프링 구성 + +납삽입 고무베어링형 면진장치의 두 전단 성분의 힘-변형 관계식은 다음과 같습니다. + +$$ +f _ {y} = r _ {y} k _ {y} \cdot d _ {y} + (1 - r _ {y}) F _ {y, y} z _ {y} +$$ + +$$ +f _ {z} = r _ {z} k _ {z} \cdot d _ {z} + (1 - r _ {z}) F _ {y, z} z _ {z} +$$ + +여기서 $k _ { _ { y } } , k _ { z }$ : 요소좌표계 y, z방향 전단 성분의 초기 강성 + +$F _ { _ { y , y } } , \ F _ { _ { y , z } }$ : 요소좌표계 y, z방향 전단 성분의 항복 강도 + + + +$r_{y}$ , $r_{z}$ : 요소좌표계 y, z방향 전단 성분의 항복 후 강성 저하율 + +$d_{y}$ , $d_{z}$ : 요소좌표계 y, z방향 전단 성분의 두 절점 사이의 변형 + +$z_{y}$ , $z_{z}$ : 요소좌표계 y, z방향 전단 성분의 이력거동 내부변수 + +$z_{y}$ , $z_{z}$ 는 이력거동을 나타내는 내부변수로서, 1축 소성에 대한 Wen(1976)의 모델을 확장시킨 Park, Wen, and Ang(1986)의 2축 소성(Biaxial Plasticity) 모델에 의해 다음의 미분방정식으로 정의됩니다. + +$$ +\left\{ \begin{array}{l} \dot {z} _ {y} \\ \dot {z} _ {z} \end{array} \right\} = \left[ \begin{array}{l l} 1 - z _ {y} ^ {2} \left\{\alpha_ {y} \operatorname{sgn} \left(\dot {d} _ {y} z _ {y}\right) + \beta_ {y} \right\} & - z _ {y} z _ {z} \left\{\alpha_ {z} \operatorname{sgn} \left(\dot {d} _ {z} z _ {z}\right) + \beta_ {z} \right\} \\ - z _ {y} z _ {z} \left\{\alpha_ {y} \operatorname{sgn} \left(\dot {d} _ {y} z _ {y}\right) + \beta_ {y} \right\} & 1 - z _ {z} ^ {2} \left\{\alpha_ {z} \operatorname{sgn} \left(\dot {d} _ {z} z _ {z}\right) + \beta_ {z} \right\} \end{array} \right] \left\{ \begin{array}{l} \frac {k _ {y}}{F _ {y , y}} \dot {d} _ {y} \\ \frac {k _ {z}}{F _ {y , z}} \dot {d} _ {z} \end{array} \right\} +$$ + +여기서 + +$\alpha_{y}$ , $\beta_{y}$ , $\alpha_{z}$ , $\beta_{z}$ : 요소좌표계 y, z방향 전단 성분의 이력곡선 형상 관련 상수 + +$\dot{d}_{y}$ , $\dot{d}_{z}$ : 요소좌표계 y, z방향 전단 성분의 변형 변화율 + +위 모델은 비선형 전단 성분이 1개인 경우 이력거동 시스템(Hysteretic System)에서 s=2 인 경우와 동일해 지며, 각 상수들의 역할도 이력거동 시스템과 동일하므로 설명은 생략합니다. + + + +# 8-5-12 마찰진자형 면진장치 (Friction Pendulum System Type Isolator) + +마찰진자형 면진장치는 납삽입고무베어링형과 같은 목적으로 사용되는 면진장치로써 고유진동수 이동과 이력거동에 의한 에너지 소산에 의해 구조물을 지반진동으로부터 보호합니다. 마찰진자형 면진장치는 마찰면의 곡률반경에 의해 복원력을발생시키며 이 곡률반경의 조정을 통해 전체구조물의 고유진동수를 원하는 값으로이동시킬 수 있습니다. 또한 이력거동에 의한 에너지 소산작용은 마찰면의 미끄러짐 현상을 통해 이루어집니다. + +마찰진자형 면진장치는 2개의 전단 성분에 대해서는 상호 연관된 2축 소성(BiaxialPlasticity)의 특성을 가지며, 축 성분에 대해서는 갭(Gap)과 동일한 비선형 특성을가지고 나머지 3개의 성분에 대해서는 상호 독립된 선형탄성 스프링의 특성을 가집니다. + +![](images/page-264_9085f407eaae8ff7c18a8d5b90188738b077e56958ed7511f8a89cbcb4f6170b.jpg) + +
+text_image + +P +R +P +P +μ +P +k +f +N1 +N2 +f +
+ +그림 2.8.19 마찰진자형 면진장치 전단스프링 + +마찰진자형 면진장치의 축 성분의 힘-변형 관계식은 다음과 같이 초기간격이 0인갭(Gap)과 같습니다. + +여기서, $f _ { x } = P = { \left\{ \begin{array} { l l } { k _ { x } d _ { x } } & { i f ~ d _ { x } < 0 } \\ { 0 } & { o t h e r w i s e } \end{array} \right. }$ + +$P$ : 마찰진자형 진동격리장치에 작용하는 축방향 하중 + +$k _ { x }$ : 선형 강성 + +$d _ { x }$ : 변형 + + + +마찰진자형 면진장치의 두 전단 성분의 힘-변형 관계식은 다음과 같습니다. + +$$ +f _ {y} = \frac {| P |}{R _ {y}} d _ {y} + | P | \mu_ {y} z _ {y} +$$ + +$$ +f _ {z} = \frac {| P |}{R _ {z}} d _ {z} + | P | \mu_ {z} z _ {z} +$$ + +여기서 P : 마찰진자형 면진장치에 작용하는 축방향 하중 + +$R_{y}$ , $R_{z}$ : 요소좌표계 y, z방향 전단 성분의 마찰면 곡률반경 + +$\mu_{y}$ , $\mu_{z}$ : 요소좌표계 y, z방향 전단 성분의 마찰면 마찰계수 + +$d_{y}$ , $d_{z}$ : 요소좌표계 y, z방향 전단 성분의 두 절점 사이의 변형 + +$z_{y}$ , $z_{z}$ : 요소좌표계 y, z방향 전단 성분의 이력거동 내부변수 + +마찰면의 마찰계수를 나타내는 $\mu_{y}$ , $\mu_{z}$ 는 2개 전단 변형의 속도와 관련되며 Constantinou, Mokha and Reinhorn(1990)에 의해 제안된 다음 식에 의해 결정됩니다. + +$$ +\mu_ {y} = \mu_ {\text { fast }, y} - \left(\mu_ {\text { fast }, y} - \mu_ {\text { slow }, y}\right) e ^ {- r | v |} +$$ + +$$ +\mu_ {z} = \mu_ {\text { fast }, z} - \left(\mu_ {\text { fast }, z} - \mu_ {\text { slow }, z}\right) e ^ {- r | v |} +$$ + +여기서 $v=\sqrt{\dot{d}_{y}^{2}+\dot{d}_{z}^{2}}$ , $r=\frac{r_{y}\dot{d}_{y}^{2}+r_{z}\dot{d}_{z}^{2}}{v^{2}}$ + +$\mu_{fast,y}$ , $\mu_{fast,z}$ : 요소좌표계 y, z방향 마찰면의 고속변형 마찰계수 + +$\mu_{slow,y}$ , $\mu_{slow,z}$ : 요소좌표계 y, z방향 마찰면의 저속변형 마찰계수 + +$r_{y}$ , $r_{z}$ : 요소좌표계 y, z방향 마찰계수 변화율 + +$\dot{d}_{y}$ , $\dot{d}_{z}$ : 요소좌표계 y, z방향 전단 성분의 변형 변화율 + +$z_{y}$ , $z_{z}$ 는 이력거동을 나타내는 내부변수로서, 1축 소성에 대한 Wen(1976)의 모델을 확장시킨 Park, Wen, and Ang(1986)의 2축 소성(Biaxial Plasticity) 모델에 의해 다음의 미분방정식으로 정의됩니다. + + + +$$ +\left\{ \begin{array}{l} \dot {z} _ {y} \\ \dot {z} _ {z} \end{array} \right\} = \left[ \begin{array}{l l} 1 - z _ {y} ^ {2} \left\{\alpha_ {y} \operatorname{sgn} \left(\dot {d} _ {y} z _ {y}\right) + \beta_ {y} \right\} & - z _ {y} z _ {z} \left\{\alpha_ {z} \operatorname{sgn} \left(\dot {d} _ {z} z _ {z}\right) + \beta_ {z} \right\} \\ - z _ {y} z _ {z} \left\{\alpha_ {y} \operatorname{sgn} \left(\dot {d} _ {y} z _ {y}\right) + \beta_ {y} \right\} & 1 - z _ {z} ^ {2} \left\{\alpha_ {z} \operatorname{sgn} \left(\dot {d} _ {z} z _ {z}\right) + \beta_ {z} \right\} \end{array} \right] \left\{ \begin{array}{l} \frac {k _ {y}}{| P | \mu_ {y}} \dot {d} _ {y} \\ \frac {k _ {z}}{| P | \mu_ {z}} \dot {d} _ {z} \end{array} \right\} +$$ + +여기서 $k_{y}, k_{z}$ : 미끄러짐 발생 이전의 요소좌표계 y, z방향 전단 성분의 초기 강성(연결부재 강성) + +$\alpha_{y}, \beta_{y}, \alpha_{z}, \beta_{z}$ : 요소좌표계 y, z방향 전단 성분의 이력곡선 형상 관련상수 + +$\dot{d}_{y}, \dot{d}_{z}$ : 요소좌표계 y, z방향 전단 성분의 두 절점 사이의 변형 변화율 + +위 모델은 항복강도에 해당하는 값이 축방향하중의 절대값과 마찰계수의 곱으로 표현된 점을 제외하면 납삽입고무베어링과 동일한 형태를 가지므로 각 상수들의 작용에 대한 설명은 생략하며 비선형 전단 성분이 1개인 경우에는 s=2 인 1축 소성 특성과 동일해집니다. + + + +# 8-5-13 Runge-Kutta Method + +경계비선형 해석에서는 상미분방정식의 수치해석 기법으로 Runge-Kutta Method을사용합니다. 상미분방정식을 풀기 위해서는 구간간격을 설정할 필요가 있습니다.시간이력해석인 경계비선형 해석에서의 구간간격은 일정한 시간증분 간격이 됩니다. 그러나, 그림 2.8.20에 나타낸 것과 같이 미분방정식의 해가 급격히 변하는 경우는 일정한 구간간격으로 해를 구할 때, 심각한 제한성을 가질 수 있습니다. 따라서, midas Civil에서는 비선형 경계요소의 수치해를 구할 때, Runge-Kutta Method의수렴성을 향상시키기 위하여, 구간간격인 입력된 시간증분 t 를 세분하는 수렴기법을 사용합니다. 여기서, 구간간격인 t 를 세분한다는 것은 전체 구조물의 운동방정식을 풀 때 시간증분 간격을 세분하는 것이 아니고, 일정한 시간증분 t 를 이용하여 전체 구조물의 운동 방정식을 풀어 변형을 구하고, Runge-Kutta Method를이용하여 비선형 경계요소의 요소내력을 구할 때, 구간간격인 t 를 세분한다는 의미입니다. + +![](images/page-267_c5934d1c8f22afda8e6507d1316d467174e2e6442d3f72266475f653b0f1712d.jpg) + +
+line + +| x | y | +|-------|-------| +| x₀ | y₀ | +| x₁ | y₁ | +| x₂ | y₂ | +
+ +그림 2.8.20 미분방정식의 초기치 문제 + + + +# 8-5-14 Cash-Karp (Adaptive Stepsize Control) + +차수가 다른 Runge-Kutta법의 예측값을 이용하여 오차를 구하여, 구간 간격을 자동으로 조절하는 방법입니다. Adaptive Stepsize Control의 기본 개념은 그림 2.8.21과 같이 오차가 너무 작으면 구간간격 크기를 크게 하고, 오차가 크면 구간간격을작게 하는 것입니다. + +![](images/page-268_d7a02f0c8f74a5192974668585e28886d614211d3cfb06a4f5d1afee1602d9e1.jpg) + +
+line + +| X | Y | +| --- | --- | +| 0 | 0.0 | +| 1 | 0.1 | +| 2 | 0.2 | +| 3 | 0.3 | +| 4 | 0.4 | +| 5 | 0.5 | +| 6 | 0.6 | +| 7 | 0.7 | +| 8 | 0.8 | +| 9 | 0.9 | +| 10 | 1.0 | +| 11 | 0.9 | +| 12 | 0.8 | +| 13 | 0.7 | +| 14 | 0.6 | +| 15 | 0.5 | +| 16 | 0.4 | +| 17 | 0.3 | +| 18 | 0.2 | +| 19 | 0.1 | +| 20 | 0.0 | +
+ +그림 2.8.21 Cash-Karp(Adaptive Stepsize Control) + +구간간격의 설정은 Press et al.(1992)에 의해 제한된 다음 식을 사용합니다. + +$$ +h _ {n e w} = h _ {p r e s e n t} \left| \frac {\Delta_ {n e w}}{\Delta_ {p r e s e n t}} \right| ^ {a} +$$ + +여기서, $h _ { n e w }$ : 새로운 구간 간격 + +$h _ { p r e s e n t }$ : 현재의 구간 간격 + +new $\Delta _ { n e w }$ : 요구되는 $\mathop { \mathbb { X } } ^ { \bullet } \mathbin { \mathbb { \bar { \geq } } } \mathfrak { t } \mathop { \mathbb { E } } \big ( \Delta _ { n e w } = \varepsilon \Delta _ { s c a l } \big )$ + +$\Delta _ { p r e s e n t }$ : 계산된 현재의 정확도 + + + +a : 계산된 현재의 정확도 + +$$ +\Delta_ {p r e s e n t} \leq \Delta_ {n e w}: a = 0. 2 +$$ + +$$ +\Delta_ {p r e s e n t} > \Delta_ {n e w}: a = 0. 2 5 +$$ + +： + +허용오차수준  는 작을수록 오차가 적어지지만, 해석시간과 수렴성을 고려하여경험치인 1.0e-8 전후의 값을 입력합니다. + + + +# 8-5-15 Fehlberg (Stepsize Sub-Division for Non-convergence Control) + +이 방법은 초기의 구간간격을 시간증분 t 로 두고 Runge Kutta Method의 4차공식과 5차공식에 의한 예측값을 이용하여 오차를 구하고, 구해진 오차가 허용오차수준 를 만족할 때까지 구간간격을 1/2씩 분할하여 해를 구하는 방법입니다. + +![](images/page-270_d9d2d9517ddfa29cf9cda520278b910d21c200999db1f80b4b805987accd6d41.jpg) + +
+line + +| Iteration | Value | +| --------- | ----- | +| 1st Iteration | 0 | +| 2nd Iteration | 0 | +| 3rd Iteration | 0 | +| 4th Iteration | 0 | +
+ +그림 2.8.22 Fehlberg(Stepsize Sub-Division for Non-convergence Control) diff --git a/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_028.md b/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_028.md new file mode 100644 index 00000000..12e35165 --- /dev/null +++ b/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_028.md @@ -0,0 +1,337 @@ + + +# 8-6 재료비선형 해석 (Material Nonlinear Analysis) + +탄성과 소성 재료거동을 비교하면, 일반적으로 탄성 거동시에는 구조물에 영구 변형이 발생하지 않는 반면, 소성 거동에서는 영구적 혹은 비가역적 변형이 발생할 수 있습니다. + +# 8-6-1 소성이론 + +정적 소성 변형율의 성분은 다음 가정에 따라 구성됩니다. + +▪ 구성 응답(Constitution Response)은 변형의 속도와 무관합니다. +▪ 탄성 응답(Elastic Response)은 소성 변형의 영향을 받지 않습니다. +▪ 총 변형율은 다음과 같이 정의합니다. + +$$ +\underset {\sim} {\varepsilon} = \underset {\sim} {\varepsilon} ^ {e} + \underset {\sim} {\varepsilon} ^ {p} \tag {4} +$$ + +여기서 ε̃ : 총 변형율 + +$\varepsilon^{e}$ : 탄성 변형율 + +$\varepsilon^{p}$ : 소성 변형율 + +그리고 수식 구성을 위하여 다음과 같은 기본 개념들이 사용됩니다. + +▪ 소성 변형의 시작을 규정하기 위한 항복 조건 (Yield Criteria) +■ 소성 변형을 정의하기 위한 흐름 법칙 (Flow Rule) +■ 소성 변형시의 항복면의 변화를 정의하는 경화 법칙 (Hardening Rule) + + + +# 항복조건 + +탄성 응답(Elastic Response)의 영역에 대한 경계를 정의하는 항복함수 (혹은 재하함수) F는 다음과 같습니다.(그림 2.8.23 참조) + +$$ +F \left(\underset {\sim} {\sigma}, \underset {\sim} {\varepsilon} ^ {p}, \kappa\right) = \sigma_ {e} \left(\underset {\sim} {\sigma}, \underset {\sim} {\varepsilon} ^ {p}\right) - \kappa \left(\varepsilon_ {p}\right) \leq 0 \tag {5} +$$ + +여기서 σ̃ : 현재의 응력 + +$\sigma_{e}$ : 등가(Equivalent) 혹은 유효(Effective) 응력 + +$\kappa : \varepsilon_{p}$ 의 함수인 경화인자 + +$\varepsilon_{p}$ : 등가(Equivalent) 소성 변형율 + +소성 이론에서 항복 함수의 값이 양이 되는 응력 상태는 존재할 수 없습니다. 항복이 발생하면, 응력 상태는 항복 함수가 0으로 감소될 때까지 소성 변형율을 축적 함으로써 수정되어야 하며, 이 과정을 소성 보정(Plastic Corrector) 단계 혹은 회귀 사상(Return Mapping) 이라고 합니다. + +![](images/page-272_01ff5d2a2e17b97d878a03f4af30a20f3542d6fd59d3c28ec5768b92cfbf28e3.jpg) + +
+text_image + +d\u03c0^p = \u03c0 \frac{\u03c0 f}{\u03c0 \u03c0} +Smooth +a +\u03c0^a +Plastic potential +g(\u03c0) = f(\u03c0) = const. +\u03c0^d +d +Corner +d\u03c0^p +
+ +그림 2.8.23 연속 흐름 법칙과 특이점 + + + +# 흐름법칙 + +흐름법칙은 소성변형을 정의하고 다음 식과 같습니다.(그림 2.8.23) + +$$ +d \underset {\sim} {\varepsilon} ^ {p} = d \lambda \frac {\partial g}{\partial \underset {\sim} {\sigma}} = d \lambda \underset {\sim} {\mathbf {b}} \tag {6} +$$ + +여기서  g : 소성 변형의 방향 $\frac { \partial g } { \partial \underset { \mathcal { \alpha } } { \sigma } }$  + +d: 소성 변형의 크기를 정의하는 소성계수 + +함수 g는 ‘소성 위치에너지(Plastic Potential)’ 함수라 하고, 일반적으로 응력불변량(Stress Invariant)의 항으로 정의됩니다. 그리고 g=F면 ‘연속 흐름(Associated Flow)법칙’이라 하고, g≠F면 ‘비관련 흐름 (Non-associated Flow) 법칙’이라고 합니다. + +midas Civil의 모든 모델은 연속 흐름 법칙을 사용합니다. 즉, 소성 변형율 벡터의방향은 항복면에 수직이므로, 위의 식 (6)은 다음과 같이 나타낼 수 있습니다. + +$$ +d \underset {\sim} {\varepsilon} ^ {p} = d \lambda \frac {\partial F}{\partial \sigma} = d \lambda \underset {\sim} {\mathbf {a}} \tag {7} +$$ + +그림 2.8.23의 모서리나 평평한 면은 소성 흐름의 방향을 단일하게 결정할 수 없는 특이점(Singular Point)을 나타내며, 이 점들에 대해서는 특별한 고려가 필요합니다. + +# 경화 법칙 (Hardening Rule) + +경화 법칙은 재료가 항복할 때 소성 변형에 따른 항복면의 변화를 정의하는 것입니다. + +경화 법칙은 유효 소성 변형율을 정의하는 방법에 따라 ‘변형율 경화(StrainHardening)’와 ‘일 경화 (Work Hardening)’로 나눠집니다. 변형율 경화는 소성 비압축성(Plastic Incompressibility)의 가정에 따라 정의되므로, 정수압의 영향을 받지 않는 재료 모델에 적합합니다. 따라서, 소성 일의 정의에 의한 일 경화가 더 일반적 + + + +인 개념입니다. + +또한, 경화 법칙은 항복면의 변화 형식에 따라, ‘등방성 경화 (Isotropic Hardening)’, ‘운동형 경화 (Kinematic Hardening)’, 그리고 ‘훈합형 경화 (Mixed Hardening)’로 나눠집니다(그림 2.8.24). + +![](images/page-274_d63aba2db249aa9e0ab7ed7096cb998b78faee301896ab85eb364f25fdfded88.jpg) + +
+text_image + +Initial Yield Surface +F(σ̃) = κ² +σ₁ +σ₂ +Translation and Expansion +F(α̃ - α̃) = κ₁² > κ² +Translation only +F̃(σ̃ - α̃) = κ² +
+ +그림 2.8.24 운동형 경화와 혼합형 경화 + + + +# 8-6-2 구성 행렬 (Constitutive Matrix) + +표준 소성 구성행렬을 구성하는 방법은 다음과 같습니다. + +응력은 변형율 변화율 벡터의 탄성 부분에 의해 결정됩니다. 즉, + +$$ +d \underline {{\sigma}} = \mathbf {D} ^ {e} \left(d \underline {{\varepsilon}} - d \underline {{\varepsilon}} ^ {p}\right) = \mathbf {D} ^ {e} \left(d \underline {{\varepsilon}} - d \lambda \underline {{\mathbf {a}}}\right) \tag {8} +$$ + +여기서, D $^{e}$ 는 탄성 구성 행렬입니다. + +응력은 항상 항복면 상에 있어야 하므로 다음의 일관성 조건(Consistence Condition)을 만족해야 합니다. + +$$ +d F = \frac {\partial F}{\partial \widetilde {\sigma}} ^ {T} d \widetilde {\sigma} + \frac {\partial F}{\partial \widetilde {\varepsilon} ^ {p}} d \widetilde {\varepsilon} ^ {p} + \frac {\partial F}{\partial \kappa} d \kappa = \widetilde {\mathbf {a}} ^ {T} \widetilde {\mathbf {D}} ^ {e} d \widetilde {\varepsilon} - \left(\widetilde {\mathbf {a}} ^ {T} \widetilde {\mathbf {D}} ^ {e} \widetilde {\mathbf {a}} + h\right) d \lambda = 0 \tag {9} +$$ + +여기서, h는 소성 경화 계수입니다. 따라서, 미소 응력 변화율은 다음과 같이 구할 수 있습니다. + +$$ +d \underline {{\sigma}} = \left(\underline {{\mathbf {D}}} ^ {e} - \frac {\mathbf {D} ^ {e} \underline {{\mathbf {a}}} \underline {{\mathbf {a}}} ^ {T} \mathbf {D} ^ {e T}}{\underline {{\mathbf {a}}} ^ {T} \underline {{\mathbf {D}}} ^ {e} \underline {{\mathbf {a}}} + h}\right) d \underline {{\varepsilon}} \tag {10} +$$ + +완전한 Newton-Raphson 반복 과정이 사용될 때는 일관성(Consistent) 구성 행렬을 사용하면, Newton-Raphson 반복 과정의 이차 수렴 특성으로 인해 더 빠른 수렴을 얻을 수 있습니다. + +$$ +d \underline {{\sigma}} = \left(\underline {{\mathbf {R}}} - \frac {\underline {{\mathbf {R}}} \underline {{\mathbf {a}}} \underline {{\mathbf {a}}} ^ {T} \underline {{\mathbf {R}}} ^ {T}}{\underline {{\mathbf {a}}} ^ {T} \underline {{\mathbf {R}}} \underline {{\mathbf {a}}} + h}\right) d \underline {{\varepsilon}} \tag {11} +$$ + +여기서, $\mathbf{R}=\left(\mathbf{I}+d\lambda\mathbf{D}^{e}\frac{\partial\mathbf{a}}{\partial\sigma}\right)^{-1}\mathbf{D}^{e}=\left(\mathbf{I}+d\lambda\mathbf{D}^{e}\mathbf{A}\right)^{-1}\mathbf{D}^{e}$ 입니다. + + + +# 8-6-3 응력 적분 + +응력의 적분을 위해서는 다음의 두 가지 방법이 사용될 수 있습니다. + + 부증분을 갖는 명시적 전방 오일러 방법(Explicit forward Euler algorithmwith sub-incrementation) (그림 2.8.25, 그림 2.8.26) + 암시적 후방 오일러 방법 (Implicit backward Euler algorithm) (그림 2.8.27) + +![](images/page-276_e041a6e6d33679a0b55f4684ab25615d521f92bdff868ef2dc48d5e3b90a7d99.jpg) + +
+text_image + +Δσₑ +B +A +X +σ +X : 이전단계의 최종 stress 상태 +A : 응력 증분과 항복면의 교차점 +B : 탄성 응력 증분이 고려된 시험응력 상태 +
+ +(a) 교차 점 A의 위치 +![](images/page-276_4196658b8f2b4ad063772837d9d2a8e982ae4f9ab2d2ffa0a51d6fce780f1de2.jpg) + +
+text_image + +Δσₑ +B +-ΔλCₑ +C +X +σ +C : 보정 후의 응력 상태 +D : 인위적 회귀 방법을 적용한 후의 응력 상태 +
+ +(b) A에서 접선방향으로 C로 이동한 후 D위치로 보정 +그림 2.8.25 명시적 전방오일러 방법 + + + +![](images/page-277_3251c3023d3c9fb7d196530d10c555b898f6aa663c5423239a13310b44c53931.jpg) + +
+text_image + +A, B, C, D +E : 인위적 +
+ +D : 각 부 증분에 의한 응력 보정 이후의응력 상태 +회귀 방법을 적용한 후의 응력 상태 + +그림 2.8.26 부 증분 +![](images/page-277_845a4ea7955c088ccc8798cd9f0b7cc9c8664e5a45623433420b71bb6349680e.jpg) + +
+text_image + +σ₁ +σₓ +σc +X : 이전단계 +B : 탄성 응력 +C : 미지의 최 +
+ +의 최종 stress 상태 +력 증분이 고려된 시험응력 상태 +최종 응력 상태 + +그림 2.8.27 암시적 후방 오일러 + +명시적 방법에서 경화 자료와 소성 흐름의 방향은 ‘교차 점’ 즉, 탄성 응력 증분이항복 면을 지나는 점(그림 2.8.25의 A)에서 계산됩니다. 반면, 암시적 방법에서는최종 응력 지점(그림 2.8.27의 B)에서 계산됩니다. + +명시적 방법은 상대적으로 단순하고, 응력을 직접적으로 적분합니다. 즉, 가우스 + + + +점(Gauss Point)에서 반복과정이 필요하지 않으나, 이 방법은 다음과 같은 단점이 있습니다. + +▪ 조건에 따라서만 안정적입니다. +■ 허용 가능한 정확도를 얻기 위해 응력 보정 중에 부-증분이 필요합니다. +▪ 항복 면으로부터 떨어진 정도를 보정하기 위해 인위적인 회귀 방법이 필요합니다. + +또한,이 방법으로는 일관성 접선 계수 행렬을 구성할 수가 없습니다. + +암시적 방법은 부-증분이나 인위적 회귀 없이도 충분히 정확한 결과를 도출하고, 조건에 관계없이 안정적입니다. 그러나, 일반적인 항복 기준에 대해, 가우스 점에서 반복과정이 필요합니다. 이 방법을 사용하면 일관성 있는 접선 행렬을 구성할 수 있으므로, Newton-Raphson 반복 과정을 사용하면 가우스 점에서 반복 과정을 수행해도 계산적으로 더 효율적입니다. + +# 명시적 방법 (전방 오일러) + +1. 변형율 증분을 계산합니다. + +$$ +d \underset {\sim} {\varepsilon} = \underset {\sim} {\mathbf {B}} d \underset {\sim} {\mathbf {u}} \tag {12} +$$ + +여기서 B: 변형율-변위 관계 행렬 + +$d_{u}$ : 변위의 변화량 + +2. 탄성 시험 응력 상태를 계산합니다. + +$$ +\begin{array}{l} d \underline {{{\sigma}}} = \underline {{{\mathbf {D}}}} ^ {e} d \underline {{{\varepsilon}}} \\ \underline {{{\tau}}} = \underline {{{\tau}}} + d \underline {{{\tau}}} \end{array} \tag {13} +$$ + +$$ +\widetilde {\sigma} _ {B} = \widetilde {\sigma} _ {X} + d \widetilde {\sigma} +$$ + +위 식과 아래식에서의 첨자는 그림 2.8.25를 참조합니다. + + + +3. 시험 응력이 항복면 내에 있으면 응력 보정은 완료되며, 항복면 밖에 있으면 소성 변형에 의해 항복면으로 돌아와야 합니다. + +4. 다음으로 교차 응력이 계산됩니다. 시험 탄성 응력 증분은 허용 가능한 응력 증분과 허용 불가능한 응력 증분으로 나눠고, 교차 응력은 다음 식을 이용하여 계산됩니다. + +$$ +\begin{array}{l} F \left(\underset {\sim} {\sigma} _ {X} + (1 - r) d \underset {\sim} {\sigma}\right) = 0 \\ r = \frac {F _ {B}}{F _ {B} - F _ {X}} \tag {14} \\ \end{array} +$$ + +5. 물리적으로, 추가적인 변형은 응력 지점이 항복면 상에서 이동하도록 합니다. 이는 허용될 수 없는 응력 증분 rdσ̃을 m개의 작은 응력 증분으로 나누어 근사됩니다 (그림 2.8.26). 부 증분의 개수는 오차의 크기에 직접적으로 관계되고, 다음과 같이 계산됩니다. + +$$ +m = \operatorname{INT} \left(8 \left(\sigma_ {e B} - \sigma_ {e A}\right) / \sigma_ {e A}\right) + 1 \tag {15} +$$ + +6. 최종 응력 상태가 항복면 상에 있지 않으면, 다음의 인위적 회귀 방법에 의해 항복면 상에 옮겨져야 합니다. + +$$ +\delta \lambda_ {C} = \frac {F _ {C}}{\underset {\sim C} {\mathbf {a}} ^ {T} \underset {\sim C} {\mathbf {D}} ^ {e} \underset {\sim C} {\mathbf {a}} + h} \tag {16} +$$ + +$$ +\boldsymbol {\sigma} _ {D} = \boldsymbol {\sigma} _ {C} - \delta \lambda_ {C} \mathbf {D} ^ {e} \mathbf {a} _ {C} +$$ + +▪ 항복면의 형상은 각 부-증분의 끝에서 경화 법칙을 사용하여 보정됩니다. +■ Unloading은 탄성으로 가정됩니다. + + + +# 암시적 후방 오일러 + +암시적 방법에서는 다음 식에 의해 최종 응력을 계산합니다. + +$$ +\underline {{{\sigma}}} _ {C} = \underline {{{\sigma}}} _ {B} - d \lambda \underline {{{\mathbf {D}}}} ^ {e} \underline {{{\mathbf {a}}}} _ {C} \tag {17} +$$ + +여기서 첨자는 그림 2.8.27을 참조합니다. + +C점에서 식 (17)의 값은 알고 있지 않으므로, Newton 반복과정을 사용하여 미지수를 푸는 방법이 사용됩니다. 따라서 어떤 벡터 r이 현재의 응력과 후방 오일러 응력 간의 차이를 나타내기 위해 설정됩니다. + +$$ +\underset {\sim} {\mathbf {r}} = \underset {\sim} {\boldsymbol {\sigma}} _ {C} - \left(\underset {\sim} {\boldsymbol {\sigma}} _ {B} - d \lambda \underset {\sim} {\mathbf {D}} ^ {e} \underset {\sim} {\mathbf {a}} _ {C}\right) \tag {18} +$$ + +이제 반복과정은 r을 0으로 감소하기 위해 도입되며, 최종 응력은 항복 기준을 만족해야 합니다. 고정된 시험 탄성 응력을 사용하여 다음과 같이 새로운 잔류량을 만들기 위해 Taylor 전개를 적용합니다. + +$$ +\mathbf {r} _ {n} = \mathbf {r} _ {o} + \dot {\sigma} + \dot {\lambda} \mathbf {D} ^ {e} \mathbf {a} \tag {19} +$$ + +여기서 $\dot{\sigma}:\sigma$ 의 변화량 + +$\dot{\lambda}:d\lambda$ 의 변화량 + +위 식의 값을 0으로 두고 $\dot{\sigma}$ 에 대해 풀면 다음과 같습니다. + +$$ +\dot {\tilde {\sigma}} = - \mathbf {r} _ {o} - \dot {\lambda} \mathbf {D} ^ {e} \mathbf {a} \tag {20} +$$ + +또한 항복 함수에 대해 Taylor 전개를 적용하면, 다음과 같습니다. diff --git a/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_029.md b/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_029.md new file mode 100644 index 00000000..08b6afdb --- /dev/null +++ b/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_029.md @@ -0,0 +1,389 @@ + + +$$ +F _ {C n} = F _ {C o} + \frac {\partial F}{\partial \widetilde {\sigma}} ^ {T} \dot {\widetilde {\sigma}} + \frac {\partial F}{\partial \varepsilon_ {p}} \dot {\varepsilon} _ {p} = F _ {C o} + \widetilde {\mathbf {a}} _ {C} ^ {T} \dot {\widetilde {\sigma}} - h \dot {\lambda} = 0 \tag {21} +$$ + +여기서, $\varepsilon_{p}$ : 유효 소성 변형율 + +따라서, $\dot{\lambda}$ 는 다음과 같이 구해지고, 최종적인 응력의 값도 얻을 수 있습니다. + +$$ +\dot {\lambda} = \frac {F _ {o} - \mathbf {a} ^ {T} \mathbf {r} _ {o}}{\mathbf {a} ^ {T} \mathbf {D} ^ {e} \mathbf {a} + h} \tag {22} +$$ + +# 8-6-4 소성 재료 모델 + +4가지 일반적인 소성 모델이 이용 가능합니다. + +- Tresca, von Mises – 금속과 같이, 소성 비압축성(Plastic Incompressibility)을 보이는 연성 재료에 대해 적합합니다(그림 2.8.28). +- Mohr-Coulomb, Drucker-Prager – 콘크리트나 암석, 지반처럼, 부피 소성 변형을 보이는 재료에 대해 적합합니다(그림 2.8.29). + +![](images/page-281_451a4e50cf2103fca70f190b78fb8b92dde882b4a38faf5d7eff3a19aa92d9ac.jpg) + +
+text_image + +von Mises yield surface +σ₃ +Hydrostatic axis +Tresca yield surface +σ₂ +σ₁ +
+ +그림 2.8.28 Tresca와 von Mises 항복 기준 + + + +![](images/page-282_f273ee0bdb43789fe0c4bb9d085cacf86f5c5d1d2f4cce3384df013983bd24d5.jpg) + +
+text_image + +Drucker-Prager +Mohr-Coulomb +-σ₁ +-σ₃ +-σ₂ +σ₁ +ρₜ +ρc +σ₂ +σ₃ +
+ +그림 2.8.29 Mohr-Coulomb과 Drucker-Prager 항복 기준 + +# Tresca 기준 + +Tresca 항복 기준은 금속과 같이 부피 소성 변형을 보이지 않고, 연성인 재료에 대해 적합합니다. 이 기준을 따르면, 최대 전단 응력이 규정된 값에 도달할 때 항복이 시작됩니다. 그러므로, 주응력(Principal Stress)이 $\sigma_{1}, \sigma_{2}, \sigma_{3} \left( \sigma_{1} \geq \sigma_{2} \geq \sigma_{3} \right)$ 이면, 항복 함수는 식 (23)과 같습니다. + +$$ +F (\widetilde {\sigma}, \kappa) = \sigma_ {1} - \sigma_ {3} - \kappa \left(\varepsilon_ {p}\right) \tag {23} +$$ + +응력 상태가 항복면 상의 특이점에 있을 때 수치적 문제가 발생할 수 있으며, Tresca 기준에 대해 이는 편각 (Lode Angle) θ 가 ± 30°에 근접할 때 발생할 수 있습니다. 따라서, 이 경우에 대해서는 응력 적분 방법이 보정되어야 합니다. midas Civil에서는 θ >29°일 때, 흐름 벡터를 형성하기 위해 Von Mises 항복 기준이 사용됩니다. + +# Von Mises 기준 + +Von Mises 기준은 금속에 대해 가장 일반적으로 수용되는 항복 기준입니다. 이 기준은 변형 에너지를 기반한 것으로, 항복 함수는 다음과 같습니다. + +$$ +F (\underset {\sim} {\sigma}, \kappa) = \sqrt {3 J _ {2}} - \kappa \left(\varepsilon_ {p}\right) \tag {24} +$$ + +여기서, J₂는 두 번째 편향 응력 상수 (Second Deviatoric Stress Invariant) 입니다. + + + +# Mohr-Coulomb 기준 + +Mohr-Coulomb 기준은 콘크리트, 지반, 암석과 같은 부피 소성 변형을 보이는 재료에 적합합니다. Mohr-Coulomb 항복 기준은 Coulomb 마찰 법칙의 일반화로서, 다음과 같이 정의됩니다. + +$$ +F \left(\underset {\sim} {\sigma}, \kappa\right) = \tau - \left(c - \sigma_ {n} \tan \phi\right) \tag {25} +$$ + +여기서 τ: 전단 응력의 크기 + +$\sigma_{n}$ : 수직 응력 + +C: 점성 + +$\phi$ : 내부 마찰각 + +점성 c와 내부 마찰각 $\phi$ 는 모두 변형율 경화 인자 K에 따릅니다. + +Tresca 기준과 마찬가지로, 응력 지점이 항복면의 특이점에 있을 때 수치적 문제가 발생합니다. Mohr-Coulomb 기준에 대해서 이는 편각 (Lode Angle) θ 가 ± 30°에 근접하거나 꼭지점에서 발생할 수 있습니다. 따라서, 이 두 경우에 대해서는 응력 적분 방법이 보정되어야 합니다. midas Civil에서는 θ >29°일 때, 흐름 벡터를 형성하기 위해 Drucker-Prager 기준이 사용됩니다. + +# Drucker-Prager 기준 + +Drucker-Prager 기준은 지반, 콘크리트, 암석 등의 부피 소성 변형을 보이는 재료에 대해 적합합니다. 이 기준은 Mohr-Coulomb 기준과 근사하고, Von Mises 기준의 확장된 형태입니다. 항복 함수는 정수압 (Hydrostatic Stress) 의 영향을 포함하고, 다음과 같이 정의됩니다. + +$$ +F (\widetilde {\sigma}, \kappa) = \frac {2 \sin \phi}{\sqrt {3} (3 - \sin \phi)} I _ {1} + \sqrt {J _ {2}} - \frac {6 c \cos \phi}{\sqrt {3} (3 - \sin \phi)} \tag {26} +$$ + +여기서, I는 첫 번째 응력 상수 (First Stress Invariant) 입니다. + +Drucker-Prager 기준에 대해서는, 응력 지점이 항복면의 꼭지점 상에 있을 때 수치적 문제가 발생합니다. + + + +# 8-6-5 경화 법칙 + +# 유효 소성 변형율(Effective Plastic Strain)의 정의 방식에 따른 분류 + +# 1. 변형율 경화 (Strain Hardening) + +변형율 경화에 대해, 유효 소성 변형율은 다음과 같이 정의됩니다. + +$$ +d \varepsilon_ {p} = \sqrt {\frac {2}{3} \left(d \underset {\sim} {\varepsilon} ^ {p}\right) ^ {T} d \underset {\sim} {\varepsilon} ^ {p}} = \sqrt {\frac {2}{3} \underset {\sim} {\mathbf {a}} ^ {T} \underset {\sim} {\mathbf {a}}} d \lambda \tag {27} +$$ + +이 유효 소성 변형율은 부피 소성 변형이 없다는 가정 하에서 소성 변형율의 랣(Norm)을 단일 축 변형율에 맞게 스케일링 한 것입니다. 따라서, 이것은 원칙적으로는 Tresca나 Von Mises에만 적용되어야 하겠지만, 수치적인 편리함으로 인해 다른 경우에도 많이 사용됩니다. + +# 2. 일 경화 (Work Hardening) + +소성 일의 증분은 다음과 같습니다. + +$$ +d W _ {p} = \underset {\sim} {\sigma} ^ {T} d \underset {\sim} {\varepsilon} ^ {p} = d \lambda \underset {\sim} {\mathbf {a}} ^ {T} \underset {\sim} {\sigma} \tag {28} +$$ + +단일 축 경우에 대해, 위의 소성 일의 증분은 다음과 같습니다. + +$$ +d W _ {p} = \sigma_ {1} d \varepsilon_ {1} = \sigma_ {e} d \varepsilon_ {p} \tag {29} +$$ + +따라서,일 경화에 대한 유효 소성 변형율은 다음과 같이 정의됩니다. + +$$ +d \varepsilon_ {p} = \frac {\mathbf {a} ^ {T} \boldsymbol {\sigma}}{\boldsymbol {\sigma} _ {e}} d \lambda \tag {30} +$$ + + + +# 항복면의 변화 형식에 따른 분류 + +# 1. 완전 소성 (Perfectly Plastic) + +완전 소성 재료에 대해, 소성 변형이 일어나고 항복면은 변하지 않습니다. 따라서 항복 함수는 다음과 같이 표현됩니다. + +$$ +F \left(\underset {\sim} {\sigma}, \kappa\right) = \sigma_ {e} \left(\underset {\sim} {\sigma}\right) - \kappa \tag {31} +$$ + +여기서 K는 상수입니다. + +# 2. 등방성 경화(Isotropic Hardening) + +등방성 경화의 경우에는, 그림 2.8.30(a)에서와 같이 항복면이 균일하게 팽창하므로, 항복 함수는 다음과 같이 표현됩니다. + +$$ +F (\underset {\sim} {\sigma}, \kappa) = \sigma_ {e} (\underset {\sim} {\sigma}) - \kappa (\varepsilon_ {p}) \tag {32} +$$ + +# 3. 운동형 경화 (Kinematic Hardening) + +운동형 경화의 경우에 항복면은 그림 2.8.30(b)에서 처럼 크기는 변하지 않고 위치만 이동되므로, 항복 함수는 다음과 같이 표현됩니다. + +$$ +F \left(\underset {\sim} {\sigma}, \underset {\sim} {\alpha}, \kappa\right) = \sigma_ {e} \left(\underset {\sim} {\sigma} - \underset {\sim} {\alpha}\right) - \kappa \tag {33} +$$ + +여기서 $\alpha$ : 항복면 중심 좌표 + +K: 상수 + +운동형 경화에서는 유발 항복면 중심 좌표 α 를 결정하는 것이 중요합니다. 경화 인자 α 를 결정하는 방법은 보통 두 가지가 있는데, 하나는 Prager의 경화 법칙이고, 다른 하나는 Ziegler의 경화 법칙입니다. Prager의 경화 법칙은 다음과 같이 표현됩니다. + +$$ +d \underset {\sim} {\alpha} = C _ {p} d \underset {\sim} {\varepsilon} ^ {p} = C _ {p} \underset {\sim} {\mathbf {a}} d \lambda \tag {34} +$$ + +여기서 $C_{p}$ 는 Prager 경화 계수입니다. + + + +![](images/page-286_8adecef3d416a749dea945fa9c4f943d1b90dd1c77d5477f2869f90b37b04101.jpg) + +
+text_image + +σ +A B +O C +ε +A' B' +
+ +(a) 등방성 경화 + +![](images/page-286_e6066b92a97946eca1e0b82bde50853e4e200a5ff2ca01b673e17c10b41c9245.jpg) + +
+text_image + +σ +A B +a +O C +ε +a' +A' +B' +
+ +(b) 운동형 경화 +그림 2.8.30 일차원에서의 경화법칙 + +이 방법은 응력의 부 공간에서 사용될 때, 몇 가지 문제가 발생할 수 있습니다. 가령 응력의 어떤 성분이 0이더라도 $d\alpha$ 는 0이 아닐 수 있기 때문에 항복면의 이동만을 나타내지 않을 수 있습니다. + +Ziegler의 경화 법칙은 중심의 이동 변화율 $d\alpha$ 가 감소된 응력(Reduced-Stress) 벡터 $\sigma-\alpha$ 의 방향으로 발생한다고 가정하므로, 이러한 문제가 발생하지 않습니다. 이 경화 법칙은 다음과 같이 표현됩니다. + +$$ +d \underline {{\alpha}} = d \mu (\underline {{\sigma}} - \underline {{\alpha}}) = C _ {z} d \varepsilon_ {p} (\underline {{\sigma}} - \underline {{\alpha}}) \tag {35} +$$ + +여기서 $C_{z}$ 는 Ziegler 경화 계수입니다. + +# 4. 혼합형 경화 (Mixed Hardening) + +혼합형 경화는 등방성 경화와 운동형 경화가 조합된 형태로 다음과 같이 표현됩니다. + +$$ +F \left(\underset {\sim} {\sigma}, \underset {\sim} {\alpha}, \kappa\right) = \sigma_ {e} \left(\underset {\sim} {\sigma} - \underset {\sim} {\alpha}\right) - \kappa \left(\varepsilon_ {p}\right) \tag {36} +$$ + + + +# 8-6-6 재료비선형 모델 사용시 주요 고려사항 + +midas Civil에 탑재되어 있는 재료비선형 모델들은 탄소성 모델로써 Von Mises,Tresca, Mohr-Coulomb, Drucker-Prager로 구성된 네 개의 모델이 있습니다. 이중Von Mises, Tresca모델은 구속압력에 독립적인 형태의 모델로써 연성재료인 강재의모델링에 적합한 재료모델입니다. Mohr-Coulomb모델과 Drucker-Prager모델은 구속압에 종속적인 특성을 가지며 콘크리트나 암반 또는 지반과 같은 취성거동을 하는재료에 적합합니다. 네 개의 각 모델들은 모두 등방성경화모델(Isotropic Hardening)과 이동경화모델(Kinematic Hardening)을 가지고 있습니다. 그러나 실무적으로 이동경화모델은 강재와 같은 연성재료에서 나타나는 거동특성으로써 Von Mises,Tresca모델에 많이 사용되며 Mohr-Coulomb, Drucker-Prager와 같은 취성모델에는일반적으로 사용하지 않습니다. midas Civil 에서는 경화거동을 Bilinear거동으로 규정하고 있으며 Cyclic 하중을 받는 강재에 등방성경화와 이동경화모델을 혼용하는Mixed 모델을 적용할 경우 다음 그림과 같은 응력경로를 보입니다. + +![](images/page-287_432b47206621bfaf46b6a4b5ca96fa09c1a42e597518cbcd7569b60b4b81f9f3.jpg) + +
+text_image + +σ +ε +
+ +그림 2.8.31 Cyclic 하중을 받는 강재의 거동 + +시공시 많이 사용되는 일반 구조용강을 해석할 경우는 다음 그림과 같이 완전소성거동을 가정하는 것이 일반적이지만 항복점을 넘어서는 경우 강성이 0이 되므로구조물의 모델링 시 특별한 주의를 요합니다. + + + +![](images/page-288_f317ad8320a023e498dbbb19396c9a5b44330cf3b8448523d7c5b80d03d010d5.jpg) + +
+line + +| ε | σ | +| ---- | ---- | +| 0 | 0 | +| ε | Yield Stress | +
+ +그림 2.8.32 완전 소성거동 + +콘크리트와 같은 취성모델들은 다음 그림과 같이 인장거동과 압축거동이 다릅니다.인장거동의 경우 균열모델을 사용하여 거동을 예측하는 것이 일반적입니다. midasCivil에서는 균열모델과 콘크리트의 압축 거동 시 관측되는 비선형 경화거동에 대한 모델은 현재 탑재되어 있지 않습니다. + +![](images/page-288_c6a1dea0463d2fb5a9be416c389967cfa6421e3c3241cf57c8369b7f27d47477.jpg) + +
+line + +| ε | σ | +| ---- | ---- | +| 0.0 | 0.0 | +| 0.5 | 0.5 | +| 1.0 | 0.0 | +
+ +그림 2.8.33 콘크리트의 인장 및 압축거동 + + + +Mohr-Coulomb이나 Drucker-Prager모델의 경우 3차원 주응력 공간 상에서 아래의두 그림과 같이 육각추나 원추형 형상의 파괴면을 갖습니다. 탄소성 모델의 수치해석 시 요구되는 응력회귀는 이러한 파괴면의 수직방향을 사용합니다. + +![](images/page-289_9338cb48adf5a7b4b7d49e3c5d9db370150286dbb918af17e5bfb7f9976bc581.jpg) + +
+text_image + +-σ₁ +hydrostatic axis +-σ₃ +-σ₂ +
+ +(a) Mohr-Coulomb failure surface + +![](images/page-289_3fd4d96049c1c924b01cebe6ee522eb9f08e1eacae7864fba6e6a6249794d996.jpg) + +
+text_image + +σ₁ +θ +rᵢ₀ +r꜀₀ +σ₂ +σ₃ +
+ +(b)  -plane + +![](images/page-289_d767cd8c7ce8c7fae5e2bd806da6ac556c39ecb72263dcfa432e068de58fecc8.jpg) + +
+text_image + +θ = -π/6 +r₀ = 2√6c cosø / 3 + sinø +√3c cotø +hydrostatic axis +r₀ = 2√6c cosø / 3 - sinø +θ = π/6 +
+ +(c) meridian plane +그림 2.8.34 Mohr-Coulomb yield surface in Π-plane & meridian plane + + + +![](images/page-290_187dd3c087c69337bdb72a062a780f859bac24bb512c55be69de4574bdb4cd47.jpg) + +
+text_image + +-σ₁ +hydrostatic axis +-σ₃ +-σ₂ +
+ +(a) Drucker-Prager failure surface + +![](images/page-290_598e7416bf3cf54da74e6e1085682f0f26b0905c55cbe05fb5a3a59a88c70dbf.jpg) + +
+text_image + +σ₁ +θ +r +r₀ +σ₂ +σ₃ +
+ +(b)  -plane + +![](images/page-290_cd9c4087ecf4f06e1cc2681dcd6c886c89e307fabd8bda4f9f398ac8e626738f.jpg) + +
+text_image + +θ = -π/6 +√6α +1 +deviatoric axis +r₀ = √2k +k +√3α +hydrostatic axis +1 +r₀ = √2k +θ = π/6 +
+ +(c) meridian plane +그림 2.8.35 Drucker-Prager yield surface in Π-plane & meridian plane diff --git a/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_030.md b/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_030.md new file mode 100644 index 00000000..38f903bb --- /dev/null +++ b/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_030.md @@ -0,0 +1,303 @@ + + +Mohr-Coulomb이나 Drucker-Prager의 경우 꼭지점에서 ${ \mathsf { C } } ^ { - 1 }$ 불연속성(미분불가능)으로 인해 회귀방향이 단일 해로 보장되지 않으면서 해가 발산합니다. 즉, 다음 그림에 표시된 것과 같이 Apex Regime에 응력상태가 존재할 경우 해는 발산합니다.현재 midas Civil 에서는 이를 고려하고 있지 않습니다. + +![](images/page-291_addd07f3b8675a9a17caf679ab6952aa1fd4eeabff7ce7776893f63b9a2869d6.jpg) + +
+text_image + +Deviatoric axis +Hydrostatic axis +Apex regime +
+ +그림 2.8.36 Apex Regime + + + +# 8-7 정적증분해석 (Pushover 해석) + +# 8-7-1 개요 + +구조물의 내진성능 평가를 위한 비선형 해석법은 비선형 정적해석법과 비선형 동적해석법으로 분류됩니다. 비선형 동적해석법은 가장 정확한 해석법이라 할 수 있지만, 일반 엔지니어가 사용하기에는 많은 시간과 노력을 필요로 합니다. 반면에 정적해석법은 지진하중에 대한 구조물의 고유한 동적 특성을 반영하기 어렵다는 단점이 있으나, 해석절차가 간단하여 사용하기 쉽고 해석결과를 개념적으로 쉽게 표현하고 이해할 수 있어서 가장 많이 사용되고 있는 방법입니다. + +비선형 정적해석을 일반적으로 Pushover 해석법이라 부릅니다. Pushover 해석은 부재의 재료비선형적인 특성을 고려하여 구조물이 항복한 이후의 거동과 한계상태를 파악하는 가장 효과적인 해석방법입니다. Pushover 해석은 최근 지진공학과 내진설계 분야에서 많은 연구와 실무 적용이 이루어지고 있는 성능에 기초한 내진설계(Performance-Based Seismic Design, PTSD)에서 대표적인 해석방법으로 적용되고 있습니다. 성능에 기초한 내진설계의 목적은 사용자 및 설계자 모두가 대상 구조물의 목표성능(Target Performance)을 명확히 설정하고, 이를 구현할 수 있도록하는 것입니다. 따라서 내진설계를 먼저 수행한 후에 Pushover 해석을 통하여 구조물의 보유능력을 파악하고 고려하는 지진하중에 대하여 미리 설정된 목표성능이 달성되는지를 평가합니다. + +일반적인 내진설계법에서 등가정적하중을 산정할 때 그림 2.8.37과 같은 방법이 주로 사용됩니다. 이 개념은 반응수정계수(R)를 통하여 설계하중을 낮게 산정하고 구조물은 설계하중 이상의 강도를 갖도록 하는 것입니다. 여기서 반응수정계수를 사용하는 이유는 지진하중에 대하여 비탄성 영역에서 발생할 수 있는 구조물의 에너지 흡수능력을 고려하기 위한 것입니다. 이러한 설계법은 하중을 대상으로 하기 때문에 하중기반설계(Force-Based Design)이라 할 수 있습니다. 그러나 강도의 단순한 비교만을 통해서는 구조물의 실제적인 거동을 예측하기가 어렵습니다. 또한 구성 부재의 강도만으로 구조물 전체적인 강도 및 변형율을 산정할 수 없습니다. 결과적으로 구조물의 성능이 명확하게 파악되지 않은 상태로 설계될 가능성이 높습니다. + + + +성능에 기초한 내진설계에서는 사용자 혹은 발주자, 그리고 설계자가 구조물의 목표성능을 미리 설정합니다. 즉, 예상되는 지진하중에 대하여 주어진 여건에서 허용할 수 있는 적절한 피해정도 혹은 에너지 흡수정도를 미리 설정하고, 이를 달성할수 있도록 하는 것입니다. 그런데 에너지 흡수정도에 따라 구조물의 거동이 달라지기 때문에 파괴에 이를 때까지 구조물의 변형성능을 예측할 수 있어야 합니다.이때 성능평가의 대상을 구조물의 손상과 직접적인 연관성이 있는 변위로써 평가하기 때문에 이를 변위기반설계(Displacement-Based Design)이라 합니다. + +![](images/page-293_fdc4dbd65995e92443f742d82217908e07adc00a7bdfeccad3665c62ac3b4a96.jpg) + +
+line + +| Displacement | Base Shear | Elastic Force Reduction | Capacity (elastic) | +| ------------ | ---------- | ------------------------ | ------------------ | +| D yield | V design | - | - | +| D_max | R | - | - | +| D | - | - | - | +
+ +그림 2.8.37 하중기반설계법에 따른 지진하중의 산정 + +구조물의 변형성능을 평가하기 위한 하나의 방법으로 Pushover 해석을 수행하면,그림 2.8.38과 같은 하중-변형에 대한 능력스펙트럼이 생성됩니다. 그리고 구조물의 에너지 흡수정도에 따라 비탄성 요구스펙트럼을 산정할 수 있습니다. 능력스펙트럼과 요구스펙트럼이 교차되는 점은 대상구조물이 해석시 고려한 지진하중에 대하여 발휘할 수 있는 비선형 최대내력 및 변위를 의미합니다. 이 교차점이 목표성능의 범위에 존재하면 목표가 달성되었다고 할 수 있습니다. + +![](images/page-293_3772cb31c96362603ef7529adc7ab5c9bc73697b79813593d7bf0ad74301652b.jpg) + +
+line + +| Spectral Displacement | Spectral Acceleration (Performance Point) | Spectral Acceleration (Demand Spectrum) | Spectral Acceleration (Capacity Spectrum) | +| --------------------- | ------------------------------------------ | ---------------------------------------- | ----------------------------------------- | +| D_design | S_a | S_a | S_a | +| S_d | S_d | S_d | S_d | +
+ +그림 2.8.38 변위기반설계법에 의한 구조물의 내진성능 평가 + + + +# 8-7-2 해석방법 + +구조물에 대한 목표성능은 최소 이상의 법규 및 내진설계기준을 만족하는 상태에서 사용자와 설계자에 의해서 결정됩니다. 그리고 구조물의 구조성능을 파악하기위하여 구조해석을 수행하게 되는데, 성능에 기초한 내진설계에서는 크게 다음과같은 4가지 해석방법을 선택하고 있습니다. + + 선형 정적해석법(Linear Static Procedure, LSP) + 선형 동적해석법(Linear Dynamic Procedure, LDP) + 비선형 정적해석법(Nonlinear Static Procedure, NSP) + 비선형 동적해석법(Nonlinear Dynamic Procedure, NDP) + +midas Civil에서는 선형 정적 및 선형 동적해석법과 비선형 정적해석법 중에서 가장 대표적인 해석방법이라고 할 수 있는 Pushover 해석을 제공하고 있습니다. 소성힌지해석법이라고 불리는 Pushover 해석은 항복 이후의 극한내력과 안정상태를매우 효과적으로 파악할 수 있는 방법입니다. 이 해석법은 고차모드와 동적특성의영향을 받지 않는 구조물에 주로 사용할 수 있습니다. Pushover 해석에서는 재료학적 비선형거동을 파악할 수 있으며, P-Delta효과를 고려할 수 있습니다. 재료의비선형 특성은 부재의 단면에 대한 하중-변위관계를 이용하는 요소모델(Stress-Resultant Stress Approach)을 도입하여 적용합니다. + +Pushover 해석은 미리 설정한 정적하중을 구조물에서 예상할 수 있는 최대 성능점까지 점진적으로 가하여 저항력과 변위와의 관계인 능력곡선(Capacity Curve)를산정합니다. 다자유도 구조물에서의 저항력과 변위의 관계는 단자유도 시스템의응답가속도와 응답변위와의 관계로 표현되는 능력스펙트럼(Capacity Spectrum)으로전환됩니다. 그리고 지진하중에 대한 응답스펙트럼은 ADRS 형식(Acceleration-Displacement Response Spectrum)으로 표현되는 요구스펙트럼(Demand Spectrum)으로 변환합니다. 이 두 개의 스펙트럼을 비교하여 구조물의 비선형 상태에서의최대 요구내력과 변형능력을 평가하고 목표성능과 비교하여 구조물의 성능수준(Performance Level)을 결정하게 됩니다. + +midas Civil에서는 기본적으로 ATC-40(1996)과 FEMA-273(1997)에서 제공하는 능력스펙트럼법(Capacity Spectrum Method, CSM)의 원리를 이용하여 구조물의 내진성능을 평가하고 있습니다. 이들 보고서에서 제시하고 있는 이론 및 적용계수 등 + + + +을 적용하여 구조물 및 부재의 성능을 평가할 수 있도록 하였습니다. + +능력스펙트럼법(CSM)의 원리는 그림 2.8.39와 같습니다. + +![](images/page-295_f51496b581d6a7ab467285d2c7b3761bd5e55452bb388e5c0e9aa15da0be0c8f.jpg) + +
+flowchart + +```mermaid +graph LR + A["Δfloor"] -->|Pushover Analysis| B["MDOF System"] + B -->|Transform| C["SDOF System"] + B -->|Vbase| D["Δfloor"] + C -->|S_a| E["Output"] +``` +
+ +(a) 구조물의 능력곡선(Capacity Curve)과 능력스펙트럼(Capacity Spectrum) 산정 +![](images/page-295_e5c92926396ee263a7e303b9a020b5379d4d95ff0e5f1fad3bc0ed39562d9110.jpg) + +
+line + +| Spectrum Type | Peak Value | Annotation | +| ------------------- | ---------- | ----------------- | +| Response Spectrum | Tn,1 | Tn,1,1 | +| Response Spectrum | Tn,2 | Tn,2,2 | +| Demand Spectrum | S_d | S_d = T_n^2 / 4π^2 S_a | +| Demand Spectrum | S_d | A_max | +| Demand Spectrum | S_d | D_max | +| Demand Spectrum | S_d | Capacity Spectrum | +| Performance Point | 5% Elastic | Performance Point | +| Demand Spectrum | Tn,1 | Tn,1,1 | +| Demand Spectrum | Tn,2 | Tn,2,2 | +| Capacity Spectrum | A_max | A_max | +| Capacity Spectrum | D_max | D_max | +
+ +(b) 요구스펙트럼(Demand Spectrum)의 산정 +(c) 성능점(Performance Point) 평가 +그림 2.8.39 능력스펙트럼법(Capacity Spectrum Method, CSM)의 원리 + +Pushover 해석에서는 구조물이 보유하고 있는 내진성능을 평가하는 것이 주요 목적이기 때문에 반드시 1차적으로 해석 및 설계가 완료된 구조물에서만 적용이 가능합니다. Pushover 해석을 통하여 얻을 수 있는 장점은 다음과 같습니다. + + 구조물의 항복 이후 거동 및 보유내력의 평가 + 구조물의 에너지 소산능력 및 변위요구량의 파악 + 구조물을 구성하는 각 구조요소의 소성화 과정을 순차적으로 산정 + 보수ㆍ보강을 통하여 구조성능을 높이고자 할 때 필요부재의 경제적인 선정 + + + +# 8-7-3 정적증분해석방법 + +# 정증증분해석의 개요 + +midas Civil의 정적증분해석의 목적은 설정한 하중에 대하여 하중 혹은 변위를 단계적으로 구조물에 작용시켜, 구조물의 내력과 변위의 관계를 추적하여 구조물의보유내력, 변형능력, 보유성능을 평가하기 위한 것입니다. + +구조물의 내력과 변위의 관계는 그림 2.8.40과 같이 나타낼 수 있습니다. 구조물에작용한 외력에 대하여 변형이 작은 범위에서 구조물은 거의 탄성거동하며, 내력과변위의 관계는 선형으로 나타납니다. 그러나 외력이 점진적으로 증가하여 요소의내력이 항복내력을 초과하면 소성힌지가 발생합니다. 소성힌지는 부재의 균열, 항복 등에 의한 것으로, 이로 인해 부재의 강성과 내력이 변화하여 비선형거동을 하게 됩니다. + +그림 2.8.40에서 점 A를 초과하면 내력과 변위는 비선형관계가 되므로, 점 A를 탄성한계라 합니다. 점 A에서 외력을 점차 증가시키면 소성힌지가 구조물 전체로 확대되어 미소한 외력의 증가에 의해 변형이 급격히 증가하는 점 B의 상태에 이르게됩니다. 점 B에서 외력을 더욱 증가시키면 더 이상 외력에 저항할 수 없는 지점인점C에 도달하게 됩니다. 이때 점 C에서의 내력을 최대보유내력이라 합니다. + +![](images/page-296_d2c16a517c85c7341011aa7f5fdedc2b4a60b922805341ce67706636876a7c22.jpg) + +
+text_image + +Elastic Range +Inelastic Range +Internal Force +A +B +C +Plastic Hinge +Displacement +
+ +그림 2.8.40 내력과 변형변위관계 + + + +수평보유내력이란 구조물이 수평력에 저항하는 능력으로, 구조물이 잠재적으로 보유하고 있는 내력을 의미합니다. + +점 C는 하중을 분할하여 점진적으로 하중을 증가시키는 하중증분해석에서 안정해를 얻을수 있는 한계점입니다. 따라서, 극한점 이후의 거동을 파악하기 위해서는변위를 점진적으로 증가시켜 해석하는 변위증분해석을 수행하여야 합니다. + +midas Civil의 정적증분해석은 하중을 분할하여 해석하는 하중증분법(Load Control)과 목표변위를 분할하여 해석하는 변위증분법(Displacement Control)을 제공합니다. + +# 비선형 증분해석 과정 + +정적증분해석에서는 요소의 소성화에 의한 강성 변화와 이로 인한 각 요소의 내력변화에 의해 불평형력(Residual Force)이 발생합니다. 이와 같은 불평형력을 해소하기 위해서는 반복해석(수렴계산)이 필요합니다. midas Civil에서는 반복해석기법으로Full Newton-Raphson Method를 사용하고 있습니다. + +Full Newton-Raphson Method에 의한 비선형 정적증분 해석과정은 다음과 같습니다. + +![](images/page-297_b177e9e05eb3315bd2b265d910c9c922fd3a577548ee1386b257a3a4cb01487a.jpg) + +
+line + +| Point | Description | Description | +|-------|---------------------------------|---------------------------------| +| A | λₙ·P₀ | Reference Load | +| B | Kₙ⁽¹⁾ | Reference Load | +| C | R⁽²⁾ | Reference Load | +| D | ΔUₙ⁽²⁾ | Reference Load | +| D | ΔUₙ⁽¹⁾ | Reference Load | +| D | δUₙ⁽²⁾ | Reference Load | +| D | δUₙ⁽³⁾ | Reference Load | +| Fₙ⁽¹⁾ | Fₙ⁽¹⁾ | Reference Load | +| Fₙ⁽¹⁾ | R⁽ⁱ⁾ | Residual Force | +| Fₙ⁽¹⁾ | Fₙ⁽ⁱ⁾ | Internal Force | +
+ +그림 2.8.41 Newton-Raphson Method + + + +(1) 현재증분(n)의 외력벡터 $\lambda_{n} \cdot P_{0}$ 를 구조물에 작용시키면 그림 2.8.41의 점 A가 얻어집니다. 이때의 비선형 정적방정식은 다음과 같이 표현됩니다. + +$$ +\boldsymbol {K} _ {n} \Delta \boldsymbol {U} _ {n} + \boldsymbol {F} _ {n - 1} = \lambda_ {n} \cdot \boldsymbol {P} _ {0} \tag {37} +$$ + +여기서 $K_{n}$ : 현재증분스텝(n)에서의 구조물의 접선 강성행렬 + +$\Delta U_{n}$ : 현재증분스텝(n)에서의 증분 변위벡터 + +$F_{n-1}$ : 직전증분스텝(n-1)까지의 내력벡터 + +$\lambda_{n}$ : 현재증분스텝(n)에서의 하중 파라메터 + +$P_{0}$ : 설정된 하중벡터 + +$\lambda_{n} \cdot P_{0}$ : 현재증분스텝(n)에서의 외력벡터 + +식 (37)은 다음과 같이 증분방정식으로 표현할 수 있습니다. + +$$ +\boldsymbol {K} _ {n} \Delta \boldsymbol {U} _ {n} = \lambda_ {n} \cdot \boldsymbol {P} _ {0} - \boldsymbol {F} _ {n - 1} +$$ + +$$ +\boldsymbol {K} _ {n} \Delta \boldsymbol {U} _ {n} = \Delta \lambda_ {n} \cdot \boldsymbol {P} _ {0} \tag {38} +$$ + +여기서 $\Delta\lambda_{n}\cdot P_{0}$ : 현재증분스텝(n)에서의 증분 외력벡터 + +식 (38)를 풀어 미지수인 증분 변위벡터 $\Delta U_{n}$ 를 구합니다. + +(2) 증분 변위벡터 $\Delta U_{n}$ 를 이용하여 각 비선형 요소의 접선강성과 내력을 구합니다. 구해진 각 요소의 접선강성을 조합하여 전체 구조물의 접선강성행렬 $K_{n}^{(i)}$ 을 구성합니다. 또한, 각 요소의 내력을 절점력으로 조합하여 내력벡터 $F_{n}^{(i)}$ 를 구합니다. 이때, 구조물의 내력과 변형량의 관계는 점 B가 됩니다. + +(3) 하중이 $\Delta\lambda_{n}\cdot P_{0}$ 만큼 증가하는 동안 비선형 요소가 항복하면, 요소강성이 변화하여 불평형력 $\boldsymbol{R}_{n}^{(i)}$ 이 발생합니다. 불평형력은 수렴계산을 통하여 해소합니다. + +$$ +\boldsymbol {K} _ {n} ^ {(i)} \delta \boldsymbol {U} _ {n} ^ {(i)} = \boldsymbol {\lambda} _ {n} \cdot \boldsymbol {P} _ {0} - \boldsymbol {F} _ {n} ^ {(i)} +$$ + +$$ +\boldsymbol {K} _ {n} ^ {(i)} \delta \boldsymbol {U} _ {n} ^ {(i)} = \boldsymbol {R} _ {n} ^ {(i)} \tag {39} +$$ + +여기서 $K_{n}^{(i)}$ : 현재증분스텝(n)의 i번째 수렴계산에서의 접선 강성행렬 + + + +$\delta \pmb { U } _ { n } ^ { ( i ) }$ : 현재증분스텝(n) 의 i번째 수렴계산에서의 변위벡터 + +$F _ { n } ^ { ( i ) }$ : 현재증분스텝(n) 의 i번째 수렴계산에서의 내력벡터 + +$\pmb { R } _ { n } ^ { ( i ) }$ : 현재증분스텝(n) 의 i번째 수렴계산에서의 불평형력 + +식 (39)를 풀어 미지수인 변위벡터 $\delta \pmb { U } _ { n } ^ { ( i ) }$ 를 구합니다. 각 요소의 내력과 접선강성을 구하고, 불평형력 $\pmb { R } _ { n } ^ { ( i ) }$ 을 구하여, 수렴조건이 만족할 때까지 (1)\~(3)의과정을 반복합니다. + +(4) 수렴조건이 만족되면(점 C) (1)로 돌아가 다음 증분의 해석을 수행합니다. + +# 불평형력과 수렴계산 + +식 (39)를 풀어 미지수인 변위벡터 $\delta \pmb { U } _ { n } ^ { ( i ) }$ 를 구합니다. 각 요소의 내력과 접선강성을 구하고, 불평형력 $\pmb { R } _ { n } ^ { ( i ) }$ 을 구하여, 수렴조건이 만족할 때까지 (1)\~(3)의 과정을반복합니다. + +# (1) 불평형력과 수렴계산 + +불평형력(Residual Force)은 각 증분에서의 요소강성 변화와 이로 인한 요소내력의 변화에 의해서, 가해지는 외력과 요소에 발생하는 내력 간의 차이를의미합니다. 불평형력은 전술한 바와 같이 Newton-Raphson Method에 의한반복해석(수렴계산)을 통하여 해소되며, 반복해석 조건에 따라서 다음과 같이처리됩니다. + +#  수렴계산을 수행하는 경우(최대반복회수를 2이상으로 설정한 경우) + +Newton- Raphson법에 의해 불평형력이 해소되어 수렴판단 조건을 만족할때까지 반복해석을 수행합니다. 단, 아래의 a, b의 불평형력은 다음 증분의외력으로 처리합니다. + +a) 최대반복회수만큼 반복해석을 수행한 경우에도 수렴판단 조건을 만족하지 못하여 잔존하는 불평형력 + +b) 수렴판단 조건을 만족했지만 여전히 잔존하는 불평형력 + + + +수렴판단 조건을 만족했다고 하더라도 불평형력이 완전히 0이 되는 것은 아닙니다. 단지 무시할 정도로 작아지는 것으로, 잔존 불평형력은 해석결과에 거의 영향을 미치지 않습니다. + +# - 수렴계산을 수행하지 않는 경우(최대반복회수를 1로 설정한 경우) + +불평형력은 다음 증분의 외력으로 처리합니다. + +따라서, 최대반복수 만큼 반복해석을 수행한 경우에도 수렴판단 조건을 만족하지 못하여 잔존하는 불평형력이 다음 증분의 외력에 더해지므로, 직전 증분에서 수렴하지 않았다고 하더라도, 현재 증분에서 수렴하면 전체 해석결과에 미치는 영향은 크지 않습니다. + +# (2) 수렴판단조건 + +반복해석을 통하여 불평형력을 해소한다고 하더라도, 불평형력이 완전히 0이 되도록 수렴시키는 것은 수치해석적으로 불가능합니다. 따라서, 불평형력이 어느 정도 이하가 되면 수렴되었다고 판단하고 다음 증분으로 넘어가기 위해서 수렴판단조건을 설정하게 합니다. + +반복해석에서 수렴을 판정하는 기준 Norm은 변위, 하중 및 에너지의 세가지 방법을 제공하며, 이 가운데 하나 또는 복수의 Norm을 선택하여 수렴판단에 사용할 수 있습니다. 각 Norm의 정의는 다음과 같습니다. + +변위 Norm + +$$ +\varepsilon_ {D} = \sqrt {\frac {\delta \boldsymbol {U} _ {n} ^ {(i) T} \cdot \delta \boldsymbol {U} _ {n} ^ {(i)}}{\Delta \boldsymbol {U} _ {n} ^ {(i) T} \cdot \Delta \boldsymbol {U} _ {n} ^ {(i)}}} \tag {40a} +$$ + +■ 하중 Norm + +$$ +\varepsilon_ {F} = \sqrt {\frac {\delta \boldsymbol {F} _ {n} ^ {(i) T} \cdot \delta \boldsymbol {F} _ {n} ^ {(i)}}{\Delta \boldsymbol {F} _ {n} ^ {(i) T} \cdot \Delta \boldsymbol {F} _ {n} ^ {(i)}}} \tag {40b} +$$ + +■ 에너지Norm + +$$ +\varepsilon_ {E} = \sqrt {\frac {\delta \boldsymbol {F} _ {n} ^ {(i) T} \cdot \delta \boldsymbol {U} _ {n} ^ {(i)}}{\Delta \boldsymbol {F} _ {n} ^ {(i) T} \cdot \Delta \boldsymbol {U} _ {n} ^ {(i)}}} \tag {40c} +$$ diff --git a/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_031.md b/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_031.md new file mode 100644 index 00000000..90bd8089 --- /dev/null +++ b/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_031.md @@ -0,0 +1,331 @@ + + +여기서 $\varepsilon _ { D }$ : 변위 Norm + +$\varepsilon _ { F }$ : 하중 Norm + +$\varepsilon _ { E }$ : 에너지 Norm + +$\Delta { \pmb U } _ { n } ^ { ( i ) }$ : 현재증분스텝(n)의 i번째 반복계산까지 누적된 증분변위벡터 + +$\delta \pmb { U } _ { n } ^ { ( i ) }$ : 현재증분스텝(n)의 i번째 수렴계산에서의 변위벡터 + +$\Delta F _ { n } ^ { ( i ) }$ : 현재증분스텝(n)의 i번째 반복계산까지 누적된 증분내력벡터 + +$\delta F _ { n } ^ { ( i ) }$ : 현재증분스텝(n)의 i번째 수렴계산에서의 내력벡터 + +# (3) 수렴판단조건의 설정 + +수렴판단 조건은 변위 Norm만을 선택하는 것이 일반적입니다. 복수의 Norm을 수렴판단 조건으로 설정하면, 하나의 조건을 선택한 경우에 비해서 수렴반복회수가 증가할 수 있습니다. + +# Substep에 의한 증분량의 자동분할 + +구조물의 비선형성이 매우 강한 경우, 수렴해석의 최대반복수까지 수렴하지 않는경우가 있습니다. midas Civil의 비선형 해석에서는 이와 같이 수렴이 되지 않는 경우에는 현재 증분량을 자동분할하여 수렴성능을 향상시키는 Substep기능을 제공합니다. + +Substep기능은 현재스텝에서 수렴되지 않은 경우, 수렴된 직전 스텝으로 되돌아가현재 증분량을 0.5만큼만 계속 분할하여 해석을 수행하는 비선형 해석기법이며, 이와 같은 분할된 스텝을 Substep이라 합니다. + +예를 들어 이전 증분 단계까지 누적된 총 증분(증분 하중파라메터 혹은 증분 변위량의 합)이 0.5이고, 현재 스텝의 증분량이 0.1인 경우, 수렴조건을 만족하지 않으면 내부적으로 증분량을 분할하여 0.05로 조절하여 해석을 수행합니다. 이 경우에도 수렴을 하지 않을 경우 0.025로 증분량을 조절하여 해석을 수행하는 방법입니다. + +Substep에 의한 증분량 자동분할과정은 다음과 같습니다. + + 최대반복수까지 수렴되지 않은 경우, 현재상태 B\*에서 직전스텝에서 수렴 + + + +된 A로 이동합니다. + + 현재 스텝의 증분량을 2분할하여 증분해석을 수행합니다. + 증분량을 2분할한 경우에도 수렴되지 않는 경우, 재차 증분량을 2분할 합니다. + +![](images/page-302_67072b4a1d6e90cc8b16d7a62086613d1f4bbdf88f0517f8d6eac8802e0a68af.jpg) + +
+flowchart + +```mermaid +graph TD + A["Point A"] -->|Converged| B["Point B"] + B -->|Diverged| C["Point B*"] + C -->|Diverged| D["Point A"] + style A fill:#f9f,stroke:#333 + style B fill:#f9f,stroke:#333 + style C fill:#f9f,stroke:#333 + style D fill:#f9f,stroke:#333 +``` +
+ +그림 2.8.42 Substep에 의한 증분량 자동분할 + +이와 같이 증분량을 Substep으로 분할하는 횟수는 Pushover탭>Control그룹>Pushover Global Control의 Nonlinear Analysis Option의 Max. Number ofSubsteps에서 지정합니다. Max. Number of Substeps의 기본값은 10이며, Substep은수렴성능 향상을 위한 임시스텝으로 Substep에 대한 결과는 출력되지 않습니다.Max. Number of Substeps을 “1”로 정의하면, 수렴하지 않는 경우, 증분량을 분할하지 않습니다. + +# 초기하중의 고려 + +횡력 혹은 수평력에 대한 구조물의 보유내력을 구하기 위하여 정적증분해석을 수행하는 경우에, 중력방향하중(고정하중, 적재하중)에 대한 해석을 선행해야 할 필요가 있습니다. 특히, 휨과 축력이 동시에 작용하는 기둥부재의 경우, 축력의 변동을 고려하여 항복모멘트를 산정하므로 반드시 초기하중을 고려하여 해석을 수행하여야 합니다. + +정적증분해석에서 초기하중은 Pushover탭>Control그룹>Pushover Global Control + + + +의 Initial Load에서 설정합니다. 정적증분해석에서 초기하중을 고려하는 경우에는Pushover탭>Load Case그룹>Pushover Load Cases>Pushover Load Cases 에서Use Initial Load를 선택합니다. + +정적증분해석에서 초기하중의 설정방법은 다음 2가지 방법을 제공합니다. + + Nonlinear Analysis for Initial Load + Import Static Analysis / Construction Stage Analysis Results + +Nonlinear Analysis for Initial Load는 설정된 초기하중에 대해서 실제 비선형해석을 수행하는 방법입니다. 정적증분해석은 비선형 증분해석이므로, 선형탄성해석과같이 하중조합에 의해 각 요소의 내력을 구할 수 없습니다. 따라서, 초기하중을 고려하는 경우에는 초기하중에 대해서도 비선형 해석을 수행하여야 합니다. 단, 이때의 증분수는 1스텝으로 고정하여 해석합니다. + +Import Static Analysis / Construction Stage Analysis Results는 초기하중과 정적증분해석의 경계조건이 다른 경우 혹은 시공단계해석의 최종단계를 초기하중으로 고려할 경우에 에 적용하는 방법입니다. 정적해석 결과 혹은 시공단계해석 결과를 초기하중으로 설정하는 경우, 선형해석에서는 입력된 초기단면력을 단순히 선형조합하여 처리 가능합니다. 하지만, 비선형 시간이력해석에서는 입력된 초기단면력을비선형 요소의 상태판정에 고려하지 않으면, 연속적으로 수행되는 하중조건 사이의 정합성을 확보할 수 없습니다. 또한, 부재에 발생하는 단면력은 가해지는 외력에 의해 발생하므로, 입력된 초기단면력을 부재내력으로 평형방정식에 그대로 반영하면, 평형조건이 성립되지 않습니다. + +정적증분해석에서는 입력된 초기단면력에 대해 가상의 변형을 구하여, 비선형 부재의 상태 판정시에 고려하는 방법으로 비선형 시간이력해석을 수행합니다. 단, 평형방정식을 구성할때는 초기단면력은 무시되며, 상세한 해석방법은 다음과 같습니다. + +1. 정적증분해석의 초기증분에 들어가기 전에, 초기강성 $K _ { 0 }$ 를 이용하여, 입력된 초기 단면력에 대해 비탄성 힌지의 가상의 변형 $D _ { i n i }$ 을 구합니다. + +a. 구해진 $D _ { i n i }$ 이 항복변형 이내에 있으면(탄성범위), 입력된 초기 단면력을 그대로 해석에 반영합니다. +b. 구해진 $D _ { i n i }$ 이 항복변형을 초과한 경우, 이력루틴에서 변형 $D _ { i n i }$ 에 대한 내력 $P _ { i n i }$ 를 구하여 해석에 반영합니다. 단, $D _ { i n i }$ 와 $P _ { i n i }$ 는 초기증분에서 1회만 구합니다. + + + +2. 평형방정식을 풀어, 증분변위 $\delta u _ { { t + \Delta t } }$ 를 구합니다. 단, 초기 단면력은 내력으로 입력되므로, 평형방적식을 구성할때는 무시됩니다. +3. 증분변위 $\delta u _ { { t + \Delta t } }$ 을 이용하여, 비선형 힌지의 변형 D 와 내력 P 를 구합니다. +4. 비선형 부재의 상태판정을 위해 이력루틴에 들어갑니다. 단, 이력루틴에 들어가기전에 비탄형 힌지의 변형과 부재력은 초기 단면력을 고려하여 다음과 같이 수정됩니다. + +$$ +\begin{array}{l} D ^ {*} = D + D _ {i n i} \\ P ^ {*} = P + P _ {i n i} \\ \end{array} +$$ + +5. 이력루틴에서 변형 $D ^ { * }$ 에 의해 강성과 내력 $\overline { { P } } ^ { * }$ 을 계산합니다. + +6. 비탄성 힌지의 해석결과를 출력합니다. + +7. 평형방정식을 구성하기 위해서, 변형과 복원력을 다음과 같이 수정합니다. + +$$ +D = D ^ {*} - D _ {i n i} +$$ + +$$ +\overline {{P}} = \overline {{P}} ^ {*} - P _ {i n i} +$$ + +평형방정식을 구성하고, 2. 로 돌아가 마직막 증분까지 이와 같은 해석을 반복합니다. + +![](images/page-304_b51b46469a98225f141f77796f2c942c91b12f1cc1c0039f12d619afebc15746.jpg) + +
+line +| Point | D | P | +|-------|-------|-------| +| K0 | Dini | P0ini | +| P1(+) | D1(+) | P0ini | +
+ +그림 2.8.43 초기단면력의 처리 + +초기하중을 고려하는 경우에는 초기하중에 의해 발생한 각 요소의 내력은 정적증분해석에 계승됩니다. 단, 초기하중에 의해 발생한 각 절점의 변위 및 반력/층전단력을 정적증분해석 결과에 포함시켜 출력할 것인가의 여부는 Pushover Load Case + + + +에서 선택가능 합니다. + +# P-Delta 효과 + +P-Delta해석은 요소가 횡력과 축력을 동시에 받을 때 2차적인 구조적 거동을 고려하기 위한 기하학적 비선형 해석의 일종입니다. P-Delta효과를 고려하면, 요소의 강성은 요소 고유의 횡방향 강성 k 에 축력에 따른 강성의 증감효과를 나타내는 기하강성 $k _ { G }$ 가 더해집니다. P-Delta효과를 고려하는 경우의 비선형 정적방정식은 다음과 같이 표현됩니다. + +$$ +\left(\boldsymbol {K} + \boldsymbol {K} _ {G}\right) _ {n} \Delta \boldsymbol {U} _ {n} + \boldsymbol {F} _ {n - 1} = \lambda_ {n} \cdot \boldsymbol {P} _ {0} \tag {41} +$$ + +여기서 K : 구조물 고유의 강성행렬 + +$\pmb { { \cal K } } _ { G } : $ 기하강성 행렬 + +정적증분해석에서 P-Delta효과는 다음 식과 같이 강성행렬의 행렬값이 양의 값을갖는 영역에서만 반영됩니다. 강성행렬의 행렬값이 0 혹은 음의 값을 나타내면 요소강성 구성시에 기하강성 $k _ { G }$ 를 무시합니다. + +$$ +\left| \boldsymbol {K} + \boldsymbol {K} _ {G} \right| > 0 \tag {42} +$$ + +강성행렬의 행렬값이 0 혹은 음의 값을 나타내는 경우는 다음과 같습니다. + + FEMA Type에서 변형이 커져서 부구배 상태에 들어가는 경우 (그림 2.8.41의 C점 이후) + Multi-Linear Type에서 소성힌지 발생으로 갱신된 강성에 기하강성을 더한이후 요소강성의 대각성분에 0 또는 음의 값이 나타나는 경우 + + + +# 해석종료조건 + + 최대 증분회수에 도달하는 경우 + Limit Inter-Story Deformation Angle구조물의 각 층에서 발생하는 층간변형각의 최대값이 설정된 한계 층간변형각을 초과하는 경우 자동종료합니다. 각 층에서 발생하는 층간변형각의 산정방법은 다음과 같습니다. + +1) 각 층에 설정된 수직부재의 층간변형각의 최대값 + +2) 층 중심에서의 층간변형각 + +3) 층의 절점변위의 평균값으로 층간변형각 산정 + + 전단성분 힌지가 최초로 항복한 경우 + +1) 구조물에 설정된 전단성분 힌지중에서 최초로 항복한 경우 자동종료합니다. + +2) 전단성분 힌지항복시 자동종료 조건은 Pushover탭>Control그룹>Pushover Global Control의 Analysis Stop에서 Shear Component Yield를설정합니다. + +3) 전단성분 힌지의 자동종료를 위해서는 요소에 전단성분 힌지를 설정할필요가 있습니다. + + 축력성분 힌지의 압괴 / 좌굴이 발생한 경우 + +1) 구조물에 설정된 축력성분 힌지중에서 최초로 압괴 / 좌굴이 발생한 경우 자동종료 합니다. + +2) 축력힌지의 압괴 / 좌굴시 자동종료는 Pushover탭>Assign그룹>DefinePushover Hinge Type/Properties에서 적용한 Material Type에 따라서 다음과 같이 처리 됩니다. + +a. RC / SRC(encased) : 축력힌지의 압축측 압괴(항복)시 자동종료합니다. + +b. Steel / SRC(filled) - Input Method>Auto-Calculation + +i) 요소가 항복하기 전에 좌굴하는 경우 : 좌굴시 자동종료 + + + +(축력항복내력보다 좌굴내력이 작은 경우) + +ii) 요소가 좌굴하지 않는 경우 : 압축측 항복이 발생한 경우에도 자동종료 하지 않고 계속 해석진행 + +\- Input Method>User Input : 좌굴유무에 관계없이 압축측 항복시 자동 종료 + +3) 축력성분 힌지의 압괴 / 좌굴시 자동종료는 Pushover탭>Control그룹>Pushover Global Control의 Analysis Stop에서 Axial ComponentCollapse/ Buckling를 설정합니다. +4) 요소에 축력성분 힌지를 설정할 필요가 있습니다. +5) 축력성분의 인장측 항복시에는 종료하지 않고 계속 해석을 진행합니다. + + Current Stiffness Ratio + +전체 구조물의 초기강성과 현재의 강성과의 비가 지정된 값에 도달한 경우 (하중증분 해석만 해당) + + 초기하중에 대한 해석시에 PMM TYPE이 설정된 기둥의 축력이 항복축력을 초과한 경우 (중력방향하중에 대해 기둥부재의 축성분이 항복했다는 의미이므로, 초기하중이 과대하게 설정되었거나, 항복곡면의 설정에 문제가있다고 판단할 수 있습니다. 따라서, 이와 같은 경우는 메시지 출력 후 강제 종료 시킵니다.) + + + +# 8-7-4 하중증분법에 의한 하중제어 방법 + +midas Civil 에서 하중증분법은 사용자가 구조물에 작용하는 최대하중을 미리 설정하고, 최대하중에 도달할 때까지 하중을 적절히 분할하여 점진적으로 증가시키는 해석방법입니다. 즉, 하중증분법에서는 증분화하는 대상이 최대하중 $P_{0}$ 가 됩니다. 각 증분에서의 하중증분량 $\Delta\lambda_{n} \cdot P_{0}$ 은 최대하중 $P_{0}$ 에 증분 파라메터 $\Delta\lambda$ 를 곱하여 산정합니다. + +$$ +\lambda_ {n} \cdot P _ {0} = \lambda_ {n - 1} \cdot P _ {0} + \Delta \lambda_ {n} \cdot P _ {0} \tag {43a} +$$ + +$$ +\lambda_ {n} = \lambda_ {n - 1} + \Delta \lambda_ {n} \tag {43b} +$$ + +$$ +\Delta \lambda_ {n} = \lambda_ {n} - \lambda_ {n - 1} \tag {43c} +$$ + +여기서, $\Delta\lambda_{n}\cdot P_{0}$ : n 스텝에서의 하중증분량 + +$P_{0}$ : 최대하중 벡터 + +$\lambda_{n}$ : n 스텝에서의 하중 파라메터 + +$\lambda_{n-1}$ : n-1 스텝에서의 하중 파라메터 + +$\Delta\lambda_{n}$ : n 스텝에서의 증분 파라메터 + +하중증분해석에 의한 Pushover해석에서는 구조물에 최대내력(구조물이 외력에 저항 할 수 있는 최대내력, Limit Point)이상의 외력이 작용하면, 하중증분해석으로 더 이상 안정해를 얻을 수 없는 해석불능 상태가 되어 발산하게 됩니다. Pushover해석에서는 이와 같은 극한점에서 해석을 자동으로 종료하는 기능을 제공합니다. + +![](images/page-308_2965fa297c3cfa73aab817d0e0bdb2d35fa102d0928b73771e43197f1cf7266c.jpg) + +
+line +| U: Displacement | P₀: Reference Load | +| -------------- | ------------------ | +| λ₁·P₀ | λ₁·P₀ | +| λ₂·P₀ | λ₂·P₀ | +| λₙ·P₀ | λₙ·P₀ | +| λₙ₋₁·P₀ | λₙ₋₁·P₀ | +| Δλₙ·P₀ | Δλₙ·P₀ | +| λₙ·P₀ | λₙ·P₀ | +| λₙ₋₁·P₀ | λₙ₋₁·P₀ | +| n step | n step | +| n-1 step | n-1 step | +| Kₙ | Kₙ | +
+ +그림 2.8.44 하중증분법 + + + +midas Civil에서 하중제어 방법은 Full Newton-Raphson 방법을 따르고 있습니다.Full Newton-Raphson 방법은 미분의 원리를 이용하여 함수값을 찾는 방법으로 해에 접근하는 속도가 빠르다는 장점이 있습니다. + +하중은 기본적으로 일정한 분포를 가진 횡하중 또는 수평하중입니다. 하중의 분포는 하중조건 중에서 사용자가 설정한 지진하중(Qud) 이외에도 사용자가 하중조건에서 설정한 임의의 하중분포도 가능합니다. 또한 특정 절점에서의 절점하중을 포함하여 하중조건에서 설정한 하중을 정적증분해석시의 하중으로 사용할 수 있습니다 + +midas Civil에서는 다음의 3가지 하중증분 제어방법을 제공합니다. + + Auto Stepping Control + Equal Step Control + Incremental Control Function + +Auto Stepping Control +![](images/page-309_4dc889c31ba0af072f79e9e05bc6f1425eaf17706fbe26842e50fd5c175d0482.jpg) + +
+line + +| Displacement | λ₁·P₀ | λₙ·P₀ | +| ------------ | ----- | ----- | +| n=1 | 1 | n=1 | +| n=1 | 1 | nstep | +
+ +그림 2.8.45 Auto Stepping Control에 의한 하중증분해석 + +자동증분 제어방법은 비선형성이 작은 구간에서는 증분간격을 크게 하고, 비선형성이 큰 구간에서는 증분간격을 작게 제어하는 방법입니다. 자동증분제어에 의한하중증분해석 방법은 다음과 같습니다. + + + +# (1) 1단계: 탄성한계 계산 $(n = 1)$ + +사용자가 설정한 수평하중을 재하하여, 수평하중에 대한 각 요소에 할당된 비선형 힌지의 내력과 힌지의 항복내력의 비율을 구합니다. 이 비율은 최초로 비선형 힌지에 항복이 발생하는 하중을 의미하며 탄성한계가 됩니다. + +$$ +\text { ratio } = \frac {P _ {\text { yild }} - P _ {\text { ini }}}{P _ {\text { crnt }} - P _ {\text { ini }}} \tag {44} +$$ + +여기서 ratio : 비선형 힌지의 항복내력과 수평하중에 의한 내력의 비율 + +$P_{yild}$ : 비선형 힌지의 항복내력 + +$P_{crnt}$ : 수평하중에 의해 발생된 비선형 힌지의 내력 + +$P_{ini}$ : 초기하중에 의해 발생된 비선형 힌지의 내력 + +탄성한계의 90%를 1번째 하중스텝의 하중 파라메터로 설정하여 해석합니다. + +$$ +\boldsymbol {K} _ {1} \Delta \boldsymbol {U} _ {1} + \boldsymbol {F} _ {i n i} = \lambda_ {1} \cdot \boldsymbol {P} _ {0} + \boldsymbol {P} _ {i n i} \tag {45} +$$ + +여기서 $K_{1}$ : 1번째 하중증분스텝의 구조물의 접선 강성행렬 + +$\Delta U_{1}$ : 1번째 하중증분스텝의 증분 변위벡터 + +$F_{ini}$ : 초기하중에 대한 내력벡터 + +$\lambda_{1}$ : 1번째 하중증분스텝의 하중 파라메터( $\lambda_{1}=0.9*ratio$ ) + +$P_{0}$ : 설정된 정적증분 하중벡터 + +$P_{ini}$ : 초기하중 벡터 + +$\lambda_{1} \cdot P_{0}$ : 1번째 하중증분스텝의 외력벡터 + +# (2) 2단계: 등차급수에 의한 증분해석 (1 < n < nstep) + +탄성한계에서 설정된 총하종까지의 하중 파라메터는 다음과 같은 등차급수에 의해 할당됩니다. + +$$ +\lambda_ {n} = \lambda_ {n - 1} + \frac {\left\{(n s t e p + 1) - n \right\}}{\sum_ {i = 1} ^ {n s t e p - 1} i} \times \left(1 - \lambda_ {1}\right) \tag {46} +$$ + +여기서 $\lambda_{n}$ : 현재스텝(n)의 하중 파라메터 + +$\lambda_{n-1}$ : 직전스텝(n-1)의 하중 파라메터 diff --git a/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_032.md b/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_032.md new file mode 100644 index 00000000..41496a41 --- /dev/null +++ b/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_032.md @@ -0,0 +1,383 @@ + + +nstep : 총 스텝수 + +i : 등차증분 step 수 + +현재스텝에서의 외력벡터는 다음과 같이 설정됩니다. + +$$ +\boldsymbol {P} _ {n} = \lambda_ {n} \cdot \boldsymbol {P} _ {0} \tag {47} +$$ + +# (3) 3단계: 최종 스텝의 증분하중 ( n nstep  ) + +최종 하중증분스텝( nstep )의 외력벡터는 다음과 같습니다. + +$$ +\boldsymbol {P} _ {n s t e p} = \lambda_ {n s t e p} \cdot \boldsymbol {P} _ {0}; \lambda_ {n s t e p} = 1. 0 \tag {48} +$$ + +![](images/page-311_c355c29748d4b2f6dbefae0451ff6b75864b56e30fe980b54186db412a491715.jpg) + +
+line +| INCREM. STEP | ITERATION | LOAD | PRAMTER | +| ------------ | --------- | ------ | ------- | +| 1 | 2 | 0.39738| | +| 2 | 3 | 0.45764| | +| 3 | 3 | 0.51473| | +| 4 | 3 | 0.56865| | +| 5 | 3 | 0.61940| | +| 6 | 2 | 0.66697| | +| 7 | 3 | 0.71138| | +| 8 | 2 | 0.75261| | +| 9 | 2 | 0.79067| | +| 10 | 2 | 0.82556| | +| 11 | 3 | 0.85727| | +| 12 | 2 | 0.88582| | +| 13 | 2 | 0.91119| | +| 14 | 3 | 0.93339| | +| 15 | 2 | 0.95242| | +| 16 | 2 | 0.96828| | +| 17 | 2 | 0.98097| | +| 18 | 2 | 0.99048| | +| 19 | 2 | 0.99683| | +| 20 | 2 | 1.00000| | +
+ +그림 2.8.46 자동증분제어에 의한 하중증분해석시의 해석예 (하중 파라메터) + +# Equal Step Control + +설정된 하중을 총 증분수로 나누어 증분해석을 합니다. 따라서, 각 증분에서의 하중 파라메터의 증분량은 동일하며, 각 증분하중도 동일한 값이 적용됩니다. + +# Incremental Control Function + +사용자가 정의한 함수룰 근거로 각 증분의 하중 파라메터를 구하여, 해석합니다.설정방법과 해석시에 고려방법은 다음과 같습니다. + + + +1. Pushover load case에서 총 스텝수 nstep 를 설정합니다. +2. Stepping Control Option>Incremental Control Function을 선택하여 증분함수를 입력합니다. + +![](images/page-312_363508893f8795637fdee7c85cee019206283e90ba2629073a7757f7237901ce.jpg) + +
+line + +| No. | Increment Function | +| ---- | ------------------ | +| 0.01 | -0.05 | +| 0.61 | 0.55 | +| 1.21 | 0.65 | +| 1.61 | 0.75 | +| 2.41 | 0.85 | +| 3.01 | 0.90 | +| 3.61 | 0.95 | +| 4.21 | 0.98 | +| 4.61 | 1.00 | +
+ +No. : 입력되는 하중 파라메터 함수의 X축 정의. 총 증분수와는 무관한 값으로 총 증분수를 변경하여도 함수를 변경할 필요는 없습니다. + +Function : 해석에 직접 사용되는 하중 파라메터 + +3. 적용예 + +Case 1 : 각 증분의 하중 파라메터를 증분제어함수에 직접입력 + +nstep =10으로 설정 +Incremental Control Function을 다음과 같이 설정 + +
No.Function
10.30
20.60
30.65
40.70
50.75
60.80
70.85
80.90
90.95
101.00
+ +위와 같이 증분함수를 입력하면, 설정된 Function을 하중 파라메터로 그대로 적용합니다. 단, 총 증분수를 10으로 설정한 경우입니다. + +Case 2 : 하중 파라메터를 총 증분수와 무관한 중분함수로 입력 + +nstep =10으로 설정 + + + +Incremental Control Function을 다음과 같이 설정 + +
No.FunctionNo.Function
00.0또는0.00.30
10.60.20.60
51.01.00.65
+ +위와 같이 증분함수를 입력한 경우에 No.항의 최종값으로 No.항을 나누면두 함수의 No.값은 같게 되므로, 위의 두 함수는 동일한 조건이 됩니다.위와 같이 증분함수를 설정한 경우, 해석시의 각 증분에서의 하중 파라메터는 다음과 같이 반영됩니다. 단, 총 증분수는 10으로 설정 + +
Step No.Load Parameter
10.30
20.60
30.65
40.70
50.75
60.80
70.85
80.90
90.95
101.00
+ + + +# Current Stiffness Ratio에 의한 하중증분해석의 자동종료 + +하중증분해석에 의한 정적증분해석에서는 구조물에 최대내력(구조물이 외력에 저항 할 수 있는 최대내력 : 그림 2.8.40의 점 C)이상의 외력이 작용하면, 하중증분해석으로 더 이상 안정해를 얻을 수 없는 해석불능 상태가 되어 발산하게 됩니다.midas Civil의 정적증분해석에서는 이와 같은 극한점에서 해석을 자동으로 종료하는 기능을 제공합니다. + +Current Stiffness Ratio는 초기상태(탄성상태)에서의 구조물의 강성행렬과 현재상태의 강성행렬을 비율로 나타내어, 구조물의 상태를 판단하는 계수입니다. + +Current Stiffness Ratio는 구조물의 상태에 따라서, 다음과 같이 표현됩니다. + + 탄성상태 Cs  100.0% + 최대내력점까지의 상태 : $0 . 0 \% < C s < 1 0 0 . 0 \%$ + 최대내력점 상태 Cs  0.0% + 최대내력점이후의 상태 : Cs  0.0% + +현재의 증분에서 Cs가 0보다 작게 되면 안정해를 구할 수 없기 때문에 직전 하중스텝으로 돌아가서 해석을 자동종료 합니다. + +![](images/page-314_13355f53d905ea40be1d322e959fbf4f3fdc89a8dce50a582cb50ce8e96d45b1.jpg) + +
+line + +| Displacement | Load | Cs (%) | +| ------------ | ---- | ------ | +| 0 | 100 | 100 | +| 30 | 50 | 30 | +| 50 | 70 | 50 | +| 70 | 80 | 70 | +| 90 | 90 | 80 | +| 100 | 100 | 100 | +
+ +그림 2.8.47 Current Stiffness Ratio + + + +# 8-7-5 목표변위에 의한 변위제어 방법 + +midas Civil에서 변위제어는 사용자가 구조물에서 발생할 수 있는 목표변위를 미리 설정하고 구조물에서 목표변위가 달성될 때까지 하중을 증가시키는 방법입니다. + +midas에서 변위제어는 사용자가 구조물에서 발생할 수 있는 목표변위를 미리 설정하고 구조물에서 목표변위를 적절히 분할하여 단계적으로 변위를 증가시켜 해석하는 방법입니다. 각 증분에서의 변위증분량은 목표변위 U 에 증분 파라메터 Δλ 를 곱하여 산정합니다. 변위제어는 최대보유내력이후의 거동을 파악할 수 있습니다. + +$$ +U _ {n} = U _ {n - 1} + \Delta \lambda_ {n} \cdot U \tag {49a} +$$ + +$$ +\Delta \lambda_ {n} \cdot U = U _ {n} - U _ {n - 1} \tag {49b} +$$ + +여기서, $U_{n}$ : n 스템까지 누적된 변위벡터 + +$U_{n}$ : n-1 스텝까지 누적된 변위벡터 + +$\Delta\lambda_{n}$ : n 스텝에서의 증분 파라메터 + +![](images/page-315_ee29deb5e94d8154a4ac54da6c18979f81da74deaf2090b6b7612c5e530d6ce2.jpg) + +
+line +| U | P₀ (Reference Load) | +|-------|----------------------| +| U₁ | Δλₙ·U | +| Uₙ₋₁ | Δλₙ·U (n-step) | +| Uₙ₋₁ | Δλₙ·U (n-step) | +| Uₙ | Δλₙ·U (n-step) | +| Uₙ₋₁ | Δλₙ·U = Uₙ | +| Uₙ | Δλₙ·U (n-step) | +| Uₙ₋₁ | Δλₙ·U (n-step) | +| Uₙ | Δλₙ·U (n-step) | +| Uₙ₋₁ | Δλₙ·U = Uₙ | +| Uₙ | Δλₙ·U (n-step) | +| Uₙ₋₁ | Δλ₁·U | +| Uₙ | Δλ₁·U (n-step) | +| Uₙ₋₁ | Δλ₁·U (n-step) | +| Uₙ | Δλ₁·U (n-step) | +| Uₙ₋₁ | Δλ₁·U = Uₙ | +| Uₙ | Δλ₁·U (n-step) | +| Uₙ₋₁ | Δλ₁·U = Uₙ | +| Uₙ | Δλ₁·U (n-step) | +| Uₙ₋₁ | Δλ₁·U = Uₙ | +| Uₙ | Δλ₁·U (n-step) | +| Uₙ₋₁ | Δρ₁·U | +| Uₙ | Δρ₁·U | +| Uₙ₋₁ | Δρ₁·U | +| Uₙ | Δρ₁·U | +| Uₙ₋₁ | Δρ₁·U | +| Uₙ | Δρ₁·U | +| Uₙ₋₁ | Δρ₁·U | +| Uₙ | Δρ₁·U | +| Uₙ₋₁ | Δρ₁·U | +
+ +그림 2.8.48 변위증분법 + + + +목표변위는 크게 Global Control과 Master Node Control로 설정할 수 있습니다.Global Control은 구조물에서 발생하는 최대변위가 사용자가 입력한 목표변위를 만족할 때까지 하중을 증가시키는 방법입니다. 이것은 하중의 방향성과 무관합니다.Master Node Control은 사용자가 특정한 절점을 지정하고 그 절점에서 사용자가지정한 방향에 대한 목표변위를 만족하도록 하중을 증가시키는 방법입니다. 성능에 기초한 내진설계에서는 대부분 최대변위가 발생할 가능성이 있는 절점과 방향을 고려하여 목표변위를 설정합니다. + +목표변위는 구조물 전체높이의 1%, 2%, 4% 정도로 가정합니다. 이 값들은 구조물의 시스템 수준에서의 최대 층간변위에 해당하는 것으로 구조물의 손상상태와 연관성이 있습니다. 즉, ATC-40이나 FEMA-273 등에서는 최대 층간변위 1%를Immediate Occupant Level, 2%를 Life Safety Level, 4%를 Collapse Prevention Level로 정의합니다. 이 값들은 부재수준에서는 다르게 적용될 수 있습니다. + +# 8-7-6 작용하중 + +작용하중은 각 층에서의 관성력을 반영할 수 있는 횡력이 되어야 합니다. 따라서최소한 2가지 이상의 횡력 분포를 작용시키도록 권장하고 있습니다. midas Civil에서는 3가지 형태의 횡하중 분포(Lateral Load Pattern)를 제공합니다. 정적하중의 형상에 따른 하중(Static Load Case), 모드형상에 따른 하중분포(Mode Shape), 각 층의 질량에 비례하는 하중분포(Uniform Acceleration)가 있습니다. 정적하중의 형상에따른 하중분포를 이용하면 사용자가 임의 형상으로 하중을 분포시킬 수 있습니다.그리고 모드형상에 따른 분포하중을 사용하기 위해서는 반드시 고유치해석이 선행되어야 합니다. + +정적증분해석에서 작용하중은 각 층에서의 관성력을 반영할 수 있는 횡력이 되어야 합니다. 따라서, 작용하중은 기본적으로 일정한 분포를 가진 횡하중 또는 수평하중을 설정하는 것이 일반적입니다. 정적증분해석시에는 최소한 2가지 이상의 횡력 분포를 작용시키도록 권장하고 있으며, midas Civil에서는 다음의 4가지 형태의횡하중 분포(Lateral Load Pattern)를 제공합니다. + + 정적하중의 형상에 따른 하중분포(Static Load Case) + 질량에 비례하는 하중분포(Uniform Acceleration) + + + + 모드형상에 따른 하중분포(Mode Shape) + 일반화 모드형상과 질량의 곱에 의한 하중분포(Normalized Mode Shape\*Mass) + +각 하중분포에 의해 실제 해석에 작용하는 하중은 다음과 같이 산정됩니다. + +# 정적하중의 형상에 따른 하중분포(Static Load Case) + +정적하중의 분포는 선형탄성에서 정의된 정적하중을 적용합니다. 하중조건 중에서사용자가 설정한 지진하중 이외에도 사용자가 하중조건에서 설정한 임의의 하중분포도 설정가능 합니다. 또한 특정 절점에서의 절점하중을 포함하여 하중조건에서설정한 하중을 정적증분해석시의 하중으로 사용할 수 있습니다 + +# 질량에 비례하는 하중분포(Uniform Acceleration) + +힘과 질량의 관계는 다음과 같이 나타낼 수 있습니다. + +$$ +m \cdot a = P \tag {50} +$$ + +여기서 m : 질량 $\left( { \frac { N } { m / \sec ^ { 2 } } } \right)$ 2 / secm    + +a : 가속도   2 m / sec + +P : 힘  N  + +질량을 정적인 하중분포로 사용하기 위해서 가속도를 다음과 같이 가정합니다. + +$$ +a \approx 1. 0 m / \sec^ {2} \tag {51} +$$ + +식(15)을 식(14)에 대입하면, 질량을 하중으로 표현할 수 있습니다. 즉, 질량에 비례하는 하중분포는 각 자유도의 질량이 그대로 하중으로 적용됩니다. + +$$ +m _ {i} \cdot 1. 0 = P _ {i} \tag {52} +$$ + +여기서 mi : i번 자유도의 질량 + +Pi : i번 자유도의 외력 + + + +# 모드형상에 따른 하중분포(Mode Shape) + +고유모드는 구조물이 자유진동(또는 변형) 할 수 있는 고유형상을 의미하며, 구조물의 동적 특성을 나타내는 중요한 지표중의 하나입니다. 특히, 구조물의 1차 모드는구조물이 진동할 때 가장 적은 에너지(또는 힘)로 변형되는 형상을 의미하기 때문에, Pushover해석에서 모드형상을 하중분포로 설정하는 경우에는 가력방향의 1차모드형상을 설정하는 것이 일반적입니다. + +정적증분해석에서 모드형상을 하중분포로 설정하는 경우, 가력방향의 모드형상을하중분포로 설정할 수 있으며, 각 모드의 조합도 가능합니다. + +모드형상을 하중분포로 설정할 경우, 실제 해석에 작용하는 하중은 다음과 같이 산정됩니다. + +모드형상은 물리적으로 구조물의 변형, 즉 각 절점의 변위를 의미하므로 변위분포를 하중분포로 변환할 필요가 있습니다. 식(16)에 나타낸 것과 같이 질량은 하중으로 표현할 수 있으므로, 식(16)의 관계를 응용하여 모드형상을 하중으로 표현합니다. + +$$ +\varphi_ {i} \frac {\sum_ {j = 1} ^ {n} m _ {j}}{\sum_ {j = 1} ^ {n} \varphi_ {j}} = P _ {i} \tag {53} +$$ + +여기서 i : i번 자유도의 모드형상(변위) + +Pi : i번 자유도의 외력 + +$\sum _ { j = 1 } ^ { n } m _ { j }$ : 모든 자유도에 대한 질량의 총합 + +$\sum _ { j = 1 } ^ { n } \varphi _ { j }$ : 모든 자유도에 대한 모드형상(변위)의 총합 + +# 일반화 모드형상과 질량의 곱에 의한 하중분포(Normalized Mode shape \* Mass) + +정적증분해석의 하중재하시에 아래식과 같이 변위를 가정하고, 질량을 곱하여 하중을 결정하는 방법입니다. + + + +$$ +P _ {i} = m _ {i} \Phi_ {i} \tag {54} +$$ + +여기서 $\Phi_{i}$ : 최대치로 일반회된 i번 자유도의 모드형상(변위) + +$P_{i}$ : i번 자유도의 외력 + +$m_{i}$ : i번 자유도의 질량 (혹은, 총 질량) + +일반화된 모드형상 $\Phi$ 는 고유치해석을 통해 얻어진 고유모드 $\varphi$ 를 최대값으로 일반화 하여 다음과 같이 구합니다. + +![](images/page-319_40c64b5d6f8b0c6d7f0368015bc2fde9cbb397de66deb179c8e2fc4934a321ce.jpg) + +
+text_image + +φ₃ +φ₂ +φ₁ +V +
+ +![](images/page-319_ec3706a1463c31772572d7dc9ba3e82489a2fa10f40cc2998f2ff5df8f2c3b2b.jpg) + +
+text_image + +Φ₃ +Φ₂ +Φ₁ +V +
+ +$$ +\varphi_ {M A X} = \max \left(\varphi_ {1}, \varphi_ {2}, \varphi_ {3} \dots \dots , \varphi_ {i}\right) +$$ + +$$ +\Phi_ {i} = \frac {\varphi_ {i}}{\varphi_ {M A X}} \tag {55} +$$ + +여기서 $\varphi_{i}$ : i번 자유도의 모드형상(변위) + +따라서, Φ의 최대값은 10이 됩니다. + + + +# 8-7-7 정적증분해석의 비선형 요소 + +midas Civil의 정적증분해석에서 제공하는 비선형요소는 2D 보요소(2-DimensionalBeam Element), 3D 보-기둥요소(3-Dimensional Beam-Column Element), 트러스요소(Truss Element) 그리고 비선형 범용연결요소 등이 있습니다. 각 요소는 다음과 같은 특징을 가지고 있습니다. + +# 정적증분해석의 비선형 요소의 개요 + +# (1) 모멘트-회전각 관계 비선형 보요소 + +요소강성 : 유연도법에 의한 정식화 + +모멘트성분 힌지 특성 : 모멘트-회전각 관계로 정의 + +비선형 힌지 : 일축(Single Component) 및 다축-힌지(P-M-M)모델 + +비선형 힌지의 위치 + + RC, Steel Type : 요소양단(축, 전단, 비틀림, 모멘트) + +골격곡선 : Bilinear, Trilinear, FEMA 타입 + +비선형 힌지의 초기강성 : 요소의 초기강성행렬(탄성상태)구성시에는 직접 반 + +영되지 않음. 비탄성 스프링 항복 후에만 해석에 영향을 미침 + +모멘트성분 힌지의 초기강성 : 6EI/L, 3EI/L, 2EI/L로 가정 + +# (2) 모멘트-곡률 관계 비선형 보요소 + +요소강성 : 유연도법에 의해 정식화(수치적분) + +모멘트성분 힌지 특성 : 모멘트-곡률관계로 정의 + +비선형 힌지 : 일축(Single Component) 및 다축-힌지(P-M-M)모델 + +골격곡선 : Bilinear, Trilinear 타입 + +Lumped Type 요소 + +요소양단의 소성화만 고려 +비선형 힌지의 초기강성 : 요소의 초기강성행렬(탄성상태)구성시에는 +직접 반영되지 않음. 비탄성 스프링 항복 후에만 해석에 영향을 미침 +비선형 힌지의 위치 : 요소양단(축, 전단, 비틀림, 모멘트) + +Distributed Type 요소 diff --git a/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_033.md b/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_033.md new file mode 100644 index 00000000..ec27f766 --- /dev/null +++ b/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_033.md @@ -0,0 +1,575 @@ + + +요소전체의 소성화 고려 +비선형 힌지의 초기강성 : 요소의 초기강성행렬(탄성상태) 구성시에직접 반영됨. +비선형 힌지의 위치 : 요소내 적분점 + +# (3) 트러스 및 범용연결요소 + +비선형 힌지 : 일축(Single Component)모델 + +비선형 힌지의 위치 : 요소중앙 + +골격곡선 : Bilinear, Trilinear, FEMA, Slip 타입 + +비선형 힌지의 초기강성 : 요소의 초기강성행렬(탄성상태)구성시 반영 + +# 2D 보요소 및 3D 보-기둥요소 + +보요소 및 보-기둥요소는 동일한 방법으로 정식화할 수 있기 때문에 그림 2.8.49와같은 절점력과 절점변위를 대상으로 수식화하며, 유연도법(Flexibility Method)에 의해 정식화 됩니다. 유연도법은 요소내력(Element Section Force)의 분포에 근거하여정식화 되므로, 강성도법에 비해서 정확한 해석이 가능합니다. 또한 변위법(강성도법)에 비하여 적은 수의 요소로 모델링 한 경우에도 강성도법과 거의 같은 정도의결과를 얻을 수 있는 수치해석적인 이점이 있습니다. + +보요소 및 보-기둥요소에서는 다음과 같은 3차원 공간에서의 하중과 변위를 사용합니다. 보요소는 축력이 작용하지 않는 경우에 사용할 수 있습니다. + +$$ +\boldsymbol {f} ^ {T} = \left\{F _ {x i}, F _ {y i}, F _ {z i}, M _ {x i}, M _ {y i}, M _ {z i}, F _ {x j}, F _ {y j}, F _ {z j}, M _ {x j}, M _ {y j}, M _ {z j} \right\} \tag {56a} +$$ + +$$ +\boldsymbol {u} ^ {T} = \left\{u _ {i}, v _ {i}, w _ {i}, \theta_ {x i}, \theta_ {y i}, \theta_ {z i}, u _ {j}, v _ {j}, w _ {j}, \theta_ {x j}, \theta_ {y j}, \theta_ {z j} \right\} \tag {56b} +$$ + +![](images/page-321_0aecaf8cc3eb382646cf202b6e6c56258e711e75e286de4e8ac0419f8e5d1e09.jpg) + +
+text_image + +Moment_z (+) +Shear_z (-) +Axial (-) +Moment_x +y +Moment_y (+) +Shear_y (-) +z +Moment_x +
+ +![](images/page-321_74fcb2a5c255e872a5bb4efc41fd37b57a6ef75bf79e664d5ec06cb21adb8a7a.jpg) + +
+text_image + +Moment_y +(-) +Shear_y +(+) +Shear_z +(+) +Axial +(+) +Moment_x +(-) +y +z +Moment_z +(-) +
+ +![](images/page-321_fb4e039b97607d39fdfe461329cf5dde93ecc012515881eb6bbfcfe525ab4cae.jpg) + +
+text_image + +i +L +j +
+ +그림 2.8.49 2D 보요소 및 3D 보-기둥요소의 절점력 및 절점변위 + + + +비선형 보요소는 모멘트 성분의 비선형 힌지 정의방법에 따라서, 모멘트-회전각 관계요소와 모멘트-곡률관계 요소로 구분됩니다. 또한, 이들 요소는 비선형 힌지의위치와 정식화방법에 따라서 집중형 힌지모델(Lumped Type Hinge Model)과 분포형힌지모델(Distributed Hinge Model)로 구분됩니다. + +# (1) 비선형 보요소의 해석 과정 + +비선형 보요소의 해석과정은 다음과 같습니다. 다음의 해석과정에서 수렴계산과정은 포함되어 있지 않습니다. 실제 해석시에는 아래의 과정에 수렴계산과정이 추가적으로 수행되므로 주의할 필요가 있습니다. + +# ① 절점변위 계산 + +식 (57)의 비선형 정적증분방정식을 이용하여 전체구조물의 절점증분변위 벡터 U 를 구합니다. 전체좌표계에서의 증분변위벡터 U 를 요소좌표계로 변환하여 요소 양절점의 증분변위 Δu 를 구합니다. 단, 요소의 좌단을 i, 우단을 j로 나타냅니다. + +$$ +\Delta \boldsymbol {u} ^ {T} = \left\{\Delta u _ {i}, \Delta v _ {i}, \Delta w _ {i}, \Delta \theta_ {x i}, \Delta \theta_ {y i}, \Delta \theta_ {z i}, \Delta u _ {j}, \Delta v _ {j}, \Delta w _ {j}, \Delta \theta_ {x j}, \Delta \theta_ {y j}, \Delta \theta_ {z j} \right\} \tag {57} +$$ + +# ② 증분절점변위를 변형으로 변환(절대변위  상대변위) + +요소증분변위 Δu 는 증분하중에 의해 발생한 양절점의 절대변위로서 강체이동모드(Rigid-Body Mode)가 포함된 변위입니다. 요소가 강체이동하면 변위가발생해도 변형은 0이 되므로, 강체이동에 의한 내력 역시 0이 됩니다. 따라서,비선형 힌지의 내력을 계산할 때는 요소의 증분절점변위 Δu 에서 강체이동모드에 의한 변위를 제외한 변위, 즉 상대변위를 이용하여 계산할 필요가 있습니다. + +비선형 보요소의 요소증분변위 Δu 에서 강체이동모드를 제외하면, 1개의 축성분과 1개의 비틀림성분 그리고 양 절점에서 각각 2개의 변형각이 얻어집니다.각 성분별 상대변위 u 는 다음과 같이 정의할 수 있습니다. + + 축성분 + +양절점의 축방향 변위의 차이가 요소의 축방향 변형입니다. + + + +$$ +\overline {{u}} = u _ {j} - u _ {i} \tag {58} +$$ + +# ■ 비틀림성분 + +양절점의 비틀림 회전각의 차이로 구합니다. + +$$ +\overline {{\theta}} _ {x} = \theta_ {x j} - \theta_ {x i} \tag {59} +$$ + +# ■ 회전성분 + +한 절점에서 요소의 회전각은 그림 2.8.50에 나타낸 것과 같이 모멘트와 전단에 의한 변형각과 강체이동에 의한 회전각으로 구성됩니다. + +$$ +\theta = \overline {{\theta}} - \theta_ {s} \tag {60} +$$ + +여기서 θ : 절점에서의 총회전각 + +$\overline{\theta}$ : 모멘트와 전단에 의한 변형각 + +$\theta_{s}$ : 강체이동에 의한 회전각, 단, $\theta_{sy} = \frac{w_{j} - w_{i}}{L}$ , $\theta_{sz} = \frac{v_{j} - v_{i}}{L}$ + +hover성분 비선형 힌지의 내력-변형관계는 모멘트와 강체이동을 제외한 변형각으로 정의해야 하므로, 각 절점에서의 변형각은 다음과 같이 정의합니다. + +$$ +\overline {{\theta}} = \theta_ {s} + \theta \tag {61} +$$ + +![](images/page-323_843e7ca153852879c38ae31b704714de3b682db2f5e78963d9de630e2a21e0ef.jpg) + +
+text_image + +Momentz +(+) +y +Shearz +(-) +Momenty +(+) +Sheary +(-) +Axial +(-) +Momentx +(+) +x +
+ +![](images/page-323_921884861d90e0490921bd88dbd8f50684fa051ac180abe750cc58b497b8e5eb.jpg) + +
+text_image + +Momentz +(-) +y +Shearz +(+) + +Momenty +(-) +Sheary +(+) + +Axial +(+) + +Momentx +(-) +
+ +![](images/page-323_4329829aadb73d8071432f420fb644fe5512f4caf06ae7889d1715d6741d2d02.jpg) + +
+text_image + +i +L +j +w_j - w_i +(-) +y +x +w_j - w_i +(-) +w_j +L_y +w_j +w_i +w_i +θ_i +θ_i +θ_i +θ_i +θ_i +θ_i +θ_i +θ_i +θ_i +θ_i +θ_i +θ_i +θ_i +θ_i +θ_i +θ_i +θ_i +θ_i +θ_i +θ_i +θ_i +θ_i +θ_i +θ_i +θ_i +θ_i +θ_i +θ_i +θ_i +θ_i +θ_i +θ_i +θ_i +θ_j +L_y +θ_j +L_y +θ_j +θ_j +θ_j +θ_j +θ_j +θ_j +θ_j +θ_j +θ_j +θ_j +θ_j +θ_j +θ_j +θ_j +θ_j +θ_j +θ_j +θ_j +θ_j +θ_j +θ_j +θ_j +θ_j +θ_j +θ_j +θ_j +θ_j +θ_j +θ_j +θ_j +θ_j +θ_j +θ_j +θ_j +
+ +그림 2.8.50 요소의 절대변위와 상대변위와의 관계 + + + +식 (58)\~(61)의 관계를 이용하여 각 성분별 상대변위의 증분벡터 $\Delta\overline{u}$ 는 다음과 같이 얻을 수 있습니다. + +$$ +\Delta \overline {{\boldsymbol {u}}} ^ {T} = \left\{\Delta \bar {u} \quad \Delta \bar {\theta} _ {y i} \quad \Delta \bar {\theta} _ {z i} \quad \Delta \bar {\theta} _ {y j} \quad \Delta \bar {\theta} _ {z j} \mid \Delta \bar {\theta} _ {x} \right\} \tag {62} +$$ + +여기서 $\Delta\overline{u} = \Delta u_{j} - \Delta u_{i}$ + +$$ +\Delta \overline {{\theta}} _ {y i} = \frac {\Delta w _ {j} - \Delta w _ {i}}{L _ {y}} + \Delta \theta_ {y i} +$$ + +$$ +\Delta \overline {{\theta}} _ {z i} = \frac {\Delta v _ {j} - \Delta v _ {i}}{L _ {z}} + \Delta \theta_ {z i} +$$ + +$$ +\Delta \bar {\theta} _ {y j} = \frac {\Delta w _ {j} - \Delta w _ {i}}{L _ {y}} + \Delta \theta_ {y j} +$$ + +$$ +\Delta \bar {\theta} _ {z j} = \frac {\Delta v _ {j} - \Delta v _ {i}}{L _ {z}} + \Delta \theta_ {z j} +$$ + +$$ +\Delta \bar {\theta} _ {x} = \Delta \theta_ {x j} - \Delta \theta_ {x i} +$$ + +# ③ 증분변형 $\Delta\overline{u}$ 를 이용하여 증분내력 $\Delta\overline{q}$ 계산 + +비선형 한지의 증분내력 $\Delta\overline{q}$ 는 요소의 증분변형 $\Delta\overline{u}$ 에 강체이동모드(Rigid-Body Mode)를 제외한 접선강성행렬 $\overline{k}_{AB}$ 를 곱하여 구합니다. 증분내력 $\Delta\overline{q}$ 는 축성분, 비틀림성분, 전단성분의 경우에는 요소중앙, 그리고 모멘트 성분의 경우에는 양단에서의 내력입니다. + +$$ +\boldsymbol {\Delta} \overline {{\boldsymbol {q}}} _ {A B} = \overline {{\boldsymbol {k}}} _ {A B} \cdot \boldsymbol {\Delta} \overline {{\boldsymbol {u}}} \tag {63} +$$ + +여기서 $\Delta\overline{q}_{AB}^{T}=\{\Delta\overline{n}\quad\Delta\overline{m}_{yi}\quad\Delta\overline{m}_{zi}\quad\Delta\overline{m}_{yj}\quad\Delta\overline{m}_{zj}\quad\Delta\overline{m}_{x}\}$ + +: 축력, 모멘트성분의 증분내력 벡터 + +$\Delta\overline{u}$ : 비선형 흰지의 증분변형 벡터 + +$\overline{k}_{AB}$ : 강체이동모드를 제외한 접선강성행렬 + +전단성분의 증분내력 $\Delta\overline{q}_{S}$ 는 증분모멘트를 이용하여 다음과 같이 계산됩니다. + +$$ +\Delta \overline {{{\boldsymbol {q}}}} _ {S} ^ {T} = \left\{\Delta \overline {{{q}}} _ {y} \Delta \overline {{{q}}} _ {z} \right\} \tag {64} +$$ + +여기서, $\Delta\overline{q}_{y}=\frac{\Delta\overline{m}_{zi}+\Delta\overline{m}_{zj}}{L_{z}}$ , $\Delta\overline{q}_{z}=\frac{\Delta\overline{m}_{yi}+\Delta\overline{m}_{yj}}{L_{y}}$ + + + +④ 증분내력 Δq 과 힌지의 유연도를 이용하여 비선형 힌지의 증분변형계산 + +증분내력 $\Delta \overline { { q } }$ 를 비선형 힌지의 증분내력 Δq 로 변환하는 방법은 비선형요소의 종류(모멘트-회전각요소와 모멘트-곡률관계요소)에 따라서 다릅니다. 이에대해서는 해당요소에 설명되어 있습니다. + +각 성분의 비선형 힌지의 증분내력 Δq 가 구해지면 비선형 힌지의 현재상태의 유연도를 이용하여 비선형 힌지의 증분변형량 Δd 를 구합니다. + +$$ +\boldsymbol {\Delta} d = f _ {n} \cdot \boldsymbol {\Delta} q \tag {65} +$$ + +여기서 $f _ { n }$ : 비선형 힌지의 유연도( $f _ { n } = 1 / k _ { n } )$ + +⑤ 성분별 비선형 힌지의 총내력과 총변형 누적 + +성분별 비선형 힌지의 총내력과 총변형량은 직전스텝까지의 내력과 변형에현재스텝의 증분내력과 증분변형을 더하여 다음과 같이 구합니다. + +$$ +d _ {n} = d _ {n - 1} + \Delta d \tag {66} +$$ + +$$ +q _ {n} = q _ {n - 1} + \Delta q \tag {67} +$$ + +여기서 $d _ { n - 1 }$ : 비선형 힌지의 직전스텝까지의 총변형 + +$q _ { n - 1 }$ : 비선형 힌지의 직전스텝까지의 총내력 + +⑥ 비선형 힌지의 유연도와 내력의 산정 + +비선형 힌지의 유연도와 내력은 그림 2.8.51에 나타낸 것과 같이 미리 설정된골격곡선을 이용하여 다음의 과정으로 산정합니다. + +1) 직전스텝(n-1)에서 현재스텝(n)으로 이동하는 사이에 비선형 힌지의 변형$d _ { n }$ 이 항복변형 $d _ { y }$ 를 초과했는지 판정합니다. 현재스텝에서 항복변형 $d _ { y }$ 를 초과했다는 것은 비선형 힌지의 내력이 항복내력을 초과한 것으로 요소의 항복을 의미합니다. + + + +2) 비선형 힌지의 변형 $d _ { n } 0 |$ 항복변형 $d _ { y } \triangleq$ 초과한 경우, 강성 $k _ { n } \in$ 미리설정한 강성저감율에 따라서 새로운 강성 $k _ { n } ^ { * }$ 로 갱신합니다. 갱신된 강성$k _ { n } ^ { * }$ 를 이용하여 유연도 $f _ { n } ^ { * } \equivq$ 구합니다. +3) 새로운 강성 $k _ { n } ^ { * } \equiv 0$ 용하여 비선형 힌지의 내력 $q _ { n } ^ { * } \frac { \circ } { \equiv }$ 구합니다. +4) 불평형력 r 을 계산합니다. 단, $r = q _ { n } - q _ { n } ^ { * }$ + +![](images/page-326_e5ad41fbdfe71911f0398969f5b49dfa7963371a73e5553e04135a5be67f089b.jpg) + +
+line + +| Point | Hinge Deform. | Hinge Force | Annotation | +|-------|---------------|-------------|------------| +| q_n | d_{n-1} | q_n | Δq | +| q_y | d_y | q_y | Δq | +| q_{n-1}| d_{n-1} | q_{n-1} | Δd | +| k_n | d_y | k_n | Δd | +| k_n* | d_n | k_n* | r | +| q_n* | d_n | q_n* | Δd | +
+ +그림 2.8.51 비선형 힌지의 내력과 변형의 관계(골격곡선) + +# ⑦ 비선형 보요소의 요소강성과 내력계산 + +골격곡선을 통하여 얻어진 비선형 힌지의 유연도와 내력을 이용하여 요소의유연도 행렬과 요소내력을 구합니다. 요소의 강성행렬은 유연도 행렬의 역행렬로 계산됩니다. + +$$ +\boldsymbol {K} _ {n} = \boldsymbol {F} _ {n} ^ {- 1} \tag {68} +$$ + +여기서 $F _ { n }$ : 비선형 보요소의 유연도 행렬 + +$\pmb { K } _ { n }$ : 비선형 보요소의 강성 행렬 + + + +# (2) 모멘트-회전각 관계 비선형 보요소 + +횡력을 받는 골조구조물의 정적증분해석에서는 보요소에 역대칭 모멘트가 작용하므로 요소양단에 모멘트가 집중되어 소성힌지가 발생합니다. 이와 같은골조구조물의 해석에서는 탄성 보요소의 양단에 모멘트-회전각 관계로 정의되는 회전 비탄성 스프링을 설정하여 요소단에서 발생하는 소성힌지를 효과적으로 모델링한 모멘트-회전각 관계 비선형 보요소가 주로 사용됩니다. 모멘트-회전각 관계요소는 모멘트 성분의 비선형 힌지가 요소양단에 설정되기 때문에, 집중형힌지모델(Lumped Type Hinge Model)이라고도 합니다. + +#  모멘트-회전각 관계 비선형 보요소의 성분별 비선형 힌지 특성 + +모멘트-회전각 관계 비선형 보요소는 소성변형이 가능한 길이가 0인 병진또는 회전 비탄성스프링을 탄성 보요소에 삽입하며 이를 제외한 나머지 부분은 탄성 보요소로 모델링합니다. + +모멘트-회전각 관계 비선형 보요소의 비선형 힌지의 설정위치는 그림2.8.48에 나타난 것과 같이 각 성분에 따라 다릅니다. 모멘트 성분은 요소양단에 방향별로 2개씩 설정되고, 축력, 비틀림성분의 경우는 요소중앙에1개씩 설정됩니다. 또한, 전단성분은 요소중앙에 방향별로 1개씩 설정됩니다. + +그림 2.8.52에서 스프링으로 표현되는 그림은 실제적인 스프링 요소의 존재를 나타내는 것이 아니라 해석방법의 의미전달을 위한 것으로 비탄성 스프링의 위치에서 소성변형이 집중되어 발생함을 의미합니다. 모멘트-회전각 관계 비선형 보요소의 각 성분별 비선형 힌지 특징은 표 2.8.2와 같습니다. + +![](images/page-327_f8ca172fdcd79742e5529ab3f36664f027180d6778b5e0ee14486178edc207d4.jpg) + +
+text_image + +Flexural spring +Torsional spring +Shear spring +
+ +그림 2.8.52 모멘트-회전각 관계요소의 비선형 힌지 위치 + + + +\*. 단, Masonry Type인 경우, 축력, 전단성분 힌지는 요소중앙에 위치 + +
성 분비선형 히지 특성초기강성힌지의설정위치
축력(Fx)축력-변형(상대변위)EA/L요소양단
전단력(Fy,Fz)전단력-전단변형율GAs
비틀림(Mx)모멘트-회전각GJ/L
모멘트(My,Mz)모멘트-회전각6EI/L3EI/L2EI/L요소양단
+ +표 2.8.2 모멘트-회전각 관계요소의 성분별 비선형 힌지 특성 + +#  모멘트-회전각 관계 비선형 보요소의 유연도 행렬 + +모멘트-회전각 관계 비선형 보요소의 요소 유연도 행렬은 비탄성 스프링의유연도 행렬과 탄성보의 유연도 행렬을 더해서 구성됩니다. 이 때 비탄성스프링의 유연도는 사용자가 정의한 집중형 힌지의 접선 유연도와 초기 유연도의 차이로 정의되며 요소가 항복하기 전에는 0입니다. 비선형 힌지의접선 유연도 행렬은 일축(Single Component) 또는 다축-힌지(P-M-M) 모델에 의거한 상태판정으로부터 결정됩니다. + +모멘트-회전각 관계 비선형 보요소의 해석과정은 다음과 같습니다. + +① (1)비선형 보요소의 해석과정의 ①\~⑥의 과정을 통하여 비선형 힌지의유연도와 내력을 산정합니다. 단, 모멘트-회전각 관계 비선형 보요소는증분내력 Δq 를 구한 지점에 비선형 힌지가 위치하므로, Δq 를 비선형힌지의 증분내력Δq 로 그대로 사용합니다. +② 그림 2.8.53(a)의 골격곡선을 통하여 얻어진 각 성분별 비선형 힌지의 유연도는 초기상태의 유연도와 비탄성스프링의 유연도로 구분하여 나타낼수 있습니다. + +$$ +f _ {n} = f _ {0} + f _ {s p r}; \frac {1}{k _ {n}} = \frac {1}{k _ {0}} + \frac {1}{k _ {s p r}} \tag {69a} +$$ + +$$ +d _ {n} = d _ {e l} + d _ {s p r} \tag {69b} +$$ + + + +여기서 $f _ { n }$ : 골격곡선을 통하여 얻어진 비선형 힌지의 유연도 + +$$ +f _ {0}: \text { 초기 유연도 } +$$ + +$$ +f _ {s p r}: \text { 비탄성 스프링의 유연도 } +$$ + +$$ +d _ {n}: \text { 비선형 흰지의 변형 } +$$ + +$$ +d _ {e l}: \text { 탄성변형 } +$$ + +$$ +d _ {s p r}: \text { 비탄성 스프링의 소성변형 } +$$ + +![](images/page-329_015f5eb9e1adc86462f2d4958b7d636406d6b0115bcefe3ffb7882a6a982b85a.jpg) + +
+line +| Hinge Deform. | Hinge Force | +| ------------- | ----------- | +| 0 | 0 | +| d_el | d_el | +| d_spr | d_spr | +| k_n = 1/f_n | 1 | +
+ +Flexibility &Deformation of Inelastic Hinge based on Skeleton Curve + +![](images/page-329_e7bf0381c386e9d56d16feda23845ac2cc46d7ed90354086f770b4255f99d959.jpg) + +
+line + +| Hinge Deform | Hinge Force | +| ------------ | ----------- | +| 0 | 0 | +| d_el | d_el | +| k₀ | 1/f₀ | +
+ +Initial Flexibility& Elastic Deformation of Inelastic Hinge + +![](images/page-329_c87c9da9c7613642e4e131b5b982944d5a93c9ea78ddbfc7cd60460625235e27.jpg) + +
+line + +| Hinge Deform | Hinge Force | +| ------------ | ----------- | +| Low | Low | +| Medium | Medium | +| High | High | +
+ +Flexibility & Deformation of Inelastic Spring +(a) +(b) +그림 2.8.53 비선형 힌지의 유연도 + +③ 비탄성 스프링의 유연도는 다음과 같이 나타낼 수 있습니다. + +$$ +f _ {s p r} = f _ {n} - f _ {0} \tag {70} +$$ + +④ 전체 비선형 보요소의 유연도행렬은 탄성보의 유연도 행렬에 비탄성 스프링의 유연도 행렬을 더해서 구합니다. + +$$ +\boldsymbol {F} _ {n} = \boldsymbol {F} _ {0} + \sum f _ {s p r} \tag {71} +$$ + +여기서 $F _ { n }$ : 비선형 보요소의 유연도 행렬 + +$$ +\boldsymbol {F} _ {0} \quad : \text { 탄성보의 유연도 행렬 } +$$ + +$$ +\sum f _ {s p r}: \text { 비탄성 스프링의 유연도 행렬(단, 탄성상태에서는 0) } +$$ + +비탄성 스프링은 비선형 힌지가 항복내력에 도달한 시점에서 발생하므로탄성범위에서 유연도는 0이 되어, 비선형 힌지가 항복하기 전에는 보요 + + + +소의 유연도 행렬은 탄성보의 유연도와 같습니다. 따라서, 사용자가 설정한 비선형 힌지의 초기강성은 힌지가 항복하기 전에는 해석결과에 영향을 미치지 않음에 주의할 필요가 있습니다. + +⑤ 모멘트-회전각 관계 비선형 보요소의 강성행렬은 비선형 보요소의 유연도 행렬의 역행렬을 취하여 구합니다. + +\- 모멘트-회전각 관계 비선형 보요소의 모멘트 성분 비선형 힌지의 초기강성 휩 변형 힌지의 모멘트-회전각 관계는 단부의 휩 모멘트 뿐만 아니라 부재 중간의 휩 모멘트 분포에 의해서도 영향을 받습니다. 따라서 휩 변형 힌지의 모멘트-회전각 관계를 결정하기 위해서는 휩 모멘트의 분포를 가정할 필요가 있습니다. 일반적으로 그림 2.8.54 \~ 그림 2.8.56과 같이 모멘트가 작용하는 단순보를 기준으로 하여 모멘트 분포의 가정에 의해 초기유연도를 정의하고 초기강성을 설정합니다. + +① 직선분포로 가정된 휨모멘트의 양단값이 크기가 같고 방향이 반대인 경우 + +![](images/page-330_2a3793b56c0b420fd5903aed02108d688469ce8d8fc2627554ce69231cd6b905.jpg) + +
+text_image + +M_a \theta_a \rightarrow M_b +\theta_b +
+ +(a) Deflection Shape + +![](images/page-330_801a0502a47a8e6b450662a25a91bdcdd773a9ac9b98cae3b7fbc024b8c04995.jpg) +(b) Moment Distribution +그림 2.8.54 역대칭 모멘트를 받는 단순보의 변형상태 + +2차원 탄성 보요소의 힘-변위의 관계는 다음과 같이 표현됩니다. + +$$ +\left\{ \begin{array}{l} V _ {a} \\ M _ {a} \\ V _ {b} \\ M _ {b} \end{array} \right\} = \frac {E I}{L ^ {3}} \left[ \begin{array}{c c c c} 1 2 & 6 L & - 1 2 & 6 L \\ 6 L & 4 L ^ {2} & - 6 L & 2 L ^ {2} \\ - 1 2 & - 6 L & 1 2 & - 6 L \\ 6 L & 2 L ^ {2} & - 6 L & 4 L ^ {2} \end{array} \right] \left\{ \begin{array}{l} v _ {a} \\ \theta_ {a} \\ v _ {b} \\ \theta_ {b} \end{array} \right\} \tag {72} +$$ + +그림 2.8.54의 경우, $v_{a} = v_{b} = 0$ 이고 $\theta_{a} = \theta_{b}$ 가 되므로, 웃 식는 다음과 같이 표현됩니다. diff --git a/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_034.md b/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_034.md new file mode 100644 index 00000000..5c6c389b --- /dev/null +++ b/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_034.md @@ -0,0 +1,456 @@ + + +$$ +\left\{ \begin{array}{l} M _ {a} \\ M _ {b} \end{array} \right\} = \frac {E I}{L ^ {3}} \left[ \begin{array}{l l} 4 L ^ {2} & 2 L ^ {2} \\ 2 L ^ {2} & 4 L ^ {2} \end{array} \right] \left\{ \begin{array}{l} \theta_ {a} \\ \theta_ {b} \end{array} \right\} \tag {73a} +$$ + +따라서, + +$$ +\left\{ \begin{array}{l} \theta_ {a} \\ \theta_ {b} \end{array} \right\} = \frac {L}{E I} \left[ \begin{array}{c c} \frac {1}{3} & - \frac {1}{6} \\ - \frac {1}{6} & \frac {1}{3} \end{array} \right] \left\{ \begin{array}{l} M _ {a} \\ M _ {b} \end{array} \right\} \tag {73b} +$$ + +여기서, $\theta_{a}=\theta_{b}$ , $M_{a}=M_{b}$ 이므로, $\theta_{a}=\frac{L}{6EI}M_{a}$ , $\theta_{b}=\frac{L}{6EI}M_{b}$ 로 나타 낼 수 있습니다. + +따라서, 역대칭 모멘트를 받는 보요소의 모멘트성분의 초기유연도와 강성은 다음과 같이 정의합니다. + +$$ +f _ {0} = \frac {L}{6 E I}, \quad k _ {0} = \frac {6 E I}{L} \tag {73c} +$$ + +# ② 한쪽 단부에만 모멘트가 작용하는 경우 + +![](images/page-331_28630419f8b02ca45cd542ff788071918160c6b3ac0a7ed8aea4bb57244628b1.jpg) +(a) Deflection Shape + +![](images/page-331_8ace83d7949e82d4d6a6f94347db3c34247f2f5ab288cf1b4eb602a95344dd09.jpg) +(b) Moment Distribution +그림 2.8.55 한쪽 단부에만 모멘트를 받는 단순보의 변형상태( $M_{b}=0$ ) + +그림 2.8.55의 경우, $v_{a} = v_{b} = 0$ 이고 $M_{b} = 0$ 가 되므로, 식(72)는 다음과 같이 표현할 수 있습니다. + +$$ +\left\{ \begin{array}{c} M _ {a} \\ 0 \end{array} \right\} = \frac {E I}{L ^ {3}} \left[ \begin{array}{c c} 4 L ^ {2} & 2 L ^ {2} \\ 2 L ^ {2} & 4 L ^ {2} \end{array} \right] \left\{ \begin{array}{c} \theta_ {a} \\ \theta_ {b} \end{array} \right\} \tag {74a} +$$ + +따라서, + +$$ +\left\{ \begin{array}{l} \theta_ {a} \\ \theta_ {b} \end{array} \right\} = \frac {L}{E I} \left[ \begin{array}{c c} \frac {1}{3} & - \frac {1}{6} \\ - \frac {1}{6} & \frac {1}{3} \end{array} \right] \left\{ \begin{array}{l} M _ {a} \\ 0 \end{array} \right\} \tag {74b} +$$ + + + +$$ +\theta_ {a} = \frac {L}{3 E I} M _ {a} \text { 로 나타낼 수 있습니다. } +$$ + +즉, 한쪽 단부에만 모멘트를 받는 보요소에 모멘트성분의 초기유연도와 강성은 다음과 같이 정의합니다. + +$$ +f _ {0} = \frac {L}{3 E I}, \quad k _ {0} = \frac {3 E I}{L} \tag {74c} +$$ + +③ 양단 모멘트의 크기 및 부호가 모두 같은 경우( $M_{b} = -M_{a}$ ) + +![](images/page-332_f0cb2ab270320ebcd3bd74e870981919b5df135493c2fd2950f237813dc040cb.jpg) +(a) Deflection Shape + +![](images/page-332_24f1afdb546af72be16fbcfe6022274d311dde9e40a41d00d8ce634343f1a721.jpg) +(b) Moment Distribution +그림 2.8.56 양단 모멘트의 크기와 부호가 같은 경우의 단순보의 변형상태 + +그림 2.8.56의 경우, $v_{a} = v_{b} = 0$ 이고 $M_{b} = -M_{a}$ 가 되므로, 식(72)는 다음과 같이 표현 할 수 있습니다. + +$$ +\left\{ \begin{array}{l} M _ {a} \\ M _ {b} \end{array} \right\} = \frac {E I}{L ^ {3}} \left[ \begin{array}{l l} 4 L ^ {2} & 2 L ^ {2} \\ 2 L ^ {2} & 4 L ^ {2} \end{array} \right] \left\{ \begin{array}{l} \theta_ {a} \\ \theta_ {b} \end{array} \right\} \tag {75a} +$$ + +따라서, + +$$ +\left\{ \begin{array}{l} \theta_ {a} \\ \theta_ {b} \end{array} \right\} = \frac {L}{E I} \left[ \begin{array}{c c} \frac {1}{3} & - \frac {1}{6} \\ - \frac {1}{6} & \frac {1}{3} \end{array} \right] \left\{ \begin{array}{l} M _ {a} \\ M _ {b} \end{array} \right\} \tag {75b} +$$ + +즉, 양쪽 모멘트의 크기와 부호가 같은 경우, 모멘트성분의 초기유연도와 강성은 다음과 같이 정의합니다. + +$$ +f _ {0} = \frac {L}{2 E I}, \quad k _ {0} = \frac {2 E I}{L} \tag {75c} +$$ + +# (3) 모멘트-곡률 관계 비선형 보요소 + +모멘트-곡률 관계 비선형 보요소는 요소 내에 여러개의 비선형 흰지를 설정합니다. 설정한 각 흰지 위치에서 탄소성 여부를 판단하여 흰지의 유연도를 계산한 후에 수치적분을 통하여 요소의 유연도행렬를 구성하고, 요소강성행렬 + + + +을 구합니다. 모멘트-곡률 관계 비선형 보요소의 비선형 힌지는 축성분의 경우 축력-변형율 관계로 정의하고, 모멘트성분은 모멘트-곡률 관계로 정의합니다. + +보-기둥요소의 비탄성 거동은 주로 요소 단부에 집중되는 경우가 많습니다.그러나, 일반적인 수치적분법으로 널리 사용되는 Gauss-Legendre적분법은 요소단부에 적분점을 설정할 수 없습니다. 따라서, midas Civil에서는 요소 단부에 적분점을 취할 수 있는 Gauss-Lobatto 수치적분법을 사용합니다. + +#  모멘트-곡률관계 비선형 보요소의 성분별 비선형 힌지 특성 + +midas Civil의 정적증분해석의 모멘트-곡률관계 비선형 보요소는 요소전체의 소성화를 고려할 수 있는 분포형모델(Distributed Type)과 요소양단에서의 소성화만을 고려하는 집중형모델(Lumped Type)을 제공합니다. + +모멘트-곡률관계 비선형 보요소의 분포형모델(Distributed Type)은 요소양단에소성힌지의 길이를 정의할 수 있습니다. 힌지의 분포를 요소전체(Entirety)로 선택하면, 요소전체의 소성화를 고려합니다. 요소양단에 소성힌지의 길이를 정의하면, 항복시에 힌지길이만 소성화하며 요소내부는 선형탄성으로처리합니다. 힌지의 분포는 각 성분별로 정의 가능합니다. + +① Moment- Curvature Distributed Type + + Hinge Distribution : Entirety Type + +- 비선형 힌지 : 요소전체에 분포하는 적분점으로 설정(1\~20개 설정가능) +- 요소전체의 소성화 고려가능 +- 성분별로 적분점의 개수를 각각 설정가능 + Hinge Distribution : I-End, J-End, I&J-End +- 비선형 힌지 : 요소양단에 위치한 소성힌지길이로 설정(각1적분점) +- 힌지길이만 소성화되며, 내부는 선형탄성으로 처리 +- 성분별로 힌지분포 설정가능 + +② Lumped Type + +- 비선형 힌지의 위치 : 요소양단(모멘트), 요소중앙(축, 전단, 비틀림) +- 요소양단의 소성화만 고려가능 +- 모멘트성분 비선형 힌지 : 요소에 3개의 적분점 설정(단, 중앙의 적분점은 탄성) + + + +\- 축, 전단, 비틀림 : 요소중앙에 1개의 적분점 설정 + +![](images/page-334_cc5fedae066b263fe032f429c870ec92bbfcbe6954ccacd9fdb11fc9c9b15274.jpg) + +
+text_image + +M +M +Rigid +Zone +Integration Point +(Inelastic Hinge) +Rigid +Zone +
+ +(a) Distributed Type + +![](images/page-334_2c69040e796e7dcb08b9a60479049d40b0acc861a53f069b51cc42f9543f0568.jpg) + +
+text_image + +M +Inelastic Hinge +M +Rigid +Zone +Integration Point +• Elastic Hinge +Rigid +Zone +
+ +(b) Lumped Type(모멘트성분) +그림 2.8.57 모멘트-곡률관계 비선형 보요소의 비선형 힌지 + +③ 소성힌지의 길이(Hinge Length) + +- Moment- Curvature Distributed Type만 정의가능 +- 요소양단에 설정 +- 소성힌지길이로 정의된 이외의 영역은 선형탄성으로 처리 +- 소성힌지길이는 요소전체길이에 대한 비율로 정의(0 < Lp/L < 1.0) +단, 양단의 소성힌지길이의 합은 요소전체의 길이를 넘을수 없습니다. +(LpI/L + LpJ/L ≤ 1.0)- 요소양단에 설정 +- Beam End Offset 등이 설정한 경우, 요소전체길이는 Offset을 제외한 순스팬 길이로 간주하여 처리됩니다. +- 소성힌지길이가 정의된 경우에는 Modified Two-Point Gauss-Radau 수치적분법에 의해 요소강성을 구성합니다. + +![](images/page-334_a6c1badcee55eb65550394f54d30ae4a8eb6e1c4a905c8744649792eb8360e58.jpg) + +
+text_image + +L_{pI}/L +Linear Elastic +0 +1.0 +
+ +(a) I-End + +![](images/page-334_40115e54d4ea6cf2f1a4bdcc96fba9b7182a359565dd6141c2f13c00b68fcc32.jpg) + +
+text_image + +Linear Elastic +0 +1.0 +L_{pJ}/L +
+ +(b) J-End +![](images/page-334_7a7dc0f1702ab98fb4692fa08ced4aaace161c47a483e6387e52270bf5811dc4.jpg) + +
+text_image + +L_{pI}/L +0 +Linear Elastic +L_{pJ}/L +1.0 +j +• Integration Point +(Inelastic Hinge) +
+ +(c) I&J-End +그림 2.8.58 모멘트-곡률관계 비선형 보요소의 소성길이 + + + +모멘트-곡률 관계 비선형 보요소의 각 성분별 비선형 힌지 특징은 표 2.8.3과같습니다. + +
성 분비선형 힌지 특성초기강성힌지의 설정위치(Lumped/Distributed)
축력(Fx)축력-변형율EA요소중앙 / 적분점 위치
전단력(Fy,Fz)전단력-전단변형율GAs요소중앙 / 적분점 위치
비틀림(Mx)모멘트-곡률GJ요소중앙 / 적분점 위치
모멘트(My,Mz)모멘트-곡률EI요소양단 / 적분점 위치
+ +표 2.8.3 모멘트-곡률관계요소의 성분별 비선형 힌지 특성 + +#  모멘트-곡률 관계 비선형 보요소의 유연도 행렬 + +모멘트-곡률 관계 비선형 보요소의 요소 유연도 행렬은 각 적분점에 위치하는 비선형 힌지의 유연도를 수치적분하여 구합니다. 비선성힌지의 접선유연도 행렬은 일축(Single Component) 또는 다축-힌지(P-M-M) 모델에 의거한 상태판정으로부터 결정됩니다. + +모멘트-곡률 관계 비선형 보요소의 해석과정은 다음과 같습니다. + +① (1)비선형 보요소의 해석과정의 ①\~③의 과정을 통하여 요소의 증분내력Δq 를 구합니다. 각 적분점에 위치한 비선형 힌지의 증분내력 Δq( ) x는 증분내력 Δq 를 내삽함수(Force Interpolation Function)를 이용하여다음과 같이 변환하여 구합니다. + +\- 축력과 모멘트성분의 비선형 힌지의 증분내력 + +$$ +\boldsymbol {\Delta} \boldsymbol {q} _ {A B} (x) = \mathbf {b} (x) \cdot \boldsymbol {\Delta} \overline {{\boldsymbol {q}}} _ {A B} \tag {76} +$$ + +여기서   TAB yi zi yj zjΔq =      n m m m m : 요소의 증분내력 + +$$ +\Delta \boldsymbol {q} _ {A B} (x) ^ {T} = \left\{\Delta n _ {\text { sec }} \quad \Delta m _ {y, \text { sec }} \quad \Delta m _ {z, \text { sec }} \right\}: \text { 비선형 히지의 증분내력 } +$$ + + + +$$ +\mathbf {b} (x) = \left[ \begin{array}{c c c c c} 1 & 0 & 0 & 0 & 0 \\ 0 & \xi - 1 & 0 & \xi & 0 \\ 0 & 0 & \xi - 1 & 0 & \xi \end{array} \right], \quad \xi = \frac {x}{L}: \text { 内 沿 求 求 } +$$ + +-전단성분의 비선형 힌지의 증분내력 + +$$ +\Delta q _ {y, \mathrm{sec}} = \frac {\Delta \bar {m} _ {z i} + \Delta \bar {m} _ {z j}}{L _ {z}} +$$ + +$$ +\Delta q _ {z, \text {sec}} = \frac {\Delta \bar {m} _ {y i} + \Delta \bar {m} _ {y j}}{L _ {y}} \tag {77} +$$ + +$$ +\Delta m _ {x, \mathrm{sec}} = \Delta \overline {{m}} _ {x} +$$ + +여기서 $\Delta\overline{m}_{yi},\Delta\overline{m}_{zi},\Delta\overline{m}_{yj},\Delta\overline{m}_{zj}$ : 요소 양단의 증분모멘트 + +$\Delta q_{y,\sec} \Delta q_{z,\sec}$ : 전단성분의 비선형 힌지의 증분내력 + +$\Delta m_{x,\sec}$ : 비틀림성분의 비선형 힌지의 증분내력 + +② 비선형 보요소의 해석과정의 ④\~⑥의 과정을 통하여 비선형 한지의 유연도 $f(x)$ 와 내력을 산정합니다. + +③ 각 적분점에서 구한 성분별 유연도 $f(x)$ 를 수치적분하여 보요소의 유연도행렬을 구성합니다. + +$$ +\boldsymbol {F} = \int_ {0} ^ {L} b ^ {T} (x) f (x) b (x) d x \tag {78} +$$ + +여기서 $f(x)$ : 위치 x 에서의 단면의 유연도 행렬 + +$b(x)$ : 위치 x 에서의 부재력 분포 함수 행렬(내삽함수) + +F : 요소 유연도 행렬 + +L : 요소 길이 + +x : 단면의 위치 + +④ 모멘트-곡률관계 비선형 보요소의 강성행렬은 비선형 보요소의 유연도 행렬의 역행렬을 취하여 구합니다. + + + +# 트러스요소 + +트러스요소는 그림 2.8.59와 같이 부재 축방향(x방향)의 압축력 및 인장력을 받을수 있는 비선형 스프링을 사용합니다. + +![](images/page-337_42cdde59a8307f3e3817256742dfbf0c94ce6fe885a7c9088b8637d4d67d5c97.jpg) + +
+text_image + +1 +F_{x1} +2 +F_{x2} +Axial spring +x +
+ +그림 2.8.59 트러스요소의 절점력 + +트러스 요소의 비선형 힌지 특징은 표 2.8.4와 같습니다. + +
성 분비선형 힌지 특성초기강성힌지의 설정위치
축력(Fx)축력-변형(상대변위)EA/L요소중앙
+ +표 2.8.4 트러스요소의 비선형 힌지 특성 + +# 비선형 범용연결 요소 + +범용연결요소(General Link)는 두 절점을 연결하는 요소로서, 세 방향의 신장 및회전을 가지는 6개의 스프링으로 구성됩니다. 정적증분해석에서는 General LinkProperties에서 Spring Type으로 설정한 후에 Pushover Hinge Properties를 할당하여 범용연결요소의 비선형특성을 정의합니다. + +비선형 범용연결요소의 각 성분별 비선형 힌지 특징은 표 2.8.5와 같습니다. + +
성 분비선형 힌지 특성초기강성힌지의 설정위치
축력(Fx)축력-변형사용자정의(EA/L)요소중앙
전단력(Fy,Fz)전단력-변형사용자정의(GAs/L)요소중앙
비틀림(Mx)모멘트-회전각사용자정의(GJ/L)요소중앙
모멘트(My,Mz)모멘트-회전각사용자정의(EI/L)요소중앙
+ +표 2.8.5 비선형 범용 연결 요소의 성분별 비선형 힌지 특성 + + + +# 8-7-8 비선형 힌지 특성 + +midas Civil의 정적증분해석은 요소에 비선형 힌지를 설정하여 힌지의 변형과 그로인한 내력으로 비선형 힌지의 항복상태를 판정합니다. 비선형 힌지 특성은 각 성분이 독립적으로 거동하는 일축-힌지모델(Single Component Type)과 축력-모멘트성분의 상호작용을 고려하는 다축-힌지모델(P-M-M Type)로 구분할 수 있습니다. 비선형 힌지는 골격곡선(Skeleton Curve)에 의해 정의됩니다. 골격곡선은 해석용 최소모델 단위인 요소단면에서의 구성재료의 응력-변형율관계, 단면의 모멘트-곡률관계, 양단의 모멘트-회전각 관계 등의 비선형 거동특성을 이상화된 곡선으로 표현한것입니다. + +비선형 힌지의 내력과 변형의 관계, 즉 골격곡선(Skeleton Curve)상의 힘과 변형의관계는 비선형 요소의 성분별 비선형 힌지 특성을 나타낸 표 2.8.2\~2.8.5를 참고하시기 바랍니다. + +# Skeleton Curve의 개요 + +midas Civil의 정적증분해석에서 제공하는 모든 골격곡선은 하중증분법(LoadControl)과 변위증분법(Displacement Control)에 모두 사용가능하며, 접선강성 행렬(Tangent Stiffness Matrix)을 사용합니다. + +# (1) Bilinear Type + +대응요소 : 보요소, 벽요소, 트러스, 범용 연결요소 + +힌지특성 : 일축-힌지 및 다축-힌지 정의 가능 + +골격곡선의 초기강성 k : (+), (-)방향 대칭으로만 설정가능 + +# (2) Trilinear Type + +대응요소 : 보요소, 벽요소, 트러스, 범용 연결요소 + +힌지특성 : 일축-힌지 및 다축-힌지 정의 가능 + +골격곡선의 초기강성 k : (+), (-)방향 대칭으로만 설정가능 + +# (3) FEMA Type + +대응요소 : 모멘트-회전각관계 보요소, 벽요소, 트러스, 범용연결요소 + +힌지특성 : 일축-힌지 및 다축-힌지 정의 가능 + + + +골격곡선의 초기강성 $k _ { 0 }$ : (+), (-)방향 대칭으로만 설정가능 + +# (4) Slip Type + +대응요소 : 트러스, 범용 연결요소 + +힌지특성 : 일축-힌지정의 + +초기 Gap 설정가능 + +# Multi-Linear Hinge Type : Bilinear , Trilinear + +Multi-Linear 힌지특성은 하중제어와 변위제어 해석에서 모두 적용될 수 있습니다. + + 하중과 변형관계는 Bilinear와 Trilinear의 두 가지 형식으로 정의 가능함 + 항복 후 강성과 균열강성은 초기강성에 대한 강성비(Stiffness Ratio)로써특성을 표현함 + 요소의 강성감소는 표현되지만 강도저하(부구배)는 표현할 수 없음 + +![](images/page-339_890041ccc85cc9bfb8cb4b4d2f753e1c0c91603a4976f46af07e10dc8df39c3a.jpg) + +
+text_image + +Force +P1(+) +α1(+) +K0 +K0 +Deform. +K0: Ini. Stiff. +α1(-) +K0 +P1(-) +
+ +(a) Bilinear Type + +![](images/page-339_c27a71a11696860db98107052ca1462c995a245eb2272fd6d0056dd5ea6254a1.jpg) + +
+text_image + +Force +P1(+) +P1(+) +α2(+) +K0 +α1(+) +K0 +K0 +Deform. +K0: Ini. Stiff. +α1(-) +K0 +P1(-) +α2(-) +K0 +P2(-) +
+ +(b) Trilinear Type +그림 2.8.60 Multi-Linear Hinge Type을 이용한 소성힌지 특성 + + + +# FEMA Hinge Type + +FEMA 힌지특성은 철근콘크리트 부재와 철골부재에 대하여 반복하중(ReversedCyclic Load)실험을 통해 저항능력을 평가한 후에 실무에 적용할 수 있도록 이상화(Idealized)한 것으로 아래 그림과 같이 나타내고 있습니다. + +midas Civil의 FEMA 힌지특성은 변위증분해석과 하중증분해석에 적용가능합니다.하중증분해석에서 일부 요소가 파괴되어 점 C이후가 되더라도, 전체구조물의 내력이 감소하지 않는다면 해석이 진행됩니다. 단, 요소의 파괴가 점차 진행되어 구조물의 전체내력이 감소하는 구간 이후에는 안정해를 구할 수 없기 때문에, midasCivil에서 제공하는 자동종료조건(Current Stiffness Ratio)에 의해 강제종료 됩니다. + +![](images/page-340_c64fb6e1f013c4ebc05616d9149dc2006fd678a8d58d38cb1dad409e2738d30d.jpg) + +
+flowchart + +```mermaid +graph TD + A["Initial Stiffness"] --> B["Yield Point"] + B --> C["Strain Hardening"] + C --> D["Initial Failure"] + D --> E["Residual Resistance"] + E --> D + style A fill:#f9f,stroke:#333 + style B fill:#f9f,stroke:#333 + style C fill:#f9f,stroke:#333 + style D fill:#f9f,stroke:#333 + style E fill:#f9f,stroke:#333 +``` +
+ +그림 2.8.61 FEMA Hinge Type을 이용한 소성힌지 특성 + +- Point A: 하중이 재하되지 않은 상태 +- Slop A-B: 부재의 초기강성(Initial Stiffness) 상태 구간, 재료특성, 부재치수, 철근량, 경계조건, 응력과 변형수준에 따라 결정 +- Point B: 공칭항복강도(Nominal Yield Strength) 상태 +- Slop B-C: 변형경화(Strain Hardening) 구간, 일반적으로 초기강성의 5-10%를가지며 인접한 부재와의 내력 재분배에 중요한 영향을 미침 +- Point C: 공칭강도(Nominal Strength), 부재내력에서 강도저하가 시작되는 시점 +- Drop C-D: 부재의 초기파괴(Initial Failure)상태, 철근콘크리트 부재의 경우에 주근이 파단(Fracture)되거나 콘크리트가 파손(Spalling)되는 상태, 철골부재의 경우 전단내력이 급격하게 감소 +- Zero D-E: 잔류저항(Residual Resitance) 상태, 공칭강도의 20% 수준에서 저항 +- Point E: 최대변형능력, 중력하중을 더 이상 받을 수 없는 상태 diff --git a/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_035.md b/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_035.md new file mode 100644 index 00000000..61f1044d --- /dev/null +++ b/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_035.md @@ -0,0 +1,309 @@ + + +# 다축-힌지 모델 : P-M-M Type + +다축-힌지 모델은 축력과 2축의 모멘트를 받는 기둥부재의 모델링에 주로 사용되는 힌지모델입니다. 다축-힌지 모델의 축력-모멘트의 관계는 항복곡면에 의해 정의되며, 축력의 변동에 따라서 항복모멘트를 산정합니다. + +![](images/page-341_368f195d8a7a4f1a730def9f463be15004f6a2c491dd2116142030657a316c7f.jpg) + +
+line + +| M | P(compression) | P(tension) | +|-------|---------------|----------| +| PC0(t)| P_max | P(tension)| +| MC0 | P_max | P(tension)| +| MY0 | P_max | P(tension)| +| MY,max| P(tension) | P(tension)| +
+ +(a) RC TYPE(Trilinear) + +![](images/page-341_b882878fa1b605a129a5c942ba40bc698c63bde543eab3e0168ec3e5adbc7162.jpg) + +
+line + +| Point | P(tension) | P(compression) | +|-------|------------|----------------| +| M | max | max | +
+ +(b) Steel Type(Bilinear) +그림 2.8.62 P-M-M Type 힌지의 항복곡면 + +2방향 모멘트 및 축력을 받는 경우에는 주어진 축력에 대한 각 축 방향의 항복 모멘트를 구한 후 다음과 같은 관계식을 사용합니다. + +![](images/page-341_6c5b4d82cdd0212a3e873bb02be3368bc79cb3fd40622de6a944eec580a9199b.jpg) + +
+text_image + +MY_z +MY'_z +M_z +M_y MY'_y MY_y +MY_y +
+ +$$ +\left(\frac {M Y _ {y} ^ {\prime}}{M Y _ {y}}\right) ^ {\alpha} + \left(\frac {M Y _ {z} ^ {\prime}}{M Y _ {z}}\right) ^ {\alpha} = 1. 0 \tag {79} +$$ + + + +α 는 1.0\~2.0값을 설정하며, 식(79)는 콘크리트와 철골부재에 모두 사용합니다. 단, H형강인 경우는 강축인 경우, α =2.0, 약축인 경우 α =1.0값을 채용합니다. 식(80)는 y축이 강축인 경우의 2방향 모멘트상관관계를 나타냅니다. + +$$ +\left(\frac {M Y _ {y} ^ {\prime}}{M Y _ {y}}\right) ^ {2. 0} + \left(\frac {M Y _ {z} ^ {\prime}}{M Y _ {z}}\right) ^ {1. 0} = 1. 0 \tag {80} +$$ + +# RC 부재의 2차 구배 강성저감율 + +RC 부재를 모멘트-회전각 관계요소로 정의하고, 모멘트성분에 Trilinear Type의 골격곡선을 정의한 경우, 균열 후의 강성은 항복시 강성저감율 $\alpha_{y}$ 를 이용하여 자동계산 됩니다. + +항복시 강성저감율 $\alpha_{y}$ 는 그림 2.8.63과 같이 표현되며, 2차구배의 강성저감율 $\alpha_{1}$ 는 식(81)로 구할 수 있습니다. + +![](images/page-342_dee87500b593a08f83000b2dd080d91d57dc9a278ff52bdbf41597bc68bfb3c3.jpg) + +
+text_image + +M +M_y +M_c +α_1·k_0 +α_2·k_0 +k_0 +α_y·k_0 +θ_c +θ_y +θ +
+ +그림 2.8.63 RC부재의 균열후의 강성과 ay와의 관계 + +$$ +\alpha_ {y} = \left(0. 0 4 3 + 1. 6 4 n p _ {t} + 0. 0 4 3 \frac {a}{D} + 0. 3 3 \eta_ {0}\right) \left(\frac {d}{D}\right) ^ {2} \tag {81} +$$ + +(일본건축학회 [철근콘크리트구조설계규준·동해석]) + +$$ +\alpha_ {1} = \frac {M _ {y} - M _ {c}}{\frac {M _ {y}}{\alpha_ {y}} - M _ {c}} \tag {82} +$$ + + + +다축-힌지(PMM Type)의 경우, 축력변동을 고려하여 항복모멘트를 강성하기 때문에강성저감율 $\alpha _ { y }$ 의 계산시에도 축력변동의 영향을 고려할 필요가 있습니다. PMMType에서 $\alpha _ { y }$ 의 계산방법은 다음과 같습니다. + +1. 항복곡면상에서 현재 스텝에서의 부재축력 $P _ { i }$ 과 균열면과의 교차점인 균열모멘트를 산정합니다. +2. 초기하중(장기하중)에 대한 부재축력 부재축력 $P _ { 0 }$ 과 점a를 지나는 직선을항복면까지 연장하여 점b를 구합니다. +3. 점b의 모멘트를 예측항복모멘트로 하여, $\alpha _ { y }$ 산정용 축력 $P _ { \alpha y } { \ttqqless }$ 구합니다. + +![](images/page-343_1a83fc7c6cc1b9244b90343adf6a61eab07f950e2883e8f757dc5b74b537d954.jpg) + +
+line + +| Point | P(tension) | P(compression) | +|-------|------------|----------------| +| a | MC | P0 | +| b | MY | Pax | +| c | MC | Pax | +
+ +그림 2.8.64 PMM TYPE의 ay계산시의 축력산정 + + + +# 8-7-9 성능점을 이용한 내진성능평가 + +midas Civil에서는 기본적으로 능력스펙트럼법(CSM)의 원리를 이용하여 구조물의보유내력과 내진성능을 평가합니다. 구조물의 보유내력은 Pushover 해석을 이용하여 능력곡선과 능력스펙트럼을 산정하여 평가할 수 있습니다. 그리고 지진하중에대한 요구스펙트럼은 유효감쇠 원리가 적용된 탄성설계스펙트럼을 이용하여 평가할 수 있습니다. 이 두 가지 스펙트럼을 하나의 좌표계로 표현하면 교차점이 발생하며 이 교차점이 바로 구조물의 비선형 최대 요구내력을 의미하는 성능점(Performance Point)으로 결정됩니다. 성능점에서의 변형정도와 보유내력을 이용하여 구조물이 보유하고 있는 내진성능과 성능수준을 평가할 수 있습니다. + +# 능력스펙트럼과 요구스펙트럼 + +구조물의 내진성능과 성능수준을 평가하기 위해 능력스펙트럼(Capacity Spectrum)과 요구스펙트럼(Demand Spectrum)을 사용합니다. Pushover 해석에서는 하중-변위관계(V -U )가 생성되며, 응답스펙트럼의 경우에는 가속도-주기(A-T)의 관계가 얻어집니다. 따라서, 두 가지를 상호 비교하기 위하여 가속도-변위 스펙트럼(Acceleration-Displacement Response Spectrum)의 관계(ADRS Format)로 다시 표현하게 됩니다. + +![](images/page-344_f58decd9bbd0614e1a281787050211fa11e9d49d41cec5e629b358ca0adeb08e.jpg) + +
+line + +| U | V | +|-------|-------| +| U | Peak | +| >U | Decreasing from Peak to plateau | +
+ +(a) 하중-변위관계의 가속도-변위 스펙트럼으로의 변환 + +![](images/page-344_96ee1cadfb440f195747e76a11165d89bfe2147d7699ccca651aa727448b3f11.jpg) + +
+line + +| Tn | Demand spectrum | +| ------ | --------------- | +| Tn1 | Tn1 | +| D | Tn2 | +
+ +(b) 가속도-주기 스펙트럼의 가속도-변위 스펙트럼으로의 변환 +그림 2.8.65 능력스펙트럼과 요구스펙트럼의 산정 + + + +하중-변위관계는 그림 2.8.65(a)와 같은 가속도-변위의 관계로 변환되며, 이는 식(83), (84)와 같은 방식으로 변환됩니다. + +$$ +A = \frac {V}{M ^ {k}} \tag {83} +$$ + +$$ +D = \frac {U}{\Gamma^ {k} \phi^ {k}} \tag {84} +$$ + +여기서 $\Gamma^{k}$ 와 $M^{k}$ 는 각각 해당방향의 k차 모드에 대한 모드참여계수와 유효질량 계수를 의미합니다. 산정식은 각각 식 (85), (86)과 같습니다. + +$$ +\text {모드참여계수} \quad \Gamma^ {k} = \frac {\sum_ {j = 1} ^ {N} m _ {j} \phi_ {j k}}{\sum_ {j = 1} ^ {N} m _ {j} \phi_ {j k} ^ {2}} \tag {85} +$$ + +$$ +\text {모드참여질량} M ^ {k} = \frac {\left(\sum_ {j = 1} ^ {N} m _ {j} \phi_ {j k}\right) ^ {2}}{\sum_ {j = 1} ^ {N} m _ {j} \phi_ {j k} ^ {2}} \tag {86} +$$ + +식 (83)과 식 (84)는 동역학 이론에서 다자유도(MDOF) 시스템과 단자유도(SDOF) 시스템의 관계를 의미합니다. 즉, A와 D는 단자유도 시스템의 응답을 의미하는 스펙트럼상에서의 응답가속도와 응답변위를 말하며 V와 U는 다자유도 시스템에서의 밑면전단력과 변위를 의미합니다. + +그리고 탄성응답스펙트럼은 단자유도 시스템에서의 변위와 가속도 관계인 식 (87)을 이용하여 그림 2.8.65(b)와 같은 방법으로 변환됩니다. + +$$ +D = \frac {T _ {n} ^ {2}}{4 \pi^ {2}} A \tag {87} +$$ + + + +# 성능점 (Perrformance Poinnt)의 평가 + +성능스펙트럼럼과 요구스펙트트럼이 만나는 점을 성능점(PPerformance Pooint)이라고 정의합니다. mmidas Civil에서 제공하는 성능능점의 평가방법법은 ATC-40의 능력스펙트럼법(CSM)에 제시된 Proceddure-A와 Proceedure-B 방법을을 모두 적용할 수 있습니다.두가지 방법법의 근본적인 원원리는 동일합니니다. 성능점을 찾는 과정에서서 유효감쇠 산정에 의한 직접 반복법을 적용하는 것이이 Procedure-A 이며, 연성비 가가정과 유효주기원리를 이이용한 방법이 PProcedure-B 입 니다. + +# (1) 등가가감쇠(Equivallent Damping) 의 산정 + +능력스스펙트럼법(CSMM)에서는 Pushoover해석에 의한한 능력스펙트럼럼을 산정한 후에 아래래 그림과 같이이 등가의 면적을을 가지는 이선선형(Bilinear)곡선선으로 표현합니다. CSM에서는 5%% 감쇠를 가지 는 탄성응답스 펙트럼과 능력스스펙트럼을 이용하여여 구조물의 등가가감쇠를 산정합합니다. 구조물물의 감쇠에 의하하여 소산되는에너지지의 양은 등가 이선형 곡선에 서의 이력거동 에 대한 면적을을 나타내며 식(88a), (88b)와 같이 산산정할 수 있습습니다. + +![](images/page-346_8a020665156e9ff0e16e2f511f5ee00d1f2e6cb87f4d74c3f4609b041f23cb6c.jpg) + +
+text_image + +Bilinear representation +of capacity spectrum +Capacity spectrum +a_{pi} +K_{initial} +K_{effective} +a_y +E_{SO} +d_y +d_{pi} +E_D +Spectral Acceleration +a_{pi} +a_y +d_y +d_{pi} +Spectral Displacement +E_D +
+ +그림림 2.8.66 이력거동동에 의한 등가감쇠의의 산정 + +$$ +\beta_ {e q} = \beta_ {0} + 0. 0 5 \tag {88a} +$$ + +$$ +\beta_ {0} = \frac {1}{4 \pi} \frac {E _ {D}}{E _ {S O}} = \frac {6 3 . 7 (a _ {y} d _ {p i} - d _ {y} a _ {p i})}{a _ {p i} d _ {p i}} \tag {88b} +$$ + +여기서서, ED = 구구조물의 감쇠에 의하여 소산되되는 에너지 + +E = 구구조물의 최대변변형에너지 + + + +식(88a)를 백분율의 형태로 표현하면 다음 식과 같이 나타낼 수 있습니다. + +$$ +\beta_ {e q} = \beta_ {0} + 5 = \frac {6 3 . 7 (a _ {y} d _ {p i} - d _ {y} a _ {p i})}{a _ {p i} d _ {p i}} + 5 \tag {89} +$$ + +여기서, $\beta _ { e q }$ 는 감쇠비(%)를 나타내며, ATC-40에서는 25%를 초과할 경우는신중한 판단이 요구되며 최대 50%를 초과할 수 없다고 설명하고 있습니다. + +# (2) 유효감쇠(Effective Damping)의 산정 + +지진하중을 받는 철근콘크리트 구조물의 이력특성은 강도저하(StiffnessDegradation)와 강성저하(Strength Deterioration), 슬립 및 핀칭(Slip or Pinching)등에 의하여 이상화된 이력모델의 특성을 나타내지는 못합니다. 그러므로ATC-40에서는 철근콘크리트 구조물에서의 이러한 이력거동의 특성을 반영하기 위하여 감쇠조정계수(Damping Modification Factor)를 사용하여 등가감쇠를조정합니다. 조정된 등가감쇠를 유효감쇠계수라고 하며 아래식과 같이 산정할수 있습니다. + +$$ +\beta_ {e q} = \kappa \beta_ {0} + 5 = \frac {6 3 . 7 \kappa (a _ {y} d _ {p i} - d _ {y} a _ {p i})}{a _ {p i} d _ {p i}} + 5 \tag {90} +$$ + +위의 식에서 좌변의 감쇠비 5%는 탄성시스템에 대한 지진요구이므로, 철근콘크리트 재료의 이력특성을 반영하는 감쇠조정계수는 등가감쇠에 적용됩니다.그리고 이러한 이력특성으로 인한 구조물의 에너지 소산능력의 저하현상을반영하기 위하여 분류된 구조물에 따라서 감쇠조정계수를 아래와 같이 세 가지로 구분하여 적용합니다. +
구조거동 형식등가감쇠 $\beta_{0}$ (%)감쇠조정계수( $K$ )
Type A (완전한 이력특성) $\leq 16.25$ $> 16.25$ $1.0$ $1.13 - \frac{0.51 \left( a_{y} d_{pi} - d_{y} a_{pi} \right)}{a_{pi} d_{pi}}$
Type B (보통의 이력특성) $\leq 25$ $> 25$ $0.67$ $0.845 - \frac{0.446 \left( a_{y} d_{pi} - d_{y} a_{pi} \right)}{a_{pi} d_{pi}}$
Type C (열악한 이력특성)모든 값 $0.33$
+ +표 2.8.6 구조물의 이력거동에 따른 감쇠조정계수 + + + +# (3) 비탄성 요구스펙트럼의 산정 + +앞서 산정한 유효감쇠계수를 적용하여 비탄성 응답스펙트럼을 고려합니다. 즉, 유효감쇠계수를 이용하여 응답스펙트럼의 조정계수인 응답감소계수(Spectrum Reduction Factor, SR)를 산정하며 응답감소계수는 가속도구간 및 속도구간으로 구분하여 그림 2.8.67과 같이 각각 다르게 적용합니다. 응답감소계수는 Newmark와 Hall(1982)의 지반운동 증폭계수를 이용한 것이며, 가속도구간의 응답감소계수(SR $_{A}$ )와 속도구간의 응답감소계수(SR $_{V}$ )는 아래 식(91)과 같이 산정합니다. ATC-40에서는 구조물의 이력거동에 따라 응답감소계수의 하한치를 표 2.8.7과 같이 제시하고 있습니다. + +![](images/page-348_9dbec6f3069908b7781fc60e8230cfe13b5fadf75daea5cc9125bd049a4a72a0.jpg) + +
+line + +| S_d Range | S_a Range | Label | +| --------------- | --------- | ------------------------- | +| 0 | 0 | Acceleration Range | +| 1 | 1 | Velocity Range | +| 2 | 2 | Elastic Response Spectrum | +| 3 | 3 | Reduced Response Spectrum | +
+ +그림 2.8.67 응답감소계수에 의한 비탄성 응답스펙트럼의 산정 + +$$ +S R _ {A} = \frac {3 . 2 1 - 0 . 6 8 \ln \left[ \frac {6 3 . 7 \kappa \left(a _ {y} d _ {p i} - d _ {y} a _ {p i}\right)}{a _ {p i} d _ {p i}} + 5 \right]}{2 . 1 2} \geq \left\{ \begin{array}{l} 0. 3 3 (\text { for Type } A) \\ 0. 4 4 (\text { for Type } B) \\ 0. 5 6 (\text { for Type } C) \end{array} \right. \tag {91a} +$$ + +$$ +S R _ {V} = \frac {2 . 3 1 - 0 . 4 1 \ln \left[ \frac {6 3 . 7 \kappa \left(a _ {y} d _ {p i} - d _ {y} a _ {p i}\right)}{a _ {p i} d _ {p i}} + 5 \right]}{1 . 6 5} \geq \left\{ \begin{array}{l l} 0. 5 0 & \text {(for Type A)} \\ 0. 5 6 & \text {(for Type B)} \\ 0. 6 7 & \text {(for Type C)} \end{array} \right. \tag {91b} +$$ + + + +
구분 $\kappa$ SRASRV
Type A (완전한 이력특성)1.000.330.50
Type B (보통의 이력특성)0.670.440.56
Type C (열악한 이력특성)0.330.560.67
+ +표 2.8.7 구조물의 이력거동에 따른 응답감소계수의 하한치 + +이상과 같은 절차를 통하여 설계지진하중 또는 선형탄성 응답스펙트럼에 대한 비탄성 요구를 산정할 수 있습니다. 이와 같이 산정된 비탄성 지진요구스펙트럼과 Pushover 해석을 통해 산정한 구조물의 능력스펙트럼과 비교하여구조물의 성능점을 산정할 수 있습니다. + +# (4) 성능점의 산정 + +Pushover 해석에 의하여 산정된 구조물의 능력스펙트럼과 비탄성 설계응답스펙트럼의 교차점을 이용하여 재현주기별 지진하중에 대한 구조물의 비탄성최대 변위와 내력을 의미하는 성능점을 산정할 수 있으며, 또한 구조물의 성능수준도 평가할 수 있습니다. + + + +# 8-7-10 성능점을 산정하는 방법 + +midas Civil에서는 크게 2가지 방법으로 능력스펙트럼법(CSM)에 의한 성능점을 산정할 수 있습니다. 이 방법은 ATC-40에서 제시하고 있는 방법으로 기본적인 원리는 유효감쇠계수를 이용하여 비탄성 요구스펙트럼을 평가하고 능력스펙트럼과의교차점을 통하여 성능점을 산정하는 방식입니다. + +# Procedure-A + +ATC-40에서 제시하는 기본적인 방법으로서 능력스펙트럼의 초기강성에 대한 기울기와 5% 탄성 설계응답스펙트럼과의 교차점을 초기 성능점이라고 가정합니다. 초기 성능점에 대한 등가감쇠를 산정하고, 유효감쇠계수가 적용된 비탄성 설계응답스펙트럼을 구한 후에 다시 교차점에서의 성능점을 산정합니다. 이러한 방법으로유효감쇠계수를 적용한 비탄성 설계응답스펙트럼과 능력스펙트럼과의 교차점에서의 응답변위와 응답가속도와의 변화가 오차범위내에 들어올 때까지 계속적인 반복작업을 통하여 최종적인 성능점을 산정합니다. Procedure-A 방법을 이용한 성능점산정의 원리는 그림 2.8.68과 같습니다. + +![](images/page-350_123f8dad8d7e09953a74e939a8feccea96cdfd935c74df53303985685531fe0b.jpg) +그림 2.8.68 Procedure-A 방법을 이용한 성능점 산정 (ATC-40) diff --git a/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_036.md b/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_036.md new file mode 100644 index 00000000..62a7d32f --- /dev/null +++ b/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_036.md @@ -0,0 +1,298 @@ + + +# Procedure-B + +ATC-40에서 성능점을 산정하는 두 번째 방법은 먼저 변위연성비를 가정한 후에이에 대한 구조물의 유효주기를 산정하고 유효주기 직선과 5% 탄성 설계응답스펙트럼과의 교차점을 초기 성능점으로 가정합니다. 가정한 변위연성비에 대한 방사선 형태의 유효주기와 비탄성 설계응답스펙트럼과의 교차점들은 궤적을 형성하게되며, 이 궤적선과 구조물의 능력스펙트럼과의 교차점이 최종적인 성능점으로 설정됩니다. Procedure-B 방법을 이용한 성능점 산정의 원리는 그림 2.8.69와 같습니다. + +![](images/page-351_4c2edbbdeb4b582adccb5ef90bb13ef6e4ea773b53d0d3f7bbd76eb2ecb27022.jpg) + +
+line + +| Spectral Displacement, inches | Spectral Acceleration, g | +| ----------------------------- | ------------------------ | +| d_y | a'_y | +| d'_* | a'_* | +
+ +그림 2.8.69 Procedure-B 방법을 이용한 성능점 산정 (ATC-40) + +이 방법의 경우에는 변위연성비를 먼저 가정하여 순차적으로 유효감쇠계수를 산정하기 때문에 교차점에서 발생하는 응답 오차에 대하여 발산할 수 있는 확률이 적습니다. 앞서 설명한 Procedure-A의 경우에는 성능점을 찾는 과정에서 수렴성이떨어질 수 있다는 단점이 있는 반면에 Procedure-B는 수렴성이 좋고 탄성 응답스펙트럼을 감쇠비의 변화에 따라서 여러번 작성할 필요없이 변화된 감쇠비와 진동주기에 따른 응답스펙트럼 값의 궤적만을 구하면 되므로 보다 간단한 방법이라고할 수 있습니다. + + + +midas Civil에서 제공하는 두가지 방법에 의한 성능점 산정 과정은 그림 2.8.70,2.8.71과 같습니다. + +![](images/page-352_8d1503bf21eb926be97a980dbead5ae790bfc6b53796528a970a76c331493a19.jpg) + +
+line + +| Metric | Value | +| --- | --- | +| Spectral Acceleration (Sa) | 0.24 | +| Spectral Displacement (Sd) | 0.52 | +| Description for Printed Output | 0.24 | +| Evaluation of Performance point by CSM Procedure-A (ATC-40) | 0.24 | +| Performance Point | 0.24 | +| Displ. Control Node: 11 Dir.: DX | 0.24 | +| Load Pattern: Acceleration | 0.24 | +| V.D. | 0.24 | +| Sa.Sd | 0.24 | +| Teff.Deff | 0.24 | +| Black | 0.24 | +| White | 0.24 | +| Graph Display Option Background Color | 0.24 | +| Change Graph Title | 0.24 | +| Change Graph Range | 0.24 | +| Save Window As *.bmp | 0.24 | +| Text Output | 0.24 | +| Draw | 0.24 | +| Show Ref. Line | 0.24 | +| Show Symbol | 0.24 | +| Close | 0.24 | +
+ +그림 2.8.70 Procedure-A 방법에 의한 성능점 평가 (midas Civil) + +![](images/page-352_862bcbd8b76498530b24adb208919e5d18f8221e798bdf38bc5ed2d23b36321c.jpg) + +
+line + +| Spectral Displacement(Sd) | Spectral Acceleration(Ga) | +| ------------------------- | ------------------------- | +| 0.00 | 0.00 | +| 0.04 | 0.14 | +| 0.08 | 0.20 | +| 0.12 | 0.22 | +| 0.16 | 0.22 | +| 0.20 | 0.22 | +| 0.24 | 0.22 | +| 0.28 | 0.22 | +| 0.32 | 0.22 | +| 0.36 | 0.22 | +| 0.40 | 0.22 | +| 0.44 | 0.22 | +| 0.48 | 0.22 | +| 0.52 | 0.22 | +
+ +그림 2.8.71 Procedure-B 방법에 의한 성능점 평가 (midas Civil) + + + +# 8-7-11 성능평가 + +구조물의 변위가 목표성능의 범위내에 포함되는 것이 확인되면, 계속해서 각 개별부재의 성능을 평가합니다. 이때 midas Civil에서는 FEMA-273이나 ATC-40에서 권장하는 방법과 유사한 방법으로 부재의 성능을 평가할 수 있도록 하였습니다. 이들 보고서에서는 성능상태를 그림 2.8.72와 같은 세가지의 단계로 분류하고 있습니다. + +![](images/page-353_0560d03c37dc3e30fafb225efbafa7b29e0d23da878f2bb084958fb1cec9f313.jpg) + +
+line + +| Point | Displacement | Force | +|-------|--------------|-------| +| A | A | 0 | +| B | ~0.5 | ~1.0 | +| C | ~1.0 | ~1.2 | +| D | ~1.5 | ~0.5 | +| E | ~2.0 | ~0.0 | +
+ +IO = 즉시사용 한계상태 (Immediate Occupancy) +LS = 안전 한계상태 (Life Safety) +CP = 붕괴방지 한계상태 (Collapse Prevention) +그림 2.8.72 부재의 성능평가 + +# 8-7-12 Pushover 해석과정 + +1. 정적해석 및 부재설계 완료 + +지진하중에 대한 구조물의 보유성능을 검토하기 위해 Pushover해석을 수행하는 경우, 우선 해석모델에 대한 정적해석과 부재설계를 완료합니다. + +2. Pushover해석 제어용 데이터 입력 + +Pushover탭>Global Control그룹>Pushover Global Control 대화상자를 호출하여 초기하중, 각 step별 내부반복 최대 계산회수 및 수렴조건을 지정합니다. + + + +3. Pushover Load Case 입력 + +Pushover탭>Load Case그룹>Pushover Load Cases 대화상자를 호출하여Pushover해석의 최대 계산회수, 증분방법, 초기상태 하중의 사용여부와Pushover 하중조건을 입력합니다. + +먼저 하중제어(Load Control)와 변위제어(Displacement Control)를 선택합니다. 그리고 초기상태하중을 고려하기 위해 자중을 입력하고, Pushover 하중조건으로는 Static Load Case, Uniform Acceleration 및 Mode Shape 등을 선택할 수 있으며, 각 하중형태는 조합이 가능합니다. + +4. Hinge Data 정의 + +Pushover탭>Assign그룹>Assign Hinge Properties>Define Pushover HingeProperties 대화상자를 호출하여 비선형요소의 종류,비선형성을 고려할 성분과 성분별 골격곡선을 정의합니다. + +5. 부재에 Hinge Data 지정 + +Pushover탭>Assign그룹>Assign Hinge Properties>Assign Pushover HingeProperties 대화상자를 호출하여 정의된 힌지 데이터를 각 부재에 할당합니다. 일반적으로 보에는 모멘트 힌지, 기둥에는 축력과 모멘트 힌지를 할당합니다. + +6. Pushover해석 수행 + +Pushover탭>Perform그룹>Perform Pushover Analysis를 클릭하여Pushover해석을 수행합니다. + +7. 해석결과 확인 + +해석이 성공적으로 완료되면, Pushover탭>Pushover Results그룹>PushoverCurve를 클릭하여 Pushover곡선을 출력하고, 각종 Design Spectrum에 대한 구조물의 성능정도를 검토합니다. 또한, Pushover탭>Pushover Results그룹>Deformations>Deformed Shape 대화상자에서 Pushover 하중조건을 선택하여 단계별 변형형상 및 힌지발생상태를 확인합니다. 이때 Animate기능을 이용하면 힌지 발생과정을 동영상으로 확인할 수 있습니다. + + + +# Chapter 9. 비선형 시간이력해석 + +# 9-1 개요 + +구조물에 지진동이 작용할때, 변형이 작은 범위에서 구조물은 탄성거동 합니다. 그러나, 외력의 증가에 의해 구조물의 변형이 커지면, 부재응력은 탄성한계를 넘게 되고, 균열, 항복 등의 현상이 발생합니다. 이때 복원력과 변형의 관계는 이력곡선을 그리며, 이와 같은 복원력특성은 탄소성 복원력특성이라 합니다. 구조물이 대지진을 받을 때, 골조가 항복하여 소성역에 들어가는 것은 피할 수 없으므로, 대지진에 대한 구조물의 안전성을 확보하기 위하여 구조물의 소성변형능력과 이력에너지 흡수능력은 매우 중요한 인자가 됩니다. 비선형 시간이력 해석(Nonlinear Time History Analysis)은 구조부재의 비선형 복원력특성을 단순화한 이력모델을 통하여 구조물의 비선형 거동을 파악하는 시간이력 해석방법입니다. 대상 구조물은 해석의 효율성을 고려하여 주요 부분에 비선형 요소를 사용하고 나머지 부분은 탄성인 것으로 가정합니다. + +# 9-1-1 비선형 운동방정식 + +비선형 요소가 포함된 구조물의 운동방정식은 다음과 같이 구성되며, 요소의 비선 형성은 접선 강성법에 의해 정식화 됩니다. 단, 비선형 연결요소는 Element Type 범용 연결요소의 Spring에 Inelastic Hinge Properties를 부여한 요소입니다. + +$$ +M \ddot {u} + C \dot {u} + K _ {S} u + f _ {I} + f _ {N} = p \tag {1} +$$ + +여기서 M: 질량행렬 + +C: 감쇠행렬 + +$K_{S}$ : 비탄성 요소 및 비선형 연결요소를 제외한 나머지 탄성 부재들의 강성행렬 + +u, ù, ù : 절점에 대한 변위, 속도 및 가속도 응답 + +p: 절점에 대한 동적하중 + +$f_{1}$ : 비탄성 요소의 전체좌표계 절점내력 + + + +$f_{N}$ : 비선형 연결요소에 포함된 비선형 스프링의 전체좌표계 절점내력 비선형 시간이력 해석은 선형 시간이력 해석과 달리 중첩의 원리가 적용될 수 없습니다. 따라서, 수치적분법에 의해 해석되어야만 하며, 비선형 동적운동방정식의 시간이력 수치적분법은 직접적분법에 의해 수행됩니다. 직접적분법에 의한 응답해석은 임의의 외력에 의한 강제진동 운동방정식을 직접 수치적분하여 2계의 연립미분방정식의 해를 구하여, 미지수인 변위, 속도 및 가속도 응답을 구하는 방법입니다. midas Civil에서는 Newmark-β 법에 의한 직접적분법으로 해석을 수행합니다. Newmark-β 법은 각 시간증분에서의 변위의 증분을 구하여 누적하는 방식으로 처리됩니다. 각 시간증분에서 발생하는 불평형력은 Newton-Raphson법에 의한 반복해석을 통해 해소합니다. + +Newmark-β의 기본 가정에 의해 시각 t 에서의 가속도와 변위를 이용하면, $t + \Delta t$ 에서의 속도와 변위는 다음과 같이 나타냅니다. + +$$ +\dot {u} _ {t + \Delta t} = \dot {u} _ {t} + (1 - \gamma) \Delta t \ddot {u} _ {t} + \gamma \Delta t \ddot {u} _ {t + \Delta t} \tag {2} +$$ + +$$ +u _ {t + \Delta t} = u _ {t} + \Delta t \dot {u} _ {t} + \left(\frac {1}{2} - \beta\right) \Delta t ^ {2} \ddot {u} _ {t} + \beta \Delta t ^ {2} \ddot {u} _ {t + \Delta t} \tag {3} +$$ + +위의 식을 변위로 정리하면 다음 식으로 나타낼 수 있습니다. + +$$ +\dot {u} _ {t + \Delta t} = \frac {\gamma}{\beta \Delta t} \Delta u _ {t + \Delta t} + \left(1 - \frac {\gamma}{\beta}\right) \dot {u} _ {t} + \left(1 - \frac {\gamma}{2 \beta}\right) \Delta t \ddot {u} _ {t} \tag {4} +$$ + +$$ +\ddot {u} _ {t + \Delta t} = \frac {1}{\beta \Delta t ^ {2}} \left(\Delta u _ {t + \Delta t} - \Delta t \dot {u} _ {t} + \left(\frac {1}{2} - \beta\right) \Delta t ^ {2} \ddot {u} _ {t}\right) \tag {5} +$$ + +변위, 속도, 가속도 증분은 다음과 같이 표현됩니다. + +$$ +\Delta u _ {t + \Delta t} = u _ {t + \Delta t} - u _ {t} \tag {6} +$$ + +$$ +\Delta \dot {u} _ {t + \Delta t} = \frac {\gamma}{\beta \Delta t} \Delta u _ {t + \Delta t} - \frac {\gamma}{\beta} \dot {u} _ {t} + \left(1 - \frac {\gamma}{2 \beta}\right) \Delta t \ddot {u} _ {t} \tag {7} +$$ + + + +$$ +\Delta \ddot {u} _ {t + \Delta t} = \frac {1}{\beta \Delta t ^ {2}} \Delta u _ {t + \Delta t} - \frac {1}{\beta \Delta t} \dot {u} _ {t} - \frac {1}{2 \beta} \ddot {u} _ {t} \tag {8} +$$ + +Newton-Raphson법에 의한 반복계산시의 증분응답은 다음과 같이 표현됩니다. + +$$ +\delta \dot {u} ^ {(i)} = \Delta \dot {u} ^ {(i)} - \Delta \dot {u} ^ {(i - 1)} = \frac {\gamma}{\beta \Delta t} \delta u ^ {(i)} \tag {9} +$$ + +$$ +\delta \ddot {u} ^ {(i)} = \Delta \dot {u} ^ {(i)} - \Delta \dot {u} ^ {(i - 1)} = \frac {1}{\beta (\Delta t) ^ {2}} \delta u ^ {(i)} \tag {10} +$$ + +따라서, 시각 $t + \Delta t$ 에서의 (i)번째 반복해석시의 변위, 속도, 가속도는 다음과 같이 표현됩니다. + +$$ +u _ {t + \Delta t} ^ {(i)} = u _ {t + \Delta t} ^ {(i - 1)} + \delta u ^ {(i)} \tag {11} +$$ + +$$ +\dot {u} _ {t + \Delta t} ^ {(i)} = \dot {u} _ {t + \Delta t} ^ {(i - 1)} + \delta \dot {u} ^ {(i)} = \dot {u} _ {t + \Delta t} ^ {(i - 1)} + \frac {\gamma}{\beta \Delta t} \delta u ^ {(i)} \tag {12} +$$ + +$$ +\ddot {u} _ {t + \Delta t} ^ {(i)} = \ddot {u} _ {t + \Delta t} ^ {(i - 1)} + \delta \ddot {u} ^ {(i)} = \ddot {u} _ {t + \Delta t} ^ {(i - 1)} + \frac {1}{\beta (\Delta t) ^ {2}} \delta u ^ {(i)} \tag {13} +$$ + +시각 $t + \Delta t$ 에서의 (i)번째 반복해석시의 비선형 운동 방정식은 다음과 같습니다. + +$$ +M \ddot {u} _ {t + \Delta t} ^ {(i)} + C \dot {u} _ {t + \Delta t} ^ {(i)} + f (u) _ {t + \Delta t} ^ {(i)} = p _ {t + \Delta t} \tag {14} +$$ + +식(14)에 식(12), (13)을 대입하면, 증분변위 $\delta u^{(i)}$ 에 관한 평형방정식은 다음과 같이 표현됩니다. + +$$ +K _ {E f f} ^ {(i)} \cdot \delta u ^ {(i)} = \Delta p _ {E f f} ^ {(i)} \tag {15} +$$ + +여기서, $K_{Eff}$ : 유효강성행렬, $K_{Eff} = \frac{1}{\beta (\Delta t)^{2}} M + \frac{1}{\beta \Delta t} C + K_{t + \Delta t}^{(i)}$ + +$\Delta p_{Eff}$ : 각 반복해석 단계에서의 유효하중벡터 + + + +$$ +\Delta p _ {E f f} = p _ {t + \Delta t} - \left(M \ddot {u} _ {t + \Delta t} ^ {(i - 1)} + C \dot {u} _ {t + \Delta t} ^ {(i - 1)} + f _ {t + \Delta t} ^ {(i - 1)}\right) +$$ + +( )i Kt t   : 비탄성 요소를 포함한 접선강성행렬 + +( )i  u : 각 반복해석 단계에서의 변위증분벡터 + +β：Newmark-β 蛭 よ + +# 9-1-2 비선형 정적해석 + +비선형 시간이력해석에서 질량과 감쇠의 효과를 배제함으로써 비선형 정적해석을수행할 수 있습니다. Time History Load Cases의 Nonlinear Static은 이와 같은 해석기능입니다. 비선형 정적해석은 중력하중에 의한 초기조건을 생성하거나Pushover 해석을 수행하는데 사용할 수 있습니다. 중력하중에 의한 초기조건 생성에 있어서 비선형 정적해석을 수행하면 이 과정에서 발생할 수 있는 비선형 거동을 비선형 시간이력 해석에 반영할 수 있게 됩니다. 따라서 비선형 요소의 상태를판정하는데 있어서 연속적으로 수행되는 하중조건 사이의 정합성을 확보할 수 있습니다. Pushover 해석은 항복이후의 극한내력과 한계상태를 매우 효과적으로 파악할 수 있는 간편한 해석방법입니다. 특히 최근에 지진공학과 내진설계 분야에서많은 연구와 실무 적용이 이루어지고 있는, 성능에 기초한 내진설계(Performance-Based Seismic Design, PBSD)에서 대표적인 해석방법으로 적용되고 있습니다. 이해석은 고차모드의 동적특성의 영향을 받지않는 구조물에 주로 사용할수 있습니다. + +비선형 정적해석에서 사용되는 해석방법은 Newton-Raphson법을 기본으로 하며,하중제어 및 변위제어법을 모두 지원합니다. 하중제어는 사용자가 부과한 정적하중을 재하 스텝 수(Increment Steps)로 나누어 재하합니다. 변위제어는 사용자가구조물에서 발생할 수 있는 목표변위를 미리 설정하고 목표변위가 달성될 때까지하중을 증가시키는 방법입니다. 목표변위는 크게 Global Control과 Master NodeControl로 설정할 수 있습니다. Global Control은 구조물에서 발생하는 최대변위가사용자가 입력한 목표변위를 만족할 때까지 하중을 증가시키는 방법입니다. 이 방법은 하중의 방향성과 무관합니다. Master Node Control은 사용자가 특정한 절점을지정하고 그 절점에서 사용자가 지정한 방향에 대한 목표변위를 만족하도록 하중을 증가시키는 방법입니다. 성능에 기초한 내진설계에서는 대부분 최대변위가 발생할 가능성이 있는 절점과 방향을 고려하여 목표변위를 설정합니다. + + + +비선형 정적해석에서는 서로 다른 제어 방법을 가진 하중조건의 연속해석도 가능합니다. 단, 1) 하중제어에 의한 하중조건을 연속으로 해석할 경우와, 2) 변위제어에의한 하중조건 후에, 하중제어에 의한 해석을 수행할 경우, 부적한 결과를 얻을 수있으므로 주의할 필요가 있습니다. 연속해석에 대한 하중조건을 정리하면 다음과같습니다. + + 하중 제어  변위 제어 (O) + 하중 제어  하중 제어 (X) + 변위 제어  변위 제어 (O) + 변위 제어  하중 제어 (X) + +하중은 시간변동 정적하중(Time Varying Static Load)을 통해 재하되며, 이때 하중함수는 Time History Functions에서 Time Function Data Type을 Normal로 입력합니다. 해석과정에서 하중제어의 경우, 하중계수는 0부터 1까지 선형증가합니다. 변위제어의 경우에는 변위 증분에 따른 하중계수를 자동 계산합니다. 비선형 정적해석에서 하중계수의 시간이력은 저장 및 출력이 가능합니다. + +# 9-1-3 비선형 시간이력해석에서의 초기 단면력의 고려 + +midas Civil의 비선형 직접적분법에 의한 시간이력해석에서 중력하중에 의한 정적해석 결과를 동적해석의 초기조건으로 반영하는 방법은, 1) 중력하중에 대해 비선형 정적해석을 수행하고 연속해서 시간이력해석을 수행하는 방법과, 2) 중력하중에대한 정적해석 결과를 초기단면력으로 입력하여 시간이력해석 결과에 반영시키는방법이 있습니다. + +정적해석 결과를 초기단면력으로 입력하는 경우, 선형해석에서는 입력된 초기단면력을 단순히 시간이력 해석결과와 조합하여 처리 가능합니다. 하지만, 비선형 시간이력해석에서는 입력된 초기단면력을 비선형 요소의 상태판정에 고려하지 않으면,연속적으로 수행되는 하중조건 사이의 정합성을 확보할 수 없습니다. 또한, 부재에발생하는 단면력은 가해지는 외력에 의해 발생하므로, 입력된 초기단면력을 부재내력으로 평형방정식에 그대로 반영하면, 평형조건이 성립되지 않습니다. + + + +midas Civil은 입력된 초기단면력에 대해 가상의 변형을 구하여, 비선형 부재의 상태판정시에 고려하는 방법으로 비선형 시간이력해석을 수행합니다. 단, 동적평형방정식을 구성할 때는 초기단면력은 무시되며, 상세한 해석방법은 다음과 같습니다.(그림 2.9.1 참조) + +1. 시간이력해석의 초기증분에 들어가기 전에, 초기강성 $K _ { 0 }$ 를 이용하여, 입력된 초기 단면력에 대해 비탄성 힌지의 가상의 변형 $D _ { i n i }$ 을 구합니다. +a) 구해진 $D _ { i n i }$ 이 항복변형 이내에 있으면(탄성범위), 입력된 초기 단면력을 그대로 해석에 반영합니다. +b) 구해진 $D _ { i n i }$ 이 항복변형을 초과한 경우, 이력루틴에서 변형 $D _ { i n i }$ 에 대한 복원력 $P _ { i n i }$ 를 구하여 해석에 반영합니다. 단, $D _ { i n i }$ 와 $P _ { i n i } \equiv \bar { \pounds } \supset$ 증분에서 1회만 구합니다. +2. 동적평형방정식을 풀어, 증분변위 $\delta u _ { t + \Delta t } \equivq$ 구합니다. 단, 초기 단면력은 내력으로 입력되므로, 동적평형방적식을 구성할때는 무시됩니다. +3. 증분변위 $\delta u _ { { t + \Delta t } }$ 을 이용하여, 수치적분법으로 $\ddot { u } _ { t + \Delta t } , \dot { u } _ { t + \Delta t } , u _ { t + \Delta t }$ 을 산출합니다. 변위를 이용하여 비선형 힌지의 변형 D 와 복원력 P 를 구합니다. +4. 비선형 부재의 상태판정을 위해 이력루틴에 들어갑니다. 단, 이력루틴에 들어가기전에 비탄형 힌지의 변형과 부재력은 초기 단면력을 고려하여 다음과 같이 수정됩니다. ${ \boldsymbol { D } } ^ { * } = { \boldsymbol { D } } \ + \boldsymbol { D _ { i n i } } \ , \ { \boldsymbol { P } } ^ { * } = { \boldsymbol { P } } \ + \boldsymbol { P _ { i n i } }$ i +5. 이력루틴에서 변형 $D ^ { * }$ 에 의해 강성과 복원력 $\overline { P } ^ { * }$ 을 계산합니다. +6. 비탄성 힌지의 해석결과를 출력합니다. +7. 동적평형방정식을 구성하기 위해서, 변형과 복원력을 다음과 같이 수정합니다. $\textit { D } = \boldsymbol { D } ^ { * } - D _ { i n i } , \ \overline { { \boldsymbol { P } } } \ = \overline { { \boldsymbol { P } } } ^ { * } - P _ { i n i }$ +8. 동적평형방정식을 구성하고, 2.로 돌아가 마직막 시간증분까지 이와 같은해석을 반복합니다. diff --git a/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_037.md b/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_037.md new file mode 100644 index 00000000..1e558970 --- /dev/null +++ b/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_037.md @@ -0,0 +1,513 @@ + + +![](images/page-361_7cd2062b703c42e7758a70b6d8efc8f53e090a1d31c924ec2526ad3eb8827be1.jpg) + +
+line + +| Point | D | P | +|-------|-------|-------| +| K₀ | D_ini | P₀ini | +| P1(+) | D1(+) | P0ini | +
+ +(a) 초기단면력이 탄성범위인 경우 +![](images/page-361_bc0e2b1c272fb7ca3c2c17f482bf7be71c548cdbf9fff46b2bb0bb9c68676305.jpg) + +
+line + +| D | P0 | P1(+) | P2(+) | +|-------|--------|--------|--------| +| D1(+) | | | | +| D2(+) | | | | +| Dini | P0 | | | +| Dini | P1(+) | | | +
+ +(b) 초기단면력이 탄성한계를 넘은 경우 +그림 2.9.1 입력된 초기단면력의 처리 + + + +# 9-1-4 비선형 시간이력해석에서의 초기강성 + +midas Civil의 비선형 시간이력해석에서 비탄성 부재의 초기강성은 Inelastic HingeProperties의 Initial Stiffness에서 다음과 같이 설정할 수 있습니다. + + Elastic : 탄성강성을 초기강성으로 사용, 단, 집중형 힌지의 휨 성분은6EI/L, 3EI/L, 2EI/L중 선택 + User : 사용자가 비선형 부재의 초기강성을 직접 입력 + Skeleton : 입력된 항복강도와 항복변형으로 초기강성 계산 + +Elastic과 User인 경우, (+), (-)측에서 동일한 초기강성을 갖습니다. + +Skeleton을 선택한 경우, (+),(-)측의 항복변형을 각각 별도로 입력할 수 있습니다.이 경우 (+),(-)측의 항복강도와 항복변형의 기울기로 각각의 초기강성을 구하여,해석에는 큰 값을 적용합니다. 단, Orient-Origin, Elastic/Bilinear, Elastic/Trilinear,Elastic/Tetralinear형 이력은 비대칭으로 입력된 (+), (-)측 초기강성을 그대로 해석에반영합니다. + +# 9-1-5 Newton-Raphson Method + +비선형 시간이력해석의 각 시간증분에서는 비선형 요소의 강성 변화와 부재력 변화에 의해서 불평형력(Residual Force)이 발생하게 됩니다. 변위 증분을 구하는 과정에서 부재력과 외력 사이의 불평형력은 다음과 같이 처리됩니다. + +# 1. 수렴계산을 수행하는 경우 + +Newton- Raphson법에 의해 불평형력이 해소될 때까지 반복해석을 수행 + +# 2. 수렴계산을 수행하지 않는 경우 + +불평형력을 다음 시간증분의 외력으로 처리 + +반복계산에 의한 불평형력의 해소방법은 그림 2.9.2에 나타낸 것과 같이 Full + + + +Newton-Rapshon 법을 이용합니다. 반복해석에서 수렴을 판정하는 기준 Norm은 변위, 하중 및 에너지의 세가지가 있으며, 이 가운데 하나 또는 복수의 Norm을 선택하여 수렴판정에 반영할 수 있습니다. 각 Norm의 정의는 다음과 같습니다. + +$$ +\varepsilon_ {D} = \sqrt {\frac {\delta u _ {n} ^ {T} \cdot \delta u _ {n}}{\Delta u _ {n} ^ {T} \cdot \Delta u _ {n}}}, \varepsilon_ {F} = \sqrt {\frac {p _ {e f f , n} ^ {T} \cdot p _ {e f f , n}}{p _ {e f f , 1} ^ {T} \cdot p _ {e f f , 1}}}, \varepsilon_ {E} = \sqrt {\frac {p _ {e f f} ^ {T} \cdot \delta u _ {n}}{p _ {e f f , 1} ^ {T} \cdot \delta u _ {1}}} +$$ + +여기서, $\varepsilon_{D}$ : 변위 Norm + +$\varepsilon_{F}$ : 하중 Norm + +$\varepsilon_{E}$ : 에너지 Norm + +$p_{eff}^{T}$ : n번째 반복계산 단계에서의 유효하중 벡터 + +$\delta u_{n}$ : n번째 반복계산 단계에서의 변위 증분 벡터 + +$\Delta u_{n}$ : n회의 반복계산에 의해 누적된 변위 증분 벡터 + +Newton-Raphson Method에 의한 수렴계산시, 비선형성이 매우 강한 경우, 반복횟수가 사용자가 입력한 최대 반복회수에 도달해도 수렴하지 않는 경우가 발생할 수 있습니다. 이와 같은 경우에는 시간증분 $\Delta t$ 를 재설정하여 해석하여야만 하지만, midas Civil에서는 반복해석시에 최대 반복회수에도 수렴되지 않는 경우, 해당 시간증분의 초기상태로 복귀하여 자동적으로 시간증분 $\Delta t$ 를 세분하여 재해석을 수행합니다. + +![](images/page-363_518d7f804d38d01796f2257450ea8db054c0cf8f5d3329a2c24f08802042bc7e.jpg) + +
+line +| Point Label | Description | Description | +|-------------|--------------------------|---------------------------------| +| t | Effective Load (t+Δt) | Effective Load (t+Δt) | +| t | Effective Load (t) | Effective Load (t+Δt) | +| t | Effective Load (t+Δt) | Effective Load (t+Δt) | +| t | Effective Load (t) | Effective Load (t+Δt) | +| t | Effective Load (t) | Effective Load (t+Δt) | +| t | Effective Load (t) | Effective Load (t+Δt) | +| t | Effective Load (t) | Effective Load (t+Δt) | +| t | Effective Load (t) | Effective Load (t+Δt) | +| t | Estimated Load (u(t+Δt)) | Estimated Load (u(t+Δt)) | +| t | Estimated Load (u(t) | Estimated Load (u(t+Δt)) | +| t | Estimated Load (u(t) | Estimated Load (u(t+Δt)) | +| t | Estimated Load (u(t) | Estimated Load (u(t+Δt)) | +| t | Estimated Load (u(t) | Estimated Load (u(t+Δt)) | +| t | Estimated Load (u(t) | Estimated Load (u(t+Δt)) | +| t | Estimated Load (-u(t+Δt)) | Estimated Load (-u(t+Δt)) | +| t | Estimated Load (-u(t) | Estimated Load (-u(t+Δt)) | +| t | Estimated Load (-u(t) | Estimated Load (-u(t+Δt)) | +| t | Estimated Load (-u(t) | Estimated Load (-u(t+Δt)) | +| t | Estimated Load (-u(t) | Estimated Load (-u(t+Δt)) | +| t | Estimated Load (-u(t) | Estimated Load (-u(t+Δt)) | +| t | Estimated Load (u(t+Δt)) | Estimated Load (u(t+Δt)) | +| t | Estimated Load (u(t) | Estimated Load (u(t+Δt)) | +| t | Estimated Load (u(t) | Estimated Load (u(t+Δt)) | +| t | Estimated Load (u(t) | Estimated Load (u(t+Δt)) | +| t | Effective Load (u(t+Δt)) | Estimated Load (u(t+Δt)) | +| t | Effective Load (u(t) | Estimated Load (u(t+Δt)) | +| t | Effective Load (u(t) | Estimated Load (u(t+Δt)) | +| t | Effective Load (u(t) | Estimated Load (u(t+Δt)) | +| t | Effective Load (u(t) | Estimated Load (u(t+Δt)) | +| t | Effective Load (u(t) | Estimated Load (u(t+Δt)) | +| t | Effective Load (-u(t+Δt)) | Estimated Load (-u(t+Δt)) | +| t | Effective Load (-u(t) | Estimated Load (-u(t+Δt)) | +| t | Effective Load (-u(t) | Estimated Load (-u(t+Δt)) | +| t | Effective Load (-u(t) | Estimated Load (-u(t+Δt)) | +| t | Effective Load (-u(t) | Estimated Load (-u(t+Δt)) | +| t | Effective Load (-u(t) | Estimated Load (-u(t+Δt)) | +| t | Effective Load (u(t+Δt)) | Estimated Load (u(t+Δt)) | +| t | Effective Load (u(t) | Estimated Load (u(t+Δt)) | +| t | Effective Load (u(t) | Estimated Load (u(t+Δt)) | +| t | Effective Load (u(t) | Estimated Load (u(t+Δt) | +| t | Effective Load (u(t) | Estimated Load (u(t+Δt) | +| t | Effective Load (u(t) | Estimated Load (u(t+Δt) | +| t | Effective Load (u(t) | Estimated Load (u(t+Δt) | +| t | Effective Load (u(t) | Estimated Load (u(t+Δt) | +| t | Effective load (u(t+Δt)) | Estimated load (u(t+Δt)) | +| t | Effective load (u(t) | Estimated load (u(t+Δt)) | +| t | Effective load (u(t) | Estimated load (u(t+Δt)) | +| t | Effective load (u(t) | Estimated load (u(t+Δt)) | +| t | Effective load (u(t) | Estimated load (u(t+Δt)) | +| t | Effective load (u(t) | Estimated load (u(t+Δt)) | +| t | Effective load (-u(t+Δt)) | Estimated load (-u(t+Δt)) | +| t | Effective load (-u(t) | Estimated load (-u(t+Δt)) | +| t | Effective load (-u(t) | Estimated load (-u(t+Δt)) | +| t | Effective load (-u(t) | Estimated load (-u(t+Δt)) | +| t | Effective load (-u(t) | Estimated load (-u(t+Δt)) | +| t | Effective load (-u(t) | Estimated load (-u(t+Δt)) | +| t | Effective load (u(t+Δt)) | Estimated load (u(t+Δt)) | +| t | Effective load (u(t) | Estimated load (u(t+Δt)) | +| t | Effective load (u(t) | Estimated load (u(t+Δt)) | +| t | Effective load (u(t) | Estimated load (u(t+Δt)) | +| t | +| t | Effective load (u(t) | Estimated load (u(t+Δt)) | +| t | Effective load (u(t) | Estimated load (u(t+Δt)) | +| t | Effective load (u(t) | Estimated load (u(t+Δt)) | +| t | Effective load (u(t) | Estimated load (u(t+Δt)) | +| t | Effective load 0 (u(t+Δt)) | Estimated load (u(t+Δt)) | +| t | Effective load 0 (u(t) | Estimated load (u(t+Δt)) | +| t | Effective load 0 (u(t) | Estimated load (u(t+Δt)) | +| t | Effective load 0 (u(t) | Estimated load (u(t+Δt)) | +| t | Effective load 0 (u(t) | Estimated load (u(t+Δt)) | +| t | Effective load 0 (u(t) | Estimated load (u(t+Δt)) | +| t | Effective load 0 (u(t) | Estimated load (u(t+Δt)) | +| t | Effective load 0 (u(t) | Estimated load (u(t+Δt)) | +| t | Effective load 0 (u(t) | Estimated load (u(t+Δt)) | +| t | Effective load 0 (u(t) | Estimated load (u(t+Δt)) | +| t | Effective load 0 (u(t) | Estimated load (u(t+Δt)) | +| t | Effective load 0 (u(t) | Estimated load (u(t+Δt)) | +| t | Effective load 0 (u(t) | Estimated load (u(t+Δt)) | +| t | Effective load 0 (u(t) | Estimated load (u(t+Δt)) | +| t | Effective load 0 (u(t) | Estimated load (u(t+Δt)) | +| t | Effective load 0 (u(t) | Estimated load (u(t+Δt)) | +| t | Effective load 0 (u(t) + Δt) | Estimated load (u(t+Δt)) | +| t | Effective load 0 (u(t) + Δt) | Estimated load (u(t+Δt)) | +| t | Effective load 0 (u(t) + Δt) | Estimated load (u(t+Δt)) | +| t | Effective load 0 (u(t) + Δt) | Estimated load (u(t+Δt)) | +| t | Effective load 0 (u(t) + Δs) | Estimated load (u(t+Δt)) | +| t | Effective load 0 (u(t) + Δs) | Estimated load (u(t+Δt)) | +| t | Effective load 0 (u(t) + Δs) | Estimated load (u(t+Δt)) | +| t | Effective load 0 (u(t) + Δs) | Estimated load (u(t+Δt)) | +| t, t+Δt | Effective Load (u(t+Δt)) | Estimated load (u(t+Δt)) | +| t, t+Δt | Effective Load (u(t) + Δt) | Estimated load (u(t+Δt)) | +| t, t+Δt | Effective Load (u(t) + Δs) | Estimated load (u(t+Δt)) | +| t, t+Δt | Effective Load (u(t) + Δs) | Estimated load (u(t+Δt)) | +| t, t+Δt | Effective Load (u(t) + Δs) | Estimated load (u(t+Δt)) | +| t, t+Δt | Effective Load (u(t) + Δs) | Estimated load (u(Δt) + Δs) | +| t, t+Δt | Effective Load (u(t) + Δs) | Estimated load (u(t) + Δs) | +| t, t+Δt | Effective Load (u(t) + Δs) | Estimated load (u(t) + Δs) | +| t, t+Δt | Effective Load (u(t) + Δs) | Estimated load (u(t) + Δs) | +| t, t+Δt | Effective Load (u(t) + Δs) | Estimated load (u(t) + Δt) | +| t, t+Δt | Effective Load (u(t) + Δs) | Estimated load (u(t) + Δt) | +| t, t+Δt | Effective Load (u(t) + Δs) | Estimated load (u(t) + Δs) | +| t, t+Δt | Effective Load (u(t) + Δs) | Estimated load (u(t) + Δs) | +| t, t+Δt | Effective Load (u(t) + Δs) | Estimated load (u(t) + Δs) | +| +| t, t+Δt | Effective Load (u(t) + Δs) | Estimated load (u(t) + Δs) | +| t, t+Δt | Effective Load (u(t) + Δs) | Estimated load (u(t) + Δs) | +| t, t+Δt | Effective Load (u(t) + Δs) | Estimated load (u(t) + Δs) | +| t, u(t+Δt) | Effective Load (u(t+Δt)) | Estimated load (u(t+Δt)) | +| t, u(t+Δt) | Effective Load (u(t) + Δt) | Estimated load (u(t+Δt)) | +| t, u(t+Δt) | Effective Load (u(t) + Δs) | Estimated load (u(t+Δt)) | +| t, u(t+Δt) | Effective Load (u(t) + Δs) | Estimated load (u(t+Δt)) | +| t, u(t+Δt) | Effective Load (u(t) + Δs) | Estimated load (u(t+Δt)) | +| t, u(t+Δt) | Effective Load (u(t) + Δs)| Estimated load (u(t+Δt)) | +| t, u(t+Δt) | Effective Load (u(t) + Δs)| Estimated load (u(t+Δt)) | +| t, u(t+Δt) | Effective Load (u(t) + Δs)| Estimated load (u(t+Δt)) | +| t, u(t+Δt) | Effective Load (u(t) + Δs)| Estimated load (U(t+Δt)) | +| t, u(t+Δt) | Effective Load (u(t) + Δs)| Estimated load (U(t+Δt)) | +| t, u(t+Δt) | Effective Load (u(t) + Δs)| Estimated load (U(t+Δt)) | +| t, u(t+Δt) | Effective Load (u(t) + Δs)| Estimated load (U(t+Δs)) | +| t, u(t+Δt) | Effective Load (u(t) + Δs)| Estimated load (U(t+Δt)) | +| t, u(t+Δt) | Effective Load (u(t) + Δs)| Estimated load (U(t+Δt)) | +| t, u(t+Δt) | Effective Load (u(t) + Δs)| Estimated load (U(t+Δt)) | +| +| t, u(t+Δt) | Effective Load (u(t) + Δs)| Estimated load (U(t+Δt)) | +| t, u(t+Δt) | Effective Load (u(t) + Δs)| Estimated load (U(t+Δt)) | +| t, u(t+Δt) | Effective Load (u(t) + Δs)| Estimated load (U(t+Δt)) | +| t, t+Δt | Effective Load (u(t) + Δs)| Estimated load (U(t+Δt)) | +| t, t+Δt | Effective Load (u(t) + Δs)| Estimated load (U(t+Δt)) | +| t, t+Δt | Effective Load (u(t) + Δs)| Estimated load (U(t+Δt)) | +| t, t+Δt | Effective Load (u(t) + Δt) | Estimated load (U(t+Δt)) | +| t, t+Δt | Effective Load (u(t) + Δs) | Estimated load (U(t+Δt)) | +| t, t+Δt | Effective Load (u(t) + Δs) | Estimated load (U(t+Δt)) | +| t, t+Δt | Effective Load (u(t) + Δs) | Estimated load (U(t+Δt)) | +| t, t+Δt | Effective Load (u(t) + Δs) | Estimated load (U(Δt) + Δs) | +| t, t+Δt | Effective Load (u(t) + Δs) | Estimated load (U(t) + Δs) | +| t, t+Δt | Effective Load (u(t) + Δs) | Estimated load (U(t) + Δs) | +| t, t+Δt | Effective Load (u(t) + Δs) | Estimated load (U(t) + Δs) | +| t, t+Δt | Effective Load (u(t) + Δs) | Estimated load (U(t) + Δs) | +| +| t, t+Δt | Effective Load (u(t) + Δs) | Estimated load (U(t) + Δs) | +| t, t+Δt | Effective Load (u(t) + Δs) | Estimated load (U(t) + Δs) | +| t, t+Δt | Effective Load (u(t) + Δs) | Estimated load (U(t) + Δs) | +| t, t+Δs | Effective Load (u(t) + Δs) | Estimated load (U(t) + Δs) | +| t, t+Δs | Effective Load (u(t) + Δs) | Estimated load (U(t) + Δs) | +| t, t+Δs | Effective Load (u(t) + Δs) | Estimated load (U(t) + Δs) | +| t, t+Δs | Effective Load (u(T) + Δs) | Estimated load (U(T) + Δs) | +| t, t+Δs | Effective Load (u(T) + Δs) | Estimated load (U(T) + Δs) | +| t, t+Δs | Effective Load (u(T) + Δs) | Estimated load (U(T) + Δs) | +| t, t+Δs | Effective Load (u(T) + Δs) | Estimated load (U(T) + Δs) | +| +| t, t+Δs | Effective Load (u(T) + Δs) | Estimated load (U(T) + Δs) | +| t, t+Δs | Effective Load (u(T) + Δs) | Estimated load (U(T) + Δs) | +| t, t+Δs | Effective Load (u(T) + Δs) | Estimated load (U(T) + Δs) | +| t, t+Δt | Effective Load (u(T) + Δs) | Estimated load (U(T) + Δs) | +| t, t+Δt | Effective Load (u(T) + Δs) | Estimated load (U(T) + Δs) | +| t, t+Δt | Effective Load (u(T) + Δs) | Estimated load (U(T) + Δs) | +| t, t+Δt | Effective Load (u(t) + Δs) | Estimated load (U(T) + Δs) | +| t, t+Δt | Effective Load (u(t) + Δs) | Estimated load (U(T) + Δs) | +| t, t+Δt | Effective Load (u(t) + Δs) | Estimated load (U(T) + Δs) | +| t, t+Δt | Effective Load (u(T) + Δs) | Estimated load (U(T) + Δs) | +| t, t+Δt | Effective Load (u(T) + Δs) | Estimated load (U(T) + Δs) | +| t, t+Δt | Effective Load (u(T) + Δs) | Extended load (u(T) + Δs) | +| t, t+Δt | Effective Load (u(T) + Δs) | Extended load (u(T) + Δs) | +| t, t+Δt | Effective Load (u(T) + Δs) | Extended load (u(T) + Δs) | +| t, t+Δt | Effective Load (u(T) + Δs) | Extended load (u(T) + Δs) | +
+ +$\Delta^{eff}\mathbf{R}_{t + \Delta t}$ ;Incremental Effective Load +$r_{t+\Delta t}$ ; Residual Force +그림 2.9.2 Newton-Raphson Method + + + +# 9-2 비탄성 요소 + +# 9-2-1 비탄성 보요소 + +midas Civil의 비탄성 보요소는 비탄성 힌지가 지정된 보요소입니다. 비탄성 보요소는 유연도법(Flexibility Method)에 의해 정식화되며, 하종이 재하되는 동안 미소변형, 평면보존의 가정을 전제로 하는 Euler Bernoulli Beam Theory를 따릅니다. 비틀림성분은 축력, 모멘트 성분과 연동하지 않는 것으로 가정합니다. + +비탄성 보요소는 기하학적 선형으로 정식화됩니다. 단, 사용자가 초기부재력을 Initial Element Force에서 입력하고, Initial Force Control Data의 Check to Reflect Initial Axial Forces into Geometric Stiffness를 선택한 경우, 입력되는 초기부재력에 의한 기하강성을 구성하여 요소강성에 더하는 방법으로 고려되며, 해석중에 기하강성은 갱신되지 않습니다. + +구조부재의 비탄성 거동을 추적하고 이를 통해 변위연성능력을 평가하기 위해서는 부재의 항복변형을 초과하는 변형영역에 대한 해석이 필수적으로 필요합니다. 하지만, 기존의 강성도법(Stiffness-Based Method)은 형상함수(Shape Function)에 기초하여 정식화 되므로, 비탄성 해석시에 실제 변형형상과 정식화에서 가정된 형상함 수간에 차이가 생길 수 있습니다. 유연도법(Flexibility Method)에 기초한 모델은 단면형상뿐만 아니라 단면력에도 형상함수를 적용하여 정식화되며, 유연도법에서의 부재내력(Elementsection Force)분포는 실제 분포와 일치하기 때문에 보다 정확한 해석이 가능합니다. 유연도법이 단면력에 대해 선형의 형상함수를 적용하는 것은 포물선 형태의 강성도 변화를 가정하는 것에 해당합니다. 이는 강성도법에서 3차 함수의 변형 형상함수를 사용하는 것이 선형의 곡률분포를 가정하는 것에 비교되며, 따라서 더 적은 수의 단면으로도 강성도법과 같은 정도의 결과를 낼 수 있는 수치적 이점을 갖습니다. 결과적으로 유연도법을 사용하면 보다 적은 수의 요소로 정확한 모델링이 가능하고 그에 따라 보다 빠른 해석 속도를 얻을 수 있는 것으로 알려져 있습니다. + +midas Civil의 비탄성 보요소는 부재의 비탄성 영역의 분포 여부 및 해석방법에 따라서, 집중형 힌지 모델과 분포형 힌지 모델로 구분됩니다. + + + +![](images/page-365_8b9d0b907fd6f66a50825f5ee725c6201f04089c9d713ee9cb5657663048bb5f.jpg) + +
+text_image + +Inelastic Hinge +M +M +Rigid +Zone +Elastic Beam +Rigid +Zone +
+ +(a) 집중형 힌지모델 + +![](images/page-365_aa6cbb09ec4473fbf0a2ad11dad52589a207f81ac462f06c91a4dedcbf7c601f.jpg) + +
+text_image + +M +Rigid +Zone +Integration Point +M +Rigid +Zone +
+ +(b) 분포형 힌지모델 +그림 2.9.3 비탄성 힌지 + +집중형 힌지모델은 지진하중이 작용하는 경우, 보요소의 역대칭 모멘트에 의해 부재단에 생기는 소성힌지를 효과적으로 모델링한 방법입니다. 따라서, 비탄성힌지는휨, 전단 성분인 경우, 요소양단에 위치하게 되고, 축성분 힌지는 요소중앙에 위치하며, 집중형 힌지모델의 휨성분이력은 휨모멘트-회전각의 관계로 표현됩니다. + +분포형 힌지모델은 부재내에 복수의 비탄성 힌지를 할당하여, 각 힌지 위치에서의탄소성판정에 의해 힌지의 강성을 갱신한 후에 수치적분에 의해 요소강성을 구성합니다. 분포형 힌지모델의 휨 성분 이력관계는 단면의 휨모멘트-곡률관계로 표현됩니다. + +집중형 힌지는 분포형 힌지에 비하여 계산량이 상대적으로 적다는 이점이 있으나그림 2.9.4에 나타낸 것과 같이 부재력의 분포를 임의로 가정해야 하므로, 이 가정을 크게 벗어나는 경우에는 부정확한 결과를 초래할 수 있습니다. 또한, 집중형 힌지는 비탄성 힌지가 부재양단에 위치하므로, 비탄성 변형 영역의 확장은 무시됩니다. 반면에 분포형 힌지는 부재내의 비탄성 힌지수에 따라 계산시간이 늘어나는단점이 있으나, 부재력 분포를 보다 정확하게 반영할 수 있으며, 요소내 임의의 단면에서 일어나는 비탄성 거동을 파악할 수 있으므로, 집중형 힌지에 비해 정확한해석이 가능합니다. + +midas Civil에서는 하나의 보요소에 속한 힌지들은 동일한 속성을 갖도록 제한하고있습니다. 따라서, 교량의 상부구조와 같이 변단면(Tapered Beam)을 갖는 부재는실제 계산시에는 양단의 강성을 평균하여 등단면 보요소(Prismatic Beam)로 처리하고 있습니다. 따라서, 단면의 변화가 매우 심한 변단면 요소의 경우는 변단면을 등단면으로 치환하여도 해석결과에 큰 영향이 미치지 않을 정도로 적절히 분할하여모델링하는 것이 바람직하다. + + + +# 집중형 힌지모델 + +집중형 힌지(Lumped Type Hinge Model)는 소성변형이 가능한 길이가 없는 병진 또는 회전 비탄성 스프링을 보요소에 삽입하여 모델링되며, Inelastic Hinge Properties의 Lumped Type으로 정의합니다. 보 요소에서 집중형 힌지를 제외한 나머지 부분은 탄성 보 요소로 모델링됩니다. 비탄성 스프링은 축변형 성분에 대해서는 부재중앙, 휨과 전단성분에 대해서는 부재 양 단부에 위치합니다. + +![](images/page-366_93cca4265a31d298cdbea24a3d0d930117cd8028a8aa57909966476ce82183d5.jpg) + +
+text_image + +cord rotation +angle: θ +M +M +Inelastic Hinge +M +Rigid +Zone +Elastic Beam +Inelastic Spring +Rigid +Zone +M +
+ +그림 2.9.4 집중형 힌지 모델 + +힌지를 정의하는 비탄성 스프링은 축방향 변형의 경우에는 힘-변위 관계로, 휨 변형의 경우에는 단부에서의 모멘트-회전각 관계로 정의됩니다. 집중형 힌지가 부여된 보요소의 강성행렬은 유연도 행렬의 역행렬로 계산되며, 전체 보요소의 유연도행렬은 비탄성 스프링의 유연도 행렬과 탄성보의 유연도 행렬을 더해서 구성됩니다. 비탄성 스프링의 유연도는 사용자가 정의한 집중형 힌지의 접선 유연도와 초기 유연도의 차이로 정의되며 항복하기 전에는 영(Zero)이고, 항복하면서 유연도가발생합니다. 비탄성힌지의 접선 유연도 행렬은 뒤에서 설명하는 일축 또는 다축-힌지 이력모델에 의해 정의됩니다. + +$$ +F _ {S} = F _ {H} - F _ {H 0} +$$ + +$$ +F = F _ {B} + \sum F _ {S} +$$ + +$$ +K = F ^ {- 1} +$$ + + + +여기서 FH : 비탄성힌지의 유연도 행렬 + +FH0 : 비탄성힌지의 초기 유연도 행렬 + +FS : 비탄성 스프링의 유연도 행렬 + +FB : 탄성 보의 유연도 행렬 + +F : 비탄성보의 요소 유연도 행렬 + +K : 비탄성보의 요소 강성 행렬 + +![](images/page-367_11e225479e583c0ee1f06714da66438d0b44195003dc09d2c56d2792b9cc4b95.jpg) + +
+line + +| θ | M | +|-------|-------| +| 0 | 0 | +| θ_e | M | +| θ_p | M | +| θ_s | 1/F_S | +
+ +Flexibility & Inelsastic Deformation of Inelastic Spring based on Hysteresis Model + +![](images/page-367_75e249025738d78c0f6c4d691c496675f0c0e494764e2e2a39a64a98d6a2ca96.jpg) +Initial Flexibility & Elastic Deformation of Inelastic Spring + ++ +![](images/page-367_fdca8b81a0017bb91167f870fdf4daaff73bbdb46e77664ad6341f50c6b5c7af.jpg) + +
+text_image + +M +θ_p +1/F_H +θ +
+ +Flexibility & Deformation of Inelastic Hinge +그림 2.9.5 집중형 비탄성힌지의 유연도 + +휨 변형 힌지의 모멘트-회전각 관계는 단부의 휨 모멘트뿐만 아니라 부재 중간의휨모멘트 분포에 의해서도 영향을 받습니다. 따라서 휨 변형 힌지의 모멘트-회전각관계를 결정하기 위해서는 휨 모멘트의 분포를 가정해야 합니다. 그림 2.9.6은 모멘트 분포 가정과 그에 따른 초기강성입니다. + +
Deflection ShapeMoment DistributionInitial Stiffness
M6EI/L
M3EI/L
M2EI/L
+ +그림 2.9.6 휨 변형에 대한 비탄성 힌지의 초기강성 (전체길이=L, 단면휨강성=EI) + + + +# 분포형 힌지모델 + +분포형 힌지 모델(Distributed Type Hinge Model)은 요소 유연도 행렬계산에 필요한재축방향 적분점에서의 단면 유연도에 의해 정의됩니다. 분포형 힌지가 부여된 보요소의 유연도 행렬은 다음 식으로 정의되며 Gauss-Lobbato 적분을 통하여 계산됩니다. 요소 강성행렬은 유연도 행렬의 역행렬로 계산됩니다. 재축방향 적분점에서의 부재단면의 유연도는 일축 또는 다축-힌지 이력모델에 의거한 상태판정으로부터 결정됩니다. 분포형 힌지모델의 각 힌지는 파이버 모델로도 모델링이 가능합니다. 분포형 힌지 모델은 Inelastic Hinge Properties의 Distributed Type으로 설정합니다. 비탄성 힌지는 축 성분의 경우에는 단면에서의 힘-변형율 관계로, 휨 성분의경우에는 모멘트-곡률 관계로 정의됩니다. + +$$ +F = \int_ {0} ^ {L} b ^ {T} (x) f (x) b (x) d x +$$ + +$$ +K = F ^ {- 1} +$$ + +여기서 f(x) : 위치 x에서의 단면의 유연도 행렬 + +b(x) : 위치 x에서의 부재력 분포 함수 행렬 + +F : 요소 유연도 행렬 + +K : 요소 강성도 행렬 + +L : 부재 길이 + +x : 단면의 위치 + +![](images/page-368_baa03d7e75e510c7ee25acc5638c43e4c877000fb479014d4bf5c0e3e38e0de3.jpg) + +
+bar +| Position | Flexibility | +| -------- | ----------- | +| Left | High | +| Center | Low | +| Right | High | +
+ +Flexibility Distribution + +![](images/page-368_50e5625b3248622ae4cd5114d0642d160850472b674c98099b1e3c8e63b0591f.jpg) + +
+line + +| Curvature | Bending Moment | +| --------- | -------------- | +| 1 | 1 | +
+ +Tri-linear Skeleton Curve +그림 2.9.7 분포형 힌지 모델 + + + +보-기둥 요소의 비탄성 거동은 주로 부재의 단부에 집중되는 경우가 많습니다. 따라서, 기존의 Gauss-Legendre 적분법으로는 부재단을 적분점으로 취할수 없기 때문에, midas Civil에서는 요소단부의 단면을 적분점으로 취할 수 있는 Gauss-Lobatto 적분법을 사용하여 분포형 힌지 요소의 부재 유연도행렬을 유도합니다. + +적분점의 개수는 요소내의 비탄성 힌지의 개수를 의미하며 1개에서 최대 20개까지설정가능합니다. 적분점의 위치는 그림 2.9.8에 나타낸 것과 같이 적분점의 갯수에의해 정해지며, 양 단부로 갈수록 적분점 사이의 간격이 좁아지게 됩니다. 단,Gauss-Lobatto법은 요소 단부에서 적분점을 취하는 관계로, 적분점이 2개의 처리가 불가능하며, 적분점이 2개인 경우에는 Classical Gauss Integration을 사용하여유연도 행렬을 구성합니다. + +또한, 적분점의 수와 정확도는 반드시 비례하는 것이 아니며, 적분점의 개수가 많을수록 힌지의 상태 판정에 요구되는 계산량이 증가하는 단점이 있습니다. 적분점의 수와 해의 정확도를 검토한 연구결과에 의하면, 적분점의 수가 5개 이상일 때결과 차이가 거의 없는 것으로 알려져 있습니다. 따라서, 요소의 길이와 요소분할수에 의한 영향은 있겠지만, 대략 5개 이하가 적절합니다. + +![](images/page-369_9b9736261657c0a3b0542f923f47e0cee2426ac996cf838ba593c7a966f21dfc.jpg) +(a) 적분점=1 + +![](images/page-369_e251ab6171aea9d5e071d392acc04ecb70251a37d2356f666fb7c3043021b624.jpg) +(b) 적분점=2 + +![](images/page-369_7930399a78395558553f9f6e4d7b26c1c85478e12e0d26853f10e6756da9bba5.jpg) +(c) 적분점=3 + +![](images/page-369_6324cb3d3a941cf71850d9ac19932082385a04fd938575cdab3832d869c605c0.jpg) +(d) 적분점=4 + +![](images/page-369_32e80ba261accd3f4c0ba9c62e132d63e74015b81a2bcc1735e8e75cd46ac4cb.jpg) +(e) 적분점=5 + +![](images/page-369_0448568b82c9b042fe7a5204df02cdb007af2ad33c492617d17db05d8f700afe.jpg) +(f) 적분점=6 +그림 2.9.8 Gauss-Lobatto Integration에서의 적분점 위치 + + + +# 9-2-2 비탄성 범용 연결 요소 + +범용연결요소(General Link)는 요소좌표계의 x, y, z의 3방향 신장 및 회전을 표현하는 6개의 스프링으로 2절점을 연결하는 요소입니다. midas Civil의 범용연결요소가운데 Inelastic Hinge Properties을 부여할 수 있는 것은 Element Type의 Spring으로 제한됩니다. 범용연결요소는 단순히 각 성분별로 탄성강성만을 갖고 있으며Inelastic Hinge Properties를 부여함으로써 비선형 요소가 되며, 이력모델에 의해 비탄성 해석을 수행합니다. + +비탄성 범용연결요소는 구조물의 특정 부분 또는 지반의 소성변형이 하나의 스프링에 집중된 것으로 모델링하는 경우에 사용되며, Inelastic Hinge Properties에서Spring Type으로 정의 됩니다. 범용연결요소는 일반구조부재와 달리 부재의 재료제원이나 단면특성을 정의할 수 없으므로, 계산에 의해 요소강성을 산정할 수 없습니다. 따라서, 사용자가 각 성분별로 강성을 정의해야만 하며, 입력된 강성은 비선형 해석시에 초기강성으로 사용됩니다. + +![](images/page-370_090251038416f0939c8df35179b22dc24e35fb90f2acadbad5c2e3311e48648d.jpg) + +
+text_image + +jointi +kₓ +kᵧ +k_z +k_θₓ +k_θᵧ +k_θ_z +joint j +
+ +그림 2.9.9 범용연결요소의 스프링 강성 diff --git a/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_038.md b/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_038.md new file mode 100644 index 00000000..e1887363 --- /dev/null +++ b/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_038.md @@ -0,0 +1,360 @@ + + +# 9-2-3 비탄성 트러스 요소 + +비탄성 트러스 요소는 축방향 강성만을 갖는 요소로서 Inelastic Hinge Properties에서 Truss Type 으로 정의됩니다. 비탄성 트러스 요소의 비선형성은 축방향 성분만이 정의 가능하며, Truss Type 비탄성 힌지는 일축힌지 이력모델에 의거한 상태판정으로부터 강성을 갱신하여, 요소강성을 재구성합니다. + +비탄성 트러스 요소는 기하학적 선형으로 정식화됩니다. 단, 비탄성 보요소와 마찬가지로 사용자가 초기부재력을 Initial Element Force에서 입력하고, Initial ForceControl Data의 Check to Reflect Initial Axial Forces into Geometric Stiffness를 선택한 경우, 입력되는 초기부재력에 의한 기하강성을 구성하여 요소강성에 더하는 방법으로 고려됩니다. 단, 해석중에 기하강성은 갱신되지 않습니다. + +![](images/page-371_5e11b54fdbe8a6453b4d16f22ef1ce9ff4a14b02d0529c035eeb28d0b5a1fc40.jpg) + +
+text_image + +i +kₓ +j +nᵢ, uᵢ +nⱼ, uⱼ +i +j +n, u +L +
+ +그림 2.9.10 비탄성 트러스요소와 축방향 강성 + + + +# 9-3 비선형 이력 모델의 개요 + +구조물이 지진하중과 같은 불규칙한 반복하중을 받아서, 균열, 항복 등이 발생하면, 현재까지의 변위이력이 이후의 복원력-변위관계에 영향을 미치기 때문에 정적하중을 받을때와 달리 매우 복잡한 거동을 나타냅니다. 부재의 1방향하중에 대한 힘과 변형의 관계를 골격곡선이라 합니다. 이력모델은 골격곡선을 기본으로 하여, 정(+), 부(-)의 반복하중이 작용할 때, 제하시(UnLoading)시와 재재하(Re-Loading)시의 힘과 변형의 관계를 규칙화한 것으로, 비선형해석시에는 부재의 복원력특성을 이력모델로 정의하는 것이 일반적입니다. + +이력모델은 해석용 최소모델 단위인 부재단면의 거동특징을 구성재료의 응력-변형율관계, 단면의 힘모멘트-곡률관계, 부재단의 힘모멘트-회전각관계 등의 간단한 힘과 변형의 관계로 이상화한 것으로, 전체 하중이력에 대해서 하중과 변형관계로 표현 가능하여야 하며, 실험시에 시험체에서 관측되는 공통된 특성을 반영할 수 있어야 합니다. + +비선형 해석시에는 사용하는 이력모델과 비선형 해석조건의 설정에 의해 해석결과가 크게 달라질 수 있으므로, 해석결과를 적절히 얻기 위해서는 모델링시에 충분한 검토를 통하여 사용재료와 부재의 복원력특성을 충실히 반영할 수 있는 이력모델을 선택하여야만 합니다. 표 2.9.1는 midas Civil에서 제공하는 이력을 용도에 따라 분류한 것입니다. + +표 2.9.1의 축력-모멘트 상호작용에 대한, P-M, P-M-M Type에 대해서는 9-5-2 P-M 및 P-M-M 상관작용에서 다루도록 합니다. + +# 9-3-1 비선형 힌지 속성 + +비탄성 힌지의 속성은 집중형(Lumped Type), 분포형(Distributed Type), 스프링형(Spring Type), 트러스형(Truss Type)으로 구분됩니다. 집중형 및 분포형은 보요소에만 적용되며 용수철형은 범용연결요소(General Link), 트러스형은 트러스요소에 적용됩니다. + + + +비탄성힌지 속성은 각 성분 별로 정의된 비선형 거동특성의 집합으로, 보요소인경우, 비틀림을 제외한 5개성분, 범용연결요소는 6개성분, 트러스 요소는 축성분만정의 가능합니다. 여기서 비탄성 힌지의 비선형 거동특성은 이력모델에 의해 정의되며, 각 성분의 특성은 독립적으로 일축-힌지 이력모델(Uni-axial HingeHysteresis Model)에 의해서 정의되거나, 축력-모멘트 성분의 상호작용을 고려한 다축-힌지 이력모델(Multi-axial Hinge Hysteresis Model)에 의해서 정의될 수 있습니다. + +\*B : Beam, T : Truss, S : Spring + +
분 류이력모델적용요소축력-모멘트상호작용주요용도
Simplified Model이동경화형(Kinematic Hardening/Trilinear)B, T, SP-M, P-M-M강재
원점지향형(Origin-oriented/Trilinear)B, T, SP-M교량 상부구조
최대점지향(Peak-oriented/Trilinear)B, T, SP-M교량 상부구조
노말 2선형(Normal Bilinear)B, T, SP-M강재, 간략모델화
Degrading Model클러프(Clough/Bilinear)형B, T, SP-M철근 콘크리트 부재
강성저감3선형(Degrading Tri-linear)형B, T, SP-M철근 콘크리트 부재
다케다(Original Takeda Trilinear)형B, T, SP-M철근 콘크리트 부재
다케다(Original Takeda Tetralinear)형B, T, SP-M철근 콘크리트 부재
수정 다케다(Modified Takeda Trilinear)형B, T, SP-M철근 콘크리트 부재
수정 다케다(Modified Takeda Tetralinear)형B, T, SP-M철근 콘크리트 부재
Nonlinear Elastic탄성 2선형(Elastic Bilinear)B, T, SP-M교량 상부구조
탄성 3선형(Elastic Trilinear)B, T, SP-M교량 상부구조
탄성 4선형(Elastic Tetralinear)B, T, SP-M교량 상부구조
Slip Model슬립(Slip Bilinear)형B, T, SP-M강재, 고무 Support
슬립(Slip Bilinear/Tension)형B, T, SP-M강재, 고무 Support
슬립(Slip Bilinear/Compression)형B, T, SP-M강재, 고무 Support
슬립(Slip Trilinear)형B, T, SP-M강재, 고무 Support
슬립(Slip Trilinear/Tension)형B, T, SP-M강재, 고무 Support
슬립(Slip Trilinear/Compression)형B, T, SP-M강재, 고무 Support
Special ModelRamberg OsgoodS-비선형 지반용
Hardin DrnevichS-비선형 지반용
+ +표 2.9.1 midas Civil의 이력모델 분류(1) + + + +\*B : Beam, T : Truss, S : Spring + +
분 류이력모델적용요소축력-모멘트상호작용주요 용도
Multi-LinearModel탄성형(Elastic)B, T, S-간략모델화
이동경화형(Plastic Kinematic)B, T, S-강재, 간략모델화
다케다형(Plastic Takeda)B, T, S-철근 콘크리트 부재
피봇형(Plastic Pivot)B, T, S-철근 콘크리트 부재
+ +표 2.9.1 midas Civil의 이력모델 분류(2) + +# 9-3-2 보요소의 항복강도 + +비탄성 흰지에 설정되는 이력모델은 항복강도와 항복 후의 강성저감율로 정의됩니다. 요소의 항복강도는 사용자가 User Input으로 입력하거나, midas Civil의 항복강도 자동계산기능을 통해서 설정가능합니다. midas Civil의 자동계산기능에 의한, 휈에 의한 보요소의 항복은 그림 2.9.11과 같이 정의됩니다. 철골 단면의 경우에 1차 항복은 중립축으로부터 가장 먼 위치의 휈 응력이 항복응력에 도달한 것으로 간주합니다. 2차 항복은 전단면의 휈응력이 항복응력에 도달한 것으로 간주합니다. RC 단면의 경우에 1차 항복은 중립축으로부터 가장 먼 위치의 휈 응력이 콘크리트의 균열응력에 도달한 것으로 간주합니다. 2차 항복은 콘크리트의 압축연단이 극한변 형율에 도달한 것으로 간주하며 이 때 철근의 응력은 항복응력보다 작거나 같습니다. SRC 단면의 경우에 콘크리트 충전강관 형태인 경우에는 철골단면, 콘크리트 피복형인 경우에는 RC단면의 계산 기준을 적용합니다. + +축력과 모멘트의 상관작용을 고려하고자 하는 P-M, P-M-M Type의 경우에는 축력에 의한 중리축의 이동을 고려하여 축력-모멘트의 상관곡선(항복곡면)을 작성해야 하며, 이 경우에도 자동계산이 가능합니다. + + + +# 9-4 일축-힌지 이력모델(Hysteresis Model for Uni-axial Hinge) + +일축-힌지(Uni-axial Hinge) 모델은 3개의 병진 및 3개의 회전 성분이 상호 독립적으로 거동하는 히지입니다. midas Civil에서 일축-힌지를 대상으로 제공되는 이력모델은 골격곡선(Skeleton Curve)에 기초하고 있는 것들로서, 표 2.9.1의 모든 이력모델이 일축-힌지로 정의가능합니다. 이들 모델은 비대칭의 단면 혹은 재료 특성에 대응할 수 있도록 1, 2차 항복 강도 및 강성저감률을 정(+), 부(-) 비대칭으로 지정할 수 있습니다. 단, 이동경화형 모델의 경우, 이력의 특성상 강성저감률은 비대칭성을 지원하지 않습니다. + +이하의 이력모델 설명에 있어서 응답점(Response Point)은 이력모델의 경로상에 위치한 하중-변형 좌표점을 의미합니다. 재하(Loading)는 하중의 절대치가 증가함을, 제하(Unloading)는 하중의 절대치가 감소함을, 재재하(Re-loading)은 제하 도중에 하중의 부호가 바뀌면서 절대치가 증가하는 것을 의미합니다. 제하점(Unloading Point)은 재하에서 제하로 바뀌는 응답점을 의미합니다. + +![](images/page-375_c5595499821201c292edcc8fc990b98bec0127e232959a82ce3c79852a7d63a4.jpg) + +
+text_image + +P (+) +Mz (+) +y +z +My (+) +
+ +![](images/page-375_24fe7c52c7424c0d28ff74dba9fb435290b528629d4c2c4c7761b4a3c64b6bf8.jpg) + +
+text_image + +1st Yielding +Compression +Tension +Dc +Dt +Strain +Fsc +Fy +N.A. +α ≤ Fy / Es ++ ++ +Fst +Fy +α ≤ Fy / Es +
+ +![](images/page-375_bdf95c05da8737027806889712bbb0b2bad1f9002264485f44db765095cb5b82.jpg) + +
+text_image + +2nd Yielding +Compression +Tension +Dc +Dt +Strain +ε ≤ Fy/Es ++ +- +Fsc +Fy +Stress +Fst +Fy +N.A. +
+ +$D_{c}$ : Center of Steel Compressive Force $D_{t}$ : Center of Steel Tensile + +(a) Steel + + + +![](images/page-376_904dde8f84a13fea7750efc74feb0e9089ce4057f98eb2a6fe75fde812365bc0.jpg) + +
+text_image + +P (+) +Mz (+) +y +z +My (+) +
+ +1st Yielding (Cracking) + +$$ +M _ {\sigma} = k \sqrt {f _ {d}} Z - \frac {Z}{A} N +$$ + +Mcr : Cracking Moment + +k : Coefficien t for Cracking Moment +(ACI =7. 5 in lb -in u nit, A IJ= 1.8 in kg f-cm un it ) +fck : Specified Comp res sive Strength of Concrete +Z : Elast ic S ection Mod ulus + +2nd Yielding +![](images/page-376_23b4167be163e18a662e2daf48f40fea6cc58dbd150e7019ae88c7bc40280887.jpg) + +
+text_image + +Compression +Tension +D_{s1} +D_{s2} +D_{s3} +D_{s4} +Strain +Stress +1: f_{ck} +f_{k1} +f_{k2} +f_{k3} +f_{k4} +f_{k5} +f_{k6} +f_{k7} +f_{k8} +f_{k9} +f_{k10} +N.A. +
+ +Dc : Center of Concrete Compres sive Force + +(b) RC + +그림 2.9.11 보 요소의 항복강도 산정 기준 + +철근 콘크리트부재의 경우, 콘크리트의 균열, 철근의 항복에 의한 강성저감이 일어납니다. 또한, 반복하중이 작용하는 경우, 항복후의 제하시에도 강성이 저하되어하중의 방향이 바뀌면 과거에 경험한 최대변위점을 지향하는 특징이 있습니다. 철근콘크리트 부재의 복원력 특성을 모델화한 이력모델은 다수 제안되어 있지만, 어느 모델에도 강성저하와 최대점지향이 필수적으로 고려되어 있습니다. 철근콘크리트 모델을 대표하는 것이 Takeda 모델이며, Clough형, 강성저감3선형 등도 사용됩니다. + +강재는 한번 어느 방향에서 소성변형을 받은 후 역방향의 하중이 작용하면, 소성변형을 받지 않은 강재에서 동일한 방향의 하중이 작용한 경우보다 작은 응력에서소성화하는 것으로 알려져 있습니다. 이것을 바우싱거 효과(Bauschinger Effect)라고 합니다. 또한 변형율이 크게 되면 응력이 증대하는 성질, 즉 변형율 경화(StrainHardening)가 일어납니다. 이와 같은 성질을 갖는 강재의 복원력 특성은 이동경화형의 Normal Bilinear 이력모델로 표현하는 것이 일반적이며, Normal Trilinear 이력모델을 적용하는 경우도 있습니다. + +콘크리트로 충진된 강재교각의 비선형 특성은 Takeda 이력 혹은, 항복점에서 강성이 변화하는 이동경화형 Normal Bilinear로 표현됩니다. Normal Bilinear 이력은 철 + + + +근 콘크리트 부재와 달리 강성저하가 일어나지 않는 이력곡선을 그리도록 정의됩니다. + +콘크리트로 충진되지 않은 강재교각은 Normal Bilinear로 표현하는 것이 일반적입니다. 한편, 교각은 중력하중에 의한 압축력의 작용으로 인해, 압축측에서 항복한후에 인장측에서 항복이 일어나게 되므로, 압축측과 인장측이 항복에 도달하는 하중이 다른 것을 고려하여 강부재의 골격곡선을 3개의 직선의 Normal Trilinear로 표현하는 경우도 있습니다. + +# 9-4-1 Normal Bilinear Type + +# 이력의 개요 + +초기 재하시의 응답점은 2선형 골격곡선상에서 이동합니다. 제하(Unloading)강성은탄성강성과 동일하며, 항복 후 강성 저감률은 정(+), 부(-) 비대칭 정의가 가능합니다. 집중형 및 분포형 힌지 요소, 스프링 요소, 트러스 요소 등에 적용가능합니다. + +![](images/page-377_365892140fb9f184892320692666ecf9aa8bca98a4c2d93a02ca1388f72f1c2b.jpg) + +
+text_image + +P +P1(+) +K2(+) +K0 +D1(-) +K0 +D1(+) +K0 +D +K2(-) +P1(-) +
+ +그림 2.9.12 Normal Bilinear 이력모델 + + + +# 골격곡선의 정의 + +이력모델의 비선형특성은 이하의 값으로 정의됩니다. + +$$ +P 1 _ {(+)}, P 1 _ {(-)} \quad : (+), (-) \text { 측 제1차항복강도 } +$$ + +$$ +D 1 _ {(+)}, D 1 _ {(-)} \quad : (+), (-) \text {측 제1차항복변형} +$$ + +$$ +K _ {0} \quad : \text { 초기강성 } +$$ + +$$ +\mathrm{w} _ {0} (+) = \mathrm{w} _ {0} (-) \quad : (+), (-) \text { 측제2강성. } +$$ + +$$ +\text { 단, } K 2 ^ {(+)} = \alpha 1 ^ {(+)} \cdot K _ {0}, \quad K 2 ^ {(-)} = \alpha 1 ^ {(-)} \cdot K _ {0} +$$ + +$$ +\alpha 1 ^ {(+)}, \alpha 1 ^ {(-)} \quad : (+), (-) \text {측 제1차항복후의 강성저감율 } +$$ + +# 노말 2선형(Normal Bilinear Type)의 이력규칙 + +1. $\left|D_{max}\right| + +# 9-4-2 Kinematic Hardening Type + +# 이력의 개요 + +초기 재하시의 응답점은 3선형 골격곡선상에서 이동합니다. 제하(Unloading)강성은탄성강성과 동일하며, 하중이 증가하면서 강도가 증가하는 경향을 보이는데, 이는금속 재료의 바우싱거효과(Bauschinger Effect)를 모델링하는데 사용되는 것입니다.따라서 콘크리트의 경우에는 에너지 소산량을 과대 평가할 수 있으므로 주의해야합니다. 항복 후 강성 저감률은 모델의 특성상, 정(+), 부(-) 대칭만이 정의가 가능하며, 집중형 힌지 및 분포형 힌지 요소, 스프링 요소, 트러스 요소 등에 적용가능합니다. + +![](images/page-379_c16d2dc6f0dd444510305cbfacea09b22cd0f8641f76b322c30553a9f016ada0.jpg) + +
+text_image + +P +P2(+) +P1(+) +K2(+) +K3(+) +K0 +K0 +K0 +D2(-) +D1(-) +D1(+) +D2(+) +D +K0 +K2(-) +K3(-) +P1(-) +P2(-) +
+ +그림 2.9.13 Kinematic Hadening 이력모델 + + + +# 골격곡선의 정의 + +이력모델의 비선형특성은 이하의 값으로 정의됩니다. + +$$ +P 1 _ {(+)}, P 1 _ {(-)} \quad : (+), (-) \text { 측 제1차항복강도 } +$$ + +$$ +P 2 _ {(+)}, P 2 _ {(-)} \quad : (+), (-) \text { 측 제2차항복강도 } +$$ + +$$ +D 1 _ {(+)}, D 1 _ {(-)} \quad : (+), (-) \text { 측 제1차항복변형 } +$$ + +$$ +D 2 _ {(+)}, D 2 _ {(-)} \quad : (+), (-) \text { 측 제2차항복변형 } +$$ + +$$ +K _ {0} \quad : \text { 초기강성 } +$$ + +$$ +: (+), (-) \text { 측 제2강성. } +$$ + +$$ +\begin{array}{l} K 2 ^ {(+)}, K 2 ^ {(-)} \quad \cdot (1), (-) \text {是} \text {제288}. \\ \text { 단, } K 2 ^ {(+)} = \alpha 1 ^ {(+)} \cdot K _ {0}, \quad K 2 ^ {(-)} = \alpha 1 ^ {(-)} \cdot K _ {0} \\ \end{array} +$$ + +$$ +\kappa_ {2} (+) - \kappa_ {2} (-) \quad : (+), (-) \text { 측 제3강성. } +$$ + +$$ +\text { 단, } K 3 ^ {(+)} = \alpha 2 ^ {(+)} \cdot K _ {0}, \quad K 3 ^ {(-)} = \alpha 2 ^ {(-)} \cdot K _ {0} +$$ + +$$ +\alpha 1 ^ {(+)}, \alpha 1 ^ {(-)} \qquad \qquad : (+), (-) \text {측 제1차항복후의 강성저감율 } +$$ + +$$ +\alpha 2 ^ {(+)}, \alpha 2 ^ {(-)} \quad : (+), (-) \text { 측 제2차항복후의 강성저감율 } +$$ + +# 이동경화형(Kinematic Hardening Type)의 이력규칙 + +1. $\left|D_{max}\right|D2$ 의 경우, Trilinear로 이동합니다. +3. 재하시는, Masing의 법칙에 의해 탄성강성의 직성상에서 이동합니다. diff --git a/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_039.md b/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_039.md new file mode 100644 index 00000000..6d3fbe6f --- /dev/null +++ b/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_039.md @@ -0,0 +1,461 @@ + + +# 9-4-3 Origin-Oriented Type + +# 이력의 개요 + +초기 재하시의 응답점은 3구배 골격곡선 상에서 이동합니다. 제1차항복 혹은 제2차항복후에 제하(Unloading)되는 경우, 원점을 지향하는 직선상을 이동합니다. 제하과정에서 재재하(Re-loading)되는 경우는, 제하시와 같은 구배의 직선상을 이동하여, 골격곡선과 만나면, 골격곡선상에서 이동합니다. 입력에 의해, 대칭 및 비대칭이 정의가능하며, 대응요소는 집중형 힌지 및 분포형 힌지 요소, 스프링 요소,트러스 요소 등입니다. 또한, Origin-Oriented Type은 (+), (-)측의 초기강성을 비대칭으로 고려할 수 있습니다. + +![](images/page-381_65cc56bae09626319df92415f6a773ca21b9e017cd28d2733ff077f5cb27ca48.jpg) + +
+flowchart + +```mermaid +graph TD + P -->|P2(+)| K3 + P -->|P1(+)| K2 + P -->|P1(-)| K0 + P -->|P2(-)| K3 + D1(-) -->|D2(-)| K0 + D1(+) -->|D1(+)| K2 + D2(+) -->|D2(-)| K0 + D2(-) -->|D2(-)| K3 + K3 -->|P1(-)| K2 + K2 -->|P2(-)| K3 + K0 -->|K0| K3 + style P fill:#f9f,stroke:#333 + style D fill:#ccf,stroke:#333 + style D1(-) fill:#cfc,stroke:#333 + style D2(-) fill:#fcc,stroke:#333 + style D1(+) fill:#cff,stroke:#333 + style D2(+) fill:#ffc,stroke:#333 + style D2(-) fill:#cfc,stroke:#333 + style K3 fill:#fcc,stroke:#333 + style K2 fill:#fcc,stroke:#333 + style K0 fill:#fcc,stroke:#333 +``` +
+ +그림 2.9.14 Origin-Oriented 이력모델 + + + +# 골격곡선의 정의 + +이력모델의 비선형특성은 이하의 값으로 정의됩니다. + +$$ +P 1 _ {(+)}, P 1 _ {(-)} \quad : (+), (-) \text { 측 제1차항복강도 } +$$ + +$$ +P 2 _ {(+)}, P 2 _ {(-)} \quad : (+), (-) \text { 측 제2차항복강도 } +$$ + +$$ +D 1 _ {(+)}, D 1 _ {(-)} \quad : (+), (-) \text {측 제1차항복변형} +$$ + +$$ +D 2 _ {(+)} , D 2 _ {(-)} \quad : (+), (-) \text { 측 제2차항복변형 } +$$ + +$$ +K _ {0} \quad : \text { 초기강성 } +$$ + +$$ +w _ {0} (+) \quad w _ {0} (-) \quad : (+), (-) \text { 측 제2강성. } +$$ + +$$ +\text { 단, } K 2 ^ {(+)} = \alpha 1 ^ {(+)} \cdot K _ {0}, \quad K 2 ^ {(-)} = \alpha 1 ^ {(-)} \cdot K _ {0} +$$ + +$$ +K 3 ^ {(+)}, K 3 ^ {(-)} \quad : (+), (-) \text { 측 제3강성. } +$$ + +$$ +\text { 단, } K 3 ^ {(+)} = \alpha 2 ^ {(+)} \cdot K _ {0}, \quad K 3 ^ {(-)} = \alpha 2 ^ {(-)} \cdot K _ {0} +$$ + +$$ +\alpha 1 ^ {(+)}, \alpha 1 ^ {(-)} \qquad \qquad : (+), (-) \text {측 제1차항복후의 강성저감율 } +$$ + +$$ +\alpha 2 ^ {(+)}, \alpha 2 ^ {(-)} \quad : (+), (-) \text {측 제2차항복후의 강성저감율 } +$$ + + + +# 9-4-4 Peak-Oriented Type + +# 이력의 개요 + +초기 재하시의 응답점은 3구배 골격곡선 상에서 이동합니다. 제1차항복 혹은 제2차항복후에 제하(Unloading)되는 경우, 반대측의 최대 변형점을 지향하는 직선상을이동합니다. 반대측이 1차 항복하지 않은 경우는 1차 항복점이 최대 변형점이 됩니다. 제하과정에서 재재하(Re-loading)되는 경우는, 제하시와 같은 구배의 직선상을 이동하여, 골격곡선과 만나면, 골격곡선상에서 이동합니다. 입력에 의해, 대칭및 비대칭이 정의가능하며, 대응요소는 집중형 힌지 및 분포형 힌지 요소, 스프링요소, 트러스 요소 등입니다. Inelastic Hinge Properties의 Directional HingeProperties에서 Input Type을 Strength-Yield Displacement를 선택하여, 정(+),부(-)축의 1차 항복변위를 이용하여 초기강성을 (+),(-)측 비대칭으로 입력하여 고려할 수있습니다. + +![](images/page-383_81dae41f1b58baf19424fda92e59fb9b7473bb0d0cbfb640535c60ab4422fdb5.jpg) + +
+flowchart + +```mermaid +graph TD + K0 --> K2 + K2 --> K3 + K3 --> P1 + K3 --> P2 + K2 --> D1 + K2 --> D2 + K0 --> D1 + K0 --> D2 + K2 --> D1 + K2 --> D2 + K3 --> D1 + K3 --> D2 + K1 --> D1 + K1 --> D2 + K2 --> D1 + K2 --> D2 + K3 --> D1 + K3 --> D2 + K1 --> D2 + K2 --> D2 + K3 --> D2 + K0 --> D1 + K0 --> D2 + K2 --> D1 + K2 --> D2 + K3 --> D1 + K3 --> D2 +``` +
+ +그림 2.9.15 Peak-Oriented 이력모델 + + + +# 골격곡선의 정의 + +이력모델의 비선형특성은 이하의 값으로 정의됩니다. + +$$ +P 1 _ {(+)}, P 1 _ {(-)} \quad : (+), (-) \text { 측 제1차항복강도 } +$$ + +$$ +P 2 _ {(+)} , P 2 _ {(-)} \qquad \quad : (+), (-) \text { 측 제2차항복강도 } +$$ + +$$ +D 1 _ {(+)}, D 1 _ {(-)} \quad : (+), (-) \text { 측 제1차항복변형 } +$$ + +$$ +D 2 _ {(+)}, D 2 _ {(-)} \quad : (+), (-) \text {측 제2차항복변형} +$$ + +$$ +K _ {0} \quad : \text { 초기강성 } +$$ + +$$ +\kappa \mathfrak {e} ^ {(+)} - \kappa \mathfrak {e} ^ {(-)} \quad : (+), (-) \text { 측 제2강성. } +$$ + +$$ +\text { 단, } K 2 ^ {(+)} = \alpha 1 ^ {(+)} \cdot K _ {0}, \quad K 2 ^ {(-)} = \alpha 1 ^ {(-)} \cdot K _ {0} +$$ + +$$ +\kappa_ {2} (+) - \kappa_ {2} (-) \quad : (+), (-) \text { 측 제3강성. } +$$ + +$$ +\text { 단, } K 3 ^ {(+)} = \alpha 2 ^ {(+)} \cdot K _ {0}, \quad K 3 ^ {(-)} = \alpha 2 ^ {(-)} \cdot K _ {0} +$$ + +$$ +\alpha 1 ^ {(+)}, \alpha 1 ^ {(-)} \quad : (+), (-) \text {측 제1차항복후의 강성저감율 } +$$ + +$$ +\alpha 2 ^ {(+)}, \alpha 2 ^ {(-)} \quad : (+), (-) \text {측 제2차항복후의 강성저감율 } +$$ + + + +# 9-4-5 Clough Type + +# 이력의 개요 + +초기재하시의 응답점은 2구배 골격곡선 상에서 이동합니다. 항복 후의 변형의 진전에 의해 재하강성이 점진적으로 감소하는 Degrading Bilinear형입니다. 콘크리트는 건조수축 등에 의해 균열이 발생하기 쉬우므로, 균열 전의 상태는 무시하고 전체 단면에 균열이 발생한 것으로 간주하여, 인장철근의 휨항복에 의한 강성변화만을 고려하도록 모델링된 이력입니다. 입력에 의해, 대칭 및 비대칭이 정의가능하며,대응요소는 집중형 힌지 및 분포형 힌지 요소, 스프링 요소, 트러스 요소 등입니다. + +![](images/page-385_28fded88a53c02201ef88d888160cdf9be344b86db0c9fcef07e378bde5de801.jpg) + +
+text_image + +P +P1(+) +K2(+) +(Dmax(-)Pmax(-)) +D1(-) +K0 +K1(+) +D1(+) +K1(-) +P1(-) +K2(+) +(Dmax(-)Pmax(-)) +
+ +![](images/page-385_373abbcf16c664c0ead2506f24cd2930cdfd035b3caa0f154688d704b0668cd4.jpg) + +
+text_image + +P +P1(+) +D1(-) +D1(+) +D +P1(-) +
+ +그림 2.9.16 Clough 이력모델 + + + +# 골격곡선의 정의 + +이력모델의 비선형특성은 이하의 값으로 정의됩니다. + +$$ +P 1 _ {(+)}, P 1 _ {(-)} \quad : (+), (-) \text { 측 제1차항복강도 } +$$ + +$$ +D 1 _ {(+)}, D 1 _ {(-)} \quad : (+), (-) \text { 측 제1차항복변형 } +$$ + +$$ +K _ {0} \quad : \text { 초기강성 } +$$ + +$$ +K 2 ^ {(+)}, K 2 ^ {(-)} \quad : (+), (-) \text { 측 제2강성. } +$$ + +$$ +\text { 단, } K 2 ^ {(+)} = \alpha 1 ^ {(+)} \cdot K _ {0} , \quad K 2 ^ {(-)} = \alpha 1 ^ {(-)} \cdot K _ {0} +$$ + +$$ +\alpha 1 ^ {(+)}, \alpha 1 ^ {(-)} \quad : (+), (-) \text { 측 제1차항복후의 강성저감율 } +$$ + +$$ +\alpha 2 ^ {(+)}, \alpha 2 ^ {(-)} \quad : (+), (-) \text { 측 제2차항복후의 강성저감율 } +$$ + +$$ +K r ^ {(+)} , K r ^ {(-)} \quad : (+), (-) \text { 측 재하시의 강성 } +$$ + +$$ +K r ^ {(+)} = K _ {0} \cdot \left| \frac {D 1 ^ {(+)}}{D _ {\max} ^ {(+)}} \right| ^ {\beta} \leq K _ {0}, K r ^ {(-)} = K _ {0} \cdot \left| \frac {D 1 ^ {(-)}}{D _ {\max} ^ {(-)}} \right| ^ {\beta} \leq K _ {0} +$$ + +여기서, $D1^{(+)}$ , $D1^{(-)}$ : (+), (-) 측 항복변형 + +$$ +D _ {\max} ^ {(+)}, D _ {\max} ^ {(-)}: (+), (-) \text {측의 최대변형} +$$ + +(항복이 발생하지 않은 영역에서는 항복변위로 대체) + +β:재하강성 산정용 정수 + +# 클러프형(Clough Type)의 이력규칙 + +1. $\left|D_{max}\right| + +# 9-4-6 Degrading Trilinear Type + +# 이력의 개요 + +골격곡선은 Trilinear로서, 1차항복후, 2차항복 이전에는 Bilinear로 거동하고, 2차항복 이후는 변형의 진전에 의해 제하강성이 점진적으로 감소하는 강성저감3선형으로 거동합니다. 입력에 의해, 대칭 및 비대칭이 정의가능하며, 대응요소는 집중형힌지 및 분포형 힌지 요소, 스프링 요소, 트러스 요소 등입니다. + +![](images/page-387_2889d453f57de3a557d464566e35efd190fc09a419758574a8dc76e61fb9ea0f.jpg) + +
+text_image + +P +P2(+) +P1(+) +D2(-) +D1(-) +D1(+) +D2(+) +D +P1(-) +P2(-) +
+ +그림 2.9.17 Degrading Trilinear 이력모델 + + + +# 골격곡선의 정의 + +이력모델의 비선형특성은 이하의 값으로 정의됩니다. + +$$ +\begin{array}{l} P 1 _ {(+)}, P 1 _ {(-)} \quad : (+), (-) \text { 측 제1차항복강도 } \\ P 2 _ {(+)}, P 2 _ {(-)} \quad : (+), (-) \text { 측 제2차항복강도 } \\ D 1 _ {(+)}, D 1 _ {(-)} \quad : (+), (-) \text { 측 제1차항복변형 } \\ D 2 _ {(+)}, D 2 _ {(-)} \quad : (+), (-) \text { 측 제2차항복변형 } \\ K _ {0} \quad : \text { 초기강성 } \\ K 2 ^ {(+)}, K 2 ^ {(-)} \quad : (+), (-) \text { 측 제2강성. } \\ \text { 단, } K 2 ^ {(+)} = \alpha 1 ^ {(+)} \cdot K _ {0} , \quad K 2 ^ {(-)} = \alpha 1 ^ {(-)} \cdot K _ {0} \\ \end{array} +$$ + +$$ +K 3 ^ {(+)}, K 3 ^ {(-)} \quad : (+), (-) \text { 측 제3강성. } +$$ + +$$ +\text { 단, } K 3 ^ {(+)} = \alpha 2 ^ {(+)} \cdot K _ {0} , \quad K 3 ^ {(-)} = \alpha 2 ^ {(-)} \cdot K _ {0} +$$ + +$$ +\alpha 1 ^ {(+)}, \alpha 1 ^ {(-)} \quad : (+), (-) \text { 측 제1차항복후의 강성저감율 } +$$ + +$$ +\alpha 2 ^ {(+)}, \alpha 2 ^ {(-)} \quad : (+), (-) \text { 측 제2차항복후의 강성저감율 } +$$ + +# 강성저감 3선형(Degrading Trilinear Type)의 이력규칙 + +1. $\left|D_{max}\right| + +# 9-4-7 Takeda Type + +# 이력의 개요 + +다케다형 이력은 철근콘크리트 부재의 실험에 의해 관찰된 복원력특성을 상세히모델링한 것으로 강성저감 Trilinear형입니다. 제하강성은 제하점의 골격곡선상에서의 위치 및 반대편 영역에서의 1차 항복 여부에 의해 결정됩니다. 입력에 의해, 대칭 및 비대칭이 정의가능하며, 대응요소는 집중형 힌지 및 분포형 힌지 요소, 스프링 요소, 트러스 요소 등입니다. + +![](images/page-389_08ab4c4801261d9e7fc17a47c7a58529ff315b51c12e24e440f9c7690bdb1827.jpg) + +
+flowchart + +```mermaid +graph TD + A["P1(-)"] --> B["D1(-)"] + B --> C["D2(-)"] + C --> D["P2(-)"] + D --> E["D2(+)"] + E --> F["P2(+)"] + F --> G["D1(+)"] + G --> H["D2(+)"] + H --> I["P1(+)"] + I --> J["D1(-)"] + J --> K["D2(-)"] + K --> L["P2(+)"] + L --> M["D2(+)"] + M --> N["P1(+)"] + N --> O["D1(-)"] + O --> P["D2(-)"] + P --> Q["P2(+)"] + Q --> R["D2(+)"] + R --> S["P1(+)"] + S --> T["D1(-)"] + T --> U["D2(-)"] + U --> V["P2(+)"] + V --> W["D2(+)"] + W --> X["P1(+)"] + X --> Y["D1(-)"] + Y --> Z["D2(-)"] + Z --> AA["P2(+)"] +``` +
+ +그림 2.9.18 Takeda 이력모델 + + + +# 골격곡선의 정의 + +이력모델의 비선형특성은 이하의 값으로 정의됩니다. + +$$ +P 1 _ {(+)}, P 1 _ {(-)} \quad : (+), (-) \text { 측 제1차항복강도 } +$$ + +$$ +P 2 _ {(+)}, P 2 _ {(-)} \quad : (+), (-) \text { 측 제2차항복강도 } +$$ + +$$ +D 1 _ {(+)}, D 1 _ {(-)} \quad : (+), (-) \text { 측 제1차항복변형 } +$$ + +$$ +D 2 _ {(+)}, D 2 _ {(-)} \quad : (+), (-) \text { 측 제2차항복변형 } +$$ + +$$ +K _ {0} \quad : \text { 초기강성 } +$$ + +$$ +: (+), (-) \text { 측 제2강성 } +$$ + +$$ +\begin{array}{l} K 2 ^ {(+)}, K 2 ^ {(-)} \\ \text { 단, } K 2 ^ {(+)} = \alpha 1 ^ {(+)} \cdot K _ {0} , \quad K 2 ^ {(-)} = \alpha 1 ^ {(-)} \cdot K _ {0} \\ \end{array} +$$ + +$$ +(+) \quad (-) \quad : (+), (-) \text { 측 제3강성. } +$$ + +$$ +\text { 단, } K 3 ^ {(+)} = \alpha 2 ^ {(+)} \cdot K _ {0}, \quad K 3 ^ {(-)} = \alpha 2 ^ {(-)} \cdot K _ {0} +$$ + +$$ +\alpha 1 ^ {(+)}, \alpha 1 ^ {(-)} \quad : (+), (-) \text { 측 제1차항복후의 강성저감율 } +$$ + +$$ +\alpha 2 ^ {(+)}, \alpha 2 ^ {(-)} \quad : (+), (-) \text { 측 제2차항복후의 강성저감율 } +$$ + +$$ +\beta : \text {제하강성 파라메터} +$$ + +$$ +\alpha : \text { 내부 루프 반복시의 강성저감율 } +$$ + +# 다케다형(Takeda Type)의 이력규칙 + +1. $\left|D_{max}\right| + +c) 반대측 최대변형점에 도달하기 전에 재재하되는 경우는 같은 제하직선을 따라서 진행되며(Rule:3), 골격곡선에 도달하면, $K2^{(+)}$ , $K2^{(-)}$ 구배로 골격곡선상에서 이동합니다.(Rule: 4) + +![](images/page-391_649217cb40ef2ca2f2972ac04f5ed293ad893bc009cdb4b67aadd72ae92c35f2.jpg) + +
+line + +| Point | Label | P | D | +|-------|-------|------|------| +| 1 | K2 | + | 0 | +| 2 | Kr1 | (+) | 0 | +| 3 | Rule 3| 4 | 0 | +| 4 | Rule 4| 4 | 0 | +| 5 | K2 | (-) | 0 | +
+ +3. a) D 가 최초로 $D2_{(\pm)}$ 를 초과한 경우, 제3구배 $K3^{(+)}$ , $K3^{(-)}$ 직선상을 진행합니다.(Rule: 13) + +b) 이 직선상에서, 제하되면 제하구배 $Kr^{(+)}$ , $Kr^{(-)}$ 로 이동합니다.(Rule: 15) 반대측이 제1차항복을 경험하기 전인 경우, 구배 $Kr^{(\pm)}$ 의 범위는 P1 까지가 되고, P1을 초과하면 제2항복점을 향하여 이동합니다.(Rule: 17) + +$$ +K r ^ {(+)} = K _ {b} ^ {(+)} * \left| \frac {D _ {\max} ^ {(+)}}{D 2 ^ {(+)}} \right| ^ {- \beta}, \quad K r ^ {(-)} = K _ {b} ^ {(-)} * \left| \frac {D _ {\max} ^ {(-)}}{D 2 ^ {(-)}} \right| ^ {- \beta} +$$ + +여기서, $K_{b}^{(+)}=\frac{P2_{(+)}-P1_{(-)}}{D2_{(+)}-D1_{(-)}}$ $K_{b}^{(-)}=\frac{P2_{(-)}-P1_{(+)}}{D2_{(-)}-D1_{(+)}}$ + +β 제하강성 파라메터 ( $\beta = 0.4, Default$ ) + + + +![](images/page-392_a6681ed2251efd4642b1ce2faa957835e85b6ca62c3a756dd127b2b7951bf37b.jpg) + +
+flowchart + +Diagram illustrating a multi-rule decision flow with labeled points P1 to P2 and directional arrows, including rule labels 13–19 and Kr(+). +
+ +4. 복원력0점을 초과하면, 반대측의 최대변형점을 향하여 이동하고(Rule:18),반대측 최대변형점을 향하는 직선상에서 제하되는 경우, 내부루프에 들어갑니다.(Rule:20) 내부루프에서는 복원력0점까지는 $K _ { u n } ^ { ( - ) }$ , $K _ { u n } ^ { ( + ) }$ 의 구배로제하되어, 복원력0점을 초과하면 반대측의 직전제하점을 이동합니다.(Rule:21) + +![](images/page-392_bebd06e822494fb2cd8d18edce7b25c2d6ea8b4f1acf6714faca3d1b5399767f.jpg) + +
+flowchart + +```mermaid +graph TD + P["P"] -->|Rule: 13| D["D"] + P -->|Rule: 14| D + P -->|Rule: 17| D + P -->|Rule: 18| D + P -->|Rule: 20| D + P -->|Rule: 21| D + P -->|Rule: 22| D + P -->|Rule: 23| D + P -->|P2(-)| D + P -->|P2(+)| D + P -->|P1(-)| D + P -->|P1(+)| D + P -->|P2(-)| D + P -->|P2(+)| D + P -->|P1(-)| D + P -->|P1(+)| D + P -->|P2(-)| D + P -->|P2(+)| D + P -->|P1(-)| D + P -->|P1(+)| D + P -->|P2(-)| D + P -->|P2(+)-| D + P -->|P1(-)| D + P -->|P1(+)-| D + P -->|P2(-)| D + P -->|P2(+)-| D + P -->|P1(-)| D + P -->|P1(+)-| D + P -->|P2(-)| D + P -->|P2(+)-| D + P -->|P1(-)| D + P -->|P1(+)-| D + P -->|P2(-)| D + P --> P2(-) +``` +
+ + + +# 9-4-8 Takeda Tetralinear Type + +# 이력의 개요 + +다케다4선형 이력은 강성저감 Tetralinear로서, 입력에 의해, 대칭 및 비대칭이 정의가능하며, 대응요소는 집중형 힌지 및 분포형 힌지 요소, 스프링 요소, 트러스요소 등입니다. + +![](images/page-393_5d1ec2269a443ca7931c773186feb6acf953bdb0d2d2b27a83b5a09b4c0fd13d.jpg) + +
+text_image + +P +P3(+) +P2(+) +P1(+) +K4(+) +D3(-) +D2(-) D1(-) D1(+) +D2(+) +D3(+) +D +P1(-) +K4(-) +P2(-) +P3(-) +
+ +그림 2.9.19 Takeda Tetralinear 이력모델 + + + +# 골격곡선의 정의 + +이력모델의 비선형특성은 이하의 값으로 정의됩니다. + +$$ +P 1 _ {(+)}, P 1 _ {(-)} \quad : (+), (-) \text { 측 제1차항복강도 } +$$ + +$$ +P 2 _ {(+)}, P 2 _ {(-)} \quad : (+), (-) \text { 측 제2차항복강도 } +$$ + +$$ +P 3 _ {(+)}, P 3 _ {(-)} \quad : (+), (-) \text { 측 제3차항복강도 } +$$ + +$$ +D 1 _ {(+)}, D 1 _ {(-)} \quad : (+), (-) \text { 측 제1차항복변형 } +$$ + +$$ +D 2 _ {(+)}, D 2 _ {(-)} \quad : (+), (-) \text { 측 제2차항복변형 } +$$ + +$$ +D 3 _ {(+)}, D 3 _ {(-)} \quad : (+), (-) \text { 측 제3차항복변형 } +$$ + +$$ +K _ {0} \quad : \text { 초기강성 } +$$ + +$$ +\mathrm{ye} ^ {(+)} - \mathrm{ye} ^ {(-)} \quad : (+), (-) \text {측 제2강성.} +$$ + +$$ +\text { 단, } K 2 ^ {(+)} = \alpha 1 ^ {(+)} \cdot K _ {0}, \quad K 2 ^ {(-)} = \alpha 1 ^ {(-)} \cdot K _ {0} +$$ + +$$ +K 3 ^ {(+)} , K 3 ^ {(-)} \qquad \qquad : (+), (-) \text { 측 제3강성. } +$$ + +$$ +\text { 단, } K 3 ^ {(+)} = \alpha 2 ^ {(+)} \cdot K _ {0}, \quad K 3 ^ {(-)} = \alpha 2 ^ {(-)} \cdot K _ {0} +$$ + +$$ +\alpha 1 ^ {(+)}, \alpha 1 ^ {(-)} \quad : (+), (-) \text {측 제1차항복후의 강성저감율 } +$$ + +$$ +\alpha 2 ^ {(+)}, \alpha 2 ^ {(-)} \quad : (+), (-) \text { 측 제2차항복후의 강성저감율 } +$$ + +$$ +\alpha 3 ^ {(+)}, \alpha 3 ^ {(-)} \quad : (+), (-) \text {측 제3차항복후의 강성저감율 } +$$ + +$$ +\beta : \text {제하강성 파라메터} +$$ + +$$ +\alpha \quad : \text { 내부 루프 반복시의 강성저감율 } +$$ + +# 다케다4선형(Takeda Tetralinear Type)의 이력규칙 + +1. 초기재하시는 Tetralinear 골격곡선상에서 이동합니다. +2. 변형 D 가 $D3_{(\pm)}$ 을 초과하기 전의 이력규칙은, 다케다형 Trilinear와 동일합니다. +3. $D$ 가 $D3_{(\pm)}$ 을 초과한후에는 제4구배 $K4^{(+)}$ , $K4^{(-)}$ 직선상에서 이동합니다. +4. 제4구배 $K4^{(+)}$ , $K4^{(-)}$ 에서 재하되는 경우도 다케다형과 동일한 재하구배로 이동합니다. + + + +# 9-4-9 Modified Takeda Type + +# 이력의 개요 + +수정 다케다형 이력은 강성저감 Trilinear로서, 입력에 의해, 대칭 및 비대칭이 정의가능하며, 대응요소는 집중형 힌지 및 분포형 힌지 요소, 스프링 요소, 트러스 요소 등입니다. + +![](images/page-395_dbeeac1295ed5a0eaaa6c774d524ffad04caf0cdcc1d2e961b3bbcb50f34df3f.jpg) + +
+flowchart + +```mermaid +graph TD + P1["Point P1(+)"] --> D1["Point D1(-)"] + P2["Point P2(+)"] --> D2["Point D2(+)"] + D1 --> P1 + D2 --> P2 + P1 --> D1 + P2 --> D2 + D1 --> P1 + D2 --> P2 + P1 --> D1 + P2 --> D2 + D1 --> P1 + D2 --> P2 + P1 --> D1 + P2 --> D2 + D1 --> P1 + D2 --> P2 + P1 --> D1 + P2 --> D2 + D1 --> P1 + D2 --> P2 + D1 --> P2 + D2 --> P1 + D1 --> P2 + D2 --> P1 + D1 --> P2 + D2 --> P1 + D1 --> P2 + D2 --> P1 + D1 --> P2 + D2 --> P1 + D1 --> P2 + D2 --> P1 + D1 --> P2 + D2 --> P1 + D1 --> P2 + D2 --> P1 +``` +
+ +그림 2.9.20 Modified Takeda 이력모델 + + + +# 골격곡선의 정의 + +이력모델의 비선형특성은 이하의 값으로 정의됩니다. + +$$ +P 1 _ {(+)}, P 1 _ {(-)} \quad : (+), (-) \text { 측 제1차항복강도 } +$$ + +$$ +P 2 _ {(+)}, P 2 _ {(-)} \quad : (+), (-) \text { 측 제2차항복강도 } +$$ + +$$ +D 1 _ {(+)}, D 1 _ {(-)} \quad : (+), (-) \text { 측 제1차항복변형 } +$$ + +$$ +D 2 _ {(+)} , D 2 _ {(-)} \qquad : (+), (-) \text { 측 제2차항복변형 } +$$ + +$$ +K _ {0} \quad : \text { 초기강성 } +$$ + +$$ +K 2 ^ {(+)}, K 2 ^ {(-)} \quad : (+), (-) \text {측 제2강성.} +$$ + +$$ +\text { 단, } K 2 ^ {(+)} = \alpha 1 ^ {(+)} \cdot K _ {0} , \quad K 2 ^ {(-)} = \alpha 1 ^ {(-)} \cdot K _ {0} +$$ + +$$ +K 3 ^ {(+)}, K 3 ^ {(-)} \quad : (+), (-) \text { 측 제3강성. } +$$ + +$$ +\text { 단, } K 3 ^ {(+)} = \alpha 2 ^ {(+)} \cdot K _ {0} , \quad K 3 ^ {(-)} = \alpha 2 ^ {(-)} \cdot K _ {0} +$$ + +$$ +\alpha 1 ^ {(+)}, \alpha 1 ^ {(-)} \quad : (+), (-) \text {측 제1차항복후의 강성저감율 } +$$ + +$$ +\alpha 2 ^ {(+)}, \alpha 2 ^ {(-)} \quad : (+), (-) \text { 측 제2차항복후의 강성저감율 } +$$ + +$$ +\beta : \text {제하강성 파라메터} +$$ + +$$ +\alpha \quad : \text { 내부 루프 반복시의 강성저감율 } +$$ + +# 수정 다케다형(Modified Takeda Type)의 이력규칙 + +1. $\left|D_{max}\right| + +b) 이 직선상에서 제하되는 경우는 반대측의 최대변형점을 향하여 이동합니다. (Rule:2) + +c) 반대측 최대변형점에 도달하기 전에 재재하되는 경우는 같은 제하직선을 따라서 진행되며(Rule:3), 골격곡선에 도달하면, $K2^{(+)}$ , $K2^{(-)}$ 구배로 골격곡선상에서 이동합니다.(Rule: 4) + +![](images/page-397_6c281edccc0fe6e3d2c21f4583872b3f20a747a5832473d78ebb03f132eb7226.jpg) + +
+flowchart + +```mermaid +graph TD + P1["Point P1(+)"] -->|Rule: 0| D1["Point D1(-)"] + P2["Point P2(+)"] -->|Rule: 1| D1 + P1 -->|Rule: 2| D1 + P2 -->|Rule: 3| D1 + P1 -->|Rule: 4| D2["Point D2(+)"] + P2 -->|Rule: 5| D2 + D1 -->|P1(-)| D2 + D1 -->|P2(-)| D2 + style P1 fill:#f9f,stroke:#333 + style P2 fill:#f9f,stroke:#333 + style D1 fill:#ccf,stroke:#333 + style D2 fill:#ccf,stroke:#333 + style D1 fill:#dfd,stroke:#333 + style D2 fill:#dfd,stroke:#333 +``` +
+ +3. i) D 가 최초로 $D2_{(\pm)}$ 를 초과한 경우, 제3구배 $K3^{(+)}$ , $K3^{(-)}$ 직선상을 진행합니다. (Rule:10) + +ii) 이 직선상에서, 제하되면 제하구배 $Kr^{(+)}$ , $Kr^{(-)}$ 로 이동한다.(Rule:11) 반대측이 제2차항복을 경험하기 전인 경우, 반대측의 제2항복점이 반대측의 최대변형점이 됩니다. + +$$ +K r ^ {(\pm)} = \max \left(K _ {0} * \left| \frac {D _ {\max} ^ {(\pm)}}{D 1 ^ {(\pm)}} \right| ^ {- \beta}, K _ {b}\right) +$$ + + + +$$ +\text { 여기서, } K _ {b} = \frac {P _ {\max} ^ {(+)} - P _ {\max} ^ {(-)}}{D _ {\max} ^ {(+)} - D _ {\max} ^ {(-)}} +$$ + +$\beta$ : 제하강성 파라메터 ( $\beta = 0.4, Default$ ) + +4. 복원력0점을 초과하면, 반대측의 최대변형점을 향하여 이동하고(Rule:14), 반대측 최대변형점을 향하는 직선상에서 제하되는 경우, 내부루프에 들어 갑니다.(Rule:15) 내부루프에서는 복원력0점까지는 $Kr^{(-)}$ , $Kr^{(+)}$ 의 구배로 제하되어, 복원력0점을 초과하면 반대측 최대점을 향하여 이동합니다.(Rule:16) + +![](images/page-398_77631913f7131708c900f65032f522429973840f883b3ddebe8e24dd91d08368.jpg) + +
+flowchart +```mermaid +graph TD + A["Rule: 10"] --> B["Prize (D_max^(+) P_max^(+))"] + C["Rule: 11"] --> D["Prize (D_max^(+) P_max^(+))"] + E["Rule: 12"] --> F["Prize (D_max^(+) P_max^(+))"] + G["Rule: 13"] --> H["Prize (D_max^(+) P_max^(+))"] + I["Rule: 14"] --> J["Prize (D_max^(+) P_max^(+))"] + K["Rule: 15"] --> L["Prize (D_max^(+) P_max^(+))"] + M["Rule: 16"] --> N["Prize (D_max^(+) P_max^(+))"] + O["Rule: 17"] --> P["Prize (D_max^(+) P_max^(+))"] + Q["Rule: 18"] --> R["Prize (D_max^(+) P_max^(+))"] + S["Rule: 19"] --> T["Prize (D_max^(+) P_max^(+))"] + U["Rule: 20"] --> V["Prize (D_max^(+) P_max^(+))"] + W["Kr(+)"] --> X["Prize (D_max^(+) P_max^(+))"] + Y["Kr(-)"] --> Z["Prize (D_max^(+) P_max^(+))"] + style A fill:#f9f,stroke:#333 + style C fill:#f9f,stroke:#333 + style E fill:#f9f,stroke:#333 + style G fill:#f9f,stroke:#333 + style I fill:#f9f,stroke:#333 + style Q fill:#f9f,stroke:#333 + style W fill:#f9f,stroke:#333 + style U fill:#f9f,stroke:#333 + style V fill:#f9f,stroke:#333 + style W fill:#f9f,stroke:#333 + style X fill:#f9f,stroke:#333 + style Y fill:#f9f,stroke:#333 + style Z fill:#f9f,stroke:#333 + style W fill:#f9f,stroke:#333 + style X fill:#f9f,stroke:#333 + style Y fill:#f9f,stroke:#333 + style Z fill:#f9f,stroke:#333 +``` +
+ + + +# 9-4-10 Modified Takeda Tetralinear Type + +# 이력의 개요 + +수정 다케다4선형 이력은 강성저감 Tetralinear로서, 입력에 의해, 대칭 및 비대칭이 정의가능하며, 대응요소는 집중형 힌지 및 분포형 힌지 요소, 스프링 요소, 트러스 요소 등입니다. + +![](images/page-399_b80a975183c986e71e2967375f1a0a2ed9ffa2ff1c89d16e764ad4fd1d16a2dc.jpg) + +
+text_image + +P +P3(+) +P2(+) +P1(+) +K4(+) +D3(-) +D2(-) +D1(-) +D1(+) +D2(+) +D3(+) +D +P1(-) +P2(-) +P3(-) +K4(-) +
+ +그림 2.9.21 Modified Takeda Tetralinear 이력모델 + + + +# 골격곡선의 정의 + +이력모델의 비선형특성은 이하의 값으로 정의됩니다. + +$P1_{(+)}, P1_{(-)}$ : (+), (-) 측 제1차항복강도 + +$P2_{(+)}, P2_{(-)}$ : $(+)$ , $(-)$ 측 제2차항복강도 + +$P3_{(+)}, P3_{(-)}$ : (+),(-)측 제3차항복강도 + +$D1_{(+)}, D1_{(-)}$ : $(+)$ , $(-)$ 측 제1차항복변형 + +$D2_{(+)}, D2_{(-)}$ : $(+)$ , $(-)$ 측 제2차항복변형 + +$D3_{(+)}, D3_{(-)}$ : $(+)$ , $(-)$ 측 제3차항복변형 + +$K_{0}$ : 초기강성 + +$\mathrm{v}_{\mathrm{e}}(+)$ , $\mathrm{v}_{\mathrm{e}}(-)$ : (+), (-) 측 제2강성. + +단, $K2^{(+)}=\alpha1^{(+)}\cdot K_{0}$ , $K2^{(-)}=\alpha1^{(-)}\cdot K_{0}$ + +$\kappa_{2}(+)$ $\kappa_{2}(-)$ : (+),(-)측 제3강성. + +단, $K3^{(+)}=\alpha2^{(+)}\cdot K_{0}$ , $K3^{(-)}=\alpha2^{(-)}\cdot K_{0}$ + +$K4^{(+)}, K4^{(-)}$ : (+), (-) 측 제4강성. + +단, $K4^{(+)}=\alpha3^{(+)}\cdot K_{0}$ , $K4^{(-)}=\alpha3^{(-)}\cdot K_{0}$ + +$\alpha1^{(+)}, \alpha1^{(-)}$ : (+),(-)측 제1차항복후의 강성저감율 + +$\alpha2^{(+)},\alpha2^{(-)}$ : $(+),(-)$ 측 제2차항복후의 강성저감율 + +$\alpha3^{(+)}$ , $\alpha3^{(-)}$ : (+), (-) 측 제3차항복후의 강성저감율 + +β : 제하강성 파라메터 + +α : 내부 루프 반복시의 강성저감율 diff --git a/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_041.md b/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_041.md new file mode 100644 index 00000000..aa54c2ca --- /dev/null +++ b/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_041.md @@ -0,0 +1,403 @@ + + +# 9-4-11 Elastic Bilinear Type + +# 이력의 개요 + +비선형탄성으로 골격곡선은 Bilinear입니다. 재하와 제하에 관계없이 루프를 그리지않는 이력으로, Bilinear골격곡선 상에서만 이동합니다. 따라서, 이력상에서의 지진에너지흡수는 기대할 수 없습니다. 입력에 의해 대칭 혹은 비대칭 정의가 가능합니다. 집중형 힌지 및 분포형 힌지 요소, 스프링 요소, 트러스 요소 등에 적용가능합니다. Inelastic Hinge Properties의 Directional Hinge Properties에서 Input Type을Strength-Yield Displacement를 선택하여, 정(+),부(-)축의 1차 항복변위를 이용하여초기강성을 (+),(-)측 비대칭으로 입력하여 고려할 수 있습니다. + +![](images/page-401_9fa6fcc2433a5b697a4b2424158dc53a340a89dc2f55d18246007bbf9a00d44b.jpg) + +
+text_image + +P +K2(+) +P1(+) +K0 +D1(-) +D1(+) +D +K0 +P1(-) +K2(-) +
+ +그림 2.9.22 Elastic Bilinear 이력모델 + + + +# 골격곡선의 정의 + +이력모델의 비선형특성은 이하의 값으로 정의됩니다. + +$$ +P 1 _ {(+)}, P 1 _ {(-)} \quad : (+), (-) \text { 측 제1차항복강도 } +$$ + +$$ +D 1 _ {(+)}, D 1 _ {(-)} \qquad : (+), (-) \text {측 제1차항복변형} +$$ + +$$ +K _ {0} \quad : \text { 초기강성 } +$$ + +$$ +\kappa \mathfrak {e} ^ {(+)} - \kappa \mathfrak {e} ^ {(-)} \quad : (+), (-) \text {측 제2강성.} +$$ + +$$ +\text { 단, } K 2 ^ {(+)} = \alpha 1 ^ {(+)} \cdot K _ {0}, \quad K 2 ^ {(-)} = \alpha 1 ^ {(-)} \cdot K _ {0} +$$ + +$$ +\alpha 1 ^ {(+)}, \alpha 1 ^ {(-)} \qquad \qquad : (+), (-) \text {측 제1차항복후의 강성저감율 } +$$ + + + +# 9-4-12 Elastic Trilinear Type + +# 이력의 개요 + +비선형탄성으로 골격곡선은 Trilinear입니다. 재하와 제하에 관계없이 루프를 그리지 않는 이력으로, Trilinear골격곡선상에서만 이동합니다. 따라서, 이력상에서의 지진 에너지흡수는 기대할 수 없습니다. 입력에 의해 대칭 혹은 비대칭 정의가 가능하며, 집중형 힌지 및 분포형 힌지 요소, 스프링 요소, 트러스 요소 등에 적용가능합니다. Inelastic Hinge Properties의 Directional Hinge Properties에서 Input Type을Strength-Yield Displacement를 선택하여, 정(+),부(-)축의 1차 항복변위를 이용하여초기강성을 (+),(-)측 비대칭으로 입력하여 고려할 수 있습니다. + +![](images/page-403_ba4b1b8b380da722adc86139af2b8643020347c72f5a813b402e75745d14ed70.jpg) + +
+line + +| Point | D | P | +|-------|-------|-------| +| P1 | D1 | P1 | +| P2 | D1 | P2 | +| P3 | D2 | P3 | +| P1 | D1 | K0 | +| P2 | D1 | K2 | +| P3 | D2 | K3 | +
+ +그림 2.9.23 Elastic Trilinear 이력모델 + + + +# 골격곡선의 정의 + +이력모델의 비선형특성은 이하의 값으로 정의됩니다. + +$$ +P 1 _ {(+)}, P 1 _ {(-)} \quad : (+), (-) \text { 측 제1차항복강도 } +$$ + +$$ +P 2 _ {(+)}, P 2 _ {(-)} \quad : (+), (-) \text { 측 제2차항복강도 } +$$ + +$$ +D 1 _ {(+)}, D 1 _ {(-)} \quad : (+), (-) \text { 측 제1차항복변형 } +$$ + +$$ +D 2 _ {(+)}, D 2 _ {(-)} \quad : (+), (-) \text { 측 제2차항복변형 } +$$ + +$$ +K _ {0} \quad : \text { 초기강성 } +$$ + +$$ +K 2 ^ {(+)}, K 2 ^ {(-)} \quad : (+), (-) \text {측 제2강성.} +$$ + +$$ +\text { 단, } K 2 ^ {(+)} = \alpha 1 ^ {(+)} \cdot K _ {0}, \quad K 2 ^ {(-)} = \alpha 1 ^ {(-)} \cdot K _ {0} +$$ + +$$ +K 3 ^ {(+)}, K 3 ^ {(-)} \quad : (+), (-) \text { 측 제3강성. } +$$ + +$$ +\text { 단, } K 3 ^ {(+)} = \alpha 2 ^ {(+)} \cdot K _ {0}, \quad K 3 ^ {(-)} = \alpha 2 ^ {(-)} \cdot K _ {0} +$$ + +$$ +\alpha 1 ^ {(+)}, \alpha 1 ^ {(-)} \quad : (+), (-) \text {측 제1차항복후의 강성저감율 } +$$ + +$$ +\alpha 2 ^ {(+)}, \alpha 2 ^ {(-)} \quad : (+), (-) \text {측 제2차항복후의 강성저감율 } +$$ + + + +# 9-4-13 Elastic Tetralinear Type + +# 이력의 개요 + +비선형탄성으로 골격곡선은 Tetralinear입니다. 재하와 제하에 관계없이 루프를 그리지 않는 이력으로, Tetralinear골격곡선상에서만 이동합니다. 따라서, 이력상에서의 지진 에너지흡수는 기대할 수 없습니다. 입력에 의해 대칭 혹은 비대칭 정의가가능합니다. 집중형 힌지 및 분포형 힌지 요소, 스프링 요소, 트러스 요소 등에 적용가능합니다. Inelastic Hinge Properties의 Directional Hinge Properties에서 InputType을 Strength-Yield Displacement를 선택하여, 정(+),부(-)축의 1차 항복변위를 이용하여 초기강성을 (+),(-)측 비대칭으로 입력하여 고려할 수 있습니다. + +![](images/page-405_938c1a1479e5e8a9ad58623b273af4e08826ddf7b3a57e11eea8bfd18b347e40.jpg) + +
+line + +| Point | D | P | +|-------|-------|-------| +| K0 | D1(-) | K0 | +| K1 | D1(+) | K1 | +| K2 | D2(+) | K2 | +| K3 | D3(+) | K3 | +| K4 | D3(-)| K4 | +
+ +그림 2.9.24 Elastic Tetralinear 이력모델 + +# 골격곡선의 정의 + +이력모델의 비선형특성은 이하의 값으로 정의됩니다. + +$$ +P 1 _ {(+)}, P 1 _ {(-)} \quad : (+), (-) \text { 측 제1차항복강도 } +$$ + +$$ +P 2 _ {(+)}, P 2 _ {(-)} \quad : (+), (-) \text { 측 제2차항복강도 } +$$ + +$$ +P 3 _ {(+)} , P 3 _ {(-)} \qquad : (+), (-) \text { 측 제3차항복강도 } +$$ + +$$ +D 1 _ {(+)}, D 1 _ {(-)} \quad : (+), (-) \text {측 제1차항복변형} +$$ + +$$ +D 2 _ {(+)}, D 2 _ {(-)} \quad : (+), (-) \text { 측 제2차항복변형 } +$$ + +$$ +D 3 _ {(+)}, D 3 _ {(-)} \quad : (+), (-) \text { 측 제3차항복변형 } +$$ + + + +$$ +\begin{array}{l} K _ {0} \quad : \text { 초기강성 } \\ K 2 ^ {(+)}, K 2 ^ {(-)} \quad : (+), (-) \text { 측 제2강성. } \\ \text { 단, } K 2 ^ {(+)} = \alpha 1 ^ {(+)} \cdot K _ {0}, \quad K 2 ^ {(-)} = \alpha 1 ^ {(-)} \cdot K _ {0} \\ K 3 ^ {(+)}, K 3 ^ {(-)} \quad : (+), (-) \text { 측 제3강성. } \\ \text { 단, } K 3 ^ {(+)} = \alpha 2 ^ {(+)} \cdot K _ {0}, \quad K 3 ^ {(-)} = \alpha 2 ^ {(-)} \cdot K _ {0} \\ K 4 ^ {(+)}, K 4 ^ {(-)} \quad : (+), (-) \text {측 제4강성.} \\ \text { 단, } K 4 ^ {(+)} = \alpha 3 ^ {(+)} \cdot K _ {0} , \quad K 4 ^ {(-)} = \alpha 3 ^ {(-)} \cdot K _ {0} \\ \alpha 1 ^ {(+)}, \alpha 1 ^ {(-)} \qquad : (+), (-) \text {측 제1차항복후의 강성저감율 } \\ \alpha 2 ^ {(+)}, \alpha 2 ^ {(-)} \quad : (+), (-) \text {측 제2차항복후의 강성저감율 } \\ \alpha 3 ^ {(+)}, \alpha 3 ^ {(-)} \quad : (+), (-) \text { 측 제3차항복후의 강성저감율 } \\ \end{array} +$$ + +# Elastic Tetralinear Type의 이력규칙 + +1. 재하와 제하에 관계없이 루프를 그리지 않는 이력으로, Tetralinear골격곡 선상에서만 이동합니다. +2. 부구배에 들어가서, 복원력이 0.0이 되는 점을 초과하면, 변형축 상에서 이동합니다. 또한, 재하될 경우는 아래 그림과 같이 이동하여, 통상의 이력 규칙을 따릅니다. + +![](images/page-406_2747f53a00a89414290e14069fc825ba869c00880d74d11ce59e8f7d64ba587e.jpg) + +
+flowchart +```mermaid +graph TD + P["Point P"] -->|1| D["Point D"] + P -->|2| D + P -->|3| D + P -->|4| D + P -->|5| D + P -->|6| D + P -->|7| D + P -->|8| D + P -->|9| D + P -->|10| D + P -->|11| D + P -->|12| D + P -->|13| D + P -->|14| D + P -->|15| D + P -->|16| D + P -->|17| D + P -->|18| D +``` +
+ + + +# 9-4-14 Slip Bilinear Type + +# 이력의 개요 + +골격곡선은 Bilinear로서 항복 후 강성 저감률은 정(+), 부(-) 비대칭 정의가 가능하며, 집중형 힌지 및 분포형 힌지 요소, 스프링 요소, 트러스 요소 등에 적용가능합니다. + +![](images/page-407_efbe64d9004f6340bf2550847ab456ee44b40ab1c6cc80ff94f8ba607e31f847.jpg) + +
+text_image + +P +P1(+) +K2(+) +δ(-) +gap +δ(+) +gap +D1(-) +K0 +K0 +D1(+) +K0 +K2(-) +P1(-) +
+ +(a) Slip Bilinear + +![](images/page-407_65b88686472d5484514413869e408df12d2283316e517ac13c5277fd3eb71acb.jpg) + +
+text_image + +P +P1(+) +K2(+) +δ(+) +δ(+) +K0 +K0 +D1(+) +D +
+ +(b) Slip Bilinear/Tension + +![](images/page-407_514e3e3905cc5769ed73ec61854255e81d57a868fe761a5533fe8b6c2e15bfcd.jpg) + +
+text_image + +P +δ pop +D1(-) +K0 +K0 +P1(-) +K2(-) +
+ +(c) Slip Bilinear/Compression +그림 2.9.25 Slip Bilinear 이력모델 + + + +# 골격곡선의 정의 + +이력모델의 비선형특성은 이하의 값으로 정의됩니다. + +$$ +P 1 _ {(+)}, P 1 _ {(-)} \quad : (+), (-) \text { 측 제1차항복강도 } +$$ + +$$ +D 1 _ {(+)}, D 1 _ {(-)} \quad : (+), (-) \text { 측 제1차항복변형 } +$$ + +$$ +K _ {0} \quad : \text { 초기강성 } +$$ + +$$ +\mathrm{w} _ {0} (+) \quad \mathrm{w} _ {0} (-) \quad : (+), (-) \text { 측 제2강성. } +$$ + +$$ +\text { 단, } K 2 ^ {(+)} = \alpha 1 ^ {(+)} \cdot K _ {0}, \quad K 2 ^ {(-)} = \alpha 1 ^ {(-)} \cdot K _ {0} +$$ + +$$ +\alpha 1 ^ {(+)}, \alpha 1 ^ {(-)} \quad : (+), (-) \text {측 제1차항복후의 강성저감율 } +$$ + +$$ +\delta_ {g a p} ^ {(+)}, \delta_ {g a p} ^ {(-)} \quad : (+), (-) \text {측 Initial Gap} +$$ + + + +# 9-4-15 Slip Trilinear Type + +# 이력의 개요 + +이력곡선은 Trilinear로서, 항복 후 강성 저감률은 정(+), 부(-) 비대칭 정의가 가능하며, 집중형 힌지 및 분포형 힌지 요소, 스프링 요소, 트러스 요소 등에 적용가능합니다. + +![](images/page-409_62e19b56370d3c369c0a5df056e8e70d420ff3110dc1a252f7320a65b237cc58.jpg) + +
+text_image + +P +P2(+) +P1(+) +K2(+) +δ(-) +δ(+) +gap +D2(-) +D1(-) +K0 +D1(+) +D2(+) +K0 +D +K0 +K2(-) +P1(-) +K3(-) +P1(-) +
+ +(a) Slip Trilinear +![](images/page-409_fcb97fbb8c139b6a27779e0145442e048f47079e03e26d3623d0102090cdba95.jpg) + +
+text_image + +P +P2(+) +P1(+) +K3(+) +K2(+) +δ_{prop}^{(+)} +K0 +D1(+) +D2(+) +K0 +D +
+ +(b) Slip Trilinear/Tension + +![](images/page-409_8a932360a494d6ebc63a5d9a2d522b31e764a117819c5395759f516a5387a223.jpg) + +
+text_image + +P +δ(−) +D2(−) D1(−) +K0 +K0 +P1(+) P1(−) +K2(+) K3(+) +
+ +(c) Slip Trilinear/Compression +그림 2.9.26 Slip Trilinear 이력모델 + + + +# 골격곡선의 정의 + +이력모델의 비선형특성은 이하의 값으로 정의됩니다. + +$$ +\begin{array}{l} P 1 _ {(+)}, P 1 _ {(-)} \quad : (+), (-) \text { 측 제1차항복강도 } \\ P 2 _ {(+)}, P 2 _ {(-)} \quad : (+), (-) \text { 측 제2차항복강도 } \\ D 1 _ {(+)}, D 1 _ {(-)} \quad : (+), (-) \text { 측 제1차항복변형 } \\ D 2 _ {(+)}, D 2 _ {(-)} \quad : (+), (-) \text { 측 제2차항복변형 } \\ K _ {0} \quad : \text { 초기강성 } \\ K 2 ^ {(+)}, K 2 ^ {(-)} \quad : (+), (-) \text { 측 제2강성. } \\ \text { 단, } K 2 ^ {(+)} = \alpha 1 ^ {(+)} \cdot K _ {0} , \quad K 2 ^ {(-)} = \alpha 1 ^ {(-)} \cdot K _ {0} \\ K 3 ^ {(+)}, K 3 ^ {(-)} \quad : (+), (-) \text { 측 제3강성. } \\ \text { 단, } K 3 ^ {(+)} = \alpha 2 ^ {(+)} \cdot K _ {0} , \quad K 3 ^ {(-)} = \alpha 2 ^ {(-)} \cdot K _ {0} \\ \alpha 1 ^ {(+)}, \alpha 1 ^ {(-)} \quad : (+), (-) \text { 측 제1차항복후의 강성저감율 } \\ \alpha 2 ^ {(+)}, \alpha 2 ^ {(-)} \quad : (+), (-) \text { 측 제2차항복후의 강성저감율 } \\ \delta_ {g a p} ^ {(+)}, \delta_ {g a p} ^ {(-)} \quad : (+), (-) \text {측 Initial Gap} \\ \end{array} +$$ diff --git a/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_042.md b/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_042.md new file mode 100644 index 00000000..fc62dc64 --- /dev/null +++ b/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_042.md @@ -0,0 +1,341 @@ + + +# 9-4-16 Ramberg-Osgood Type + +# 이력의 개요 + +Ramberg-Osgood 이력모델은 금속재료의 비선형해석을 위해서 제안된 것으로, 이력곡선은 Masing의 법칙을 따르고 있습니다. 지반 비선형 Ramberg-Osgood 모델의 골격곡선과 이력곡선은 다음과 같이 표현됩니다. + +$$ +\text { 골격곡선 }: \gamma = \frac {\tau}{\mathbf {G} _ {0}} \left(1 + \alpha | \tau | ^ {\beta}\right) +$$ + +$$ +\text { 이력곡선 }: \frac {\gamma \pm \gamma_ {0}}{2} = \frac {\tau \pm \tau_ {0}}{2 \boldsymbol {G} _ {0}} \left(1 + \alpha \left| \frac {\tau \pm \tau_ {0}}{2} \right| ^ {\beta}\right) +$$ + +$$ +\beta = \frac {2 \pi \boldsymbol {h} _ {\max}}{2 - \pi \boldsymbol {h} _ {\max}}, \alpha = \left(\frac {2}{\gamma_ {r} \boldsymbol {G} _ {0}}\right) ^ {\beta} +$$ + +여기서, $\gamma$ : 전단변형율, + +τ: 전단응력, + +$G_{0}$ : 전단탄성계수 + +$\gamma_{r}$ : 기준 전단변형율 + +α,β : Ramberg-Osgood 모델의 파라메터 + + + +# 골격곡선의 정의 + +이력모델의 비선형특성은 이하의 값으로 정의됩니다. + +
$\mathbf{G}_{0}$ ,:초기강성(전단탄성계수)
$\gamma_{r}$ :기준 변형율
$\mathbf{h}_{\max}$ :최대감쇠정수
+ +![](images/page-412_f28dc016521443848cb904d338d9a792e6933041f4d39a8c1f9505a502208393.jpg) + +
+text_image + +Skeleton Curve +G₀ +(τ₀,γ₀) +Hysteresis Curve +(-τ₀,-γ₀) +
+ +그림 2.9.27 Ramberg-Osgood 이력모델 + + + +# 9-4-17 Hardin-Drnevich Type + +# 이력의 개요 + +Hardin-Drnevich 모델의 이력곡선은 훑의 정적인 응력-변형율관계에 주로 사용되는 쌍곡선 모델을 그대로 골격곡선에 사용합니다. + +$$ +\text { 골격곡선 }: \tau = \frac {\mathbf {G} _ {0} \cdot \gamma}{1 + \left| \gamma / \gamma_ {r} \right|} +$$ + +$$ +\text { 이력곡선 }: \tau - \tau_ {m} = \frac {\mathbf {G} _ {0} (\gamma - \gamma_ {m})}{1 + \left| \gamma - \gamma_ {m} \right| / 2 \gamma_ {r}} (\text { 하강곡선 }) +$$ + +$$ +\tau + \tau_ {m} = \frac {\mathbf {G} _ {0} (\gamma + \gamma_ {m})}{1 + \left| \gamma + \gamma_ {m} \right| / 2 \gamma_ {r}} (\text {상승곡선}) +$$ + +여기서, $\gamma$ : 전단변형율, + +τ: 전단응력, + +$G_{0}$ : 전단탄성계수 + +$\gamma_{r}$ : 기준 전단변형율 + +$(\gamma_{m},\tau_{m})$ : 이력곡선상의 반전점 + + + +# 골격곡선의 정의 + +이력모델의 비선형특성은 이하의 값으로 정의됩니다. + +$G _ { 0 }$ : 초기강성(전단탄성계수) + +$\gamma _ { r }$ + +![](images/page-414_401506e1b77a524c4640815e96a2db6751866c48722d33a9d49c6490c78af4e6.jpg) + +
+text_image + +Skeleton Curve +γ₀ +G₀ +(τ₀,γ₀) +τ₀ +Hysteresis Curve +(-τ₀,-γ₀) +
+ +그림 2.9.28 Hardin-Drnevich 이력모델 + + + +# 9-5 다축-힌지 이력모델(Hysteresis Model for Multi-axial Hinge) + +지진과 같은 복잡한 형태의 하중에 대해서 축력과 2축 힌을 받는 기둥은 3개 성분 사이에 복잡한 상호작용이 존재합니다. 이와 같은 상호작용을 보다 상세하게 모델링하기 위해서는 하나의 기둥을 입체요소로 세분하여 해석할 수 있으나, 상당한 계산량이 요구되기 때문에 요소의 갯수를 줄이기 위해 다축-힌지 모델(Hysteresis Model for Multi-axial Hinge)이 일반적으로 사용되고 있습니다. 다축-힌지 모델은 요소내에 복수의 비선형 힌지를 할당하여 힌지의 상태에 의해 부재의 비탄성거동을 해석하는 모델로서, 파이버 모델(Fiber Model)과 다축-힌지 이력모델로 구분할 수 있습니다. + +# ▪ 파이버 모델(Fiber Model) + +파이버 모델은 부재내에 비탄성 거동을 모니터링 하는 단면을 섬유로 세분하여, 단일 보요소로 모델링하는 모델입니다. 따라서, 복수개의 요소로 분할하지 않고도, 정밀한 비선형 거동을 파악할 수 있는 장점이 있습니다. 그러나, 대규모 구조물의 비선형 시간이력해석시에 모든 구조부재를 파이버 모델로 모델링할 경우, 계산시간이 과도하게 소요되며 및 메모리 문제가 발생할 수 있습니다. + +# ▪ 다축-힌지 이력모델 + +다축-힌지 이력모델은 축력과 2축 휈성분을 항복곡면에 의해 정의하고, 소성이론(Plasticity Theory)에 의해, 축력과 2축 휈성분들 사이의 상호작용을 고려하는 모델입니다. 축력과 휈성분은 연성하지만 각각의 성분은 이력모델로서 정의되므로, 단면을 세분하는 파이버 모델에 비해 힌지의 상태판정에 소요되는 계산량이 대폭 감소되어 대규모 구조물의 비선형 시간이력해석에도 적용가능합니다. + +midas Civil에서는 다축-힌지 모델로서 파이버 모델과 소성이론을 응용한 이동경화형 이력모델을 제공합니다. + + + +# 9-5-1 이동경화형 (Kinematic Hardening Type) + +다축-힌지를 대상으로 하는 이동경화형 이력모델은 2개의 항복면을 사용한 이동경화 법칙을 따릅니다. 이는 기본적으로 일축-힌지를 대상으로 하는 3선형 이동경화형 Trilinear이력을 축성분과 2축 휨성분으로 확장시킨 것입니다. 힌지의 상태 판정및 그에 따른 유연도 행렬 계산은 정해진 항복면에 대한 하중점의 상대적 위치관계에 의해 결정됩니다. 제하강성은 탄성강성과 동일하며, 2개의 항복면은 항복에의해 위치만 이동하고 형태나 크기 변화는 없다고 가정합니다. + +항복의 판정은 그림 2.9.29에 나타낸 것과 같이 하중점이 1차 항복면 내부에 위치한 경우에는 탄성상태로 간주합니다. 재하 과정에서 하중점이 1차 항복면과 만나면 1차 항복이 발생한 것으로 간주하며 계속해서 하중점이 2차 항복면에 도달하면2차 항복이 발생한 것으로 간주합니다. + +![](images/page-416_9bda33487b66bb64eb43c618b2ec2ccaa390d00fa83efa9a2737d0118dca016e.jpg) + +![](images/page-416_1ba8030fe8b37c82cf69f0ef6bd6778c72f545a0f79b4700e06105da427a407b.jpg) +(a) Elastic Loading + +![](images/page-416_7439c37c0240e77e41486dc8ffc36eeede7486d97551cb749e41f034dbd84178.jpg) + +![](images/page-416_af1d36a9d2b82054b1e3a6f22d7a46245c438463b83263167fe07d6e47cd732b.jpg) +(b) Post Crack + +![](images/page-416_aab9ce68548bd360f3ea6301296f062526f3ec398907d023301fd0b2a1096056.jpg) + +![](images/page-416_49e5adfffa00e7a2ac14ae9449179e395ecf9b223d8968b856b4ab271b4448ce.jpg) +(c) Post Yielding + +![](images/page-416_45df0ca71a71c7872b1b943753183443af297bbd8c75c70936083c6d7d30a178.jpg) + +![](images/page-416_c91877dd34d6591960efdf1031ee74894c58d4420c5db8b48c44a941cf8779e2.jpg) +(d) UnLoading +그림 2.9.29 항복면의 이동 및 강성변화 + +힌지의 유연도 행렬은 세 개의 직렬 연결된 스프링의 유연도의 합으로 가정됩니다.직렬 연결된 스프링은 각각 탄성 스프링과 두개의 비탄성 스프링으로 구성되며 초기에는 탄성 스프링만 유연도를 갖고 나머지는 강체(Rigid)로 가정합니다. 하중점이 각각의 항복면과 접할때 마다 관련된 비탄성 스프링의 유연도가 발생하는 것으로 간주합니다. N-차 항복 후에 유연도 행렬의 계산식은 다음과 같습니다. 여기서항복면과 관련된 항목은 현재의 하중점이 접하고 있는 항복면에 대해서만 계산됩니다. + +$$ +F _ {s} = K _ {s, (0)} ^ {- 1} + \sum_ {i = 1} ^ {N} \frac {a _ {(i)} a _ {(i)} ^ {T}}{a _ {(i)} ^ {T} K _ {s , (i)} a _ {(i)}} +$$ + + + +여기서, + +$$ +K _ {s, (i)} = \left[ \begin{array}{c c c} k _ {1, (i)} & 0 & 0 \\ 0 & k _ {2, (i)} & 0 \\ 0 & 0 & k _ {3, (i)} \end{array} \right] +$$ + +$$ +\frac {1}{k _ {n , (i)}} = \left(\frac {1}{r _ {n , (i)}} - \frac {1}{r _ {n , (i - 1)}}\right) \frac {1}{k _ {n , (0)}} \quad (n = 1, 2, 3; i = 1, 2) +$$ + +i : 현재의 하중점이 접하고 있는 항복면의 차수 + +Fs : 힌지의 접선 유연도 행렬 + +a(i) : i-번째 항복면의 하중점 위치에서의 법선 벡터 + +kn,(i) : n-번째 성분의 i-번째 직렬 스프링 강성(i=0인 경우에는 탄성강성) + +rn,(i) : n-번째 성분의 i-번째 항복 시 강성 저감률(i=0인 경우에는 1.0) + +rn,(i) : n-번째 성분의 i-번째 항복 시 강성 저감률(i=0인 경우에는 1.0) + +상기 식의 유연도 행렬 Fs는 탄성상태에서는 대각행렬로서 3개 성분이 완전히 독립적이며 항복 변형 중에는 대각성분에 의해 3개 성분 사이의 상관작용이 발생합니다. 하중점이 도달 항복면의 외측으로 이동하게 되면 항복면은 하중점과 접해있는 상태를 유지하도록 함께 이동합니다. 이동 방향은 변형된 Mroz의 경화법칙(Hardening Rule)을 따릅니다. 하중점이 항복면상에서 내측으로 이동하게 되면 제하로 판정하며 제하강성은 탄성강성과 동일하다. 제하과정에서 항복면은 이동하지않습니다. + +Sc : conjugate loading point + +S : translation of loading point + +: translation of the 1st yield surface center + +: translation of the 2nd yield surface center + +![](images/page-417_d5735f7d39c558c010423cd45f2e3c96a6e6f5a3f234b8a6be8660a31c34c109.jpg) + +
+text_image + +Mz +S +C1 +Sc +My +
+ +(a) 1차 항복 이후의 경화 상태 + +![](images/page-417_f3d59ff9a8e9397c72ef7dda624166ba19e5c0df196db06956204b417c6d36ce.jpg) + +
+text_image + +Mz +S +C1 +C2 +My +
+ +(b) 2차 항복 이후의 경화 상태 +그림 2.9.30 경화법칙 + + + +# 9-5-2 P-M 및 P-M-M 상관작용 + +기둥이나 교각과 같이 축력과 휨 모멘트가 동시에 작용하는 부재의 경우, 축력과모멘트의 상관작용에 의해 각 성분이 독립적으로 작용할 때와는 다른 항복강도를갖게 됩니다. 특히 3차원 시간이력해석에서는 2방향 지진에 의해 기둥 부재 내부의 2축 휨 모멘트 및 축력 사이에 복잡한 상관작용이 발생하고 이는 구조물의 동적응답에 큰 영향을 미칠 수 있습니다. midas Civil의 비선형 시간이력 해석에서는P-M 상관작용 또는 P-M-M 상관작용을 고려한 해석을 수행할 수 있습니다. + +# P-M 상관작용 + +P-M 상관작용은 축력의 영향을 고려하여 힌지의 휨 항복강도를 산정함으로써 반영됩니다. 이 때 2축 휨 모멘트의 상관작용은 무시됩니다. 각각의 시간증분에 대한힌지 상태판정에 있어서는 축력과 두개의 휨 모멘트는 모두 상호 독립적인 것으로간주됩니다. 축력을 고려한 휨 모멘트 항복강도의 재산정을 위해서는, 비선형 정적해석을 수행하여 정적해석에 의한 축력을 계산하고, 연속해서 시간이력 해석을 수행하도록 각각의 하중을 별도의 하중조건으로 만든 뒤 재하순서 및 하중의 연속성을 부과하여 해석을 수행하여 합니다. + +대상 요소는 P-M 상관작용이 적용되는 힌지 속성이 부여된 비탄성 보요소입니다.이 때 초기단면력은 시간변동 정적하중(Time Varying Static Load)에 포함되는 모든 정적하중에 대한 선형탄성 해석결과의 조합으로 가정되며 조합에 사용되는 계수는 시간변동 정적하중에 입력하는 Scale Factor에 의해 정의됩니다. + +MC : 1st Yield Moment MY : 2nd Yield Moment S : Loading Point by Static Loads +![](images/page-418_54e39d77cad01f1c260c7ac1b486ec9a77a7e47892064b34f14c9286421dcb24.jpg) + +
+text_image + +2nd Yield Surface +1st Yield Surface +P +S +MC MY M +
+ +(a) + +![](images/page-418_0b76eaeca839274fae52730fe7c8ff47ff67b16ffbedb7e473e2564a4600862c.jpg) + +
+text_image + +P +S +MC MY M +
+ +(b) + +![](images/page-418_ef8992fce7c20639e8ef0b7b829003629cfe4d91252168b370aa47b347e21737.jpg) + +
+text_image + +P +S +MC MY M +
+ +그림 2.9.31 P-M 상관작용에 의한 휨 항복강도 산정 + + + +휨에 대한 항복강도의 계산은 위와 같이 계산된 단면력의 2차원 상관곡선의 상대적 위치에 의해 결정되며 이는 그림 2.9.31에 나타냅니다. 초기 단면력이 상관곡선내측에 있으면, 이 하중점의 축력에 해당되는 휨 항복강도를 상관곡선으로부터 계산합니다. 하중점이 상관곡선 외측에 있으면, 하중점과 원점을 잇는 직선이 항복면과 교차하는 점에서 휨 항복강도를 계산합니다. + +# P-M-M 상관작용 + +P-M-M 상관작용은 다축-힌지 이력모델을 사용함으로써 비선형 시간이력해석에 반영될 수 있습니다. 다축-힌지 이력모델은 축력 및 2축 휨 모멘트의 상호작용을 소성이론을 응용해서 구현한 것으로서, P-M-M 상관을 고려하면, 변동축력에 의한 휨항복강도를 변화시키면서 각 시간증분마다 3개 성분의 변동을 통합적으로 고려한상태판정을 수행합니다. midas Civil에서는 이동경화형(Kinematic Hardening type)이지원됩니다. + +![](images/page-419_1611f5a14ea69749f10c3aad0d65aa03a7ba68fd3ef1b181f58fd860cd4e1299.jpg) + +
+text_image + +P +M +
+ +(a) P-M Type(초기축력) + +![](images/page-419_0042f6b4da274ec67e39d6798568b553f3ac7651877c8129c3569d42bbd10af4.jpg) + +
+text_image + +P +M +
+ +(b) P-M-M Type(변동축력) +그림 2.9.32 P-M과 P-M-M 상관에서의 축력관계 + + + +# 9-5-3 항복면의 근사화 + +힌지의 항복강도 산정 또는 상태판정에 있어서 P-M 또는 P-M-M 상관작용을 고려하기 위해서는 P-M 상관곡선으로부터 3차원 상의 항복면을 정의할 필요가 있습니다. 그러나 제한된 P-M 상관곡선의 데이터로부터 정확한 3차원 상의 항복면을 정의하는 것은 어렵기 때문에 이를 단순한 수식으로 근사화할 수 있습니다. midasCivil에서 P-M 상관곡선은 다음 식을 통해 근사화 됩니다. + +$$ +\left| \frac {M}{M _ {\max}} \right| ^ {\gamma} + \left| \frac {P - P _ {\mathrm{bal}}}{P _ {\max} - P _ {\mathrm{bal}}} \right| ^ {\beta} = 1. 0 +$$ + +여기서 M : 하중점의 요소좌표계 y축 또는 z축에 대한 모멘트 성분(My 또는 Mz) + +$M _ { \mathrm { { m a x } } } .$ 요소좌표계 y축 또는 z축에 대한 최대 휨 항복강도 $( M _ { y , \operatorname* { m a x } }$ 또는 $M _ { z , \mathrm { m a x } } )$ + +$P$ : 하중점의 축력 성분 + +$P _ { \mathsf { b a l } }$ : y-축 또는 z-축에 대한 균형파괴시 축하중 $( P _ { \mathsf { b a l } , y } , P _ { \mathsf { b a l } , z } )$ + +$P _ { \mathrm { m a x } }$ : 축 항복강도로서 정(+), 부(-) 비대칭 가능. + +$\boldsymbol { \mathsf { Y } }$ : 곡면 차수 + +$\beta$ : 요소좌표계 y축/z축에 대한 곡면 차수로서 정(+), 부(-) 비대칭 가능 + +midas Civil에서 M-M 상관곡선은 다음 식을 통해 근사화 됩니다. + +$$ +\left| \frac {M _ {y}}{M _ {y , \max}} \right| ^ {\alpha} + \left| \frac {M _ {z}}{M _ {z , \max}} \right| ^ {\alpha} = 1. 0 +$$ + +여기서 $M _ { y , \mathrm { m a x } }$ : 요소좌표계 y-축에 대한 최대 휨 항복강도 + +$M _ { z , \mathrm { m a x } }$ : 요소좌표계 z-축에 대한 최대 휨 항복강도 + +α : 곡선 차수 + +3차원 항복면은 상기의 근사화된 상관곡선을 만족할 수 있는 다음 수식을 사용합니다. diff --git a/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_043.md b/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_043.md new file mode 100644 index 00000000..ceedfdf5 --- /dev/null +++ b/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_043.md @@ -0,0 +1,406 @@ + + +$$ +\begin{array}{l} f \left(P, M _ {y}, M _ {z}\right) = \left\{\left(\frac {M _ {y}}{M _ {y , \max}}\right) ^ {\gamma} + \left\{g _ {y} \left(M _ {y}, M _ {z}\right) \right\} ^ {\frac {\gamma}{\alpha}} \cdot \left(\frac {P - P _ {b a l , y}}{P _ {\max} - P _ {b a l , y}}\right) ^ {\beta_ {y}} \right\} ^ {\frac {\alpha}{\gamma}} \\ + \left\{\left(\frac {M _ {z}}{M _ {z , \max}}\right) ^ {\gamma} + \left\{g _ {z} \left(M _ {y}, M _ {z}\right) \right\} ^ {\frac {\gamma}{\alpha}} \cdot \left(\frac {P - P _ {b a l , z}}{P _ {\max} - P _ {b a l , z}}\right) ^ {\beta_ {z}} \right\} ^ {\frac {\alpha}{\gamma}} = 1 \\ \end{array} +$$ + +여기서 + +$$ +g _ {y} \left(M _ {y}, M _ {z}\right) = \frac {\left(\frac {M _ {y}}{M _ {y , \max}}\right) ^ {\alpha}}{\left(\frac {M _ {y}}{M _ {y , \max}}\right) ^ {\alpha} + \left(\frac {M _ {z}}{M _ {z , \max}}\right) ^ {\alpha}} +$$ + +$$ +g _ {z} \left(M _ {y}, M _ {z}\right) = \frac {\left(\frac {M _ {z}}{M _ {z , \max}}\right) ^ {\alpha}}{\left(\frac {M _ {y}}{M _ {y , \max}}\right) ^ {\alpha} + \left(\frac {M _ {z}}{M _ {z , \max}}\right) ^ {\alpha}} +$$ + +근사적 상관곡선 차수 $\beta_{y}$ , $\beta_{z}$ 및 $\gamma$ 는 사용자 입력 또는 최적 값에 대한 자동계산이 가능합니다. 최적 값은 $\gamma$ 를 1.0 에서 3.0까지 0.1씩 증가시켜 가면서 주어진 $\gamma$ 에 대해 $\beta_{y}$ 및 $\beta_{z}$ 의 값을 계산하고 이 조합들 가운데 오차를 최소로 하는 것으로 합니다. $\beta_{y}$ 및 $\beta_{z}$ 는 각각 $P-M_{y}$ 평면 및 $P-M_{z}$ 평면이 항복면과 교차해서 만들어지는 근사적 상관곡선과 실제 계산된 상관곡선의 면적이 일치하도록 하는 값으로 산정됩니다. 오차는 상관곡선 산정의 기준 축력에서 근사적 상관곡선과 실제 상관곡선의 모멘트 차의 절대합으로 정의됩니다. + +3차원 항복면에는 삼선형(Tri-linear) 골격곡선에 상응하는 두 개의 항복면이 있으며 편의상 안쪽에 놓여지는 것을 1차 항복면, 바깥쪽에 놓여 지는 것을 2차 항복면이라고 명칭합니다. RC단면의 경우에 1차 항복면은 단면의 균열에 대응되며 2차 항복면은 단면의 항복에 대응됩니다. 이 가운데 1차 항복면은 균열곡선을 그림 2.9.33과 같이 근사화하여 사용합니다. 먼저 2차 항복면을 동일 면적을 갖도록 2개의 직선으로 근사화 합니다. 다음으로는 두 직선 중 사선과 원래의 균열곡선이 형성하는 삼각형에 접하면서 둘러싸일 수 있도록 1차 항복면의 파라미터를 계산합니다. + + + +![](images/page-422_fe45cb06c9581de7ef91acb992d8d49f26fc7bec88b8f9212f368bec16a72807.jpg) + +
+text_image + +P +yc₂ yc₁ +yt₂ yt₁ +x₁ +x₂ +M +Approximated Crack Surface +Crack Surface +Yield Surface +Approximated Yield Surface +x₁ = 0.8 · x₂ +yc₁ = 0.9 · yc₂ +yt₁ = 0.9 · yt₂ +
+ +그림 2.9.33 RC 단면의 균열곡면 근사화 + + + +# 9-6 파이버 모델 + +비탄성 힌지의 종류는 일축힌지 이력모델, 소성이론에 근거한 다축힌지 이력모델, 그리고 파이버모델 등으로 분류할 수 있습니다. 일축힌지 이력모델은 축력 혹은 2 축 휩 등의 효과 등을 반영하지 않고 경험적으로 정해진 이력 특성을 사용하는 힌지모델이며, 부재 힌지의 이력 특성이 구조체에 미치는 영향이 크지 않거나 간편한 방법으로 결과를 얻고자 할 때 유용한 방법입니다. 반면에 다축힌지 이력모델은 소성이론의 항복면을 통하여 축력과 2축 휩의 효과를 반영할 수 있는 모델이지만, 다양한 이력거동의 특성을 재현하는데는 한계가 있습니다. 파이버모델도 다축힌지 이력모델과 같이 축력과 2축 휩의 효과를 반영할 수 있는 모델이나, 전단력의 영향이 크지 않은 선부재(Line Element)의 구조적 특성을 효율적으로 이용하는 방법입니다. 휩모멘트를 받는 단면은 변형 후에도 평면을 유지한다는 특성을 이용하여 정식화 됩니다. 따라서 단면에서 발생하는 변형률은 중립축으로부터의 거리에 비례하므로, 여기에 곡률을 곱하면 변형률을 구합니다. + +파이버모델에서는 분포형 힌지 모델의 각 적분점의 단면을 그림 2.9.34와 같은 형태의 격자(Fiber) 혹은 층(Layer)의 셀(Cell)로 분할한 후, 각 셀은 동일한 응력을 갖는다는 가정을 사용합니다. 이 때 각 셀은 콘크리트, 철골 혹은 철근 등 사용자의 선택에 따라 다양한 재료를 사용할 수 있으며, 임의 형상의 단면을 사용하는 것도 가능합니다. + +![](images/page-423_432a3763d337a5001f0d009536cfcaffaca271112fb7f8aa4b7409bb76033e73.jpg) + +
+natural_image + +Grid pattern with alternating light blue and white circles (no text or symbols) +
+ +격자모델(Fiber Model) + +![](images/page-423_f71dbfe42940721932157e429bea83d04b2745f3e85622ccd1799fd6d8989ce3.jpg) + +
+natural_image + +Simple diagram of six light blue circles arranged in two horizontal rows (no text or symbols) +
+ +층모델 (Layer Model) +그림 2.9.34 파이버모델의 셀 분할 방법 + + + +각 셀은 독립된 재료 모델을 선정할 수 있습니다. 단면력(모멘트, 축력)은 각 셀의응력을 적분하며, 단면의 강성(Stiffness)은 단면 유연도(Sectional Flexibility)의 역행렬로서 얻어지게 됩니다. 그리고 요소 혹은 부재의 강성은 선정된 적분점(집중형혹은 분포형) 들을 대상으로 적분을 수행함에 의해서 얻어집니다. 따라서 파이버모델은 휨부재의 역학적 특성을 정확하게 반영하기 때문에, 해석결과의 정확도는 매우 높다고 할 수 있습니다. 단 단면을 여러 개의 셀로서 분하여야 하기 때문에 해석에 소요되는 시간이 길어지게 됩니다. + +Fiber 모델은 다음과 같은 기본 가정하에 정식화 됩니다. + +1. 단면은 변형과정에서 평면을 유지하며 부재 축과 수직을 이루는 것으로 가정합니다. 따라서 철근과 콘크리트 사이의 부착-미끄러짐(Bond-Slip)은 고려되지 않습니다. +2. 단면의 도심축은 보요소의 전체 길이에 걸쳐 직선인 것으로 가정합니다. + +파이버 모델의 해석 알고리즘은 다음과 같습니다. 각 요소의 적분점 위치에 파이버 모델이 정의된 단면이 존재한다고 가정하고. 적분점 개수는 최대 20개까지 가능합니다. 적분 방식은 기본적으로 부재 양 끝단의 결과까지 확인이 가능한Gauss-Lobatto 방식을 사용하며, 적분점이 2개일 때만 일반 Gauss 적분법을 사용합니다. 이전 시간증분에서 얻어진 양단의 부재력을 변환과정을 통해 강체거동(Rigid Body Modes)을 제외한 5개의 자유도에 해당하는, 축력과 양단 2개의 모멘트(Generalized Element Forces)로 변환합니다. 이렇게 얻어진 축력과 양단의 두 모멘트를 내삽 함수(Force Interpolation Function)를 사용하여 각 단면 위치에서의 힘을계산합니다. + + + +![](images/page-425_ab691234dfe5ab7e47d23f643486b4c7f5b5fce053fec574484e0d0f6963db7a.jpg) + +
+text_image + +Y +Q₄, q₄ +x +Q₅, q₅ +y +Q₃, q₃ +Q₂, q₂ +Q₁, q₁ +X +z +Mᵧ(x), χᵧ(x) +N(x), ε(x) +M₂(x), χ₂(x) +Z +
+ +그림 2.9.35 부재의 임의 단면에서의 부재력과 변형 + +$$ +\begin{array}{l} \text { Element Force Vector } \quad : \boldsymbol {Q} = \left\{\mathrm{Q} _ {1}, \mathrm{Q} _ {2}, \mathrm{Q} _ {3}, \mathrm{Q} _ {4}, \mathrm{Q} _ {5} \right\} ^ {T} \\ \text { Element Deformation Vector }: \boldsymbol {q} = \left\{\mathrm{q} _ {1}, \mathrm{q} _ {2}, \mathrm{q} _ {3}, \mathrm{q} _ {4}, \mathrm{q} _ {5} \right\} ^ {T} \\ \text { Section Force Vector } \quad : \quad \boldsymbol {D} (\mathrm{x}) = \left\{\mathrm{M} _ {\mathrm{z}} (\mathrm{x}) \quad \mathrm{M} _ {\mathrm{y}} (\mathrm{x}) \quad \mathrm{N} (\mathrm{x}) \right\} ^ {T} \\ \text { Section Deformation Vector } \quad : \quad \boldsymbol {d} (\mathrm{x}) = \left\{\chi_ {\mathrm{z}} (\mathrm{x}) \quad \chi_ {\mathrm{y}} (\mathrm{x}) \quad \varepsilon (\mathrm{x}) \right\} ^ {T} \\ \end{array} +$$ + +$$ +\Delta \boldsymbol {D} ^ {i} (\mathrm{x}) = \boldsymbol {b} (\mathrm{x}) \cdot \Delta \boldsymbol {Q} ^ {i} +$$ + +여기서, $\boldsymbol{b}(\mathbf{x})$ 는 내삽함수(Force Interpolation Function)로서 다음과 같습니다. + +$$ +\boldsymbol {b} (\mathrm{x}) = \left[ \begin{array}{c c c c c} \left(\frac {\mathrm{x}}{\mathrm{L}} - 1\right) & \left(\frac {\mathrm{x}}{\mathrm{L}}\right) & 0 & 0 & 0 \\ 0 & 0 & \left(\frac {\mathrm{x}}{\mathrm{L}} - 1\right) & \left(\frac {\mathrm{x}}{\mathrm{L}}\right) & 0 \\ 0 & 0 & 0 & 0 & 1 \end{array} \right] +$$ + +여기서, L : 부재 길이 + + + +단면의 힘과 유연도(Flexibility)를 연산하여 단면의 변형을 계산합니다. 이렇게 얻은단면의 축, 휨 변형으로부터 섬유 각각의 축 변형률을 계산하게 되며 그 관계식은다음과 같습니다. + +![](images/page-426_8cd5be57f6deebe52515a123290f19255416f155464c061396070b782f1e157a.jpg) + +
+text_image + +ECS z-axis +ECS y-axis +ECS x-axis +x +ECS z-axis +y_i +i-th fiber +z_i +ECS y-axis +
+ +그림 2.9.36 Fiber 모델에서의 단면 분할 + +$$ +\varepsilon_ {\mathrm{i}} = \left[ \begin{array}{l l l} \mathrm{z} _ {\mathrm{i}} & - \mathrm{y} _ {\mathrm{i}} & 1 \end{array} \right] \left\{ \begin{array}{l} \chi_ {\mathrm{y}} (\mathrm{x}) \\ \chi_ {\mathrm{z}} (\mathrm{x}) \\ \varepsilon_ {\mathrm{x}} (\mathrm{x}) \end{array} \right\} +$$ + +여기서, x : 단면의 위치 + +χy(x) : 위치 x에서의 단면의 요소좌표계 y-축에 대한 곡률 + +χz(x) : 위치 x에서의 단면의 요소좌표계 z-축에 대한 곡률 + +x(x) : 위치 x에서의 단면의 축방향 변형율 + +yi : 단면 상에서의 i-번째 섬유의 y-축 위치 + +zi : 단면 상에서의 i-번째 섬유의 z-축 위치 + +I : i-번째 섬유의 변형율 + +각 섬유의 축변형률  에 대응되는 섬유의 응력과 접선 강성을 재료별로 정의된구성관계식(Constitutive Relation)으로부터 얻으며, 섬유의 상태를 판정합니다. 하나의 단면내의 각 섬유들의 응력들을 적분하여 단면의 축력 및 휨 모멘트를 계산하며, 각 섬유들의 접선 강성을 적분하여 단면의 유연도를 얻습니다. 한 부재 내 단면들의 유연도를 적분하여 부재의 유연도를 갱신합니다. 이러한 과정의 강성행렬과 유연도 행렬은 다음과 같습니다. + + + +$$ +\boldsymbol {k} ^ {j} (\mathrm{x}) = \left[ \begin{array}{l l l} \sum_ {\mathrm{i} = 1} ^ {\mathrm{n} (\mathrm{x})} \mathrm{E} _ {\mathrm{i}} ^ {\mathrm{j}} \cdot \mathrm{A} _ {\mathrm{i}} \cdot \mathrm{y} _ {\mathrm{i}} ^ {2} & - \sum_ {\mathrm{i} = 1} ^ {\mathrm{n} (\mathrm{x})} \mathrm{E} _ {\mathrm{i}} ^ {\mathrm{j}} \cdot \mathrm{A} _ {\mathrm{i}} \cdot \mathrm{y} _ {\mathrm{i}} \cdot \mathrm{z} _ {\mathrm{i}} & - \sum_ {\mathrm{i} = 1} ^ {\mathrm{n} (\mathrm{x})} \mathrm{E} _ {\mathrm{i}} ^ {\mathrm{j}} \cdot \mathrm{A} _ {\mathrm{i}} \cdot \mathrm{y} _ {\mathrm{i}} \\ - \sum_ {\mathrm{i} = 1} ^ {\mathrm{n} (\mathrm{x})} \mathrm{E} _ {\mathrm{i}} ^ {\mathrm{j}} \cdot \mathrm{A} _ {\mathrm{i}} \cdot \mathrm{y} _ {\mathrm{i}} \cdot \mathrm{z} _ {\mathrm{i}} & \sum_ {\mathrm{i} = 1} ^ {\mathrm{n} (\mathrm{x})} \mathrm{E} _ {\mathrm{i}} ^ {\mathrm{j}} \cdot \mathrm{A} _ {\mathrm{i}} \cdot \mathrm{z} _ {\mathrm{i}} ^ {2} & \sum_ {\mathrm{i} = 1} ^ {\mathrm{n} (\mathrm{x})} \mathrm{E} _ {\mathrm{i}} ^ {\mathrm{j}} \cdot \mathrm{A} _ {\mathrm{i}} \cdot \mathrm{z} _ {\mathrm{i}} \\ - \sum_ {\mathrm{i} = 1} ^ {\mathrm{n} (\mathrm{x})} \mathrm{E} _ {\mathrm{i}} ^ {\mathrm{j}} \cdot \mathrm{A} _ {\mathrm{i}} \cdot \mathrm{y} _ {\mathrm{i}} & \sum_ {\mathrm{i} = 1} ^ {\mathrm{n} (\mathrm{x})} \mathrm{E} _ {\mathrm{i}} ^ {\mathrm{j}} \cdot \mathrm{A} _ {\mathrm{i}} \cdot \mathrm{z} _ {\mathrm{i}} & \sum_ {\mathrm{i} = 1} ^ {\mathrm{n} (\mathrm{x})} \mathrm{E} _ {\mathrm{i}} ^ {\mathrm{j}} \cdot \mathrm{A} _ {\mathrm{i}} \end{array} \right] +$$ + +$$ +\boldsymbol {f} ^ {j} (\mathrm{x}) = \left[ \boldsymbol {k} ^ {j} (\mathrm{x}) \right] ^ {- 1} +$$ + +여기서, $k^{j}(x)$ : j-step, 거리 x에 위치한 단면의 접선강성 + +$f^{j}(x)$ : j-step, 거리 x에 위치한 단면의 유연도(Flexibility) + +n(x) : 거리 x에 위치한 단면 내 총 섬유 개수 + +$E^{j}_{i}(x)$ : j-step, 거리 x에 위치한 단면 내의 ‘i’ 섬유의 접선강성. + +$A_{i}$ : ‘i’ 섬유의 단면적 + +$y_{i}, z_{i}$ : ‘i’ 섬유의 단면 내 위치 + +한편 불평형력 산정에 필요한 단면 내력(Section Resisting Force)은 다음과 같이 +현 증분에서 각 섬유가 발현하고 있는 응력을 적분하여 얻습니다. + +$$ +\boldsymbol {D} _ {\boldsymbol {R}} ^ {j} (\mathrm{x}) = \left\{- \sum_ {\mathrm{i} = 1} ^ {\mathrm{n} (\mathrm{x})} \sigma_ {\mathrm{i}} ^ {\mathrm{j}} \mathrm{A} _ {\mathrm{i}} \mathrm{y} _ {\mathrm{i}} \quad \sum_ {\mathrm{i} = 1} ^ {\mathrm{n} (\mathrm{x})} \sigma_ {\mathrm{i}} ^ {\mathrm{j}} \mathrm{A} _ {\mathrm{i}} \mathrm{z} _ {\mathrm{i}} \quad \sum_ {\mathrm{i} = 1} ^ {\mathrm{n} (\mathrm{x})} \sigma_ {\mathrm{i}} ^ {\mathrm{j}} \mathrm{A} _ {\mathrm{i}} \right\} ^ {\boldsymbol {T}} +$$ + +이와 같은 과정이 각 Newton-Raphson Iteration 내에서 사용자가 정의한 수렴조건이 만족될 때까지 수행합니다. 한편 파이버 모델에서 단면의 비선형 거동 특성은 비선형 섬유의 응력-변형율 관계에 의해서 정의됩니다. 부재의 비선형적 거동은 모두 섬유의 응력-변형율 관계로부터 구현되기 때문에 midas Civil에서는 다양한 강 섬유 및 콘크리트 섬유의 재료 모델을 제공하고 있습니다. 이하에서 각 재료의 구성모델(Constitutive Model)에 관해 설명하고 있습니다. + + + +# 9-6-1 강 섬유 구성 모델 + +# Modified Menegotto & Pinto Steel Model + +Menegotto & Pinto(1973)1)가 제안한 모델을 Filippou(1983)2) 등이 수정한 모델로 수치적인 효율성이 높고 실험적 결과와 높은 일치성을 보이는 모델로 평가 받고 있습니다. 구성모델은 기본적으로 2선형 이동경화(Kinematic Hardening) 법칙에 따라설정된 점근선으로 접근하는 곡선형상을 갖습니다. 각각 제하(Unloading) 경로 및변형율-경화(Strain Hardening) 구간에 대응되는 두 점근선 사이의 전이 구간은 곡선 형상을 갖습니다. 이 전이구간은 두 점근선의 교점과 제하(Unloading)되는 방향의 최대 변형점이 서로 멀리 떨어져 있을수록 부드러운 곡선이 되고 이러한 특성을 통해 Bauschinger Effect를 정밀하게 모사할 수 있습니다. + +$$ +\hat {\sigma} = b \cdot \hat {\varepsilon} + \frac {(1 - b) \cdot \hat {\varepsilon}}{\left(1 + \hat {\varepsilon} ^ {R}\right) ^ {1 / R}} +$$ + +여기서, $\hat { \varepsilon } = \frac { \varepsilon - \varepsilon _ { r } } { \varepsilon _ { 0 } - \varepsilon _ { r } } \ , \ \hat { \sigma } = \frac { \sigma - \sigma _ { r } } { \sigma _ { 0 } - \sigma _ { r } } \ , \ R = R _ { 0 } - \frac { a _ { 1 } \cdot \xi } { a _ { 2 } + \xi }$ + + : 강 섬유의 변형율 + + : 강 섬유의 응력 + +$( \varepsilon _ { \Gamma } , \sigma _ { \Gamma } )$ : 제하점으로서 초기 탄성상태에서는 (0, 0)으로 가정. + +$( \varepsilon 0 , \sigma 0 )$ : 현재의 재하 또는 제하 경로를 정의하는 두 점근선의 교점 + +b : 강성 저감률 + +R0, a1, a2 : 상수 (곡선형태를 결정하는 값으로 실험으로부터 얻은 최적값을 default로 사용) + + :하중이 재하/제하되는 방향으로 최대변형율과 0의 차(절대값) + +단, 최대 변형율의 초기치는 (Fy/E)와 같다고 설정 (그림 2.9.37 참조) + + + +![](images/page-429_206a0e82d267fefe12c530129ebc772909ded02e4306aadac27cd33b1fd1cbba.jpg) + +
+text_image + +(ε₀, σ₀)₂ +Fᵧ +b·E +E +(εᵣ, σᵣ)₂ +(εᵣ, σᵣ)₁ +ξ₁ +ξ₂ +
+ +그림 2.9.37 강 섬유 구성모델 + +# Bilinear Steel Model + +일반적인 이선형(Bilinear) 모델로서 항복이전과 항복이후의 강성을 달라집니다. 항복이전 재하(Loading), 제하(Unloading)는 탄성강성을 사용하며, 항복이후에는 감소된 강성으로 재하가 진행됩니다. 항복이후 제하, 재재하(Re-loading)는 탄성강성으로 진행됩니다. + +# Trilinear Steel Model + +일반적인 삼선형(Trilinear) 모델로서 탄성과 1차 항복이후와 2차 항복 이후의 강성을 달리 정의할 수 있습니다. 범용성을 갖추기 위해 압축부와 인장부의 1, 2차 항복 변형률과 기울기를 다르게 정의하여 비대칭 이력을 정의할 수 있습니다. 항복이전 재하(Loading), 제하(Unloading)는 탄성강성을 사용하며, 항복이후에는 감소된강성으로 재하가 진행됩니다. 항복이후 제하, 재재하(Re-loading)는 탄성강성으로진행됩니다. + +# Asymmetrical Bilinear Steel Model + +본 모델은 철근에서 발생할 수 있는 거의 모든 현상을 모사할 수 있도록 고안된모델로서 모든 강성은 다르게 정의할 수 있으며, 인장측으로는 항복, 파단을 고려하고, 압축측으로는 항복, 좌굴 후 파괴를 고려할 수 있도록 고안되었습니다. + + + +![](images/page-430_b6d2665a18f254158734df756897e74e19c5a775fe69ada8442cfd67cdd3b1ee.jpg) + +
+flowchart + +```mermaid +graph TD + A["Region ①"] -->|σy| B["Region ②"] + A -->|σy| C["Region ③"] + A -->|σy| D["Region ④"] + A -->|σy| E["Region ⑤"] + A -->|σy| F["Region ⑥"] + A -->|σy| G["Region ⑦"] + A -->|σy| H["Region ⑧"] + A -->|σy| I["Region ⑨"] + A -->|σy| J["Region ⑩"] + A -->|σy| K["Region ⑪"] + A -->|σy| L["Region ⑫"] + A -->|σy| M["Region ⑬"] + A -->|σy| N["Region ⑭"] + A -->|σy| O["Region ⑮"] + A -->|σy| P["Region ⑯"] + A -->|σy| Q["Region ⑰"] + A -->|σy| R["Region ⑱"] + A -->|σy| S["Region ⑲"] + A -->|σy| T["Region ⑳"] + A -->|σy| U["Region ⑪"] + A -->|σy| V["Region ⑫"] + A -->|σy| W["Region ⑬"] + A -->|σy| X["Region ⑭"] + A -->|σy| Y["Region ⑮"] + A -->|σy| Z["Region ⑯"] + A -->|σy| AA["Region ⑮"] + A -->|σy| AB["Region ⑭"] + A -->|σy| AC["Region ⑮"] + A -->|σy| AD["Region ⑯"] + A -->|σy| AE["Region ⑮"] + A -->|σy| AF["Region ⑭"] + A -->|σy| AG["Region ⑮"] + A -->|σy| AH["Region ⑯"] + A -->|σy| AI["Region ⑮"] + A -->|σy| AJ["Region ⑭"] + A -->|σy| AK["Region ⑮"] + A -->|σy| AL["Region ⑯"] + A -->|σy| AM["Region ⑮"] + A -->|σy| AN["Region ⑭"] + A -->|σy| AO["Region ⑮"] + A -->|σy| AP["Region ⑯"] + A -->|σy| AQ["Region ⑮"] + A -->|σy| AR["Region ⑭"] + A -->|σy| AS["Region ⑮"] + A -->|σy| AT["Region ⑯"] + A -->|σy| AU["Region ⑮"] + A -->|σy| AV["Region ⑭"] + A -->|σy| AW["Region ⑮"] + A -->|σy| AX["Region ⑯"] + A -->|σy| AY["Region ⑮"] + A -->|σy| AZ["Region ⑭"] + A -->|σy| BA["Region ⑮"] + A -->|σy| BB["Region ⑯"] + A -->|σy| BC["Region ⑮"] + A -->|σy| BD["Region ⑭"] + A -->|σy| BE["Region ⑮"] + A -->|σy| BF["Region ⑯"] + A -->|σy| BG["Region ⑮"] + A -->|σy| BH["Region ⑭"] + A -->|σy| BI["Region ⑮"] + A -->|σy| BJ["Region ⑯"] + A -->|σy| BK["Region ⑮"] + A -->|σy| BL["Region ⑭"] + A -->|σy| BM["Region ⑮"] + A -->|σy| BN["Region ⑯"] + A -->|σy| BO["Region ⑮"] + A -->|σy| BP["Region ⑭"] + A -->|σy| BQ["Region ⑮"] + A -->|σy| BR["Region ⑯"] + A -->|σy| BS["Region ⑮"] + A -->|σy| BT["Region ⑭"] + A -->|σy| BU["Region ⑮"] + A -->|σy| BV["Region ⑯"] + A -->|σy| BW["Region ⑮"] + A -->|σy| BX["Region ⑭"] + A -->|σy| BY["Region ⑮"] + A -->|σy| BZ["Region ⑯"] + A -->|σy| CA["Region ⑮"] + A -->|σy| CB["Region ⑭"] + A -->|σy| CC["Region ⑮"] + A -->|σy| CD["Region ⑯"] + A -->|σy| CE["Region ⑮"] + A -->|σy| CF["Region ⑭"] + A -->|σy| CG["Region ⑮"] + A -->|σy| BH["Region ⑯"] + A -->|σy| BI["Region ⑮"] + A -->|σy| BJ["Region ⑭"] + A -->|σy| BK["Region ⑮"] + A -->|σy| BL["Region ⑯"] + A -->|σy| BM["Region ⑮"] + A -->|σy| BN["Region ⑭"] + A -->|σy| BO["Region ⑮"] + A -->|σy| BP["Region ⑮"] + A -->|σy| BQ["Region ⑭"] + A -->|σy| BR["Region ⑮"] + A -->|σy| CA["Region ⑯"] + A -->|σy| CB["Region ⑮"] + A -->|σy| CC["Region ⑭"] + A -->|σy| DQ["Region ⑮"] + A -->|σy| BE["Region ⑭"] + A -->|σy| BF["Region ⑮"] + A -->|σy| BG["Region ⑭"] + A -->|σy| BH["Region ⑮"] + A -->|σy| BI["Region ⑭"] + A -->|σy| BJ["Region ⑮"] + A -->|σy| BK["Region ⑭"] + A -->|σy| BL["Region ⑮"] + A -->|σy| BN["Region ⑭"] + A -->|σy| BO["Region ⑮"] + A -->|σy| BP["Region ⑮"] + A -->|σy| BQ["Region ⑭"] + A -->|σy| BC["Region ⑮"] + A -->|σy| BD["Region ⑭"] + A -->|σy| BE["Region ⑮"] + A -->|σy| BF["Region ⑭"] + A -->|σy| BG["Region ⑮"] + A -->|σy| BH["Region ⑮"] + A -->|σy| BI["Region ⑭"] + A -->|σy| BJ["Region ⑮"] + A -->|σy| BK["Region ⑭"] + A -->|σy| BL["Region ⑮"] + A -->|σy| BN["Region ⑭"] + A -->|σy| BO["Region ⑮"] +``` +
+ +그림 2.9.38 Hysteresis Rule of Asymmetrical Bilinear Steel Model + +1. 탄성 상태 +2. 항복 이후 상태, E2나 E4의 기울기로 진행 +3. 인장 항복이후 제하(Unloading)가 발생하여 기울기 E3의 직선과 만나 압축항복이 발생한 상태, E4의 기울기 +4. 항복이후 제하(Unloading)가 진행되고 있는 상태, E1 의 기울기 +5. 압축이 좌굴 변형률, 1ε 을 초과하여 진행되는 상태. E5 의 기울기 +6. 압축 좌굴 발생 이후, 재재하(Re-loading)가 진행되는 상태, 인장 항복이전에는 인장 항복점을 향하고, 인장 항복이 이전에 발생하였으면 이전 최대변형률 점을 향해 진행. +7. 압축좌굴이 완전히 발생하여 더 이상 저항을 못하는 상태 +8. 인장 파단이 발생하여 더 이상 저항을 못하는 상태 + +1∼6 단계 진행 중에 제하(Unloading)가 발생한 상태, E1 의 기울기 diff --git a/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_044.md b/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_044.md new file mode 100644 index 00000000..363b7216 --- /dev/null +++ b/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_044.md @@ -0,0 +1,371 @@ + + +# Park Steel Model + +Kent & Park(1973)3) 에 의해 수행된 반복하중을 받는 의 실험을 통하여 제안된 모델입니다. 본 모델은 의 탄성구간, 소성구간과 변형도-경화(StrainHardening) 구간의 모사가 가능하며, Ramberg-Osgood 식에 의해 BauschingerEffect를 정밀하게 나타내어 실험적 결과와 높은 일치성을 보이는 모델입니다. + +# 1) 재하시의 거동 + +재하시의 거동은 다음과 같이 구분됩니다. 재하시의 변형도-경화구간에서의응력-변형도 관계는 Thompson & Park(1980)4) 이 제안한 식을 적용합니다. + +![](images/page-431_caf1dbd5e7daa70c8fc58f32b226da3d7e175fe483ce5bd5cee8ebaf8b6f71a0.jpg) + +
+line + +| Point | Steel Strain | Steel Stress | +|-------|--------------|--------------| +| A | ε_y | f_y | +| B | ε_sh | f_y | +| C | ε_su | f_u | +
+ +그림 2.9.39 Stress-Strain curve for steel with loading of the same sign + +. 탄성 $\daleth \sqsubseteq { \mathsf { f } } ( 0 \mathrm { - } \mathsf { A } ) : 0 \leq \varepsilon \leq \varepsilon _ { y }$ + +$$ +f = E _ {s} \cdot \varepsilon_ {y} +$$ + + + +. 소성구간(A-B) : $\varepsilon_{y} < \varepsilon < \varepsilon_{sh}$ + +$$ +f = f _ {y} +$$ + +. 변형도-경화(strain hardening)구간(B-C) : $\varepsilon_{sh} \leq \varepsilon < \varepsilon_{su}$ + +$$ +f = f _ {y} \left(\frac {m \left(\varepsilon - \varepsilon_ {s h}\right) + 2}{6 0 \left(\varepsilon - \varepsilon_ {s h}\right) + 2} + \frac {\left(\varepsilon - \varepsilon_ {s h}\right) (6 0 - m)}{2 (3 0 r + 1) ^ {2}}\right) +$$ + +$$ +m = \frac {\left(f _ {u} / f _ {y}\right) (3 0 r + 1) ^ {2} - 6 0 r - 1}{1 5 r ^ {2}} +$$ + +$$ +r = \varepsilon_ {u} - \varepsilon_ {s h} +$$ + +여기서, + +ε : 강 섬유의 변형도 + +f : 강 섬유의 응력도 + +$E_{s}$ 강 섬유의 초기강성(탄성계수) + +$\varepsilon_{v}$ : 강 섬유의 항복 변형도 + +$\mathcal{E}_{sh}$ :강 섬유의 변형도-경화시작시의 변형도 + +$\mathcal{E}_{su}$ : 강 섬유의 종국변형도(파단시) + +$f_{y}$ : 강 섬유의 항복응력도 + +$f_{u}$ : 강 섬유의 극한응력도 + +2) 제하 및 재재하시의 거동 + +제하시 및 재재하시의 거동은 Ramberg-Osgood 관계에 의해서 정의되며, Newton's Method에 의한 반복계산을 통하여 응력을 구합니다. + + + +![](images/page-433_20cc589df011c9451b75895eaefc1e22f6245c8bb5dcdd186fe50c7624dd4584.jpg) + +
+line + +| Steel Strain | Steel Stress | +| ------------ | ------------ | +| E_s | 2.20 | +| E_y | 4.49 | +
+ +그림 2.9.40 Stress-Strain curves for steel with reversed loading + +$$ +\varepsilon - \varepsilon_ {s i} = \frac {f}{E _ {s}} \left(1 + \left| \frac {f}{f _ {c h}} \right| ^ {R - 1}\right) \quad \text { Ramberg - Osgood Function } +$$ + +$$ +f _ {c h} = f _ {y} \left\{\frac {0 . 7 4 4}{\log_ {e} \left(1 + 1 0 0 0 \varepsilon_ {i p}\right)} - \frac {0 . 0 7 1}{\left(1 - e ^ {1 0 0 0 \varepsilon_ {i p}}\right)} + 0. 2 4 1 \right\} +$$ + +$$ +R = \frac {4 . 4 9}{\log_ {e} (1 + n)} - \frac {6 . 0 3}{e ^ {n} - 1} + 0. 2 9 7 \quad (n = 1 \text { 인 경우 }) +$$ + +$$ +R = \frac {2 . 2 0}{\log_ {e} (1 + n)} - \frac {0 . 4 6 9}{e ^ {n} - 1} + 3. 0 4 \quad (n = 2 \text { 인 경우 }) +$$ + +여기서, + +$$ +f _ {c h} \quad : \text { Ramberg - Osgood 함수의 특성응력도 } +$$ + +$$ +\mathcal {E} _ {i p}: \text { 이전 재하시의 소성변형도 } (0 < \varepsilon_ {i p} < 0. 7 0 9 7) +$$ + +$$ +R: \text { Ramberg - - Osgood Parameter } +$$ + +$$ +n: \text { Loading Run Number } +$$ + + + +(단, 압축측인 경우 1, 인장측인 경우 2의 고정값 사용) + +$\varepsilon _ { s i }$ si : 재하시점에서 응력 0에 대한 변형도 + +εiPp $\mathcal { E } _ { i p }$ .7， + +![](images/page-434_800b493f5aa3af5abece36f3090c5bf62f57badfd6ff049ca6408b9dc6ab5aa3.jpg) + +
+line + +| Steel Strain | Park's Result (N/mm²) | Midas Result (N/mm²) | +| ------------ | --------------------- | -------------------- | +| -0.005 | -350 | -350 | +| 0 | 0 | 0 | +| 0.005 | 300 | 300 | +| 0.01 | 250 | 250 | +| 0.015 | 200 | 200 | +| 0.02 | 100 | 100 | +| 0.025 | 0 | 0 | +
+ +![](images/page-434_9c798272b72b4a159fa482b6db3b9ee5555e53d08abde13c061a916239c863ea.jpg) + +
+line + +| Steel Strain | Park's Result (N/mm²) | Midas Result (N/mm²) | +| ------------ | --------------------- | -------------------- | +| 0.000 | 0 | 0 | +| 0.005 | 330 | 330 | +| 0.010 | -200 | -200 | +| 0.015 | 300 | 300 | +| 0.020 | 350 | 350 | +
+ +그림 2.9.41 Stress-Strain curves of Park Steel Model + + + +# 9-6-2 콘크리트 구성 모델 + +# Modified Kent & Park Concrete Model + +단조증가 압축력을 받는 콘크리트에 대해서 Kent와 Park(1971) $^{5)}$ 가 제안한 모델을 Scott(1982) $^{6)}$ 등이 수정한 모델입니다. 아래와 같은 포락곡선(Envelope Curve)식을 사용하며 콘크리트의 인장강도는 무시하고 있습니다. 본 모델은 명료함과 정확성의 적절한 조화를 이루고 있고, 횡 구속(Confinement Effect)에 의한 콘크리트 압축 강도의 증가 효과를 고려하는 재료 모델로서 널리 알려져 사용되고 있습니다. + +$$ +\sigma_ {c} = \left\{ \begin{array}{l l} K f _ {c} ^ {\prime} \left[ 2 \left(\frac {\varepsilon}{\varepsilon_ {0}}\right) - \left(\frac {\varepsilon}{\varepsilon_ {0}}\right) ^ {2} \right] & \text { for } \varepsilon \leq \varepsilon_ {0} \\ K f _ {c} ^ {\prime} \left[ 1 - Z \left(\varepsilon - \varepsilon_ {0}\right) \right] \geq 0. 2 K f _ {c} ^ {\prime} & \text { for } \varepsilon_ {0} \leq \varepsilon \leq \varepsilon_ {u} \end{array} \right. +$$ + +여기서, ε : 콘크리트 섬유의 변형율 +σ : 콘크리트 섬유의 응력 +ε₀ : 최대응력 발생시의 변형율 +εu : 종국 변형율 +K : 횡구속에 의한 강도 증가율 +Z : 변형율 연화(Strain Softening) 시의 기울기 +$f_{c}'$ : 콘크리트 실린더 압축강도(MPa) + +5); Kent, D.C., and Park, R., "Flexural Members with Confined Concrete", Journal of the Structural Division, ASCE, 97(ST7), 1971. +6) ; Scott, B.D., Park, R. and Priestley, M.J.N., "Stress-Strain Behavior of Concrete Confined by Overlapping Hoops at Low and High Strain Rates", ACI Journal, Vol.79, No.1, 1982, pp. 13-27. + + + +![](images/page-436_1400cbca8dd63fcc9b8472693cd13e03a77de570590de91ea6590e07d424e9fd.jpg) + +
+line + +| compressive strain | compressive stress | +| ------------------ | ------------------ | +| ε₀ | K·f_c' | +| ε_p | 0.2K·f_c' | +| ε_r | 0.2K·f_c' | +| ε_u | 0.2K·f_c' | +
+ +그림 2.9.42 Modified Kent & Park 콘크리트 섬유 구성모델 + +종국 변형율을 초과한 콘크리트는 압괴(Crushing)가 발생한 것으로 가정하여 더 이상의 하중을 받지 못하는 것으로 해석합니다. Kent와 Park은 직사각형 단면의 기둥에 대해서 상기의 포락곡선을 정의하는 파라미터를 계산하기 위해 다음과 같은 식을 사용합니다. + +$$ +\varepsilon_ {0} = 0. 0 0 2 K +$$ + +$$ +K = 1 + \frac {\rho_ {s} f _ {y h}}{f _ {c} ^ {\prime}} +$$ + +$$ +Z = \frac {0 . 5}{\frac {3 + 0 . 2 9 f _ {c} ^ {\prime}}{1 4 5 f _ {c} ^ {\prime} - 1 0 0 0} + 0 . 7 5 \rho_ {s} \sqrt {\frac {h ^ {\prime}}{s _ {h}}} - 0 . 0 0 2 K} +$$ + +여기서, $\begin{array} { r } { \pmb { f } _ { y h } : } \end{array}$ 횡 보강근(Stirrup)의 항복강도(MPa) + +$\rho \pmb { \mathscr { s } } \mathrm { : \ }$ + +h’ : 콘크리트 코어의 폭(직사각형의 경우 짧은쪽) + +(콘크리트 코어는 횡 보강근의 바깥쪽으로 둘러싸인 영역으로 정의) + +sk : 횡 보강근의 간격 + + + +Scott 등(1982)은 횡구속이 존재하는 직사각형 기둥에 대해서 다음과 같은 종국변형율의 식을 제안하였습니다. + +$$ +\varepsilon_ {u} = 0. 0 0 4 + 0. 9 \rho_ {s} \left(f _ {y h} / 3 0 0\right) +$$ + +상기의 포락곡선에서 제하(Unloading)가 발생하는 경우에 제하 경로는 다음 식에 의해서 정의되는 변형율 축선상의 점 ( $\varepsilon_{p}$ , 0)을 향하게 되며 이 점에 도달하면 변형율 축선상을 따라서 인장 영역으로 움직입니다. + +$$ +\frac {\varepsilon_ {p}}{\varepsilon_ {0}} = 0. 1 4 5 \cdot \left(\frac {\varepsilon_ {r}}{\varepsilon_ {0}}\right) ^ {2} + 0. 1 3 \cdot \left(\frac {\varepsilon_ {r}}{\varepsilon_ {0}}\right) \quad f o r \left(\frac {\varepsilon_ {r}}{\varepsilon_ {0}}\right) < 2 +$$ + +$$ +\frac {\varepsilon_ {p}}{\varepsilon_ {0}} = 0. 7 0 7 \cdot \left(\frac {\varepsilon_ {r}}{\varepsilon_ {0}} - 2\right) + 0. 8 3 4 \quad \text { for } \left(\frac {\varepsilon_ {r}}{\varepsilon_ {0}}\right) \geq 2 +$$ + +여기서, $\varepsilon_{r}$ : 제하 발생점의 변형율 + +$\varepsilon_{p}$ : 제하 경로상의 목표점의 변형율 + +만약 다시 압축변형율이 증가하게 되면 이제까지의 제하 경로를 그대로 거슬러 올라가서 포락곡선에 도달하게 됩니다. + +# 일본 콘크리트 표준시방서 모델 + +일본 콘크리트 표준시방서에서 제시하고 있는 콘크리트 모델로 다음과 같은 특징이 있습니다. 압축 최대 응력점을 넘은 경우 연화영역을 가지게 되며, 잔류 소성 변형을 고려하고 있습니다. 제하(Unloading), 재재하(Re-loading)의 경우 강성 저감 효과를 반영하고 있고, 일반적인 보부재의 경우에 인장 응력의 응력-변형 관계는 무시합니다. 이러한 특성을 바탕으로 압축강도가 50 N/mm² 이하의 경우에는 다음 그림과 같은 응력-변형 이력 관계를 가지게 됩니다. + + + +$$ +\sigma_ {c} ^ {\prime} = E _ {0} K \left(\varepsilon_ {c} ^ {\prime} - \varepsilon_ {p} ^ {\prime}\right) \geq 0 +$$ + +$$ +E _ {0} = \frac {2 \cdot f _ {c} ^ {\prime}}{\varepsilon_ {p e a k} ^ {\prime}} +$$ + +$$ +\boldsymbol {K} = \exp \left\{- 0. 7 3 \frac {\varepsilon_ {\max} ^ {\prime}}{\varepsilon_ {p e a k} ^ {\prime}} \left(1 - \exp \left(- 1. 2 5 \frac {\varepsilon_ {\max} ^ {\prime}}{\varepsilon_ {p e a k} ^ {\prime}}\right)\right) \right\} +$$ + +$$ +\varepsilon_ {p} ^ {\prime} = \varepsilon_ {\max} ^ {\prime} - 2. 8 6 \cdot \varepsilon_ {p e a k} ^ {\prime} \left(1 - \exp \left(- 0. 3 5 \frac {\varepsilon_ {\max} ^ {\prime}}{\varepsilon_ {p e a k} ^ {\prime}}\right)\right) +$$ + +여기서, $\varepsilon'_{peak}$ : 압축강도에 대응하는 변위 + +$\varepsilon'_{max}$ : 이전에 받았던 압축변위의 최대치 + +$\varepsilon'_{p}$ : 잔류 소성변위 + +K : 강성 잔존률 + +![](images/page-438_1b3564b01798434ba65da99114866e861ef860ec02bdea52e67c78adc1299b92.jpg) + +
+line +| ε | σ | +|-------|-------| +| 0 | 0 | +| ε | E | +| ε | E K | +| ε | K | +
+ +그림 2.9.43 일본 콘크리트 표준시방서 콘크리트 섬유 구성모델 + + + +# 일본 도로교 시방서 콘크리트 모델 + +일본 도로교 시방서(동해설), V 내진 설계편의 콘크리트 모델로 다음과 같은 특징이 있습니다. 압축 최대 응력점을 넘은 경우 연화영역을 가지게 되며, 극한 압축변형률을 초과할 경우 더 이상 저항을 하지 않는다고 가정합니다. 지진하중의 종류에 따라 극한 압축변형률이 변화하며, 구속철근의 양을 고려하여 연화구간의 기울기, 최대 압축강도와 극한 압축변형률이 조정됩니다. 한편 잔류 소성 변형을 고려하고 있으며, 제하(Unloading), 재재하(Re-loading)의 경우 초기강성으로 거동한다고 가정합니다. 인장측 응력-변형 관계를 가지며 최대 인장강도에 대응되는 변형률을 초과하는 경우 더 이상 저항을 하지 않습니다. + +![](images/page-439_fb4ded7af2530e7a3d64f9355b01a614af70b8ff9ead94cc532c106db80aba18.jpg) + +
+line + +| ε | σ | +| ---- | ----- | +| ε | 0.8σ | +| ε | 0.6σ | +
+ +그림 2.9.44 일본 도로교 시방서 콘크리트 섬유 구성모델 + +$$ +\sigma_ {c} = \left\{ \begin{array}{l l} E _ {c} \varepsilon_ {c} \left(1 - \frac {1}{n} \left(\frac {\varepsilon_ {c}}{\varepsilon_ {c c}}\right) ^ {n - 1}\right) & (0 \leq \varepsilon_ {c} \leq \varepsilon_ {c c}) \\ \sigma_ {c c} - E _ {d e s} (\varepsilon_ {c} - \varepsilon_ {c c}) & (\varepsilon_ {c c} \leq \varepsilon_ {c} \leq \varepsilon_ {c u}) \end{array} \right. +$$ + +$$ +\textbf {n} = \frac {E _ {c} \varepsilon_ {c c}}{E _ {c} \varepsilon_ {c c} - \sigma_ {c c}} +$$ + + + +$$ +\sigma_ {c c} = \sigma_ {c k} + 3. 8 \alpha \rho_ {s} \sigma_ {s y} +$$ + +$$ +\varepsilon_ {c c} = 0. 0 0 2 + 0. 0 3 3 \beta \frac {\rho_ {s} \sigma_ {s y}}{\sigma_ {c k}} +$$ + +$$ +{E _ {d e s}} = {1 1. 2 \frac {\sigma_ {c k} ^ {2}}{\rho_ {s} \sigma_ {s y}}} +$$ + +$$ +\varepsilon_ {c u} = \left\{ \begin{array}{l l} \varepsilon_ {c c} & \text {(Type I)} \\ \varepsilon_ {c c} + \frac {0 . 2 \sigma_ {c c}}{E _ {d e s}} & \text {(Type II)} \end{array} \right. +$$ + +$$ +\rho_ {s} = \frac {4 A _ {h}}{s d} \leq 0. 0 1 8 +$$ + +여기서, $\sigma_{c}$ : 콘크리트의 응력 + +$\sigma_{cc}$ : 횡구속 철근으로 구속된 콘크리트의 강도 + +$\sigma_{ck}$ : 콘크리트의 설계 기준 강도 + +$\varepsilon_{c}$ : 콘크리트의 변형률 + +$\varepsilon_{cc}$ : 최대 압축 응력에 대응되는 변형률 + +$\varepsilon_{cu}$ : 횡구속 철근으로 구속된 콘크리트의 극한 변형률 + +$E_{c}$ : 콘크리트의 탄성계수 + +$E_{des}$ : 연화구간의 하강 구배 + +$\rho_{s}$ : 횡구속 철근의 체적비 + +$A_{h}$ : 횡구속 철근 한 개 당 단면적 + +s : 횡구속 철근간 간격 + +d : 횡구속 구속장으로, 띠철근이나 중간 띠철근에 의해 분할 구속된 내부 콘크리트의 변 길이 중 가장 긴 값 + +$\sigma_{sy}$ : 횡구속 철근의 항복점 + +α,β : 단면 보정 계수(원형단면=1, 사각,사다리콜·중공단면 =0.2, 0.4) diff --git a/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_045.md b/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_045.md new file mode 100644 index 00000000..95923cd2 --- /dev/null +++ b/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_045.md @@ -0,0 +1,297 @@ + + +# 일본 나고야 공단 콘크리트 모델 + +일본 나고야 공단의 콘크리트 모델로 다음과 같은 특징이 있습니다. 강성 교각에 충전된 콘크리트의 응력-변형 관계이므로 최대 압축 강도에 대응하는 변형률을 초과한 이후는 최대압축강도를 그대로 유지한다고 가정합니다. 극한 변형률을 초과한 경우에는 더 이상 저항을 하지 않고, 잔류 소성 변형을 고려하고 있으며, 제하 (Unloading), 재재하(Re-loading)의 경우 초기강성으로 거동합니다. 시방규정에는 인장강도를 무시하도록 하고 있으나, 범용성을 갖추기 위하여 인장측 응력-변형 관계를 임의로 정의할 수 있도록 구현하였습니다. 최대 인장 변형률을 초과하는 경우 더 이상 저항을 하지 않는다고 가정합니다. + +![](images/page-441_1e016bcf41cc7e987e4c999966b0457b8dd9dc95ff7ff2d00a21c8a31b742a2a.jpg) + +
+line + +| ε | σ | +| ---- | ---- | +| ε | 0 | +| ε | σ | +
+ +그림 2.9.45 일본 나고야 공단 콘크리트 섬유 구성모델 + +$$ +\sigma_ {c} = \sigma_ {c k} \left[ 2 \left(\varepsilon_ {c} / \varepsilon_ {0}\right) - \left(\varepsilon_ {c} / \varepsilon_ {0}\right) ^ {2} \right] +$$ + +여기서, $\sigma_{ck}$ : 콘크리트의 설계 기준 강도 + +$\sigma_{bt}$ : 콘크리트의 인장 강도 + +$\varepsilon_{cu}$ : 콘크리트의 극한 변형률 + +$\varepsilon_{0}$ : 최대 압축 응력에 대응되는 변형률 + +# Tri-linear Concrete Model + +1, 2차 압축 항복까지 구현 가능하고, 인장 강도를 가지는 일반적인 모델로서 사용 +자의 의도에 따라 임의적인 정의가 가능한 모델입니다. 잔류 소성 변형을 고려하 + + + +고 있으며, 제하(Unloading), 재재하(Re-loading)의 경우 초기강성으로 거동한다고가정합니다. + +# 중국 콘크리트 시방서 모델 (GB50010-2002) + +중국 콘크리트 시방서(GB50010-2002)의 단축 콘크리트 응력-변형도 모델입니다.본 모델은 압축측과 인장측에 각각 최대 응력점을 가지며, 최대 응력점을 넘는 경우에 연화영역을 가집니다. 중국 콘크리트 시방서 모델의 적용범위는 다음과 같습니다. + +콘크리트 강도 등급 : C20\~C80 +콘크리트 질량밀도 : 2200\~2400kg/m3 +정상적인 온도, 습고환경, 정상적인 재하속도 + +구조해석방법과 극한상태 검토의 필요성에 따라서, 단축강도( \* \* , f f )는 각 표준치( $f _ { c k } \mathrm { ~ , ~ } f _ { t k } \mathrm { ~ ) ~ }$ , 설계치 혹은 평균치( $f _ { c m } , f _ { t m }$ )를 사용할 수 있습니다. + +강도의 평균치는 다음과 같이 계산합니다. + +$$ +f _ {c m} = \frac {f _ {c k}}{1 - 1 . 6 4 5 \delta_ {c}} +$$ + +$$ +f _ {t m} = \frac {f _ {t k}}{1 - 1 . 6 4 5 \delta_ {t}} +$$ + +여기서, $\delta _ { c } , \delta _ { t }$ : 콘크리트 압축강도 + +(인장강도의 돌변계수로서 시험통계에 의해 결정) + + + +![](images/page-443_5c40d6df7215a99930507169408870a30203e95d890c7a04dadcb8be69cbcaed.jpg) + +
+line + +| x = ε/ε_c | y = σ/f_c* | +| --------- | ---------- | +| 0.0 | 0.0 | +| 1.0 | 1.0 | +| ε_u / ε_c | 0.5 | +
+ +그림 2.9.46 콘크리트 압축 응력-변형 곡선 + +콘크리트 단축 압축의 응력-변형 곡선의 방정식은 다음 식과 같이 결정할 수 있습니다. + +$$ +\left\{ \begin{array}{l} \varepsilon_ {c} \leq \varepsilon ; y = \alpha_ {a} x + (3 - 2 \alpha_ {a}) x ^ {2} + (\alpha_ {a} - 2) x ^ {3} \\ \varepsilon < \varepsilon_ {c}; y = \frac {x}{\alpha_ {d} (x - 1) ^ {2} + x} \end{array} \right. +$$ + +$$ +\text { 여기서, } x = \frac {\varepsilon}{\varepsilon_ {c}}, y = \frac {\sigma}{f _ {c} ^ {*}} +$$ + +$f_{c}^{*}$ : 콘크리트의 단축 압축강도( $f_{ck}$ , $f_{c}$ or $f_{cm}$ ) + +$\varepsilon_{c}$ : $f_{c}^{*}$ 에 대응하는 최대점의 압축변형 + +$$ +\varepsilon_ {c} = \left(7 0 0 + 1 7 2 \sqrt {f _ {c} ^ {*}}\right) \times 1 0 ^ {- 6} +$$ + +$\alpha_{a}$ : 단축 압축의 응력-변형 곡선의 상승구간의 파라메터 + +$$ +\alpha_ {a} = 2. 4 - 0. 0 1 2 5 f _ {c} ^ {*} +$$ + +$\alpha_{d}$ : 단축 압축의 응력-변형 곡선의 하강구간의 파라메터 + +$$ +\alpha_ {d} = 0. 1 5 7 f _ {c} ^ {* 0. 7 8 5} - 0. 9 0 5 +$$ + +$\varepsilon_{u}$ 는 응력-변형도 곡선의 하강구간에서 응력이 $0.5 \cdot f_{c}^{*}$ 위치에서의 변형을 의미합니다. + + + +$$ +\frac {\varepsilon_ {u}}{\varepsilon_ {c}} = \frac {1}{2 \alpha_ {d}} \left(1 + 2 \alpha_ {d} + \sqrt {1 + 4 \alpha_ {d}}\right) +$$ + +\*. $E_{u}$ 는 응력-변형곡선 하강구간에서 응력이 $0.5f_{c}^{*}$ 일때의 콘크리트 압축변형 +
$f_c^*(N/mm^2)$ 15202530354045505560
$\mathcal{E}_c(x10^{-6})$ 1370147015601640172017901850192019802030
$\alpha_a$ 2.212.152.092.031.961.901.841.781.711.65
$\alpha_d$ 0.410.741.061.361.651.942.212.482.743.00
$\mathcal{E}_u/\mathcal{E}_c$ 4.23.02.62.32.12.01.91.91.81.8
+ +표 2.9.2 콘크리트 단축 압축 응력-변형곡선 파라메터 값 + +콘크리트 단축 인장의 응력-변형 곡선의 방정식은 다음 식과 같이 결정할 수 있습니다. + +![](images/page-444_07d88d32ecd8ba444da376f78cc52fa5b42d295cf30451b7dc65ee5d43551676.jpg) + +
+line + +| x = ε/εₜ | y = σ/fₜ* | +| -------- | --------- | +| 0.0 | 0.0 | +| 1.0 | 1.0 | +| >1.0 | Decreasing | +
+ +그림 2.9.47 콘크리트의 단축 인장응력-변형 곡선 + +$$ +\left\{ \begin{array}{l} \varepsilon \leq \varepsilon_ {t}; y = 1. 2 x - 0. 2 x ^ {6} \\ \varepsilon_ {t} < \varepsilon ; y = \frac {x}{\alpha_ {t} (x - 1) ^ {1 . 7} + x} \end{array} \right. +$$ + +여기서, $x=\frac{\varepsilon}{\varepsilon_{t}}$ , $y=\frac{\sigma}{f_{t}^{*}}$ + + + +$$ +f _ {t} ^ {*}: \text { 콘크리트의 단축 인장강도 } (f _ {t k}, f _ {t} \text { or } f _ {t m}) +$$ + +$$ +\varepsilon_ {t}: f _ {t} ^ {*} \text { 에 대응하는 최대점의 인장변형 } +$$ + +$$ +\varepsilon_ {t} = f _ {t} ^ {* 0. 5 4} \times 6 5 \times 1 0 ^ {- 6} +$$ + +$$ +\alpha_ {t}: \text { 단축 인장의 응력 - 변형 곡선의 하강구간의 파라메터 값 } +$$ + +$$ +\alpha_ {t} = 0. 3 1 2 f _ {t} ^ {* 2} +$$
$f_{t}^{*}$ (N/mm $^{2}$ )1.01.52.02.53.03.54.0
$\mathcal{E}_{t}$ (x10 $^{-6}$ )658195107118128137
$\alpha_{t}$ 0.310.701.251.952.813.825.00
+ +표 2.9.3 콘크리트 단축 인장 응력-변형곡선 파라메터 값 + + + +# Mander 콘크리트 모델 + +횡방향으로 배근된 구속철근은 콘크리트의 극한강도와 극한변형률을 크게 증가시키는 효과를 나타냅니다. Mander(1988)7) 는 Sheikh & Uzumeri8)가 제안한 유효구속단면적 개념 뿐만 아니라, 3차원응력상태를 고려한 콘크리트의 파괴기준을 적용한 최대압축응력도의 평가식을 제안하고, 원형단면, 정방향단면, 장방형단면에대한 실험을 통하여 제안모델의 적용성을 검토하였습니다. + +Mander모델은 콘크리트의 단면형상에 관계없이 적용할 수 있고, 종방향 철근간격및 구속철근의 양, 구속철근의 항복강도 및 배근형태 등에 의한 콘크리트의 횡구속효과를 고려할 수 있습니다. + +![](images/page-446_edcfde15abafc3b58e273ab8f301ad27576cbd674d413ab647bda5ed90ae5377.jpg) + +
+area + +| Material Type | Compressive Stress (ε_c) | Compressive Strain (ε_t) | +|-----------------------|---------------------------|---------------------------| +| Confined Concrete | 2.0 | 0.0 | +| Unconfined Concrete | 0.0 | 0.0 | +| Cover Concrete | 0.0 | 0.0 | +
+ +그림 2.9.48 Stress-Strain Curves of Confined and Unconfined Concrete + +종방향 콘크리트 압축응력은 다음과 같이 정의됩니다. + +$$ +f _ {c} = \frac {f _ {c c} ^ {\prime} x r}{r - 1 + x ^ {r}} +$$ + + + +여기서, $f_{cc}^{'}$ : 구속 콘크리트의 압축강도 + +$f_{co}^{'}$ : 횡구속되지 않은 콘크리트 압축강도 + +$$ +x = \frac {\varepsilon_ {c}}{\varepsilon_ {c c}} +$$ + +$E_{cc}$ : 횡구속된 콘크리트의 최대압축응력에 대응하는 변형률 + +(longitudinal compressive concrete strain) + +$$ +\varepsilon_ {c c} = \varepsilon_ {c o} \left[ 1 + 5 \left(\frac {f _ {c c} ^ {\prime}}{f _ {c o} ^ {\prime}} - 1\right) \right] +$$ + +$\varepsilon_{co}$ : 구속되지 않은 콘크리트강도에 상응하는 변형률 + +(단, 일반적으로 $E_{co}=0.002$ 로 추측가능) + +$$ +r = \frac {E _ {c}}{E _ {c} - E _ {\mathrm{sec}}} +$$ + +$E_{c}$ : 콘크리트의 탄성계수, $E_{c}=5,000\sqrt{f_{co}^{'}}$ MPa + +$$ +E _ {\mathrm{sec}} = \frac {f _ {c c} ^ {\prime}}{\varepsilon_ {c c}} +$$ + +구속 콘크리트의 압축강도 $f_{cc}^{'}$ 는 다음과 같이 정의됩니다. + +$$ +f _ {c c} ^ {\prime} = f _ {c o} ^ {\prime} \left(- 1. 2 5 4 + 2. 2 5 4 \sqrt {1 + \frac {7 . 9 4 f _ {l} ^ {\prime}}{f _ {c o} ^ {\prime}}} - 2 \frac {f _ {l} ^ {\prime}}{f _ {c o} ^ {\prime}}\right) +$$ + +여기서, $f_{l}^{\prime}$ : 콘크리트의 측면구속응력, $f_{l}^{\prime} = \frac{1}{2} k_{e} \rho_{s} f_{yh}$ + + + +# Chapter 10. 시공단계해석 + +# 10-1 개요 + +현수교, 사장교 또는 PSC교량과 같은 토목구조물은 시공중과 시공후의 구조계가 달라지며 시공중에도 가교각 및 임시케이블의 설치와 제거, 상판과 주탑의 지지조건 변화 등에 따라 구조계가 계속 변화합니다. 또한 단계적인 시공에 의해 인접 부재간의 재령이 다르므로 부재의 탄성계수나 강도 등의 재료적 특성도 달라지게 됩니다. 그리고 콘크리트의 크리프(Creep), 건조수축(Shrinkage), 강도증가(Aging) 및 PS 텐던의 이완 등 재료의 시간의존적 특성에 의한 영향으로, 시공중이나 시공이 완료된 후에도 처짐이 변하고 응력이 재분배되어 구조물의 거동이 매우 복잡해 집니다. 이와 같이 시공의 진행에 따라 계속적으로 구조계가 변화할 경우에 부재에 따라서는 시공이 완료된 후 하중이 재하되는 시점이 아니라 시공중에 최대응력이 발생할 수도 있으므로, 구조물의 각 시공단계에 따른 응력의 변화를 예측하기 위해서는 정확히 시공단계를 고려한 시간의존해석이 필요합니다. + +midas Civil을 사용하여 시공단계해석을 수행할 때 고려하는 내용은 다음과 같습니다. + +# ▪ 시간 의존적 재료의 특성 + +서로 다른 재령을 갖는 콘크리트 부재의 크리프 +서로 다른 재령을 갖는 콘크리트 부재의 건조수축 +시간이 흐름에 따른 콘크리트 부재의 강도발현 + +# ▪ 시공단계의 표현 + +임의의 재령을 가지는 부재의 생성 및 소멸 +임의의 재하시점을 가지는 하중의 재하 및 소거 +경계조건의 변화 + + + +midas Civil에서 시공단계를 고려한 시간 의존해석을 수행하기 위한 절차는 다음과같습니다. + +1. 구조물을 모델링 합니다. 이때 임의의 시공단계에서 함께 생성 또는 소멸시킬 요소, 하중 및 경계조건들을 그룹으로 지정합니다. +2. 크리프나 건조수축과 같은 시간의존적인 재질의 특성을 정의합니다. 이때시간의존적인 재질은 ACI나 CEB-FIP등과 같은 규준을 선택하여 생성하거나 사용자가 직접 정의할 수 있습니다. +3. 정의한 시간의존재질을 일반재질과 연결합니다. 이를 통하여 시간의 흐름에 따른 콘크리트 부재의 재질 변화를 자동으로 계산하여 고려합니다. +4. 실제 시공시 고려해야할 시공의 순서를 생각하여 시공단계 및 Time Step을생성합니다. +5. 미리 만들어놓은 요소그룹, 경계조건그룹, 하중그룹을 이용하여 시공단계를정의합니다. +6. 원하는 방법으로 해석조건을 지정하고, 구조해석을 수행합니다. +7. 시공단계 해석결과와 완성계 해석결과를 필요한 방법으로 조합합니다. + + + +# 10-2 시간의존적 재질 + +midas Civil에서는 콘크리트의 시간의존적 특징중에서 크리프(Creep), 건조수축(Shrinkage), 강도증가(Aging) 등을 고려할 수 있습니다. + +# 10-2-1 크리프(Creep) 및 건조수축(Shrinkage) + +그림 2.10.1과 같이 실제 구조물에서 크리프는 건조수축과 함께 발생됩니다. 따라서 건조수축, 탄성변형, 크리프를 각각 분리해서 생각할 수는 없습니다. 그러나 실제 해석 및 설계에서는 편의상 이들을 분리하여 고려합니다. + +그림 2.10.1에서 참된 탄성변형(True Elastic Strain)이란 시간과 더불어 증가되는 콘크리트의 강도에 의한 탄성계수의 증대로 인해 감소되는 탄성변형을 나타낸 것입니다. 일반적인 경우는 걸보기 탄성변형을 탄성변형으로 보지만 midas Civil에서는 해석시 콘크리트의 강도발현을 고려할 수 있으므로 참된탄성변형으로 해석할 수도 있습니다. + +크리프 변형률은 작용시킨 응력에 비례하며, 동일한 응력하에서는 고강도 콘크리트가 저강도 콘크리트보다 작은 크리프 변형률을 나타냅니다. 크리프 변형률은 탄성변형률의 1.5\~3배 정도에 이르며, 재하 후 첫 몇 개월 동안에 총 크리프 변형률의 1/2이 진행됩니다. 약 5년 후에는 대부분 Creep이 발생합니다. + +![](images/page-450_863d0be89da1abf6f5997a3a6c5461838ed5945a1b7db2907b5e1cc64d0f6ab4.jpg) + +
+line + +| Time Point | True elastic strain | Shrinkage | Creep | +| ---------- | ------------------- | --------- | ----- | +| to | Low | Low | Low | +| Peak | Low | Low | High | +
+ +그림 2.10.1 시간경과에 따른 콘크리트의 변형율 diff --git a/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_046.md b/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_046.md new file mode 100644 index 00000000..dd849a36 --- /dev/null +++ b/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_046.md @@ -0,0 +1,345 @@ + + +콘크리트의 크리프는 다음과 같은 요인에 의하여 변화합니다. + +1. 물-시멘트비의 증가는 크리프의 증대를 가져옵니다. +2. 응력을 받을 때 콘크리트 재령이 클수록 크리프는 감소합니다. +3. 콘크리트 주위의 온도가 높을수록, 또 습도는 낮을수록 크리프 변형은 커집니다. +4. 이 밖에 시멘트의 종류, 골재의 품질, 공시체의 치수 등에도 영향을 받습니다. + +크리프 현상은 대부분의 재료가 가지고 있는 성질이지만, 특히 콘크리트는 다른재료에 비하여 그 값이 커서 프리스트레스의 시간적 감소 원인의 하나가 되기 때문에 설계에서 무시할 수 없습니다. 보통의 콘크리트 구조물에서는 주로 자중에의하여 크리프 현상이 일어나지만 PSC 구조물에서는 프리스트레스에 의하여 추가로 크리프 현상이 일어납니다. + +콘크리트 시편에 일정한 축방향 응력  =1을 콘크리트 재령 $t _ { \mathbf { \rho } _ { 0 } }$ 일에 재하하였을때, 재령 t 일에서 발생하는 1축 변형율을 ${ \cal J } ( t , t _ { 0 } )$ 라고 가정합니다. + +$$ +\varepsilon (t) = \varepsilon_ {i} (t _ {0}) + \varepsilon_ {c} (t, t _ {0}) = \sigma \cdot J (t, t _ {0}) \tag {1} +$$ + +여기서, ${ \cal J } ( t , t _ { 0 } )$ 는 단위 응력이 작용할 때의 총 변형율을 의미하며 크리프함수(Creep Function)라고 정의합니다. + +![](images/page-451_eb6e770168920d4192a4b429170f3285c7fe20d36689a7ed236c17855d341bdc.jpg) + +
+bar +| Time | σ (unit) | +|---|---| +| t0 | 1 | +| t | 1 | +
+ +![](images/page-451_c22fe356fac49e25b5c97359ddb13ac590a35a79b5ff156109c5fcfbd4b0000f.jpg) + +
+line + +| Time | τ(t) | +|------|----------| +| ω | 0 | +| t | 1/(E(t₀))| +
+ +그림 2.10.2 크리프 함수 및 특성 크리프의 정의 + + + +그림 2.10.2에서 보듯이 크리프 함수 ${ \cal J } ( t , t _ { 0 } )$ 를 재하시의 초기탄성변형과 크리프변형의 합으로 나타내면 식 (2)와 같습니다. + +$$ +J (t, t _ {0}) = \frac {1}{E (t _ {0})} + C (t, t _ {0}) \tag {2} +$$ + +여기서, ( ) E t 는 하중 재하시의 탄성계수를 나타내며 ( ) C t, t 는 재령 t에서의 크리프 변형을 나타내는데 이를 특성 크리프(Specific Creep)라고 합니다. 또한 크리프함수 J ( , ) t t 를 탄성변형과의 비율로 나타내어 식 (3)과 같이 표현할 수 있습니다. + +$$ +J (t, t _ {0}) = \frac {1 + \phi (t , t _ {0})}{E (t _ {0})} \tag {3} +$$ + +여기서, 0( , ) t t 는 크리프 계수(Creep Coefficient)로서 탄성변형과 크리프 변형과의비율을 나타내며, 위의 두 식으로부터 특성 크리프와 크리프 계수는 다음과 같은관계가 성립합니다. + +$$ +\phi (t, t _ {0}) = E (t _ {0}) \cdot C (t, t _ {0}) \tag {4} +$$ + +$$ +C (t, t _ {0}) = \frac {\phi (t , t _ {0})}{E (t _ {0})} \tag {5} +$$ + +midas Civil에서는 크리프 계수나 건조수축 변형률의 계산식으로 CEB-FIP나 ACI등에서 정하고 있는 식들을 사용할 수 있고, 사용자가 실험에 의한 값을 직접 입력하여 사용할 수 있습니다. + +사용자정의는 크리프 계수(Creep Coefficient), 크리프 함수(Creep Function), 특성크리프(Specific Creep)의 세가지 값 중 사용자가 원하는 형식으로 입력이 가능합니다. + + + +![](images/page-453_82d47a46ab9bde1ea5822df42c206b2d631ab3b4396334d9b70576d055e2433d.jpg) + +
+line + +| Time (day) | Value | +| ---------- | --------- | +| 1 | 13,34 | +| 2 | 17,78 | +| 3 | 23,71 | +| 4 | 31,62 | +| 5 | 42,17 | +| 6 | 56,23 | +| 7 | 74,99 | +| 8 | 100,00 | +
+ +그림 2.10.3 사용자 정의 크리프 계수 지정 대화상자 + +콘크리트의 크리프 함수는 하중이 가해지는 시간에 따라서 각기 다른 형상을 나타내게 됩니다. 즉, 요소의 재령이 커지면 콘크리트의 강도증가(Aging) 효과에 의하여 탄성계수가 증가하기 때문에 콘크리트의 즉시 변형은 하중의 재하시기가 늦을수록 작아집니다. 그리고 하중의 재하시간으로부터 임의의 시간 후의 변형은 하중의 재하시기가 늦은 시험체의 경우 더 작아지게 됩니다. + +그림 2.10.4는 이러한 관계를 나타내고 있습니다. 이렇게 재하시간이 늦어질수록즉시 변형과 크리프 변형이 감소하는 것은 콘크리트의 수화정도와 강도발현 때문입니다. 따라서 사용자 정의로 크리프 함수를 입력할 때에는 콘크리트의 강도발현특성이 잘 반영될 수 있도록 크리프 함수에서 재하시간의 범위가 시간의존해석에서 존재하는 요소의 재령(재하시간)을 포함해야하고, 서로 다른 재하시간의 크리프함수를 많이 입력할수록 정확한 해석결과를 얻을 수 있습니다. + + + +![](images/page-454_b3e17b11411c92ab7961fb765c6e5bd43fd9b9b5988c8fceeba3c35d6a9d156f.jpg) + +
+line + +| Time t, τ | Creep Function J(τ, t - τ) | +| --------- | -------------------------- | +| τ₁ | Low | +| τ₂ | Medium | +| τ₃ | High | +| τ₄ | Very High | +
+ +그림 2.10.4 하중 재하 시간의 차이에 따른 크리프 함수 + +
지속하중을 재하 할 때의 콘크리트 재령4~7142890365
크리프 계수조강 시멘트3.83.22.82.01.1
보통 시멘트4.03.43.02.21.3
+ +표 2.10.1 보통 콘크리트의 크리프 계수 + +건조수축은 부재에 발생하는 응력과는 무관한 시간의 함수이며, 일반적으로 시간$t _ { 0 }$ 에서 t 까지 발생한 건조수축에 의한 변형율을 다음과 같이 나타냅니다. + +$$ +\varepsilon_ {s} (t, t _ {0}) = \varepsilon_ {s o} \cdot f (t, t _ {0}) \tag {6} +$$ + +여기서, $\varepsilon _ { s o }$ 는 최종시의 건조수축계수, $f ( t , t _ { 0 } )$ 는 시간의 함수, t는 관측시점, $t _ { 0 } \equiv$ 건조수축 발생시점을 의미합니다. + + + +# 10-2-2 크리프의 계산 방법 + +크리프는 응력이 발생한 상태에서 시간이 경과함에 따라 추가적인 응력의 증가없이 변형이 발생하는 현상으로 응력의 이력과 시간이 중요한 요인으로 작용합니다.크리프의 특성은 하중이 재하된 시점에서 가장 크게 발생하고 시간이 지날수록 급격하게 감소하는 경향을 보입니다. 크리프를 정확하게 고려하기 위해서는 응력의시간에 대한 이력과 시간에 따른 크리프 계수를 사용해야 합니다. 하지만 모든 부재 응력의 이력을 저장하고 모든 응력이력에 대하여 크리프를 계산하는 것은 데이터 저장량과 계산량을 크게 증가시키기 때문에 프로그램내에서 크리프를 적절하게계산할 수 있는 방법들을 사용하고 있습니다. 크리프는 비역학적(Non-mechanical)변형이므로 구속조건에 따라서 응력이 발생하지 않고 변형만 발생할 수도 있습니다. + +일반적으로 크리프를 고려하는 방법의 하나는 요소별 크리프 계수를 각 단계마다직접 입력하여 현상태까지 발생한 요소의 응력을 직접 사용하는 방법이고, 다른하나는 크리프의 특성함수를 수식화하여 응력과 시간에 대한 적분개념을 사용하여계산하는 방법입니다. 전자의 경우는 각 단계마다 요소별 크리프 계수를 산정하여입력해야 하고 후자의 경우는 프로그램 내부에서 규준에 따른 크리프 계수를 사용하여 응력 이력과의 적분식으로 크리프량을 계산하여 사용합니다. + +midas Civil에서는 위의 두 가지 방법을 모두 적용할 수 있도록 하였고 한 요소에두 가지 방법이 모두 입력된 경우에는 요소별 크리프 계수를 입력한 방법을 적용하도록 하였습니다. 전체적으로는 한가지 방법을 사용하는 것이 타당하지만 마지막 단계에서의 20 \~ 30년 정도의 시간을 도입하거나 특정한 요소에 대하여 크리프하중을 고려하고자 할 경우에는 두 가지 방법을 적절하게 병행하여 사용할 수 있습니다. + +요소별 크리프 계수를 산정하여 직접 입력하는 방법은 계수 산정을 어떻게 하느냐에 따라 결과가 상당히 달라질 수 있으므로 응력이력과 시간에 대한 충분한 자료를 가지고 크리프 계수를 산정해야 근사적인 값을 구할 수 있습니다. 그러나 경험이나 실험 등으로 각 단계에서의 크리프 계수를 알고 있다면 직접 입력하여 사용하는 것이 효율적일 수 있습니다. + + + +각 시공단계마다 요소별 크리프 계수를 입력한 크리프 하중그룹을 활성화시키면입력된 크리프 계수와 현재까지 발생한 응력을 사용하여 크리프 하중을 계산하게됩니다. 이 방법은 사용자가 크리프계수를 직접 입력하여 하중의 크기를 쉽게 이해할 수 있고 사용이 간편한 장점이 있지만 크리프 계수를 산정해야하는 어려움을가지고 있습니다. + +크리프 계수를 사용하여 크리프 하중을 계산하는 방법은 다음과 같습니다. + +$$ +\varepsilon_ {c} \left(t, t _ {0}\right) = \phi \left(t, t _ {0}\right) \varepsilon \left(t _ {0}\right): \text { 크리프 변형률 } \tag {7} +$$ + +$$ +P = \int_ {A} E (t) \varepsilon_ {c} \left(t, t _ {0}\right) d A: \text {크리프 변형에 의한 하중} \tag {8} +$$ + +여기서 $ { \varepsilon } ( t _ { 0 } )$ : 시간 $t _ { 0 }$ 에서의 응력에 의한 변형률 + +$$ +\phi (t, t _ {0}) \quad : \text { 시간 } t _ {0} \text { 에서 } t \text { 까지의 크리프 계수 } +$$ + +다음은 크리프의 특성함수를 수식화하여 응력과 시간에 대한 적분을 사용하는 방법입니다. 임의의 시간 $t _ { 0 }$ 에서의 전체 크리프량을 시간 t 까지의 각 단계마다 발생하는 응력에 의한 크리프량의 중첩적분으로 나타내면 다음식과 같습니다. + +$$ +\varepsilon_ {c} (t) = \int_ {0} ^ {t} C (t _ {0}, t - t _ {0}) \frac {\partial \sigma (t _ {0})}{\partial t _ {0}} d t _ {0} \tag {9} +$$ + +여기서 $\varepsilon _ { c } \left( t \right)$ : 시간 t 에서의 크리프 변형률 + +$$ +C (t _ {0}, t - t _ {0}) : \text { 특성크리프(Specific Creep) } +$$ + +$$ +t _ {0} \quad : \text { 하중재하시점 } +$$ + +위의 식에서 응력이 각 단계에서 일정하다고 가정하면 식 (10)과 같이 전체 변형률을 단계별로 구분된 변형률의 합으로 표현할 수 있습니다. + +$$ +\varepsilon_ {c, n} = \sum_ {j = 1} ^ {n - 1} \Delta \sigma_ {j} C (t _ {j}, t _ {n - j}) \tag {10} +$$ + + + +위 식을 사용하여 시간 $t_{n} \sim t_{n-1}$ 사이에서 발생하는 크리프 변형률의 증분( $\Delta \varepsilon_{c,n}$ )을 정리하여 나타내면 식 (11)과 같습니다. + +$$ +\Delta \varepsilon_ {c, n} = \varepsilon_ {c, n} - \varepsilon_ {c, n - 1} = \sum_ {j = 1} ^ {n - 1} \Delta \sigma_ {j} C (t _ {j}, t _ {n - j}) - \sum_ {j = 1} ^ {n - 2} \Delta \sigma_ {j} C (t _ {j}, t _ {n - j}) \tag {11} +$$ + +특성크리프를 다음과 같이 Dirichlet 급수의 Degenerate Kernel로 표현하면 응력의 전체 이력을 저장할 필요없이 크리프에 의한 증분변형률을 계산할 수 있습니다. + +$$ +C (t _ {0}, t - t _ {0}) = \sum_ {i = 1} ^ {m} a _ {i} (t _ {0}) \Big [ 1 - e ^ {- (t - t _ {0}) / \Gamma_ {i}} \Big ] \tag {12} +$$ + +여기서 + +$a_{i}(t_{0})$ : 하중재하시간 $t_{0}$ 에 관련된 특성크리프 곡선의 초기형상에 관련된 계수 + +$\Gamma_{i}$ : 시간의 경과에 따른 특성크리프 곡선의 형상에 관한 값 + +위의 특성크리프 수식을 도입하여 증분변형률을 다시 정리하면 식 (13)과 같습니다. + +$$ +\Delta \varepsilon_ {c, n} = \sum_ {i = 1} ^ {m} \left[ \sum_ {j = 1} ^ {n - 2} \Delta \sigma_ {j} a _ {i} \left(t _ {j}\right) e ^ {- \left(t - t _ {0}\right) / \Gamma_ {i}} + \sigma_ {n - 1} a _ {i} \left(t _ {n - 1}\right) \right] \left[ I - e ^ {- \left(t - t _ {0}\right) / \Gamma_ {i}} \right] \tag {13} +$$ + +$$ +\Delta \varepsilon_ {c, n} = \sum_ {i = 1} ^ {m} A _ {i, n} \left[ 1 - e ^ {- (t - t _ {0}) / \Gamma_ {i}} \right] +$$ + +여기서 $A_{i,n}=\sum_{j=1}^{n-2}\Delta\sigma_{j}a_{i}(t_{j})e^{-(t-t_{0})/\Gamma_{i}}+\Delta\sigma_{n-1}a_{i}(t_{n-1})$ + +$$ +A _ {i, n} = A _ {i, n - 1} e ^ {- (t - t _ {n - 1}) / \Gamma_ {i}} + \Delta \sigma_ {n - 1} a _ {i} (t _ {n - 1}) +$$ + +$$ +A _ {i, 1} = \Delta \sigma_ {0} a _ {i} (t _ {0}) +$$ + +위와 같은 방법으로 각 단계마다 요소의 증분변형률은 이전단계에서 발생하는 응력과 이전단계까지 수정된 응력의 누적값을 사용하여 계산할 수 있습니다. + + + +이 방법은 응력의 변화를 고려한 비교적 정확한 해석을 할 수 있고, 사용자로 하여금 필요한 물성치만 입력하면 크리프계수를 별도로 계산하지 않아도 내부적으로자동 계산되는 장점을 가지고 있습니다. 그러나 설계기준에서 제안한 식을 사용하기 때문에 사용자가 경험에 의한 값들을 요소에 직접 입력할 수 없고 특정한 요소에 특정한 크리프 값을 입력할 수 없는 문제가 있습니다. 그리고 이 방법은 해석의 시간간격의 영향을 상당히 받게 됩니다. + +일반적인 시공단계는 소요 시간이 크지 않아서 해석에 문제가 없지만 한 개의 단계에서 큰 시간 간격이 입력될 경우에는 내부적으로 시간간격을 만들어서 크리프의 효과를 적절하게 계산할 수 있도록 해야 합니다. 크리프의 특성상 시간간격은로그(Log) 스케일로 분할하는 것이 바람직하며 midas Civil에서는 간격수만 입력하면 자동으로 로그 스케일로 분할하는 기능을 가지고 있습니다. 타당한 시간간격의개수는 정해져 있지 않지만 많이 세분하면 할수록 정해에 수렴하게 되므로 큰 시간 간격이 도입되는 단계에서는 적당한 간격으로 분할해주는 것이 바람직합니다. + +# 10-2-3 건조수축의 개념 + +건조수축은 콘크리트 부재가 시간에 따라 수축하는 현상으로 각종 설계기준에서규정하는 건조수축 특성곡선을 사용하여 해석에 반영하고 있습니다. 프레임 부재의 경우에는 길이방향의 건조수축만 고려하지만 면이나 입체의 경우에는 2축이나3축까지 포함하고 있습니다. + +midas Civil 프로그램에서 건조수축 해석은 CEB-FIP Code, ACI Code, 도로교설계기준, 실험데이터를 사용한 사용자 정의 등을 사용한 건조수축 특성 곡선을 사용하여 수행하고 있습니다. 건조수축 특성 곡선을 사용하여 시공단계의 시간 경과에대하여 변형량을 계산하여 해당 단계에서의 건조수축 변형률로 사용합니다. + +$$ +\varepsilon_ {s h} (t _ {2}, t _ {1}) = \varepsilon_ {s h} (t _ {2}, t _ {0}) - \varepsilon_ {s h} (t _ {1}, t _ {0}) +$$ + +여기서 $\mathcal { E } _ { s h } ( t _ { 2 } , t _ { 1 } )$ : 시공단계 t1 에서 t2 까지의 건조수축 변형률 + +$\varepsilon _ { s h } ( t _ { 1 } , t _ { 0 } )$ : 부재의 재령 t0 에서 t1 까지의 건조수축 변형률 + +$\varepsilon _ { s h } ( t _ { 2 } , t _ { 0 } )$ : 부재의 재령 t0 에서 t2 까지의 건조수축 변형률 + + + +건조수축에 의한 하중은 탄성계수, 단면적, 건조수축 변형률의 곱으로 계산하고 축방향에 대해서만 생성합니다. + +$$ +F _ {p r i m a r y} = E A \varepsilon_ {s h} +$$ + +건조수축 변형은 온도, 크리프에 의한 변형과 같이 비역학적인(Non-mechanical) 변형이기 때문에 부재력 계산시의 변형률은 하중에 의한 변형률에서 건조수축에 의한 변형률을 감하여 계산합니다. + +$$ +F _ {\text { secondary }} = E A (\varepsilon - \varepsilon_ {s h}) = F - F _ {\text { primary }} +$$ + +그러므로 축 방향 구속이 없는 구조물에서의 건조수축에 의한 효과는 부재력을 만들지 않고 변위만을 발생시키게 됩니다. 외부하중이 없더라도 구속조건에 의한 건조수축에 의해 발생하는 부재력은 크리프 변형을 유발할 수 있습니다. 건조수축변형은 구속조건과 시간에 영향을 받습니다. + +# 10-2-4 시간에 따른 탄성계수의 변화 + +콘크리트의 압축강도와 탄성계수는 시간에 따라 변화하기 때문에 상당한 시간이경과한 후에야 비로소 콘크리트 구조물의 고유한 강도를 발휘하게 됩니다. 실제PSC 구조물이나 교량의 시공에서 콘크리트의 초기 재령을 정확하게 예측하여, 계획된 구조물의 형상과 강도를 지니도록 하기 위해서는 이러한 Aging 효과를 합리적으로 모사하는 것이 필수적이라 할 수 있습니다. 한국 도로교 시방서의 제안 식은 ACI Code 와 유사하고 콘크리트의 압축강도와 탄성계수 식은 다음과 같습니다. + +$$ +f _ {c k} (t) = \frac {t}{a + b t} f _ {9 1} +$$ + +단위질량(mc)이 1450 \~ 2500kg/m3 인 콘크리트의 경우 + +$$ +E _ {c} (t) = 0. 0 7 7 m _ {c} ^ {1. 5} \sqrt [ 3 ]{f _ {c u} (t)} \quad (\mathrm{MPa}) +$$ + + + +다만, 보통골재를 사용한 콘크리트 $( \mathsf { m } _ { \mathsf { c } } { = } 2 3 0 0 \mathsf { k g } / \mathsf { m } ^ { 3 } )$ 의 경우는 + +$$ +E _ {c} (t) = 8, 5 0 0 \sqrt [ 3 ]{f _ {c u} (t)} \quad (\mathrm{MPa}) +$$ + +여기서 $f _ { 9 1 }$ : 91 일 평균압축강도 + +$f _ { c k } ( t )$ : 임의 시간 t 일의 압축강도 + +$E _ { c } ( t )$ : 재령 28일의 탄성계수 + +$$ +f _ {c u} (t) = f _ {c k} (t) + 8 \quad (\mathrm{MPa}) +$$ + +# 10-2-5 강도발현 함수 + +midas Civil에서는 콘크리트 부재의 재령에 따른 탄성계수의 변화를 고려함으로써강도발현 효과를 포함하여 해석할 수 있습니다. 그림 2.10.5와 같이 ACI, CEB-FIP,또는 콘크리트구조설계기준 등의 규준에 따른 콘크리트의 강도발현 함수를 정의할수 있고, 사용자가 직접 입력할 수도 있습니다. midas Civil은 이렇게 정의된 강도발현 함수를 참조하여, 각각의 시공단계에 정의된 시간의 경과에 따른 콘크리트의강도변화를 자동으로 계산해서 해석을 수행합니다. + +그림 2.10.5에서 정의한 시간의존재질(크리프, 건조수축, 강도발현)은 일반재질과의연결을 통해서 해석에 적용할 수 있습니다. + +![](images/page-460_1325d1a95f85471dd253820e2faf3114461efbf7c416e85dbb44d53b5341e1af.jpg) + +
+text_image + +Add/Modify Time Dependent Material (Comp. Strength) +Name +TdMat1 +Scale Factor +1.0 +Graph Options +X-axis log scale +Y-axis log scale +Type +Code +User +Development of Strength +Code: ACI +f(t)=teq × f28/(a+b×teq) +Concrete Compressive Strength at 28 Days(f28) : +28 +kN/m² +Concrete Compressive Strength Factor(a, b) +a : 4.5 +b : 0.85 +Redraw Graph +0 +2 +4 +6 +8 +10 +12 +14 +16 +18 +20 +22 +24 +26 +28 +Time (day) +20 +28 +OK +Cancel +
+ +그림 2.10.5 규준에 따른 콘크리트의 강도발현 함수정의 diff --git a/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_047.md b/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_047.md new file mode 100644 index 00000000..746d73c0 --- /dev/null +++ b/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_047.md @@ -0,0 +1,295 @@ + + +# 10-3 시공단계의 정의 및 구성 + +midas Civil에는 기본단계(Base Stage)와 시공단계(Construction Stage) 그리고 최종 시공단계(Final Stage)의 세 종류의 Stage가 존재하며 각 Stage의 특성은 다음과 같습니다. + +# ■ 기본단계 (Base Stage) + +시공단계가 정의되지 않은 상태에서는 일반적인 해석이 수행되며, 시공단계가 정의되면 해석은 수행되지 않고 구조모델링 및 요소그룹, 경계조건그룹, 하중그룹의 정의와 구성이 이루어지는 단계. + +# ■ 시공단계 (Construction Stage) + +시공단계 하중에 대한 해석이 실제로 이루어지는 단계이며, 해당 단계에서 활성화되어 있는 경계조건그룹과 하중그룹에 해당하는 경계조건 및 하중 조건을 입력할 수 있는 단계. + +# ▪ 최종시공단계 (Final Stage) + +시공단계의 최종 단계이며, 시공단계 하중 외에 일반하중 및 이동하중 해석, 응답스펙트럼 해석 등의 특수 해석이 수행되어지는 단계. + +각 시공단계는 요소그룹, 경계조건그룹, 하중그룹의 활성화(Activation)와 비활성화(Deactivation) 정의에 의하여 구성됩니다. 따라서, 각 그룹들은 동일한 시공단계에서 활성화 또는 비활성화되는 요소, 경계조건, 하중조건들의 집합이어야 합니다. + +각각의 시공단계별로 반영할 수 있는 내용은 다음과 같습니다. + +1. 임의의 재령을 가지는 부재의 생성 및 소멸 +2. 임의의 재하시점을 가지는 하중의 재하 및 소거 +3. 경계조건의 변화 + + + +midas Civil에서 사용되는 시공단계 구성의 개념도는 그림 2.10.6과 같습니다. 시공단계는 각 단계별 기간(Duration)만 가지고 쉽게 정의될 수 있습니다. 기간이 ‘0’인시공단계도 가능하며, 시공단계가 정의되면 기본적으로 First Step과 Last Step이생성됩니다. 실질적인 요소, 경계조건 및 하중의 생성과 소멸은 각각의 Step에서이루어집니다. + +![](images/page-462_dc8e601bebe63c0676d9b1aef33b84c2b83b266b537b36308edf8fb7f5a49df1.jpg) + +
+flowchart + +```mermaid +graph LR + A["0일"] --> B["10일"] + B --> C["20일"] + C --> D["30일"] + D --> E["40일"] + F["시공단계기간"] --> G["First Step"] + F --> H["Additional Step"] + F --> I["Last Step"] + F --> J["Last Step"] + G --> K["요소, 하중, 경계조건의 생성 & 소멸"] + H --> L["지연시간을 가지는 하중의 생성 & 소멸"] + I --> M["요소, 하중, 경계조건의 생성 & 소멸"] + J --> N["..."] + style A fill:#f9f,stroke:#333 + style B fill:#f9f,stroke:#333 + style C fill:#f9f,stroke:#333 + style D fill:#f9f,stroke:#333 + style E fill:#f9f,stroke:#333 + style F fill:#f9f,stroke:#333 + style G fill:#ccf,stroke:#333 + style H fill:#ccf,stroke:#333 + style I fill:#ccf,stroke:#333 + style J fill:#ccf,stroke:#333 + style K fill:#cfc,stroke:#333 + style L fill:#cfc,stroke:#333 + style M fill:#cfc,stroke:#333 +``` +
+ +그림 2.10.6 시공단계 구성의 개념 + +기본적으로 요소의 생성 및 소멸, 경계조건의 변화, 하중의 재하 및 소거 등 모든변경사항은 매 시공단계의 First Step에서 이루어집니다. 따라서 실제 시공중에 여러가지 원인에 의하여 구조계의 변화가 발생하면, 구조계의 변화가 발생하는 시기를 반영하는 시공단계를 생성시켜야 합니다. 즉 구조계의 변화가 잦을수록 시공단계의 수는 많아지게 됩니다. + +요소 및 경계조건 등 구조계의 변화는 매 시공단계의 First Step에서만 이루어 집니다. 그러나 하중의 변화는 해석의 편의를 위하여 시공단계 내에 추가적인 Step을 만들어서 그 Step에 하중을 재하 및 소거함으로써 반영할 수 있도록 하였습니다. 즉, 임의의 시공단계 내에서 지연시간을 가지는 하중을 가할 수 있는데 이 기능을 사용하면 구조계의 변화 없이 가설재의 설치나 소거로 인한 하중의 변화를새로운 시공단계를 만들지 않고 쉽게 고려할 수 있습니다. + +또한 시공단계 내에 추가적인 Step을 많이 정의하면 크리프와 건조수축을 고려한시간의존해석시 보다 정확한 해석결과를 얻을 수 있습니다. 그러나 추가 Step을 + + + +너무 많이 정의하면 해석시간이 증가하여 비효율적일 수 있으므로 주의해야 합니다. 특히 시공단계해석조건(Analysis탭>Analysis Control그룹>Construction StageAnalysis Control)에서 시간의존적인 특성(크리프, 건조수축, 탄성계수의 변화)을 고려하지 않도록 설정하고 해석을 수행하면 추가 Step이 많더라도 해석결과에는 영향을 주지 않습니다. + +임의의 시공단계에서 지정한 재령을 가진 요소가 생성된 후 매 시공단계마다 지속기간만큼의 시간이 흐르게 됩니다. 특정 시공단계에서 요소의 재료적 특성은 시간이 흐름에 따라서 변화하게 되는데, midas Civil에서는 이렇게 변화되는 재료적 특성을 매 시공단계 마다 입력하지 않고 요소의 재령만 입력하면 미리 정의한 시간의존재질(Properties탭>Time Dependent Material그룹)을 참조하여 내부적으로 자동계산하여 고려합니다. + +동일한 시공단계에서 동일한 재령을 가진 두개의 요소를 생성시키면, 그 두개의요소에는 항상 같은 시간이 흐르게 됩니다. 그러나 같이 생성된 요소라도 특정한요소만 시간이 흐르게 할 필요가 있는 경우가 있습니다. 이때에는 시간하중(Load탭>Load Type그룹> Construction Stage>Construction Stage Data그룹>C.SLoads>Time Loads for Construction Stage) 기능을 사용하면 임의의 시공단계에서특정 요소에만 시간의 흐름이 적용되도록 할 수 있습니다. + +임의의 시공단계에 요소를 생성시킬 경우에 생성될 요소의 재령을 지정하여 주어야합니다. 재령이 ‘0’인 요소를 생성시킨다는 것은 콘크리트의 타설 순간부터 묘사를 하는 것입니다. 그러나 일반적으로 구조물을 모형화하여 해석할 때 거푸집 등의 가설구조물은 모델에 포함시키지 않기 때문에 경화되지 않은 상태의 콘크리트를 해석한다는 것은 의도하지 않은 결과를 가져올 수 있습니다. 특히 재령이 ‘0’인요소를 생성시키고 시간에 따른 강도발현을 고려하여 해석을 한다고 하면 콘크리트 타설 후 24시간까지는 강도를 발현하지 못하므로 의미없는 큰 변위가 계산될수 있습니다. 따라서 시공단계를 모형화할 때는 일반적으로 거푸집 안의 경화되기전의 콘크리트는 가설 구조물과 함께 하중으로 고려하고, 거푸집을 제거한 후에실질적인 요소가 생성된다고 생각하는 것이 올바른 해석 방법입니다. + +임의의 시공단계에 요소가 생성되는 경우에 이전 시공단계의 하중이력에 의해서구조물에 발생한 변위나 내부응력은 영향을 미치지 않습니다. 즉, 새롭게 생성되는요소는 그 시공단계에서 구조물이 어떠한 하중을 받고 있는지에 관계없이 요소의 + + + +내부 응력이 ‘0’인 상태에서 생성됩니다. + +요소를 소멸시킬 때 지정하는 응력의 재분배율이 100%인 경우에는, 소멸되는 요소의 내부응력이 남아있는 구조물로 모두 재분배가 되어, 구조물을 이루고 있는다른 요소의 응력이 변화하게 됩니다. 그러나 응력의 재분배율이 0%인 경우에는소멸되는 요소의 내부 응력이 남아있는 구조물로 전혀 전달되지 않으므로, 다른요소의 응력이 변화되지 않습니다. 이 응력 재분배율을 적당히 조절하면 요소가소멸되면서 남아있는 요소에 전달할 응력의 양을 조절할 수 있습니다. 이 기능은시공단계별 해석에서 각 단계별로 요소가 사라졌다 할지라도 응력의 이완이 완전히 끝나지 않은 상태 등을 반영할 때 사용됩니다. + +경계조건을 활성화시킬 때 옵션에서 "Original"을 선택하면, 경계조건이 활성화되는절점의 이전 시공단계에서의 변위를 반대방향으로 하여 강제변위하중을 부여하여절점의 위치를 원래의 위치에 오도록 한 후 경계조건을 생성시키게 됩니다. 반면활성화 옵션을 "Deformed"로 선택하면 경계조건이 활성화될 절점의 초기 위치가아닌 변형된 위치에 경계조건을 생성해주게 됩니다. + +시공단계를 고려한 시간의존해석에서는 앞 단계에서 발생한 구조계의 변화 및 하중이력이 뒤에 있는 시공단계의 해석 결과에 영향을 미치게 됩니다. 따라서 midasCivil에서는 각각의 시공단계별 해석모델을 독립모델로 만들어서 해석을 수행하는것이 아니라 시공단계별로 구조계 또는 하중의 변화된 것들만 입력하여 해석을 한후 앞 단계의 해석결과에 누적하여 해석결과를 출력하는 누적모델 개념을 사용하고 있습니다. + +따라서 임의의 시공단계에 하중을 재하하면 이후의 시공단계에서는 재하된 하중을소거하지 않는한 계속해서 하중이 가해진 상태가 됩니다. 요소의 생성도 임의의시공단계에서 필요한 모든 요소를 생성시키는 것이 아니라 그 시공단계에 필요한요소만 생성시킵니다. 요소는 한번 생성이 되면 다시 생성할 수 없으며, 이미 생성된 요소만을 제거할 수 있습니다. + +시공단계해석에 사용되는 하중조건이 “Construction Stage Load”인 경우에는 시공단계에만 사용되지만 기타의 하중조건들은 시공단계가 끝난 후 일반해석에 적용된다.시공단계해석에 사용되는 하중조건은 여러개가 있다고 하더라도 그림 2.10.7과 같 + + + +이 하나의 해석결과로 조합됩니다. 이것은 시공단계해석에서는 시간의존재질의 비선형성으로 인해 하중조건 사이의 선형조합이 불가능하기 때문입니다. 시공단계해석을 수행하면 그림 2.10.7과 같이 누적된 시공단계 해석 결과와 최대값 결과, 최소값 결과가 생성됩니다. 이렇게 생성된 시공단계해석결과는 완성계 모델에 대한일반해석결과와 조합될 수 있습니다. + +시공단계해석을 수행하다 보면 가장 마지막 시공단계(완성계)가 아닌 임의의 중간단계에서 구조적으로 중요한 시공단계가 발생할 수 있습니다. 이러한 중간시공단계에는 특별한 하중을 고려하여 여러 가지 해석을 수행할 필요가 있습니다. midasCivil에서는 "Final Stage"지정 기능을 이용해서 임의의 중간 시공단계를 완성계인것처럼 설정할 수 있습니다. "Final Stage"로 지정된 시공단계는 프로그램 내부에서는 완성계로 고려되므로 일반하중을 가하여 해석을 수행할 수 있고, 시간이력해석,응답스펙트럼해석 등 midas Civil의 다양한 해석기능을 적용할 수 있습니다. + +![](images/page-465_6040a28ef5ad24a3ca04f76438e068f072b12914fc43a18e23deec23ea9654ab.jpg) + +
+flowchart + +```mermaid +graph TD + A["Base Stage"] --> B["Construction Stage 1"] + B --> C["Construction Stage 2"] + C --> D["Construction Stage 3"] + D --> E["Final Stage"] + E --> F["Load Case 1 (시공단계)"] + F --> G["Load Case 2 (시공단계)"] + G --> H["시공단계 해석"] + H --> I["시공단계 해석"] + I --> J["시공단계 해석"] + J --> K["해석결과 (Min)"] + J --> L["해석결과 (Sum)"] + J --> M["해석결과 (Max)"] + K --> N["Load Case 3 (Dead Load)"] + L --> O["Load Case 4 (Live Load)"] + M --> P["Load Comb (LC3+LC4+CSLC)"] + P --> Q["Final Stage"] + Q --> R["해석결과 (Min)"] + Q --> S["해석결과 (Sum)"] + Q --> T["해석결과 (Max)"] + R --> U["해석결과 (Min)"] + R --> V["해석결과 (Sum)"] + R --> W["해석결과 (Max)"] + S --> X["해석결과 (Min)"] + S --> Y["해석결과 (Sum)"] + S --> Z["해석결과 (Max)"] + T --> AA["해석결과 (Min)"] + T --> AB["해석결과 (Sum)"] + T --> AC["해석결과 (Max)"] + U --> AD["해석결과 (Min)"] + V --> AE["해석결과 (Sum)"] + W --> AF["해석결과 (Max)"] + X --> AG["해석결과 (Min)"] + Y --> AH["해석결과 (Sum)"] + Z --> AI["해석결과 (Max)"] + AD --> AJ["Load Comb (LC3+LC4+CSLC)"] + AE --> AJ + AF --> AJ + AG --> AJ + AH --> AJ + AI --> AJ + AJ --> AK["Final Stage"] +``` +
+ +그림 2.10.7 시공단계 해석결과의 하중조합 개념도 + + + +# 10-4 비선형 시공단계 해석 + +대변형이 발생하는 구조물의 시공단계 해석인 경우에는 구조물의 기하비선형성을 고려한 시공단계 해석이 필요합니다. 비선형성을 고려한 시공단계 해석 방법으로는 각 단계를 독립적인 구조물로 가정하고 해석을 수행하는 방법과 이전단계의 해석결과를 반영하여 해석을 수행하는 방법이 있습니다. 각 방법에 대한 자세한 설명은 다음과 같습니다. + +# 10-4-1 시공단계를 독립적으로 해석하는 방법 + +각 시공단계들을 독립적인 모델로 가정할 수 있는 구조물에 적용이 가능합니다. 각 단계의 구조물, 하중, 경계조건만으로 모델을 구성하여 기하비선형 해석을 수행하고 현수 구조물 해석에 적합합니다. 각 단계를 독립적으로 가정하기 때문에 이전 단계의 영향을 받지 않으며 시간의존 특성을 반영할 수 없습니다. 각 단계의 해석은 기하비선형 정적 해석 방법을 사용합니다. + +현수교의 역방향 시공단계 해석에 적용이 가능하며 대변형 해석용도의 초기 부재력을 사용하면 하중평형을 이루는 완성계 상태 구현이 가능합니다. 완성계를 첫 번째 단계로 하고 시공의 역방향으로 시공단계를 구성하면 현수교의 역방향 해석이 가능합니다. + +주요 해석절차를 정리하면 아래와 같습니다. + +1. 각 시공단계의 독립적인 해석모델을 구성합니다. 시공단계 별로 활성화된 구조물, 하중, 경계조건 그룹 정보를 사용하여 독립적인 해석 모델을 구성합니다. +2. 초기부재력이 입력된 경우에는 사용 옵션에 따라 외력과 내력을 생성합니다. +대변형 해석용도의 초기 부재력 중에서 기하강성 계산을 위한 초기 부재력 (Initial Forces for Geometric Stiffness)을 사용하면 하중의 추가 입력없이 완성계를 구성할 수 있습니다. 이 경우의 완성계의 외력과 내력은 초기 부재력을 사용하여 계산합니다. 각 시공단계는 부재를 제거하거나 하중을 추가함 + + + +으로써 구성됩니다. + +대변형 해석용도의 초기 부재력 중에서 부재의 평형절점력(EquilibriumElement Nodal Forces)을 사용하면 외부하중을 사용하여 완성계를 구현할수 있습니다. 이 경우의 외력은 사용자가 입력한 외부하중이 되고 내력은절점 평형력을 사용하여 계산합니다. 각 시공단계는 하중이나 부재를 제거함으로써 구성됩니다. + +3. 각 단계별 결과를 정리합니다. + +콘크리트의 시간의존특성은 각 단계별 독립적인 해석방법으로 인해 반영할 수 없습니다. + +# 10-4-2 시공단계를 이전단계에 누적하여 해석하는 방법 + +시공단계가 일반 선형 시공단계와 같은 방식으로 구성이 되면서 기하비선형성을고려하여 해석을 수행하는 방법입니다. 각 단계는 이전단계의 평형상태에 구조물,하중, 경계조건이 + +추가되는 방법으로 구성됩니다. 이전 단계에서 수렴된 하중과 부재내력으로 사용하여 현단계의 초기값으로 사용하여 해석을 수행합니다. 크리프나 건조수축과 같은 콘크리트의 시간의존 특성을 반영할 수 있습니다. 각 단계의 해석은 기하비선형 정적 해석 방법을 사용합니다. + +대변형이 발생하는 구조물의 순방향 시공단계 해석에 적용이 가능하고 주요 해석절차를 정리하면 다음과 같습니다. + +1. 이전 시공단계의 평형상태를 사용하여 현단계의 초기 상태를 계산합니다.이전 단계의 부재력, 변위, 하중을 사용하여 현 단계의 초기 상태를 계산합니다. 부재력과 하중을 사용하여 외력과 내력을 계산하고 변위를 사용하여현단계의 초기 변위 상태를 계산합니다. +2. 현 단계에 추가된 부재와 하중을 사용하여 현단계의 해석모델을 구성합니다. 이전 단계의 변위를 사용하여 현 단계에서 추가된 부재의 초기 접선변위를 계산합니다. 현단계에 추가된 하중을 이전 단계의 외력에 더하여 + + + +외력을 구성합니다. 텐던하중/크리프/건조수축 변형에 의한 하중은 내력으로 포함합니다. + +3. 현 단계의 외력과 내력에 대한 비선형 해석을 수행합니다. 정적 기하비선형 해석방법을 사용하여 평형상태 해석을 수행합니다. +4. 현 단계의 결과를 저장합니다. 현 단계의 결과와 다음 단계의 해석에 필요한 데이터를 저장합니다. + +비선형 시공단계 누적 모델해석에서 적용할 수 있는 트러스, 보요소로 사용이 제한이 되어 있습니다. + +![](images/page-468_33d12f996cca59896eb8c246a7eba3ffe9f733b806213ae2459f1725e203b25c.jpg) + +
+flowchart + +```mermaid +graph TD + A["시공단계 정보 처리"] --> B["현단계의 초기상태를 계산"] + B --> C["현단계의 해석모델 구성"] + C --> D["기하비선형 해석 수행 (Newton-Rapson)"] + D --> E["현단계 해석결과 저장"] + B -->|초기 외력과 내력 계산 +이전단계의 평형상태(하중/부재력) +추가된 절점의 접선변위 계산 +이전단계의 변위 사용| C + C -->|현단계의 외력과 내력 계산 +초기 외력과 내력 +현단계하중(크리프/건조수축 포함) +현단계 경계조건| D +``` +
+ +2.10.8 비선형 시공단계 누적 모델 해석 순서도 + + + +# 10-5 현수교 평형상태 해석 + +midas Civil에서 현수교의 초기형상 결정을 위한 해석 단계는 크게 두 가지로 구분할 수 있습니다. 첫 번째 단계는 케이블 시스템만의 형상 결정 단계이고 두 번째 단계는 케이블 시스템과 보강형, 주탑 시스템 모두 고려한 전체 구조계의 형상 결정 단계입니다. + +타정식 현수교의 초기형상의 경우는 첫번째 단계인 케이블 시스템만의 형상 해석 만으로도 충분하나 주탑의 초기부재력 산출이나 보다 엄밀한 해석을 위해서, 자정식 현수교의 경우에는 주 케이블이 보강형에 축력을 발생시켜 보강형에 축방향 변위가 발생하므로 이 영향을 고려하기 위해서 두번째 단계인 전체 구조계에 대한 형상 결정이 필요합니다. + +# 10-5-1 케이블 시스템만의 현수교 평형상태 결정(현수교 위저드) + +현수교의 평형상태 계산을 위한 첫 번째 단계는 현수교 위저드에서 계산이 수행됩니다. + +하중 평형식을 사용하여 케이블 절점좌표와 케이블의 장력을 계산하는 방법을 정리하면 다음과 같습니다. + +교량의 자중과 주 케이블 부재의 장력 사이의 평형방정식으로부터 케이블 좌표와 케이블 부재의 장력을 산정합니다. 이 방식은 수직, 수평 모두에서 새그(Sag)를 갖는 모노-듀오 현수교의 형상결정도 가능한데, 다음과 같은 기본 가정 하에 해석을 수행하게 됩니다. + +- 행어는 교축 직각방향에 대해서만 경사를 이루고 교축에 대해서는 수직이다. +- 주 케이블의 수평장력 중 교축방향 성분은 전 경간에 대해 일정하다. +- 주 케이블-행어 연결절점와 절점 사이의 케이블 부재는 포물선형태가 아닌 직선형태라 가정한다. +- 주 케이블 양단 좌표, 중앙경간의 sag, 행어의 보강형 정착점, 보강형의 고정하중 등은 알고 있는 값으로 가정한다. + + + +기본적으로 수직, 수평면에 케이블을 투영하여 각각의 평면상에서 장력과 고정하중의 평형 관계로부터 해석을 수행하게 됩니다. + +# 10-5-2 수직면 내에서의 해석 + +아래의 그림에서 주 케이블의 수직면 투영 형상을 나타내고 있습니다. 한 경간 내행어의 전체 개수를 N-1 개라고 하면 다음과 같이 전체구간을 N개의 구간으로 분할 할 수 있습니다. + +![](images/page-470_f6edb52665a0a1e4d914ca710d210c02543d7e59ef769173c86022562de346f9.jpg) + +
+text_image + +Y→X +Z +0 +1 +2 +T_i +Wci +T_{i+1} +Wsi +i +N-1 +N +d1 +d2 +dN +Where, x_i - x_{i-1} = d_i +
+ +그림 2.10.9 X-Z 평면에 투영된 주 케이블 형상과 힘의 평형 + +여기서 $W _ { s i }$ 는 보강형과 행어에 의해 케이블에 재하 되는 분포하중이며, $W _ { c i }$ 는 케이블 자중에 의한 수직하중을 의미한다. 힘의 평형조건에 의해 i번째 절점위치에서 다음과 같은 관계식을 얻을 수 있습니다. + +$$ +T _ {i} \frac {d _ {i}}{l _ {i}} = T _ {i + 1} \frac {d _ {i + 1}}{l _ {i + 1}} (i = 1, 2, \dots , N - 1) +$$ + +$$ +T _ {1} \frac {d _ {1}}{l _ {1}} = T _ {2} \frac {d _ {2}}{l _ {2}} = \Lambda = T _ {N} \frac {d _ {N}}{l _ {N}} = T _ {x} \tag {14} +$$ + +여기서, $T _ { i }$ : 절점 i-1과 절점 i 사이 케이블 요소의 장력 diff --git a/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_048.md b/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_048.md new file mode 100644 index 00000000..67cf2d96 --- /dev/null +++ b/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_048.md @@ -0,0 +1,409 @@ + + +$$ +l _ {i}: \text { 요소의 길이 } +$$ + +$$ +T _ {x} \colon \text { 케이블 부재의 수평장력 } +$$ + +교량의 횡단면, 즉 Y-Z평면상에서의 힘의 평형은 다음 그림과 같다. + +![](images/page-471_0f7814e1ef1455bc1b482a9737ceae9689733a64affc8719d73f028a61d54fb4.jpg) + +
+text_image + +Main Cable +Hanger, hi +X +Y +Z +(x_i, z_i) +W_{0i} \times (y_{0i} - y_i) / (z_{0i} - z_i) +W_{0i} \rightarrow P_i +y_{0i} - y_i +z_{0i} - z_i +(y_{0i}, z_{0i}) +Stiffening Truss +
+ +그림 2.10.10 Y-Z 평면에서의 평형상태 + +위 평형관계식은 다음과 같이 정리할 수 있습니다. + +$$ +T _ {i} \frac {z _ {i} - z _ {i - 1}}{l _ {i}} - T _ {i + 1} \frac {z _ {i + 1} - z _ {i}}{l _ {i + 1}} = P _ {i} \frac {z _ {G i} - z _ {i}}{h _ {i}} + W _ {c i} (i = 1, 2, \dots , N - 1) \tag {15} +$$ + +여기서 $P_{i}$ 는 i 번 행어의 장력이고, $h_{i}$ 는 행어의 길이 + +식 (15)와 식 (16)으로 부터 다음과 같이 총 N-1개의 연립방정식을 얻을 수 있습니다. + +$$ +T _ {x} \left(- \frac {z _ {i - 1} - z _ {i}}{d _ {i}} + \frac {z _ {i} - z _ {i + 1}}{d _ {i + 1}}\right) = P _ {i} \frac {z _ {G i} - z _ {i}}{h _ {i}} + W _ {c i} = W _ {s i} + W _ {c i} \tag {16} +$$ + +$$ +(i = 1, 2, \dots , N - 1) +$$ + + + +여기서 Wsi 는 보강형과 행어에 의해 케이블에 재하되는 분포하중이며, Wci 는 케이블 자중에 의한 수직하중을 의미합니다. 위의 식에서 미지수는iz ( i 1 , 2 ,..., N 1 )   와 Tx 로 총 N개이기 때문에 해를 구하기 위해 한 개의조건이 더 필요하게 됩니다. 추가적인 조건으로 기지 값인 중앙 경간의 새그(Sag) ,f 에 대한 다음과 같은 관계식을 사용하도록 합니다. + +$$ +\mathrm{z} _ {\frac {\mathrm{N}}{2}} = \frac {1}{2} \left(\mathrm{z} _ {\mathrm{N}} + \mathrm{z} _ {0}\right) + \mathrm{f} \tag {17} +$$ + +# 10-5-3 수평면 내에서의 해석 + +수평면내에서도 수직면내에서의 해석과 동일한 방식으로 힘의 평형관계로부터 다음과 같이 N-1개의 연립방정식을 얻을 수 있다. + +$$ +\mathrm{T} _ {\mathrm{x}} \left(- \frac {\mathrm{y} _ {\mathrm{i} - 1} - \mathrm{y} _ {\mathrm{i}}}{\mathrm{d} _ {\mathrm{i}}} + \frac {\mathrm{y} _ {\mathrm{i}} - \mathrm{y} _ {\mathrm{i} + 1}}{\mathrm{d} _ {\mathrm{i} + 1}}\right) = \mathrm{P} _ {\mathrm{i}} \frac {\mathrm{y} _ {\mathrm{Gi}} - \mathrm{y} _ {\mathrm{i}}}{\mathrm{h} _ {\mathrm{i}}} = \mathrm{W} _ {\mathrm{si}} \frac {\mathrm{y} _ {\mathrm{Gi}} - \mathrm{y} _ {\mathrm{i}}}{\mathrm{z} _ {\mathrm{Gi}} - \mathrm{z} _ {\mathrm{i}}} \tag {18} +$$ + +$$ +(\mathrm{i} = 1, 2, \dots , \mathrm{N} - 1) +$$ + +여기서 수평장력( Tx )는 수직면 내에서의 해석에서 얻은 값이며, 주 케이블 양 끝단의 y 좌표인 y , y 0 N 은 기지 값이므로 총 N-1개의 미지수인 iy( i 1 , 2 ,..., N 1 )   는 연립방정식을 풀어서 얻을 수 있습니다. + +이상과 같이 평형식을 사용하여 초기 기하형상을 유도하면 각 케이블의 절점 좌표와, 보강형의 좌표, 주 케이블의 수평장력이 결정할 수 있습니다. + +# 10-5-4 비선형 해석을 사용한 케이블 시스템의 평형상태 계산 + +현수교 위저드에서 간략한 평형식으로부터 계산한 케이블의 형상과 변형전 길이를초기값으로 가정하여 케이블 시스템으로 구성하고 비선형 해석을 수행하여 정확한절점좌표와 케이블의 변형전 길이를 산정합니다. 주 케이블의 양 끝단과 주탑지점,행어의 아래 끝단 부분을 모두 고정지점으로 처리합니다. 산정된 변형전 길이를케이블 요소에 적용하면 불균형 하중을 발생시켜 케이블 구조계의 변형이 발생합니다. 이 케이블 좌표들의 변화를 검토하여 수렴여부를 판단합니다. 수렴상태가 아 + + + +니면 케이블 시스템 절점 좌표와 변형전 길이를 업데이트하여 수렴조건을 만족할때까지 반복 수행합니다. + +![](images/page-473_dd3e2b33da6f3cec20f88db917a3b157461dd9709f17ef2fa07bc2dc61e39d62.jpg) + +
+line + +| Point | Series 1 | Series 2 | +|-------|----------|----------| +| 1 | 0.5 | 0.3 | +| 2 | 0.7 | 0.4 | +| 3 | 0.9 | 0.5 | +| 4 | 0.6 | 0.6 | +| 5 | 0.8 | 0.7 | +| 6 | 0.4 | 0.8 | +| 7 | 0.6 | 0.9 | +| 8 | 0.7 | 1.0 | +| 9 | 0.5 | 0.9 | +| 10 | 0.8 | 0.8 | +| 11 | 0.6 | 0.7 | +| 12 | 0.7 | 0.8 | +| 13 | 0.9 | 0.9 | +| 14 | 0.5 | 0.8 | +| 15 | 0.7 | 0.9 | +| 16 | 0.8 | 1.0 | +| 17 | 0.6 | 0.9 | +| 18 | 0.8 | 0.8 | +| 19 | 0.7 | 0.7 | +| 20 | 0.9 | 0.8 | +| 21 | 0.5 | 0.7 | +| 22 | 0.7 | 0.8 | +| 23 | 0.8 | 0.9 | +| 24 | 0.6 | 0.8 | +| 25 | 0.8 | 0.9 | +| 26 | 0.7 | 0.8 | +| 27 | 0.9 | 0.9 | +| 28 | 0.5 | 0.8 | +| 29 | 0.7 | 0.9 | +| 30 | 0.8 | 1.0 | +| 31 | 0.6 | 0.9 | +| 32 | 0.8 | 0.8 | +| 33 | 0.7 | 0.7 | +| 34 | 0.9 | 0.8 | +| 35 | 0.5 | 0.7 | +| 36 | 0.7 | 0.8 | +| 37 | 0.8 | 0.9 | +| 38 | 0.6 | 0.8 | +| 39 | 0.8 | 0.9 | +| 40 | 0.7 | 0.8 | +| 41 | 0.9 | 0.9 | +| 42 | 0.5 | 0.8 | +| 43 | 0.7 | 0.9 | +| 44 | 0.8 | 1.0 | +| 45 | 0.6 | 0.9 | +| 46 | 0.8 | 0.8 | +| 47 | 0.7 | 0.7 | +| 48 | 0.9 | 0.8 | +| 49 | 0.5 | 0.7 | +| 50 | 0.7 | 0.8 | +| 51 | 0.8 | 0.9 | +| 52 | 0.6 | 0.8 | +| 53 | 0.8 | 0.9 | +| 54 | 0.7 | 0.8 | +| 55 | 0.9 | 0.9 | +| 56 | 0.5 | 0.8 | +| 57 | 0.7 | 0.9 | +| 58 | 0.8 | 1.0 | +| 59 | 0.6 | 0.9 | +| 60 | 0.8 | 0.8 | +| 61 | 0.7 | 0.7 | +| 62 | 0.9 | 0.8 | +| 63 | 0.5 | 0.7 | +| 64 | 0.7 | 0.8 | +| 65 | 0.8 | 0.9 | +| 66 | 0.6 | 0.8 | +| 67 | 0.8 | 0.9 | +| 68 | 0.7 | 0.8 | +| 69 | 0.9 | 0.9 | +| 70 | 0.5 | 0.8 | +| 71 | 0.7 | 0.9 | +| 72 | 0.8 | 1.0 | +| 73 | 0.6 | 0.9 | +| 74 | 0.8 | 0.8 | +| 75 | 0.7 | 0.7 | +| 76 | 0.9 | 0.8 | +| 77 | 0.5 | 0.7 | +| 78 | 0.7 | 0.8 | +| 79 | 0.8 | 0.9 | +| 80 | 0.6 | 0.8 | +| 81 | 0.8 | 0.9 | +| 82 | 0.7 | 0.8 | +| 83 | 0.9 | 0.9 | +| 84 | 0.5 | 0.8 | +| 85 | 0.7 | 0.9 | +| 86 | 0.8 | 1.0 | +| 87 | 0.6 | 0.9 | +| 88 | 0.8 | 0.8 | +| 89 | 0.7 | 0.7 | +| 90 | 0.9 | 0.8 | +| 91 | 0.5 | 0.7 | +| 92 | 0.7 | 0.8 | +| 93 | 0.8 | 0.9 | +| 94 | 0.6 | 0.8 | +| 95 | 0.8 | 0.9 | +| 96 | 0.7 | 0.8 | +| 97 | 0.9 | 0.9 | +| 98 | 0.5 | 0.8 | +| 99 | 0.7 | 0.9 | +| 100 | 0.8 | 1.0 | +
+ +그림 2.10.11 케이블시스템만의 해석을 위한 지점처리 + +![](images/page-473_7b50f060898bfde6d8f248faebdf7e8c4e2410f41cde47ae0459003dffaa32e8.jpg) + +
+flowchart + +```mermaid +graph TD + A["힘의 평형식을 사용한 초기 기하형상 계산 (절점좌표, 케이블의 수평장력)"] --> B["케이블만으로 구성된 케이블 시스템 구성 (절점좌표, 주케이블의 수평장력, 행어의 장력)"] + B --> C["케이블 시스템의 기하비선형 평형상태 계산 (절점변위, 케이블의 장력 계산)"] + C --> D["절점 좌표, 케이블 변형전 길이 업데이트"] + D --> E{변위 수렴 정도 판단} + E -->|Yes| F["절점좌표와 케이블 변형전 길이 저장"] + E -->|No| G["End"] +``` +
+ +그림 2.10.12 케이블 시스템만의 현수교 평형상태 결정 순서도 + + + +# 10-5-5 현수교 전체구조물의 평형상태 결정 + +하중평형식과 케이블시스템만의 비선형해석 단계를 거쳐서 계산한 3차원 현수교의케이블 좌표와 변형전 길이를 기반으로 현수교 전체구조물의 평형상태를 계산하기위한 방법입니다. 타정식 현수교인 경우에는 케이블시스템으로만 계산한 평형상태에서 추가적인 변화가 있을 경우에 이 방법을 적용하여 좀더 정확한 평형상태를계산합니다. 자정식 현수교의 경우에는 주 케이블의 양단부가 고정되어 있지 않고보강형에 연결되기 때문에 케이블시스템만으로 구한 평형상태를 적용할 수 없습니다. 자정식의 경우에는 케이블시스템만으로 계산한 평형상태의 정보를 기본값으로하여 전체구조물에 대한 평형상태해석을 수행해야 합니다. 현수교에서의 평형상태라고 하는 것은 하중과 케이블의 변형전 길이와 보강형과 주탑의 내력이 평형을이루어 추가적인 변형이 발생하지 않는 상태를 의미합니다. 전체구조물의 평형상태해석 과정에서 업데이트 되는 항목으로는 주 케이블의 절점좌표, 케이블의 변형전 길이, 보강형과 주탑의 내력 등이 있습니다. 현수교 전체구조물의 평형상태는특정한 유일한 상태에서만 수렴이 되는 것이 아니기 때문에 수렴된 결과를 적용할것인지에 대한 사용자의 판단이 필요합니다. 현수교의 전체구조물의 평형상태 계산을 위한 계산 절차는 다음과 같습니다. + +![](images/page-474_db6aa1a9404c1ae42fd30185ac38f9bad500110c5b8835758a3ae88d26f2a02d.jpg) + +
+flowchart + +```mermaid +graph TD + A["전체 구조물(케이블 + 보강형 + 주탑) 구성
- 절정 좌표, 케이블의 변형전길이, 하중
- 이전 단계에서 계산된 부재내력을 초기내력 계산"] --> B["전체구조물의 기하비선형 평형상태 계산
- 절정변위, 부재내력 계산"] + B --> C["절정 좌표, 부재내력. 케이블 변형전 길이 엄데이트"] + C --> D{변위 수렴 정도 판단} + D --> E["절정좌표와 케이블 변형전 길이 저장"] +``` +
+ +그림 2.10.13 현수교 전체구조물 평형상태 결정 순서도 + + + +# Chapter 11. 수화열해석 + +콘크리트 구조물의 대형화 및 시공방법의 발전에 의한 대량 급속시공의 증가에 따라 시멘트의 수화열에 의한 온도 응력의 발생이 구조물에 균열을 발생시켜 구조물의 내구성 뿐만 아니라 구조적인 안정을 저해하는 요인이 됩니다. + +이러한 문제를 해결하기 위해 매스콘크리트 구조물의 타설시 온도와 응력의 분포를 계산하여 균열을 적절히 제어하고자 하는 목적으로 수화열해석을 수행합니다. + +수화열해석을 수행해야 할 매스콘크리트 구조물의 치수는 구조형식, 사용재료, 시공조건에 따라 다르지만 대략 슬래브는 80 100cm 이상이고, 하단이 구속되어 있는 벽체는 두께 50cm 이상을 기준으로 합니다. + +수화열에 의한 온도균열은 초기에 표면부와 중심부의 온도차이에 의해 발생하는 표면균열과 콘크리트 타설이 끝난 후 시멘트 수화열에 의한 온도상승이 최고치에 달한 후 온도강하에 의한 수축이 외적으로 구속되어 발생하는 관통균열로 구분할 수 있습니다. 이러한 수화열 해석은 크게 시멘트의 수화과정에서 발생하는 발열, 대류, 전도 등에 의한 온도분포해석과 발생한 온도, 재령에 의한 탄성계수의 변화, 크리프 및 건조수축 등에 의한 응력해석으로 구분할 수 있으며 각 해석에서 고려되는 사항들은 다음과 같습니다. + +# 11-1 열전달해석 (Heat Transfer Analysis) + +시멘트의 수화과정에서 발생하는 발열, 전도, 대류 등에 의한 시간에 따른 절점온도 변화를 계산하게 됩니다. 열전달 해석에서 사용되는 주요개념과 midas Civil에서 고려하는 사항들은 다음과 같습니다. + +# 11-1-1 전도 (Conduction) + +유체의 경우는 분자의 운동이나 직접적인 충돌, 고체의 경우는 전자의 이동에 의하여 고온구역에서 저온구역으로 에너지 교환이 일어나는 방식의 열전달 입니다. + + + +전도에 의해 전달되는 열전달율은 열속(Heat Flux)에 수직한 면적과 그 방향 온도구배의 곱에 비례합니다. (Fourier's Law) + +$$ +Q _ {x} = - k A \frac {\partial T}{\partial x} +$$ + +여기서 Qx : 열전달률 + +$A$ : 면적 + +$k$ : 열전도율 + +$\frac { \partial T } { \partial x }$ : 온도구배 + +일반적으로 포화된 콘크리트의 열전도율은 1.21\~ 3.11 정도이며 열전도율의 단위는2 kcal m h C/    입니다. 콘크리트의 열전도율은 온도가 증가하면서 감소하는 경향을 보이지만 대기온도의 범위에서는 큰 영향을 보이지 않습니다. + +# 11-1-2 대류 (Convection) + +유체가 고체 위 또는 유로 내를 흐를 때 유체와 고체 표면의 온도가 다르면 표면에 대한 유체의 상대운동의 결과로 유체와 고체의 표면 사이에서 열이 전달됩니다.이러한 열전달 방법을 대류라고 합니다. + +유체를 표면위로 강제로 흐르게 하는 경우처럼 유체유동을 인위적으로 일으킬 때의 열전달을 강제대류(Forced Convection)에 의한 열전달이라 하고, 유체의 유동이유체내의 온도차에 의해 생기는 밀도차에 의한 부력효과 때문에 일어나는 열전달을 자유대류(Free Convection)에 의한 열전달 이라고 합니다. 이러한 열전달에서는온도장(Temperature Field)이 유체 유동의 영향을 받기 때문에 실제 경우에 온도분포와 대류 열전달을 결정하는 것은 매우 복잡합니다. + +일반적으로 온도가 T 인 고체표면과 그 위로 흐르는 평균온도가 T 인 유체 사이의 열전달 계산을 간단히 하기 위해 열전달계수 h 를 다음과 같이 정의합니다. + + + +$$ +q = h _ {c} (T - T _ {\infty}) +$$ + +열전달계수( $h_c$ )는 흐름의 종류, 물체의 기하학적 형상 및 흐름의 접촉면적, 유체의 물리적 성질, 대류접촉면의 평균온도, 위치 등에 따라 복잡하게 변화하기 때문에 정식화하기 매우 어렵습니다. + +일반적으로 매스콘크리트의 온도해석에서 사용되는 대류 문제는 콘크리트 표면과 대기의 열교환 형태로 이루어지므로 대기 풍속의 함수로 다음의 경험식을 사용하기도 합니다. + +$$ +h _ {c} = h _ {n} + h _ {f} = 5. 2 + 3. 2 v \quad (m / \sec) +$$ + +열전달계수(대류계수)의 단위는 kcal/m²·h·°C 입니다. + +# 11-1-3 발열 (Heat Source) + +수화과정에서 발생하는 열량을 모델하기 위한 것으로 매스콘크리트의 수화발열에 의한 단위시간당 단위부피의 내부 발열량은 단열온도 상승식을 미분하고 비열과 밀도를 곱하여 얻을 수 있습니다. + +단위시간당 단위부피의 내부발열량 (kcal/m³·h) + +$$ +g = \frac {1}{2 4} \rho c K \alpha e ^ {- \alpha t / 2 4} +$$ + +단열온도 상승식(°C) + +$$ +T = K (1 - e ^ {- \alpha t}) +$$ + +여기서 T : 단열온도(°C) + +K : 단열최고상승온도(°C) + +α : 반응속도 + +t : 시간(days) + + + +# 11-1-4 파이프쿨링 (Pipe Cooling) + +파이프쿨링은 콘크리트 구조물 속에 파이프를 매설하고, 파이프 속으로 온도가 낮은 유체를 흐르게 하여 열교환을 발생시켜 수화열로 인한 온도상승을 감소시키는방법입니다. + +열교환의 형태는 유체와 파이프 표면사이의 대류에 의한 것이며, 파이프내의 유체온도는 파이프를 통과하면서 상승하게 됩니다. 유체와 파이프 사이의 대류에 의한열전달량은 다음과 같습니다. + +$$ +q _ {c o n v} = h _ {p} A _ {s} (T _ {s} - T _ {m}) = h _ {p} A _ {s} \left(\frac {T _ {s , i} + T _ {s , o}}{2} - \frac {T _ {m , i} + T _ {m , o}}{2}\right) +$$ + +여기서 $h _ { { } _ { p } }$ : 파이프의 유수대류계수( 2 kcal m h C /    ) + +As : 파이프의 표면적(m2 ) + +Ts , Tm : 파이프 표면과 냉각수의 온도( C ) + +# 11-1-5 초기온도(Initial Temperature) + +콘크리트 타설시의 온도로 물, 시멘트, 골재의 평균온도이며, 해석의 초기조건이됩니다. + +# 11-1-6 외기온도(Ambient Temperature) + +콘크리트의 타설 후 양생과정에서의 외기온도를 의미합니다. 일정한 온도나 Sine함수 또는 시간에 대한 온도 형태의 입력이 가능합니다. + + + +# 11-1-7 고정온도(Prescribed Temperature) + +열전달해석의 경계조건을 구성하게 되며 항상 일정한 온도를 유지하게 됩니다. 대류조건이나 고정온도가 입력되지 않은 절점은 열의 전달이 전혀 없는 단열상태로해석을 하게 됩니다. 일반적으로 대칭모델을 사용할 경우 대칭면에서 단열경계 조건을 사용하게 됩니다. + +아래 식은 열전달 해석에 사용되는 기본 평형 방정식이고, 해석결과로는 각 시간별 절점 온도가 됩니다. + +$$ +C T + (K + H) T = F _ {Q} + F _ {h} + F _ {q} +$$ + +$$ +C = \left[ \int_ {V} \rho c N _ {i} N _ {j} d x d y d z \right]: \text { Capacitance (Mass) } +$$ + +$$ +K = \left[ \int_ {V} \left(k _ {x x} \frac {\partial N _ {i}}{\partial x} \frac {\partial N _ {j}}{\partial x} + k _ {y y} \frac {\partial N _ {i}}{\partial y} \frac {\partial N _ {j}}{\partial y} + k _ {z z} \frac {\partial N _ {i}}{\partial z} \frac {\partial N _ {j}}{\partial z}\right) d x d y d z \right] +$$ + +$$ +H = \left[ \int_ {S} h N _ {i} N _ {j} d S _ {h} \right]: \text { Convection } +$$ + +$$ +F _ {Q} = \int_ {V} N _ {i} Q d x d y d z: \text { Heat load due to Heat Source / Sink } +$$ + +$$ +F _ {h} = \int_ {S} h T _ {\infty} N _ {i} d S _ {h} \quad : \text { Heat load due to Convection } +$$ + +$$ +F _ {q} = - \int_ {S} q N _ {i} d S _ {q}: \text { Heat load due to Heat Flux } +$$ + +T : Nodal Temperature + +여기서 ρ : 밀도 + +c : 비열 + +$k_{xx}$ $k_{yy}$ $k_{zz}$ : 열전도율 + +h : 대류계수 + +Q : 발열량 + +q : 열속 + + + +# 11-2 열응력해석(Thermal Stress Analysis) + +열전달해석에서 얻어진 절점온도 분포와 시간과 온도에 따른 재질의 변화, 시간에 따른 건조수축, 시간과 응력에 따른 크리프 등을 고려하여 매스콘크리트의 각 단계별 응력을 계산합니다. 열응력 해석에서 사용되는 주요개념과 midas Civil에서 고려하는 사항들은 다음과 같습니다. + +# 11-2-1 온도와 시간에 의한 등가재령, 적산온도 + +콘크리트의 경화 과정에서 발생하는 재질특성의 변화는 온도와 시간의 함수 형태로 나타나게 됩니다. 이러한 현상을 반영하기 위해 등가재령과 적산온도라는 개념을 사용하였습니다. + +탄성계수 계산시에는 일본 콘크리트 표준시방서(JSCE, 2002) 및 일본 도로교 시방서(Japanese standard, 2002)를 따르는 경우에는 절대재령을 사용하고 나머지 규준에서는 등가재령을 사용하였습니다. 그리고, 크리프와 건조수축 계산시에는 일본 콘크리트 표준시방서 및 일본 도로교 시방서를 따르는 경우에는 등가재령을 사용하고 나머지 규준에서는 절대재령을 사용하였습니다. + +등가재령은 기본적으로 CEB-FIP MODEL CODE를 사용하여 산정하였고 일본 도로교 시방서를 따르는 경우에만 해당 규준에서 정의하는 방법을 사용하였습니다. 숙성도 이론에 근거하는 적산온도는 Ohzagi식을 사용하였습니다. diff --git a/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_049.md b/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_049.md new file mode 100644 index 00000000..21ff1257 --- /dev/null +++ b/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_049.md @@ -0,0 +1,535 @@ + + +# CEB-FIP MODEL CODE에서의 등가재령 + +$$ +t _ {e q} = \sum_ {i = 1} ^ {n} \Delta t _ {i} \exp \left[ 1 3. 6 5 - \frac {4 0 0 0}{2 7 3 + T \left(\Delta t _ {i}\right) / T _ {0}} \right] +$$ + +여기서 $t_{eq}$ : 등가재령 (days) + +$\Delta t_{i}$ : 각 해석단계에서의 시간간격 (days) + +$T(\Delta t_{i})$ : 각 해석단계에서의 온도 (°C) + +$T_{0}$ :1 + +# 일본 도로교 시방서(Japanese standard, 2002)에서의 등가재령 + +$$ +t _ {e q} = \frac {T (\Delta t _ {i}) + 1 0}{3 0} \Delta t _ {i} +$$ + +여기서 $t_{eq}$ : 등가재령 (days) + +$\Delta t_{i}$ : 각 해석단계에서의 시간간격 (days) + +$T(\Delta t_{i})$ : 각 해석단계에서의 온도 (°C) + +# Ohzagi 식에 의한 적산온도 + +$$ +M = \sum_ {i = 1} ^ {n} \Delta t _ {i} \cdot \beta \cdot \left(T (\Delta t _ {i}) + 1 0\right) +$$ + +$$ +\beta = 0. 0 0 0 3 (T (\Delta t _ {i}) + 1 0) ^ {2} + 0. 0 0 6 (T (\Delta t _ {i}) + 1 0) + 0. 5 5 +$$ + +여기서 M : 적산온도 (°C) + +$\Delta t_{i}$ : 각 해석단계에서의 시간간격 (days) + +$T(\Delta t_{i})$ : 각 해석단계에서의 온도 (°C) + + + +# 11-2-2 등가재령과 적산온도를 사용한 콘크리트 압축강도 계산방법 + +콘크리트표준시방서(2003) + +$$ +f _ {c u} (t) = \frac {t}{a + b t} d (i) f _ {c k} +$$ + +여기서 a, b : 시멘트종류에 따른 계수 + +d(i) : 재령 28일에 대한 재령 91일의 강도 증가율 + +ACI CODE + +$$ +\sigma_ {c} (t) = \frac {t}{a + b t _ {e q}} \sigma_ {c (2 8)} +$$ + +여기서 a, b : 시멘트종류에 따른 계수 + +$\sigma_{c(28)}$ : 28일 압축강도 + +CEB-FIP MODEL CODE + +$$ +\sigma_ {c} (t) = \exp \left\{s \left[ 1 - \left(\frac {2 8}{t _ {e q} / t _ {1}}\right) ^ {1 / 2} \right] \right\} \sigma_ {c (2 8)} +$$ + +여기서 S : 시멘트종류에 따른 계수 + +$\sigma_{c(28)}$ : 28일 압축강도 + +$t_{1}$ : 1 day + + + +# Ohzagi 식 + +$$ +\sigma_ {c} (t) = \sigma_ {c (2 8)} \cdot y +$$ + +여기서 2 y ax bx c    + +$$ +x = 2. 3 8 9 \ln \left(\frac {M}{3 . 5}\right) - 1. 0 +$$ + +a, b, c :시멘트 종류에 따른 계수 + +c(28) ：28 + +# 11-2-3 온도의 변화량에 의한 변형 (Temperature) + +열전달해석을 통해서 구해진 각 단계별 절점온도의 변화를 사용하여 온도에 의한변형과 응력을 계산하였습니다. + +# 11-2-4 건조수축에 의한 변형 (Shrinkage) + +콘크리트의 초기 양생이 끝나게 되면 거푸집을 떼어내게 되는데 이때부터 건조수축이 시작되고 이로 인해 변형과 응력이 추가로 발생하게 됩니다. midas Civil에서는 ACI CODE와 CEB-FIP MODEL CODE를 사용하여 시멘트의 종류, 구조물의 형상, 시간에 따른 건조 수축량을 계산하여 열응력 해석에 포함하였습니다. + +# 11-2-5 크리프에 의한 변형 (Creep) + +콘크리트에 응력이 발생하게 되면 시간이 흐름에 따라 크리프 변형이 발생하게 되고 구조물에 추가적인 변형과 응력을 유발하게 됩니다. midas Civil 에서는 ACICODE와 CEB-FIP MODEL CODE를 사용하여 크리프에 의한 효과를 고려할 수 있도록 하였습니다. 이때, 온도에 의해 발생하는 응력이 각 시간 구간에 대해 선형적으로 변화한다고 가정하여 크리프 변형을 계산합니다. + + + +# 11-3 수화열 해석과정 + +1. Properties 탱>Time Dependent Material그룹>Time Dependent Material(Creep/Shrinkage)과 Time Dependent Material(Comp. Strength)를 선택하여 시간의존적 부재재질을 입력하고, Properties 탱>Time Dependent Material그룹>Time Dependent Material Link에서 일반 부재재질과 시간의존적 부재재질을 연결합니다. +2. Load탭>Load Type그룹>Heat of Hydration>Heat of Hydration Analysis +Data그룹의 하위메뉴에서 수화열 해석에 필요한 데이터를 입력합니다. +3. Analysis 탱>Analysis Control그룹>Heat of Hydration Analysis Control에서 시간이산계수, 초기온도, 응력출력위치, 크리프와 건조수축 고려여부를 입력합니다. +4. Analysis 탑>Perform그룹>Perform Analysis 메뉴로 해석을 수행합니다. +5. 해석이 완료되면 해석결과를 등고선도, 그래프, 동영상 등으로 확인합니다. + +![](images/page-484_9dff79a6f83766f9e43eaa9364a3902b0116ee8c907689945f4fcc2af1f3ef29.jpg) + +
+text_image + +MDAS/Crwi +View Structure Node/Element Properties Boundary Load Analysis Results PSC Pushover MODES Query Tools +Static Loads Seismic Settlement/Etc. +Temp./Prestress Construction Stage Load Tables +Moving Load Heat of Hydration Connection Prescribed Assign Heat Pipe Define CS for Hydration +Load Type Boundary - Source - Cooling +Heat of Hydration analyze Data +Hi Base Hi +Model View +For Help, press F1 None: 0.0, 0.0 0.0, 0.0 +End m > > non +
+ +그림 2.11.1 분할타설을 고려한 Extradosed PSC Box 주두부 수화열해석 모델 + + + +![](images/page-485_a8a5742e002ce05dcd843ae8d80a8b70512ba159d2bb44b4cfc63d310fa93aa0.jpg) + +
+text_image + +Add/Modify Time Dependent Material (Comp. Strength) +Name: Grade C4000 +Scale Factor: 1.0 +Graph Options +X-axis log scale +Y-axis log scale +Type +Code +User +Development of Strength +Code: CEB-FIP +t/(t+(t+Δt)+e+(t+Δt)/t+Δt) +Mean compressive strength of concrete +at the age of 20 days (t(t+Δt+Δt) +4000 +t(t+Δt+Δt) +Cement Type(s) +N, R: 0.25 +Redraw Graph +Show Time Dependent Material Function +Creep Function Data Type +Creep Coefficient +Shrinkage Drain +Start Loading: 10 +Day +End Loading: 10000 +Day +Num. of Steps: 24 +Graph Options +X-axis log scale +Y-axis log scale +Add/Modify Time Dependent Material (Creep / Shrinkage) +Name: Grade C4000 +Code: CEB-FIP(1990) +CEB-FIP(1990) +Characteristic compressive strength of concrete +at the age of 26 days (t(t)) : 4000 +Relative Humidity of ambient environment (40 - 90) : 70 +Notational size of member : 25 +h = 2 × Ac / u (Ac : Section Area, u : Perimeter in contact with atmosphere) +Type of cement +Rapid hardening high strength cement (RS) +Normal or rapid hardening cement (N, R) +Slowly hardening cement (SL) +Age of concrete at the beginning of shrinkage : 3 +Add/Modify Ambient Temperature Functions +Function Name +AMB1 +Function Type +Constant +Sine Function +User +Scale Factor +Graph Options +X-axis log scale +Y-axis log scale +Constant +Temperature : 20 +C1 +Redraw Graph +OK +Cancel +Show Result... OK +Close +
+ +그림 2.11.2 열특성 및 시간의존적 재료특성 대화상자 + +![](images/page-485_cd341db889c05521ba2a6ec48874c2a073bcf2a1e744bc2b1aeb878d7b413f03.jpg) + +
+text_image + +Construction Stage for Hydration +Stage +CS1 +CS2 +CS3 +Add +Insert Prev +Insert Next +Modify/Show +Delete +Close +
+ +![](images/page-485_33c246c198aa4782a97ac33b4e2bbd1df209b60828cb4ffe2de8aa242566edc9.jpg) + +
+text_image + +Compose Construction Stage for Hydration +Stage +Name : CS1 +Current Stage Information... +Initial Temperature : 20 [T] +Step +Time( hr ) : ( Example: 1, 3, 7, 14 ) +Auto Generate +Duration : 0 hr +Step Number : 1 +Generate Step +Step Time(hr) +1 10 +2 20 +3 30 +4 50 +5 80 +6 120 +7 170 +Add +Modify +Delete +Clear +Element Boundary Load +Group List ... +Phase2 +Phase3 +Activation +Group List +Name +Phase1 +Add Delete +OK Cancel Apply +
+ +그림 2.11.3 분할타설을 고려하기 위한 Construction Stage 대화상자 +(각 시공단계의 요소, 경계조건 등을 정의) + + + +![](images/page-486_7dfcdcaa60d66b6b1544964a92223530dbc8225f7ab2fb4320aea884fc985a53.jpg) + +
+line + +Stress & Allowable Tensile Stress +| Time (h) | 34000 - Max (mm) | 44000 - Max (mm) | +|---|---|---| +| 0 | -1.0 | -1.0 | +| 20 | 1.0 | 1.0 | +| 40 | 3.0 | 3.0 | +| 60 | 5.0 | 5.0 | +| 80 | 7.0 | 7.0 | +| 100 | 9.0 | 9.0 | +| 120 | 11.0 | 11.0 | +| 140 | 13.0 | 13.0 | +| 160 | 15.0 | 15.0 | +| 180 | 17.0 | 17.0 | +| 200 | 18.0 | 18.0 | +| 220 | 19.0 | 19.0 | +| 240 | 20.0 | 20.0 | +| 260 | 20.5 | 20.5 | +| 280 | 21.0 | 21.0 | +| 300 | 21.5 | 21.5 | +| 320 | 22.0 | 22.0 | +| 340 | 22.5 | 22.5 | +| 360 | 23.0 | 23.0 | +| 380 | 23.5 | 23.5 | +| 400 | 24.0 | 24.0 | +| 420 | 24.5 | 24.5 | +| 440 | 25.0 | 25.0 | +| 460 | 25.5 | 25.5 | +| 480 | 26.0 | 26.0 | +| 500 | 26.5 | 26.5 | +| 520 | 27.0 | 27.0 | +| 540 | 27.5 | 27.5 | +| 560 | 28.0 | 28.0 | +
+ +![](images/page-486_217a4e7772ef9ba86953189d6b4cde46e68f55fff7b0543d8a7ec93330cbfc43.jpg) + +
+line + +Temperature +| Time (h) | A2071 - Max Temperature (°C) | A488 - Max Temperature (°C) | +|---|---|---| +| 0 | 20 | 20 | +| 20 | 30 | 22 | +| 40 | 40 | 24 | +| 60 | 45 | 26 | +| 80 | 48 | 27 | +| 100 | 50 | 28 | +| 120 | 48 | 27 | +| 140 | 45 | 26 | +| 160 | 42 | 25 | +| 180 | 38 | 24 | +| 200 | 35 | 23 | +| 220 | 32 | 22 | +| 240 | 30 | 21 | +| 260 | 28 | 20 | +| 280 | 26 | 19 | +| 300 | 24 | 18 | +| 320 | 22 | 17 | +| 340 | 20 | 16 | +| 360 | 18 | 15 | +| 380 | 16 | 14 | +| 400 | 14 | 13 | +| 420 | 12 | 12 | +| 440 | 10 | 11 | +| 460 | 8 | 10 | +| 480 | 6 | 9 | +| 500 | 4 | 8 | +
+ +1st Stage + +![](images/page-486_d60bc51813070b373ff9a18a4ac3fd7c37a5584170274db1183c4970665fb35f.jpg) + +![](images/page-486_514de689747d1880646f2c33dd401138c349d1519ea8af532c67b90c8bc208b2.jpg) + +
+line + +Total Precision: Average Temperature +| Time (h) | N1104 - Max (°C) | N0340 - A (°C) | +|---|---|---| +| 0 | 25 | 25 | +| 50 | 40 | 28 | +| 100 | 55 | 29 | +| 150 | 60 | 28 | +| 200 | 62 | 27 | +| 250 | 60 | 26 | +| 300 | 55 | 25 | +| 350 | 50 | 24 | +| 400 | 45 | 23 | +| 450 | 40 | 22 | +| 500 | 35 | 21 | +| 550 | 30 | 20 | +| 600 | 25 | 19 | +| 650 | 20 | 18 | +| 700 | 15 | 17 | +| 750 | 10 | 16 | +| 800 | 5 | 15 | +| 850 | 0 | 14 | +| 900 | -5 | 13 | +| 950 | -10 | 12 | +| 1000 | -15 | 11 | +
+ +2nd Stage + +![](images/page-486_8cd6bb2589cb70991c474a2621a550a1ae53d67b00f9ba0eb379f706995db3f8.jpg) + +
+line + +Stress & Allowable Tensile Stress +| Time (h) | ME201 - Max. (Stress/Weight) | ME202 - Max. (Stress/Weight) | +|---|---|---| +| 0 | 0 | 0 | +| 50 | 5 | -5 | +| 100 | 10 | -10 | +| 150 | 15 | -15 | +| 200 | 20 | -10 | +| 250 | 22 | 0 | +| 300 | 24 | 5 | +| 350 | 25 | 10 | +| 400 | 26 | 15 | +| 450 | 26 | 18 | +| 500 | 26 | 20 | +| 550 | 26 | 20 | +| 600 | 26 | 20 | +| 650 | 26 | 20 | +| 700 | 26 | 20 | +| 750 | 26 | 20 | +| 800 | 26 | 20 | +| 850 | 26 | 20 | +| 900 | 26 | 20 | +| 950 | 26 | 20 | +| 1000 | 26 | 20 | +
+ +![](images/page-486_51f560048d1603a7108fe4caed40c917b5c1b634058901bcc272eaef2e5f3d89.jpg) + +
+line + +Temperature +| Time (h) | H5201 - Max (°C) | H5304 - Max (°C) | +|---|---|---| +| 0 | 18 | 18 | +| 50 | 36 | 28 | +| 100 | 46 | 29 | +| 200 | 36 | 26 | +| 300 | 28 | 22 | +| 400 | 22 | 20 | +| 600 | 20 | 19 | +| 800 | 19 | 19 | +| 1000 | 19 | 19 | +
+ +3rd Stage +그림 2.11.4 각 시공단계별 해석결과 그래프 + + + +# Chapter 12. PSC 해석 + +# 12-1 프리스트레스트 콘크리트의 해석 + +프리스트레스트 콘크리트 구조물의 거동은 작용하는 유효 프리스트레스에 따라 크게 변화됩니다. 따라서 프리스트레스트 콘크리트 구조물을 해석할 때에는 각각의 시공단계마다 가해지는 다양한 하중이력을 거치는 동안 PS 텐던의 인장력 변화를 정확히 계산해내는 것이 중요합니다. PS 텐던의 인장력은 텐던을 긴장하는 방법에 따른 여러 가지 원인에 의해서 손실됩니다. + +프리텐션방식의 경우 인장력의 손실을 단계별로 보면 인장력이 도입되기 전까지는 콘크리트의 건조수축 및 텐던의 이완에 의해 손실이 발생하고, 인장력이 도입될 때에는 콘크리트의 탄성변형에 의하여 손실이 발생하며, 인장력이 도입된 이후에는 콘크리트의 크리프, 건조수축 및 텐던의 이완과 하중 및 온도의 변화에 의하여 손실이 발생하게 됩니다. + +포스트텐션방식의 경우에는 인정력을 도입할 때 텐던과 쉬스사이의 마찰에 의한 손실, 정착구의 활동에 의한 손실이 발생하고, 인장력이 도입된 후에는 콘크리트의 크리프, 건조수축 및 텐던의 이완과 하중 및 온도의 변화에 의하여 손실이 발생하게 됩니다. + +midas Civil을 사용하여 프리스트레스트 콘크리트의 해석을 수행할 때 고려할 수 있는 인장력의 손실은 다음과 같습니다. + +- 프리스트레스를 도입할 때 일어나는 즉시손실(Instantaneous Loss) +■ 프리스트레스 도입 후에 일어나는 시간적 손실(Time Dependent Loss) + +midas Civil에서는 보다 정확한 해석을 위하여 PS 텐던이 설치되어 긴장되고 정착되기 이전까지는 단면적, 힘강성 등의 단면특성치를 계산할 때 총단면(Gross Cross Section)에서 덕트의 면적을 제외한 순단면(Net Cross Section)을 사용하고, 텐던이 정착된 이후에는 텐던의 단면적을 고려한 환산단면을 사용합니다. + + + +텐던을 고려한 환산단면은 텐던의 강성이 콘크리트보다 훨씬 크기 때문에 단면의도심이 변화하고, 변경된 도심을 기준으로한 텐던의 편심을 계산하여 텐던의 인장력을 계산하게 됩니다. + +midas Civil에서는 PS 텐던을 해석모델 상에서 고려할 때 트러스 등과 같은 요소로서 모형화하지 않고 텐던에 의한 프리스트레스를 등가의 하중으로 계산하여 고려합니다. 이때 위에 설명한 바와 같이 텐던의 강성은 단면계산시 포함됩니다. 등가하중을 계산하기 위한 텐던의 인장력은 매 시공단계마다 여러가지 원인에 의한 프리스트레스 손실을 고려하여 계산되므로 해석모델에 재하되는 등가하중에는 앞서설명한 프리스트레스 손실이 모두 반영됩니다. + +midas Civil에서 프리스트레스트 콘크리트의 해석을 수행하기 위한 절차는 다음과같습니다. + +1. 구조물을 모델링합니다. +2. 시간의존재질 및 시공단계를 정의한 후 시공단계별로 요소, 경계조건, 하중의 변화를 정의하여 시공단계를 생성합니다. +3. PS 텐던의 단면적, 재질, 극한강도, 덕트의 직경, 마찰계수 등 사용할 텐던의 특징을 정의합니다. +4. 원하는 부재에 텐던을 할당하고 텐던의 배치형상(Profile)을 정의합니다. +5. 텐던에 작용하는 인장력을 정의하고, 긴장하기를 원하는 시공단계에서 긴장력을 입력합니다. +6. 해석을 수행합니다. + + + +# 12-2 프리스트레스의 손실 + +PS 텐던에 가해진 인장응력은 여러 가지 원인에 의해서 감소하게 됩니다. 텐던의 인장응력이 감소하면 콘크리트에 도입된 프리스트레스트도 감소하게 됩니다. 이러한 프리스트레스 감소의 원인은 다음과 같습니다. + +\- 프리스트레스를 도입할 때 일어나는 즉시손실(Instantaneous Loss) + +정착장치의 활동(Anchorage Slip) + +PS 텐던과 쉬스 사이의 마찰 + +콘크리트의 탄성변형(Elastic Shortening) + +\- 프리스트레스 도입 후에 일어나는 시간적 손실(Time Dependent Loss) + +콘크리트의 크리프 + +콘크리트의 건조수축 + +PS강재의 이완(Relaxation) + +포스트텐션 방식에서는 이상의 여섯 가지의 프리스트레스 손실 원인을 모두 고려하지만, 프리텐션 방식에 있어서는 PS 텐던과 쉬스 사이의 마찰은 고려하지 않습니다. 프리스트레스의 즉시손실 및 시간적손실을 합한 텐던 인장력의 총 손실량은 재킹 힘(Original Jacking Force)의 15\~20%의 범위입니다. PSC 부재의 콘크리트 응력계산에서 가장 중요한 것은 즉시손실 후의 긴장력과 시간적 손실까지 끝난 후에 최종적으로 텐던에 작용하는 긴장력인 $P_{e}$ (Effective Prestress Force)입니다. $P_{i}$ 와 $P_{e}$ 의 관계를 다음과 같이 나타낼 수 있습니다. + +$$ +P _ {e} = R P _ {i} +$$ + +여기서, R을 프리스트레스의 유효율(Effective Ratio)이라고 합니다. 이 유효율의 일반적인 값은 프리텐션 방식의 경우 R=0.80, 포스트텐션 방식의 경우 R=0.85입니다. + +다음은 위에서 설명한 프리스트레스 손실을 midas Civil에서 고려하는 방법에 대해서 설명한 것입니다. + + + +# 12-2-1 정착장치의 활동에 의한 손실 + +PS 텐던의 긴장이 완료된 후 인장단을 정착시킬 때 정착장치에 따라서 약간의 정착부 이동이 발생하게 됩니다. 이로 인하여 PS 텐던의 인장단 부근에서 인장력의손실이 발생하게 되는데 이것을 정착장치의 활동에 의한 손실이라고 합니다. 이러한 손실은 포스트텐션 방식뿐 아니라 프리텐션 방식에서도 발생하는데 어떤 경우든 긴장작업시 초과긴장(Overstressing)함으로써 보정할 수 있습니다. + +일반적으로 PS 텐던과 쉬스 사이에 마찰이 있기 때문에 정착장치의 활동으로 인한 인장력의 손실은 정착장치 근처, 즉 인장단에 가까운 부위에 한정되며, 인장단에서 멀어지면 그 영향이 미치지 않게 됩니다. + +그림 2.12.1에서 정착부에서 정착장치의 활동에 의하여 영향을 받는 긴장재의 길이( setl )는 마찰손실의 함수로써 마찰손실이 크면 짧아지고 마찰손실이 작으면 길어지게 됩니다. 정착장치의 활동량을 l 이라 하고, 여기에 강재의 단면적( Ap )과탄성계수( Ep )를 곱하면 그림 2.12.1의 삼각형부분의 면적과 같게 되므로 식 (1)이성립됩니다. + +$$ +\text { 삼각형의 면적 } (0. 5 \Delta P l _ {s e t}) = A _ {p} E _ {p} \Delta l \tag {1} +$$ + +긴장재의 단위길이에 대한 마찰손실을 p 라고 하면, 인장력의 손실 P 는 그림2.12.1로부터 식 (2)와 같이 나타낼 수 있습니다. + +$$ +\Delta P = 2 p l _ {s e t} \tag {2} +$$ + +따라서, 정착부에서 정착장치의 활동의 영향을 받는 긴장재의 길이( setl )는 식(1)과(2)로부터 식 (3)과 같이 유도할 수 있습니다. + +$$ +l _ {s e t} = \sqrt {\frac {A _ {p} E _ {p} \Delta l}{p}} \tag {3} +$$ diff --git a/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_050.md b/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_050.md new file mode 100644 index 00000000..3bbd6076 --- /dev/null +++ b/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_050.md @@ -0,0 +1,274 @@ + + +![](images/page-491_3f601dbcfb2b35fae27b12863c15bd5c8cfd12003cf22e4859cd49ac9921d7b0.jpg) + +
+text_image + +Pre-stressing force +Jacking tension force +ΔP +P +1 +P +1 +Tension after release +l_set +Distance from the anchorage +
+ +그림 2.12.1 정착장치의 활동이 긴장력에 미치는 영향 + +그림 2.12.1에서 텐던의 인장력의 분포가 직선으로 나타나지만 실제로는 곡선의형태로 분포하게 되므로, midas Civil에서는 이러한 인장력의 곡선분포를 고려하여정착장치의 활동에 의한 프리스트레스의 손실을 계산하고 있습니다. + +# 12-2-2 텐던과 쉬스 사이의 마찰에 의한 손실 + +포스트텐션방식에서는 PS 텐던과 쉬스 사이의 마찰로 인하여 PS 강재의 인장력이긴장재의 끝으로부터 멀어질수록 작아집니다. 이러한 마찰손실은 긴장재의 각도변화(Curvature Effect)에 의한 손실인 곡률마찰(Curvature Friction) 손실과 긴장재의길이의 영향(Length Effect)에 의한 손실인 파상마찰(Wobble Friction) 손실로 나눌수 있으며 각각 단위각도당 마찰계수  (Radian)와 단위길이당 마찰계수 k를 사용하여 나타낼 수 있습니다. + +긴장단에서 P0 로 긴장했을 경우 텐던의 곡선길이  만큼 떨어진 곳에서의 총 각변화가  이면 그 지점에서의 인장력 P 는 식 (4)와 같이 나타낼 수 있습니다. + +$$ +P _ {x} = P _ {0} e ^ {- (\mu \alpha + k l)} \tag {4} +$$ + + + +
긴장재의 종류k(/m)(μ/Rad)
부착시킨 긴장재강선0.0033~0.00500.15~0.25
강봉0.0003~0.00200.08~0.30
고강도 스트랜드0.0015~0.00660.15~0.25
부착시키지않은 긴장재수지방수피복강선0.0033~0.00660.05~0.15
고강도 스트랜드0.0033~0.00660.05~0.15
그리스 도포강선0.0010~0.00660.05~0.15
고강도 스트랜드0.0010~0.00660.05~0.15
+ +표 2.12.1 k 및  의 값 (도로교 및 콘크리트 설계기준) + +
긴장재의 종류덕트의 종류k/(m)( $\mu$ /Rad)
PS강선및 PS 강연선금속쉬스0.00660.30
아연도금 금속쉬스0.00500.25
그리스 또는 아스팔트로 코팅하고 또 피복된 것0.00660.30
아연도금된 강성덕트0.00070.25
PS 강봉금속 쉬스0.00100.20
아연도금 금속쉬스0.00070.15
+ +표 2.12.2 k 및  의 값 (철도교 설계기준) + +# 12-2-3 콘크리트의 탄성변형에 의한 손실 + +콘크리트에 프리스트레스를 도입하면 콘크리트는 압축되어 그만큼 부재의 길이가줄어들게 되고 콘크리트에 정착된 텐던의 길이 역시 줄어들게 되어 텐던의 인장응력이 감소하게 됩니다. 이러한 탄성변형에 의한 손실은 프리텐션방식이나 포스트텐션방식에서 모두 발생하지만 그 형태는 조금 다르게 나타납니다. + +프리텐션방식의 경우에는 인장재(고정지주)의 긴장력(Jacking Force)을 부재에 가하는 순간 탄성수축이 일어나고 텐던의 길이가 짧아져서 프리스트레스의 손실이 일어나게 됩니다. 즉, 그림 2.12.2에 나타난 것처럼 인장재에 가하는 긴장력( Pj )과실제로 부재에 가해지는 긴장력( Pj )은 다르게 됩니다. + + + +그러나 포스트텐션방식의 경우 따로 인장대가 있는 것이 아니라 경화한 콘크리트부재를 받침으로 하여 텐던을 긴장합니다. 따라서 콘크리트 부재가 단축하는 것은프리텐션방식과 같지만 이 때 텐던의 인장력은 콘크리트 부재가 탄성 단축된 후에측정되기 때문에 콘크리트의 탄성변형으로 인한 인장력의 감소는 있을 수 없습니다. midas Civil에서는 임의의 시공단계에서 요소가 생성된 후 긴장력이 가해지는포스트텐션방식과는 달리 시공단계에서 모형화가 불가능한 프리텐션방식의 경우에는 콘크리트 탄성변형에 의한 프리스트레스의 손실은 고려하지 않습니다. 따라서프리텐션방식으로 긴장을 할 때는 하중값 입력시에 인장대에 가하는 긴장력( Pj )이아니라 실제로 부재에 가해지는 긴장력( P )을 입력해 주어야 합니다. + +대부분의 포스트텐션 부재는 많은 수의 긴장재가 배치되어 미리 정해 놓은 긴장순서에 따라 긴장하고 정착하는 것이 일반적이므로, 콘크리트의 탄성수축도 순차적으로 일어납니다. 따라서 그림 2.12.3(b)의 Tendon 1과 같이 제일 먼저 정착하는텐던은 그 시점에서 인장력의 감소가 없지만 두번째 텐던을 정착하면 이 탄성수축때문에 그림 2.12.3(c)과 같이 첫번째 텐던의 인장력이 감소하게 됩니다. midasCivil은 매 시공단계마다 텐던의 긴장에 의한 탄성수축으로 발생하는 프리스트레스의 손실 뿐 아니라 외부하중에 의한 탄성수축으로 발생하는 프리스트레스의 손실모두 고려하고 있습니다. + +![](images/page-493_aa83be6100e226d44525f4c85734f2f79dcf1877240903b84d2875673e2f2af5.jpg) + +
+text_image + +Jacking tension force +before release +Pre-tension force +after release +Pj +P1 +Pre-stress tension +anchorage abutment +
+ +그림 2.12.2 탄성수축으로 인한 인장력의 감소 (프리텐션 부재) + + + +![](images/page-494_015fe6f7d334ce96bcf0ba61e9a690371d0964dcdf7acd1780d8ad250b790c00.jpg) + +
+text_image + +Tendon 1 +Tendon 2 +
+ +(a) + +![](images/page-494_a16a6b79d183636ea20554e99bbc985e47a4acafc9974124770d0cfd77b21547.jpg) + +
+text_image + +First tendon tensioning +
+ +(b) + +![](images/page-494_a39ec0300c1dfba5ccb6c3dc7ce16bdf0f1fa480f40223e52b78242cd7554be8.jpg) + +
+text_image + +Second tendon tensioning +
+ +(c） +그림 2.12.3 프리스트레스의 순차적인 도입에 의한 인장력의 감소 +(포스트텐션 부재) + +# 12-2-4 시간의존적 손실 + +프리스트레스는 콘크리트의 크리프와 건조수축 및 PS 텐던의 이완(Relaxation) 때문에 시간의 경과와 더불어 감소하게 됩니다. midas Civil에서는 각각의 시공단계마다 콘크리트 부재의 시간의존적인 특성을 고려하여 크리프 및 건조수축에 의해 발생하는 변형을 계산하고 있습니다. 또한 계산된 콘크리트 부재의 변형으로 발생하는 PS 텐던의 인장응력의 감소도 고려합니다. 계산된 프리스트레스의 시간적 손실은 매 시공단계마다 그래프를 통하여 쉽게 확인할 수 있습니다. + +이완(Relaxation)이란 PS 텐던에 인장응력을 작용시켜서 그 변형률을 일정하게 유지하면, PS 텐던에 준 인장응력이 시간의 경과와 더불어 점차 감소하는 현상으로정의되며, 이완에 의한 손실은 재하된 초기응력의 크기, 경과된 시간, 제품의 성질에 따라 각각 다르게 나타납니다. midas Civil에서는 PS 텐던의 이완을 고려하기위하여 널리 사용되고 있는 Magura1) 의 식과 CEB-FIP의 식을 사용합니다. + + + +# Magura식을 사용한 Relaxation 계산식 + +$$ +\frac {f _ {s}}{f _ {s i}} = 1 - \frac {\log t}{C} \left(\frac {f _ {s i}}{f _ {y}} - 0. 5 5\right) \quad \text { 여기서, } \frac {f _ {s i}}{f _ {y}} \geq 0. 5 5 \tag {5} +$$ + +여기서, $f_{si}$ 는 초기응력, $f_{s}$ 는 재하 후 t시간 후의 응력, $f_{y}$ 는 항복응력(0.1% Offset Yield Stress), C는 제품에 관한 상수로서 일반강재인 경우 10, 저 름랙세이션 강재를 사용한 경우 45를 사용합니다. 이 식은 이완의 정의에서 설명한 것처럼 텐던의 응력이 일정하게 유지될 때를 가정하고 있습니다. 그러나 실제 PS 텐던의 인장력은 시간에 따라서 크리프, 건조수축, 외부하중의 변화에 의하여 불연속적으로 변화하기 때문에 식 (5)를 직접 적용하기에는 어려움이 있습니다. 따라서 midas Civil에서는 매 시공단계마다 이완에 의한 손실을 제외한 다른 요인에 의한 텐던의 인장력의 변화를 계산한 후에 각각의 시공단계에 해당하는 가상의 초기응력 (Fictitious Initial Prestress) $^{2}$ 을 구한 후 이완에 의한 손실을 계산합니다. + +# CEB-FIP식을 사용한 Relaxation 계산식 + +이 식은 시간에 따라서 초기응력의 일정한 비율이 감소하는 것으로 계산합니다. + +$$ +\Delta f _ {n} = f _ {s i} \times \left(\frac {R P}{1 0 0}\right) \left(D L _ {n} - D L _ {n - 1}\right) +$$ + +$$ +- t _ {n}, t _ {n - 1} \leq 1 0 0 0 (\text { days }) +$$ + +$$ +D L _ {n} = \frac {1}{1 6} \ln \left(\frac {t _ {n} - t _ {s}}{1 0} + 1\right), \quad D L _ {n - 1} = \frac {1}{1 6} \ln \left(\frac {t _ {n - 1} - t _ {s}}{1 0} + 1\right) +$$ + +$$ +- 1 0 0 0 < t _ {n}, t _ {n - 1} \leq 5 0 0 0 0 0 (\text { days }) +$$ + +$$ +D L _ {n} = \left(\frac {t _ {n} - t _ {s}}{5 0 0 0 0 0}\right) ^ {0. 2}, \quad D L _ {n - 1} = \left(\frac {t _ {n - 1} - t _ {s}}{5 0 0 0 0 0}\right) ^ {0. 2} +$$ + +$$ +- t _ {n}, t _ {n - 1} \geq 5 0 0 0 0 0 (\text { days }) +$$ + +$$ +D L _ {n} = D L _ {n - 1} = 1. 0 +$$ + + + +여기서, $f_{si}$ : 초기응력 + +RP : Relaxation 발생 비율 (%) + +$t_{n}$ , $t_{n-1}$ : 시공단계 n, n-1 의 시간 (days) + +$t_{s}$ : 긴장력 도입 시간 (days) + +# 12-3 프리스트레스 하중 + +midas Civil에서 PS 텐던에 의해 구조물에 가해지는 프리스트레스 하중을 등가의 하중으로 치환하는 방법은 그림 2.12.4와 같습니다. + +![](images/page-496_3fbc6141f9bd8e742909e06a4c96e826257c45b341bd09c6287ccaf3079d4d01.jpg) +그림 2.12.4 텐던의 프리스트레스에 의한 등가하중 + +그림 2.12.4은 하나의 보요소에 텐던이 배치되어 있는 형상을 나타낸 것입니다. 설명의 편의상 2차원으로 표현하였지만 요소좌표계 xy평면에 대해서도 동일한 방법으로 계산됩니다. 그림에서 보는 바와 같이 midas Civil에서는 내부적으로 하나의 보요소를 4등분을 하여 등가하중을 계산합니다. 이때 4등분한 요소 내에서의 텐던 형상은 그림 2.12.4(b)와 같이 실제적인 텐던의 형상을 따르며, 다만 구간내에서 등가하중은 그림 2.12.4(c)와 같이 일정하다고 가정합니다. 텐던에 가해지는 인장 + + + +력 $P_{i}$ 와 $P_{j}$ 는 마찰에 의한 손실 때문에 다르게 되므로, i 와 j 단 양쪽의 3개의 집중하중( $p_{x}$ , $m_{y}$ , $p_{z}$ ) 만으로는 내부적으로 평형이 될 수 없습니다. 따라서 부재 내에서 자체로 평형을 이루도록 등분포하중을 고려합니다. 식 (6)과 (7)에 의해서 양단의 집중하중을 계산하고 식 (8)과 (9)를 이용해서 요소 내부의 등분포하중을 계산합니다. + +$$ +p _ {x} ^ {i} = p ^ {i} \cos \theta^ {i} +$$ + +$$ +p _ {z} ^ {i} = p ^ {i} \sin \theta^ {i} \tag {6} +$$ + +$$ +m _ {y} ^ {i} = p _ {x} ^ {i} \cdot e _ {z} ^ {i} +$$ + +$$ +p _ {x} ^ {j} = p ^ {j} \cos \theta^ {j} +$$ + +$$ +p _ {z} ^ {j} = p ^ {j} \sin \theta^ {j} \tag {7} +$$ + +$$ +m _ {y} ^ {j} = p _ {x} ^ {j} \cdot e _ {z} ^ {j} +$$ + +$$ +\sum F _ {x} = p _ {x} ^ {i} + w _ {x} l - p _ {x} ^ {j} = 0 +$$ + +$$ +\sum F _ {z} = - p _ {z} ^ {i} + w _ {z} l + p _ {z} ^ {j} = 0 \tag {8} +$$ + +$$ +\sum M _ {y} ^ {j} = m _ {y} ^ {i} - p _ {z} ^ {i} l + w _ {z} \frac {l ^ {2}}{2} + m _ {y} ^ {j} + m _ {y} l = 0 +$$ + +$$ +w _ {x} = \frac {p _ {x} ^ {j} - p _ {x} ^ {i}}{l} +$$ + +$$ +w _ {z} = \frac {p _ {z} ^ {i} - p _ {z} ^ {j}}{l} \tag {9} +$$ + +$$ +m _ {y} = p _ {z} ^ {i} - w _ {z} \frac {l}{2} - \frac {m _ {y} ^ {i} + m _ {y} ^ {j}}{l} +$$ + +midas Civil에서는 매 시공단계마다 크리프, 건조수축, 텐던의 이완 등 프리스트레스의 시간적 손실 뿐 아니라 외부하중이나 온도의 변화로 인한 구조물의 변형으로 인해 이미 설치된 텐던에서 발생하는 프리스트레스의 손실도 고려하고 있습니다. 먼저 시공단계 해석시 발생하는 변형에 의한 텐던의 인장력의 변화를 계산하고 계산된 인장력의 변화량을 앞서 설명한 방법대로 등가하중으로 치환하여 요소에 부여합니다. + + + +# Chapter 13. 이동하중해석 + +midas Civil의 이동하중 해석기능은 구조물의 설계시 정적차량이동하중(StaticVehicle Moving Load) 조건을 반영하는데 사용되며 주요한 기능은 다음과 같습니다. + + 이동하중에 따른 처짐, 부재내력, 반력 등에 대한 영향선(Influence Line)과영향면 (Influence Surface)의 산출 + 산출된 영향선과 영향면을 이용하여 주어진 차량이동하중조건에 대한 최대최소 절점변위, 부재내력, 지점반력의 산출 + +구조물에 대한 이동하중해석은 차량하중이 이동하면서 유발하는 하중에 대한 해석을 수행하는 것으로 차량하중의 이동경로 전체에 대해 해석을 수행하여 최대값,최소값을 계산하고 이동하중조건 결과로 사용합니다. + +구조물에 대한 이동하중해석을 수행하기 위해서는 재하해야 할 차량하중, 차량하중이 재하되는 차선이나 차선면, 차량하중의 재하방법 등을 정의하고, 차선이나 차선면에 단위하중을 재하하여 영향선이나 영향면을 계산합니다. + +영향선은 구조물의 차선을 따라가면서 단위하중을 재하하여 정적해석을 수행하고각 성분들의 해석결과를 차선상에 나타낸 것입니다. 영향면은 구조물의 차선면내에 위치한 판요소의 절점에 단위하중을 재하하여 해석한 결과를 하중작용점에 나타낸 것입니다. 영향선이나 영향면의 계산이 가능한 결과에는 구조모델에 포함된절점의 변위와 트러스요소, 보요소, 판요소의 부재력과 지점반력이 포함됩니다. + +차량이동하중해석의 해석과정을 간략하게 정리하면 다음과 같습니다. + +1. 차량하중, 이동하중재하방법, 차선이나 차선면 등을 정의합니다. +2. 차선이나 차선면을 사용하여 단위하중조건을 만들고 각 단위하중에 대한정적해석을 수행하여 각 성분별 영향선이나 영향면을 계산합니다. +3. 차량하중의 재하방법에 따라 영향선 또는 영향면을 사용하여 차량의 이동에 따른 해석결과들을 산출합니다. + + + +이러한 과정을 거쳐 산출되는 해석결과들은 한 개의 이동하중조건에 대해 최대와최소의 두 가지 해석결과를 갖게 되고 다른 하중조건들과의 조합이 가능합니다.이동하중조건이 최대와 최소의 두 가지 해석결과를 갖기 때문에 하중조합을 하게되면 조합된 결과 역시 최대와 최소의 두 가지 해석 결과를 갖게 됩니다. 해석의결과로는 절점변위, 지점반력, 트러스, 보, 판요소의 부재내력 등이 출력되며, 이외의 입력된 요소에 대해서는 강성만 고려되고 해석결과는 산출하지 않습니다 + +차량이동하중해석에서 사용하는 영향선 또는 영향면의 단위하중은 전체좌표계의 -Z방향으로 작용하게 되며 이동하중해석 조건의 수는 무제한으로 사용이 가능합니다. + +영향선해석과 영향면해석은 동시에 수행될 수 없으며, 표 2.13.1은 각각의 특징과용도를 정리한 것입니다. + +
구 분영향선 해석영향면 해석
용 도주형에 의한 거동이 지배적인 교량 또는 교량의 입면 2차원 해석에 적용(강박스교 등)교축직각방향으로 차량이동하중에 의한 구조적 거동의 변화가 큰 경우 (슬래브교, 라멘교 등)
영향 해석결과의 표현차선요소(보요소)상의 영향선으로 표현차선면요소(판요소)상의 영향면으로 표현
해석성분절점변위, 지점반력, 부재내력절점변위, 지점반력, 부재내력
해석대상요소트러스, 보, 판요소 (기타요소는 해석과정에서 강성효과만 기여)트러스, 보, 판요소 (기타요소는 해석과정에서 강성효과만 기여)
하중재하방법차륜하중 및차선집중하중차선요소(보요소)에 집중하중으로재하차선을 구성하는 절점에 집중하중으로 재하
차선분포하중차선요소(보요소)에 분포하중으로재하차선면요소(판요소)에 압력하중으로 재하
+ +표 2.13.1 영향선 및 영향면 해석의 용도 및 특징 + + + +영향선은 보요소상의 차선(Traffic Lane)을 따라가면서 단위하중(수직력 또는 편심비틀림모멘트)을 재하하여 해석되며, 모델에 포함된 모든 절점과 트러스요소, 보요소, 판요소, 그리고 지지점에 대해 절점변위, 요소내력, 반력성분으로 산출됩니다. + +영향면은 차선면(Traffic Surface Lane)내에 위치한 판요소 절점에 단위하중(수직력)을 재하한 해석 결과로부터 산출됩니다. + +그리고 midas Civil은 산출된 영향선 또는 영향면을 이용하여 한국 도로교설계기준,AASHTO1), Caltrans2), AREA3) 또는 사용자가 임의로 입력한 차량하중(Vehicle LiveLoad) 및 재하조건에 대해서 변위, 반력, 트러스요소, 보요소, 판요소, Link요소에대한 최대최소 설계치(Design Quantities)를 계산하여 출력하며, 보요소에 대해서는보 부재력중 축력과 강약축에 대한 모멘트의 각 성분별 최대최소치 뿐 아니라 해당 최대최소부재력과 동시에 발생하는 기타 내력성분도 산출합니다. + +midas Civil은 입력된 다수의 차선과 차선면에 대해 양방향 주행조건, 편심비틀림조건을 포함한 모든 재하가능 조건과 각 지간(Span)별로 입력된 충격계수(ImpactFraction)를 고려하여 차량하중을 재하하게 되며, 재하되는 모든 차량하중(차륜, 차선, 기타)에 대해 가장 불리한 조건에 대한 해석결과를 산출합니다. + +해석모델에 트러스요소, 보요소, 판요소 이외의 요소(평면응력요소, 입체요소 등)가포함되면 이 요소들은 구조물의 해석시 강성효과에만 반영되고 차량하중에 의한부재력은 산출되지 않습니다. + +이와 같은 제약은 차량하중이 재하되면 프로그램 내부에서 하중재하점의 개수만큼에 해당하는 하중조건이 발생하게 되고, 이를 해석적으로 소화하기 위해서는 해석소요시간에 대한 문제뿐 아니라 대용량의 데이터에 따른 문제를 해소하기 위한 것입니다. diff --git a/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_051.md b/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_051.md new file mode 100644 index 00000000..2384544c --- /dev/null +++ b/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_051.md @@ -0,0 +1,1849 @@ + + +midas Civil에서 이동하중해석을 수행하기 위한 절차는 다음과 같습니다. + +1. 구조물을 모델링 합니다. 이 때 차선을 배치하고자 하는 차선요소는 보요소(변단면 보요소)를 사용해야 하고, 차선면을 배치하고자 하는 차선면요소는 판요소를 사용하여 모델링 하여야 합니다. +2. 구조물 모델상에 차량이동경로 및 설계차선수, 차선폭 등을 고려하여 차선이나 차선면을 배치합니다. +3. 연속교일때 내부지지점의 위치를 지정합니다. 이 과정은 한국 도로교설계 기준과 AASHTO Standard에서 정하고 있는 “연속보에서 차선하중으로 최대 부모멘트를 구할때는 고려하는 지점의 좌우 두 시간의 분포차선 하중과 두 시간에 가장 불리한 위치에 같은 크기의 집중하중을 각각 두어야 하며 …” 조건을 고려하기 위한 것입니다. +4. 차선이나 차선면상에 재하할 차량하중을 정의합니다. 차량하중은 한국 도로교설계기준 또는 AASHTO Standard 등에서 정하고 있는 표준차량하중을 사용할 수 있고, 별도로 사용자가 원하는 차륜하중 또는 차선하중을 만들어서 입력할 수도 있습니다. +5. 요구되는 설계조건을 고려하여 차량하중을 재하 하고자 하는 차선 또는 차선면과 재하조건을 입력합니다. +6. 해석을 수행합니다. +7. 차량하중조건과 기타 정적 동적하중조건에 대한 해석결과를 조합합니다. + +영향선이나 영향면만을 구할 경우에는 두번째 단계까지만 필요하며, 세번째 단계인 지지점의 입력단계는 최대 부모멘트 연산용 집중차선하중을 지점을 기준으로 좌우 두 시간에 동시에 같은 크기로 재하하는 경우에만 사용되기 때문에 설계시 그러한 조건을 고려하지 않을 경우에는 별도로 입력할 필요가 없습니다. + +# 13-1 차선과 차선면 + +midas Civil에서 차량하중은 전체좌표계 Z축의 반대방향으로 입력되기 때문에 구조물의 모델은 증력방향이 전체좌표계 -Z축의 방향이 되도록 입력되어야 합니다. + +차량하중은 지정된 차선(Traffic Lane)이나 차선면(Traffic Surface Lane)을 통하여 구 + + + +조물에 재하 됩니다. 그리고 차선이나 차선면은 설계기준에서 요구하는 설계차선수 및 설계차선폭을 고려하여 교축직각방향으로 여러 개를 배치할 수 있습니다.일반적인 경우에 차선간의 간격이나 차선과 차선요소간의 간격은 상호 평행하게배치되지만 곡선형 입체교차로에서와 같이 2개 이상의 도로가 만나는 교차구간에대해서 반드시 평행되게 배치할 필요는 없습니다. + +midas Civil을 이용하여 구조물을 모델링할 때 상부구조를 단일차선 요소열로 모델링할 수도 있고, 필요에 따라 격자형 모델을 이용할 경우에는 각 종형을 여러 개의 차선요소열로 모델링할 수도 있습니다. 또한 슬래브나 라멘 형태의 교량인 경우에는 판요소를 사용하여 모델링 하면 됩니다. + +# 13-1-1 차선 (Traffic Lane) + +영향선해석에서 차선은 그림 2.13.1과 같이 한 개의 보요소(이하 변단면 보요소 포함) 또는 상호 연결 입력되어 있는 일련의 보요소열 상에 놓이거나 보요소열과 편심을 이루면서 배치됩니다. 여기서 차선을 배치하는데 사용된 보요소열을 차선요소열이라 합니다. + +차선요소열에서 임의 요소의 j (또는 N2)절점의 위치는 반드시 후행하는 요소의 i(또는 N1)절점으로 사용되거나 동일위치에 있도록 배치되어야 하며, 불가피한 경우 비록 위치가 일치하지 않더라도 동일 차선요소열의 인접 차선요소간의 길이방향 이격거리를 최소화하는 것이 해석결과의 정확도 확보에 유리합니다. 가령 일정간격을 가진 2개 이상의 차륜집중하중이 배치될 때 차륜간격보다 인접 차선요소간의 이격거리가 클 경우에는 일부 차륜집중하중을 고려할 수 없게 됩니다. 그러나차선의 폭방향 및 수직방향 이격거리는 크게 영향을 미치지 않습니다. 차선요소의요소좌표계 z축은 전체좌표계 Z축과 평행하거나 평행에 근접하도록 배치되어야 하며, 차선요소의 요소좌표계 x축은 전체좌표계 Z축에 평행하게 배치될 수 없습니다. + +영향선에서 모든 차량하중은 차선의 중심선에 재하되어 차선요소로 전달됩니다.그리고 영향선을 구하기 위해 차선과 차선요소의 위치가 일치하면 단위수직하중만차선을 통해 차선요소로 입력되고, 차선의 위치가 차선요소와 횡방향으로 편심배치되어 있는 경우에는 단위수직하중과 추가로 단위비틀림 모멘트가 차선요소에 입력됩니다. + + + +차선과 차선요소 사이의 편심량(Eccentricity)은 차선요소의 중심에서 요소좌표계(+)y축 방향(또는 (-)y축 방향)으로 차선위치까지의 수직거리입니다. 그리고 편심량의 부호는 해당차선의 하중이 요소좌표계 x축에 대해 정방향의 비틀림을 유발하게되는 경우, 즉 해당차선이 요소좌표계 x축을 기준으로 (-)y축 방향쪽으로 위치하면양(+)이 되고, (+)y축 방향쪽에 위치하면 음(-)이 됩니다. (“보요소” 참조) + +![](images/page-503_d78540bb1c1ea41403f5ffb1cc7851ab0fcb333118b5d2bbfcb198f1062068fe.jpg) + +
+text_image + +traffic lane (centerline) with negative eccentricity +ECC(-) +ECC(+) +i +E1 +j +i +E2 +j +i +traffic lane (centerline) +with positive eccentricity +traffic lane +element +Y +Z +X +Ee-1 +j +En +i +j +En +Y +Z +X +
+ +(a) 차선요소와 차선의 배치개념도 +![](images/page-503_c0308b244fd3abd2e16365d997519de04d274ec0ba30aeb92ec11224f35e8cac.jpg) + +
+text_image + +z +y +vehicle load +i +traffic lane element +ECC (positive) +j +positive torsion +about ECS x-axis +x +z +y +vehicle load +i +traffic lane element +ECC (negative) +j +negative torsion about +ESC x-axis +x +
+ +(b) 편심량의 부호 +그림 2.13.1 차선, 차선요소 및 편심량의 상관관계 + + + +편심량은 차선요소 별로 각각 별도로 주어질 수 있기 때문에 차선이 차선요소 열을 따라가면서 가변적으로 배치될 수 있습니다. + +그림 2.13.1와 같이 차선이 입력되면 영향선을 구하기 위해 차선요소에 단위 수직하중과 단위 비틀림모멘트(편심이 주어졌을 경우)를 작용시키게 되는데, 이때 하중작용점의 위치는 차선요소의 양단부 절점과 요소길이의 1/4, 1/2, 3/4 위치로 프로그램 내부에서 자동 설정됩니다. 그리고 이 하중은 i쪽 절점부터 j쪽 절점으로 진행하면서 순차적으로 가해집니다. + +차량이동하중을 고려할 때 모든 해석결과치의 정확도는 상기의 하중작용점(보요소의 양절점과 길이를 4등분한 점)의 간격에 직접적인 영향을 받기 때문에 보다 정밀한 해석을 요하는 부위나 차선요소(보요소)의 길이가 너무 긴 경우 요소를 보다세분하는 것이 바람직합니다. + +# 13-1-2 차선면 (Traffic Surface Lane) + +차선면(Traffic Surface Lane)은 이동하중의 2방향 분포효과가 큰 라멘이나 슬래브구조물에서 차량이 이동하는 영역을 지정하는데 사용되며, 차선면요소와 차선절점열로 구성됩니다. 차선면은 구조물의 영향면해석과 이를 사용한 차량이동하중해석에 사용되며, 그림 2.13.2와 같이 일련의 차선면요소와 차선절점열로 구분됩니다. + +영향면이란 하중의 재하가 2방향 분포를 갖는 구조물에서 하중재하가 가능한 모든위치에 단위하중을 작용하여 발생하는 결과값들을 하중재하의 위치에 나타낸 것으로, 특정한 성분(변위, 반력, 부재력 등)에 대한 하중의 재하위치별 영향을 의미하며, 이동하중해석에 필수적으로 사용됩니다. 이러한 영향면을 구하기 위한 해석과정을 영향면 해석이라 하고, 이에 필요한 입력은 하중의 재하가 가능한 영역이 됩니다. midas Civil에서는 차량이 직접 이동하는 차선면과 사용자가 필요한 영역에대해 추가적으로 입력된 판요소를 영향면 해석을 위한 하중재하영역으로 간주합니다. midas Civil은 영향면영역으로 입력한 모든 판요소의 각 절점에 각각 단위 연직하중을 가하여 정적해석을 수행하고, 이 해석결과로부터 각 성분별(변위, 반력, 부재력) 영향면을 구합니다. + + + +차선면요소는 차량이 이동하는 차선면 영역을 지정하는 것으로 차선폭, 차선절점열과 편심거리로 입력되며 중복입력이 가능합니다. 또한 각 요소별로 경간길이에따른 충격계수의 입력이 가능하고 차선하중(DL)의 분포압력하중 재하에 사용됩니다. + +![](images/page-505_f8f984e985cafdd40245402b2213289dc9b8cdeda39fc24e42bcb75fef7316b3.jpg) + +
+text_image + +range of loading effect in +influence surface analysis +traffic lane node +axis of bridge +centerline of traffic surface lane +eccentricity +width of traffic +lane +traffic surface lane element +• • • • +• • • • +• • • • +• • • • +• • • • +• • • • +• • • • +• • • • +• • • • +• • • • +• • • • +• • • • +• • • • +• • • • +• • • • +• • • • +• • • • +• • • • +• • • • +• • • • +• • •• +• • • • +• • • • +• • • • +• • • • +• • • • +• • • • +• • • • +• • • • +• • • • +• • • • +• • • • +• • • • +• • • • +• • • • +• • • • +• • • • +• • • • +• • • • +• • • • +• • • +• • • • +• • • • +• • • • +• • • • +• • • • +• • • • +• • • • +• • • • +• • • • +• • • • +• • • • +• • • • +• • • • +• • • • +• • • • +• • • • +• • • • +• • • • +• • • • +•• • • • +• • • • • +• • • • • +• • • • • +• • • • • +• • • • • +• • • • • +• • • • • +• • • • • +• • • • • +• • • • • +• • • • • +• • • • • +• • • • • +• • • • • +• • • • • +• • • • • +• • • • • +
+ +그림 2.13.2 차선면의 차선면요소와 차선절점열 + +차선절점열과 편심거리는 집중차량하중이 이동하는 선을 구성합니다. 차선면의 중심이 교축방향을 기준으로 차선절점열의 오른쪽에 있으면 양(+)의 편심거리를 갖게 되고, 차선절점열의 왼쪽에 놓이게 되면 음(-)의 편심거리를 갖게 됩니다. 차선절점열은 차선면내의 절점으로 입력되며, 입력순서로 차량진행방향을 결정하기 때문에 반드시 입력순서를 고려하여야 하고, 중복입력은 허용하지 않습니다. 각 절점별로 경간 길이에 따른 충격계수의 입력이 가능합니다. 차선절점열은 편심거리와차선폭을 사용하여 차선면을 구성하는데 기준이 되므로, 가능한한 차선면의 중심에 가까운 절점열을 사용하는 것이 바람직합니다. + +지점과 인접한 요소를 입력하여 설계기준에서 요구하는 차선하중(DL)의 추가적인부(-)모멘트 계산이 가능합니다 + +midas Civil은 차선 또는 차선면이 입력되면 상기의 과정을 통해 다음과 같은 5가지 종류의 설계 변수에 대해 영향선 또는 영향면을 산출합니다. + + + +![](images/page-506_386c3c65b0095ecddc41a823445750b298908fb3ec294dfdc79769bb6cf7c45c.jpg) + +
+text_image + +MAX. 0.6277 Cable +Pylon +MIN. -0.3723 +
+ +(a) 점 A 위치의 전단력 영향선도 +![](images/page-506_404d9b230d6dc8f9d4ddcb9f4f3688d85465fa77d214952b041d0d39313a2439.jpg) + +
+area + +| Point | Value | +|-------|-----------| +| A | 20.7436 | +| MIN | -2.4523 | +
+ +(b) 점 A 위치의 굽힘모멘트 영향선도 + +![](images/page-506_487c71e7c7083b01b456a1559606da467e0949d226ba8364e4bbc424fbb57f03.jpg) + +
+text_image + +MIN. 0 B +MAX. -0.0001288 +
+ +(c) 점 B 위치에서의 수직변위 영향선도 +![](images/page-506_a603c5c65c1f2c83ea1a7ee2afb551315f3effe3be30cfd25682e30b5e3d6116.jpg) + +
+area + +| Point | Value | +|-------|--------| +| MAX | 0.8933 | +
+ +(d) 지지점 C 위치에서의 수직반력 영향선도 +그림 2.13.3 3개의 지지점을 가진 사장교의 각 성분별 영향선도 + + + +![](images/page-507_266242620fc8ed998e4e1679dea8f4195a41be0b438c5de16a289a4a7af5a164.jpg) + +
+text_image + +MIDAS/Ctrl +View +Structure +Node/Element +Properties +Boundary +Load +Analysis +Results +PSC +Pushover +MDCS +Query +Tools +Help +Load +Combination +Reactions +Stresses +Deformations +Diagram +Hy Results +Reduction Moment +Mode Shapes +Local Damping Ratio... +Modal Results of RS +Mode shape +Influ. Lines +Influ. Surfaces +Moving Tracer +T.H Results +T.H Graph/Text +Stage/Step Graph +Cable Control +Camber/Reaction +Tendon Loss Graph +Bridge Order +Diagram +Text +Output +Results +Tables +Combination +Results +Detail +Mode shape +Moving Load +Time History +Bridge +Text +Tables +Tree Menu +Inf: L... +Inf: Sw... +MVL T... +Batch... +Displacements +Line/Surface Lanes +LANE all +Key Node: +706 +Components +DX +DY +DZ +RX +RY +RZ +Type of Display +Contour +Legend +3D Cont +Values +Animate +Include Impact factor +Write to File... +Apply +Close +Apply +Close +MIDAS/Ctrl1 +POST-PROCESSOR +INFUS: SURFACE +2-DIRECTION +2.39440e-005 +1.54901e-005 +7.03552e-006 +0.00000e+000 +-9.87427e-006 +-1.83291e-005 +-2.67839e-005 +-3.52386e-005 +-4.36934e-005 +-5.21482e-005 +-6.06030e-005 +-6.90576e-005 +LANE all +MAX: None +MIN: None +FLUX: slab +UNIT: m +DATE: 04/04/2012 +VIEW-DIRECTION +X1=0.738 +Y1=0.470 +Z1=0.485 +Start Page +Model View/ +Message Window +Command Message +Analysis Message +Tree Menu +Task Pane +For Help, press F1 +Name +U: 6, 6, 0 +G: 6, 6, 0 +kN +m +non +2 +
+ +(a) 좌측 경간 중앙 절점의 변위(Dz)에 대한 영향선도 + +![](images/page-507_3c2befe4deb591904f6e62dc3d02a5e92396a0880b6f99f8d4a5afab448648d7.jpg) + +
+text_image + +MIDAS/CVI +View +Structure +Node/Element +Properties +Boundary +Load +Analysis +Results +PSC +Pushover +MODS +Query +Tools +Help +Load +Combination +Reactions +Stresses +Deformations +Diagram +Local Direction.. +HY Results +Reduction Moment +Mode Shapes +Modal Damping Ratio.. +Nodal Results of RS +Influ. Lines +Influ. Surfaces +Moving Tracer +T.H Results +T.H Graph/Text +Stage/Step Graph +Cable Control +Camber/Reaction +Tendon Loss Graph +Bridge Order +Diagram +Text +Output +Results +Tables +Combination +Results +Detail +Mode shape +Moving Load +Time History +Bridge +Text +Tables +Tree Menu +Inf. L... +Inf. S... +MVL T... +Batch... +Reactions +Line/Surface Lanes +LANE all +Key Node: +727 +Components +FX +FY +FZ +MX +MY +MZ +Local (if defined) +Type of Display +Contour +Legend +3D Cont +Values +Animate +Include Impact factor +Write to File... +Apply +Close +Base +MIDAS/C1=11 +POST-PROCESSOR +INFLU +SURFACE +FORCE-Z +1.000000e+000 +9.08781e-001 +8.17562e-001 +7.26343e-001 +6.35124e-001 +5.43904e-001 +4.52685e-001 +3.61466e-001 +2.70247e-001 +1.79028e-001 +0.00000e+000 +-3.41026e-003 +LANT +a11 +MAX : None +MIN : None +FILK : elab +UNIT: IN +DATE: 04/04/2012 +VIEW-DIRECTION +11-0.738 +Y1-0.479 +Z1: 0.465 +Start Page +Message Window +Command Message +Analysis Message +For Help, press F1 +Node-B +U: 0, 9.5, 0 +G: 0, 9.5, 0 +IN +m +non +0 +2 +3 +
+ +(b) 중앙교각부의 중앙 절점 반력(Fz)에 대한 영향면도 + + + +![](images/page-508_5c501454e7759bea4b6a65c22f82ac6d370d221388799eef22d4baee5d100c1e.jpg) + +
+text_image + +MODA/CWI +View +Structure +Node/Element +Properties +Boundary +Load +Analysis +Results +PSC +Pushover +MODS +Query +Tools +Help +Load +Combination +Reactions +Stresses +Deformations +Diagram +Hy Results +Reduction Moment +Mode Shapes +Local Damping Ratio... +Modal Results of RS +Influ. Lines +Influ. Surfaces +Moving Tracer +T.H Results +T.H Graph/Text +Stage/Step Graph +Cable Control +Camber/Reaction +Tendon Loss Graph +Bridge Order +Diagram +Text +Output +Tables +Results +Tables +Tree Menu +Inf1 +Inf2 +MVL T... +Batch... +Plate Forces/Moments +Line/Surface Lanes +LANE all +Key Element: +727 +Parts +cent +k +Components +Fox +Mox +Vox +Fxy +Mxy +Fxy +Type of Display +Contour +3D Cont +Values +Include Impact factor +Legend +Apply +Close +Type +Move to File... +Apply +Close +RIDAS/CVI1 +POST-PROCESSOR +INPUT: SURFACE +RIGHT-Box +3.70168e-001 +3.33552e-001 +2.96936e-001 +2.60320e-001 +2.23703e-001 +1.87087e-001 +1.50471e-001 +1.13855e-001 +7.72385e-002 +4.06223e-002 +0.00000e+000 +-3.26102e-002 +LAME all +MAX: None +MIN: None +FILE: slab +UNIT: KM-a/a +DATE: 04/04/2012 +VIEW-DIRECTION +X1=0.738 +Y1=0.470 +Z1=0.485 +Start Page +Message Window +Command Message +Analysis Message +Tree Menu +Task Pane +Plate-1021 +U: 0, 1, 5, 0 +G: 0, 1, 5, 0 +k +m +non +/ / 2 +
+ +(c) 좌측 경간 중앙부 판요소의 모멘트(Mxx)에 대한 영향면도 + +![](images/page-508_addab9d1b8e537afab12d008849b350e7a131ab2b0d5ed59b055214f1acceb82.jpg) + +
+text_image + +MIDAS/Ctrl +View +Structure +Node/Element +Properties +Boundary +Load +Analysis +Results +PSC +Pushover +MODS +Query +Tools +Help +Load +Combination +Reactions +Stresses +Deformations +Diagram +Hy Results +Reduction Moment +Mode Shapes +Local Damping Ratio... +Modal Results of RS +Mode shape +Influ. Lines +Influ. Surfaces +Moving Tracer +T.H Results +T.H Graph/Text +Stage/Step Graph +Cable Control +Camber/Reaction +Tendon Loss Graph +Bridge Order +Diagram +Text +Output +Results +Tables +Combination +Results +Detail +Mode shape +Moving Load +Time History +Bridge +Text +Tables +Tree Menu +Inf. L... +Inf. S... +MVL T... +Batch... +Plate Forces/Moments +Line/Surface Lanes +LANE all +Key Element: +Y27 +Parts +cent +i +k +Components +Fox +Fyy +Fxy +Mox +Myy +Mxy +Vox +Vyy +Type of Display +Contour +Legend +3D Cont +Values +Animate +Include Impact factor +Write to File... +Apply +Close +MIDAS/Ctrl+1 +POST-PROCESSOR +INFLU +SURFACE +SHEAR-Vox +3.19645e-001 +2.57196e-001 +1.94748e-001 +1.32299e-001 +6.98504e-002 +0.00000e+000 +-5.50468e-002 +-1.17495e-002 +-1.79944e-002 +-2.42393e-001 +-3.04841e-001 +-3.67290e-001 +LANT +a11 +MAX : None +MIN : None +FILIX: elab +UNIT: kW/m +DATE: 04/04/2012 +VIEW-DIRECTION +31-0.738 +Y1-0.470 +Z1: 0.465 +Start Page +Model View/ +Message Window +Command Message +Analysis Message +Help, press F1 +Path-1081 +U: 0, 1, 5, 0 +G: 0, 1, 5, 0 +IN +m +non +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +12 +13 +14 +15 +16 +17 +18 +19 +20 +21 +22 +23 +24 +25 +26 +27 +28 +29 +30 +31 +32 +33 +34 +35 +36 +37 +38 +39 +40 +41 +42 +43 +44 +45 +46 +47 +48 +49 +50 +51 +52 +53 +54 +55 +56 +57 +58 +59 +60 +61 +62 +63 +64 +65 +66 +67 +68 +69 +70 +71 +72 +73 +74 +75 +76 +77 +78 +79 +80 +81 +82 +83 +84 +85 +86 +87 +88 +89 +90 +91 +92 +93 +94 +95 +96 +97 +98 +99 +100 +101 +102 +103 +104 +105 +106 +107 +108 +109 +110 +111 +112 +113 +114 +115 +116 +117 +118 +119 +120 +121 +122 +123 +124 +125 +126 +127 +128 +129 +130 +131 +132 +133 +134 +135 +136 +137 +138 +139 +140 +141 +142 +143 +144 +145 +146 +147 +148 +149 +150 +151 +152 +153 +154 +155 +156 +157 +158 +159 +160 +161 +162 +163 +164 +165 +166 +167 +168 +169 +170 +171 +172 +173 +174 +175 +176 +177 +178 +179 +180 +181 +182 +183 +184 +185 +186 +187 +188 +189 +190 +191 +192 +193 +194 +195 +196 +197 +198 +199 +200 +201 +202 +203 +204 +205 +206 +207 +208 +209 +210 +211 +212 +213 +214 +215 +216 +217 +218 +219 +220 +221 +222 +223 +224 +225 +226 +227 +228 +229 +230 +231 +232 +233 +234 +235 +236 +237 +238 +239 +240 +241 +242 +243 +244 +245 +246 +247 +248 +249 +250 +251 +252 +253 +254 +255 +256 +257 +258 +259 +260 +261 +262 +263 +264 +265 +266 +267 +268 +269 +270 +271 +272 +273 +274 +275 +276 +277 +278 +279 +280 +281 +282 +283 +284 +285 +286 +287 +288 +289 +290 +291 +292 +293 +294 +295 +296 +297 +298 +299 +300 +301 +302 +303 +304 +305 +306 +307 +308 +309 +310 +311 +312 +313 +314 +315 +316 +317 +318 +319 +320 +321 +322 +323 +324 +325 +326 +327 +328 +329 +330 +331 +332 +333 +334 +335 +336 +337 +338 +339 +340 +341 +342 +343 +344 +345 +346 +347 +348 +349 +350 +351 +352 +353 +354 +355 +356 +357 +358 +359 +360 +361 +362 +363 +364 +365 +366 +367 +368 +369 +370 +371 +372 +373 +374 +375 +376 +377 +378 +379 +380 +381 +382 +383 +384 +385 +386 +387 +388 +389 +390 +391 +392 +393 +394 +395 +396 +397 +398 +399 +400 +401 +402 +403 +404 +405 +406 +407 +408 +409 +410 +411 +412 +413 +414 +415 +416 +417 +418 +419 +420 +421 +422 +423 +424 +425 +426 +427 +428 +429 +430 +431 +432 +433 +434 +435 +436 +437 +438 +439 +440 +441 +442 +443 +444 +445 +446 +447 +448 +449 +450 +451 +452 +453 +454 +455 +456 +457 +458 +459 +460 +461 +462 +463 +464 +465 +466 +467 +468 +469 +470 +471 +472 +473 +474 +475 +476 +477 +478 +479 +480 +481 +482 +483 +484 +485 +486 +487 +488 +489 +490 +491 +492 +493 +494 +495 +496 +497 +498 +499 +500 +501 +502 +503 +504 +505 +506 +507 +508 +509 +510 +511 +512 +513 +514 +515 +516 +517 +518 +519 +520 +521 +522 +523 +524 +525 +526 +527 +528 +529 +530 +531 +532 +533 +534 +535 +536 +537 +538 +539 +540 +541 +542 +543 +544 +545 +546 +547 +548 +549 +550 +551 +552 +553 +554 +555 +556 +557 +558 +559 +560 +561 +562 +563 +564 +565 +566 +567 +568 +569 +570 +571 +572 +573 +574 +575 +576 +577 +578 +579 +580 +581 +582 +583 +584 +585 +586 +587 +588 +589 +590 +591 +592 +593 +594 +595 +596 +597 +598 +599 +600 +601 +602 +603 +604 +605 +606 +607 +608 +609 +610 +611 +612 +613 +614 +615 +616 +617 +618 +619 +620 +621 +622 +623 +624 +625 +626 +627 +628 +629 +630 +631 +632 +633 +634 +635 +636 +637 +638 +639 +640 +641 +642 +643 +644 +645 +646 +647 +648 +649 +650 +651 +652 +653 +654 +655 +656 +657 +658 +659 +660 +661 +662 +663 +664 +665 +666 +667 +668 +669 +670 +671 +672 +673 +674 +675 +676 +677 +678 +679 +680 +681 +682 +683 +684 +685 +686 +687 +688 +689 +690 +691 +692 +693 +694 +695 +696 +697 +698 +699 +700 +701 +702 +703 +704 +705 +706 +707 +708 +709 +710 +711 +712 +713 +714 +715 +716 +717 +718 +719 +720 +721 +722 +723 +724 +725 +726 +727 +728 +729 +730 +731 +732 +733 +734 +735 +736 +737 +738 +739 +740 +741 +742 +743 +744 +745 +746 +747 +748 +749 +750 +751 +752 +753 +754 +755 +756 +757 +758 +759 +760 +761 +762 +763 +764 +765 +766 +767 +768 +769 +770 +771 +772 +773 +774 +775 +776 +777 +778 +779 +780 +781 +782 +783 +784 +785 +786 +787 +788 +789 +790 +791 +792 +793 +794 +795 +796 +797 +798 +799 +800 +801 +802 +803 +804 +805 +806 +807 +808 +809 +810 +811 +812 +813 +814 +815 +816 +817 +818 +819 +820 +821 +822 +823 +824 +825 +826 +827 +828 +829 +830 +831 +832 +833 +834 +835 +836 +837 +838 +839 +840 +841 +842 +843 +844 +845 +846 +847 +848 +849 +850 +851 +852 +853 +854 +855 +856 +857 +858 +859 +860 +861 +862 +863 +864 +865 +866 +867 +868 +869 +870 +871 +872 +873 +874 +875 +876 +877 +878 +879 +880 +881 +882 +883 +884 +885 +886 +887 +888 +889 +890 +891 +892 +893 +894 +895 +896 +897 +898 +899 +900 +901 +902 +903 +904 +905 +906 +907 +908 +909 +910 +911 +912 +913 +914 +915 +916 +917 +918 +919 +920 +921 +922 +923 +924 +925 +926 +927 +928 +929 +930 +931 +932 +933 +934 +935 +936 +937 +938 +939 +940 +941 +942 +943 +944 +945 +946 +947 +948 +949 +950 +951 +952 +953 +954 +955 +956 +957 +958 +959 +960 +961 +962 +963 +964 +965 +966 +967 +968 +969 +970 +971 +972 +973 +974 +975 +976 +977 +978 +979 +980 +981 +982 +983 +984 +985 +986 +987 +988 +989 +990 +991 +992 +993 +994 +995 +996 +997 +998 +999 +1000 +Start Page +Model View/ +Message Window +Command Message +Analysis Message +For Help, press F1 +Path-1081 +U: 0, 1, 5, 0 +G: 0, 1, 5, 0 +IN +m +non +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +12 +13 +14 +15 +16 +17 +18 +19 +20 +21 +
+ +(d) 좌측 경간 중앙부 판요소의 전단력(Vxx)에 대한 영향면도 +그림 2.13.4 2경간 슬래브교에 대한 각 성분별 영향면도 + + + +▪ 전체좌표계를 기준으로 모든 절점에서 6개 자유도성분에 대한 변위 영향선 또는 영향면 +▪ 전체좌표계를 기준으로 각 지지점에서 6개 자유도성분에 대한 반력 영향선 또는 영향면 +■ 요소좌표계를 기준으로 모든 트러스요소의 축력에 대한 영향선 또는 영향면 +- 요소좌표계를 기준으로 모든 보요소(또는 변단면 보요소)의 양 절점과 요소의 길이를 4등분한 위치(5개소)에서 6개 내력성분에 대한 영향선 또는 영향면 +- 요소좌표계를 기준으로 모든 판요소의 단위길이에 대한 8개의 부재내력에 대한 영향선 또는 영향면 + +상기의 영향선 또는 영향면은 후처리모드(Post-Processing Mode)를 통해 화면상에 도화 처리되거나 프린터로 출력됩니다. + +midas Civil은 차량이동하중에 의한 구조적 응답을 계산하는데 영향선 또는 영향면을 사용하고, 하중작용점 사이의 구간에 대해서는 선형보간한 값을 사용합니다. + + + +# 13-2 차량이동하중 + +midas Civil에서 차량하중을 입력하는 방법은 다음과 같이 두 가지가 있습니다. + +1. 차륜하중과 차선하중을 사용자가 직접 입력하는 방법 +2. 한국 도로교설계기준, AASHTO, Caltrans Standard 등에서 정하고 있는 각종 표준차량하중을 입력하는 방법 + +첫째 방법은, 차륜하중과 차선하중을 사용자가 직접 입력하는 방법입니다. 차륜의 입력시에는 그림 2.13.5와 같이 집중차륜하중과 차축간의 간격을 지정하게 됩니다. 그리고 최후륜축과 인접한 차축간의 간격이 가변적일 때는 차축간격을 입력하는 마지막항에 가변범위의 최소간격과 최대간격을 동시에 입력합니다. 차선하중은 그림 2.13.6과 같이 위치가변성 차선집중하중과 차선분포하중으로 구성됩니다. 차선집중하중에는 최대최소모멘트 계산용 하중(PLM), 최대최소전단력 계산용 하중(PLV), 그리고 모멘트 및 전단력에 대한 구분없이 모든 해석결과에 적용되는 하중(PL) 등이 있습니다. 차선분포하중은 차선 전체길이에 걸쳐서 작용할 수 있는 것으로 가정되며, 발생가능한 조건중 가장 불리한 설계치를 산출하는 구간에만 프로그램 내부에서 조정, 재하됩니다. 대부분의 설계기준에서는 차륜하중과 차선하중이 동시에 작용하지 않는 것으로 정하고 있으나, midas Civil에서는 사용자의 필요에 따라 동시에 작용하는 조건을 고려할 수도 있습니다. + +둘째 방법은, 각종 설계기준에서 정하고 있는 표준차량하중의 명칭을 지정함으로써 프로그램 내부에 등록되어 있는 데이터베이스로부터 표준차량하중을 자동으로 재하하도록 하는 방법입니다. midas Civil의 차량하중 데이터베이스에 대한 내용은 다음 표와 그림을 참조하기 바랍니다. diff --git a/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_052.md b/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_052.md new file mode 100644 index 00000000..3601b8aa --- /dev/null +++ b/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_052.md @@ -0,0 +1,697 @@ + + +
규 준표준차량하중 명칭
한국 도로교설계기준DB-24, DB-18, DB-13.5, DL-24, DL-18, DL-13.5
한국 철도설계기준L-25, L-22, L-18, L-15, S-25, S-22, S-18, S-15, HL, EL25, EL22, EL18
AASHTO StandardH15-44, HS15-44, H15-44L, HS15-44L, H20-44, HS20-44, H20-44L, HS20-44L, HS-25, HS-25L, AML
AASHTO LRFDHL-93TRK, HL-93TDM, HS20-FTG
Caltrans StandardP5, P7, P9, P11, P13, P15
PENNDOTPHL-93TRK, PHL-93TDM, PHS20-FTG, P-82, ML-80, TK-527
ChinaCH-CD, CH-CL, CH-RQ
IndiaClass A, Class B, Class 70R, Class 40R, Class AA, Footway
TaiwanHS20-44(MS18), HS15-44(MS13.5), H20-44(M18), H15-44(M13.5), H10-44(M9), HS-20-44(MS18), HS-15-44(MS13.5), H-20-44(M18), H-15-44(M13.5), H-10-44(M9), C-AML
CanadaCL-625 Truck, CL-625 Lane
BSHA & HB
EurocodeLoad Model 1~4, Fatigue Load Model
RussiaSK, SK Fatigue, AK, N14, N11, Subway Trains, Trancars, NK-80, NG-60
기타차량하중CE-80(Cooper E80 Train Load)UIC80(UIC80 Train Load)
+ +표 2.13.2 각 규준별 표준차량하중 명칭 + + + +P# : Concentrated wheel load +![](images/page-512_c9bd69a17bfad337002f51e01ba2316fb0f93b457e2b5fc6fae0d5b77b4cfd28.jpg) + +
+text_image + +P1 P2 P3 Pn-1 Pn +traffic lane +From minimum Dn-1 to +D1 D2 +
+ +그림 2.13.5 집중차륜하중의 정의방법 + +![](images/page-512_90d7c7833e9a52a345b7404cc361c6ce4fb45d5197a668e0258e389051951366.jpg) + +
+flowchart + +```mermaid +graph TD + A["traffic lane"] --> B["concentrated traffic lane load"] + B --> C["variable location"] + C --> D["PL"] + D --> E["variable location"] + E --> F["PLM"] + F --> G["variable location"] + G --> H["PLV"] + H --> I["W distributed traffic lane load"] + I --> J["∞"] +``` +
+ +Influence Line + +![](images/page-512_60ae8e67ff4cb417d2b1ac252bd839a11b67aa4cea3ec69016d6bc158a92e8df.jpg) + +
+text_image + +concentrated traffic +lane load +variable location +PL +variable location +PLM +variable location +PLV +W distributed +traffic lane load +W width of +traffic lane +centerline of traffic surface +traffic surface lane +∞ +
+ +(b) influence Surface +그림 2.13.6 차선하중의 정의방법 + + + +![](images/page-513_5c5cfcf123fff939859352ae37ae07390b9846fc41d27006a847f62dcb168f76.jpg) + +
+text_image + +P1 +P2 +P3 +4.2 m +4.2m ~ 9.0m +DB load +
+ +P1 : front axle load + +P2 : middle axle load + +P3 : rear axle load + +Ps : concentrated lane load for shear calculation + +Pm : concentrated lane load for moment calculation + +W : distributed lane load(for influence line) + +w : distributed lane load (for influence surface) + +![](images/page-513_e8d303f414b5b9422fc1d1049b8b5fc30aaf51974b6d9e1bd782ef4adf215369.jpg) + +
+text_image + +[ For influence line ] +variable location ←→ Ps +variable location ←→ Pm +W +variable range loading application(∞) +DL load +
+ +![](images/page-513_5803887d5ef3739b19aabc1cddb32bc7b62d52a54e6a23bf9ec0195e6a46407b.jpg) + +
+text_image + +[For influence surface] +variable location ←→ +Ps +Pm +variable location +w +width of traffic lane +variable range of loading application(∞) +DL load +
+ +
Load ClassP1 [kN]P2 [kN]P3 [kN]PS [kN]PM [kN]W [kN/m]w [kN/m2]
DB-24 or DL-244819219215610812.7012.70/3
DB-18 or DL-1836144144117819.509.50/3
DB-13.5 or DL-13.52710810887860807100710/3
+ +그림 2.13.7 한국 도로교설계기준의 DB 및 DL 하중 + + + +![](images/page-514_da68e7e902d18b950704d5ee75f85274cdc312fe7d6a4ce400ef8a1e284cd977.jpg) + +
+text_image + +18 kips +14 +32 kips +
+ +H20-44 Truck load + +![](images/page-514_5d2dead4609ac9ce991cdb2fe6df2e9778b57afe36f4aa486e7516a866eea5c4.jpg) + +
+text_image + +8 kips +14 +32 kips +14' ~ 30' +32 kips +
+ +H20-44 Truck load + +![](images/page-514_0188e4811cd80640b5daa503b09fe5efaf6035d98f90d89912cc5fb3614eb004.jpg) + +
+line + +| Variable Location | Value (kips) | +| ----------------- | ------------ | +| 18 kips | 18 | +| 26 kips | 26 | +| Total | 0.640 | +
+ +![](images/page-514_384ceff6a242bc1caf29a42a120cb1a745d6afd36bc2bf9bbfae73d36479332f.jpg) + +
+text_image + +[For influence surface] +18 kips +26 kips +variable location +variable location +0.064 kips/ft +variable (∞) +
+ +H20-44L or HS20-44L Lane Load + +![](images/page-514_af837d259e4d1e87c3e55746d60d7f503d31e5ae70ac7969f1b0e5d7aaa70458.jpg) + +
+text_image + +24kips +24kips +4' +
+ +AML Load +그림 2.13.8 AASHTO Standard H 또는 HS Vehicle Loads, Alternative Military Load + + + +![](images/page-515_1d76ece60f2f0452cca3687c0871d6cb1c73528595c836dfd64fc76ef13ec171.jpg) + +
+text_image + +26 +48 +48 +18' +P5 +18' +Permit +
+ +![](images/page-515_6622901604e009a512ff18bbd13b72b065f9970e9cfa5a13d6756e29450908fc.jpg) + +
+text_image + +26 +48 +48 +48 +18' +18' +18' +P7 Permit +
+ +![](images/page-515_1f8e7848ec0ce22c3809705515aac673b2556ccb7f2a4c2f115f69a253c28021.jpg) + +
+text_image + +26 +48 +48 +48 +48 +18' +18' +18' +18' +P9 Permit +
+ +![](images/page-515_4cb7ea59794de3e52b579d25c9819e62c9e8230ccd4f09e412bc253bca4cec32.jpg) + +
+other + +| Position | Value | +|---|---| +| 1 | 26 | +| 2 | 48 | +| 3 | 48 | +| 4 | 48 | +| 5 | 48 | +| 6 | 48 | +| 7 | 48 | +| 8 | 18' | +| 9 | 18' | +| 10 | 18' | +| 11 | 18' | +| 12 | 18' | +| 13 | 18' | +
+ +![](images/page-515_1b098c08bf338b2d4c01a04411f0651a0757f45c09ba7c9c6dc27ceb4c584697.jpg) + +
+line + +| Position | Value | +|---|---| +| 1 | 26 | +| 2 | 48 | +| 3 | 48 | +| 4 | 48 | +| 5 | 48 | +| 6 | 48 | +| 7 | 48 | +| 8 | 48 | +| 9 | 48 | +| 10 | 18' | +| 11 | 18' | +| 12 | 18' | +| 13 | 18' | +| 14 | 18' | +| 15 | 18' | +| 16 | 18' | +| 17 | 18' | +| 18 | 18' | +| 19 | 18' | +| 20 | 18' | +| 21 | 18' | +| 22 | 18' | +| 23 | 18' | +| 24 | 18' | +| 25 | 18' | +| 26 | 18' | +| 27 | 18' | +| 28 | 18' | +| 29 | 18' | +| 30 | 18' | +| 31 | 18' | +| 32 | 18' | +| 33 | 18' | +| 34 | 18' | +| 35 | 18' | +| 36 | 18' | +| 37 | 18' | +| 38 | 18' | +| 39 | 18' | +| 40 | 18' | +| 41 | 18' | +| 42 | 18' | +| 43 | 18' | +| 44 | 18' | +| 45 | 18' | +| 46 | 18' | +| 47 | 18' | +| 48 | 18' | +| 49 | 18' | +| 50 | 18' | +| 51 | 18' | +| 52 | 18' | +| 53 | 18' | +| 54 | 18' | +| 55 | 18' | +| 56 | 18' | +| 57 | 18' | +| 58 | 18' | +| 59 | 18' | +| 60 | 18' | +| 61 | 18' | +| 62 | 18' | +| 63 | 18' | +| 64 | 18' | +| 65 | 18' | +| 66 | 18' | +| 67 | 18' | +| 68 | 18' | +| 69 | 18' | +| 70 | 18' | +| 71 | 18' | +| 72 | 18' | +| 73 | 18' | +| 74 | 18' | +| 75 | 18' | +| 76 | 18' | +| 77 | 18' | +| 78 | 18' | +| 79 | 18' | +| 80 | 18' | +| 81 | 18' | +| 82 | 18' | +| 83 | 18' | +| 84 | 18' | +| 85 | 18' | +| 86 | 18' | +| 87 | 18' | +| 88 | 18' | +| 89 | 18' | +| 90 | 18' | +| 91 | 18' | +| 92 | 18' | +| 93 | 18' | +| 94 | 18' | +| 95 | 18' | +| 96 | 18' | +| 97 | 18' | +| 98 | 18' | +| 99 | 18' | +| 100 | 18' | +
+ +![](images/page-515_37c9f3eba5ccdfae191443220c743a0a4549b2b798793fcc4cacdf3c81a9c70b.jpg) +그림 2.13.9 Caltrans Standard Permit Loads + + + +
Class $P_1$ [kN] $P_2$ [kN] $P_3$ [kN] $P_4$ [kN] $P_5$ [kN] $P_6$ [kN]W[kN/m]
L-15751501007515010050
L-18901801209018012060
L-22110220440/3110220440/3220/3
L-25125250500/3125250500/3250/3
+ +(a) 표준열차하중 (L하중) + +![](images/page-516_c5c67e6b9a53e632ba8d4898534db39689c06398dbff31d260537130c12507ca.jpg) + +
+text_image + +P₁ +2 m +P₂ +
+ +
Class $P_{1}$ [kN]
S-151650/9
S-18220
S-222420/9
S-252750/9
+ +(b) 표준열차하중 (S하중) + +![](images/page-516_afcc47e72f4de47e0de5217661e4e28deedbcbfaaa51c3e4da25be3ff8cd4a70.jpg) + +
+bar + +| Position | Load (kN/m) | Distance (m) | +|---|---|---| +| Top Left | 80 | ∞ | +| Top Right | 250 | 0.8 | +| Top Left | 250 | 1.6 | +| Top Right | 250 | 1.6 | +| Top Left | 250 | 1.6 | +| Top Right | 250 | 0.8 | +| Top Left | 80 | ∞ | +| Top Right | 250 | ∞ | +
+ +(c) HL 표준열차하중 (고속철도) +그림 2.13.10 한국 표준열차하중 + + + +그림 2.13.11 는 특수목적 차량의 모델링에 사용할 수 있는 Permit 차량입니다. 차량의 차축의 개수나 바퀴의 개수는 사용자가 임의로 선언할 수 있습니다. Permit차량은 Permit 이동하중 조건을 사용하여 재하할 수 있습니다. 차량의 바퀴의 위치마다 자동으로 내부적인 차선을 만들어 영향선 해석을 수행하고 차량의 진행에 따른 결과를 계산하여 최대/최소값으로 이동하중조건의 결과를 생성합니다. Permit하중은 각 횡방향 차선에 대하여 독립적인 영향선을 계산하고 이를 활용하여 결과를 구합니다. Permit 차량은 항상 일체된 차량으로 가정하기 때문에 일반 차량과달리 영향선의 부호에따라 일부 차축의 하중을 재하하지 않는 경우는 발생하지 않습니다. + +![](images/page-517_8abd5ba9a46d70c3d40dc9dad8ce28182791fb290554f4ca61c72a3e36b8e681.jpg) + +
+text_image + +Center of Vehicle +Eccentricity +Center of Ref. Lane +Axle 1 +Axle 2 +Wheel +Axle 3 +
+ +![](images/page-517_d97af68c055f4f7da75ba0c021cd2e793658c7b0a5e42899c10325352ba7912b.jpg) + +
+flowchart + +```mermaid +graph TD + A["Center of Vehicle"] --> B["P1"] + A --> C["P2"] + B --> D["P1"] + B --> E["P2"] + C --> F["P1"] + C --> G["P2"] + D --> H["P1"] + D --> I["P2"] + E --> J["P1"] + E --> K["P2"] + F --> L["P1"] + F --> M["P2"] + G --> N["P1"] + G --> O["P2"] + H --> P["P1"] + H --> Q["P2"] + I --> R["P1"] + I --> S["P2"] + J --> T["P1"] + J --> U["P2"] + K --> V["P1"] + K --> W["P2"] + L --> X["P1"] + L --> Y["P2"] + M --> Z["P1"] + M --> AA["P2"] + N --> AB["P1"] + N --> AC["P2"] + O --> AD["P1"] + O --> AE["P2"] + P --> AF["P1"] + P --> AG["P2"] + Q --> AH["P1"] + Q --> AI["P2"] + R --> AJ["P1"] + R --> AK["P2"] + S --> AL["P1"] + S --> AM["P2"] + T --> AN["P1"] + T --> AO["P2"] + U --> AP["P1"] + U --> AQ["P2"] + V --> AR["P1"] + V --> AS["P2"] + W --> AT["P1"] + W --> AU["P2"] + X --> AV["P1"] + X --> AW["P2"] + Y --> AX["P1"] + Y --> AY["P2"] + Z --> AZ["P1"] + Z --> BA["P2"] + AA --> BB["P1"] + AA --> BC["P2"] + AB --> BD["P1"] + AB --> BE["P2"] + AC --> BF["P1"] + AC --> BG["P2"] + AD --> BH["P1"] + AD --> BI["P2"] + AE --> BJ["P1"] + AE --> BK["P2"] + AF --> BL["P1"] + AF --> BM["P2"] + AG --> BN["P1"] + AG --> BO["P2"] + AH --> BP["P1"] + AH --> BQ["P2"] + AI --> BR["P1"] + AI --> BS["P2"] + AJ --> BT["P1"] + AJ --> BU["P2"] + AK --> BV["P1"] + AK --> BW["P2"] + AL --> BX["P1"] + AL --> BY["P2"] + AM --> BZ["P1"] + AM --> CA["P2"] + AN --> CB["P1"] + AN --> CC["P2"] + AO --> CD["P1"] + AO --> CE["P2"] + AP --> CF["P1"] + AP --> CG["P2"] + AQ --> CH["P1"] + AQ --> CI["P2"] + AR --> CJ["P1"] + AR --> CK["P2"] + AS --> CL["P1"] + AS --> CM["P2"] + AT --> CN["P1"] + AT --> CO["P2"] + AU --> CP["P1"] + AU --> CQ["P2"] + AV --> CR["P1"] + AV --> CS["P2"] + AW --> CT["P1"] + AW --> CU["P2"] + AX --> CV["P1"] + AX --> CW["P2"] + AU --> CX["P1"] + AU --> CY["P2"] + AV --> CZ["P1"] + AV --> DA["P2"] + AW --> DB["P1"] + AW --> DC["P2"] + AX --> DD["P1"] + AX --> DE["P2"] + AY --> EY["P1"] + AY --> ZY["P2"] + AZY --> ZY + BAY --> ZY + BBY --> ZY + BCY --> ZY + CCY --> ZY + DZY --> ZY + DZY --> ZY + EY --> ZY + EY --> ZY + ZY --> ZY + ZY --> ZY + ZY --> ZY + ZY --> ZY + ZY --> ZY + ZY --> ZY + ZY --> ZY + ZY --> ZY + ZY --> ZY + ZY --> ZY + ZY --> ZY + ZY --> ZY + ZY --> ZY + ZY --> ZY + ZY + ZY + ZY + ZY + ZY + ZY + ZY + ZY + ZY + ZY + ZY + ZY + ZY + ZY + ZY + ZY + ZY + ZY + ZY + ZY + ZY + ZY + ZY + ZY + ZY +``` +
+ +그림 2.13.11 Caltrans Permit Load(User Defined) + + + +상기와 같은 방법을 통해 차량하중이 입력되면 차량하중은 전체좌표계 -Z축 방향으로 재하되고, 계산된 영향선 또는 영향면을 이용하여 지정된 이동하중조건에 따라 최대최소 설계변수(부재내력, 절점변위, 지점반력)를 산출합니다. + +영향선 또는 영향면을 이용하여 최대최소 설계변수를 산출하는 개념은 다음과 같습니다. 차량집중하중의 경우는 임의 위치의 설계변수를 구하기 위하여 해당 영향선 또는 영향면의 값과 차량집중하중의 값을 곱하게 되고, 차선분포하중의 경우는차선 전구간의 영향선(영향면)을 양(+)과 음(-)의 구간 또는 영역으로 구분하여 구분된 영향선 또는 영향면 영역을 해당구간 또는 영역에 대하여 적분한 값과 차선분포하중값을 곱함으로써 최대최소 설계변수를 구하게 됩니다. (그림 2.13.12 참조) + +![](images/page-518_1ee6c91a54effd00c0083695210eeb98f82135c4cdb29f36c6719a642ef23e85.jpg) + +
+text_image + +concentrated vehicle load +I_max +influence line for bending moment at point A +A +maximum positive moment at point A = P × I_max +maximum negative moment at point A = P × I_min +I_min +P +
+ +(a) 집중차량하중(P)의 재하시 최대최소 휨모멘트의 연산방법 + +![](images/page-518_3aced3673c9bd7273490213f18d39465a4606b47166d6a15aed744a157cf1b22.jpg) + +
+text_image + +Influence line for bending moment at point B +distributed lane load +A2 +A1 +A3 +A4 +B +
+ +(b) 차선분포하중(W)의 재하시 최대최소 휨모멘트 연산방법 +그림 2.13.12 집중 및 분포차량하중의 재하시 최대최소 설계변수의 연산방법 + + + +그리고 차선하중에서 지지점이 입력되면 한국 도로교설계기준과 AASHTO Code에서 정하고 있는 “연속보에서 차선하중으로 최대 부모멘트를 구할 때는 고려하는지점의 좌우 두 지간에 등분포 차선하중과 두 지간에서 가장 불리한 위치에 같은크기의 집중하중을 각각 두어야 하며…”의 규정에 따라 최대최소모멘트 계산용 차선집중하중이 재하 됩니다. (그림 2.13.13 참조) + +![](images/page-519_b51369b5602db937ef99cd03f6558ae32157462ca18ff292d0367040d8b5c9f2.jpg) + +
+text_image + +concentrated design lane loads placed at +the maximum points of influence line to +find the maximum negative moment +Pm +Pm +distributed design lane loads +placed over the negative (-) +loading condition +A +B +C +D +E +influence line for +negative moment at +support B +location of maximum negative +moment influence line in span A-B +location of maximum negative moment +influence line in span B-C +
+ +그림 2.13.13 연속보에서 최대부모멘트 연산용 차선하중의 재하방법 + + + +차량의 바퀴간의 횡방향폭을 반영하여 이동하중 해석을 하고자하는 경우에는 차선의 선언시에 Wheel Spacing 을 입력하면 됩니다. Wheel Spacing 을 입력한 경우에는 Wheel Spacing을 고려하여 그림 2.13.14과 같이 영향선을 생성하고, 각 Wheel하중을 재하하여 해석을 수행합니다. Wheel Spacing 이 영인 경우에는 1개의 Line에 대한 영향선을 사용합니다. 입력창에서는 Code에 따라 Default 값을 보여주고있습니다. + +![](images/page-520_30bab4381a9a878787a152890579fa56173d1974777bcaaa9ca2f2ed9e7e9729.jpg) + +
+text_image + +Center +of +Lane +
+ +그림 2.13.14 Wheel Spacing 을 입력한 경우의 차량하중의 재하방법 diff --git a/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_053.md b/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_053.md new file mode 100644 index 00000000..4521a7cc --- /dev/null +++ b/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_053.md @@ -0,0 +1,432 @@ + + +다축의 집중차륜하중이 한 조로 재하될 때 해석작업의 효율성을 고려하여 midasCivil에서 사용할 수 있는 재하방법은 다음과 같습니다 + +1. 이 방법은 다축 집중차륜하중을 구성하고 있는 개개의 집중하중을 순차적으로 차선을 따라 이동하면서 매 하중작용점에 재하하게 되고, 다축의 집중차륜하중 중의 나머지 집중하중이 하중작용점 사이에 위치할 경우에는인접한 두 개의 하중작용점에서의 영향선(또는 영향면) 값을 보간하여 계산하는 방법입니다. 따라서 이 방법은 주어진 영향선(또는 영향면)의 정확도 범위 내에서 정확한 설계치를 산출합니다. 그러나 이 방법은 차선을 따라가면서 모든 하중작용점에 재하하기 때문에 해석소요시간이 길어지는 단점이 있습니다. 사용자지침서에서는 이 방법을 ‘E’(Exact)라고 약칭합니다. (그림 2.13.15 참조) +2. 방법 1에서는 개개의 집중하중을 차선을 따라 이동하면서 매 작용점에 재하하는 반면, 이 방법은 영향선(또는 영향면) 내에서 최대최소 설계변수가발생한 위치에만 재하시키게 되며 나머지는 방법 1과 같습니다. 사용자지침서에서는 이 방법을 ‘Q’(Quick)라고 약칭합니다. (그림 2.13.16 참조) +3. 방법 1에서는 다축 집중차륜하중을 구성하고 있는 개개의 집중하중을 순차적으로 매번 차선을 따라 이동재하하는 반면, 이 방법에서는 1개의 기준차축을 선정하여 매 하중작용점에 재하하고 나머지 차축에 의한 영향은 보간한 값을 사용합니다. 여기서 기준차축은 차량무게중심에서 가장 가까운 위치에 있는 차축으로 midas Civil 내부에서 자동 설정됩니다. 사용자지침서에서는 이 방법을 ‘P’(Pivot)라고 약칭합니다. (그림 2.13.17 참조) + +상기 재하방법중 예비설계 단계에서는 방법 2를 이용하고, 방법 1 또는 3은 최종설계 단계에서 사용하는 것이 효과적입니다. + +또한, 2개 이상의 집중차륜하중이 한 조로 재하될 때, 앞뒤의 하중값 또는 차축간격이 서로 상이해서 비대칭으로 재하될 경우에는 차량의 주행방향에 따라 차량하중이 미치는 영향이 다르기 때문에 양방향 주행효과를 고려하여 차선요소에 재하됩니다. + + + +![](images/page-522_82ab7b3c6633a96b5afd48a86050a709c03e487901bbc072625129299fc11d0a.jpg) + +
+flowchart + +```mermaid +graph TD + A["A moving vehicle load composed of two axle loads ①&②"] --> B["Stage 1"] + B --> C["Starting point of traffic lane"] + C --> D["Stage 2"] + D --> E["Stage 3"] + E --> F["Stage 4"] + F --> G["Last stage"] + + B --> H["Point ①: Axle load ① applied to the starting point of load application"] + B --> I["Point ②: Axle load ② applied to the starting point"] + D --> J["Point ①: Axle load ① applied between the starting point and the 2nd point of loading application"] + D --> K["Point ②: Axle load ② applied between the starting point and the 2nd point of loading application"] + + E --> L["Point ①: Axle load ① applied to the line 2nd point of loading application"] + E --> M["Point ②: Axle load ② applied between the starting point and the 2nd point of loading application"] + + F --> N["Point ①: Axle load ① applied between the 2nd & 3rd points of loading application"] + F --> O["Point ②: Axle load ② applied to the 2nd point of loading application"] + + G --> P["Point ①: Axle load ② applied to the end point"] + G --> Q["Point ②: Axle load ② applied to the end point"] +``` +
+ +그림 2.13.15 ‘E’(Exact) 방법에 따른 집중차륜하중의 재하개념 + + + +A moving vehicle load composed of two axle loads & +![](images/page-523_2ec73a95965663b5b84a81c2d96f4606d8bcb7f5d578308c08a372bc895be4ff.jpg) + +
+flowchart + +```mermaid +graph TD + A["Stage 1"] --> B["influence line for moment"] + B --> C["Location of maximum positive moment"] + C --> D["Node"] + D --> E["Location of maximum positive moment"] + E --> F["Traffic lane"] + C --> G["Axle load ① applied to the location of maximum positive moment"] + C --> H["Axle load ② applied next to the location of maximum positive moment"] +``` +
+ +![](images/page-523_b294e61e4e7d99703e60265b5dc15a81ad02452fcfddcbe291a03e045d085d5a.jpg) + +
+text_image + +Stage 2 +② +① +Axle load ② applied to the location +of maximum positive moment Axle +load ① applied next to the location of +maximum positive moment +
+ +![](images/page-523_03eab31deae6e6cecca2c7c0cb0d9277e98171896a59dd72a58ac59177fe67c5.jpg) + +
+text_image + +Stage 3 +① +② +① applied to the location +of maximum negative moment Axle load +② applied next to the location +of maximum negative moment +
+ +![](images/page-523_334b7561c38c6a9e7cc7f25e9a8546f22654ba494f95ed48e68e23f2c71a165d.jpg) + +
+text_image + +Stage 4 +① +② +① applied to the +location of maximum negative +moment Axle load ① applied +next to the location of maximum +negative +
+ +그림 2.13.16 ‘Q’(Quick) 방법에 따른 집중차륜하중의 재하개념 + + + +A moving vehicle load composed of two axle loads & if axle load  is reference axle + +![](images/page-524_4c5400fd0bad0db30ea670d31ab6da0c7b9a86ecf5bd2abe641d15a88bca33cc.jpg) +그림 2.13.17 ‘P’(Pivot ) 방법에 따른 집중차륜하중의 재하개념 + + + +# 13-3 차량하중의 재하조건 + +교량구조물의 해석시 가장 불리한 조건의 설계변수(부재내력, 변위, 지지점반력 등)를 도출하기 위해서는 모든 차량이동하중 재하조건이 고려되어야 합니다. 즉, 설계차량하중의 그룹과 차선수가 여러 개일 경우, 고려해야할 설계차량하중이 동시에 재하되어야 하는 것인지, 아니면 여러 설계차량하중 그룹 중에서 가장 불리한 조건의 차량하중만 고려되어야 하는지, 그리고 동시에 재하가능한 임의의 차량하중이 여러 차선중 특정 차선에만 재하되는 것인지, 또한 여러 차선에 차량하중을 동시에 재하할 경우 감소율은 동시 재하차선수에 따라 얼마나 적용할 것인지 등 설계변수에 영향을 미치는 모든 조건을 고려하여야 합니다. + +midas Civil는 이와 같은 설계조건을 고려하여 순열조합방법을 통해 모든 발생가능한 조건에 대한 최대최소 설계변수를 산출합니다. + +midas Civil에서 최대최소 설계변수를 산출하는데 필요한 데이터는 다음과 같습니다. + +■ 설계차량하중 그룹과 재하할 차선번호 +▪ 동시에 재하할 수 있는 최소최대차선수 +▪ 동시에 재하할 차선수에 따른 차량하중 감소율 +- Permit Load 인 경우에는 특성에 맞는 입력방법 사용 + +![](images/page-525_29184c0dc0d63001d5dac56884f5dac1b663daf58863a0917a99f6ef376ac3cb.jpg) + +
+line + +| CL Section | Distance (m) | +| ---------- | ------------ | +| E1 | 3.5 | +| E2 | 1.75 | +| E3 | 1.75 | +| E4 | 1.75 | +| E5 | 1.75 | +| E6 | 1.75 | +| E7 | 1.75 | +| E8 | 1.75 | +| E9 | 1.75 | +| E10 | 1.75 | +| E11 | 1.75 | +| E12 | 1.75 | +| E13 | 1.75 | +| E14 | 1.75 | +| E15 | 1.75 | +| E16 | 1.75 | +| CL of 2 | 3.5 | +| CL of 3 | 3.5 | +| CL of 4 | 3.5 | +
+ +(a) 평면도 + +![](images/page-525_e9bc5ae4c44a4f05973163e06e301ec1c95a9f77e53b310ed7b99540cf545d7c.jpg) +(b) 입면도 +그림 2.13.18 구조물 모델 + + + +다음은 midas Civil에서 이동하중 순열조합에 대한 개념을 설명하기 위한 예입니다. + +[ 예제1 ] 한국도로교설계기준의 DB-24, DL-24 하중조건을 고려하여 4개 차선을가진 구조물의 해석 + +# 1. 이동하중으로 “한국도로교설계기준 을 선택합니다. + +Main Menu에서 Load탭>Load Type그룹>Moving Load>Moving LoadCode>Korea를 선택합니다. + +# 2. 차선을 입력합니다. + +Moving Load Analysis Data그룹>Traffic Line Lanes을 선택하면 아래와 같이 Traffic Line Lanes의 입력상황을 보여주는 대화상자가 활성화됩니다.새로운 차선을 정의하기 위해선 아래 그림과 같은 차선입력 대화상자를 호출합니다. Lane Name에 차선 이름을 입력하고, 차선을 이루는 일련의 보요소를 선택한 후 편심거리(Eccentricity)와 충격계수를 입력하여 차선을 정의합니다. + +![](images/page-526_255d0b75069e59480a79cca6b78276fb263538341a4e4ba6cf438a4a9dd0db29.jpg) + +
+text_image + +Traffic Line Lanes +Lane Name +Lane1 +Lane2 +Lane3 +Lane4 +Add +Modify +Delete +Copy +Close +
+ +![](images/page-526_f6303c718fe4a989a1118feb4b9bb327ffa500280c41284abbbbd4bd80e07733.jpg) + +
+text_image + +Define Design Traffic Lin... +Lane Name : Lane1 +Traffic Lane Properties +Start End +a : Eccentricity +Eccentricity : -2,5 m +Wheel Spacing: 1,8 m +Impact Factor : 0,0 +Vehicular Load Distribution +Lane Element Cross Beam +Cross Beam Group +Cross Beam +Skew +Start End 0 [deg] +Moving Direction +Forward Backward Both +Selection by +2 Points Picking Number +0, 6, 1,5 m +10, 0, 3 m +Operations +Add Insert Delete +No Elem Eccen, Impact Span +(m) Factor Start +1 4 -2,5 0 +2 20 -2,5 0 +3 21 -2,5 0 +4 22 -2,5 0 +OK Cancel Apply +
+ +차선입력 대화상자 + + + +# 3. 차량하중을 입력합니다. + +Moving Load Analysis Data그룹>Vehicles의 Add Standard에서 원하는 기준을 선택한 후(여기에서는 Korean Standard Load), 원하는 하중을 선택하여입력합니다. + +![](images/page-527_8acbdcd99eb7427238c47ccc7962a950d5e5a900f5634a050f865c4eca75050b.jpg) + +
+text_image + +Vehicles +Vehicle Name Type +DB-24 Standard +DL-24 Standard +Add Standard +Add User Defined +Modify +Delete +Close +
+ +![](images/page-527_476e8fa51a73371d88b0e19444454ed4133b9da4f8925f81d32b313d3ab0ca1f.jpg) + +
+text_image + +Define Standard Vehicular Load +Standard Name +Korean Standard Load(Specification for Roadway Bridges) +Vehicular Load Properties +Vehicular Load Name : DB-24 +Vehicular Load Type : DB-24 +P1 P2 P3 +D1 D2 +No Load(N) Spacing(m) +1 47071.9 4.2 +2 188288 4.2 +3 188288 9 +W 0 N/m +Ps 0 N +Pm 0 N +dW1 0 N/m +dD1 0 m +dW2 0 N/m +dD2 0 m +OK Cancel Apply +
+ +차량하중의 정의 + +DB-24와 DL-24 중에서 불리한 조건을 고려하기 위해 다음 그림과 같이 동일한 차량하중그룹(Class 1)을 사용합니다. + +![](images/page-527_ffa1515fbd6ba2fab088f3e9c369a7414fb99689889d4809b48082b5fa63ff1b.jpg) + +
+text_image + +Vehicle Classes +Class Name +Class1 +Add +Modify +Delete +Close +
+ +![](images/page-527_ba73b008509de3411cc421e0c29db252399db2979fb6c7f1e339c61a61960b78.jpg) + +
+text_image + +Vehicle Class Data +Vehicle Class Name : Class1 +Vehicle Load +Selected Load +DB-24 +DL-24 +OK Cancel Apply +
+ +차량하중그룹입력 + + + +# 4. 차량하중의 재하방법을 입력합니다. + +Main Menu의 Analysis탭>Analysis Control그룹>Moving Load AnalysisControl 에서 그림과 같이 ‘Exact’을 선택하고 차량하중의 재하방법을지정합니다. + +![](images/page-528_bd87833eb841ef1b3b42f63596d2be25cb3e1c575f992b37e362dbb93612f536.jpg) + +
+text_image + +Moving Load Analysis Control Data +Truck/Train Load Control Option +Analysis Method +● Exact +● Pivot +● Quick +Load Point Selection +● Influence Line Dependent Point +● All Points +Influence Generating Points +● Number/Line Element : 3 +● Distance between Points : 0 m +Analysis Results +Plate +● Center +● Center + Nodal +□ Stress Calculation +Frame +● Normal +● Normal + Concurrent Force +□ Combined Stress +Calculation Filters +● Reactions +● All +● Group : +● Displacements +● All +● Group : +● Forces/Moments +● All +● Group : +OK Cancel +
+ +차량하중의 재하방법 + + + +5. 차량하중이 동시에 재하되는 차선수에 대한 차량하중 감소율을 입력합니다.아래 그림과 같이 동시에 재하되는 차선수가 1에서 4까지 변할 때의 차량하중감소율을 입력합니다. + +![](images/page-529_46024cdf078f6289e37bf47008aadd695632820b151e575c7942691d3c19875c.jpg) + +
+text_image + +Define Moving Load Case +Load Case Name : Moving Load 1 +Description : Moving Load 1 +Load Case for Permit Vehicle +Multi Lane Factor in KS Rail Load +Multiple Presence Factor +Num of Loaded Lanes Scale Factor +1 1 +2 1 +3 0,9 +> 3 0,75 +Sub-Load Cases +Loading Effect +Combined Independent +Vehicle class Scale Lane1 +VC:Class1 1 Lane1 +Add Modify Delete +OK Cancel Apply +
+ +차량하중이 동시에 재하되는 차선수에 대한 차량하중 감소율 + +다음과 같이 차량하중그룹, 재하될 차선, 재하되는 최대ㆍ최소 차선 수를사용하여 차량하중 재하조건을 입력합니다 + +![](images/page-529_a127e81d376ff7818a769d8ecc1aa41657dec7259b34f5f9ec50f0c609e6ceb3.jpg) + +
+text_image + +Sub - Load Case +Load Case Data +Vehicle Class : VC:Class1 +Scale Factor : 1 +Min. Number of Loaded Lanes : 1 +Max. Number of Loaded Lanes : 4 +Assignment Lanes +List of Lanes +Selected Lanes +Lane1 +Lane2 +Lane3 +Lane4 +-> +<- +OK Cancel +
+ +차량하중그룹과 차선을 사용한 차량하중 재하조건의 입력 + +이상의 입력된 조건에 따라 midas Civil에서 자동으로 순열 조합되는 조건의 개수는 아래의 표와 같이 15가지가 되고, 최대 최소설계변수는 이 15가지 조건에 대해 가장 불리한 값으로 각각 산출됩니다 + + + +“DB-24 or DL-24”는 두가지 차량하중조건에 대해불리한 최대•최소 설계변수를 산출한다는 의미이다. + +
순열조합조건번호재하될 차선번호차량하중감소율
#1#2#3#4
1DB-24 or DL-241.0
2DB-24 or DL-241.0
3DB-24 or DL-241.0
4DB-24 or DL-241.0
5DB-24 or DL-24DB-24 or DL-241.0
6DB24 or DL-24DB-24 or DL-241.0
7DB-24 or DL-24DB-24 or DL-241.0
8DB-24 or DL-24DB-24 or DL-241.0
9DB-24 or DL-24DB-24 or DL-241.0
10DB-24 or DL-24DB-24 or DL-241.0
11DB-24 or DL-24DB-24 or DL-24DB-24 or DL-240.9
12DB-24 or DL-24DB-24 or DL-24DB-24 or DL-240.9
13DB-24 or DL-24DB-24 or DL-24DB-24 or DL-240.9
14DB-24 or DL-24DB-24 or DL-24DB-24 or DL-240.9
15DB-24 or DL-24DB-24 or DL-24DB-24 or DL-24DB-24 or DL-240.75
diff --git a/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_054.md b/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_054.md new file mode 100644 index 00000000..fde8bcb0 --- /dev/null +++ b/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_054.md @@ -0,0 +1,285 @@ + + +[ 예제2 ]. “예제1”의 모델을 이용하여 Caltrans Combination Group Ipw에 따라 임의의 1개 차선에 P13 하중과, 나머지 1개 차선에 AASHTO의 HS Vehicle Load를 재하하는 이동하중에 대한 해석 + +1. “예제1”과 같이 차선요소에 차선을 배치합니다. +2. 다음과 같이 차량하중을 입력하고 차량하중그룹을 Class 1(HS20-44, HS20-44L)과 Class 2(P13)로 분류합니다. + +![](images/page-531_8aef28cd7a2088783adc92f884cf0bb9f397c0c484fe995c80110541d98705de.jpg) + +
+text_image + +Vehicles +Vehicle Name Type +HS20-44 Standard +HS20-44L Standard +P13 Standard +Add Standard +Add User Defined +Modify +Delete +Close +
+ +![](images/page-531_04852ae456f3ef09685b532ae4c55fddc61581d690153a432a3f960558001af3.jpg) + +
+text_image + +Vehicle Classes +Class Name +Class1 +Class2 +Add +Modify +Delete +Close +
+ +차량하중과 차량하중그룹의 입력 + +3. 하중재하방법을 “예제 1”과 같이 Exact 방법을 사용합니다 +4. 다음과 같이 차량하중그룹, 재하될 차선, 재하되는 최대ㆍ최소 차선 수를사용하여 차량하중 재하조건을 입력합니다. + +![](images/page-531_f9882313ee1ede9df471d0d957f79fbd4b32f80a9599238aae81a2929f11391a.jpg) + +
+text_image + +Sub - Load Case +Load Case Data +Vehicle Class : VC:Class1 +Scale Factor : 1 +Min. Number of Loaded Lanes : 0 +Max. Number of Loaded Lanes : 4 +Assignment Lanes +List of Lanes +Selected Lanes +Lane1 +Lane2 +Lane3 +Lane4 +-> +<-> +OK Cancel +
+ +![](images/page-531_0ac6e54a4797a0678d30d424e7b071764ceb73c0d82da2c9137eef3c4fbe2d81.jpg) + +
+text_image + +Sub - Load Case +Load Case Data +Vehicle Class : VC:Class2 +Scale Factor : 1 +Min, Number of Loaded Lanes : 1 +Max, Number of Loaded Lanes : 4 +Assignment Lanes +List of Lanes +Selected Lanes +Lane1 +Lane2 +Lane3 +Lane4 +-> +<-> +OK Cancel +
+ +차량하중그룹과 차선을 사용한 차량하중 재하조건의 입력 + + + +5. Caltrans Combination Group Ipw에서는 P13 하중은 반드시 임의 1개 차선에 재하되어야 하고, 나머지 임의 1개 차선에 HS 하중을 재하하도록 정하고 있으며, HS 하중이 재하되지 않는 조건에 대해서도 검토하도록 하고 있습니다. 따라서 2단계에서 HS 하중과 P13 하중에 대해서는 별도의 차량하중그룹을 1, 2로 각각 부여하고, 4단계에서 HS 하중에 대해서는 최대·최소재하가능 차선수를 0, 1로 그리고 P13 하중의 경우는 반드시 임의 1개 차선에 재하되어야 하기 때문에 1, 1로 입력합니다. 이상의 조건에 따라midas Civil에서 자동으로 순열조합되는 조건의 개수는 아래 표와 같이 P13하중만 재하되는 조건 4가지 그리고 P13과 HS 하중이 동시에 재하되는조건 12가지로 재하가능한 하중조건은 모두 16가지가 됩니다. + + + +“HS20-44 or HS20-44L”는 두가지 차량하중에 대해 불리한 최대•최소 설계변수를 산출한다는 의미이다. + +
순열조합 조건번호재하될 차선번호차량하중감소율
#1#2#3#4
1P131.0
2P131.0
3P131.0
4P131.0
5P13HS20-44 or HS20-44L1.0
6P13HS20-44 or HS20-44L1.0
7P13HS20-44 or HS20-44L1.0
8HS20-44 or HS20-44LP131.0
9P13HS20-44 or HS20-44L1.0
10P13HS20-44 or HS20-44L1.0
11HS20-44 or HS20-44LP131.0
12HS20-44 or HS20-44LP131.0
13P13HS20-44 or HS20-44L1.0
14HS20-44 or HS20-44LP131.0
15HS20-44 or HS20-44LP131.0
16HS20-44 or HS20-44LP131.0
+ + + +특수목적 차량인 Permit Vehicle을 사용하여 이동하중조건을 생성하는 경우에는 하중의 특성에 맞는 데이터 입력이 필요하다. midas Civil 에서 Permit Vehicle 하중을 사용하여 하중조건을 입력하는 방법은 아래와 같다. Permit Vehicle의 재하시에는 Wheel Line 마다 독립적인 영향선을 필요하기 때문에 내부적으로 하중형태에따른 차선을 자동생성한다. 이러한 하중특성 때문에 차량, 차선, 그리고 이동하중조건이 일대일 대응되도록 한다. Permit Vehicle의 재하위치를 횡방향으로 이동하기위해서는 추가적인 하중조건을 생성해야 한다. Permit Vehicle은 특정한 차량을 기준으로 정의하기 때문에 영향선의 부호를 고려한 재하는 하지 않습니다. PermitVehicle의 재하는 그림 2.13.15의 Exact 방법을 사용합니다. + +![](images/page-534_b4da31bee1018cad1d1980020d607d8d0f76f093e3406163eac2434be2c5811a.jpg) + +
+text_image + +Define Moving Load Case +Load Case Name : MV1_PERMIT +Description : +Load Case for Permit Vehicle +Permit Vehicle +Vehicle Permit Truck +Ref. Lane Lane1 +Eccentricity 0 m +Scale Factor 1 +OK Cancel Apply +
+ +그림 2.13.19 Permit Vehicle 을 사용한 이동하중조건 데이터 입력 + + + +# Chapter 14. 구조물의 지점침하를 자동 고려한 해석 + +일반구조물에서 지점침하조건을 고려한 해석을 수행하기 위해서는 동시에 침하가발생하는 지점들을 한 개의 침하그룹으로 하고, 각 침하그룹을 단위하중조건으로하여 정적해석을 수행합니다. 그 후 단위하중조건들을 사용하여 침하가 가능한 경우의 수에 대한 조합을 만들고, 그 결과들을 비교하여 최대 최소값을 구하여 이 값들을 최종 설계조건으로 사용하게 됩니다. 이러한 일련의 작업들은 상당한 양의해석, 하중조합, 결과들의 비교를 통한 최대 최소값의 선택 등의 작업을 필요로 합니다. midas Civil은 이와 같은 지점침하조건을 프로그램 내부에서 자동적으로 처리하는 기능을 내장하고 있으며 해석절차는 다음과 같습니다. + +1. 사용자에 의해 사전에 입력된 지점침하가 동시에 발생가능한 지점침하그룹과 침하의 크기를 사용하여 내부적인 단위하중조건을 생성합니다. +2. 각 단위하중조건에 대한 정적해석을 수행합니다. +3. 지점침하가 가능한 모든 경우의 수를 만들고, 그에 따라 해석결과들을 조합하여 최대　최소값을 산출합니다. + +이러한 과정을 거쳐 산출되는 해석결과들은 다른 하중조건의 결과들과 조합이 가능합니다. 해석의 결과로는 절점변위, 지점반력, 트러스, 보, 판요소의 부재내력,Link의 부재력 등이 출력되며, 그 이외에 입력된 요소에 대해서는 강성만 고려되고해석결과는 산출하지 않습니다. + +침하가 가능한 지점은 절점으로 입력되고 침하의 크기는 절점에 따라 다르게 입력할 수 있습니다. 지점침하의 방향은 전체좌표계 Z축 방향을 따라야 하며 지점침하그룹의 개수는 10개 이하로 제한되나, 1개 지점침하 그룹내에 입력되는 절점의 수는 제한이 없습니다. + + + +# Chapter 15. 강합성단면의 합성전후 해석 + +강합성단면의 합성전 후 단면성질의 변화를 고려하기 위해서는 합성전 모델과 합성후 모델에 대한 구조해석을 독립적으로 수행한 다음, 두 모델의 해석결과를 조합하여 설계에 적용하는 과정을 거치게 됩니다. midas Civil은 이와 같은 해석과정을 자동 처리하는 기능을 내장하고 있으며 해석 알고리즘은 다음과 같습니다. + +1. 사용자에 의해 사전에 구분 입력된 합성전 단면성질과 해당조건을 고려하여 정적 해석을 수행하고, 그 해석된 결과를 저장합니다. +2. 합성후 단면특성을 사용하여 합성전 하중조건을 제외한 정적, 동적, 차량이동, 지점 침하 하중조건 등에 대한 해석을 수행하고 그 결과를 저장합니다. + +합성전 하중조건에 사용되는 것은 정적하중 조건이어야 하고 하중조건의 개수는15개 이하로 제한됩니다. + + + +# Chapter 16. 최적화기법을 사용한 미지하중의 해 + +대공간 구조물의 설계과정에는 그림 2.16.1과 같이 주어진 설계조건을 만족하는데필요한 미지의 하중조건을 구하는 문제를 자주 접하게 됩니다. midas Civil에서는이러한 문제들을 해결할 수 있도록 최적화기법을 도입하여 주어진 구속조건과 선택한 목적함수에 대하여 최적의 변수값들을 산정해주는 기능을 내장하고 있습니다.제한조건의 입력은 평형조건(Equality Condition) 및 부등평형조건(InequalityCondition)에 대하여 가능하고, 목적함수(Object Function)의 종류로는 변수의 절대값의 합( ), 변수 제곱의 합( )이 있고 변수들의 절대값의 최대값2iXiX1i ( ( , , ) M ax X X X  )의 선택이 가능합니다. + +그림 2.16.1(a)는 장경간 보의 설계시 요구되는 모멘트의 분포를 인위적으로 만들거나 보에 일정량의 초기변위를 부여하는데 필요한 Jack-up 하중을 구하는 문제입니다. + +그림 2.16.1(b)는 장경간 구조물의 시공단계 해석에서 구조물이 특정 변형형상을가지도록 하는데 필요한 Leveling 하중을 구하는 문제입니다. + +그림 2.16.1(c)는 사장교의 설계시 사하중 또는 활하중조건에 대해 주탑 상부의 횡변위의 크기를 일정한 값 이하로 제한하고, B점과 C점에서의 수직변위가 양(+)의값을 갖도록 제한하는데 필요한 케이블의 인장력을 구하는 문제입니다. + +위와 같은 문제들은 평형조건(Equality Condition)이나 부등평형조건(InequalityCondition)을 만들게 되며 midas Civil에서는 최적화기법을 사용하여 해석을 수행하게 됩니다. + + + +다음은 그림 2.16.1(a)와 같은 구조물에서 평형조건(Equality Condition)을 사용하여A점과 B점에 Jack-up 하중을 구하는 문제의 해석절차입니다. + +1. 구하고자 하는 하중 대신 가상의 단위하중을 가하여 요구되는 미지하중의개수만큼 단위하중조건을 생성합니다. +그림 2.16.1 (a)에서는 Jack-up 하중을 구하고자 하는 지점과 방향으로 단위하중을 가하는 단위하중조건을 생성합니다. +2. 설계하중조건을 고려한 하중조건을 생성하고 해석을 수행합니다. +그림 2.16.1(a)의 경우에는 설계하중인 균일분포하중을 가하는 하중조건을생성하고 정적해석을 수행합니다. +3. 제한조건들을 사용하여 평형조건(Equality Condition)을 구성합니다. + +그림 2.16.1(a)의 경우에는 다음과 같은 평형조건(Equality Condition)이 구성됩니다. + +$$ +M _ {A 1} P _ {1} + M _ {A 2} P _ {2} + M _ {A D} = M _ {A} +$$ + +$$ +M _ {B 1} P _ {1} + M _ {B 2} P _ {2} + M _ {B D} = M _ {B} +$$ + +여기서 + +$M _ { A i }$ : Pi 작용방향으로 단위하중을 가한 단위하중조건의 A점 모멘트 + +$M _ { B i }$ : Pi 작용방향으로 단위하중을 가한 단위하중조건의 B점 모멘트 + +$M _ { A D }$ : 설계하중조건의 A점 모멘트 + +$M _ { { \scriptscriptstyle B D } }$ : 설계하중조건의 B점 모멘트 + +$M _ { A } :$ 설계하중조건과 미지하중 1 2P P, 의 조합에서 가져야하는 A점 모멘트 + +$M _ { B }$ : 설계하중조건과 미지하중 1 2P P, 의 조합에서 가져야하는 B점 모멘트 + +4. 선형대수법을 사용하여 평형조건(Equality Condition)을 만족하는 해를 구합니다. 평형조건(Equality Condition)에서 미지수의 개수와 조건식의 수가 같은 경우에는 행렬식이나 선형대수법을 사용하여 쉽게 해를 구할 수 있습니다. + +$$ +\left\{ \begin{array}{l} P _ {1} \\ P _ {2} \end{array} \right\} = \left[ \begin{array}{c c} M _ {A 1} & M _ {A 2} \\ M _ {B 1} & M _ {B 2} \end{array} \right] ^ {- 1} \left\{ \begin{array}{l} M _ {A} - M _ {A D} \\ M _ {B} - M _ {B D} \end{array} \right\} +$$ + + + +다음은 그림 2.16.1(c)와 같은 구조물에서 부등평형조건(Inequality Condition)을 만족하는 케이블의 인장력을 구하는 문제에 대한 해석절차입니다. + +1. 구하고자 하는 하중 대신 가상의 단위하중을 가하여 요구되는 미지하중의 개수만큼 단위하중조건을 생성합니다. + +그림 2.16.1(c)의 구조물에서는 인장력을 구하고자 하는 케이블 부채에 단위크기의 Pre-tension 하중을 도입하는 단위하중조건을 미지의 인장력 개수 만큼 생성합니다. + +2. 설계하중조건을 고려한 하중조건을 생성하고 해석을 수행합니다. + +그림 2.16.1(c)의 경우에는 설계하중인 균일분포하중을 가하는 하중조건을 생성하고 정적해석을 수행합니다. + +3. 제한조건들을 사용하여 부등평형조건(Inequality Condition)을 구성합니다. +그림 2.16.1(c)의 경우에는 다음과 같은 부등평형조건(Inequality Condition)이 구성됩니다. + +$$ +\delta_ {A 1} T _ {1} + \delta_ {A 2} T _ {2} + \delta_ {A 3} T _ {3} + \delta_ {A D} \leq \delta_ {A} +$$ + +$$ +\delta_ {B 1} T _ {1} + \delta_ {B 2} T _ {2} + \delta_ {B 3} T _ {3} + \delta_ {B D} \geq 0 +$$ + +$$ +\delta_ {C 1} T _ {1} + \delta_ {C 2} T _ {2} + \delta_ {C 3} T _ {3} + \delta_ {C D} \geq 0 +$$ + +$$ +T _ {i} \geq 0 (i = 1, 2, 3) +$$ + +여기서 + +$\delta_{A1}$ : $T_{i}$ 작용방향으로 Pre-tension 을 도입한 단위하중조건의 A점 수평변위 + +$\delta_{B1}$ : $T_{i}$ 작용방향으로 Pre-tension 을 도입한 단위하중조건의 B점 수직변위 + +$\delta_{C1}$ : $T_{i}$ 작용방향으로 Pre-tension 을 도입한 단위하중조건의 C점 수직변위 + +$\delta_{AD}$ : 설계하중조건의 A점 수평변위 + +$\delta_{BD}$ : 설계하중조건의 B점 수직변위 + +$\delta_{CD}$ : 설계하중조건의 B점 수직변위 + +$\delta_{A}$ :설계하중조건과 케이블 인장력 하중의 조합조건에서 가져야 되는 A점 수평변위 + + + +4. 최적화기법을 사용하여 부등평형조건(Inequality Condition)을 만족하는 해를 구합니다. 조건에 따라 부등평형조건(Inequality Condition)을 만족하는미지계수는 많은 해를 가질 수 있습니다. 많은 해들 중에서 필요로 하는해를 선택해야 하는데 midas Civil에서는 목적함수(Object Function)를 최소로 하는 변수를 부등평형조건(Inequality Condition)의 해로 사용하게 됩니다. midas Civil에서의 목적함수(Object Function)는 변수의 선형 합, 변수의 제곱의 합 그리고 변수들의 절대값의 최대값 등 3가지를 선택할 수있습니다. 그리고 특정한 변수에 가중치를 입력하여 변수의 중요도를 조정할 수 있고 변수의 사용 가능한 범위를 입력할 수도 있습니다. + +위에서 설명한 최적화기법을 사용하여 필요로 하는 설계 변수들을 구하는 방법을사용할 경우에는 구조물에 대한 충분한 이해를 필요로 합니다. 평형조건(EqualityCondition)이나 부등평형조건(Inequality Condition)은 경우에 따라 해를 갖지 않을 수도 있기 때문에 적절한 설계조건의 입력 및 목적함수의 선택을 필요로 합니다. + +![](images/page-540_435165be0d057c1d5b1d40c4526f5148791531b335351e1ddbdabc3eea828de2.jpg) + +
+text_image + +Z +Design condition: M_A = M_1 +Unknown design variables: P_1, P_2 +Design load +A +P_1 +B +P_2 +X +
+ +![](images/page-540_c0e0d3299781c8f20d3e4e2b9134df61e0d1376d817843a02bdebc95a717ac40.jpg) + +
+text_image + +Moment diagram +M₁ +M₂ +
+ +(a) 주어진 하중조건에서 A점에서의 모멘트가 M1이 되고, B점에서의 모멘트가 M2가 되는 +Jack-up 하중 P1, P2를 구하고자 할 경우 diff --git a/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_055.md b/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_055.md new file mode 100644 index 00000000..722096be --- /dev/null +++ b/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_055.md @@ -0,0 +1,331 @@ + + +Design conditions : $\delta _ { A Z } = \delta _ { D Z } = \delta _ { G Z }$ + +$$ +R _ {B} = R _ {C} = R _ {\xi} = R _ {f} +$$ + +![](images/page-541_3a011434d482d9c46c5089ce3277e23bc7276db4cef8741e225d60f6c6e7fcf7.jpg) + +
+text_image + +Unknown design variables: P₁, P₂ +Design load +Z +P₂ +A B C D E F G +P₁ P₁ +X +
+ +(b) 주어진 하중조건에서 A, D, G점의 수직변위가 같고, +B, C, F, F에서의 반력이 같게 되는 Leveling 하중 P1, P2를 구하고자 할 경우 + +Design conditions : AX ≤ A + +$$ +\delta_ {\mathrm{BZ}} \geq 0 +$$ + +$$ +\delta_ {\mathrm{CZ}} \geq 0 +$$ + +![](images/page-541_3edab63c71bcbee57b3ada4a98cf1a051a2cfb69834e4fff464a19da8f390e84.jpg) + +
+text_image + +A +Cable +Unknown design variables: T₁, T₂, T₃ +Z +T₁ +T₂ +T₃ +Design load +B +C +X +
+ +(c) 주어진 하중조건에서 A점의 횡력변위가 $\delta _ { A }$ 보다 작고, B, C점에서의수직변위가 0보다 크게 되는 초기케이블인장력 T1, T2, T1을 구하고자 할 경우 +그림 2.16.1 각종 설계조건을 만족하기 위한 미지의 하중조건을 구하는 문제 + + + +# Chapter 17. 입의형상 기둥의 부재설계 + +# 17-1 확대모멘트 계산 + +기둥은 계수축하중 $P_{u}$ 와 확대된 최대모멘트 $M_{c}$ 에 대하여 설계됩니다. 여기서 확대된 최대모멘트 $M_{c}$ 는 장주효과의 근사방법인 모멘트 확대계수 설계법을 사용하여 계산됩니다. 한국 도로교설계기준 (2005, 2010)에서 철근콘크리트 기둥의 확대모멘트 계산방법은 다음과 같습니다. + +# 17-1-1 구조물의 횡구속 여부 판정 + +층안정지수(Q)를 계산하여 구조물의 횡구속 여부를 판정합니다. 구조물의 한 층의 안정지수(Q)가 0.05이하이면 구조물의 그 층은 횡변위가 방지되어 있다고 말할 수 있습니다. + +$$ +Q = \frac {\sum P _ {u} \Delta_ {o}}{V _ {u} l _ {c}} \leq 0. 0 5 \rightarrow \text { 펉구속 골조 } +$$ + +$$ +Q = \frac {\sum P _ {u} \Delta_ {o}}{V _ {u} l _ {c}} > 0. 0 5 \rightarrow \text { 비횡구속 골조 } +$$ + +여기서, $\sum P_{u}$ : 해당층의 총 연직계수축력 + +$V_{u}$ : 해당층의 전단력 + +$\Delta_{o} V_{u}$ : $V_{u}$ 에 의한 해당층의 상·하부의 1차 상대처짐 + +$l_{c}$ : 골조에서 절점간 거리로 측정된 압축부재의 길이 + +# 17-1-2 장주효과의 고려 + +$kl_{u}/r$ 의 값에 따라 장주효과의 고려 여부를 판단합니다. + +-횡구속 구조인 경우 + + + +$$ +\frac {k l _ {u}}{r} \leq 3 4 - 1 2 \frac {M _ {1}}{M _ {2}} \quad \rightarrow \text { 장주효과 무시 } +$$ + +$$ +3 4 - 1 2 \frac {M _ {1}}{M _ {2}} \leq \frac {k l _ {u}}{r} \leq 1 0 0 \rightarrow \text { 장주효과 고려 } +$$ + +$$ +\frac {k l _ {u}}{r} > 1 0 0 \quad \rightarrow P - \Delta \text { 해석 } +$$ + +여기서, $M_{1}/M_{2} \geq -0.5$ 이어야 하며, 기둥이 단일곡률로 휘는 경우 $M_{1}/M_{2}$ 는 정(+)입니다. + +\- 비횡구속 구조인 경우 + +$$ +\frac {k l _ {u}}{r} < 2 2 \quad \rightarrow \text { 장주효과 무시 } +$$ + +$$ +2 2 \leq \frac {k l _ {u}}{r} \leq 1 0 0 \rightarrow \text { 장주효과 고려 } +$$ + +$$ +\frac {k l _ {u}}{r} > 1 0 0 \quad \rightarrow P - \Delta \text { 해석 } +$$ + +$kl_{u}/r$ 의 값이 100을 초과하는 모든 압축부재에 대해서는 $P-\Delta$ 해석을 해야 합니다. + +# 17-1-3 횡구속 구조물의 확대모멘트 + +횡구속 구조물의 확대모멘트 $M_{c}$ 는 다음과 같이 계산됩니다. + +$$ +M _ {c} = \delta_ {n s} M _ {2} +$$ + +여기서, $M_{2}$ : 기둥의 상·하부 단모멘트 중 큰 값, $M_{2,\min}=P_{u}(15+0.05h)$ 이상 + +$\delta_{ns}$ : 횡구속 골조에서 압축부재의 양단 사이의 부재곡률의 영향을 반영한 +모멘트 확대계수 + +$$ +\delta_ {n s} = \frac {C _ {m}}{1 - \frac {P _ {u}}{0 . 7 5 P _ {c}}} \geq 1. 0 +$$ + + + +$$ +P _ {c} = \frac {\pi^ {2} E I}{\left(k l _ {u}\right) ^ {2}} +$$ + +이 경우, $EI$ 값의 산정은 다음의 식을 이용하여도 좋으며, $\beta_{d}=0.6$ 으로 가정하여 $EI=0.25E_{c}I_{g}$ 로 사용할 수도 있습니다. + +$$ +E I = \frac {0 . 2 E _ {c} I _ {g} + E _ {s} I _ {s e}}{1 + \beta_ {d}} \quad \text {또는} E I = \frac {0 . 4 E _ {c} I _ {g}}{1 + \beta_ {d}} +$$ + +$\beta_{d}$ = 축방향 계수고정하중에 의한 최대 계수축력 / 전체 계수축력, $C_{m}$ 은 등가모멘트 계수로서 아래 식을 따릅니다. + +$$ +C _ {m} = 0. 6 + 0. 4 \frac {M _ {1}}{M _ {2}} \geq 0. 4 \text {(횡하중이 없는 경우)} +$$ + +$$ +C _ {m} = 1. 0 \quad (\text { 햜하종이 있는 경우 }) +$$ + +# 17-1-4 비횡구속 구조물의 확대모멘트 + +비횡구속 구조물의 확대모멘트는 모멘트 확대계수 $\delta_{s}$ 를 고려하여 계산된 횡변위가 가능한 모멘트 $\delta_{s}M_{s}$ 와 횡변위가 방지된 모멘트 $M_{ns}$ 의 합으로 계산됩니다. + +$$ +M _ {1} = M _ {1 n s} + \delta_ {s} M _ {1 s} +$$ + +$$ +M _ {2} = M _ {2 n s} + \delta_ {s} M _ {2 s} +$$ + +확대된 횡변위가 가능한 모멘트 $\delta_{s}M_{s}$ 는 다음 방법으로 계산됩니다. + +$$ +\delta_ {s} M _ {s} = \frac {M _ {s}}{1 - Q} \geq M _ {s} (\text { 단 }, \delta_ {s} \leq 1. 5) +$$ + + + +$$ +\delta_ {s} M _ {s} = \frac {M _ {s}}{1 - \frac {\sum P _ {u}}{0 . 7 5 \sum P _ {c}}} \geq M _ {s} \quad (\text {단,} \delta \mathrm{s} > 1. 5) +$$ + +여기서, ΣPu: 해당층의 총 연직계수축력 + +ΣPc:해당층의 횡변위를 지지하는 기둥들의 임계축력의 합 + +$$ +P _ {c} = \frac {\pi^ {2} E I}{\left(k l _ {u}\right) ^ {2}}, E I = \frac {0 . 2 E _ {c} I _ {g} + E _ {s} I _ {s e}}{1 + \beta_ {d}} \text {또는} E I = \frac {0 . 4 E _ {c} I _ {g}}{1 + \beta_ {d}} +$$ + +$\beta_{d}$ = 해당층의 최대계수지속전단력 / 해당층의 전체 계수전단력단, + +$\frac{l_{u}}{r}>\frac{35}{\sqrt{P_{u}/\left(f_{ck}A_{g}\right)}}$ 인 경우에는 최대모멘트가 기둥 단부가 아닌 기둥의 양단 사이 + +에서 발생하게 되며, 최대모멘트의 값은 다음과 같습니다. + +$$ +M _ {c} = \delta_ {n s} M _ {2} = \frac {C _ {m}}{1 - \frac {P _ {u}}{0 . 7 5 P _ {c}}} \left(M _ {2 n s} + \delta_ {s} M _ {2 s}\right) +$$ + +단, $\delta_{ns}$ 는 1.0 이상입니다. + + + +# 17-2 기둥부재 설계 + +기둥부재는 부재 방향에 대하여 압축 또는 인장력이 작용하고 동시에 휨 모멘트를 받는 부재이며, 작용하는 하중의 형태에 따라 다음과 같이 구분할 수 있습니다. + +▪ 중심 축하중을 받는 기둥 +- 축하중 및 1축 힘모멘트를 받는 기둥 +- 축하중 및 2축 힘모멘트를 받는 기둥 + +한국 도로교설계기준 (2005, 2010)에서 극한강도 설계법에 의한 기둥부재 설계시, 기둥부재는 세장비에 따라 단주 또는 장주로 구분하여 단면설계(강도검증)를 수행합니다. 그러므로 세장비가 정해진 한계를 초과하는 장주는 모멘트 확대계수를 구하여 계수 hover모멘트에 곱함으로써 설계용 계수모멘트를 산출합니다. 그리고 이 hover모멘트 값을 적용하여 단면설계(강도검증)를 수행합니다. + +편심이 없는 순수 축하중을 받는 압축재의 최대 축하중강도는 다음 식과 같이 구할 수 있습니다. + +$$ +P _ {o} = 0. 8 5 f _ {c k} (A _ {g} - A _ {s t}) + f _ {y} A _ {s t} +$$ + +여기서 $P_{0}$ : 편심이 없을 때의 공칭 축하중 + +$f_{ck}$ : 콘크리트의 설계기준 압축강도 + +$A_{g}$ : 전단면적 + +$A_{st}$ : 기둥 주철근의 단면적 + +$f_{y}$ : 철근의 항복강도 + +그러나 압축재의 설계축하중 ( $\phi P_{n}$ )은 압축재에 존재할 수 있는 예측치 못한 편심하중에 대비해야 합니다. 따라서 순수 압축재에서 단면의 축하중 설계강도를 최대 공칭축하중( $\phi P_{o}$ )의 80～85% 감소하도록 제한하고 있습니다. + +띠철근 기둥 : $\phi P_{n(\max)} = 0.80\phi P_{o}$ + +나선철근 기둥 : $\phi P_{n(\max)} = 0.85\phi P_{o}$ + + + +따라서 편심이 없는 순수 축하중을 받는 압축재의 계수축하중은 다음 식을 만족하도록 설계합니다. + +$$ +\phi P _ {n (\max)} > P _ {u} +$$ + +축하중과 1축 휨모멘트를 동시에 받는 기둥부재는 힘의 평형조건식과 변형율의 적합조건을 만족하여야 하며, 다음과 같은 기본조건을 만족시키도록 설계합니다. + +$$ +\phi P _ {n (\max)} > P _ {u}, \phi \mathbf {M} _ {n (\max)} > M _ {u} +$$ + + + +midas Civil 에서는 설계단면에 대한 정확한 소요철근량 산출을 위하여 축력-모멘트상관도 분석을 수행합니다. 그리고 계수축력과 계수휨모멘트에 의한 편심거리를고려하여 단면설계(강도검증)를 수행하므로 기둥부재가 축인장을 받는 경우에도 설계가 가능합니다. + +![](images/page-548_85f495337ba66929e19d2a1ffca3fd46e5413b6e6a60b29deeb305d636aa2568.jpg) + +
+line + +| Region | Description | Value | +|--------|--------------------------------------|-----------| +| Region 1 | Region 1 (design axial load strength) | φPn(max) | +| Region 1 | φPn(max) = 0.80φPo (tied reinforcement) | φPn(max) | +| Region 1 | φPn(max) = 0.85φPo (spiral reinforcement) | φPn(max) | +| Region 2 | Region 2 (compression failure) | εmin | +| Region 2 | balanced failure condition | εb | +| Region 3 | Region 3 (tensile failure) | Mb | +| Region 3 | Pure bending | Pt(max) | +| Region 3 | Pt(max) | Pt | +
+ +그림 2.17.1 축하중과 1축 휨모멘트의 상관도 + + + +축하중과 2축 휨모멘트를 동시에 받는 기둥부재는 공칭 축하중(Pn) 및 공칭 휨모멘트(Mny, Mnz)에 의한 3차원 축력-모멘트 상관도 분석을 수행합니다. 그리고 이 결과를 근거하여 정밀해에 의한 정확한 소요철근량을 산출하며 다음과 같은 기본조건을 만족시키도록 설계합니다. + +$$ +\phi P _ {n} \geq P _ {u}, \phi M _ {n y} \geq M _ {u y}, \phi M _ {n z} \geq M _ {u z} +$$ + +![](images/page-549_50558aed1f709a6f635540c62836d09e0a851b3ec369a7eabcd44d110c18776f.jpg) + +
+text_image + ++Pn +Pn(max) +ey = 0 +ez = Muy/Pu +Mbz +0 +e = Mu/Pu +Mnz +-Pn +Mny +
+ +그림 2.17.2 축하중과 2축 휩모멘트의 상관도 + + + +# 17-3 임의 단면에 대한 3차원 축력-모멘트 상관도 분석 + +임의 단면의 3차원 축력-모멘트 상관도 분석은 다음과 같이 수행합니다. + +계산시간의 단축을 위하여 단면의 대칭 여부를 미리 판별하여 대칭시에는 대칭부분만 계산을 수행하며, 타 영역에 대해서는 대칭되는 계산된 값을 적용합니다. + +대칭형태는 축대칭, 방사대칭, 역대칭으로 구분되며 분류기준은 콘크리트 단면과 철근의 배근형태를 축을 중심으로 단면1차모멘트와 단면적을 이용하여 자동으로 판별합니다. + +![](images/page-550_83cf9dd24188da2e4642af0ca9ba3bc8d2a902caa579c8406dd5959d003e8a27.jpg) +그림 2.17.3 대칭 형태에 따른 계산 수행 범위 + +임의 단면의 경우 형태가 일정하지 않으므로 직사각형 형태의 단면에 주로 적용되는 Whitney가 제안한 등가 직사각형 응력분포대신 Parabolic-Plateau형식의 응력분포를 적용하여 축력-모멘트 상관도를 계산합니다. + +![](images/page-550_9b959dbb7044bce6e0f2956ff3fa0d71fc05e9e54fe1c15ed2161c811773abf8.jpg) + +
+line + +| Strain | Stress | +| ------ | ------ | +| 0 | 0.85 | +| E₀ | 0.85 | +| Eᵤ | 0.85 | +
+ +그림 2.17.4 Parabolic-Plateau형식의 응력분포도 diff --git a/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_056.md b/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_056.md new file mode 100644 index 00000000..cd366e1a --- /dev/null +++ b/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_056.md @@ -0,0 +1,3582 @@ + + +임의 형태의 단면에 비선형 응력-변형율을 적용하므로 중립축과 평행하게 단면을미소요소로 자른 후 각 미소요소에 작용하는 응력은 동일하다고 가정하여 각 요소의 도심에서의 변위를 이용하여 응력-변형율에서 응력을 산출, 각각의 요소에 대한작용력을 계산합니다. 계산된 작용력들을 각 요소의 도심에 작용하는 것으로 보고 모든 요소의 작용력 및 철근의 작용력을 취합하여 공칭 축하중(Pn) 및 공칭 휨모멘트 $( M _ { n y } , M _ { n z } ) \boldsymbol { \Xi }$ 산출합니다. + +해석시 단면의 중심을 도심을 기준으로 하므로 공칭강도 설계시의 중심점을 소성중심이 아닌 도심을 기준으로 계산을 수행합니다. + +$$ +P _ {n} = \Sigma (C _ {c i}) + \Sigma (F _ {s i}) +$$ + +$$ +M _ {n y} = \Sigma (C _ {c i} \cdot z _ {c i}) + \Sigma (F _ {s i} \cdot z _ {s i}) +$$ + +$$ +M _ {n z} = \Sigma (C _ {c i} \cdot y _ {c i}) + \Sigma (F _ {s i} \cdot y _ {s i}) +$$ + +여기서, $C _ { c i } \colon$ 각각의 미소요소의 작용력 + +$F _ { s i } \colon$ 각각의 철근의 작용력 + +$Z _ { C I } , \ y _ { C I }$ : 단면의 도심에 대한 미소요소 Ci 의 도심의 좌표 + +$Z s i , \ : y _ { s i } :$ 단면의 도심에 대한 철근 Si 의 중심좌표 + +![](images/page-551_40876c1069521a0f2641bcd568cdbac63e8236988d86a9c20778bd8ce5165c0a.jpg) + +
+text_image + +미소요소 C₁의 도심 +단면의 도심 +0.85f +ck +C₁ +C₂ +C₃ +C₄ +C₅ +C₆ +C₇ +C₈ +C₉ +C₁₀ +C₁₁ +C₁₂ +C₁₃ +C₁₄ +C₁₅ +C₁₆ +C₁₇ +C₁₈ +C₁₉ +C₂₀ +C₂₁ +C₂₂ +C₂₃ +C₂₄ +C₂₅ +C₂₆ +C₂₇ +C₂₈ +C₂₉ +C₃₀ +C₃₁ +C₃₂ +C₃₃ +C₃₄ +C₃₅ +C₃₆ +C₃₇ +C₃₈ +C₃₉ +C₄₀ +C₄₁ +C₄₂ +C₄₃ +C₄₄ +C₄₅ +C₄₆ +C₄₇ +C₄₈ +C₄₉ +C₅₀ +C₅₁ +C₅₂ +C₅₃ +C₅₄ +C₅₅ +C₅₆ +C₅₇ +C₅₈ +C₅₉ +C₆₀ +C₆₁ +C₆₂ +C₆₃ +C₆₄ +C₆₅ +C₆₆ +C₆₇ +C₆₈ +C₆₉ +C₇₀ +C₇₁ +C₇₂ +C₇₃ +C₇₄ +C₇₅ +C₇₆ +C₇₇ +C₇₈ +C₇₉ +C₈₀ +C₈₁ +C₈₂ +C₈₃ +C₈₄ +C₈₅ +C₈₆ +C₈₇ +C₈₈ +C₈₉ +C₉₀ +C₉₁ +C₉₂ +C₉₃ +C₉₄ +C₉₅ +C₉₆ +C₉₇ +C₉₈ +C₉₉ +C₁₀ +C₁₁ +C₁₂ +C₁₃ +C₁₄ +C₁₅ +C₁₆ +C₁₇ +C₁₈ +C₁₉ +C₂₀ +C₂₁ +C₂₂ +C₂₃ +C₂₄ +C₂₅ +C₂₆ +C₂₇ +C₂₈ +C₂₉ +C₈₀ +C₈₁ +C₈₂ +C₈₃ +C₈₄ +C₈₅ +C₈₆ +C₈₇ +C₈₈ +C₈₉ +C₉₀ +C₉₁ +C₉₂ +C₉₃ +C₉₄ +C₉₅ +C₉₆ +0.85f +ck +
+ +그림 2.17.5 임의단면에 적용되는 비선형·응력 변형율 + + + +각각의 하중조합에 대해 발생하는 $P _ { u } , \ M _ { u y } , \ M _ { u z } 0 \lVert$ 대한 검토는 계산된 3차원 축력-모멘트 상관도에서 원점을 기준으로 Pu, Muy, Muz 방향으로 3차원 직선을 그려서교차하는 평면으로부터 해당  Pn,  $M _ { n y } ,$  $M _ { n z } { \frac { \equiv } { \equiv } }$ 산출해 내며, 해당 하중조합의축력-모멘트 상관도는 교차점의 양 옆에 있는 2개의 계산되어진 3차원 축력-모멘트 상관도와의 인접된 비율로 산출합니다. + +![](images/page-552_2d3ace239d988d0cf6cf79f2585a2bb3f51d9d8ab71a02f8fd79ccc37305ef8d.jpg) + +
+text_image + +P +인접한 2개의 +계산되어진 +축력-모멘트 상관도 +P_u, M_uy, M_uz 방향과 교 +차하는 3차원 축력-모 +멘트 상관도 상의 평면 +0.701 +P +M +My +Mz +인접비율로 산출된 +축력-모멘트 상관도 +
+ +그림 2.17.6 ΦPn, ΦMny, ΦMnz 및 인접비율을 이용한 해당 하중조합의 축력-모멘트 상관도 + + + +최대 위험 하중조합에 대해서는 평면상의 교차점이 아닌 정밀한 $\phi \ P _ { n } , \phi \ M _ { n y } ,$ 멘트 상관도를 산출합니다. + +![](images/page-553_3600a4c744c352aa356b0e40350acdfbd5d7f77bf794e7f700de4ab2c9c7f534.jpg) + +
+text_image + +P +순수 압축 +지철근 기둥 : +Pn(max) = 0.80 P0 +순수 인장 +M +순수 파괴 구간 +순수 파괴 구간 +z +y +z +y +z +y +z +y +z +y +z +y +z +y +z +y +z +y +z +y +z +y +z +y +z +y +z +y +z +y +z +y +z +y +z +y +z +y +z +y +z +y +z +y +z +y +z +y +z +y +z +z +y +z +y +z +y +z +y +z +y +z +y +z +y +z +y +z +y +z +y +z +y +z +y +z +y +z +y +z +y +z +y +z +y +z +y +z +y +z +y +z +y +z +y +z +y +z +y +z +x +y +z +y +z +y +z +y +z +y +z +y +z +y +z +y +z +y +z +y +z +y +z +y +z +y +z +y +z +y +z +y +z +y +z +y +z +y +z +y +z +y +z +y +z +y +z +y +z +y +y +z +y +z +y +z +y +z +y +z +y +z +y +z +y +z +y +z +y +z +y +z +y +z +y +z +y +z +y +z +y +z +y +z +y +z +y +z +y +z +y +z +y +z +y +z +y +z +y +x +y +z +y +z +y +z +y +z +y +z +y +z +y +z +y +z +y +z +y +z +y +z +y +z +y +z +y +z +y +z +y +z +y +z +y +z +y +z +y +z +y +z +y +z +y +z +y +z +x +x +y +z +y +z +y +z +y +z +y +z +y +z +y +z +y +z +y +z +y +z +y +z +y +z +y +z +y +z +y +z +y +z +y +z +y +z +y +z +y +z +y +z +y +z +y +z +y +z +z +x +y +z +y +z +y +z +y +z +y +z +y +z +y +z +y +z +y +z +y +z +y +z +y +z +y +z +y +z +y +z +y +z +y +z +y +z +y +z +y +z +y +z +y +z +y +z +y +x +y +x +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +x +y +x +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +x +y +x +y +x +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +Y +Y +Y +Y +y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +y +x +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +Y +Y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +Y +Y +X +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +X +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +y +x +y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +X +X +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +y +X +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +Y +X +y +x +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +x +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +s +y +x +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +S +y +x +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +N +y +x +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +n +y +x +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +m +y +x +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +M +y +x +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +k +y +x +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +l +y +x +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +t +y +x +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +i +y +x +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +I +y +x +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +1 +y +x +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +0 +y +x +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +8 +y +x +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +5 +y +x +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +6 +y +x +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +3 +y +x +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +4 +y +x +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +2 +y +x +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +o +y +x +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +r +y +x +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +b +y +x +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +c +y +x +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +a +y +x +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +e +y +x +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +g +y +x +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +u +y +x +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +d +y +x +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +p +y +x +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +h +y +x +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +f +y +x +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +v +y +x +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +z +y +x +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +P +y +x +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +R +y +x +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +B +y +x +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +C +y +x +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +F +y +x +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +G +y +x +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +L +y +x +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +y +
+ +그림 2.17.7 기둥강도 상관도 ( P-M 상관도) + + + +# 17-4 임의 단면에 대한 기둥의 전단설계 + +한국 도로교설계기준 (2005, 2010)에서는 전단을 받는 단면은 다음의 식을 만족하도록 설계합니다. + +$$ +V _ {u} \leq \phi V _ {n} +$$ + +여기서 $V_{u}$ 는 해당 단면의 계수전단력이며, $V_{n}$ 은 다음식에 의해 계산되는 공칭 전단 강도입니다. + +$$ +V _ {n} = V _ {c} + V _ {s} +$$ + +$V_{c}$ : 콘크리트에 의한 공칭전단강도 + +$V_{s}$ : 전단철근에 의한 공칭전단강도 + +압축 선단부에서 최 외측 인장철근 사이의 길이를 유효높이 d로 산정하여 사이의 단면적만 전단면적으로 적용하며, 전단면적을 d를 기준으로 동일한 면적의 직사각형으로 환산하여 적용합니다. + +따라서 인장철근의 도심을 기준으로 유효높이를 산출하고 정밀하게 전단면적을 산출하는 정형단면의 전단설계에 비해 큰 콘크리트의 전단강도를 나타낼 수 있으므로 사용상에 주의가 필요합니다. + +![](images/page-554_d3d36589650cbe82966b9981eb2f532f7e0f732ddff80bf2662cdb520ac6243d.jpg) + +
+text_image + +compression +tension +y +d +b_w +y +Z +Z +
+ +그림 2.17.8 환산 전단단면적 + + + +축방향력을 받는 부재의 경우 콘크리트에 의한 전단강도 + +압축력을 받는 경우: 1 16 14 uc ck wgN V f b d A      $\natural \leq Z \leqq \leq \natural \equiv \ \forall \models \ \exists \leq \ \exists \leq \ { \mathcal { T } } \colon \ V _ { c } = { \frac { 1 } { 6 } } { \left( 1 + { \frac { N _ { u } } { 1 4 A _ { g } } } \right) } \sqrt { f _ { c k } } b _ { w } d$ + +$\underline { { 0 } } | \underline { { \bar { x } } } | \geq | \underline { { \underline { { \partial } } } } | \equiv \underline { { \underline { { \forall } } } } \underline { { \underline { { \breve { \mathbf { \delta } } } } } } | \underline { { \underline { { \circ } } } } | \mathrm { \mathcal { \ : \ : } V } _ { c } = \frac { 1 } { 6 } \left( 1 + \frac { N _ { u } } { 3 . 5 A _ { g } } \right) \sqrt { f _ { c k } } b _ { w } d$ + +여기서 Nu는 인장력일 때, 부(-)이며, $N _ { u } / A _ { g } \underline { { \circ } } |$ 단위는 MPa입니다. + +부재축의 직각으로 설치되는 스트럽의 간격은 철근 콘크리트부재의 경우 0.5d 이하, 600mm 이하로 배치하여야 합니다. + +철근이 부담하는 전단강도 Vs가 $( \sqrt { ( } t _ { c k } ) / 3 ) b _ { w } \cdot d$ 를 초과하는 경우 규정된 철근간격을 절반으로 감소시켜야 하며, 철근이 부담해야 하는 전단강도 Vs는 2(√$( f _ { c k } ) / 3 ) b _ { w }$ · d이하로 하여야 합니다. + +압축부재의 경우에는 단면의 최소치수 이하 및 300mm이하 이어야 합니다. + +계수전단력 $V _ { u } \mathcal { I } \vdash$ 콘크리트에 의한 설계전단강도  Vc/2를 초과하는 경우 최소 단면적의 전단철근을 배치합니다. + +$$ +A _ {v} = 0. 3 5 \cdot b _ {w} \cdot s / f _ {y} +$$ + +여기서, $b _ { w } \mathcal { \underline { { { Q } } } } \}$ s의 단위는 mm입니다. + + + +# Chapter 18. Wave Load 하중 생성 + +# 18-1 개요 + +midas Civil에서 제공하는 Wave Load는 해양 구조물에 작용하는 파력에 대해 입력한 파랑정보를 통해 정적/동적해석을 위한 파랑하중을 생성하는 기능입니다. 일반적으로 해양구조물의 부재에 걸리는 파력은 Morrison의 식을 이용하여 산정할 수 있습니다. + +18-1-1 Wave Parameters의 용어정리 +![](images/page-556_b7934afe3ec4ba3d9ec6b5c49f0ba96490769b42c23feec330b40cf6e7a3ee53.jpg) + +
+text_image + +crest +L = Length +SWL +η +x +H = Height +h = depth +trough +bottom +C +Exaggerated Vertical Scale +
+ +그림 2.18.1 Wave Parameters의 정의 + +(1) H = 파고 +(2) C = L/T = : 파속 +(3) L = gT2/2π : 파장(천해 = $\frac{g}{2\pi}$ T² tanh $\frac{2\pi h}{L}$ , 심해 : tanh $\frac{2\pi h}{L}$ = 1 +(4) $\eta =$ 파형 $(\eta =0,$ 정수면 $)$ , $\eta(x,t) = \frac{1}{g} \frac{\partial \phi}{\partial t}$ +(5) h = 수심 +(6) k = : 파수 (Wave Number) +(7) $\sigma = \frac{2\pi}{\mathrm{T}}\Rightarrow \mathrm{T} = \frac{2\pi}{\sqrt{\mathrm{gk}\tanh\mathrm{kh}}}$ +(8) $\sigma^2 = \mathrm{gk}\tanh \mathrm{kh}$ : dispersion relationship, $\Rightarrow C^2 = \frac{L^2}{T^2} = \frac{g}{k}\tanh gh$ + + + +상대수심 h/L의 크기로 파의 종류를 분류할 수 있습니다. + +(심해파: h/L>1/2, 천해파: 1/25 + +# 18-1-3 해양파랑의 공학적 성질 + +# (1) 수립자 운동 + +연직 및 수평방향 수립자 속도성분을 공간의 함수로 나타내 보면, 상호간 90°의 위상차를 갖습니다. + +![](images/page-558_9db1a145d7ed0902693f8bee523d2ad64117af400597e2277b22916a60009804.jpg) + +
+text_image + +z +Direction of progressive +wave propagation +x = L/2 +x = L +x +
+ +그림 2.18.2 진행파의 수립자 속도 + +# (2) 수립자 궤적 : 타원 방정식 + +![](images/page-558_f4ddfdc3a07b06db60505db17ce64db0d9284bac8d781dcfb292ed27c5f78518.jpg) + +
+text_image + +water particle +of interest +ξ +ξ +(x, z) +A +(ξ/A)² + (ξ/B)² = 1 +
+ +그림 2.18.3 타원형 수립자의 궤적 + +천해역(상대수심 h/L<1/20)에서는 $A=\frac{HT}{4\pi}\sqrt{\frac{g}{h}}$ , $B=\frac{H}{2}\left(1+\frac{z}{h}\right)$ 이고 A는 z의 함수가 아니므로 수평방향의 수립자 이동거리는 수심에 따라 모두 일정합니다. + +심해역(상대수심 h/L>1/2)에서는 $A=\frac{H}{2}e^{kz}$ , B=A 이므로 원운동을 하며 수심에 따라 지수함수적으로 감소합니다. + + + +![](images/page-559_ec09125067db6ed06e0b1aa9b5d1e54713ddff81f3ec67b68fb86d5e6b4ee73a.jpg) + +$$ +k h < \frac {\pi}{1 0} +$$ + +$$ +(\frac {h}{L} < \frac {1}{2 0}) +$$ + +$$ +\frac {\pi}{1 0} < k h < \frac {\pi}{2} +$$ + +$$ +\left(\frac {1}{2 0} < \frac {h}{L} < \frac {1}{2}\right) +$$ + +$$ +k h > \frac {\pi}{2} +$$ + +$$ +(\frac {h}{L} > \frac {1}{2}) +$$ + +그림 2.18.4 상대수심에 따른 진행파의 수립자 궤적 + +Morrison equation에서는 기본적으로 입사하는 Wave에 비해 Pile의 지름이 크지 않다( $\frac { D } { \lambda } < 0 . 2 { \bf \Gamma } ) \underline { { \boldsymbol { \pi } } }$ 가정하고 Diffraction을 고려하지 않으며, Member 간 Interaction 또한 고려하지 않습니다. 하지만, Diffraction 효과의 경우 Pile의 지름이 파장에 비해커지게 되면 $\cdot \frac { D } { \lambda } > 0 . 2 )$ Drag force는 Inertia force 에 비해 무시할만하게 되고, 이 때는 Pile의 지름 효과를 고려해야 합니다.. 이 경우에 대해서는 MacCamy andFuchs(1954)나, Mogridge and Jamieson(1976) 등이 원형 실린더에 대해 해석해를구한바 있으며 임의 단면에 대해서는 수치해석이 요구된다. 이 결과들을 살펴보면Morrison equation에서 Inertia Coefficient를 조정하고 Phase lag를 줌으로써 보정을할 수 있음을 알 수 있습니다. 앞서 주어진 Morrison eq.은 Member에 대해Normal 방향 성분만을 나타내고 있는데, Tangential 성분(대개 Inertia term은 제외한drag term만 사용)은 Normal 방향 성분에 비해 그 크기가 작기 때문에 보통Normal 방향 성분만 고려를 합니다. + + + +# 18-2 파랑이론 + +파랑이론의 기본적인 가정은 비회전성, 비압축성 유체운동을 가정하므로 유체운동은 라플라스 방정식으로 $\nabla^{2}\emptyset = \frac{\partial^{2}\emptyset}{\partial x^{2}} + \frac{\partial^{2}\emptyset}{\partial y^{2}} + \frac{\partial^{2}\emptyset}{\partial z^{2}}$ 로 표현할 수 있습니다. + +비점성 비회전유체의 경계조건 + +(1) 운동학적 경계조건(Kinematic Boundary Condition, KBC) + +: 수립자의 운동에 관한 조건, 어떤 경계면에서도 경계면을 통한 흐름은 있을 수 없습니다 + +① 운동학적 자유수면 경계조건(KFSBC) + +② 해저면 경계조건(KBBC) + +(2) 역학적 경계조건(Dynamic Boundary Condition, DBC) + +:대기와 해수면과의 경계면인 자유수면은 압력이 일정하게 유지되어야 합니다. + +① 역학적 자유수면 경계조건(DFSBC) : 자유수면상의 압력은 파형을 따라 일정해야 합니다. + +(3) 측면 경계조건(Lateral Boundary Condition, LBC) + +# 18-2-1 Airy wave theory : 선형 파동이론 + +중력파(Gravity wave) 이론 중에서 가장 기본적인 모델로서 Linear wave 라고도 합니다. 해양파는 실제 불규칙한 운동을 하지만, 이 불규칙한 해양파는 Unidirectional, Monochromatic 성질을 가지는 정현파(Regular wave)의 중첩으로 나타낼 수 있기 때문에 이 정현파 모델을 통해서 파의 메커니즘을 논할 수가 있습니다. + +중력파 모델의 유도는 몇 가지 기본적인 가정 ( Unidirectional, Monochromatic, Progressive, Infinitely even bottom, Inviscid fluid, Incompressible fluid, Surface tension 무시, Irrotational motion)을 통해 속도 포텐설을 상정함으로써 2-D ‘potential boundary value problem’으로 정식화 할 수 있습니다. + +이 때 지배방정식은 Laplace equation ( $\nabla^{2}\Phi=0$ )으로 유도되고, Bottom Condition, Radiation Condition, Free surface condition을 만족하는 Potential을 구하게 되면 유체 Particle의속도, 가속도 및 압력 등을 구할 수 있게 됩니다. + +하지만, 위에서 언급했던 경계조건 중에서 Free surface condition이 Non-linear로 주어지므로 Exact-solution은 구할 수 없습니다. 따라서 Perturbation method를 사용 diff --git a/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_057.md b/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_057.md new file mode 100644 index 00000000..4de2d47d --- /dev/null +++ b/.raw/MidasCivilAnalysisReference/MidasCivilAnalysisReference_057.md @@ -0,0 +1,373 @@ + + +해 Approximated solution을 구하게 되는데, 이 경계 조건 적용 과정에서 Linear term 만을 고려함으로써 Airy wave 모델이 유도가 됩니다. 따라서 Airy wave theory 는 수심이나 파장에 비해 파고가 상대적으로 작을 때 적용 타당성을 가지게 됩니다. 최종적으로 얻을 수 있는 결과 중 뒤에서 Wave force를 얻는데 필요한 결과를 정리하면 다음과 같습니다. + +![](images/page-561_9c3811bf14a64bf9562a2cfde48eea0d5fe8b470302bdccf991d3576cc1bef47.jpg) + +
+text_image + +H +y +x +h +
+ +\- Wave height : H + +\- Wave length : λ + +\- Wave period: $T = \frac{2\pi}{\omega}$ ( $\omega =$ circular frequency) + +\- Wave number: $k = \frac{2\pi}{\lambda}$ + +\- Water depth : h + +Surface Wave form $\zeta = \frac{H}{2}\cos (kx - \omega t)$ + +일반 Finite depth 환경일 때 + +Dispersion relation $\omega^{2}=gk\tanh kh$ + +Particle velocity + +$$ +u = \frac {\omega H}{2} \frac {\cosh k y}{\sinh k h} \cos (k x - \omega t), v = \frac {\omega H}{2} \frac {\sinh k y}{\sinh k h} \sin (k x - \omega t) +$$ + +Particle acceleration + +$$ +a _ {x} = \frac {\omega^ {2} H}{2} \frac {\cosh k y}{\sinh k h} \sin (k x - \omega t), a _ {y} = - \frac {\omega^ {2} H}{2} \frac {\sinh k y}{\sinh k h} \cos (k x - \omega t) +$$ + +Airy wave theory를 적용할 때 한 가지 주의해야 할 점은 자유표면 경계 조건의 선형화 과정에서 정수면에 대해 테일러 전개를 했기 때문에, 특정 깊이(y)에서의 입자의 속도 및 가속도를 구할 때 구하고자 하는 위치가 아니라 그때의 입자 궤적이 가지는 mean position 값을 대입해야 한다는 것입니다. 일반 finite depth 환경에서 + + + +입자의 궤적은 속도 성분의 적분을 통해 다음과 같이 타원 형태로 구해집니다 (무한수심일 경우 원 궤적을 가지게 됩니다) + +Particle trajectory + +$$ +\left(\frac {\xi}{\frac {H}{2} \frac {\cosh k y}{\sinh k h}}\right) ^ {2} + \left(\frac {\zeta}{\frac {H}{2} \frac {\sinh k y}{\sinh k h}}\right) ^ {2} = 1 +$$ + +![](images/page-562_a68ec97481e3ade03d1ea5ecaa47b3bdb58c31c75dc7f0b6b36450634d4c3f01.jpg) + +
+text_image + +H/2 +apparent position +H/2 coth kh +mean position +y +x +
+ +# 18-2-2 Stokes 5th wave theory: 비선형 파동이론(5차항까지 stokes 급수전개) + +18-2-1 에서 살펴봤던 linear wave는 여러 면에서 아주 유용하지만 경계조건을 선형화 시키기 위해 small-amplitude를 가정함으로써 실제 파형을 반영하기에는 부족한 점이 많습니다. Linear theory의 확장은 Stokes(2nd order)나 Hunt(3rd oder) 등이해석해를 구한바 있고, Skjelbreia & Hendrickson (1961)이 5th order theory를 정리했습니다. 현재 전세계적으로 선급 등이나 여러 협회에서 선박이나 해양구조물의 파랑하중 등을 구할 때는 5th order 모델을 사용하도록 규정하고 있습니다. + +Linear wave는 1st order approximation을 함으로써 sinusoidal 파형을 가지게 됩니다. 반면 higher-order theory는 finite-amplitude theory로서 좀 더 실제 파형에 가까운 모델을 만들 수 있습니다. 아래 그림에서 볼 수 있듯이 higher-order 모델의 파형은 linear에 비해 crest가 좀 더 경사가 지는 반면, trough는 좀 더 평평한 형태를가지고 있습니다. + + + +![](images/page-563_8bb449e53089556354d4bbc3b4828c003894af7ad706dd02dc3c6fa279d798a1.jpg) + +
+line + +| Line Type | Description | +| --------------------- | ------------------- | +| linear theory | flatter trough | +| higher-order theory | steeper crest | +
+ +Stokes 5th order wave에서 구해지는 유체의 속도, 가속도 등은 다음과 같습니다. + +Surface wave form + +$$ +\begin{array}{l} k \varsigma = \delta \cos \theta + \left(\delta^ {2} B _ {2 2} + \delta^ {4} B _ {2 4}\right) \cos 2 \theta \\ + \left(\delta^ {3} B _ {3 3} + \delta^ {5} B _ {3 5}\right) \cos 3 \theta + \delta^ {4} B _ {4 4} \cos 4 \theta + \delta^ {5} B _ {5 5} \cos 5 \theta \\ \end{array} +$$ + +Dispersion relation + +$$ +\omega^ {2} = g k \tanh k h \left(1 + \delta^ {2} C _ {1} + \delta^ {4} C _ {2}\right) +$$ + +$$ +\mathrm{O} (\varepsilon) \mathrm{O} (\varepsilon^ {3}) \mathrm{O} (\varepsilon^ {5}) +$$ + +여기서 $B_{ij}$ , $C_{i}$ 등은 Coefficient이고, $\delta = \mathrm{O}(\varepsilon)$ 는 Quantitative order를 나타내며 Wave number k 의 함수입니다. 한편, Liner wave에서는 파의 주기, 파고, 수심 등이 주어지면 Dispersion relation으로부터 Wave number가 바로 구해졌지만, 여기서는 Wave form 식과 Dispersion relation 식을 Iterative 하게 풀어야만 Wave number k 와 $\delta$ 를 구할 수 있습니다 + +Particle velocity + +$$ +\begin{array}{l} \mathrm{u} = \frac {\partial \phi}{\partial \mathrm{x}} = C _ {\mathrm{P}} \left[ \left(\bar {\delta} A _ {1 1} + \bar {\delta} A _ {1 3} + \bar {\delta} A _ {1 5}\right) \cosh \mathrm{ky} \cos \theta \right] \\ + 2 \left(\bar {\delta} ^ {2} A _ {2 2} + \bar {\delta} ^ {4} A _ {2 4}\right) \cosh 2 k y \cos 2 \theta \\ + 3 \left(\bar {\delta} ^ {3} A _ {3 3} + \bar {\delta} ^ {5} A _ {3 5}\right) \cosh 3 k y \cos 3 \theta \\ + 4 \bar {\delta} ^ {4} A _ {4 4} \cosh 4 k y \cos 4 \theta \\ + 5 \bar {\delta} ^ {5} A _ {5 5} \cosh 5 k y \cos 5 \theta \\ \end{array} +$$ + + + +Particle acceleration + +$$ +\begin{array}{l} \frac {\partial \mathrm{u}}{\partial \mathrm{t}} = \mathrm{wC} _ {\mathrm{P}} \left[ \left(\bar {\delta} \mathrm{A} _ {1 1} + \bar {\delta} ^ {2} \mathrm{A} _ {1 3} + \bar {\delta} ^ {5} \mathrm{A} _ {1 5}\right) \cosh \mathrm{ky} \cos \theta \right] \\ + 4 \left(\bar {\sigma} ^ {2} A _ {2 2} + \bar {\sigma} ^ {4} A _ {2 4}\right) \cosh 2 k y \sin 2 \theta \\ + 9 \left(\bar {\delta} ^ {3} A _ {3 3} + \bar {\delta} ^ {5} A _ {3 5}\right) \cosh 3 k y \sin 3 \theta \\ + 1 6 \delta^ {4} \mathrm{A} _ {4 4} \cosh 4 \mathrm{ky} \sin 4 \theta \\ + 2 5 \delta^ {5} \mathrm{A} _ {5 5} \cosh 5 \mathrm{ky} \sin 5 \theta \\ \end{array} +$$ + +z방향 속도와 가속도도 마찬가지 방법으로 구할 수 있습니다. + +# 18-2-3 Stream Function : 흐름함수를 이용한 비선형 파동이론 + +앞서 18-2-.2에서 살펴본 higher-order Stokian Wave theory(3rd, 5th order)는 유도과정뿐만 아니라 식 자체도 상당히 복잡해서 그 이상의 Higher-order로 전개하는것은 현실적으로 어려움이 많습니다. 이러한 이유로 Computer를 이용해 ‘어떤’Order로도 전개가 가능한 Wave theory의 필요성이 대두되었습니다. 처음으로Chappelear(1961)가 Velocity potentail을 이용하여 만든 Theory가 나온 이후,Dean(1965)이 Stream function을 이용해 Chappelear의 이론 보다 계산이 간단한Stream function Wave theory를 발표했습니다. 이후에 Cokelet(1977)이 Breaking 직전의 Wave height 범위까지 아주 정확한 계산이 가능한 이론을 발표했지만, 현재Design에는 적용되지 않고 있는 것으로 보입니다. 여기에서는 가장 일반적으로 이용되는 Dean의 이론을 이용 했습니다. + +Stream function theory는 앞서의 Wave theory에서 기본적으로 가정했던 Velocitypotential을 대신 Stream function을 사용합니다. Stream function을 이용해 x, y 방향Fluid velocity 성분을 다음과 같이 정의할 수 있습니다. + +수립자 속도 : $u = \frac { \hat { \sigma } \psi } { \hat { \sigma } z } , \nu = \frac { \hat { \sigma } \psi } { \hat { \sigma } x }$ , v  + +이렇게 정의된 Stream function은 앞서의 Velocity potential 과 마찬가지로 지배방정식인 Laplace equation을 만족하고, 이에 맞는 Stream Function 형태의 경계 조건 + + + +들 역시 얻 을 수 있습니다다. 편의상 Wavve celerity C로 이동하는 좌표표계를 잡으면Water wavee의 Stream funnction은 시간 teerm이 사라진 xx, z 만의 함수수가 되며, 경계조건을 만족족하는 Nth-orde r stream functioon은 다음과 같같은 형태를 가지지게 됩니다. + +$$ +\Psi (\mathrm{x}, \mathrm{z}) = \mathrm{Cz} + \sum_ {\mathrm{n} = 1} ^ {\mathrm{N}} \mathrm{X} (\mathrm{N}) \sinh \mathrm{nkz} \cos \mathrm{nkx} +$$ + +이 형태의 해해는 한가지 경 계조건-Dynam ic free surface Condition-을 만족하지 못하는데, 이를 ‘Approximatelly’ 만족하도록 계수 X n( ) 을 정해야 하하며, 이는 곧X n( ) 에 대대한 최적화 문제제가 됩니다. 한한편, Stream fuunction은 curreent를 고려하여서도 계산이이 가능한데, Cuurrent가 있을 때 Particle 속도도에 Current 속속도 보정만을하는 다른 모모델보다는 훨씬씬 정교하다고 할 수 있습니다다. + +# 18-2-4 Cn oidal / Solitarry wave : 천해해역, 주기파(자코코비안 타원적분분항으로 표현) + +Shallow waater 환경이 되면 파 자체의 거동뿐만 아니라, 깊이에 따른른 Pressure와velocity 변화화도 상당히 복 잡해집니다. 앞앞서 살펴본 Stookian wave proofile들 같은 경우, Shallow 환경을 가정하하고 그에 맞게 전개를 하려면 짧은 파장이나나 작은 파고를가진다고 가가정해야 하므로로, Shallow 환환경에서 깊이에에 비해 긴 파파장의 파(Longwave)를 Moodeling하기 위해해서는 앞에서와와는 다른 Pertuurbation 과정이이 필요합니다.Solitary waave는 Cnoidal wave의 아주 특수한 경우인인데, 이론 자체체는 Solitary가1870년대에 먼저 발표되었었습니다. Finitte Amplitude가 형태적 변화 없이 그대로Translation하하는 Steady w ave 모델이 다 른 선형화 과정정 없이 얻어졌 는데, 이 모델의 Solution이 Solitary wavve입니다. Solitaary wave는 Wavve에서 아주 멀멀리 떨어진 곳의 물은 교 란되지 않는다고고 가정하여 이이론적으로 무한한 파장을 갖고 , 파의 위상이(+)값만을 가가진다.(즉, 파고고가 정수면 위에에만 존재합니다다) + +![](images/page-565_d3a885364da98400d626ad4fa2aad6cca4a12d7431845d3ef9c786af40feed4c.jpg) + +
+line +| √(3/4 a/h) x/h | η/a | +| -------------- | --- | +| 0.2 | 1.0 | +| 0.4 | 0.9 | +| 0.6 | 0.8 | +| 0.8 | 0.6 | +| 1.0 | 0.4 | +| 1.2 | 0.3 | +| 1.4 | 0.2 | +| 1.6 | 0.1 | +| 1.8 | 0.05 | +| 2.0 | 0.02 | +| 2.2 | 0.01 | +| 2.4 | 0.005 | +
+ + + +Cnoidal theory는 처음에 Solitary와 같은 Approximations을 통해 얻어졌으나, Solution의 형태가 Periodic으로 주어집니다. “Cnoidal”이라는 이름은 Surface elevation이 Jacobian elliptic function의 제곱에 비례하고, Sinusoidal한 특성을 가지고 있기 때문에 그 합성어 형태로 지어졌다고 합니다. Cnoidal solution의 파 형태를 살펴보면, Shallow water에서 실제 Wave의 특징인 길고 평평한 Trough와 좁은 crest 형태를 잘 나타내고 있음을 알 수 있습니다. 앞서 언급했듯 Cnoidal wave에서 속도, 가속도, Wave profile 등은 Jacobian elliptic function 의 급수로 표현이 되는데, 이 함수의 Parameter m (0 ≤ m ≤ 1) 이 10이 되면 Solitary wave가 됩니다. + +# 18-2-5 Solitary wave + +The solitary wave 이론은 Cnoidal theory의 제한된 경우에 해당됩니다. +0 ≤ m ≤ 1에서 m=1일 경우 solitary wave가 됩니다. + + + +# 18-3 파랑이론의 적용한계 + +앞서 살펴본 Wave theory들은 각각의 기본 가정에 바탕을 두고 있으므로 이 이론들을 모든 Wave modeling에 그대로 적용할 수는 없으며, 파장, 파고, 수심 등에 따라 타당성을 가지는 모델을 알맞게 적용해야 합니다. 다음 두 도표는 Wave theory 적용에 있어 각 이론이 타당성을 가지는 범위를 나타내고 있습니다. + +실제 적용에서뿐만 아니라 검증을 위한 예제 환경 모델링 시에도 각 Wave 이론이 적용될 수 있는 타당한 파고, 파장, 수심 등을 가정하여 모델링해야 합니다. + +![](images/page-567_f817c02a69e10d34802ea87fc548285365a364ede8e30ad23b1f0145d54b4117.jpg) + +
+line + +| x | y | Label | +| ------- | ------- | -------------- | +| 0.001 | 0.0001 | SHALLOW | +| 0.01 | 0.001 | AIRY | +| 0.1 | 0.01 | CNOIDAL | +| 0.4 | 0.04 | STOKES V | +
+ +![](images/page-567_2f4f94b38a82d765bbb7ae6d2b569a3e47969e287f36582e4f3f44ca0110bde5.jpg) + +
+line + +| Level | Value | +| ----------- | ------ | +| CNOIDAL | 0.0001 | +| BREAKING LIMIT | 0.01 | +| STREAM FUNCTION | 0.001 | +| AIRY | 0.0001 | +| STOKES II | 0.01 | +| STOKES III | 0.01 | +| STOKES IV | 0.01 | +| DEEP | 0.05 | +
+ +그림 2.18.5 Dean Graph와 Le Mehaute Graph + +참고문헌 + +[1] T. H. Dawson, Offshore Structural Engineering, Prentice-Hall, Inc., 1983. +[2] R. G. Dean and R. A. Dalrymple, Water Wave Mechanics for Engineers and Scientists, 2nd ver., World Scientific, 1991. +[3] J. D. Fenton, The Cnoidal Theory of Water Waves, in Developments in Offshore Engineering, Gulf Publishing Company, pp 55-100, 1999. +[4] J. D. Fenton, A high-order cnoidal wave theory, Journal of Fluid Mechanics, Vol. 94, pp. 129-161, 1979. +[5] W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C, 2nd edition, Cambridge University Press, 1992. + + + +# 18-4 Flow Chart + +midas Civil에서 입력한 파랑정보를 통해 파랑하중을 생성하는 흐름은 다음과 같습니다. + +![](images/page-568_68c2db8b26e4413ac8fab8241320e7ede496bac1656fdf7a6d25c2fe30355ce7.jpg) + +
+flowchart + +```mermaid +graph TD + A["Main
-read element info.
-read wave info.
-read current info."] --> B["Get wave load
-wave load 계산
-current load"] + B --> C["Get node load
-각 joint에 걸리는
load summation"] + C --> D["Get total load
-sum of total
joint loads"] + D --> E["Output
시간(phase)에 따른 total wave load 변화 (Dynamic)
원하는 시점(ex. Max base shear)에서 각 joint load (Static)"] + E --> A + F["Input file"] --> A + G["Phase (ωt) specified"] --> B + H["Phase (ωt) specified"] --> B + I["Phase (ωt) specified"] --> B + J["Phase (ωt) specified"] --> B + K["Phase (ωt) specified"] --> B + L["Phase (ωt) specified"] --> B + M["Phase (ωt) specified"] --> B + N["Phase (ωt) specified"] --> B + O["Phase (ωt) specified"] --> B + P["Phase (ωt) specified"] --> B + Q["Phase (ωt) specified"] --> B + R["Phase (ωt) specified"] --> B + S["Phase (ωt) specified"] --> B + T["Phase (ωt) specified"] --> B + U["Phase (ωt) specified"] --> B + V["Phase (ωt) specified"] --> B + W["Phase (ωt) specified"] --> B + X["Phase (ωt) specified"] --> B + Y["Phase (ωt) specified"] --> B + Z["Phase (ωt) specified"] --> B + AA["Phase (ωt) specified"] --> B + AB["Phase (ωt) specified"] --> B + AC["Phase (ωt) specified"] --> B + AD["Phase (ωt) specified"] --> B + AE["Phase (ωt) specified"] --> B + AF["Phase (ωt) specified"] --> B + AG["Phase (ωt) specified"] --> B + AH["Phase (ωt) specified"] --> B + AI["Phase (ωt) specified"] --> B + AJ["Phase (ωt) specified"] --> B + AK["Phase (ωt) specified"] --> B + AL["Phase (ωt) specified"] --> B + AM["Phase (ωt) specified"] --> B + AN["Phase (ωt) specified"] --> B + AO["Phase (ωt) specified"] --> B + AP["Phase (ωt) specified"] --> B + AQ["Phase (ωt) specified"] --> B + AR["Phase (ωt) specified"] --> B + AS["Phase (ωt) specified"] --> B + AT["Phase (ωt) specified"] --> B + AU["Phase (ωt) specified"] --> B + AV["Phase (ωt) specified"] --> B + AW["Phase (ωt) specified"] --> B + AX["Phase (ωt) specified"] --> B + AY["Phase (ωt) specified"] --> B + AZ["Phase (ωt) specified"] --> B + BA["Phase (ωt) specified"] --> B + BB["Phase (ωt) specified"] --> B + BC["Phase (ωt) specified"] --> B + BD["Phase (ωt) specified"] --> B + BE["Phase (ωt) specified"] --> B + BF["Phase (ωt) specified"] --> B + BG["Phase (ωt) specified"] --> B + BH["Phase (ωt) specified"] --> B + BI["Phase (ωt) specified"] --> B + BJ["Phase (ωt) specified"] --> B + BK["Phase (ωt) specified"] --> B + BL["Phase (ωt) specified"] --> B + BM["Phase (ωt) specified"] --> B + BN["Phase (ωt) specified"] --> B + BO["Phase (ωt) specified"] --> B + BP["Phase (ωt) specified"] --> B + BQ["Phase (ωt) specified"] --> B + BR["Phase (ωt) specified"] --> B + BS["Phase (ωt) specified"] --> B + BT["Phase (ωt) specified"] --> B + BU["Phase (ωt) specified"] --> B + BV["Phase (ωt) specified"] --> B + BW["Phase (ωt) specified"] --> B + BX["Phase (ωt) specified"] --> B + BY["Phase (ωt) specified"] --> B + BZ["Phase (ωt) specified"] --> B + CA["Phase (ωt) specified"] --> B + CB["Phase (ωt) specified"] --> B + CC["Phase (ωt) specified"] --> B + CD["Phase (ωt) specified"] --> B + CE["Phase (ωt) specified"] --> B + CF["Phase (ωt) specified"] --> B + DG["Phase (ωt) specified"] --> B + DH["Phase (ωt) specified"] --> B + DI["Phase (ωt) specified"] --> B + DJ["Phase (ωt) specified"] --> B + DK["Phase (ωt) specified"] --> B + DL["Phase (ωt) specified"] --> B + DJ["Phase (ωt) specified"] --> B + DK["Phase (ωt) specified"] --> B + DL["Phase (ωt) specified"] --> B + DJ["Phase (ωt) specified"] --> B + DK["Phase (ωt) specified"] --> B + BE["Phase (ωt) specified"] --> B + BF["Phase (ωt) specified"] --> B + BG["Phase (ωt) specified"] --> B + BH["Phase (ωt) specified"] --> B + BI["Phase (ωt) specified"] --> B + BJ["Phase (ωt) specified"] --> B + BK["Phase (ωt) specified"] <-- B + BL["Phase (ωt) specified"] <-- B + BM["Phase (ωt) specified"] <-- B + BN["Phase (ωt) specified"] <-- B + BO["Phase (ωt) specified"] <-- B + BP["Phase (ωt) specified"] <-- B + BZ["Phase (ωt) specified"] <-- B + BW["Phase (ωt) specified"] <-- B + BX["Phase (ωt) specified"] <-- B + BY["Phase (ωt) specified"] <-- B + BZ["Phase (ωt) specified"] <-- B + BZ["Phase (ωt) specified"] <-- B + BZ["Phase (ωt) specified"] <-- B + BZ["Phase (ωt) specified"] <-- B + BZ["Phase (ωt) specified"] <-- B + BZ["Phase (ωt) specified"] <-- B + BZ["Phase (ωt) specified"] <-- B + BZ["Sequence (ωt) specified"] --> BZ + BZ["Sequence (ωt) specified"] --> BZ + BZ["Sequence (ωt) specified"] --> BZ + BZ["Sequence (ωt) specified"] --> BZ + BZ["Sequence (ωt) specified"] --> BZ + BZ["Sequence (ωt) specified"] --> BZ + BZ["Sequence (ωt) specified"] --> BZ + BZ[Sequence (ωt)| BZ + BZ[Sequence (ωt)| BZ + BZ[Sequence (ωt)| BZ + BZ[Sequence (ωt)| BZ + BZ[Sequence (ωt)| BZ + BZ[Sequence (ωt)| BZ + BZ[Sequence (ωt)| BZ + BZ[Sequence (ωt)| BZ + BZ[Sequence (ωt)| BZ +``` +
diff --git a/.raw/MidasCivilAnalysisReference/images/page-001_9d9d07a2fdc50e84d4b5539b3fde05f70467736df5c1373d697a90e0aaa40602.jpg b/.raw/MidasCivilAnalysisReference/images/page-001_9d9d07a2fdc50e84d4b5539b3fde05f70467736df5c1373d697a90e0aaa40602.jpg new file mode 100644 index 00000000..411be26f Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-001_9d9d07a2fdc50e84d4b5539b3fde05f70467736df5c1373d697a90e0aaa40602.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-004_daaff903e0a0a470fb141f69ceedcf34a60aefe60d14e1a35022cffe3eea09c7.jpg b/.raw/MidasCivilAnalysisReference/images/page-004_daaff903e0a0a470fb141f69ceedcf34a60aefe60d14e1a35022cffe3eea09c7.jpg new file mode 100644 index 00000000..916eed81 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-004_daaff903e0a0a470fb141f69ceedcf34a60aefe60d14e1a35022cffe3eea09c7.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-019_f3768e4767a79825ff3cecb12ded7391e8d1297e37b8824b213c1fb5b15afc7d.jpg b/.raw/MidasCivilAnalysisReference/images/page-019_f3768e4767a79825ff3cecb12ded7391e8d1297e37b8824b213c1fb5b15afc7d.jpg new file mode 100644 index 00000000..92b89bb2 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-019_f3768e4767a79825ff3cecb12ded7391e8d1297e37b8824b213c1fb5b15afc7d.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-022_2f7d4eb903cf2dc9bbaf473441bcd6e85566ac9e4399a0d26621e2d3be0d4102.jpg b/.raw/MidasCivilAnalysisReference/images/page-022_2f7d4eb903cf2dc9bbaf473441bcd6e85566ac9e4399a0d26621e2d3be0d4102.jpg new file mode 100644 index 00000000..f030e9cb Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-022_2f7d4eb903cf2dc9bbaf473441bcd6e85566ac9e4399a0d26621e2d3be0d4102.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-022_7e88388a9d95fb73fccfee28bfe20b9338dc4867bb987c3a6bfe899406e254a2.jpg b/.raw/MidasCivilAnalysisReference/images/page-022_7e88388a9d95fb73fccfee28bfe20b9338dc4867bb987c3a6bfe899406e254a2.jpg new file mode 100644 index 00000000..a8da264e Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-022_7e88388a9d95fb73fccfee28bfe20b9338dc4867bb987c3a6bfe899406e254a2.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-022_b85c414a1f543c7a534c456716cb78be860d7d2b62bbcb27255c76c7071ec75f.jpg b/.raw/MidasCivilAnalysisReference/images/page-022_b85c414a1f543c7a534c456716cb78be860d7d2b62bbcb27255c76c7071ec75f.jpg new file mode 100644 index 00000000..f14ed3e9 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-022_b85c414a1f543c7a534c456716cb78be860d7d2b62bbcb27255c76c7071ec75f.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-023_34919fd38ebece97bdc78c845e013083325912cbbe4a1f5a3ced584225c34ffb.jpg b/.raw/MidasCivilAnalysisReference/images/page-023_34919fd38ebece97bdc78c845e013083325912cbbe4a1f5a3ced584225c34ffb.jpg new file mode 100644 index 00000000..e0eb4f2b Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-023_34919fd38ebece97bdc78c845e013083325912cbbe4a1f5a3ced584225c34ffb.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-023_8a05876f23d0fed75b81c84c4efcedd60b125c49c736482b5047d4c4647cf879.jpg b/.raw/MidasCivilAnalysisReference/images/page-023_8a05876f23d0fed75b81c84c4efcedd60b125c49c736482b5047d4c4647cf879.jpg new file mode 100644 index 00000000..5adcf4fd Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-023_8a05876f23d0fed75b81c84c4efcedd60b125c49c736482b5047d4c4647cf879.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-024_2691b8dd21d4ba4bc1257517e73e95f80aa4cd59e9a19cdce62f58e2d0a91147.jpg b/.raw/MidasCivilAnalysisReference/images/page-024_2691b8dd21d4ba4bc1257517e73e95f80aa4cd59e9a19cdce62f58e2d0a91147.jpg new file mode 100644 index 00000000..2132046c Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-024_2691b8dd21d4ba4bc1257517e73e95f80aa4cd59e9a19cdce62f58e2d0a91147.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-024_646e43bc60ced79c4417069b2ece4cd25498ded156a0ef341c8851b4098cc22f.jpg b/.raw/MidasCivilAnalysisReference/images/page-024_646e43bc60ced79c4417069b2ece4cd25498ded156a0ef341c8851b4098cc22f.jpg new file mode 100644 index 00000000..ca659bd5 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-024_646e43bc60ced79c4417069b2ece4cd25498ded156a0ef341c8851b4098cc22f.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-025_21e250cad96a512afd51f94043c6df26bbfe6aa0bde2869773b97fc974b92cbe.jpg b/.raw/MidasCivilAnalysisReference/images/page-025_21e250cad96a512afd51f94043c6df26bbfe6aa0bde2869773b97fc974b92cbe.jpg new file mode 100644 index 00000000..1b2c2c58 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-025_21e250cad96a512afd51f94043c6df26bbfe6aa0bde2869773b97fc974b92cbe.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-025_7e48e29670efde6c12578cacda61e8de4fbbed5e426a8883044c75b85f54ab13.jpg b/.raw/MidasCivilAnalysisReference/images/page-025_7e48e29670efde6c12578cacda61e8de4fbbed5e426a8883044c75b85f54ab13.jpg new file mode 100644 index 00000000..d86fc327 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-025_7e48e29670efde6c12578cacda61e8de4fbbed5e426a8883044c75b85f54ab13.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-027_3c024e022d2e3b8380b9cc398c79e7c706e07f5ceb84a01f6f019636360a9941.jpg b/.raw/MidasCivilAnalysisReference/images/page-027_3c024e022d2e3b8380b9cc398c79e7c706e07f5ceb84a01f6f019636360a9941.jpg new file mode 100644 index 00000000..271158a7 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-027_3c024e022d2e3b8380b9cc398c79e7c706e07f5ceb84a01f6f019636360a9941.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-027_4fce5a675ce95e8631372522df83a20ee1c79495a001ac4075633fd3bb56bf60.jpg b/.raw/MidasCivilAnalysisReference/images/page-027_4fce5a675ce95e8631372522df83a20ee1c79495a001ac4075633fd3bb56bf60.jpg new file mode 100644 index 00000000..f36c8fd0 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-027_4fce5a675ce95e8631372522df83a20ee1c79495a001ac4075633fd3bb56bf60.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-027_da07defa25edaf0c7fc20be8e1a115531c45eace71ac829a4eeacad87c7d3653.jpg b/.raw/MidasCivilAnalysisReference/images/page-027_da07defa25edaf0c7fc20be8e1a115531c45eace71ac829a4eeacad87c7d3653.jpg new file mode 100644 index 00000000..288f6df0 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-027_da07defa25edaf0c7fc20be8e1a115531c45eace71ac829a4eeacad87c7d3653.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-027_f5622565a73e43212539851fe5fd5f7310fe9f2a2e26179c6630dba11c034f85.jpg b/.raw/MidasCivilAnalysisReference/images/page-027_f5622565a73e43212539851fe5fd5f7310fe9f2a2e26179c6630dba11c034f85.jpg new file mode 100644 index 00000000..30e6fed6 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-027_f5622565a73e43212539851fe5fd5f7310fe9f2a2e26179c6630dba11c034f85.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-028_238080cb5ec57e69e5a15d0238ef4b463ee7c0d5b54664c9cf0f84b3a9c36acd.jpg b/.raw/MidasCivilAnalysisReference/images/page-028_238080cb5ec57e69e5a15d0238ef4b463ee7c0d5b54664c9cf0f84b3a9c36acd.jpg new file mode 100644 index 00000000..64d5bb31 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-028_238080cb5ec57e69e5a15d0238ef4b463ee7c0d5b54664c9cf0f84b3a9c36acd.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-028_3d8e4f101ff36f42b031d13f7a59b476ddb52a85e6b4fad2541d1a7285c49e4e.jpg b/.raw/MidasCivilAnalysisReference/images/page-028_3d8e4f101ff36f42b031d13f7a59b476ddb52a85e6b4fad2541d1a7285c49e4e.jpg new file mode 100644 index 00000000..05607546 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-028_3d8e4f101ff36f42b031d13f7a59b476ddb52a85e6b4fad2541d1a7285c49e4e.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-028_8ae1567f4f0496d9728ecc437109b1b66a358d18d014855613eb90c32adafafb.jpg b/.raw/MidasCivilAnalysisReference/images/page-028_8ae1567f4f0496d9728ecc437109b1b66a358d18d014855613eb90c32adafafb.jpg new file mode 100644 index 00000000..8f3f4cf4 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-028_8ae1567f4f0496d9728ecc437109b1b66a358d18d014855613eb90c32adafafb.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-028_8d3a8adc50abdd06df7cbd1c9248345b0579f4af1f224b9ab022e4d1872c12b0.jpg b/.raw/MidasCivilAnalysisReference/images/page-028_8d3a8adc50abdd06df7cbd1c9248345b0579f4af1f224b9ab022e4d1872c12b0.jpg new file mode 100644 index 00000000..8cf11b94 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-028_8d3a8adc50abdd06df7cbd1c9248345b0579f4af1f224b9ab022e4d1872c12b0.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-028_c735a9aec2feac9d6a1748cf92494d61c816f00715c7c801a3967f51b913ba0c.jpg b/.raw/MidasCivilAnalysisReference/images/page-028_c735a9aec2feac9d6a1748cf92494d61c816f00715c7c801a3967f51b913ba0c.jpg new file mode 100644 index 00000000..8bf06266 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-028_c735a9aec2feac9d6a1748cf92494d61c816f00715c7c801a3967f51b913ba0c.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-028_dff747947e559eeba97fa22c5c23235f0e597d4bbf0b4747e5a813f579b9db5e.jpg b/.raw/MidasCivilAnalysisReference/images/page-028_dff747947e559eeba97fa22c5c23235f0e597d4bbf0b4747e5a813f579b9db5e.jpg new file mode 100644 index 00000000..9f6fa508 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-028_dff747947e559eeba97fa22c5c23235f0e597d4bbf0b4747e5a813f579b9db5e.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-028_ebe154abbdc4139cc7a540fc177d324ed8deb5b676c06234ebe7a85d03df556c.jpg b/.raw/MidasCivilAnalysisReference/images/page-028_ebe154abbdc4139cc7a540fc177d324ed8deb5b676c06234ebe7a85d03df556c.jpg new file mode 100644 index 00000000..aa66f97f Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-028_ebe154abbdc4139cc7a540fc177d324ed8deb5b676c06234ebe7a85d03df556c.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-029_23227befed8d207c472ccaaf89ff4566937603ebd1da1962b478d930bf72fdf1.jpg b/.raw/MidasCivilAnalysisReference/images/page-029_23227befed8d207c472ccaaf89ff4566937603ebd1da1962b478d930bf72fdf1.jpg new file mode 100644 index 00000000..04341f06 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-029_23227befed8d207c472ccaaf89ff4566937603ebd1da1962b478d930bf72fdf1.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-029_2e2dfcbaa16250b406f3a148b34b07a38fdeaf30a5b2ab51ebbf2fb1ab11a3c9.jpg b/.raw/MidasCivilAnalysisReference/images/page-029_2e2dfcbaa16250b406f3a148b34b07a38fdeaf30a5b2ab51ebbf2fb1ab11a3c9.jpg new file mode 100644 index 00000000..996588d4 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-029_2e2dfcbaa16250b406f3a148b34b07a38fdeaf30a5b2ab51ebbf2fb1ab11a3c9.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-029_56fae780a96d795d54f46af5107b1f17370fb6a3d7c442d2661001848396f142.jpg b/.raw/MidasCivilAnalysisReference/images/page-029_56fae780a96d795d54f46af5107b1f17370fb6a3d7c442d2661001848396f142.jpg new file mode 100644 index 00000000..7dd22d42 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-029_56fae780a96d795d54f46af5107b1f17370fb6a3d7c442d2661001848396f142.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-029_6c5554162e7a77f9f215a703bc3d8b0f2ebe05d23dbda6b8fc91d76d0c88419c.jpg b/.raw/MidasCivilAnalysisReference/images/page-029_6c5554162e7a77f9f215a703bc3d8b0f2ebe05d23dbda6b8fc91d76d0c88419c.jpg new file mode 100644 index 00000000..77c4050c Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-029_6c5554162e7a77f9f215a703bc3d8b0f2ebe05d23dbda6b8fc91d76d0c88419c.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-029_ae7baf2679898f33801a05b198cb573f7f82ba214b178651c30b4c45a966b96f.jpg b/.raw/MidasCivilAnalysisReference/images/page-029_ae7baf2679898f33801a05b198cb573f7f82ba214b178651c30b4c45a966b96f.jpg new file mode 100644 index 00000000..c27e1fc6 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-029_ae7baf2679898f33801a05b198cb573f7f82ba214b178651c30b4c45a966b96f.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-029_cb9c31c209345f0d470cbcdcde07da3f36d577b589d9941f9c0c75380fdde8f3.jpg b/.raw/MidasCivilAnalysisReference/images/page-029_cb9c31c209345f0d470cbcdcde07da3f36d577b589d9941f9c0c75380fdde8f3.jpg new file mode 100644 index 00000000..dd5fafe5 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-029_cb9c31c209345f0d470cbcdcde07da3f36d577b589d9941f9c0c75380fdde8f3.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-029_cf543706f9f4f6c0e77787cc8ef3fa2136db9ff6b82e74576b899819ed456adc.jpg b/.raw/MidasCivilAnalysisReference/images/page-029_cf543706f9f4f6c0e77787cc8ef3fa2136db9ff6b82e74576b899819ed456adc.jpg new file mode 100644 index 00000000..f32359fd Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-029_cf543706f9f4f6c0e77787cc8ef3fa2136db9ff6b82e74576b899819ed456adc.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-029_d591e1a98a24cdb5cb9dac96caa090a35fc61f98473acd0a2d968794cf894948.jpg b/.raw/MidasCivilAnalysisReference/images/page-029_d591e1a98a24cdb5cb9dac96caa090a35fc61f98473acd0a2d968794cf894948.jpg new file mode 100644 index 00000000..08e740ca Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-029_d591e1a98a24cdb5cb9dac96caa090a35fc61f98473acd0a2d968794cf894948.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-029_f1f96ad6bed50bd1e44c514481eec7ccf4838c608b7bb4e1c0f921f4364c44ae.jpg b/.raw/MidasCivilAnalysisReference/images/page-029_f1f96ad6bed50bd1e44c514481eec7ccf4838c608b7bb4e1c0f921f4364c44ae.jpg new file mode 100644 index 00000000..fb49084b Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-029_f1f96ad6bed50bd1e44c514481eec7ccf4838c608b7bb4e1c0f921f4364c44ae.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-030_09addca9c1c972e1939c9c4f57ebeb4fd39a6281cf5fcffce8848d337b1a2d33.jpg b/.raw/MidasCivilAnalysisReference/images/page-030_09addca9c1c972e1939c9c4f57ebeb4fd39a6281cf5fcffce8848d337b1a2d33.jpg new file mode 100644 index 00000000..1e677a09 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-030_09addca9c1c972e1939c9c4f57ebeb4fd39a6281cf5fcffce8848d337b1a2d33.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-030_229ca30cab63298087d2cb2fa5ae16f23706dba21552a8ad5162ef5d53df0294.jpg b/.raw/MidasCivilAnalysisReference/images/page-030_229ca30cab63298087d2cb2fa5ae16f23706dba21552a8ad5162ef5d53df0294.jpg new file mode 100644 index 00000000..1b0eaa78 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-030_229ca30cab63298087d2cb2fa5ae16f23706dba21552a8ad5162ef5d53df0294.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-030_9e2ad6f56ff56acefc412fa17588cbd3d581751bb250373486d0572f707ef8bf.jpg b/.raw/MidasCivilAnalysisReference/images/page-030_9e2ad6f56ff56acefc412fa17588cbd3d581751bb250373486d0572f707ef8bf.jpg new file mode 100644 index 00000000..7262fcdb Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-030_9e2ad6f56ff56acefc412fa17588cbd3d581751bb250373486d0572f707ef8bf.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-030_b19fb0751269781019eae0ebbb165f90fd4b9bd5bd9e372fd9f5d6e31779bb87.jpg b/.raw/MidasCivilAnalysisReference/images/page-030_b19fb0751269781019eae0ebbb165f90fd4b9bd5bd9e372fd9f5d6e31779bb87.jpg new file mode 100644 index 00000000..3e80f63a Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-030_b19fb0751269781019eae0ebbb165f90fd4b9bd5bd9e372fd9f5d6e31779bb87.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-030_c6c1a29b15fd6a8d3fe2d2dc1a5fa14ec74951a9b0c759c5125cef7f64a20ea8.jpg b/.raw/MidasCivilAnalysisReference/images/page-030_c6c1a29b15fd6a8d3fe2d2dc1a5fa14ec74951a9b0c759c5125cef7f64a20ea8.jpg new file mode 100644 index 00000000..94485b9b Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-030_c6c1a29b15fd6a8d3fe2d2dc1a5fa14ec74951a9b0c759c5125cef7f64a20ea8.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-030_e96297a0ba91e0123237760f43b23c92bdc58d7b142ddb8ef5ab071c74989805.jpg b/.raw/MidasCivilAnalysisReference/images/page-030_e96297a0ba91e0123237760f43b23c92bdc58d7b142ddb8ef5ab071c74989805.jpg new file mode 100644 index 00000000..2e300a92 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-030_e96297a0ba91e0123237760f43b23c92bdc58d7b142ddb8ef5ab071c74989805.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-030_eb38ff37efa13451fce56ff4b34d83f8bee87aba3f8410a10e2567918472ab0c.jpg b/.raw/MidasCivilAnalysisReference/images/page-030_eb38ff37efa13451fce56ff4b34d83f8bee87aba3f8410a10e2567918472ab0c.jpg new file mode 100644 index 00000000..83878ee8 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-030_eb38ff37efa13451fce56ff4b34d83f8bee87aba3f8410a10e2567918472ab0c.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-030_f98ba1ee684b8afecd07b8ea2e78715da5ea0928099dece583bb3b628e82bda0.jpg b/.raw/MidasCivilAnalysisReference/images/page-030_f98ba1ee684b8afecd07b8ea2e78715da5ea0928099dece583bb3b628e82bda0.jpg new file mode 100644 index 00000000..3604332a Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-030_f98ba1ee684b8afecd07b8ea2e78715da5ea0928099dece583bb3b628e82bda0.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-032_0ded6619645f01c9fc4fd949a8ab2629c3b0153d068d7409ddee58220ab70a7e.jpg b/.raw/MidasCivilAnalysisReference/images/page-032_0ded6619645f01c9fc4fd949a8ab2629c3b0153d068d7409ddee58220ab70a7e.jpg new file mode 100644 index 00000000..d8137052 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-032_0ded6619645f01c9fc4fd949a8ab2629c3b0153d068d7409ddee58220ab70a7e.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-032_a8ee8c223860ca09f573aa76835448d38d48ff1644b2735ec0b22c55141896f7.jpg b/.raw/MidasCivilAnalysisReference/images/page-032_a8ee8c223860ca09f573aa76835448d38d48ff1644b2735ec0b22c55141896f7.jpg new file mode 100644 index 00000000..70b23941 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-032_a8ee8c223860ca09f573aa76835448d38d48ff1644b2735ec0b22c55141896f7.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-034_88a8b5d5c9184fcd91ab0f1e2f1bc69c1e7031a5161103ec2355806f6946cf9d.jpg b/.raw/MidasCivilAnalysisReference/images/page-034_88a8b5d5c9184fcd91ab0f1e2f1bc69c1e7031a5161103ec2355806f6946cf9d.jpg new file mode 100644 index 00000000..1cc95ddd Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-034_88a8b5d5c9184fcd91ab0f1e2f1bc69c1e7031a5161103ec2355806f6946cf9d.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-035_c35d20dde99c1208a95764cdca9b21fc1fb005e546cf3a1d1b27cdc33dbd79ab.jpg b/.raw/MidasCivilAnalysisReference/images/page-035_c35d20dde99c1208a95764cdca9b21fc1fb005e546cf3a1d1b27cdc33dbd79ab.jpg new file mode 100644 index 00000000..e2915c5f Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-035_c35d20dde99c1208a95764cdca9b21fc1fb005e546cf3a1d1b27cdc33dbd79ab.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-036_6919f021027bc1aca0e9a91d022457d1f1db0ae3262c980b9b177fa186fa5443.jpg b/.raw/MidasCivilAnalysisReference/images/page-036_6919f021027bc1aca0e9a91d022457d1f1db0ae3262c980b9b177fa186fa5443.jpg new file mode 100644 index 00000000..4d47767a Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-036_6919f021027bc1aca0e9a91d022457d1f1db0ae3262c980b9b177fa186fa5443.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-036_de50dd18c71c3855c9149f996cd31a21f81e935b7dc00b91e1781e406670b32b.jpg b/.raw/MidasCivilAnalysisReference/images/page-036_de50dd18c71c3855c9149f996cd31a21f81e935b7dc00b91e1781e406670b32b.jpg new file mode 100644 index 00000000..67b7ebf3 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-036_de50dd18c71c3855c9149f996cd31a21f81e935b7dc00b91e1781e406670b32b.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-038_8e1cea396a0724f919ac55eeb47499529be69bae47c06e6580ba2f6f1cec367d.jpg b/.raw/MidasCivilAnalysisReference/images/page-038_8e1cea396a0724f919ac55eeb47499529be69bae47c06e6580ba2f6f1cec367d.jpg new file mode 100644 index 00000000..c2f16459 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-038_8e1cea396a0724f919ac55eeb47499529be69bae47c06e6580ba2f6f1cec367d.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-038_b2bc54418bb878e4838704f3bda76895ef8113aec8728a07b106493ee8f99c15.jpg b/.raw/MidasCivilAnalysisReference/images/page-038_b2bc54418bb878e4838704f3bda76895ef8113aec8728a07b106493ee8f99c15.jpg new file mode 100644 index 00000000..cc92fac5 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-038_b2bc54418bb878e4838704f3bda76895ef8113aec8728a07b106493ee8f99c15.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-039_1f985dc8243010c5dda82d5c735c9ecba89a74656c4a929f737604d0e55e1f73.jpg b/.raw/MidasCivilAnalysisReference/images/page-039_1f985dc8243010c5dda82d5c735c9ecba89a74656c4a929f737604d0e55e1f73.jpg new file mode 100644 index 00000000..52382dfd Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-039_1f985dc8243010c5dda82d5c735c9ecba89a74656c4a929f737604d0e55e1f73.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-039_37d3cb9d141600a144d408bdc9d9469e6a4903834b81e84aceaa20f47cf28041.jpg b/.raw/MidasCivilAnalysisReference/images/page-039_37d3cb9d141600a144d408bdc9d9469e6a4903834b81e84aceaa20f47cf28041.jpg new file mode 100644 index 00000000..978986e6 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-039_37d3cb9d141600a144d408bdc9d9469e6a4903834b81e84aceaa20f47cf28041.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-040_41d5a9cdb1bc3cd2c7a9b02a810ca81a9985eb37f81baee05a57fb5d8585fde1.jpg b/.raw/MidasCivilAnalysisReference/images/page-040_41d5a9cdb1bc3cd2c7a9b02a810ca81a9985eb37f81baee05a57fb5d8585fde1.jpg new file mode 100644 index 00000000..e6349179 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-040_41d5a9cdb1bc3cd2c7a9b02a810ca81a9985eb37f81baee05a57fb5d8585fde1.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-040_7d09c8679826be36081e55a2f91e12dd1d4f644725c32e1109ccde52e9538422.jpg b/.raw/MidasCivilAnalysisReference/images/page-040_7d09c8679826be36081e55a2f91e12dd1d4f644725c32e1109ccde52e9538422.jpg new file mode 100644 index 00000000..555b0466 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-040_7d09c8679826be36081e55a2f91e12dd1d4f644725c32e1109ccde52e9538422.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-040_8b9f7566b6e4166ec090d218b41f8083d74ec4ff925cb5e400bb569d635985e9.jpg b/.raw/MidasCivilAnalysisReference/images/page-040_8b9f7566b6e4166ec090d218b41f8083d74ec4ff925cb5e400bb569d635985e9.jpg new file mode 100644 index 00000000..7be7dc00 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-040_8b9f7566b6e4166ec090d218b41f8083d74ec4ff925cb5e400bb569d635985e9.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-040_e1ef1e99bc297c9a9e24fbe55dab64f40a3169e4258ebd27ac15ee81cac061ef.jpg b/.raw/MidasCivilAnalysisReference/images/page-040_e1ef1e99bc297c9a9e24fbe55dab64f40a3169e4258ebd27ac15ee81cac061ef.jpg new file mode 100644 index 00000000..d48e831d Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-040_e1ef1e99bc297c9a9e24fbe55dab64f40a3169e4258ebd27ac15ee81cac061ef.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-041_0cb2a53f430595f0627694514553e9be8a555d4c89859be188a19416ff6b1aa7.jpg b/.raw/MidasCivilAnalysisReference/images/page-041_0cb2a53f430595f0627694514553e9be8a555d4c89859be188a19416ff6b1aa7.jpg new file mode 100644 index 00000000..9fba81c9 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-041_0cb2a53f430595f0627694514553e9be8a555d4c89859be188a19416ff6b1aa7.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-041_332f6b92c967ad97a6d98f4e66a502702fbf38b4d5e89e0f6efc9eb2827c5b4f.jpg b/.raw/MidasCivilAnalysisReference/images/page-041_332f6b92c967ad97a6d98f4e66a502702fbf38b4d5e89e0f6efc9eb2827c5b4f.jpg new file mode 100644 index 00000000..f9d46cb3 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-041_332f6b92c967ad97a6d98f4e66a502702fbf38b4d5e89e0f6efc9eb2827c5b4f.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-041_a61117c569826167a7a9129a01ca2d3646e4de0d8bb10c0d675c166fd4e815ab.jpg b/.raw/MidasCivilAnalysisReference/images/page-041_a61117c569826167a7a9129a01ca2d3646e4de0d8bb10c0d675c166fd4e815ab.jpg new file mode 100644 index 00000000..78ac5ee3 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-041_a61117c569826167a7a9129a01ca2d3646e4de0d8bb10c0d675c166fd4e815ab.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-041_d42a82eefdc867cb740392582446f30550d029222eefd1c77b7fe317e4e9c871.jpg b/.raw/MidasCivilAnalysisReference/images/page-041_d42a82eefdc867cb740392582446f30550d029222eefd1c77b7fe317e4e9c871.jpg new file mode 100644 index 00000000..06d311e6 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-041_d42a82eefdc867cb740392582446f30550d029222eefd1c77b7fe317e4e9c871.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-042_6bbeb739eb3670ec438016c941677e44692009a9f295cfc5b414d909c797565e.jpg b/.raw/MidasCivilAnalysisReference/images/page-042_6bbeb739eb3670ec438016c941677e44692009a9f295cfc5b414d909c797565e.jpg new file mode 100644 index 00000000..e3698400 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-042_6bbeb739eb3670ec438016c941677e44692009a9f295cfc5b414d909c797565e.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-042_ade53cd04392763f440474eeb8701090f823904b23ea3af239cb82635b99e02d.jpg b/.raw/MidasCivilAnalysisReference/images/page-042_ade53cd04392763f440474eeb8701090f823904b23ea3af239cb82635b99e02d.jpg new file mode 100644 index 00000000..407c0d58 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-042_ade53cd04392763f440474eeb8701090f823904b23ea3af239cb82635b99e02d.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-042_bb7b91554814ec66e1b9be54895d92f54ac306d536f20ba6ce4f306215f2fdfc.jpg b/.raw/MidasCivilAnalysisReference/images/page-042_bb7b91554814ec66e1b9be54895d92f54ac306d536f20ba6ce4f306215f2fdfc.jpg new file mode 100644 index 00000000..c7fa3ed9 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-042_bb7b91554814ec66e1b9be54895d92f54ac306d536f20ba6ce4f306215f2fdfc.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-042_d26527bef04edd3bd1f4597d27747767a8b66a075bb8464e64af0cd337e72748.jpg b/.raw/MidasCivilAnalysisReference/images/page-042_d26527bef04edd3bd1f4597d27747767a8b66a075bb8464e64af0cd337e72748.jpg new file mode 100644 index 00000000..a28605f5 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-042_d26527bef04edd3bd1f4597d27747767a8b66a075bb8464e64af0cd337e72748.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-043_32080caa04ec56e2f301cecaf5d50045cbc7cfa9ec17fd3c335330ed388df73f.jpg b/.raw/MidasCivilAnalysisReference/images/page-043_32080caa04ec56e2f301cecaf5d50045cbc7cfa9ec17fd3c335330ed388df73f.jpg new file mode 100644 index 00000000..99b227f7 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-043_32080caa04ec56e2f301cecaf5d50045cbc7cfa9ec17fd3c335330ed388df73f.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-043_9a4f8576c96207f7b12fc9e72e24b144443e7afc1202d74c7d046ddc9f65bb61.jpg b/.raw/MidasCivilAnalysisReference/images/page-043_9a4f8576c96207f7b12fc9e72e24b144443e7afc1202d74c7d046ddc9f65bb61.jpg new file mode 100644 index 00000000..127fbc79 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-043_9a4f8576c96207f7b12fc9e72e24b144443e7afc1202d74c7d046ddc9f65bb61.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-044_736eff59cc4118578cbf5fd3bb3f6115e273ae7af4ede4fcf08536c6375272ea.jpg b/.raw/MidasCivilAnalysisReference/images/page-044_736eff59cc4118578cbf5fd3bb3f6115e273ae7af4ede4fcf08536c6375272ea.jpg new file mode 100644 index 00000000..2f5cbfc6 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-044_736eff59cc4118578cbf5fd3bb3f6115e273ae7af4ede4fcf08536c6375272ea.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-044_73c197e349ce94121cb4cce185d80b5fe54c383c3495e0bc17d7b2be68480173.jpg b/.raw/MidasCivilAnalysisReference/images/page-044_73c197e349ce94121cb4cce185d80b5fe54c383c3495e0bc17d7b2be68480173.jpg new file mode 100644 index 00000000..ecbf68bd Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-044_73c197e349ce94121cb4cce185d80b5fe54c383c3495e0bc17d7b2be68480173.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-045_30a17ccecf0303f57917e5e25c16f13b931afde9a216c418123d917f2d918947.jpg b/.raw/MidasCivilAnalysisReference/images/page-045_30a17ccecf0303f57917e5e25c16f13b931afde9a216c418123d917f2d918947.jpg new file mode 100644 index 00000000..226f9470 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-045_30a17ccecf0303f57917e5e25c16f13b931afde9a216c418123d917f2d918947.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-045_a06cfa2fae9bf4ea92460c53116c09fa1952a574d5d613281f748a5120cd9f12.jpg b/.raw/MidasCivilAnalysisReference/images/page-045_a06cfa2fae9bf4ea92460c53116c09fa1952a574d5d613281f748a5120cd9f12.jpg new file mode 100644 index 00000000..f70716db Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-045_a06cfa2fae9bf4ea92460c53116c09fa1952a574d5d613281f748a5120cd9f12.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-046_cf4a14e7aabdfaee985c5fa2763909fa29bf489c90968c7abdc67268ffa1683c.jpg b/.raw/MidasCivilAnalysisReference/images/page-046_cf4a14e7aabdfaee985c5fa2763909fa29bf489c90968c7abdc67268ffa1683c.jpg new file mode 100644 index 00000000..29c2cd4f Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-046_cf4a14e7aabdfaee985c5fa2763909fa29bf489c90968c7abdc67268ffa1683c.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-048_6bdf5048281a6eff23fa9ff9f0cfcb22a3d2b146e48e5d608734273440cf1ba3.jpg b/.raw/MidasCivilAnalysisReference/images/page-048_6bdf5048281a6eff23fa9ff9f0cfcb22a3d2b146e48e5d608734273440cf1ba3.jpg new file mode 100644 index 00000000..d4b73541 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-048_6bdf5048281a6eff23fa9ff9f0cfcb22a3d2b146e48e5d608734273440cf1ba3.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-048_9139140b1fa77709f27ed5dfa924bc88005dd2cddea9473a402430896cccd21d.jpg b/.raw/MidasCivilAnalysisReference/images/page-048_9139140b1fa77709f27ed5dfa924bc88005dd2cddea9473a402430896cccd21d.jpg new file mode 100644 index 00000000..9fdba7b7 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-048_9139140b1fa77709f27ed5dfa924bc88005dd2cddea9473a402430896cccd21d.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-049_e9cb6693dbdcdf5c4226579221c1bc60ac8d0aebfc46b54145f529a3447b60a6.jpg b/.raw/MidasCivilAnalysisReference/images/page-049_e9cb6693dbdcdf5c4226579221c1bc60ac8d0aebfc46b54145f529a3447b60a6.jpg new file mode 100644 index 00000000..4c936f7d Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-049_e9cb6693dbdcdf5c4226579221c1bc60ac8d0aebfc46b54145f529a3447b60a6.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-049_ed8a9df00858c8baba840ac944a0b8a0c98bc9168de7df748e6c57eaf3526953.jpg b/.raw/MidasCivilAnalysisReference/images/page-049_ed8a9df00858c8baba840ac944a0b8a0c98bc9168de7df748e6c57eaf3526953.jpg new file mode 100644 index 00000000..4d86c3c7 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-049_ed8a9df00858c8baba840ac944a0b8a0c98bc9168de7df748e6c57eaf3526953.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-050_20e2205c4bf851e9219b342e99938a5fadb7b731ebf9edd5fe58baefb29fbedc.jpg b/.raw/MidasCivilAnalysisReference/images/page-050_20e2205c4bf851e9219b342e99938a5fadb7b731ebf9edd5fe58baefb29fbedc.jpg new file mode 100644 index 00000000..b7c5658e Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-050_20e2205c4bf851e9219b342e99938a5fadb7b731ebf9edd5fe58baefb29fbedc.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-050_24318f7b1cf8fcba1680bec5efc72a4cd2b4f3d052abc1a6323f13d6a35c0994.jpg b/.raw/MidasCivilAnalysisReference/images/page-050_24318f7b1cf8fcba1680bec5efc72a4cd2b4f3d052abc1a6323f13d6a35c0994.jpg new file mode 100644 index 00000000..9eac8b51 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-050_24318f7b1cf8fcba1680bec5efc72a4cd2b4f3d052abc1a6323f13d6a35c0994.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-050_83bcd0a03bd1dd4095165951d7a890764d963be836702813b7c356d6d680ff7e.jpg b/.raw/MidasCivilAnalysisReference/images/page-050_83bcd0a03bd1dd4095165951d7a890764d963be836702813b7c356d6d680ff7e.jpg new file mode 100644 index 00000000..f3be7ab5 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-050_83bcd0a03bd1dd4095165951d7a890764d963be836702813b7c356d6d680ff7e.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-050_d81e17e63d567225847d8d35fcda0181c14162d4678bc52a0aec8b403588143a.jpg b/.raw/MidasCivilAnalysisReference/images/page-050_d81e17e63d567225847d8d35fcda0181c14162d4678bc52a0aec8b403588143a.jpg new file mode 100644 index 00000000..4189cb25 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-050_d81e17e63d567225847d8d35fcda0181c14162d4678bc52a0aec8b403588143a.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-051_4a0a5af8ba5e35d841efe1811b0ae1edd1a80fe3c85619bfc0b36db1b27e06ee.jpg b/.raw/MidasCivilAnalysisReference/images/page-051_4a0a5af8ba5e35d841efe1811b0ae1edd1a80fe3c85619bfc0b36db1b27e06ee.jpg new file mode 100644 index 00000000..5df15fd1 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-051_4a0a5af8ba5e35d841efe1811b0ae1edd1a80fe3c85619bfc0b36db1b27e06ee.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-051_734d44d6ebfa0ef6f11a72402fcfad437ec6c23af4418283ff3932155df36625.jpg b/.raw/MidasCivilAnalysisReference/images/page-051_734d44d6ebfa0ef6f11a72402fcfad437ec6c23af4418283ff3932155df36625.jpg new file mode 100644 index 00000000..cdb89092 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-051_734d44d6ebfa0ef6f11a72402fcfad437ec6c23af4418283ff3932155df36625.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-051_7a232068999ecf7916f220af24536eb269308ef97f8c01fb8631d78c36346c65.jpg b/.raw/MidasCivilAnalysisReference/images/page-051_7a232068999ecf7916f220af24536eb269308ef97f8c01fb8631d78c36346c65.jpg new file mode 100644 index 00000000..2be2f1fa Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-051_7a232068999ecf7916f220af24536eb269308ef97f8c01fb8631d78c36346c65.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-051_82236043434b2fa43209c15457f534935b6ec3567e9787acc1fd6d006fe39b79.jpg b/.raw/MidasCivilAnalysisReference/images/page-051_82236043434b2fa43209c15457f534935b6ec3567e9787acc1fd6d006fe39b79.jpg new file mode 100644 index 00000000..2da6c3b8 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-051_82236043434b2fa43209c15457f534935b6ec3567e9787acc1fd6d006fe39b79.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-051_8efea8bc80b6ceb18acb94baf1889cb3776ca7d6f8195cbde4a50f29597be333.jpg b/.raw/MidasCivilAnalysisReference/images/page-051_8efea8bc80b6ceb18acb94baf1889cb3776ca7d6f8195cbde4a50f29597be333.jpg new file mode 100644 index 00000000..1b5e2a90 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-051_8efea8bc80b6ceb18acb94baf1889cb3776ca7d6f8195cbde4a50f29597be333.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-051_d08b4a7a2dc12d9a76b7583218343b6b92de30386c205a1c5ff66cb53638bb50.jpg b/.raw/MidasCivilAnalysisReference/images/page-051_d08b4a7a2dc12d9a76b7583218343b6b92de30386c205a1c5ff66cb53638bb50.jpg new file mode 100644 index 00000000..bbf974cf Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-051_d08b4a7a2dc12d9a76b7583218343b6b92de30386c205a1c5ff66cb53638bb50.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-051_d3bc5c0c097f92f6824db0b8648d2d543f94077e35d7762ddc66d25eace5687e.jpg b/.raw/MidasCivilAnalysisReference/images/page-051_d3bc5c0c097f92f6824db0b8648d2d543f94077e35d7762ddc66d25eace5687e.jpg new file mode 100644 index 00000000..9817d076 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-051_d3bc5c0c097f92f6824db0b8648d2d543f94077e35d7762ddc66d25eace5687e.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-051_efd051425f58cc453461716bb0c49f9aa32eeb8a3e62d709a19615bfc6aa892f.jpg b/.raw/MidasCivilAnalysisReference/images/page-051_efd051425f58cc453461716bb0c49f9aa32eeb8a3e62d709a19615bfc6aa892f.jpg new file mode 100644 index 00000000..12e63b22 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-051_efd051425f58cc453461716bb0c49f9aa32eeb8a3e62d709a19615bfc6aa892f.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-052_0b697ec51c9df455fdde04a2e23b162cfe85dbf4e7b914c753508fbb9ef93dab.jpg b/.raw/MidasCivilAnalysisReference/images/page-052_0b697ec51c9df455fdde04a2e23b162cfe85dbf4e7b914c753508fbb9ef93dab.jpg new file mode 100644 index 00000000..0ffd058e Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-052_0b697ec51c9df455fdde04a2e23b162cfe85dbf4e7b914c753508fbb9ef93dab.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-052_28527dad12f58192e1d7172232a6f64fbbcb88250ad844821abda428a24432ea.jpg b/.raw/MidasCivilAnalysisReference/images/page-052_28527dad12f58192e1d7172232a6f64fbbcb88250ad844821abda428a24432ea.jpg new file mode 100644 index 00000000..db70568e Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-052_28527dad12f58192e1d7172232a6f64fbbcb88250ad844821abda428a24432ea.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-052_88c3e5956fe494064a462c0307df9c192668587e9101b1554430072ffb7662e8.jpg b/.raw/MidasCivilAnalysisReference/images/page-052_88c3e5956fe494064a462c0307df9c192668587e9101b1554430072ffb7662e8.jpg new file mode 100644 index 00000000..fb3de1bd Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-052_88c3e5956fe494064a462c0307df9c192668587e9101b1554430072ffb7662e8.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-052_8ed39d8c6a214ca8637b01a9fe8766c94922c867b0fc81944088d2f5a2d24eb3.jpg b/.raw/MidasCivilAnalysisReference/images/page-052_8ed39d8c6a214ca8637b01a9fe8766c94922c867b0fc81944088d2f5a2d24eb3.jpg new file mode 100644 index 00000000..9f0503be Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-052_8ed39d8c6a214ca8637b01a9fe8766c94922c867b0fc81944088d2f5a2d24eb3.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-053_f07c0b98c84afa0cc8b529a08a1c80c2830332447211798318de5469418273ae.jpg b/.raw/MidasCivilAnalysisReference/images/page-053_f07c0b98c84afa0cc8b529a08a1c80c2830332447211798318de5469418273ae.jpg new file mode 100644 index 00000000..306d5f03 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-053_f07c0b98c84afa0cc8b529a08a1c80c2830332447211798318de5469418273ae.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-054_130eb60266cc4dbed9cd42d3302b11b560f80a7321bad533bcb1e2acf8af2aa8.jpg b/.raw/MidasCivilAnalysisReference/images/page-054_130eb60266cc4dbed9cd42d3302b11b560f80a7321bad533bcb1e2acf8af2aa8.jpg new file mode 100644 index 00000000..71aa920c Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-054_130eb60266cc4dbed9cd42d3302b11b560f80a7321bad533bcb1e2acf8af2aa8.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-054_21f2c5a33262bd26f0dcfb4cc9f24ddf62c8df604e1314880879fadaccdf8f46.jpg b/.raw/MidasCivilAnalysisReference/images/page-054_21f2c5a33262bd26f0dcfb4cc9f24ddf62c8df604e1314880879fadaccdf8f46.jpg new file mode 100644 index 00000000..0d8e5a2d Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-054_21f2c5a33262bd26f0dcfb4cc9f24ddf62c8df604e1314880879fadaccdf8f46.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-054_2ed9467266920246cf0b8dc11d9926a1034a2881de5d8989a55faa21ab50d3d4.jpg b/.raw/MidasCivilAnalysisReference/images/page-054_2ed9467266920246cf0b8dc11d9926a1034a2881de5d8989a55faa21ab50d3d4.jpg new file mode 100644 index 00000000..8b77c78d Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-054_2ed9467266920246cf0b8dc11d9926a1034a2881de5d8989a55faa21ab50d3d4.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-054_3a9a3acaa0f73362216eadf648281cdca8c1cd47654e2065ba230bb8547c6008.jpg b/.raw/MidasCivilAnalysisReference/images/page-054_3a9a3acaa0f73362216eadf648281cdca8c1cd47654e2065ba230bb8547c6008.jpg new file mode 100644 index 00000000..a95299c1 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-054_3a9a3acaa0f73362216eadf648281cdca8c1cd47654e2065ba230bb8547c6008.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-054_faa848eea1545b6eb2820e052016068f01c1e283310efaa6969b7f8054b6921b.jpg b/.raw/MidasCivilAnalysisReference/images/page-054_faa848eea1545b6eb2820e052016068f01c1e283310efaa6969b7f8054b6921b.jpg new file mode 100644 index 00000000..14c06d02 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-054_faa848eea1545b6eb2820e052016068f01c1e283310efaa6969b7f8054b6921b.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-055_10516abd191c6aaf96a9b9710c03099f62fe10391ac273879c64db194fc16f7d.jpg b/.raw/MidasCivilAnalysisReference/images/page-055_10516abd191c6aaf96a9b9710c03099f62fe10391ac273879c64db194fc16f7d.jpg new file mode 100644 index 00000000..01f43db7 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-055_10516abd191c6aaf96a9b9710c03099f62fe10391ac273879c64db194fc16f7d.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-055_3fb1bb97088b5ba80053fcd322f78750a4da2a020d1b3332bcf663b5253817fa.jpg b/.raw/MidasCivilAnalysisReference/images/page-055_3fb1bb97088b5ba80053fcd322f78750a4da2a020d1b3332bcf663b5253817fa.jpg new file mode 100644 index 00000000..da5b5d24 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-055_3fb1bb97088b5ba80053fcd322f78750a4da2a020d1b3332bcf663b5253817fa.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-057_ac00094b8247f94aea221fdcc1c875c38b8e6de06b02a03eaa243d98618c4fde.jpg b/.raw/MidasCivilAnalysisReference/images/page-057_ac00094b8247f94aea221fdcc1c875c38b8e6de06b02a03eaa243d98618c4fde.jpg new file mode 100644 index 00000000..42985097 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-057_ac00094b8247f94aea221fdcc1c875c38b8e6de06b02a03eaa243d98618c4fde.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-058_d024cc4d1fea99300a41342e44a1525b2306327d5ef2f80ba0d91e308dc2ab77.jpg b/.raw/MidasCivilAnalysisReference/images/page-058_d024cc4d1fea99300a41342e44a1525b2306327d5ef2f80ba0d91e308dc2ab77.jpg new file mode 100644 index 00000000..bfa2ea62 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-058_d024cc4d1fea99300a41342e44a1525b2306327d5ef2f80ba0d91e308dc2ab77.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-059_352579de9600c71406f2a000789e17dd8f360df9837f5238ce51fea4cc752b68.jpg b/.raw/MidasCivilAnalysisReference/images/page-059_352579de9600c71406f2a000789e17dd8f360df9837f5238ce51fea4cc752b68.jpg new file mode 100644 index 00000000..4c936f7d Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-059_352579de9600c71406f2a000789e17dd8f360df9837f5238ce51fea4cc752b68.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-059_7aac4086e8a712f361d222ca4bd04e4eed8dce4efe92ea1179526ec9abf13b61.jpg b/.raw/MidasCivilAnalysisReference/images/page-059_7aac4086e8a712f361d222ca4bd04e4eed8dce4efe92ea1179526ec9abf13b61.jpg new file mode 100644 index 00000000..9c3d46f6 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-059_7aac4086e8a712f361d222ca4bd04e4eed8dce4efe92ea1179526ec9abf13b61.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-061_2341288f4cd01c03d6390d181d0ddc52693dc0389b079fff84fc03077b6ac6bc.jpg b/.raw/MidasCivilAnalysisReference/images/page-061_2341288f4cd01c03d6390d181d0ddc52693dc0389b079fff84fc03077b6ac6bc.jpg new file mode 100644 index 00000000..4b0227b6 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-061_2341288f4cd01c03d6390d181d0ddc52693dc0389b079fff84fc03077b6ac6bc.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-061_7a978bd94f10b4c78cc2f1950cbc4eb4818874f45c6ac463ec5ff3ab50eb3954.jpg b/.raw/MidasCivilAnalysisReference/images/page-061_7a978bd94f10b4c78cc2f1950cbc4eb4818874f45c6ac463ec5ff3ab50eb3954.jpg new file mode 100644 index 00000000..35d562f7 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-061_7a978bd94f10b4c78cc2f1950cbc4eb4818874f45c6ac463ec5ff3ab50eb3954.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-061_959d29ff7b510100b80dca7073975416186c76bfb2b783c86ce6d7bb07ef6750.jpg b/.raw/MidasCivilAnalysisReference/images/page-061_959d29ff7b510100b80dca7073975416186c76bfb2b783c86ce6d7bb07ef6750.jpg new file mode 100644 index 00000000..45ddf4ab Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-061_959d29ff7b510100b80dca7073975416186c76bfb2b783c86ce6d7bb07ef6750.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-061_a61df68e25b064da71477ca00903e4241792723459aa518f742ccd98b81ef534.jpg b/.raw/MidasCivilAnalysisReference/images/page-061_a61df68e25b064da71477ca00903e4241792723459aa518f742ccd98b81ef534.jpg new file mode 100644 index 00000000..95978ded Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-061_a61df68e25b064da71477ca00903e4241792723459aa518f742ccd98b81ef534.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-061_f85e4616056bf3a935a07e678fc010e30269c09fff661cf588b446488671ad72.jpg b/.raw/MidasCivilAnalysisReference/images/page-061_f85e4616056bf3a935a07e678fc010e30269c09fff661cf588b446488671ad72.jpg new file mode 100644 index 00000000..bc46155e Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-061_f85e4616056bf3a935a07e678fc010e30269c09fff661cf588b446488671ad72.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-062_0071286ab54449189132ae6b1ec9f0b923488fa4d43af2a5d75b4ee705592a24.jpg b/.raw/MidasCivilAnalysisReference/images/page-062_0071286ab54449189132ae6b1ec9f0b923488fa4d43af2a5d75b4ee705592a24.jpg new file mode 100644 index 00000000..814b5f43 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-062_0071286ab54449189132ae6b1ec9f0b923488fa4d43af2a5d75b4ee705592a24.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-062_03ba7d28b3ac0c5781d527a0544a0139554588990f89edccc740672cf5d5eb1d.jpg b/.raw/MidasCivilAnalysisReference/images/page-062_03ba7d28b3ac0c5781d527a0544a0139554588990f89edccc740672cf5d5eb1d.jpg new file mode 100644 index 00000000..87abe37d Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-062_03ba7d28b3ac0c5781d527a0544a0139554588990f89edccc740672cf5d5eb1d.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-064_5a49020b6bb0d7a336656310a51f00f579a412cd73a2199a439ae106b2afebf5.jpg b/.raw/MidasCivilAnalysisReference/images/page-064_5a49020b6bb0d7a336656310a51f00f579a412cd73a2199a439ae106b2afebf5.jpg new file mode 100644 index 00000000..76fa5c7e Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-064_5a49020b6bb0d7a336656310a51f00f579a412cd73a2199a439ae106b2afebf5.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-064_7bf972e23dcc7987be94a4cab0b872c1832cd8edf60cf849edbfa3a24a4a7259.jpg b/.raw/MidasCivilAnalysisReference/images/page-064_7bf972e23dcc7987be94a4cab0b872c1832cd8edf60cf849edbfa3a24a4a7259.jpg new file mode 100644 index 00000000..3977062b Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-064_7bf972e23dcc7987be94a4cab0b872c1832cd8edf60cf849edbfa3a24a4a7259.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-064_8ff7d6bcd8b34f4a8bdb451b53ea42cf9fb6e3f827a73a54450d9b8eb55b5361.jpg b/.raw/MidasCivilAnalysisReference/images/page-064_8ff7d6bcd8b34f4a8bdb451b53ea42cf9fb6e3f827a73a54450d9b8eb55b5361.jpg new file mode 100644 index 00000000..eb1b99c2 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-064_8ff7d6bcd8b34f4a8bdb451b53ea42cf9fb6e3f827a73a54450d9b8eb55b5361.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-065_26239696d41bd001de020a11188a6346447c054249851d10171d197d97c1adbd.jpg b/.raw/MidasCivilAnalysisReference/images/page-065_26239696d41bd001de020a11188a6346447c054249851d10171d197d97c1adbd.jpg new file mode 100644 index 00000000..2c40223e Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-065_26239696d41bd001de020a11188a6346447c054249851d10171d197d97c1adbd.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-065_352c5af19c2dd1f33aeb3c0353ec5f614ba91c05374c35e07e3ac99945b1a932.jpg b/.raw/MidasCivilAnalysisReference/images/page-065_352c5af19c2dd1f33aeb3c0353ec5f614ba91c05374c35e07e3ac99945b1a932.jpg new file mode 100644 index 00000000..ac31c5e7 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-065_352c5af19c2dd1f33aeb3c0353ec5f614ba91c05374c35e07e3ac99945b1a932.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-065_bc67b3c7b3749d73f0efd08b488d235417065e732527c6f15bfe99cf02fb86b8.jpg b/.raw/MidasCivilAnalysisReference/images/page-065_bc67b3c7b3749d73f0efd08b488d235417065e732527c6f15bfe99cf02fb86b8.jpg new file mode 100644 index 00000000..9fa9d155 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-065_bc67b3c7b3749d73f0efd08b488d235417065e732527c6f15bfe99cf02fb86b8.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-065_ea90809b6f3031b3601684c160c5c7f97cfcd33cb30a49ab886426b631698b8d.jpg b/.raw/MidasCivilAnalysisReference/images/page-065_ea90809b6f3031b3601684c160c5c7f97cfcd33cb30a49ab886426b631698b8d.jpg new file mode 100644 index 00000000..e5d43911 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-065_ea90809b6f3031b3601684c160c5c7f97cfcd33cb30a49ab886426b631698b8d.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-066_3e9d7cf239484e7c99bd5a1e3296f630641ad4695932340a88261545d3e7e8e0.jpg b/.raw/MidasCivilAnalysisReference/images/page-066_3e9d7cf239484e7c99bd5a1e3296f630641ad4695932340a88261545d3e7e8e0.jpg new file mode 100644 index 00000000..28b60322 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-066_3e9d7cf239484e7c99bd5a1e3296f630641ad4695932340a88261545d3e7e8e0.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-066_9f35030da3fef667716f6042eb9599ac74c700b5fd6092aaf35e1a5fa745c97d.jpg b/.raw/MidasCivilAnalysisReference/images/page-066_9f35030da3fef667716f6042eb9599ac74c700b5fd6092aaf35e1a5fa745c97d.jpg new file mode 100644 index 00000000..d2cee569 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-066_9f35030da3fef667716f6042eb9599ac74c700b5fd6092aaf35e1a5fa745c97d.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-066_c75b84c91f51782630690eb0406117207ce47de6f4a3e3e11e118f2c4a57d912.jpg b/.raw/MidasCivilAnalysisReference/images/page-066_c75b84c91f51782630690eb0406117207ce47de6f4a3e3e11e118f2c4a57d912.jpg new file mode 100644 index 00000000..2df1e90f Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-066_c75b84c91f51782630690eb0406117207ce47de6f4a3e3e11e118f2c4a57d912.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-066_e3bba604994fd082d955e5a2dc19c2bd3c9d8c8f8f5616aa75740d0613011d15.jpg b/.raw/MidasCivilAnalysisReference/images/page-066_e3bba604994fd082d955e5a2dc19c2bd3c9d8c8f8f5616aa75740d0613011d15.jpg new file mode 100644 index 00000000..95c59e8e Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-066_e3bba604994fd082d955e5a2dc19c2bd3c9d8c8f8f5616aa75740d0613011d15.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-066_f786186604062e36e4666cf81711b025a84258e522eb06abefc1b3a397770db9.jpg b/.raw/MidasCivilAnalysisReference/images/page-066_f786186604062e36e4666cf81711b025a84258e522eb06abefc1b3a397770db9.jpg new file mode 100644 index 00000000..923fa19f Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-066_f786186604062e36e4666cf81711b025a84258e522eb06abefc1b3a397770db9.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-067_0c9e47417d59b5aaf57942ce3879e2ce0d684c71b0c5a6309829d8e15917d9bd.jpg b/.raw/MidasCivilAnalysisReference/images/page-067_0c9e47417d59b5aaf57942ce3879e2ce0d684c71b0c5a6309829d8e15917d9bd.jpg new file mode 100644 index 00000000..efc0ce9b Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-067_0c9e47417d59b5aaf57942ce3879e2ce0d684c71b0c5a6309829d8e15917d9bd.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-067_55721316182253d4b27cab30740630ab9fa5906c12c8d6b2c3b76ea932d3b866.jpg b/.raw/MidasCivilAnalysisReference/images/page-067_55721316182253d4b27cab30740630ab9fa5906c12c8d6b2c3b76ea932d3b866.jpg new file mode 100644 index 00000000..4d1e0608 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-067_55721316182253d4b27cab30740630ab9fa5906c12c8d6b2c3b76ea932d3b866.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-067_af04ae1b3ee3fcf55a9bdf1b678c802266eb99efc3b8b3b013ea5c5fe72f86b5.jpg b/.raw/MidasCivilAnalysisReference/images/page-067_af04ae1b3ee3fcf55a9bdf1b678c802266eb99efc3b8b3b013ea5c5fe72f86b5.jpg new file mode 100644 index 00000000..f4f5eade Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-067_af04ae1b3ee3fcf55a9bdf1b678c802266eb99efc3b8b3b013ea5c5fe72f86b5.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-067_c2d293a1c6ef6342edee71535b47637ca93b2e74de9ac051bdcbb25b4b6bd655.jpg b/.raw/MidasCivilAnalysisReference/images/page-067_c2d293a1c6ef6342edee71535b47637ca93b2e74de9ac051bdcbb25b4b6bd655.jpg new file mode 100644 index 00000000..68b4af97 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-067_c2d293a1c6ef6342edee71535b47637ca93b2e74de9ac051bdcbb25b4b6bd655.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-068_a79ec86baaa9c9b1508087ad92a28e019f4f69ee14c3664c2ede011a315a7f52.jpg b/.raw/MidasCivilAnalysisReference/images/page-068_a79ec86baaa9c9b1508087ad92a28e019f4f69ee14c3664c2ede011a315a7f52.jpg new file mode 100644 index 00000000..7f6f7618 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-068_a79ec86baaa9c9b1508087ad92a28e019f4f69ee14c3664c2ede011a315a7f52.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-069_22bce5bf7e9d21dacc63f9d96de20a2a4bdb81a37ae48de359bb7c9ca3f16cbd.jpg b/.raw/MidasCivilAnalysisReference/images/page-069_22bce5bf7e9d21dacc63f9d96de20a2a4bdb81a37ae48de359bb7c9ca3f16cbd.jpg new file mode 100644 index 00000000..83a31ea8 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-069_22bce5bf7e9d21dacc63f9d96de20a2a4bdb81a37ae48de359bb7c9ca3f16cbd.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-069_354aa5f2c6fce4e21728d22302d7f858c3ffe823113b13bb696b039517c93e19.jpg b/.raw/MidasCivilAnalysisReference/images/page-069_354aa5f2c6fce4e21728d22302d7f858c3ffe823113b13bb696b039517c93e19.jpg new file mode 100644 index 00000000..7f104d05 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-069_354aa5f2c6fce4e21728d22302d7f858c3ffe823113b13bb696b039517c93e19.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-070_2386f02c195cdd4f6d327d86b5314f8f602f914a059bf492c72d2808d681d5fc.jpg b/.raw/MidasCivilAnalysisReference/images/page-070_2386f02c195cdd4f6d327d86b5314f8f602f914a059bf492c72d2808d681d5fc.jpg new file mode 100644 index 00000000..9fc0e7d7 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-070_2386f02c195cdd4f6d327d86b5314f8f602f914a059bf492c72d2808d681d5fc.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-070_5eaf37e05681119836280307e6e7b9c93327da0a3718c81baa2658997a2e586e.jpg b/.raw/MidasCivilAnalysisReference/images/page-070_5eaf37e05681119836280307e6e7b9c93327da0a3718c81baa2658997a2e586e.jpg new file mode 100644 index 00000000..fe946b72 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-070_5eaf37e05681119836280307e6e7b9c93327da0a3718c81baa2658997a2e586e.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-070_a74937ca1e3f67d48b18db11de0184af444a33632696fcb5d74783c87deaeb2a.jpg b/.raw/MidasCivilAnalysisReference/images/page-070_a74937ca1e3f67d48b18db11de0184af444a33632696fcb5d74783c87deaeb2a.jpg new file mode 100644 index 00000000..ae9db1dd Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-070_a74937ca1e3f67d48b18db11de0184af444a33632696fcb5d74783c87deaeb2a.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-070_bfb3d42420478afe6a0dd0e8ef1f6c4b90e7d64dd3796cd8fac6fae213943f4b.jpg b/.raw/MidasCivilAnalysisReference/images/page-070_bfb3d42420478afe6a0dd0e8ef1f6c4b90e7d64dd3796cd8fac6fae213943f4b.jpg new file mode 100644 index 00000000..b2156872 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-070_bfb3d42420478afe6a0dd0e8ef1f6c4b90e7d64dd3796cd8fac6fae213943f4b.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-070_c4162fd1f4ed38ad247be9999ce8d425686b68d7e93aa0b98e54aed3207761c4.jpg b/.raw/MidasCivilAnalysisReference/images/page-070_c4162fd1f4ed38ad247be9999ce8d425686b68d7e93aa0b98e54aed3207761c4.jpg new file mode 100644 index 00000000..2f5474e8 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-070_c4162fd1f4ed38ad247be9999ce8d425686b68d7e93aa0b98e54aed3207761c4.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-070_df836a996980e0bd904219ef23bd82b8821e550e2de81e101a0846130399aa7d.jpg b/.raw/MidasCivilAnalysisReference/images/page-070_df836a996980e0bd904219ef23bd82b8821e550e2de81e101a0846130399aa7d.jpg new file mode 100644 index 00000000..80a8a12c Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-070_df836a996980e0bd904219ef23bd82b8821e550e2de81e101a0846130399aa7d.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-071_36d8d6e3871984d2986f87e082f1ee3c05264cd0a9fb0427f43f3cf64ac0baa2.jpg b/.raw/MidasCivilAnalysisReference/images/page-071_36d8d6e3871984d2986f87e082f1ee3c05264cd0a9fb0427f43f3cf64ac0baa2.jpg new file mode 100644 index 00000000..3e192fbf Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-071_36d8d6e3871984d2986f87e082f1ee3c05264cd0a9fb0427f43f3cf64ac0baa2.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-071_5ba47eb18611096aa2bda5a62f709d9917132bf2e11fa582e741b6882fb6c743.jpg b/.raw/MidasCivilAnalysisReference/images/page-071_5ba47eb18611096aa2bda5a62f709d9917132bf2e11fa582e741b6882fb6c743.jpg new file mode 100644 index 00000000..60ef7193 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-071_5ba47eb18611096aa2bda5a62f709d9917132bf2e11fa582e741b6882fb6c743.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-071_c9393d2349116d25129025c55a36400deab7b59225369a68a64a378f12a65b56.jpg b/.raw/MidasCivilAnalysisReference/images/page-071_c9393d2349116d25129025c55a36400deab7b59225369a68a64a378f12a65b56.jpg new file mode 100644 index 00000000..0d46e2e3 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-071_c9393d2349116d25129025c55a36400deab7b59225369a68a64a378f12a65b56.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-071_cfb91026e5919b458db2439c573ed6f84a07f237df57989a9f010684474c7e46.jpg b/.raw/MidasCivilAnalysisReference/images/page-071_cfb91026e5919b458db2439c573ed6f84a07f237df57989a9f010684474c7e46.jpg new file mode 100644 index 00000000..f9563e8a Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-071_cfb91026e5919b458db2439c573ed6f84a07f237df57989a9f010684474c7e46.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-071_de97f32558a577a1b276bf76c08e6d01fbe92a38a241b1cfecd6e9c606d37abc.jpg b/.raw/MidasCivilAnalysisReference/images/page-071_de97f32558a577a1b276bf76c08e6d01fbe92a38a241b1cfecd6e9c606d37abc.jpg new file mode 100644 index 00000000..d0460a6c Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-071_de97f32558a577a1b276bf76c08e6d01fbe92a38a241b1cfecd6e9c606d37abc.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-071_f97c5eb90cf46cca9a19fa2828d6c5c0e549d2a54c91c8dad574b8a575106143.jpg b/.raw/MidasCivilAnalysisReference/images/page-071_f97c5eb90cf46cca9a19fa2828d6c5c0e549d2a54c91c8dad574b8a575106143.jpg new file mode 100644 index 00000000..a1924ee3 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-071_f97c5eb90cf46cca9a19fa2828d6c5c0e549d2a54c91c8dad574b8a575106143.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-071_fe074d3a01a1811b2050490f7cc9484d1c93d17e435ac864a369ce3f420e786c.jpg b/.raw/MidasCivilAnalysisReference/images/page-071_fe074d3a01a1811b2050490f7cc9484d1c93d17e435ac864a369ce3f420e786c.jpg new file mode 100644 index 00000000..63861871 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-071_fe074d3a01a1811b2050490f7cc9484d1c93d17e435ac864a369ce3f420e786c.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-072_5e0d1c1decec0a0ddf8effd094207078d3bbe0d8801bda12b5581d02fc1e604d.jpg b/.raw/MidasCivilAnalysisReference/images/page-072_5e0d1c1decec0a0ddf8effd094207078d3bbe0d8801bda12b5581d02fc1e604d.jpg new file mode 100644 index 00000000..c5c19b7b Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-072_5e0d1c1decec0a0ddf8effd094207078d3bbe0d8801bda12b5581d02fc1e604d.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-072_e42797f4bded525cbf67409615b97189508891cc40241638af12174fbf43dad8.jpg b/.raw/MidasCivilAnalysisReference/images/page-072_e42797f4bded525cbf67409615b97189508891cc40241638af12174fbf43dad8.jpg new file mode 100644 index 00000000..7f775d22 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-072_e42797f4bded525cbf67409615b97189508891cc40241638af12174fbf43dad8.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-074_0998dce0600aa7f3c8677ae785ccd4168df12641341c07397cf510a2eabddf77.jpg b/.raw/MidasCivilAnalysisReference/images/page-074_0998dce0600aa7f3c8677ae785ccd4168df12641341c07397cf510a2eabddf77.jpg new file mode 100644 index 00000000..0a5e912c Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-074_0998dce0600aa7f3c8677ae785ccd4168df12641341c07397cf510a2eabddf77.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-074_3ea7f382d8f9991dd757262458f92126a3e125f480cc01f293de909ea913a90c.jpg b/.raw/MidasCivilAnalysisReference/images/page-074_3ea7f382d8f9991dd757262458f92126a3e125f480cc01f293de909ea913a90c.jpg new file mode 100644 index 00000000..542d2d20 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-074_3ea7f382d8f9991dd757262458f92126a3e125f480cc01f293de909ea913a90c.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-074_e7fda3dac02264bfb1245f110a87ecacf8902b998d9b0c4b79543ea2e78fdba4.jpg b/.raw/MidasCivilAnalysisReference/images/page-074_e7fda3dac02264bfb1245f110a87ecacf8902b998d9b0c4b79543ea2e78fdba4.jpg new file mode 100644 index 00000000..53858cc5 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-074_e7fda3dac02264bfb1245f110a87ecacf8902b998d9b0c4b79543ea2e78fdba4.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-075_76e5d33f6c788ed8aa8486c12b52d1da7cfa72e6cc46aa88b0705bcd13afba15.jpg b/.raw/MidasCivilAnalysisReference/images/page-075_76e5d33f6c788ed8aa8486c12b52d1da7cfa72e6cc46aa88b0705bcd13afba15.jpg new file mode 100644 index 00000000..8580be67 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-075_76e5d33f6c788ed8aa8486c12b52d1da7cfa72e6cc46aa88b0705bcd13afba15.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-075_7e0c8c4d9166ea537277505f45fab883bbb2f8c376c8e5e26e9b04d24b7f5cdd.jpg b/.raw/MidasCivilAnalysisReference/images/page-075_7e0c8c4d9166ea537277505f45fab883bbb2f8c376c8e5e26e9b04d24b7f5cdd.jpg new file mode 100644 index 00000000..5f1d0af2 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-075_7e0c8c4d9166ea537277505f45fab883bbb2f8c376c8e5e26e9b04d24b7f5cdd.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-076_94366b1cea405aec7f8a625ae5159c4297be697b5d06e389eea0b99a0109bc65.jpg b/.raw/MidasCivilAnalysisReference/images/page-076_94366b1cea405aec7f8a625ae5159c4297be697b5d06e389eea0b99a0109bc65.jpg new file mode 100644 index 00000000..e12f6e5c Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-076_94366b1cea405aec7f8a625ae5159c4297be697b5d06e389eea0b99a0109bc65.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-076_a0713726c502fa385a8a5598195de49725a9203428aafdb598d16202aba7e980.jpg b/.raw/MidasCivilAnalysisReference/images/page-076_a0713726c502fa385a8a5598195de49725a9203428aafdb598d16202aba7e980.jpg new file mode 100644 index 00000000..5188634f Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-076_a0713726c502fa385a8a5598195de49725a9203428aafdb598d16202aba7e980.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-076_c590a2e012be12fbb985ae731f1be08d19c445b1e4f29da53a782b8165888601.jpg b/.raw/MidasCivilAnalysisReference/images/page-076_c590a2e012be12fbb985ae731f1be08d19c445b1e4f29da53a782b8165888601.jpg new file mode 100644 index 00000000..03c1086b Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-076_c590a2e012be12fbb985ae731f1be08d19c445b1e4f29da53a782b8165888601.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-077_11db6e5ee6a8b4ee7b1cff9b1c69f0c15400be0fc7b3b0ceddd0ec35283150e2.jpg b/.raw/MidasCivilAnalysisReference/images/page-077_11db6e5ee6a8b4ee7b1cff9b1c69f0c15400be0fc7b3b0ceddd0ec35283150e2.jpg new file mode 100644 index 00000000..1f32e2c6 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-077_11db6e5ee6a8b4ee7b1cff9b1c69f0c15400be0fc7b3b0ceddd0ec35283150e2.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-077_5b09c7622fa899df8d32f89180ba2ffa9e35ad6f133a36ceadf8f1919e0b3480.jpg b/.raw/MidasCivilAnalysisReference/images/page-077_5b09c7622fa899df8d32f89180ba2ffa9e35ad6f133a36ceadf8f1919e0b3480.jpg new file mode 100644 index 00000000..42962a80 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-077_5b09c7622fa899df8d32f89180ba2ffa9e35ad6f133a36ceadf8f1919e0b3480.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-077_7ae8fd27a3fcdded4d27f5e72bc5601bc45c3db4b6e16e45384888fafd7b3a9b.jpg b/.raw/MidasCivilAnalysisReference/images/page-077_7ae8fd27a3fcdded4d27f5e72bc5601bc45c3db4b6e16e45384888fafd7b3a9b.jpg new file mode 100644 index 00000000..cae9b7df Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-077_7ae8fd27a3fcdded4d27f5e72bc5601bc45c3db4b6e16e45384888fafd7b3a9b.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-077_8d623a34a4a69f3568867797566a2948b46385039da58f3ca2a681645f00306d.jpg b/.raw/MidasCivilAnalysisReference/images/page-077_8d623a34a4a69f3568867797566a2948b46385039da58f3ca2a681645f00306d.jpg new file mode 100644 index 00000000..afdc6f4f Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-077_8d623a34a4a69f3568867797566a2948b46385039da58f3ca2a681645f00306d.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-077_949821a4debfda567455174ad8737ab7bcb44c0fcf79b25ae87725d3266657c1.jpg b/.raw/MidasCivilAnalysisReference/images/page-077_949821a4debfda567455174ad8737ab7bcb44c0fcf79b25ae87725d3266657c1.jpg new file mode 100644 index 00000000..1dfd549c Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-077_949821a4debfda567455174ad8737ab7bcb44c0fcf79b25ae87725d3266657c1.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-077_9be5961037650891b8c7974b0a8e0907cfafe1c144f2e78f49fae535b2f3d5e5.jpg b/.raw/MidasCivilAnalysisReference/images/page-077_9be5961037650891b8c7974b0a8e0907cfafe1c144f2e78f49fae535b2f3d5e5.jpg new file mode 100644 index 00000000..36078c70 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-077_9be5961037650891b8c7974b0a8e0907cfafe1c144f2e78f49fae535b2f3d5e5.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-077_b13e9fb1c9b01ef8e83f72dc332207479da616810a5bfe8a34260f3d83d42880.jpg b/.raw/MidasCivilAnalysisReference/images/page-077_b13e9fb1c9b01ef8e83f72dc332207479da616810a5bfe8a34260f3d83d42880.jpg new file mode 100644 index 00000000..c13d58d9 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-077_b13e9fb1c9b01ef8e83f72dc332207479da616810a5bfe8a34260f3d83d42880.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-077_bc254a61e8f49523a9220cf99831a3d225f5bbfc749e353b2acb9eb8de982784.jpg b/.raw/MidasCivilAnalysisReference/images/page-077_bc254a61e8f49523a9220cf99831a3d225f5bbfc749e353b2acb9eb8de982784.jpg new file mode 100644 index 00000000..b7df102e Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-077_bc254a61e8f49523a9220cf99831a3d225f5bbfc749e353b2acb9eb8de982784.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-077_d17fbb8117b2a0127145357b300330bc2b219b5bc9ff7d7131f0661cc18907e5.jpg b/.raw/MidasCivilAnalysisReference/images/page-077_d17fbb8117b2a0127145357b300330bc2b219b5bc9ff7d7131f0661cc18907e5.jpg new file mode 100644 index 00000000..0332c3c6 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-077_d17fbb8117b2a0127145357b300330bc2b219b5bc9ff7d7131f0661cc18907e5.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-077_d80d075f6644c20e60ae3c914de1eefc581a5ebc1d5583c38b21dd66fe232f1e.jpg b/.raw/MidasCivilAnalysisReference/images/page-077_d80d075f6644c20e60ae3c914de1eefc581a5ebc1d5583c38b21dd66fe232f1e.jpg new file mode 100644 index 00000000..1c818767 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-077_d80d075f6644c20e60ae3c914de1eefc581a5ebc1d5583c38b21dd66fe232f1e.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-077_da390190ca200c34654a5b11f36125dcad67cd5a9835fed98429e0315c98fb78.jpg b/.raw/MidasCivilAnalysisReference/images/page-077_da390190ca200c34654a5b11f36125dcad67cd5a9835fed98429e0315c98fb78.jpg new file mode 100644 index 00000000..dd5f9ac1 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-077_da390190ca200c34654a5b11f36125dcad67cd5a9835fed98429e0315c98fb78.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-077_ff45521790ff632cb44d60c9211a26806bdf89d89494566ce540e043f58bf99d.jpg b/.raw/MidasCivilAnalysisReference/images/page-077_ff45521790ff632cb44d60c9211a26806bdf89d89494566ce540e043f58bf99d.jpg new file mode 100644 index 00000000..c793979e Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-077_ff45521790ff632cb44d60c9211a26806bdf89d89494566ce540e043f58bf99d.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-078_60bf9d59a6a65fb40eca619204911009179bd66c91ec4b2f1e57c40ff55dcd98.jpg b/.raw/MidasCivilAnalysisReference/images/page-078_60bf9d59a6a65fb40eca619204911009179bd66c91ec4b2f1e57c40ff55dcd98.jpg new file mode 100644 index 00000000..0f0f3315 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-078_60bf9d59a6a65fb40eca619204911009179bd66c91ec4b2f1e57c40ff55dcd98.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-078_6868f2d4499f81d4f850ce71765c566384ff2523664c1b90b28b26ff441772ec.jpg b/.raw/MidasCivilAnalysisReference/images/page-078_6868f2d4499f81d4f850ce71765c566384ff2523664c1b90b28b26ff441772ec.jpg new file mode 100644 index 00000000..1f9e2ace Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-078_6868f2d4499f81d4f850ce71765c566384ff2523664c1b90b28b26ff441772ec.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-079_59304e44f3eba685857da3d9d10694660c452560b1b2ae1f95fe741086b92251.jpg b/.raw/MidasCivilAnalysisReference/images/page-079_59304e44f3eba685857da3d9d10694660c452560b1b2ae1f95fe741086b92251.jpg new file mode 100644 index 00000000..b38b5968 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-079_59304e44f3eba685857da3d9d10694660c452560b1b2ae1f95fe741086b92251.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-079_d65892710e0b9d38c679078ede3444950b93d3d71771571640cb99df3bdcc1bc.jpg b/.raw/MidasCivilAnalysisReference/images/page-079_d65892710e0b9d38c679078ede3444950b93d3d71771571640cb99df3bdcc1bc.jpg new file mode 100644 index 00000000..b44c5dcb Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-079_d65892710e0b9d38c679078ede3444950b93d3d71771571640cb99df3bdcc1bc.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-079_e0a8e098a17e8a3ec2b3d36a6081deee9af132a00973d392722949a6db55bd89.jpg b/.raw/MidasCivilAnalysisReference/images/page-079_e0a8e098a17e8a3ec2b3d36a6081deee9af132a00973d392722949a6db55bd89.jpg new file mode 100644 index 00000000..e6a506cf Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-079_e0a8e098a17e8a3ec2b3d36a6081deee9af132a00973d392722949a6db55bd89.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-080_05b344e4071ae36ff5e4c04b06bd26671f86c0792c9ab67acc556ef9106ab230.jpg b/.raw/MidasCivilAnalysisReference/images/page-080_05b344e4071ae36ff5e4c04b06bd26671f86c0792c9ab67acc556ef9106ab230.jpg new file mode 100644 index 00000000..db5a72b4 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-080_05b344e4071ae36ff5e4c04b06bd26671f86c0792c9ab67acc556ef9106ab230.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-080_6475518e5a2292e3f03f520b66c032fcb595c0030feb2c90fcae8a7e7336e2ce.jpg b/.raw/MidasCivilAnalysisReference/images/page-080_6475518e5a2292e3f03f520b66c032fcb595c0030feb2c90fcae8a7e7336e2ce.jpg new file mode 100644 index 00000000..b9cea11f Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-080_6475518e5a2292e3f03f520b66c032fcb595c0030feb2c90fcae8a7e7336e2ce.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-080_6cc5888129f2d9d908a51acd62b3f436eaf4cc50cc052aeef6b89f098c1e5a31.jpg b/.raw/MidasCivilAnalysisReference/images/page-080_6cc5888129f2d9d908a51acd62b3f436eaf4cc50cc052aeef6b89f098c1e5a31.jpg new file mode 100644 index 00000000..d127ec95 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-080_6cc5888129f2d9d908a51acd62b3f436eaf4cc50cc052aeef6b89f098c1e5a31.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-080_9b4fd65f06f79a03486e98bbc243fbd5c8b74e845387e8770f4b0cfa393591d3.jpg b/.raw/MidasCivilAnalysisReference/images/page-080_9b4fd65f06f79a03486e98bbc243fbd5c8b74e845387e8770f4b0cfa393591d3.jpg new file mode 100644 index 00000000..1bd3f32c Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-080_9b4fd65f06f79a03486e98bbc243fbd5c8b74e845387e8770f4b0cfa393591d3.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-080_cb3b5d800ac5a72ddbbb66c95f0a37488edb3eac88b4533207443c3e9d4ed62b.jpg b/.raw/MidasCivilAnalysisReference/images/page-080_cb3b5d800ac5a72ddbbb66c95f0a37488edb3eac88b4533207443c3e9d4ed62b.jpg new file mode 100644 index 00000000..7ce6ef94 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-080_cb3b5d800ac5a72ddbbb66c95f0a37488edb3eac88b4533207443c3e9d4ed62b.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-081_8859480baf1575c713f3f88c2b8bc16509551dba2c60fca5429a85b66874d712.jpg b/.raw/MidasCivilAnalysisReference/images/page-081_8859480baf1575c713f3f88c2b8bc16509551dba2c60fca5429a85b66874d712.jpg new file mode 100644 index 00000000..a553ea1e Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-081_8859480baf1575c713f3f88c2b8bc16509551dba2c60fca5429a85b66874d712.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-081_c197644b6d4d4018f1a66ed9e2cc795bfe31929c1a39b222da01caa9e0d1ea21.jpg b/.raw/MidasCivilAnalysisReference/images/page-081_c197644b6d4d4018f1a66ed9e2cc795bfe31929c1a39b222da01caa9e0d1ea21.jpg new file mode 100644 index 00000000..5092b655 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-081_c197644b6d4d4018f1a66ed9e2cc795bfe31929c1a39b222da01caa9e0d1ea21.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-081_f95fe44980972f5bc3309a167ca25a820d21747c78542feb53befc1ac3925632.jpg b/.raw/MidasCivilAnalysisReference/images/page-081_f95fe44980972f5bc3309a167ca25a820d21747c78542feb53befc1ac3925632.jpg new file mode 100644 index 00000000..eb31b81a Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-081_f95fe44980972f5bc3309a167ca25a820d21747c78542feb53befc1ac3925632.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-082_2f7d641e00fc24653afff4104c21aa7c2702efdc6ecd540980282d9f09af8654.jpg b/.raw/MidasCivilAnalysisReference/images/page-082_2f7d641e00fc24653afff4104c21aa7c2702efdc6ecd540980282d9f09af8654.jpg new file mode 100644 index 00000000..f02a56c9 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-082_2f7d641e00fc24653afff4104c21aa7c2702efdc6ecd540980282d9f09af8654.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-082_97bad629df8fc7f048098bb1580219bf8ac947cf16f8aa0ffd0a11c45a09b392.jpg b/.raw/MidasCivilAnalysisReference/images/page-082_97bad629df8fc7f048098bb1580219bf8ac947cf16f8aa0ffd0a11c45a09b392.jpg new file mode 100644 index 00000000..11f41ff7 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-082_97bad629df8fc7f048098bb1580219bf8ac947cf16f8aa0ffd0a11c45a09b392.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-084_94e239d668d740ca9131dd24fed42501ef79fcc5fd3e58a05f9b3592f8c1405a.jpg b/.raw/MidasCivilAnalysisReference/images/page-084_94e239d668d740ca9131dd24fed42501ef79fcc5fd3e58a05f9b3592f8c1405a.jpg new file mode 100644 index 00000000..50508e16 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-084_94e239d668d740ca9131dd24fed42501ef79fcc5fd3e58a05f9b3592f8c1405a.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-085_34d6a439458d379a4971fbd2909cfca755951f1fa8aa18bc0474480c7cbbf5dd.jpg b/.raw/MidasCivilAnalysisReference/images/page-085_34d6a439458d379a4971fbd2909cfca755951f1fa8aa18bc0474480c7cbbf5dd.jpg new file mode 100644 index 00000000..04672805 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-085_34d6a439458d379a4971fbd2909cfca755951f1fa8aa18bc0474480c7cbbf5dd.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-085_5e7cbd9c28f120f793eaf54964d5c2085a6e8a9db618063315c0f70769c1507c.jpg b/.raw/MidasCivilAnalysisReference/images/page-085_5e7cbd9c28f120f793eaf54964d5c2085a6e8a9db618063315c0f70769c1507c.jpg new file mode 100644 index 00000000..0e871df9 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-085_5e7cbd9c28f120f793eaf54964d5c2085a6e8a9db618063315c0f70769c1507c.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-085_94914a67bd38e93f81f65019b193aa51e53163f4adc353bce36dc284eb7550da.jpg b/.raw/MidasCivilAnalysisReference/images/page-085_94914a67bd38e93f81f65019b193aa51e53163f4adc353bce36dc284eb7550da.jpg new file mode 100644 index 00000000..dc15f628 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-085_94914a67bd38e93f81f65019b193aa51e53163f4adc353bce36dc284eb7550da.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-085_e4b898f67e38c2edeff237a0b401a70f7a69c9aace1f87bd0506843c71c05582.jpg b/.raw/MidasCivilAnalysisReference/images/page-085_e4b898f67e38c2edeff237a0b401a70f7a69c9aace1f87bd0506843c71c05582.jpg new file mode 100644 index 00000000..b7f02515 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-085_e4b898f67e38c2edeff237a0b401a70f7a69c9aace1f87bd0506843c71c05582.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-085_faa813ef7857b723ce294d70bc0c284eb91d240ce30aeafb6c2f9dc670eaea09.jpg b/.raw/MidasCivilAnalysisReference/images/page-085_faa813ef7857b723ce294d70bc0c284eb91d240ce30aeafb6c2f9dc670eaea09.jpg new file mode 100644 index 00000000..2c8b5f8b Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-085_faa813ef7857b723ce294d70bc0c284eb91d240ce30aeafb6c2f9dc670eaea09.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-086_a93d7a19bba34565a78eb15f7428e964d918fd314eb455daf22f07d3884c9eb7.jpg b/.raw/MidasCivilAnalysisReference/images/page-086_a93d7a19bba34565a78eb15f7428e964d918fd314eb455daf22f07d3884c9eb7.jpg new file mode 100644 index 00000000..a1ddaa61 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-086_a93d7a19bba34565a78eb15f7428e964d918fd314eb455daf22f07d3884c9eb7.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-086_ad71f553193cdb5488039a5da5e81f9eaf820defee1c8e22136a22785a9a93b9.jpg b/.raw/MidasCivilAnalysisReference/images/page-086_ad71f553193cdb5488039a5da5e81f9eaf820defee1c8e22136a22785a9a93b9.jpg new file mode 100644 index 00000000..605eed4b Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-086_ad71f553193cdb5488039a5da5e81f9eaf820defee1c8e22136a22785a9a93b9.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-091_1043621a4bc00fa01b3d34f1cbc487849e9a22bddd14dd8cbf6ba9c6a5ecc82c.jpg b/.raw/MidasCivilAnalysisReference/images/page-091_1043621a4bc00fa01b3d34f1cbc487849e9a22bddd14dd8cbf6ba9c6a5ecc82c.jpg new file mode 100644 index 00000000..3054edf3 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-091_1043621a4bc00fa01b3d34f1cbc487849e9a22bddd14dd8cbf6ba9c6a5ecc82c.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-091_10eca8ae215c1972d5dd9de89c6eadc6e91d75fd2152894a599c05b550410f68.jpg b/.raw/MidasCivilAnalysisReference/images/page-091_10eca8ae215c1972d5dd9de89c6eadc6e91d75fd2152894a599c05b550410f68.jpg new file mode 100644 index 00000000..3a7c8dab Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-091_10eca8ae215c1972d5dd9de89c6eadc6e91d75fd2152894a599c05b550410f68.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-091_1b60de9d206aa96e89b72ca5a6e48cdcd4c074e43cb520830b2ec30f525df9a3.jpg b/.raw/MidasCivilAnalysisReference/images/page-091_1b60de9d206aa96e89b72ca5a6e48cdcd4c074e43cb520830b2ec30f525df9a3.jpg new file mode 100644 index 00000000..9a67f85b Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-091_1b60de9d206aa96e89b72ca5a6e48cdcd4c074e43cb520830b2ec30f525df9a3.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-091_1f3ce5d2b5b05acd78d9eb29b2588033b4c22720c677c1ecaca21e104a7ad42b.jpg b/.raw/MidasCivilAnalysisReference/images/page-091_1f3ce5d2b5b05acd78d9eb29b2588033b4c22720c677c1ecaca21e104a7ad42b.jpg new file mode 100644 index 00000000..637f5205 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-091_1f3ce5d2b5b05acd78d9eb29b2588033b4c22720c677c1ecaca21e104a7ad42b.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-091_36b6d14a395c8149378eb6b31e31e5f0515cd0f4bb15656ad5e918db360dc48a.jpg b/.raw/MidasCivilAnalysisReference/images/page-091_36b6d14a395c8149378eb6b31e31e5f0515cd0f4bb15656ad5e918db360dc48a.jpg new file mode 100644 index 00000000..fb4e14c7 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-091_36b6d14a395c8149378eb6b31e31e5f0515cd0f4bb15656ad5e918db360dc48a.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-091_99f83215705d171ef7b2c0935d618980a2fd3ac45575455d727f9da2e99bfea6.jpg b/.raw/MidasCivilAnalysisReference/images/page-091_99f83215705d171ef7b2c0935d618980a2fd3ac45575455d727f9da2e99bfea6.jpg new file mode 100644 index 00000000..4b89f5b5 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-091_99f83215705d171ef7b2c0935d618980a2fd3ac45575455d727f9da2e99bfea6.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-092_18b5622a939b3548c0f8651270eb808a0d0b9b26b8644e8c914caa295cb90363.jpg b/.raw/MidasCivilAnalysisReference/images/page-092_18b5622a939b3548c0f8651270eb808a0d0b9b26b8644e8c914caa295cb90363.jpg new file mode 100644 index 00000000..b4fb99cd Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-092_18b5622a939b3548c0f8651270eb808a0d0b9b26b8644e8c914caa295cb90363.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-092_c4185d227a64bdc5423a425a4aef7c4c022f7955eb4c3c5131868d343ed5733e.jpg b/.raw/MidasCivilAnalysisReference/images/page-092_c4185d227a64bdc5423a425a4aef7c4c022f7955eb4c3c5131868d343ed5733e.jpg new file mode 100644 index 00000000..1b2292fc Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-092_c4185d227a64bdc5423a425a4aef7c4c022f7955eb4c3c5131868d343ed5733e.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-095_49ecf5f5b6c9d701381809740ee61340f7d7a76c4df607b46c173f8e4a1193d2.jpg b/.raw/MidasCivilAnalysisReference/images/page-095_49ecf5f5b6c9d701381809740ee61340f7d7a76c4df607b46c173f8e4a1193d2.jpg new file mode 100644 index 00000000..812c468b Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-095_49ecf5f5b6c9d701381809740ee61340f7d7a76c4df607b46c173f8e4a1193d2.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-095_7d804cf2be5eccf844340311ff4e6f24f2cd147c79972aebe2addd8ee350158c.jpg b/.raw/MidasCivilAnalysisReference/images/page-095_7d804cf2be5eccf844340311ff4e6f24f2cd147c79972aebe2addd8ee350158c.jpg new file mode 100644 index 00000000..966a03a1 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-095_7d804cf2be5eccf844340311ff4e6f24f2cd147c79972aebe2addd8ee350158c.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-095_a634b89a8dfeaed898e6faccbed09dbe86d7cf969bca17fabd2662fc55b122fb.jpg b/.raw/MidasCivilAnalysisReference/images/page-095_a634b89a8dfeaed898e6faccbed09dbe86d7cf969bca17fabd2662fc55b122fb.jpg new file mode 100644 index 00000000..c2dd2a20 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-095_a634b89a8dfeaed898e6faccbed09dbe86d7cf969bca17fabd2662fc55b122fb.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-095_b067f8a628c78c08921434fcfa393db3a538c01cc1ac3c4d59d3f054ab276074.jpg b/.raw/MidasCivilAnalysisReference/images/page-095_b067f8a628c78c08921434fcfa393db3a538c01cc1ac3c4d59d3f054ab276074.jpg new file mode 100644 index 00000000..9a6f6e5a Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-095_b067f8a628c78c08921434fcfa393db3a538c01cc1ac3c4d59d3f054ab276074.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-095_d8e67f34e0db551f8cda61cb5274109b97dff09c382743776402d9f2d1b42182.jpg b/.raw/MidasCivilAnalysisReference/images/page-095_d8e67f34e0db551f8cda61cb5274109b97dff09c382743776402d9f2d1b42182.jpg new file mode 100644 index 00000000..f9972dac Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-095_d8e67f34e0db551f8cda61cb5274109b97dff09c382743776402d9f2d1b42182.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-097_3775f151192e3b180ecf5ce645eb249e46d66bfad17e8d18f6a7675773907448.jpg b/.raw/MidasCivilAnalysisReference/images/page-097_3775f151192e3b180ecf5ce645eb249e46d66bfad17e8d18f6a7675773907448.jpg new file mode 100644 index 00000000..980d54dc Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-097_3775f151192e3b180ecf5ce645eb249e46d66bfad17e8d18f6a7675773907448.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-100_f18f7374967857b56ec5f25de1594f01eceeafd3c32aa17bd82b8bd27769b7c7.jpg b/.raw/MidasCivilAnalysisReference/images/page-100_f18f7374967857b56ec5f25de1594f01eceeafd3c32aa17bd82b8bd27769b7c7.jpg new file mode 100644 index 00000000..a6d4d6e6 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-100_f18f7374967857b56ec5f25de1594f01eceeafd3c32aa17bd82b8bd27769b7c7.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-103_45a33610dc4a91c40f4a50f4bd9e3518267f4dae3533174d6804a9a1efffc12a.jpg b/.raw/MidasCivilAnalysisReference/images/page-103_45a33610dc4a91c40f4a50f4bd9e3518267f4dae3533174d6804a9a1efffc12a.jpg new file mode 100644 index 00000000..cf088150 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-103_45a33610dc4a91c40f4a50f4bd9e3518267f4dae3533174d6804a9a1efffc12a.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-104_d81079a8ebe680c105fafe34f2027d7e40b376b980bbbb275dd68dcda3e2e798.jpg b/.raw/MidasCivilAnalysisReference/images/page-104_d81079a8ebe680c105fafe34f2027d7e40b376b980bbbb275dd68dcda3e2e798.jpg new file mode 100644 index 00000000..ea1a502e Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-104_d81079a8ebe680c105fafe34f2027d7e40b376b980bbbb275dd68dcda3e2e798.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-106_20f00979de1f48e04b54f3df078056cfb66f35471649d00597f1c4463d0b2d96.jpg b/.raw/MidasCivilAnalysisReference/images/page-106_20f00979de1f48e04b54f3df078056cfb66f35471649d00597f1c4463d0b2d96.jpg new file mode 100644 index 00000000..d5c79e0a Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-106_20f00979de1f48e04b54f3df078056cfb66f35471649d00597f1c4463d0b2d96.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-106_ce2ce8eecb3ac84056764bc6d89c2cb6fe04b3015922608eb351bd4894fee2dd.jpg b/.raw/MidasCivilAnalysisReference/images/page-106_ce2ce8eecb3ac84056764bc6d89c2cb6fe04b3015922608eb351bd4894fee2dd.jpg new file mode 100644 index 00000000..c8f78f31 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-106_ce2ce8eecb3ac84056764bc6d89c2cb6fe04b3015922608eb351bd4894fee2dd.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-108_0543db136fb994674a2a8f2811f1b1c10e429d4d1cd994ec8f93ab4636e96a94.jpg b/.raw/MidasCivilAnalysisReference/images/page-108_0543db136fb994674a2a8f2811f1b1c10e429d4d1cd994ec8f93ab4636e96a94.jpg new file mode 100644 index 00000000..a6bec8e2 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-108_0543db136fb994674a2a8f2811f1b1c10e429d4d1cd994ec8f93ab4636e96a94.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-108_4392261c17bcd12320a94cbe4bb73d0aa15d71698966c08c13fd6396c6406f8c.jpg b/.raw/MidasCivilAnalysisReference/images/page-108_4392261c17bcd12320a94cbe4bb73d0aa15d71698966c08c13fd6396c6406f8c.jpg new file mode 100644 index 00000000..9ef9f1fd Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-108_4392261c17bcd12320a94cbe4bb73d0aa15d71698966c08c13fd6396c6406f8c.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-108_496e1c64d89d4ffa81cd78e810ce5af16df5aa7d7b4d6bbe9645516ed1bed025.jpg b/.raw/MidasCivilAnalysisReference/images/page-108_496e1c64d89d4ffa81cd78e810ce5af16df5aa7d7b4d6bbe9645516ed1bed025.jpg new file mode 100644 index 00000000..1d8c5089 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-108_496e1c64d89d4ffa81cd78e810ce5af16df5aa7d7b4d6bbe9645516ed1bed025.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-108_63fd2e35f2fd83092ce88d0a892fbff7c04c697858798729c1552d68af68fcd9.jpg b/.raw/MidasCivilAnalysisReference/images/page-108_63fd2e35f2fd83092ce88d0a892fbff7c04c697858798729c1552d68af68fcd9.jpg new file mode 100644 index 00000000..fc827dcd Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-108_63fd2e35f2fd83092ce88d0a892fbff7c04c697858798729c1552d68af68fcd9.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-108_650f7d0231a3f756cd817e767218fe955f85bbe5414107ac855e839ce7c9d523.jpg b/.raw/MidasCivilAnalysisReference/images/page-108_650f7d0231a3f756cd817e767218fe955f85bbe5414107ac855e839ce7c9d523.jpg new file mode 100644 index 00000000..42c107a0 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-108_650f7d0231a3f756cd817e767218fe955f85bbe5414107ac855e839ce7c9d523.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-108_741655832cc3d71d54c5f945f34fa720095ae7233068db63213eb142f7f12251.jpg b/.raw/MidasCivilAnalysisReference/images/page-108_741655832cc3d71d54c5f945f34fa720095ae7233068db63213eb142f7f12251.jpg new file mode 100644 index 00000000..4cc7c011 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-108_741655832cc3d71d54c5f945f34fa720095ae7233068db63213eb142f7f12251.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-108_a910aa23cdf5003e863cccdde0bb31223eb5a2a9a9af9d433b42e6809c4f5995.jpg b/.raw/MidasCivilAnalysisReference/images/page-108_a910aa23cdf5003e863cccdde0bb31223eb5a2a9a9af9d433b42e6809c4f5995.jpg new file mode 100644 index 00000000..44d94531 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-108_a910aa23cdf5003e863cccdde0bb31223eb5a2a9a9af9d433b42e6809c4f5995.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-108_d0afc91ab79f37f50137f7a3234c816896434b394789947eb9e012322472e3ce.jpg b/.raw/MidasCivilAnalysisReference/images/page-108_d0afc91ab79f37f50137f7a3234c816896434b394789947eb9e012322472e3ce.jpg new file mode 100644 index 00000000..da4f706d Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-108_d0afc91ab79f37f50137f7a3234c816896434b394789947eb9e012322472e3ce.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-108_dca801fce4f9de5a44a13f95d203cb478e5d706ff3103657ecdebb237b7b5afc.jpg b/.raw/MidasCivilAnalysisReference/images/page-108_dca801fce4f9de5a44a13f95d203cb478e5d706ff3103657ecdebb237b7b5afc.jpg new file mode 100644 index 00000000..c0c47525 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-108_dca801fce4f9de5a44a13f95d203cb478e5d706ff3103657ecdebb237b7b5afc.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-109_81cecce88f8069c983f73be90afcf6a730eacd4c1b6c57473509cae32f3c02f5.jpg b/.raw/MidasCivilAnalysisReference/images/page-109_81cecce88f8069c983f73be90afcf6a730eacd4c1b6c57473509cae32f3c02f5.jpg new file mode 100644 index 00000000..041ccd7c Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-109_81cecce88f8069c983f73be90afcf6a730eacd4c1b6c57473509cae32f3c02f5.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-109_d05a7fa239840b0cb3521f577820a465a8be493d66c62cd9eeb1c9647e024734.jpg b/.raw/MidasCivilAnalysisReference/images/page-109_d05a7fa239840b0cb3521f577820a465a8be493d66c62cd9eeb1c9647e024734.jpg new file mode 100644 index 00000000..02141377 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-109_d05a7fa239840b0cb3521f577820a465a8be493d66c62cd9eeb1c9647e024734.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-109_d5737afc8ce0feb1d41ecfa182e7f860839f2da8075a962d6dbbf64d14cd192c.jpg b/.raw/MidasCivilAnalysisReference/images/page-109_d5737afc8ce0feb1d41ecfa182e7f860839f2da8075a962d6dbbf64d14cd192c.jpg new file mode 100644 index 00000000..54878b19 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-109_d5737afc8ce0feb1d41ecfa182e7f860839f2da8075a962d6dbbf64d14cd192c.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-110_278d58a31f638767250a99728cb7597939cb709051f22860f984ce157234c2a3.jpg b/.raw/MidasCivilAnalysisReference/images/page-110_278d58a31f638767250a99728cb7597939cb709051f22860f984ce157234c2a3.jpg new file mode 100644 index 00000000..8b06d8da Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-110_278d58a31f638767250a99728cb7597939cb709051f22860f984ce157234c2a3.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-110_765993ef539ba5a1fb8fd2283a0085b1a0170cd64ef25b79cbbcb241db1a5e14.jpg b/.raw/MidasCivilAnalysisReference/images/page-110_765993ef539ba5a1fb8fd2283a0085b1a0170cd64ef25b79cbbcb241db1a5e14.jpg new file mode 100644 index 00000000..bf2f6138 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-110_765993ef539ba5a1fb8fd2283a0085b1a0170cd64ef25b79cbbcb241db1a5e14.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-111_165e4d1b53f69b2d35d026e1dd6fd668718bafbfe7be4be4cc4a8035c8fc3155.jpg b/.raw/MidasCivilAnalysisReference/images/page-111_165e4d1b53f69b2d35d026e1dd6fd668718bafbfe7be4be4cc4a8035c8fc3155.jpg new file mode 100644 index 00000000..0df9c586 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-111_165e4d1b53f69b2d35d026e1dd6fd668718bafbfe7be4be4cc4a8035c8fc3155.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-111_1d58c80a2f9d98514ebb3ac98a936fb3259793c223f660105ec9bbfa5d3157d3.jpg b/.raw/MidasCivilAnalysisReference/images/page-111_1d58c80a2f9d98514ebb3ac98a936fb3259793c223f660105ec9bbfa5d3157d3.jpg new file mode 100644 index 00000000..d3968314 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-111_1d58c80a2f9d98514ebb3ac98a936fb3259793c223f660105ec9bbfa5d3157d3.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-111_3858826f73985a28d11760680d04f3419a470f1f519516d59475f042eec01435.jpg b/.raw/MidasCivilAnalysisReference/images/page-111_3858826f73985a28d11760680d04f3419a470f1f519516d59475f042eec01435.jpg new file mode 100644 index 00000000..51c782b4 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-111_3858826f73985a28d11760680d04f3419a470f1f519516d59475f042eec01435.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-111_9c67197fcf9c0e6a1274af5deec6e9e4f43e66876531b0c7ed6d5d915485372c.jpg b/.raw/MidasCivilAnalysisReference/images/page-111_9c67197fcf9c0e6a1274af5deec6e9e4f43e66876531b0c7ed6d5d915485372c.jpg new file mode 100644 index 00000000..a7a804bb Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-111_9c67197fcf9c0e6a1274af5deec6e9e4f43e66876531b0c7ed6d5d915485372c.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-111_bfebb7c3e5e4061c1d28581a95df6b7b019a24e1dedac52b57a29baee9fc5ca8.jpg b/.raw/MidasCivilAnalysisReference/images/page-111_bfebb7c3e5e4061c1d28581a95df6b7b019a24e1dedac52b57a29baee9fc5ca8.jpg new file mode 100644 index 00000000..8a6789ac Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-111_bfebb7c3e5e4061c1d28581a95df6b7b019a24e1dedac52b57a29baee9fc5ca8.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-111_c393c8ca86c8f7036341096679bc5a557c0c3faf01fe6aeba2d0c5afb601bcd1.jpg b/.raw/MidasCivilAnalysisReference/images/page-111_c393c8ca86c8f7036341096679bc5a557c0c3faf01fe6aeba2d0c5afb601bcd1.jpg new file mode 100644 index 00000000..a1b4a550 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-111_c393c8ca86c8f7036341096679bc5a557c0c3faf01fe6aeba2d0c5afb601bcd1.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-112_8d664dc092135adcf1bda0bcb2369ebc9ec07090ec891cb74abefe8d7436d5d5.jpg b/.raw/MidasCivilAnalysisReference/images/page-112_8d664dc092135adcf1bda0bcb2369ebc9ec07090ec891cb74abefe8d7436d5d5.jpg new file mode 100644 index 00000000..f3234240 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-112_8d664dc092135adcf1bda0bcb2369ebc9ec07090ec891cb74abefe8d7436d5d5.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-112_baf7529d6a5641e09a1d00a5c75ec02d6a251a0a3516d4a0d07021975fc06d6a.jpg b/.raw/MidasCivilAnalysisReference/images/page-112_baf7529d6a5641e09a1d00a5c75ec02d6a251a0a3516d4a0d07021975fc06d6a.jpg new file mode 100644 index 00000000..085c2f90 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-112_baf7529d6a5641e09a1d00a5c75ec02d6a251a0a3516d4a0d07021975fc06d6a.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-112_bfd47b83e6815256eeec355ae2f9c989c36ae190d5127623ce25e32994e47af9.jpg b/.raw/MidasCivilAnalysisReference/images/page-112_bfd47b83e6815256eeec355ae2f9c989c36ae190d5127623ce25e32994e47af9.jpg new file mode 100644 index 00000000..1370fec5 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-112_bfd47b83e6815256eeec355ae2f9c989c36ae190d5127623ce25e32994e47af9.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-112_cf6befe7afd37218166f67a07a228ecd521f1715c62bc3956d1a9b6bb1400c92.jpg b/.raw/MidasCivilAnalysisReference/images/page-112_cf6befe7afd37218166f67a07a228ecd521f1715c62bc3956d1a9b6bb1400c92.jpg new file mode 100644 index 00000000..2d9900c8 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-112_cf6befe7afd37218166f67a07a228ecd521f1715c62bc3956d1a9b6bb1400c92.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-112_f3cde8678bf59affe96b2fb57ef650e24a774844fffa9d08d437123ab02092cc.jpg b/.raw/MidasCivilAnalysisReference/images/page-112_f3cde8678bf59affe96b2fb57ef650e24a774844fffa9d08d437123ab02092cc.jpg new file mode 100644 index 00000000..21f9a31b Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-112_f3cde8678bf59affe96b2fb57ef650e24a774844fffa9d08d437123ab02092cc.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-113_15d2298bbbf92022aa56864d7d7ae66a99c724cb6aa1b3aea4ad9d8283ab57b6.jpg b/.raw/MidasCivilAnalysisReference/images/page-113_15d2298bbbf92022aa56864d7d7ae66a99c724cb6aa1b3aea4ad9d8283ab57b6.jpg new file mode 100644 index 00000000..40d041f2 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-113_15d2298bbbf92022aa56864d7d7ae66a99c724cb6aa1b3aea4ad9d8283ab57b6.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-113_831bde407cfdd77e046822367b000635c81ce0ca581c2717785042e6d55217d2.jpg b/.raw/MidasCivilAnalysisReference/images/page-113_831bde407cfdd77e046822367b000635c81ce0ca581c2717785042e6d55217d2.jpg new file mode 100644 index 00000000..c48e58a1 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-113_831bde407cfdd77e046822367b000635c81ce0ca581c2717785042e6d55217d2.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-113_8c4fd8aacdc35ab5baeb62c1005f322495f63cd6e423415158151a6b6170f46c.jpg b/.raw/MidasCivilAnalysisReference/images/page-113_8c4fd8aacdc35ab5baeb62c1005f322495f63cd6e423415158151a6b6170f46c.jpg new file mode 100644 index 00000000..568a7fe9 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-113_8c4fd8aacdc35ab5baeb62c1005f322495f63cd6e423415158151a6b6170f46c.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-113_ea5b1a689cb5394ce6f1a126a272daa17d0a2bc452825a3f8f8b1d707a3c407c.jpg b/.raw/MidasCivilAnalysisReference/images/page-113_ea5b1a689cb5394ce6f1a126a272daa17d0a2bc452825a3f8f8b1d707a3c407c.jpg new file mode 100644 index 00000000..4efefa09 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-113_ea5b1a689cb5394ce6f1a126a272daa17d0a2bc452825a3f8f8b1d707a3c407c.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-113_f9bc340488b7a8dfafe36feabfb949629c400321215411e35c5f15e387491d6b.jpg b/.raw/MidasCivilAnalysisReference/images/page-113_f9bc340488b7a8dfafe36feabfb949629c400321215411e35c5f15e387491d6b.jpg new file mode 100644 index 00000000..15dfbf7a Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-113_f9bc340488b7a8dfafe36feabfb949629c400321215411e35c5f15e387491d6b.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-113_febce336ef3775661ba341f64173df59b69d36d832bc3d2530d44d3f6b52430b.jpg b/.raw/MidasCivilAnalysisReference/images/page-113_febce336ef3775661ba341f64173df59b69d36d832bc3d2530d44d3f6b52430b.jpg new file mode 100644 index 00000000..c89e4cdb Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-113_febce336ef3775661ba341f64173df59b69d36d832bc3d2530d44d3f6b52430b.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-114_550df1d2f4a6d98cccce0e19ff1db690a02ebb47fa681e28bf1b9315bb552be4.jpg b/.raw/MidasCivilAnalysisReference/images/page-114_550df1d2f4a6d98cccce0e19ff1db690a02ebb47fa681e28bf1b9315bb552be4.jpg new file mode 100644 index 00000000..a1731a83 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-114_550df1d2f4a6d98cccce0e19ff1db690a02ebb47fa681e28bf1b9315bb552be4.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-114_9878712cb637f736e87039f9201a193d1a2b0df63bf07cee677bd93a6b94877a.jpg b/.raw/MidasCivilAnalysisReference/images/page-114_9878712cb637f736e87039f9201a193d1a2b0df63bf07cee677bd93a6b94877a.jpg new file mode 100644 index 00000000..033780b8 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-114_9878712cb637f736e87039f9201a193d1a2b0df63bf07cee677bd93a6b94877a.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-114_b794fd795e31705592465f7aad76ccd1d232d8d6579236aa24b81a1c2a5499cc.jpg b/.raw/MidasCivilAnalysisReference/images/page-114_b794fd795e31705592465f7aad76ccd1d232d8d6579236aa24b81a1c2a5499cc.jpg new file mode 100644 index 00000000..3a5d16ef Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-114_b794fd795e31705592465f7aad76ccd1d232d8d6579236aa24b81a1c2a5499cc.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-114_cc777fc713a37f4d8f3da67eeb6efc9d955452761e104818387794aa05dfc3c1.jpg b/.raw/MidasCivilAnalysisReference/images/page-114_cc777fc713a37f4d8f3da67eeb6efc9d955452761e104818387794aa05dfc3c1.jpg new file mode 100644 index 00000000..864c0050 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-114_cc777fc713a37f4d8f3da67eeb6efc9d955452761e104818387794aa05dfc3c1.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-115_dd2cc8459ea4fbabd7fa2447abc1c912fd7401b590982843121d9932a8eae77f.jpg b/.raw/MidasCivilAnalysisReference/images/page-115_dd2cc8459ea4fbabd7fa2447abc1c912fd7401b590982843121d9932a8eae77f.jpg new file mode 100644 index 00000000..a7e17ff8 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-115_dd2cc8459ea4fbabd7fa2447abc1c912fd7401b590982843121d9932a8eae77f.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-115_ee2e8069d47db4fb988472e3428ec01da7b12bf8d55068f9c4183ab452b3b655.jpg b/.raw/MidasCivilAnalysisReference/images/page-115_ee2e8069d47db4fb988472e3428ec01da7b12bf8d55068f9c4183ab452b3b655.jpg new file mode 100644 index 00000000..1a86f4f4 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-115_ee2e8069d47db4fb988472e3428ec01da7b12bf8d55068f9c4183ab452b3b655.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-116_345d06dce0eb4db55081f257b2b184f18331d67b5e6edc20392146a4fbc48a6c.jpg b/.raw/MidasCivilAnalysisReference/images/page-116_345d06dce0eb4db55081f257b2b184f18331d67b5e6edc20392146a4fbc48a6c.jpg new file mode 100644 index 00000000..04b94936 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-116_345d06dce0eb4db55081f257b2b184f18331d67b5e6edc20392146a4fbc48a6c.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-116_5b4208ca92138cab56f6c44cb51d2e142d388793fcc01c987d9caa042b53315b.jpg b/.raw/MidasCivilAnalysisReference/images/page-116_5b4208ca92138cab56f6c44cb51d2e142d388793fcc01c987d9caa042b53315b.jpg new file mode 100644 index 00000000..1dce1524 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-116_5b4208ca92138cab56f6c44cb51d2e142d388793fcc01c987d9caa042b53315b.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-116_6521f9efbd1f8d42fffa48bca1d0ace72841a94d4e1b2de327f778a5086d8825.jpg b/.raw/MidasCivilAnalysisReference/images/page-116_6521f9efbd1f8d42fffa48bca1d0ace72841a94d4e1b2de327f778a5086d8825.jpg new file mode 100644 index 00000000..69e2302a Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-116_6521f9efbd1f8d42fffa48bca1d0ace72841a94d4e1b2de327f778a5086d8825.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-116_76125e620fb697981f6dc1d408faf0efd0f4d5f64e898e9e0811cf184eb2290b.jpg b/.raw/MidasCivilAnalysisReference/images/page-116_76125e620fb697981f6dc1d408faf0efd0f4d5f64e898e9e0811cf184eb2290b.jpg new file mode 100644 index 00000000..359efe85 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-116_76125e620fb697981f6dc1d408faf0efd0f4d5f64e898e9e0811cf184eb2290b.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-117_099c9492f8f0734bcf81bb3b1c24a00ccd3352d0880919e4178d6cb9ed97ca90.jpg b/.raw/MidasCivilAnalysisReference/images/page-117_099c9492f8f0734bcf81bb3b1c24a00ccd3352d0880919e4178d6cb9ed97ca90.jpg new file mode 100644 index 00000000..83fded2e Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-117_099c9492f8f0734bcf81bb3b1c24a00ccd3352d0880919e4178d6cb9ed97ca90.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-117_3f6a1dd3bdbe8b7edc7870bd66a411fd2b4236942ca9dad647ed210e816e2286.jpg b/.raw/MidasCivilAnalysisReference/images/page-117_3f6a1dd3bdbe8b7edc7870bd66a411fd2b4236942ca9dad647ed210e816e2286.jpg new file mode 100644 index 00000000..916e4db1 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-117_3f6a1dd3bdbe8b7edc7870bd66a411fd2b4236942ca9dad647ed210e816e2286.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-117_484e2815541db18fb76aaf4b684e062c79d00be3312cef4b587f5ff7416ae37f.jpg b/.raw/MidasCivilAnalysisReference/images/page-117_484e2815541db18fb76aaf4b684e062c79d00be3312cef4b587f5ff7416ae37f.jpg new file mode 100644 index 00000000..10acfc46 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-117_484e2815541db18fb76aaf4b684e062c79d00be3312cef4b587f5ff7416ae37f.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-117_58b80f0b3ab05791e4af440e434450d81cc5bab00ea58fe14a5bdb12ffe1f351.jpg b/.raw/MidasCivilAnalysisReference/images/page-117_58b80f0b3ab05791e4af440e434450d81cc5bab00ea58fe14a5bdb12ffe1f351.jpg new file mode 100644 index 00000000..75d5775b Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-117_58b80f0b3ab05791e4af440e434450d81cc5bab00ea58fe14a5bdb12ffe1f351.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-117_fa16e26bba5738f2f78520f27aab0937f57706b85aab8c06696b844aadeafa98.jpg b/.raw/MidasCivilAnalysisReference/images/page-117_fa16e26bba5738f2f78520f27aab0937f57706b85aab8c06696b844aadeafa98.jpg new file mode 100644 index 00000000..1ea2f192 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-117_fa16e26bba5738f2f78520f27aab0937f57706b85aab8c06696b844aadeafa98.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-118_3a2e941f5f203f3d670861ebab7d2ca034d805a3cc8e67769363994f19d92f4d.jpg b/.raw/MidasCivilAnalysisReference/images/page-118_3a2e941f5f203f3d670861ebab7d2ca034d805a3cc8e67769363994f19d92f4d.jpg new file mode 100644 index 00000000..2efffd4f Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-118_3a2e941f5f203f3d670861ebab7d2ca034d805a3cc8e67769363994f19d92f4d.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-118_7f8cdce052cea00faf202e5564f654d8bacf30745781a0f30db79442bce05b52.jpg b/.raw/MidasCivilAnalysisReference/images/page-118_7f8cdce052cea00faf202e5564f654d8bacf30745781a0f30db79442bce05b52.jpg new file mode 100644 index 00000000..ab6a6a38 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-118_7f8cdce052cea00faf202e5564f654d8bacf30745781a0f30db79442bce05b52.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-118_81d01df31d5b3a2b0c067287c900a9ba56af1e78137e4bf84e980fd816180ebd.jpg b/.raw/MidasCivilAnalysisReference/images/page-118_81d01df31d5b3a2b0c067287c900a9ba56af1e78137e4bf84e980fd816180ebd.jpg new file mode 100644 index 00000000..60595f60 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-118_81d01df31d5b3a2b0c067287c900a9ba56af1e78137e4bf84e980fd816180ebd.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-118_86a71919026cf2ff6cb5d3dac6a6fea4a8bd93e7b4823cbfd64565a82ae13de3.jpg b/.raw/MidasCivilAnalysisReference/images/page-118_86a71919026cf2ff6cb5d3dac6a6fea4a8bd93e7b4823cbfd64565a82ae13de3.jpg new file mode 100644 index 00000000..18306375 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-118_86a71919026cf2ff6cb5d3dac6a6fea4a8bd93e7b4823cbfd64565a82ae13de3.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-119_05db4ef3e0ba35437049f3d771117b1ff630eddc42364f0c65fbb49591012e5c.jpg b/.raw/MidasCivilAnalysisReference/images/page-119_05db4ef3e0ba35437049f3d771117b1ff630eddc42364f0c65fbb49591012e5c.jpg new file mode 100644 index 00000000..201338d3 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-119_05db4ef3e0ba35437049f3d771117b1ff630eddc42364f0c65fbb49591012e5c.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-119_0ca1147aff6b561a0aa1e637f52bdb2cf844c501614d7688987c0b381f75e0d7.jpg b/.raw/MidasCivilAnalysisReference/images/page-119_0ca1147aff6b561a0aa1e637f52bdb2cf844c501614d7688987c0b381f75e0d7.jpg new file mode 100644 index 00000000..cd1e76a4 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-119_0ca1147aff6b561a0aa1e637f52bdb2cf844c501614d7688987c0b381f75e0d7.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-119_6ed9a4ae820f8833f3bc4aef5e48f99d80e61872c7cc75a952da1f198217ced6.jpg b/.raw/MidasCivilAnalysisReference/images/page-119_6ed9a4ae820f8833f3bc4aef5e48f99d80e61872c7cc75a952da1f198217ced6.jpg new file mode 100644 index 00000000..cc84c7a5 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-119_6ed9a4ae820f8833f3bc4aef5e48f99d80e61872c7cc75a952da1f198217ced6.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-119_a0d28e663a343d488b131052405832e027b40a1a970a5b94bd51922ecce9ae83.jpg b/.raw/MidasCivilAnalysisReference/images/page-119_a0d28e663a343d488b131052405832e027b40a1a970a5b94bd51922ecce9ae83.jpg new file mode 100644 index 00000000..9745f4f2 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-119_a0d28e663a343d488b131052405832e027b40a1a970a5b94bd51922ecce9ae83.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-120_1d5a383ef5eaddd880ab40c2067231881291a7068ac6eab5b7c6e96a721a3108.jpg b/.raw/MidasCivilAnalysisReference/images/page-120_1d5a383ef5eaddd880ab40c2067231881291a7068ac6eab5b7c6e96a721a3108.jpg new file mode 100644 index 00000000..47123093 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-120_1d5a383ef5eaddd880ab40c2067231881291a7068ac6eab5b7c6e96a721a3108.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-120_dcc29f5ce3168107f61f9e42fa9c2466371c244c6c125a4d8c1a957a1f7f8f68.jpg b/.raw/MidasCivilAnalysisReference/images/page-120_dcc29f5ce3168107f61f9e42fa9c2466371c244c6c125a4d8c1a957a1f7f8f68.jpg new file mode 100644 index 00000000..611879bb Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-120_dcc29f5ce3168107f61f9e42fa9c2466371c244c6c125a4d8c1a957a1f7f8f68.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-120_f4388acb4faf3f0c639c6bcc3627237c06a2aedda5806ed685fb3387be3040ee.jpg b/.raw/MidasCivilAnalysisReference/images/page-120_f4388acb4faf3f0c639c6bcc3627237c06a2aedda5806ed685fb3387be3040ee.jpg new file mode 100644 index 00000000..f8d7b296 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-120_f4388acb4faf3f0c639c6bcc3627237c06a2aedda5806ed685fb3387be3040ee.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-121_22e5a859a28121fe56f913b111b982a795ce87bf3474a89c65d9237c75df074b.jpg b/.raw/MidasCivilAnalysisReference/images/page-121_22e5a859a28121fe56f913b111b982a795ce87bf3474a89c65d9237c75df074b.jpg new file mode 100644 index 00000000..4551b40f Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-121_22e5a859a28121fe56f913b111b982a795ce87bf3474a89c65d9237c75df074b.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-121_39c48fc743a8582c82822950488dd66c57e6003350be3f4ff27d6f06472f1afb.jpg b/.raw/MidasCivilAnalysisReference/images/page-121_39c48fc743a8582c82822950488dd66c57e6003350be3f4ff27d6f06472f1afb.jpg new file mode 100644 index 00000000..31c176e1 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-121_39c48fc743a8582c82822950488dd66c57e6003350be3f4ff27d6f06472f1afb.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-121_469b2eedfc5562e6cc25efa78b3871f50edcafd8e07d3b45f9a51b7223cfdb99.jpg b/.raw/MidasCivilAnalysisReference/images/page-121_469b2eedfc5562e6cc25efa78b3871f50edcafd8e07d3b45f9a51b7223cfdb99.jpg new file mode 100644 index 00000000..103d8be9 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-121_469b2eedfc5562e6cc25efa78b3871f50edcafd8e07d3b45f9a51b7223cfdb99.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-121_9ceb8514fccbd859d2fb4a346b038c0d43806c1440704c307096269df5832e01.jpg b/.raw/MidasCivilAnalysisReference/images/page-121_9ceb8514fccbd859d2fb4a346b038c0d43806c1440704c307096269df5832e01.jpg new file mode 100644 index 00000000..a9c53526 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-121_9ceb8514fccbd859d2fb4a346b038c0d43806c1440704c307096269df5832e01.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-122_23e858e771b5f98a80330657962986d1c1f07ddcefc477347c9d1cb4b0507570.jpg b/.raw/MidasCivilAnalysisReference/images/page-122_23e858e771b5f98a80330657962986d1c1f07ddcefc477347c9d1cb4b0507570.jpg new file mode 100644 index 00000000..706a592c Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-122_23e858e771b5f98a80330657962986d1c1f07ddcefc477347c9d1cb4b0507570.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-122_db9284da7f88d9123b8b7b53151dd9839ac283ae8a60613910ca538f6a10b48c.jpg b/.raw/MidasCivilAnalysisReference/images/page-122_db9284da7f88d9123b8b7b53151dd9839ac283ae8a60613910ca538f6a10b48c.jpg new file mode 100644 index 00000000..cbde5a59 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-122_db9284da7f88d9123b8b7b53151dd9839ac283ae8a60613910ca538f6a10b48c.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-122_f129298830a1b2bed20bee1d1da5d671d76fa266ad6f56904d73a2c652150cd1.jpg b/.raw/MidasCivilAnalysisReference/images/page-122_f129298830a1b2bed20bee1d1da5d671d76fa266ad6f56904d73a2c652150cd1.jpg new file mode 100644 index 00000000..1d8203f4 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-122_f129298830a1b2bed20bee1d1da5d671d76fa266ad6f56904d73a2c652150cd1.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-123_0a96debf79d1c75927f57a452f2d249b2e32fa01f8decb28158f37267e395571.jpg b/.raw/MidasCivilAnalysisReference/images/page-123_0a96debf79d1c75927f57a452f2d249b2e32fa01f8decb28158f37267e395571.jpg new file mode 100644 index 00000000..23f54df5 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-123_0a96debf79d1c75927f57a452f2d249b2e32fa01f8decb28158f37267e395571.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-123_6f9ecd244d5a9bfe167cac7f7ff6077f3e428ad2ff139fc65d6b44d64bc6a1f6.jpg b/.raw/MidasCivilAnalysisReference/images/page-123_6f9ecd244d5a9bfe167cac7f7ff6077f3e428ad2ff139fc65d6b44d64bc6a1f6.jpg new file mode 100644 index 00000000..adca5161 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-123_6f9ecd244d5a9bfe167cac7f7ff6077f3e428ad2ff139fc65d6b44d64bc6a1f6.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-123_e7489ca524e0b21c9892950a78477b51f09f4753c0eb52b76db9c9a7a24e21ab.jpg b/.raw/MidasCivilAnalysisReference/images/page-123_e7489ca524e0b21c9892950a78477b51f09f4753c0eb52b76db9c9a7a24e21ab.jpg new file mode 100644 index 00000000..e7ddd089 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-123_e7489ca524e0b21c9892950a78477b51f09f4753c0eb52b76db9c9a7a24e21ab.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-125_b20948b2204987833ab1d3fe09e4d32a6fca5cc5fca6ab0c29ec4cfe552f3476.jpg b/.raw/MidasCivilAnalysisReference/images/page-125_b20948b2204987833ab1d3fe09e4d32a6fca5cc5fca6ab0c29ec4cfe552f3476.jpg new file mode 100644 index 00000000..20813657 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-125_b20948b2204987833ab1d3fe09e4d32a6fca5cc5fca6ab0c29ec4cfe552f3476.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-127_7ed146049321d04f92e82e2c9c680a88e525d2ab565c6828a9cfc1f9c9aa173f.jpg b/.raw/MidasCivilAnalysisReference/images/page-127_7ed146049321d04f92e82e2c9c680a88e525d2ab565c6828a9cfc1f9c9aa173f.jpg new file mode 100644 index 00000000..8ae3a4ab Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-127_7ed146049321d04f92e82e2c9c680a88e525d2ab565c6828a9cfc1f9c9aa173f.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-127_c1a79c56d71725a1a7f41994a0e689efb5e4296790a0d193e88b9161dd480e32.jpg b/.raw/MidasCivilAnalysisReference/images/page-127_c1a79c56d71725a1a7f41994a0e689efb5e4296790a0d193e88b9161dd480e32.jpg new file mode 100644 index 00000000..469c5736 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-127_c1a79c56d71725a1a7f41994a0e689efb5e4296790a0d193e88b9161dd480e32.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-129_12b23b002d2e15e2e5a2e989beafc0c0d4c820f778ed887b4eaf416ca421ea59.jpg b/.raw/MidasCivilAnalysisReference/images/page-129_12b23b002d2e15e2e5a2e989beafc0c0d4c820f778ed887b4eaf416ca421ea59.jpg new file mode 100644 index 00000000..2344b108 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-129_12b23b002d2e15e2e5a2e989beafc0c0d4c820f778ed887b4eaf416ca421ea59.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-129_63e5dcb87354b8d5cb7946544e55b8d8cfd34738b3e7b8f828b959e60d9c1192.jpg b/.raw/MidasCivilAnalysisReference/images/page-129_63e5dcb87354b8d5cb7946544e55b8d8cfd34738b3e7b8f828b959e60d9c1192.jpg new file mode 100644 index 00000000..f662d8b7 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-129_63e5dcb87354b8d5cb7946544e55b8d8cfd34738b3e7b8f828b959e60d9c1192.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-130_16d25830df34f37224526d92db3e479bf34897e5ad8191e644bb90398592884c.jpg b/.raw/MidasCivilAnalysisReference/images/page-130_16d25830df34f37224526d92db3e479bf34897e5ad8191e644bb90398592884c.jpg new file mode 100644 index 00000000..d83417a9 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-130_16d25830df34f37224526d92db3e479bf34897e5ad8191e644bb90398592884c.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-132_89e328b3a1e89c248391d6f3772981f1c9772bf35bddcfc692cc305923d9bbfa.jpg b/.raw/MidasCivilAnalysisReference/images/page-132_89e328b3a1e89c248391d6f3772981f1c9772bf35bddcfc692cc305923d9bbfa.jpg new file mode 100644 index 00000000..5c336523 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-132_89e328b3a1e89c248391d6f3772981f1c9772bf35bddcfc692cc305923d9bbfa.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-133_591284347aceb816adcfc4db1df875f8749149ab9ee044c134716849f4888a62.jpg b/.raw/MidasCivilAnalysisReference/images/page-133_591284347aceb816adcfc4db1df875f8749149ab9ee044c134716849f4888a62.jpg new file mode 100644 index 00000000..a49b1cc0 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-133_591284347aceb816adcfc4db1df875f8749149ab9ee044c134716849f4888a62.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-134_60b061932153112d7909a4286a0fe90bf631f7227243a90f2ff7e29cf6573619.jpg b/.raw/MidasCivilAnalysisReference/images/page-134_60b061932153112d7909a4286a0fe90bf631f7227243a90f2ff7e29cf6573619.jpg new file mode 100644 index 00000000..223b8fdd Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-134_60b061932153112d7909a4286a0fe90bf631f7227243a90f2ff7e29cf6573619.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-134_6eabfddac55286296fce3d48b7bcd79c6b4e7ffbe17c27ef86fc202d04066aa6.jpg b/.raw/MidasCivilAnalysisReference/images/page-134_6eabfddac55286296fce3d48b7bcd79c6b4e7ffbe17c27ef86fc202d04066aa6.jpg new file mode 100644 index 00000000..38c4e9e5 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-134_6eabfddac55286296fce3d48b7bcd79c6b4e7ffbe17c27ef86fc202d04066aa6.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-134_b188ff5277de47f404aa5dd0b5902c84bc830d470f0b5293a384fa166a69a767.jpg b/.raw/MidasCivilAnalysisReference/images/page-134_b188ff5277de47f404aa5dd0b5902c84bc830d470f0b5293a384fa166a69a767.jpg new file mode 100644 index 00000000..24f68aaa Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-134_b188ff5277de47f404aa5dd0b5902c84bc830d470f0b5293a384fa166a69a767.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-134_ba3ce05f6a76f441380f85c472e493271a4eb6bfa53e0b07d4cd9c509e13f456.jpg b/.raw/MidasCivilAnalysisReference/images/page-134_ba3ce05f6a76f441380f85c472e493271a4eb6bfa53e0b07d4cd9c509e13f456.jpg new file mode 100644 index 00000000..cb35316a Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-134_ba3ce05f6a76f441380f85c472e493271a4eb6bfa53e0b07d4cd9c509e13f456.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-134_d09e966b5e89ffca947570f2ba3ed7adfb0e1aee16760908e9645d35fef9d63d.jpg b/.raw/MidasCivilAnalysisReference/images/page-134_d09e966b5e89ffca947570f2ba3ed7adfb0e1aee16760908e9645d35fef9d63d.jpg new file mode 100644 index 00000000..9c7d1e35 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-134_d09e966b5e89ffca947570f2ba3ed7adfb0e1aee16760908e9645d35fef9d63d.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-134_d8bd22a42e15eb535735b211b025b18b7e23324ffa0b33987b1306bae15e5e0d.jpg b/.raw/MidasCivilAnalysisReference/images/page-134_d8bd22a42e15eb535735b211b025b18b7e23324ffa0b33987b1306bae15e5e0d.jpg new file mode 100644 index 00000000..042c4ba5 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-134_d8bd22a42e15eb535735b211b025b18b7e23324ffa0b33987b1306bae15e5e0d.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-135_0e8c655a9c568cfb764aac1f31fdea201a091210b24b026f9d2006a4996a2ae1.jpg b/.raw/MidasCivilAnalysisReference/images/page-135_0e8c655a9c568cfb764aac1f31fdea201a091210b24b026f9d2006a4996a2ae1.jpg new file mode 100644 index 00000000..689f50a0 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-135_0e8c655a9c568cfb764aac1f31fdea201a091210b24b026f9d2006a4996a2ae1.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-135_23bf7caf8bcbb38a5771b6d83a69c6e48b0594f4b2449ab20dc3022e3711b201.jpg b/.raw/MidasCivilAnalysisReference/images/page-135_23bf7caf8bcbb38a5771b6d83a69c6e48b0594f4b2449ab20dc3022e3711b201.jpg new file mode 100644 index 00000000..af1c1045 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-135_23bf7caf8bcbb38a5771b6d83a69c6e48b0594f4b2449ab20dc3022e3711b201.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-135_e48538ba9e9a6c610e815d0b2a38f699f39f86cf42bf84667df11387c18fea11.jpg b/.raw/MidasCivilAnalysisReference/images/page-135_e48538ba9e9a6c610e815d0b2a38f699f39f86cf42bf84667df11387c18fea11.jpg new file mode 100644 index 00000000..d2facd54 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-135_e48538ba9e9a6c610e815d0b2a38f699f39f86cf42bf84667df11387c18fea11.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-136_63ab3629a0640a714b5dfa9f0b4e377cf4a7c82657f6690b6cf62a5f9323bebc.jpg b/.raw/MidasCivilAnalysisReference/images/page-136_63ab3629a0640a714b5dfa9f0b4e377cf4a7c82657f6690b6cf62a5f9323bebc.jpg new file mode 100644 index 00000000..26d73c7a Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-136_63ab3629a0640a714b5dfa9f0b4e377cf4a7c82657f6690b6cf62a5f9323bebc.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-137_02b73c9d9eef2214adfc26a2bfea4b6adc3824f2b6518ccfd4939bcf8a8fe869.jpg b/.raw/MidasCivilAnalysisReference/images/page-137_02b73c9d9eef2214adfc26a2bfea4b6adc3824f2b6518ccfd4939bcf8a8fe869.jpg new file mode 100644 index 00000000..811e9f16 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-137_02b73c9d9eef2214adfc26a2bfea4b6adc3824f2b6518ccfd4939bcf8a8fe869.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-139_cc0f03adbc645b7b1a16e128a53ffe29b85e4273a843bb6a800241ac783d2071.jpg b/.raw/MidasCivilAnalysisReference/images/page-139_cc0f03adbc645b7b1a16e128a53ffe29b85e4273a843bb6a800241ac783d2071.jpg new file mode 100644 index 00000000..52ced5b8 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-139_cc0f03adbc645b7b1a16e128a53ffe29b85e4273a843bb6a800241ac783d2071.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-140_1e0311c3f78b2c8caa2b8ec110ca7eed3f89ba45d25d5d77cbaf2bc27fe0de6e.jpg b/.raw/MidasCivilAnalysisReference/images/page-140_1e0311c3f78b2c8caa2b8ec110ca7eed3f89ba45d25d5d77cbaf2bc27fe0de6e.jpg new file mode 100644 index 00000000..49634a85 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-140_1e0311c3f78b2c8caa2b8ec110ca7eed3f89ba45d25d5d77cbaf2bc27fe0de6e.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-141_71a4663434cbbd0f76766263975eb682b04388304b5b66ec6d8d618c6e021f3d.jpg b/.raw/MidasCivilAnalysisReference/images/page-141_71a4663434cbbd0f76766263975eb682b04388304b5b66ec6d8d618c6e021f3d.jpg new file mode 100644 index 00000000..91f20d73 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-141_71a4663434cbbd0f76766263975eb682b04388304b5b66ec6d8d618c6e021f3d.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-141_9f66eea46704687d122c49581db22c37985b0f251e30dfd05255a233aeb89194.jpg b/.raw/MidasCivilAnalysisReference/images/page-141_9f66eea46704687d122c49581db22c37985b0f251e30dfd05255a233aeb89194.jpg new file mode 100644 index 00000000..b3eee128 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-141_9f66eea46704687d122c49581db22c37985b0f251e30dfd05255a233aeb89194.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-141_b9f308dcc7db8a450d5486bc5d0587b221c62bf6c520148d3ff4fe9704c2654f.jpg b/.raw/MidasCivilAnalysisReference/images/page-141_b9f308dcc7db8a450d5486bc5d0587b221c62bf6c520148d3ff4fe9704c2654f.jpg new file mode 100644 index 00000000..e786e405 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-141_b9f308dcc7db8a450d5486bc5d0587b221c62bf6c520148d3ff4fe9704c2654f.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-142_e7dfe90bd61415c961d2a2207c80e95a6db9c2c3e7002aefb8582c563fe10aee.jpg b/.raw/MidasCivilAnalysisReference/images/page-142_e7dfe90bd61415c961d2a2207c80e95a6db9c2c3e7002aefb8582c563fe10aee.jpg new file mode 100644 index 00000000..bcfd2029 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-142_e7dfe90bd61415c961d2a2207c80e95a6db9c2c3e7002aefb8582c563fe10aee.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-143_8c61fffdcd95606255ffbc0fc84e912bdc4a3ffd7e5caa008c16892f3b5f6c7f.jpg b/.raw/MidasCivilAnalysisReference/images/page-143_8c61fffdcd95606255ffbc0fc84e912bdc4a3ffd7e5caa008c16892f3b5f6c7f.jpg new file mode 100644 index 00000000..c530155c Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-143_8c61fffdcd95606255ffbc0fc84e912bdc4a3ffd7e5caa008c16892f3b5f6c7f.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-144_1028014255611eb677af5fddbea499b1da5599d59893c42e2a73141a558ef4f2.jpg b/.raw/MidasCivilAnalysisReference/images/page-144_1028014255611eb677af5fddbea499b1da5599d59893c42e2a73141a558ef4f2.jpg new file mode 100644 index 00000000..318598bd Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-144_1028014255611eb677af5fddbea499b1da5599d59893c42e2a73141a558ef4f2.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-144_258a988efde2bcb940608ec8e133e8e3309cd504933f5cf6110ae88c2681e556.jpg b/.raw/MidasCivilAnalysisReference/images/page-144_258a988efde2bcb940608ec8e133e8e3309cd504933f5cf6110ae88c2681e556.jpg new file mode 100644 index 00000000..365a1e1a Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-144_258a988efde2bcb940608ec8e133e8e3309cd504933f5cf6110ae88c2681e556.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-144_2c3ec786c5114ff77e3b67b011ba40a165e220989e00ed7026aca454ae14d047.jpg b/.raw/MidasCivilAnalysisReference/images/page-144_2c3ec786c5114ff77e3b67b011ba40a165e220989e00ed7026aca454ae14d047.jpg new file mode 100644 index 00000000..c6db1d16 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-144_2c3ec786c5114ff77e3b67b011ba40a165e220989e00ed7026aca454ae14d047.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-144_4037051e459329f6d72e7b7991a4f425ea99f8201dc4035bfdcd0f2fb4fd31de.jpg b/.raw/MidasCivilAnalysisReference/images/page-144_4037051e459329f6d72e7b7991a4f425ea99f8201dc4035bfdcd0f2fb4fd31de.jpg new file mode 100644 index 00000000..f8779118 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-144_4037051e459329f6d72e7b7991a4f425ea99f8201dc4035bfdcd0f2fb4fd31de.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-144_764b8f87dd9691b38ea0b81b7b64fe989c1a84e0d9e4928b257d2d1cb00291c1.jpg b/.raw/MidasCivilAnalysisReference/images/page-144_764b8f87dd9691b38ea0b81b7b64fe989c1a84e0d9e4928b257d2d1cb00291c1.jpg new file mode 100644 index 00000000..127a6354 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-144_764b8f87dd9691b38ea0b81b7b64fe989c1a84e0d9e4928b257d2d1cb00291c1.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-144_7a94e5398a85f6ab5739b41324b89168a8b3cd749e3a32dc68ec245b80d21a3f.jpg b/.raw/MidasCivilAnalysisReference/images/page-144_7a94e5398a85f6ab5739b41324b89168a8b3cd749e3a32dc68ec245b80d21a3f.jpg new file mode 100644 index 00000000..d6aef5db Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-144_7a94e5398a85f6ab5739b41324b89168a8b3cd749e3a32dc68ec245b80d21a3f.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-144_7cdf33c92e922ab986227c239cdca3756de55e13e66daf2ac64b230ca127eccc.jpg b/.raw/MidasCivilAnalysisReference/images/page-144_7cdf33c92e922ab986227c239cdca3756de55e13e66daf2ac64b230ca127eccc.jpg new file mode 100644 index 00000000..e4b6874f Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-144_7cdf33c92e922ab986227c239cdca3756de55e13e66daf2ac64b230ca127eccc.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-144_d2c068ef4aad084496fc76fe21ceca9e97a7fbf3d79b5dc628d3af4451e54e25.jpg b/.raw/MidasCivilAnalysisReference/images/page-144_d2c068ef4aad084496fc76fe21ceca9e97a7fbf3d79b5dc628d3af4451e54e25.jpg new file mode 100644 index 00000000..e8723edd Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-144_d2c068ef4aad084496fc76fe21ceca9e97a7fbf3d79b5dc628d3af4451e54e25.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-144_dd3e667ce4102e14c5f8bec6967b99700bd7b4b878d627b7745d31bd5da26f0a.jpg b/.raw/MidasCivilAnalysisReference/images/page-144_dd3e667ce4102e14c5f8bec6967b99700bd7b4b878d627b7745d31bd5da26f0a.jpg new file mode 100644 index 00000000..e45a9ed7 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-144_dd3e667ce4102e14c5f8bec6967b99700bd7b4b878d627b7745d31bd5da26f0a.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-146_6836ce8f86636835d324e2e7db65870787bea965055158e36c21bc3297bedd22.jpg b/.raw/MidasCivilAnalysisReference/images/page-146_6836ce8f86636835d324e2e7db65870787bea965055158e36c21bc3297bedd22.jpg new file mode 100644 index 00000000..2f39dd47 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-146_6836ce8f86636835d324e2e7db65870787bea965055158e36c21bc3297bedd22.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-146_a66f5099279f685c57fbce2362dbb30c290d4e3f18fea530782dd4827971b3e1.jpg b/.raw/MidasCivilAnalysisReference/images/page-146_a66f5099279f685c57fbce2362dbb30c290d4e3f18fea530782dd4827971b3e1.jpg new file mode 100644 index 00000000..fec49c4d Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-146_a66f5099279f685c57fbce2362dbb30c290d4e3f18fea530782dd4827971b3e1.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-146_b2e8d6f29539c39c64dd98411d196407c13c4b4f68530d5a5a63fa4611a65229.jpg b/.raw/MidasCivilAnalysisReference/images/page-146_b2e8d6f29539c39c64dd98411d196407c13c4b4f68530d5a5a63fa4611a65229.jpg new file mode 100644 index 00000000..033c80d8 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-146_b2e8d6f29539c39c64dd98411d196407c13c4b4f68530d5a5a63fa4611a65229.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-147_2cb17d024b5ed6c15a7e1c65c373809323fe350fb52b1fc9f2141247ec160b08.jpg b/.raw/MidasCivilAnalysisReference/images/page-147_2cb17d024b5ed6c15a7e1c65c373809323fe350fb52b1fc9f2141247ec160b08.jpg new file mode 100644 index 00000000..2c9dcc68 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-147_2cb17d024b5ed6c15a7e1c65c373809323fe350fb52b1fc9f2141247ec160b08.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-147_d90f29f44652c295fbd67a7ade16a9e6a9ed8709d3455326e14d4bed92d248da.jpg b/.raw/MidasCivilAnalysisReference/images/page-147_d90f29f44652c295fbd67a7ade16a9e6a9ed8709d3455326e14d4bed92d248da.jpg new file mode 100644 index 00000000..4aec033a Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-147_d90f29f44652c295fbd67a7ade16a9e6a9ed8709d3455326e14d4bed92d248da.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-148_015bc1ca29202c93cf44dfbc791d907c741a31feb68df4f2821007eeadee0848.jpg b/.raw/MidasCivilAnalysisReference/images/page-148_015bc1ca29202c93cf44dfbc791d907c741a31feb68df4f2821007eeadee0848.jpg new file mode 100644 index 00000000..6d2ced9c Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-148_015bc1ca29202c93cf44dfbc791d907c741a31feb68df4f2821007eeadee0848.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-148_323d313e467e52be05b5ff4fb93f935297b11474546f0d7e875a63fd872c10d0.jpg b/.raw/MidasCivilAnalysisReference/images/page-148_323d313e467e52be05b5ff4fb93f935297b11474546f0d7e875a63fd872c10d0.jpg new file mode 100644 index 00000000..c77d61be Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-148_323d313e467e52be05b5ff4fb93f935297b11474546f0d7e875a63fd872c10d0.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-148_3c43a0d5261a5fa082de0eb652d2b90e19815ec399764eab97a520d1a7fcfc3b.jpg b/.raw/MidasCivilAnalysisReference/images/page-148_3c43a0d5261a5fa082de0eb652d2b90e19815ec399764eab97a520d1a7fcfc3b.jpg new file mode 100644 index 00000000..563e2b26 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-148_3c43a0d5261a5fa082de0eb652d2b90e19815ec399764eab97a520d1a7fcfc3b.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-148_67004aed6c7c9945833f873592ddbab876e2ae063026542083ebbc29aa55846e.jpg b/.raw/MidasCivilAnalysisReference/images/page-148_67004aed6c7c9945833f873592ddbab876e2ae063026542083ebbc29aa55846e.jpg new file mode 100644 index 00000000..30546278 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-148_67004aed6c7c9945833f873592ddbab876e2ae063026542083ebbc29aa55846e.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-148_76c9c08ace7d145120af9935a58c2c5ab0aa0736c808b4910d6736f575602c1d.jpg b/.raw/MidasCivilAnalysisReference/images/page-148_76c9c08ace7d145120af9935a58c2c5ab0aa0736c808b4910d6736f575602c1d.jpg new file mode 100644 index 00000000..c3ded0f6 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-148_76c9c08ace7d145120af9935a58c2c5ab0aa0736c808b4910d6736f575602c1d.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-151_7c7795b645f597e4dbe06a01846be4d583a28758ec9e05c00a6deb6ecdb0aa01.jpg b/.raw/MidasCivilAnalysisReference/images/page-151_7c7795b645f597e4dbe06a01846be4d583a28758ec9e05c00a6deb6ecdb0aa01.jpg new file mode 100644 index 00000000..2310169c Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-151_7c7795b645f597e4dbe06a01846be4d583a28758ec9e05c00a6deb6ecdb0aa01.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-152_03ccfd3acf6c5ff18bd07ce1a8e34336751cb0650f85cb26102ec1b718dedf5a.jpg b/.raw/MidasCivilAnalysisReference/images/page-152_03ccfd3acf6c5ff18bd07ce1a8e34336751cb0650f85cb26102ec1b718dedf5a.jpg new file mode 100644 index 00000000..f2765bf9 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-152_03ccfd3acf6c5ff18bd07ce1a8e34336751cb0650f85cb26102ec1b718dedf5a.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-152_6fd9112427ce367ebc2abb833a32263d275b25320c169b608c9721293ea538ee.jpg b/.raw/MidasCivilAnalysisReference/images/page-152_6fd9112427ce367ebc2abb833a32263d275b25320c169b608c9721293ea538ee.jpg new file mode 100644 index 00000000..31f60663 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-152_6fd9112427ce367ebc2abb833a32263d275b25320c169b608c9721293ea538ee.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-152_a5ec3d0899e0b54059425fd6d4dc591a84d4963d18c2f4a129b5c77be1a1263f.jpg b/.raw/MidasCivilAnalysisReference/images/page-152_a5ec3d0899e0b54059425fd6d4dc591a84d4963d18c2f4a129b5c77be1a1263f.jpg new file mode 100644 index 00000000..44bd3ca0 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-152_a5ec3d0899e0b54059425fd6d4dc591a84d4963d18c2f4a129b5c77be1a1263f.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-152_ab1003d0518f6dafa11396afdb47f5467749ef89db47f3d94345d76af2b2d1fb.jpg b/.raw/MidasCivilAnalysisReference/images/page-152_ab1003d0518f6dafa11396afdb47f5467749ef89db47f3d94345d76af2b2d1fb.jpg new file mode 100644 index 00000000..757da0b1 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-152_ab1003d0518f6dafa11396afdb47f5467749ef89db47f3d94345d76af2b2d1fb.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-153_026038e388d25ff1bbd03d2caa5dc7948831774b155da7a691b73586193ced9a.jpg b/.raw/MidasCivilAnalysisReference/images/page-153_026038e388d25ff1bbd03d2caa5dc7948831774b155da7a691b73586193ced9a.jpg new file mode 100644 index 00000000..62c1360f Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-153_026038e388d25ff1bbd03d2caa5dc7948831774b155da7a691b73586193ced9a.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-154_20e81430706f91b91c6fcef0675fd0227c85ed646fd12196ef46704269b5b049.jpg b/.raw/MidasCivilAnalysisReference/images/page-154_20e81430706f91b91c6fcef0675fd0227c85ed646fd12196ef46704269b5b049.jpg new file mode 100644 index 00000000..5bd30441 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-154_20e81430706f91b91c6fcef0675fd0227c85ed646fd12196ef46704269b5b049.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-155_33db6eb0665835fe26364c500186f6861fd99e12fc0eca939453af48b85fd89d.jpg b/.raw/MidasCivilAnalysisReference/images/page-155_33db6eb0665835fe26364c500186f6861fd99e12fc0eca939453af48b85fd89d.jpg new file mode 100644 index 00000000..2d39756a Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-155_33db6eb0665835fe26364c500186f6861fd99e12fc0eca939453af48b85fd89d.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-155_bfcb91747c53d9dd697ce35bcb2889058f053e3f6319a587d7bbfca8ba4d65b8.jpg b/.raw/MidasCivilAnalysisReference/images/page-155_bfcb91747c53d9dd697ce35bcb2889058f053e3f6319a587d7bbfca8ba4d65b8.jpg new file mode 100644 index 00000000..491b4d6c Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-155_bfcb91747c53d9dd697ce35bcb2889058f053e3f6319a587d7bbfca8ba4d65b8.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-156_418c11a8956584029af40096c122a48a108702159e3e9d91f0c6b70014a5a94b.jpg b/.raw/MidasCivilAnalysisReference/images/page-156_418c11a8956584029af40096c122a48a108702159e3e9d91f0c6b70014a5a94b.jpg new file mode 100644 index 00000000..6f458c01 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-156_418c11a8956584029af40096c122a48a108702159e3e9d91f0c6b70014a5a94b.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-156_7dd7c38ec042d6e1981a16e0348867ed2317bb520696d20e3e7cd5d87e2a6ea0.jpg b/.raw/MidasCivilAnalysisReference/images/page-156_7dd7c38ec042d6e1981a16e0348867ed2317bb520696d20e3e7cd5d87e2a6ea0.jpg new file mode 100644 index 00000000..282fa85a Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-156_7dd7c38ec042d6e1981a16e0348867ed2317bb520696d20e3e7cd5d87e2a6ea0.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-157_79d573c4f700b4b2d3777dd349e4840d00d67e61399807486a672bc87428837b.jpg b/.raw/MidasCivilAnalysisReference/images/page-157_79d573c4f700b4b2d3777dd349e4840d00d67e61399807486a672bc87428837b.jpg new file mode 100644 index 00000000..d68fdb3a Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-157_79d573c4f700b4b2d3777dd349e4840d00d67e61399807486a672bc87428837b.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-157_aa19d04d0e78b74b50f93f0edc04c6b469efc9da6fbcd8d6cfc91ee243efab65.jpg b/.raw/MidasCivilAnalysisReference/images/page-157_aa19d04d0e78b74b50f93f0edc04c6b469efc9da6fbcd8d6cfc91ee243efab65.jpg new file mode 100644 index 00000000..63cebbe7 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-157_aa19d04d0e78b74b50f93f0edc04c6b469efc9da6fbcd8d6cfc91ee243efab65.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-158_52ee3171262615e29efd499abc7cd1f70f7c34e75b017ad544b8b72c089ba380.jpg b/.raw/MidasCivilAnalysisReference/images/page-158_52ee3171262615e29efd499abc7cd1f70f7c34e75b017ad544b8b72c089ba380.jpg new file mode 100644 index 00000000..21f39c10 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-158_52ee3171262615e29efd499abc7cd1f70f7c34e75b017ad544b8b72c089ba380.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-160_00a59acdd28b4a198114ec2c3161ecf8d2f3c66a0a890cbf907de2bed39e1f8c.jpg b/.raw/MidasCivilAnalysisReference/images/page-160_00a59acdd28b4a198114ec2c3161ecf8d2f3c66a0a890cbf907de2bed39e1f8c.jpg new file mode 100644 index 00000000..3daac36c Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-160_00a59acdd28b4a198114ec2c3161ecf8d2f3c66a0a890cbf907de2bed39e1f8c.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-160_14c380e14e0be6b6c0c39e783286a0b9e1d14c571ccbb26c5de6502af53e8d8f.jpg b/.raw/MidasCivilAnalysisReference/images/page-160_14c380e14e0be6b6c0c39e783286a0b9e1d14c571ccbb26c5de6502af53e8d8f.jpg new file mode 100644 index 00000000..4d99cdc0 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-160_14c380e14e0be6b6c0c39e783286a0b9e1d14c571ccbb26c5de6502af53e8d8f.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-160_9475088ac54558d362a327e37a50f07ed5b985901e423b900d0a012fbcc32409.jpg b/.raw/MidasCivilAnalysisReference/images/page-160_9475088ac54558d362a327e37a50f07ed5b985901e423b900d0a012fbcc32409.jpg new file mode 100644 index 00000000..adbf5e8d Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-160_9475088ac54558d362a327e37a50f07ed5b985901e423b900d0a012fbcc32409.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-160_c75f2d854a02bde8e6335cda91ed6b89fc17efbbc6144edf58a4b97285742bc3.jpg b/.raw/MidasCivilAnalysisReference/images/page-160_c75f2d854a02bde8e6335cda91ed6b89fc17efbbc6144edf58a4b97285742bc3.jpg new file mode 100644 index 00000000..c24dbb1f Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-160_c75f2d854a02bde8e6335cda91ed6b89fc17efbbc6144edf58a4b97285742bc3.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-160_ce363f62f4bb28207b08be1e3aaa0aec00f6b7b2d817b8d828718c1946644896.jpg b/.raw/MidasCivilAnalysisReference/images/page-160_ce363f62f4bb28207b08be1e3aaa0aec00f6b7b2d817b8d828718c1946644896.jpg new file mode 100644 index 00000000..f7f8bd31 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-160_ce363f62f4bb28207b08be1e3aaa0aec00f6b7b2d817b8d828718c1946644896.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-160_d8e8136da082047cc529e12294d765bc084d287733dd28ed7092a12b97e025a0.jpg b/.raw/MidasCivilAnalysisReference/images/page-160_d8e8136da082047cc529e12294d765bc084d287733dd28ed7092a12b97e025a0.jpg new file mode 100644 index 00000000..cdea8593 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-160_d8e8136da082047cc529e12294d765bc084d287733dd28ed7092a12b97e025a0.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-161_407d9be8cbd560f76ca178b819c621454164dd1ac3aad66f2214e1da72e528bc.jpg b/.raw/MidasCivilAnalysisReference/images/page-161_407d9be8cbd560f76ca178b819c621454164dd1ac3aad66f2214e1da72e528bc.jpg new file mode 100644 index 00000000..f2d2464d Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-161_407d9be8cbd560f76ca178b819c621454164dd1ac3aad66f2214e1da72e528bc.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-161_4bbb5cd7813a39e649467e19ac46670be2a09e3c6ad14f2ff78fcd39beb31fa1.jpg b/.raw/MidasCivilAnalysisReference/images/page-161_4bbb5cd7813a39e649467e19ac46670be2a09e3c6ad14f2ff78fcd39beb31fa1.jpg new file mode 100644 index 00000000..b2c8e14c Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-161_4bbb5cd7813a39e649467e19ac46670be2a09e3c6ad14f2ff78fcd39beb31fa1.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-161_632198676404f8a45a678ac688882904815b19c6e5e4ed685ef5eb59196bc77b.jpg b/.raw/MidasCivilAnalysisReference/images/page-161_632198676404f8a45a678ac688882904815b19c6e5e4ed685ef5eb59196bc77b.jpg new file mode 100644 index 00000000..c2ee064c Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-161_632198676404f8a45a678ac688882904815b19c6e5e4ed685ef5eb59196bc77b.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-161_8cc6ba1bee76375a1cd8e547054f82cc45307f13fee4bba71cd18efc6af059e1.jpg b/.raw/MidasCivilAnalysisReference/images/page-161_8cc6ba1bee76375a1cd8e547054f82cc45307f13fee4bba71cd18efc6af059e1.jpg new file mode 100644 index 00000000..a7d62296 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-161_8cc6ba1bee76375a1cd8e547054f82cc45307f13fee4bba71cd18efc6af059e1.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-161_b04751d570b627a43b1899278b5aee225772168256efcbc26bb3761f70bf8e94.jpg b/.raw/MidasCivilAnalysisReference/images/page-161_b04751d570b627a43b1899278b5aee225772168256efcbc26bb3761f70bf8e94.jpg new file mode 100644 index 00000000..da5d08cf Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-161_b04751d570b627a43b1899278b5aee225772168256efcbc26bb3761f70bf8e94.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-161_b70935e7e578efc0b0a44041e1d0d72b327db9dc7d45683aba76195871effde2.jpg b/.raw/MidasCivilAnalysisReference/images/page-161_b70935e7e578efc0b0a44041e1d0d72b327db9dc7d45683aba76195871effde2.jpg new file mode 100644 index 00000000..2a527394 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-161_b70935e7e578efc0b0a44041e1d0d72b327db9dc7d45683aba76195871effde2.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-161_f286d8e807d11ca28cccef2c44ecbb312e6e84dd7b9a792b2a5783a85f256a48.jpg b/.raw/MidasCivilAnalysisReference/images/page-161_f286d8e807d11ca28cccef2c44ecbb312e6e84dd7b9a792b2a5783a85f256a48.jpg new file mode 100644 index 00000000..866ab766 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-161_f286d8e807d11ca28cccef2c44ecbb312e6e84dd7b9a792b2a5783a85f256a48.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-161_fa0a6b5d482f3e62b37bbe1a920f064f1d4e43b862de355083f37e957fba868d.jpg b/.raw/MidasCivilAnalysisReference/images/page-161_fa0a6b5d482f3e62b37bbe1a920f064f1d4e43b862de355083f37e957fba868d.jpg new file mode 100644 index 00000000..09da08b4 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-161_fa0a6b5d482f3e62b37bbe1a920f064f1d4e43b862de355083f37e957fba868d.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-161_fbf99483b7563fb1801b9b357f37fea920622d10b3fb3921cb8ab2e216bc387a.jpg b/.raw/MidasCivilAnalysisReference/images/page-161_fbf99483b7563fb1801b9b357f37fea920622d10b3fb3921cb8ab2e216bc387a.jpg new file mode 100644 index 00000000..1c7fec61 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-161_fbf99483b7563fb1801b9b357f37fea920622d10b3fb3921cb8ab2e216bc387a.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-162_0662989bbefd65fd54ed99f6731bab6fa78a9a6d2e98bd0654e8952904acc838.jpg b/.raw/MidasCivilAnalysisReference/images/page-162_0662989bbefd65fd54ed99f6731bab6fa78a9a6d2e98bd0654e8952904acc838.jpg new file mode 100644 index 00000000..8d219991 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-162_0662989bbefd65fd54ed99f6731bab6fa78a9a6d2e98bd0654e8952904acc838.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-162_15a7e9cad926ca5f86525ccf21af2e64baaa4f5a4e7195f017731231b79ff22b.jpg b/.raw/MidasCivilAnalysisReference/images/page-162_15a7e9cad926ca5f86525ccf21af2e64baaa4f5a4e7195f017731231b79ff22b.jpg new file mode 100644 index 00000000..d97f4872 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-162_15a7e9cad926ca5f86525ccf21af2e64baaa4f5a4e7195f017731231b79ff22b.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-162_4f4b2e56ecb409dd75724fc72c2e3a7bfdae1980d7359ce25c273a83dfb00f56.jpg b/.raw/MidasCivilAnalysisReference/images/page-162_4f4b2e56ecb409dd75724fc72c2e3a7bfdae1980d7359ce25c273a83dfb00f56.jpg new file mode 100644 index 00000000..f2d2464d Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-162_4f4b2e56ecb409dd75724fc72c2e3a7bfdae1980d7359ce25c273a83dfb00f56.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-162_631bdfdfa4498a73b3001739b644810b336c8fab0730fdc1fa3cffda5b175063.jpg b/.raw/MidasCivilAnalysisReference/images/page-162_631bdfdfa4498a73b3001739b644810b336c8fab0730fdc1fa3cffda5b175063.jpg new file mode 100644 index 00000000..062ab10b Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-162_631bdfdfa4498a73b3001739b644810b336c8fab0730fdc1fa3cffda5b175063.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-162_641b889bd291f0c712f4977639f309e03e1007935fa7c061887db40ee7c5e7f6.jpg b/.raw/MidasCivilAnalysisReference/images/page-162_641b889bd291f0c712f4977639f309e03e1007935fa7c061887db40ee7c5e7f6.jpg new file mode 100644 index 00000000..9b3c760b Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-162_641b889bd291f0c712f4977639f309e03e1007935fa7c061887db40ee7c5e7f6.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-162_834dc4c076d8780882187c4bf9fe18691bd2bc385df85d038173ce1dc46d8744.jpg b/.raw/MidasCivilAnalysisReference/images/page-162_834dc4c076d8780882187c4bf9fe18691bd2bc385df85d038173ce1dc46d8744.jpg new file mode 100644 index 00000000..cabc2745 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-162_834dc4c076d8780882187c4bf9fe18691bd2bc385df85d038173ce1dc46d8744.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-163_3bfb04e1aa9387ca3bc42fd742bf2f2c86de0e7f74ef7af86bc168178364f153.jpg b/.raw/MidasCivilAnalysisReference/images/page-163_3bfb04e1aa9387ca3bc42fd742bf2f2c86de0e7f74ef7af86bc168178364f153.jpg new file mode 100644 index 00000000..22f0d510 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-163_3bfb04e1aa9387ca3bc42fd742bf2f2c86de0e7f74ef7af86bc168178364f153.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-163_867b19d6abf207efd8d88334c63a6f13ae641fb1ea5304fa9346c7b273ea3318.jpg b/.raw/MidasCivilAnalysisReference/images/page-163_867b19d6abf207efd8d88334c63a6f13ae641fb1ea5304fa9346c7b273ea3318.jpg new file mode 100644 index 00000000..c80c9a23 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-163_867b19d6abf207efd8d88334c63a6f13ae641fb1ea5304fa9346c7b273ea3318.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-163_86eac0d9c8f3fbde71a542045db93298fe73dad8d2ca17ffac48249ae10419e1.jpg b/.raw/MidasCivilAnalysisReference/images/page-163_86eac0d9c8f3fbde71a542045db93298fe73dad8d2ca17ffac48249ae10419e1.jpg new file mode 100644 index 00000000..87cd871a Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-163_86eac0d9c8f3fbde71a542045db93298fe73dad8d2ca17ffac48249ae10419e1.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-164_38b5ebd933852092a4ee50e5e5c3b84173fab177ae7f5aca497b9839e47be3d3.jpg b/.raw/MidasCivilAnalysisReference/images/page-164_38b5ebd933852092a4ee50e5e5c3b84173fab177ae7f5aca497b9839e47be3d3.jpg new file mode 100644 index 00000000..97feae22 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-164_38b5ebd933852092a4ee50e5e5c3b84173fab177ae7f5aca497b9839e47be3d3.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-164_3b6e55ae98acab5f03f45e634f8df6881c6d104b477f4656a61c0cc13755926a.jpg b/.raw/MidasCivilAnalysisReference/images/page-164_3b6e55ae98acab5f03f45e634f8df6881c6d104b477f4656a61c0cc13755926a.jpg new file mode 100644 index 00000000..e7d04e00 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-164_3b6e55ae98acab5f03f45e634f8df6881c6d104b477f4656a61c0cc13755926a.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-165_5634678b0c56eb4ac41ab1279797b10f68f1799e5ec158d6baded4738e8c2ff1.jpg b/.raw/MidasCivilAnalysisReference/images/page-165_5634678b0c56eb4ac41ab1279797b10f68f1799e5ec158d6baded4738e8c2ff1.jpg new file mode 100644 index 00000000..fc11fc52 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-165_5634678b0c56eb4ac41ab1279797b10f68f1799e5ec158d6baded4738e8c2ff1.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-165_f286257f178c84356b8eabcc0c243dd5898d1570e8b300a340f1fe6fc1266517.jpg b/.raw/MidasCivilAnalysisReference/images/page-165_f286257f178c84356b8eabcc0c243dd5898d1570e8b300a340f1fe6fc1266517.jpg new file mode 100644 index 00000000..20c0bebf Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-165_f286257f178c84356b8eabcc0c243dd5898d1570e8b300a340f1fe6fc1266517.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-166_4770fe9f45bb85e82635bf163eebe261fd198c39047a3a1c42642085b76a6316.jpg b/.raw/MidasCivilAnalysisReference/images/page-166_4770fe9f45bb85e82635bf163eebe261fd198c39047a3a1c42642085b76a6316.jpg new file mode 100644 index 00000000..120c0d50 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-166_4770fe9f45bb85e82635bf163eebe261fd198c39047a3a1c42642085b76a6316.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-166_7c0db9b7bc40a0a4fabaca7270ddd38df7195e9fbffc5eba7a68b2b6ecd26de9.jpg b/.raw/MidasCivilAnalysisReference/images/page-166_7c0db9b7bc40a0a4fabaca7270ddd38df7195e9fbffc5eba7a68b2b6ecd26de9.jpg new file mode 100644 index 00000000..921d4473 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-166_7c0db9b7bc40a0a4fabaca7270ddd38df7195e9fbffc5eba7a68b2b6ecd26de9.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-166_c598de14e3935269a997b8066fc5dbf76214f9116ed49a7f2847d00940c6e031.jpg b/.raw/MidasCivilAnalysisReference/images/page-166_c598de14e3935269a997b8066fc5dbf76214f9116ed49a7f2847d00940c6e031.jpg new file mode 100644 index 00000000..e46535b2 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-166_c598de14e3935269a997b8066fc5dbf76214f9116ed49a7f2847d00940c6e031.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-166_eb6a1ef3c8257d9cebc4547743d0bdb06e87948334ac1f2a1765155a847a323b.jpg b/.raw/MidasCivilAnalysisReference/images/page-166_eb6a1ef3c8257d9cebc4547743d0bdb06e87948334ac1f2a1765155a847a323b.jpg new file mode 100644 index 00000000..680027a5 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-166_eb6a1ef3c8257d9cebc4547743d0bdb06e87948334ac1f2a1765155a847a323b.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-168_2f41c05451cb42adfef4b1498ab171bea4546fe682834fc06a048b1a6d181a65.jpg b/.raw/MidasCivilAnalysisReference/images/page-168_2f41c05451cb42adfef4b1498ab171bea4546fe682834fc06a048b1a6d181a65.jpg new file mode 100644 index 00000000..b73ca0aa Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-168_2f41c05451cb42adfef4b1498ab171bea4546fe682834fc06a048b1a6d181a65.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-168_cf1e1364063655360c43af06448523913a394e3f0ba0eac34b8b3bcabfbd5a37.jpg b/.raw/MidasCivilAnalysisReference/images/page-168_cf1e1364063655360c43af06448523913a394e3f0ba0eac34b8b3bcabfbd5a37.jpg new file mode 100644 index 00000000..0fb26e6f Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-168_cf1e1364063655360c43af06448523913a394e3f0ba0eac34b8b3bcabfbd5a37.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-170_41b8e46ff046435c4b8537d2056fa9d6fb995ccfc025d6c30f501334a656b3d2.jpg b/.raw/MidasCivilAnalysisReference/images/page-170_41b8e46ff046435c4b8537d2056fa9d6fb995ccfc025d6c30f501334a656b3d2.jpg new file mode 100644 index 00000000..814cf59d Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-170_41b8e46ff046435c4b8537d2056fa9d6fb995ccfc025d6c30f501334a656b3d2.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-170_a2e5ec25f807545e0d9108db67d8d289be6574df82e403a7a17a8542d4255806.jpg b/.raw/MidasCivilAnalysisReference/images/page-170_a2e5ec25f807545e0d9108db67d8d289be6574df82e403a7a17a8542d4255806.jpg new file mode 100644 index 00000000..a2b2991f Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-170_a2e5ec25f807545e0d9108db67d8d289be6574df82e403a7a17a8542d4255806.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-176_38754192ddbb1bb1b59bfad861e93de8666f137e0c2fe54b65b3fa0ba8ad3995.jpg b/.raw/MidasCivilAnalysisReference/images/page-176_38754192ddbb1bb1b59bfad861e93de8666f137e0c2fe54b65b3fa0ba8ad3995.jpg new file mode 100644 index 00000000..ccb8de04 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-176_38754192ddbb1bb1b59bfad861e93de8666f137e0c2fe54b65b3fa0ba8ad3995.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-177_c14e87021946103c23241cc3f4f290a954e3895738749cc12e0e0ef8b16473d7.jpg b/.raw/MidasCivilAnalysisReference/images/page-177_c14e87021946103c23241cc3f4f290a954e3895738749cc12e0e0ef8b16473d7.jpg new file mode 100644 index 00000000..0ffda4c3 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-177_c14e87021946103c23241cc3f4f290a954e3895738749cc12e0e0ef8b16473d7.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-178_7d752e0d6e1eb3bba3f861d186ef391857174b64755e9fa713b38878e1fc3e26.jpg b/.raw/MidasCivilAnalysisReference/images/page-178_7d752e0d6e1eb3bba3f861d186ef391857174b64755e9fa713b38878e1fc3e26.jpg new file mode 100644 index 00000000..86a76fe5 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-178_7d752e0d6e1eb3bba3f861d186ef391857174b64755e9fa713b38878e1fc3e26.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-178_825c48ef5cdd5f7b1bcc61e1522f92fb8447d8f0ba628572478901070f86adad.jpg b/.raw/MidasCivilAnalysisReference/images/page-178_825c48ef5cdd5f7b1bcc61e1522f92fb8447d8f0ba628572478901070f86adad.jpg new file mode 100644 index 00000000..9af5420b Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-178_825c48ef5cdd5f7b1bcc61e1522f92fb8447d8f0ba628572478901070f86adad.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-178_a30b94431f2ec2fa5b5c73bed6915fce076f18cde62373ea4839f1f458579be7.jpg b/.raw/MidasCivilAnalysisReference/images/page-178_a30b94431f2ec2fa5b5c73bed6915fce076f18cde62373ea4839f1f458579be7.jpg new file mode 100644 index 00000000..8890e504 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-178_a30b94431f2ec2fa5b5c73bed6915fce076f18cde62373ea4839f1f458579be7.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-178_b4d1d06897d3defa80e1856e8d5deb2552d2a5ed63b09d0c152b4d547bb4cff7.jpg b/.raw/MidasCivilAnalysisReference/images/page-178_b4d1d06897d3defa80e1856e8d5deb2552d2a5ed63b09d0c152b4d547bb4cff7.jpg new file mode 100644 index 00000000..a626200d Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-178_b4d1d06897d3defa80e1856e8d5deb2552d2a5ed63b09d0c152b4d547bb4cff7.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-179_15872902395d4cfd5615a3fc0a52c186fa261347b2705543fae6b12fb1ffe4b0.jpg b/.raw/MidasCivilAnalysisReference/images/page-179_15872902395d4cfd5615a3fc0a52c186fa261347b2705543fae6b12fb1ffe4b0.jpg new file mode 100644 index 00000000..d88bb804 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-179_15872902395d4cfd5615a3fc0a52c186fa261347b2705543fae6b12fb1ffe4b0.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-179_2d014fd59d897fe80ae479239d5947d9c63614482f0ec03cbc0ea816585f9e5b.jpg b/.raw/MidasCivilAnalysisReference/images/page-179_2d014fd59d897fe80ae479239d5947d9c63614482f0ec03cbc0ea816585f9e5b.jpg new file mode 100644 index 00000000..decd1e27 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-179_2d014fd59d897fe80ae479239d5947d9c63614482f0ec03cbc0ea816585f9e5b.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-179_54d52560112aa8d2d11bcc67193a571d73594ad47db1673b68329660f3b18fc5.jpg b/.raw/MidasCivilAnalysisReference/images/page-179_54d52560112aa8d2d11bcc67193a571d73594ad47db1673b68329660f3b18fc5.jpg new file mode 100644 index 00000000..35ca7dc6 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-179_54d52560112aa8d2d11bcc67193a571d73594ad47db1673b68329660f3b18fc5.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-179_b557e1cf26f49b3375eee1afcda7e78d6f99fdd2bba8c1b98288532ac0ac5810.jpg b/.raw/MidasCivilAnalysisReference/images/page-179_b557e1cf26f49b3375eee1afcda7e78d6f99fdd2bba8c1b98288532ac0ac5810.jpg new file mode 100644 index 00000000..314df32a Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-179_b557e1cf26f49b3375eee1afcda7e78d6f99fdd2bba8c1b98288532ac0ac5810.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-179_cd8281d69551d7908f9e6d8ab351a3ee3eba86ccca0d3c1b1f58ae735187c173.jpg b/.raw/MidasCivilAnalysisReference/images/page-179_cd8281d69551d7908f9e6d8ab351a3ee3eba86ccca0d3c1b1f58ae735187c173.jpg new file mode 100644 index 00000000..2f826390 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-179_cd8281d69551d7908f9e6d8ab351a3ee3eba86ccca0d3c1b1f58ae735187c173.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-179_f50253c21439439dcafa7eb0fd094fae446f8643dce08958a3fff68ec9d17bdc.jpg b/.raw/MidasCivilAnalysisReference/images/page-179_f50253c21439439dcafa7eb0fd094fae446f8643dce08958a3fff68ec9d17bdc.jpg new file mode 100644 index 00000000..e0d7a719 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-179_f50253c21439439dcafa7eb0fd094fae446f8643dce08958a3fff68ec9d17bdc.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-180_165c9bb190f66eb11684ef40daeae2da62b608eccf7c5ae54af4ca022682285d.jpg b/.raw/MidasCivilAnalysisReference/images/page-180_165c9bb190f66eb11684ef40daeae2da62b608eccf7c5ae54af4ca022682285d.jpg new file mode 100644 index 00000000..1bf21d93 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-180_165c9bb190f66eb11684ef40daeae2da62b608eccf7c5ae54af4ca022682285d.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-180_5190c457e3cdfb375e43a4b5a6b52fa6afe91a1f5ff9c187b39aac860fb988b3.jpg b/.raw/MidasCivilAnalysisReference/images/page-180_5190c457e3cdfb375e43a4b5a6b52fa6afe91a1f5ff9c187b39aac860fb988b3.jpg new file mode 100644 index 00000000..5a3d806f Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-180_5190c457e3cdfb375e43a4b5a6b52fa6afe91a1f5ff9c187b39aac860fb988b3.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-181_2cf806344391fa8da054942adfae94e6c246cc1fb9d33134cda65e5548612f32.jpg b/.raw/MidasCivilAnalysisReference/images/page-181_2cf806344391fa8da054942adfae94e6c246cc1fb9d33134cda65e5548612f32.jpg new file mode 100644 index 00000000..257ae594 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-181_2cf806344391fa8da054942adfae94e6c246cc1fb9d33134cda65e5548612f32.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-181_35f6aaaa0ada205f029fac7b09cb4c7e7af5a23de9eea6aed3a5ee24c8c54f0c.jpg b/.raw/MidasCivilAnalysisReference/images/page-181_35f6aaaa0ada205f029fac7b09cb4c7e7af5a23de9eea6aed3a5ee24c8c54f0c.jpg new file mode 100644 index 00000000..b8503998 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-181_35f6aaaa0ada205f029fac7b09cb4c7e7af5a23de9eea6aed3a5ee24c8c54f0c.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-182_2698856accb757fdd85f82ef08b415c11f847db806f12038a10059e55bd07d3f.jpg b/.raw/MidasCivilAnalysisReference/images/page-182_2698856accb757fdd85f82ef08b415c11f847db806f12038a10059e55bd07d3f.jpg new file mode 100644 index 00000000..9e55a622 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-182_2698856accb757fdd85f82ef08b415c11f847db806f12038a10059e55bd07d3f.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-182_4a2c7f5243951c3f5e8854ed0ebad0e8ff8a9c9664e0c653aad1b993bd20a150.jpg b/.raw/MidasCivilAnalysisReference/images/page-182_4a2c7f5243951c3f5e8854ed0ebad0e8ff8a9c9664e0c653aad1b993bd20a150.jpg new file mode 100644 index 00000000..75d8a7b3 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-182_4a2c7f5243951c3f5e8854ed0ebad0e8ff8a9c9664e0c653aad1b993bd20a150.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-182_4b73f97e48b8dfa10494fffa6ad2ae7aeb97cfdd6b35b0a474e2d20e28a704cd.jpg b/.raw/MidasCivilAnalysisReference/images/page-182_4b73f97e48b8dfa10494fffa6ad2ae7aeb97cfdd6b35b0a474e2d20e28a704cd.jpg new file mode 100644 index 00000000..f81f6089 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-182_4b73f97e48b8dfa10494fffa6ad2ae7aeb97cfdd6b35b0a474e2d20e28a704cd.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-182_5e48c1981e94fcb019c2d140c3d76aabe171f856b8f8e01e115162fa9b714df1.jpg b/.raw/MidasCivilAnalysisReference/images/page-182_5e48c1981e94fcb019c2d140c3d76aabe171f856b8f8e01e115162fa9b714df1.jpg new file mode 100644 index 00000000..f96f310c Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-182_5e48c1981e94fcb019c2d140c3d76aabe171f856b8f8e01e115162fa9b714df1.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-182_7f26c83ee7a7d9876709e38d4cf99d9abd2413ec21b59a0ff7a26c76627b16ae.jpg b/.raw/MidasCivilAnalysisReference/images/page-182_7f26c83ee7a7d9876709e38d4cf99d9abd2413ec21b59a0ff7a26c76627b16ae.jpg new file mode 100644 index 00000000..6722522f Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-182_7f26c83ee7a7d9876709e38d4cf99d9abd2413ec21b59a0ff7a26c76627b16ae.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-182_98636e383463e9d9aca59b8967cfe10345f709e3402892f3f964b3ac12321ec0.jpg b/.raw/MidasCivilAnalysisReference/images/page-182_98636e383463e9d9aca59b8967cfe10345f709e3402892f3f964b3ac12321ec0.jpg new file mode 100644 index 00000000..36f0c057 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-182_98636e383463e9d9aca59b8967cfe10345f709e3402892f3f964b3ac12321ec0.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-182_deecea543fed7024aa5bebf99c9bcce95c91dbd6f90fe10bb887c44646af2935.jpg b/.raw/MidasCivilAnalysisReference/images/page-182_deecea543fed7024aa5bebf99c9bcce95c91dbd6f90fe10bb887c44646af2935.jpg new file mode 100644 index 00000000..08a3698e Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-182_deecea543fed7024aa5bebf99c9bcce95c91dbd6f90fe10bb887c44646af2935.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-183_0fa80c0fa1d240b75f5dbbb87258e521103c7d5817cb33a20887a15be696ba04.jpg b/.raw/MidasCivilAnalysisReference/images/page-183_0fa80c0fa1d240b75f5dbbb87258e521103c7d5817cb33a20887a15be696ba04.jpg new file mode 100644 index 00000000..a70f1307 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-183_0fa80c0fa1d240b75f5dbbb87258e521103c7d5817cb33a20887a15be696ba04.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-183_3278e236b1310c47f0f06f6551a18e8cddce5053bfcc5b24de4ab4d638449535.jpg b/.raw/MidasCivilAnalysisReference/images/page-183_3278e236b1310c47f0f06f6551a18e8cddce5053bfcc5b24de4ab4d638449535.jpg new file mode 100644 index 00000000..98159b29 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-183_3278e236b1310c47f0f06f6551a18e8cddce5053bfcc5b24de4ab4d638449535.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-183_da1a55b219b1d2ec0459ea2fcc412e5967ba72e5fa32602eaf815af178900655.jpg b/.raw/MidasCivilAnalysisReference/images/page-183_da1a55b219b1d2ec0459ea2fcc412e5967ba72e5fa32602eaf815af178900655.jpg new file mode 100644 index 00000000..8164b884 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-183_da1a55b219b1d2ec0459ea2fcc412e5967ba72e5fa32602eaf815af178900655.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-184_341e4ffb840e65996b619ef42630f588a41777e2ee8ecff7656a7971f6619953.jpg b/.raw/MidasCivilAnalysisReference/images/page-184_341e4ffb840e65996b619ef42630f588a41777e2ee8ecff7656a7971f6619953.jpg new file mode 100644 index 00000000..d083958d Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-184_341e4ffb840e65996b619ef42630f588a41777e2ee8ecff7656a7971f6619953.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-184_3455fa2e8bf4be0c28c1ae8164585317da6ca2ef46774a24f4e373c45027b246.jpg b/.raw/MidasCivilAnalysisReference/images/page-184_3455fa2e8bf4be0c28c1ae8164585317da6ca2ef46774a24f4e373c45027b246.jpg new file mode 100644 index 00000000..f1d8da7b Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-184_3455fa2e8bf4be0c28c1ae8164585317da6ca2ef46774a24f4e373c45027b246.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-184_6246889c211b903278bbb2b239e54ea37112bdc9b6b6b9737d81dce472a3c4d9.jpg b/.raw/MidasCivilAnalysisReference/images/page-184_6246889c211b903278bbb2b239e54ea37112bdc9b6b6b9737d81dce472a3c4d9.jpg new file mode 100644 index 00000000..d5491e5d Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-184_6246889c211b903278bbb2b239e54ea37112bdc9b6b6b9737d81dce472a3c4d9.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-184_c117cf222d7a669d3d9165f1e8daba9ffecd392c79f2f413eb54350f45de06f4.jpg b/.raw/MidasCivilAnalysisReference/images/page-184_c117cf222d7a669d3d9165f1e8daba9ffecd392c79f2f413eb54350f45de06f4.jpg new file mode 100644 index 00000000..287e21d5 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-184_c117cf222d7a669d3d9165f1e8daba9ffecd392c79f2f413eb54350f45de06f4.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-184_e0bca137932001af9c794d8d01eecb03e3bac333e2d7f8572079d847a5d4606d.jpg b/.raw/MidasCivilAnalysisReference/images/page-184_e0bca137932001af9c794d8d01eecb03e3bac333e2d7f8572079d847a5d4606d.jpg new file mode 100644 index 00000000..9ee79f2f Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-184_e0bca137932001af9c794d8d01eecb03e3bac333e2d7f8572079d847a5d4606d.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-184_fcac368b38d5602bde587169ffd0ec72dd7906d4ca98284977b6c6649be61e48.jpg b/.raw/MidasCivilAnalysisReference/images/page-184_fcac368b38d5602bde587169ffd0ec72dd7906d4ca98284977b6c6649be61e48.jpg new file mode 100644 index 00000000..eb14cf89 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-184_fcac368b38d5602bde587169ffd0ec72dd7906d4ca98284977b6c6649be61e48.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-185_5a24bcecaacbd5be41b9257569ba603a8e6b1008e100876b3a6b887256e37813.jpg b/.raw/MidasCivilAnalysisReference/images/page-185_5a24bcecaacbd5be41b9257569ba603a8e6b1008e100876b3a6b887256e37813.jpg new file mode 100644 index 00000000..17a6f961 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-185_5a24bcecaacbd5be41b9257569ba603a8e6b1008e100876b3a6b887256e37813.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-185_8ce1f0f6175bb264ca830c839144f91ff1cb3e15e842c2e53418265aed666445.jpg b/.raw/MidasCivilAnalysisReference/images/page-185_8ce1f0f6175bb264ca830c839144f91ff1cb3e15e842c2e53418265aed666445.jpg new file mode 100644 index 00000000..cc1c71d7 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-185_8ce1f0f6175bb264ca830c839144f91ff1cb3e15e842c2e53418265aed666445.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-185_9e109fbc5a5cf405439f19ed865c9f2002b9038078e48fd408c8c0fbf5cec05f.jpg b/.raw/MidasCivilAnalysisReference/images/page-185_9e109fbc5a5cf405439f19ed865c9f2002b9038078e48fd408c8c0fbf5cec05f.jpg new file mode 100644 index 00000000..59cbe4fb Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-185_9e109fbc5a5cf405439f19ed865c9f2002b9038078e48fd408c8c0fbf5cec05f.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-185_c3ea9e80379504908ecb24f0b0c0ee1fa224464d953a6a06af0f6fc1a6215bcc.jpg b/.raw/MidasCivilAnalysisReference/images/page-185_c3ea9e80379504908ecb24f0b0c0ee1fa224464d953a6a06af0f6fc1a6215bcc.jpg new file mode 100644 index 00000000..f52e0157 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-185_c3ea9e80379504908ecb24f0b0c0ee1fa224464d953a6a06af0f6fc1a6215bcc.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-186_07c83894b5f7a9b77ccf76b9d80036a2a579eb92d3de01ba07a4b870c0280a99.jpg b/.raw/MidasCivilAnalysisReference/images/page-186_07c83894b5f7a9b77ccf76b9d80036a2a579eb92d3de01ba07a4b870c0280a99.jpg new file mode 100644 index 00000000..af80289a Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-186_07c83894b5f7a9b77ccf76b9d80036a2a579eb92d3de01ba07a4b870c0280a99.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-186_9b1af79e89d3949169d15e01bb3480a5473dd599fb5cc0ea91a4cd5478a7bdd8.jpg b/.raw/MidasCivilAnalysisReference/images/page-186_9b1af79e89d3949169d15e01bb3480a5473dd599fb5cc0ea91a4cd5478a7bdd8.jpg new file mode 100644 index 00000000..d7017f51 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-186_9b1af79e89d3949169d15e01bb3480a5473dd599fb5cc0ea91a4cd5478a7bdd8.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-190_4923fff3a7693fc4e33873aa16550e7b42486f892634ef54345afe1df11b76e4.jpg b/.raw/MidasCivilAnalysisReference/images/page-190_4923fff3a7693fc4e33873aa16550e7b42486f892634ef54345afe1df11b76e4.jpg new file mode 100644 index 00000000..6c79d1d7 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-190_4923fff3a7693fc4e33873aa16550e7b42486f892634ef54345afe1df11b76e4.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-190_c1ea60296a0072f616b9e69f1fc716d3ade69c33ff7b832d94658a3d1d4d2404.jpg b/.raw/MidasCivilAnalysisReference/images/page-190_c1ea60296a0072f616b9e69f1fc716d3ade69c33ff7b832d94658a3d1d4d2404.jpg new file mode 100644 index 00000000..d614604d Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-190_c1ea60296a0072f616b9e69f1fc716d3ade69c33ff7b832d94658a3d1d4d2404.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-192_10a75b06d8b95eced68c19bdf93bfb9439b0cf2d6b75e34e5976a5b39b073d04.jpg b/.raw/MidasCivilAnalysisReference/images/page-192_10a75b06d8b95eced68c19bdf93bfb9439b0cf2d6b75e34e5976a5b39b073d04.jpg new file mode 100644 index 00000000..580915a0 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-192_10a75b06d8b95eced68c19bdf93bfb9439b0cf2d6b75e34e5976a5b39b073d04.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-192_14ca4c3710764a0bf5dc91c5c026193e5d477140643c29e1b9834640113634e1.jpg b/.raw/MidasCivilAnalysisReference/images/page-192_14ca4c3710764a0bf5dc91c5c026193e5d477140643c29e1b9834640113634e1.jpg new file mode 100644 index 00000000..34c252fb Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-192_14ca4c3710764a0bf5dc91c5c026193e5d477140643c29e1b9834640113634e1.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-192_6fda72965df7e9945592934dc9cbe1250d41e5cea3f1b04ecaec7460d8179a40.jpg b/.raw/MidasCivilAnalysisReference/images/page-192_6fda72965df7e9945592934dc9cbe1250d41e5cea3f1b04ecaec7460d8179a40.jpg new file mode 100644 index 00000000..4da9e7e0 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-192_6fda72965df7e9945592934dc9cbe1250d41e5cea3f1b04ecaec7460d8179a40.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-192_85cb491b08cfee96c598bab2618b3f205930b346f532f0781620412d2b72b881.jpg b/.raw/MidasCivilAnalysisReference/images/page-192_85cb491b08cfee96c598bab2618b3f205930b346f532f0781620412d2b72b881.jpg new file mode 100644 index 00000000..548d4445 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-192_85cb491b08cfee96c598bab2618b3f205930b346f532f0781620412d2b72b881.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-192_af21c184646bc80f6a9f013d412ac6d6c515b123f522f5448b1c7ebea9f6657d.jpg b/.raw/MidasCivilAnalysisReference/images/page-192_af21c184646bc80f6a9f013d412ac6d6c515b123f522f5448b1c7ebea9f6657d.jpg new file mode 100644 index 00000000..cfc6ebd5 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-192_af21c184646bc80f6a9f013d412ac6d6c515b123f522f5448b1c7ebea9f6657d.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-193_1b0376b80eef5ad08f1bbb69cd4f4d1c2bbfee5db8c6dd1436873f3e12d11f44.jpg b/.raw/MidasCivilAnalysisReference/images/page-193_1b0376b80eef5ad08f1bbb69cd4f4d1c2bbfee5db8c6dd1436873f3e12d11f44.jpg new file mode 100644 index 00000000..35733733 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-193_1b0376b80eef5ad08f1bbb69cd4f4d1c2bbfee5db8c6dd1436873f3e12d11f44.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-193_2c4d5cb2bc8fc6eb3f06186979b8a0661d41459b80943acc25c54fce083682b2.jpg b/.raw/MidasCivilAnalysisReference/images/page-193_2c4d5cb2bc8fc6eb3f06186979b8a0661d41459b80943acc25c54fce083682b2.jpg new file mode 100644 index 00000000..f977ac9f Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-193_2c4d5cb2bc8fc6eb3f06186979b8a0661d41459b80943acc25c54fce083682b2.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-193_addb01bd469c5f2f887cccc1061ca53c43abaa0cebafbbb3712abddb388aa353.jpg b/.raw/MidasCivilAnalysisReference/images/page-193_addb01bd469c5f2f887cccc1061ca53c43abaa0cebafbbb3712abddb388aa353.jpg new file mode 100644 index 00000000..69296521 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-193_addb01bd469c5f2f887cccc1061ca53c43abaa0cebafbbb3712abddb388aa353.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-193_daba94466bcb7db8364fafcc05e3d734c8b405e17ea8f3ee88d271a2daae5187.jpg b/.raw/MidasCivilAnalysisReference/images/page-193_daba94466bcb7db8364fafcc05e3d734c8b405e17ea8f3ee88d271a2daae5187.jpg new file mode 100644 index 00000000..d2f85ffb Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-193_daba94466bcb7db8364fafcc05e3d734c8b405e17ea8f3ee88d271a2daae5187.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-193_e6ed42e0fae6c24640f9bc1bcbefde5b4c206e9e1110f9dac787d718ce79a6fc.jpg b/.raw/MidasCivilAnalysisReference/images/page-193_e6ed42e0fae6c24640f9bc1bcbefde5b4c206e9e1110f9dac787d718ce79a6fc.jpg new file mode 100644 index 00000000..6dfb1d61 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-193_e6ed42e0fae6c24640f9bc1bcbefde5b4c206e9e1110f9dac787d718ce79a6fc.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-193_e940d939326294f20793b6a424858aab84bf2d58a0acd12f1e0c0dbc98a66ba2.jpg b/.raw/MidasCivilAnalysisReference/images/page-193_e940d939326294f20793b6a424858aab84bf2d58a0acd12f1e0c0dbc98a66ba2.jpg new file mode 100644 index 00000000..47866bdc Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-193_e940d939326294f20793b6a424858aab84bf2d58a0acd12f1e0c0dbc98a66ba2.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-194_00625d2a6a9aa64b8e0e0e98c65e769284c133e0d7f814f7bc3965db7ff621c1.jpg b/.raw/MidasCivilAnalysisReference/images/page-194_00625d2a6a9aa64b8e0e0e98c65e769284c133e0d7f814f7bc3965db7ff621c1.jpg new file mode 100644 index 00000000..1c4f64c2 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-194_00625d2a6a9aa64b8e0e0e98c65e769284c133e0d7f814f7bc3965db7ff621c1.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-194_5a7ff461e5128e6cb65384d4e40ed0abe79751614c493f08c858fc5d61db36d4.jpg b/.raw/MidasCivilAnalysisReference/images/page-194_5a7ff461e5128e6cb65384d4e40ed0abe79751614c493f08c858fc5d61db36d4.jpg new file mode 100644 index 00000000..03765ecd Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-194_5a7ff461e5128e6cb65384d4e40ed0abe79751614c493f08c858fc5d61db36d4.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-194_70e92a84716c6268bfa2da38882c7e8ad1860b5be9f6275171cd15acc7a78e0b.jpg b/.raw/MidasCivilAnalysisReference/images/page-194_70e92a84716c6268bfa2da38882c7e8ad1860b5be9f6275171cd15acc7a78e0b.jpg new file mode 100644 index 00000000..4cbcc79a Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-194_70e92a84716c6268bfa2da38882c7e8ad1860b5be9f6275171cd15acc7a78e0b.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-194_a219847656a5a6585c27bde7a84d18b7c646ac177356c3e7cabb541ba8a953a2.jpg b/.raw/MidasCivilAnalysisReference/images/page-194_a219847656a5a6585c27bde7a84d18b7c646ac177356c3e7cabb541ba8a953a2.jpg new file mode 100644 index 00000000..4930ba9e Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-194_a219847656a5a6585c27bde7a84d18b7c646ac177356c3e7cabb541ba8a953a2.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-194_cfc92db660864def0277a1ec721ddd6926a080f511c6acd41f4e72b264cd7292.jpg b/.raw/MidasCivilAnalysisReference/images/page-194_cfc92db660864def0277a1ec721ddd6926a080f511c6acd41f4e72b264cd7292.jpg new file mode 100644 index 00000000..c2a3c3cc Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-194_cfc92db660864def0277a1ec721ddd6926a080f511c6acd41f4e72b264cd7292.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-194_e43e326fa83e2f1a98ce2333dada1e75874b32c16a6929c2316371a223589564.jpg b/.raw/MidasCivilAnalysisReference/images/page-194_e43e326fa83e2f1a98ce2333dada1e75874b32c16a6929c2316371a223589564.jpg new file mode 100644 index 00000000..57633cf1 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-194_e43e326fa83e2f1a98ce2333dada1e75874b32c16a6929c2316371a223589564.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-195_0e3df5664dadc1e617fb6cc4d9c05c767e621b91264b03cb9c80c4d376cf5327.jpg b/.raw/MidasCivilAnalysisReference/images/page-195_0e3df5664dadc1e617fb6cc4d9c05c767e621b91264b03cb9c80c4d376cf5327.jpg new file mode 100644 index 00000000..19ff21e5 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-195_0e3df5664dadc1e617fb6cc4d9c05c767e621b91264b03cb9c80c4d376cf5327.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-195_780c73f23f94b0a1dfb3927c1d866643e67646670f3bde5edcec6eb51e768544.jpg b/.raw/MidasCivilAnalysisReference/images/page-195_780c73f23f94b0a1dfb3927c1d866643e67646670f3bde5edcec6eb51e768544.jpg new file mode 100644 index 00000000..acca8731 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-195_780c73f23f94b0a1dfb3927c1d866643e67646670f3bde5edcec6eb51e768544.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-195_b0ed6f048df7f9a9e0fcef73f698abfbf916de2326f0fb14d4b3ff689acdc8d0.jpg b/.raw/MidasCivilAnalysisReference/images/page-195_b0ed6f048df7f9a9e0fcef73f698abfbf916de2326f0fb14d4b3ff689acdc8d0.jpg new file mode 100644 index 00000000..eadb50c8 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-195_b0ed6f048df7f9a9e0fcef73f698abfbf916de2326f0fb14d4b3ff689acdc8d0.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-195_d9b62b670c115599e8fd4599e87297a2a91d106c57869af925ab8dfba84406b4.jpg b/.raw/MidasCivilAnalysisReference/images/page-195_d9b62b670c115599e8fd4599e87297a2a91d106c57869af925ab8dfba84406b4.jpg new file mode 100644 index 00000000..2fe84b1a Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-195_d9b62b670c115599e8fd4599e87297a2a91d106c57869af925ab8dfba84406b4.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-196_52f4f90cf19a6884fed67042d2b555aceda34f2235d1a1e7f6b8850214cbce52.jpg b/.raw/MidasCivilAnalysisReference/images/page-196_52f4f90cf19a6884fed67042d2b555aceda34f2235d1a1e7f6b8850214cbce52.jpg new file mode 100644 index 00000000..f28d7128 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-196_52f4f90cf19a6884fed67042d2b555aceda34f2235d1a1e7f6b8850214cbce52.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-196_73932ace819244dba99b88fd772f02b83786a4612db3185ca8e9f96802fba5f0.jpg b/.raw/MidasCivilAnalysisReference/images/page-196_73932ace819244dba99b88fd772f02b83786a4612db3185ca8e9f96802fba5f0.jpg new file mode 100644 index 00000000..a43f3e4c Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-196_73932ace819244dba99b88fd772f02b83786a4612db3185ca8e9f96802fba5f0.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-196_9114e8eff3e3b16dd6f3960a130a82e61c27b9e0c21ee7495637d0c63a5f3563.jpg b/.raw/MidasCivilAnalysisReference/images/page-196_9114e8eff3e3b16dd6f3960a130a82e61c27b9e0c21ee7495637d0c63a5f3563.jpg new file mode 100644 index 00000000..5f737e9a Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-196_9114e8eff3e3b16dd6f3960a130a82e61c27b9e0c21ee7495637d0c63a5f3563.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-196_c3d5092f6715d909eff7db7a75b7ea79ce645e6587c28ee9ffabce447e9b7257.jpg b/.raw/MidasCivilAnalysisReference/images/page-196_c3d5092f6715d909eff7db7a75b7ea79ce645e6587c28ee9ffabce447e9b7257.jpg new file mode 100644 index 00000000..2a28d9f1 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-196_c3d5092f6715d909eff7db7a75b7ea79ce645e6587c28ee9ffabce447e9b7257.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-198_b7d07fcef952bd5739a79076e281149e636e35aa4d73f073a39e1c46704f96ea.jpg b/.raw/MidasCivilAnalysisReference/images/page-198_b7d07fcef952bd5739a79076e281149e636e35aa4d73f073a39e1c46704f96ea.jpg new file mode 100644 index 00000000..409a9912 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-198_b7d07fcef952bd5739a79076e281149e636e35aa4d73f073a39e1c46704f96ea.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-199_4d7b1b4024249e8941faffb84a028b68543046daa071ee50b2c785f4e5f38fc0.jpg b/.raw/MidasCivilAnalysisReference/images/page-199_4d7b1b4024249e8941faffb84a028b68543046daa071ee50b2c785f4e5f38fc0.jpg new file mode 100644 index 00000000..ecc885c6 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-199_4d7b1b4024249e8941faffb84a028b68543046daa071ee50b2c785f4e5f38fc0.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-199_ea354ae5feed7e9cf66270ba193e4662828cd9070170a31853617297da89ec10.jpg b/.raw/MidasCivilAnalysisReference/images/page-199_ea354ae5feed7e9cf66270ba193e4662828cd9070170a31853617297da89ec10.jpg new file mode 100644 index 00000000..bce10bb5 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-199_ea354ae5feed7e9cf66270ba193e4662828cd9070170a31853617297da89ec10.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-200_666fa427c3025d191d502de749e714f35d555f53e03e13ea7d9a30a9b4d1753c.jpg b/.raw/MidasCivilAnalysisReference/images/page-200_666fa427c3025d191d502de749e714f35d555f53e03e13ea7d9a30a9b4d1753c.jpg new file mode 100644 index 00000000..6c9e7d51 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-200_666fa427c3025d191d502de749e714f35d555f53e03e13ea7d9a30a9b4d1753c.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-200_ad8b613b36c39e2d7f538aea2ddbbe17086011ac65743085b22f793eb2a1027b.jpg b/.raw/MidasCivilAnalysisReference/images/page-200_ad8b613b36c39e2d7f538aea2ddbbe17086011ac65743085b22f793eb2a1027b.jpg new file mode 100644 index 00000000..f6f8c28b Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-200_ad8b613b36c39e2d7f538aea2ddbbe17086011ac65743085b22f793eb2a1027b.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-200_fe7f965916ba1715b66dd282caadfc106c861545f56d3e978e9a04ee3512778d.jpg b/.raw/MidasCivilAnalysisReference/images/page-200_fe7f965916ba1715b66dd282caadfc106c861545f56d3e978e9a04ee3512778d.jpg new file mode 100644 index 00000000..386196eb Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-200_fe7f965916ba1715b66dd282caadfc106c861545f56d3e978e9a04ee3512778d.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-201_5c39fe2b8e1b30e6dd3261e19aebe047c7ff6dec633c43f9dc901e4d7c9d3524.jpg b/.raw/MidasCivilAnalysisReference/images/page-201_5c39fe2b8e1b30e6dd3261e19aebe047c7ff6dec633c43f9dc901e4d7c9d3524.jpg new file mode 100644 index 00000000..b628e53a Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-201_5c39fe2b8e1b30e6dd3261e19aebe047c7ff6dec633c43f9dc901e4d7c9d3524.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-201_bf2134dfa5165f70e89cb7b0f1ceed2b06aea3a19ac75dac56c6f6795a90d414.jpg b/.raw/MidasCivilAnalysisReference/images/page-201_bf2134dfa5165f70e89cb7b0f1ceed2b06aea3a19ac75dac56c6f6795a90d414.jpg new file mode 100644 index 00000000..e36bfab9 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-201_bf2134dfa5165f70e89cb7b0f1ceed2b06aea3a19ac75dac56c6f6795a90d414.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-204_f9a06fe858c2c5be3f7e9c833a108759ac16324047c7ef09ad2a98a0efd06518.jpg b/.raw/MidasCivilAnalysisReference/images/page-204_f9a06fe858c2c5be3f7e9c833a108759ac16324047c7ef09ad2a98a0efd06518.jpg new file mode 100644 index 00000000..8cd8928d Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-204_f9a06fe858c2c5be3f7e9c833a108759ac16324047c7ef09ad2a98a0efd06518.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-205_2240069aec3462f0ef1cf0c318115f225fd97b5beed34d2fcad7e56bcd23ebb1.jpg b/.raw/MidasCivilAnalysisReference/images/page-205_2240069aec3462f0ef1cf0c318115f225fd97b5beed34d2fcad7e56bcd23ebb1.jpg new file mode 100644 index 00000000..30577df2 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-205_2240069aec3462f0ef1cf0c318115f225fd97b5beed34d2fcad7e56bcd23ebb1.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-205_e4e94fba892e9d0c386a40c3d6bb06b609e78961604aa336b60f478accda7010.jpg b/.raw/MidasCivilAnalysisReference/images/page-205_e4e94fba892e9d0c386a40c3d6bb06b609e78961604aa336b60f478accda7010.jpg new file mode 100644 index 00000000..48011044 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-205_e4e94fba892e9d0c386a40c3d6bb06b609e78961604aa336b60f478accda7010.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-206_3842e35d7a2023b45aa645f02aa457f3e88d194430f1b9e433a5893a84ae00b2.jpg b/.raw/MidasCivilAnalysisReference/images/page-206_3842e35d7a2023b45aa645f02aa457f3e88d194430f1b9e433a5893a84ae00b2.jpg new file mode 100644 index 00000000..a37698bb Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-206_3842e35d7a2023b45aa645f02aa457f3e88d194430f1b9e433a5893a84ae00b2.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-206_84a8298c426f3e67370e418035da2eecf1b2c0c10078547bb5483e5505bec52e.jpg b/.raw/MidasCivilAnalysisReference/images/page-206_84a8298c426f3e67370e418035da2eecf1b2c0c10078547bb5483e5505bec52e.jpg new file mode 100644 index 00000000..966b26df Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-206_84a8298c426f3e67370e418035da2eecf1b2c0c10078547bb5483e5505bec52e.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-207_31e6e80f3a31ab56f20d6998ac11793a24d60e86c226b5b1af5daeb51921beec.jpg b/.raw/MidasCivilAnalysisReference/images/page-207_31e6e80f3a31ab56f20d6998ac11793a24d60e86c226b5b1af5daeb51921beec.jpg new file mode 100644 index 00000000..863a8295 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-207_31e6e80f3a31ab56f20d6998ac11793a24d60e86c226b5b1af5daeb51921beec.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-207_bc6de16defb1eee9b5b2e4268b82321c6b1931392a7d7b4de858d8a29f0d82c7.jpg b/.raw/MidasCivilAnalysisReference/images/page-207_bc6de16defb1eee9b5b2e4268b82321c6b1931392a7d7b4de858d8a29f0d82c7.jpg new file mode 100644 index 00000000..f2c41ce1 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-207_bc6de16defb1eee9b5b2e4268b82321c6b1931392a7d7b4de858d8a29f0d82c7.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-207_cfc297a74e8ca1d381a88bab743a3aaeea886669f67b29a989b8ca2037c0fd2a.jpg b/.raw/MidasCivilAnalysisReference/images/page-207_cfc297a74e8ca1d381a88bab743a3aaeea886669f67b29a989b8ca2037c0fd2a.jpg new file mode 100644 index 00000000..b02eb585 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-207_cfc297a74e8ca1d381a88bab743a3aaeea886669f67b29a989b8ca2037c0fd2a.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-208_04ea839b5f769bd2d7f6dffe7c86952b2828c78dca9a8927fdd8e7e9d8184a06.jpg b/.raw/MidasCivilAnalysisReference/images/page-208_04ea839b5f769bd2d7f6dffe7c86952b2828c78dca9a8927fdd8e7e9d8184a06.jpg new file mode 100644 index 00000000..6e048d76 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-208_04ea839b5f769bd2d7f6dffe7c86952b2828c78dca9a8927fdd8e7e9d8184a06.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-208_0a6d14c072a9991608a33cb280677958de18b4081235330aad475308874c71fc.jpg b/.raw/MidasCivilAnalysisReference/images/page-208_0a6d14c072a9991608a33cb280677958de18b4081235330aad475308874c71fc.jpg new file mode 100644 index 00000000..7b97ff36 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-208_0a6d14c072a9991608a33cb280677958de18b4081235330aad475308874c71fc.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-208_23ebb946396351d337198c16df2f0ce1e43e572988e178881d1c6dd3dfefb737.jpg b/.raw/MidasCivilAnalysisReference/images/page-208_23ebb946396351d337198c16df2f0ce1e43e572988e178881d1c6dd3dfefb737.jpg new file mode 100644 index 00000000..96417768 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-208_23ebb946396351d337198c16df2f0ce1e43e572988e178881d1c6dd3dfefb737.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-208_4335e70f27060ad8eab7eb8c1a5ce458e05afaaf8eda1809b83199a5e9f338f1.jpg b/.raw/MidasCivilAnalysisReference/images/page-208_4335e70f27060ad8eab7eb8c1a5ce458e05afaaf8eda1809b83199a5e9f338f1.jpg new file mode 100644 index 00000000..c0b39ec2 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-208_4335e70f27060ad8eab7eb8c1a5ce458e05afaaf8eda1809b83199a5e9f338f1.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-208_9d81323f6af8aa262d71deaf911f2aad436612e6576a2e97169f7a8a3df96b69.jpg b/.raw/MidasCivilAnalysisReference/images/page-208_9d81323f6af8aa262d71deaf911f2aad436612e6576a2e97169f7a8a3df96b69.jpg new file mode 100644 index 00000000..c9ac45c8 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-208_9d81323f6af8aa262d71deaf911f2aad436612e6576a2e97169f7a8a3df96b69.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-208_a997934e4fcb37bdc77f58ab403b5b5202beed1890669cb1b7dea021d307f7ba.jpg b/.raw/MidasCivilAnalysisReference/images/page-208_a997934e4fcb37bdc77f58ab403b5b5202beed1890669cb1b7dea021d307f7ba.jpg new file mode 100644 index 00000000..2b079d84 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-208_a997934e4fcb37bdc77f58ab403b5b5202beed1890669cb1b7dea021d307f7ba.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-208_ce24a0e370d821a597069c2e7d4ab401c19d5729b721a577ca405a2f80b8f681.jpg b/.raw/MidasCivilAnalysisReference/images/page-208_ce24a0e370d821a597069c2e7d4ab401c19d5729b721a577ca405a2f80b8f681.jpg new file mode 100644 index 00000000..6cea2cb7 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-208_ce24a0e370d821a597069c2e7d4ab401c19d5729b721a577ca405a2f80b8f681.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-208_ff21aff950e057c990f7cd311bebc9886cf9f9e7b441912a60ed1b63909778f7.jpg b/.raw/MidasCivilAnalysisReference/images/page-208_ff21aff950e057c990f7cd311bebc9886cf9f9e7b441912a60ed1b63909778f7.jpg new file mode 100644 index 00000000..7b309116 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-208_ff21aff950e057c990f7cd311bebc9886cf9f9e7b441912a60ed1b63909778f7.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-209_636e8fde130c06228e09dbe29264b97ff7e0f13044df1d8d2b2bc3843c0a5a80.jpg b/.raw/MidasCivilAnalysisReference/images/page-209_636e8fde130c06228e09dbe29264b97ff7e0f13044df1d8d2b2bc3843c0a5a80.jpg new file mode 100644 index 00000000..8f60624a Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-209_636e8fde130c06228e09dbe29264b97ff7e0f13044df1d8d2b2bc3843c0a5a80.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-210_56a2e78c8bf8a604e991587ca7f742c2af766663a5fd0c4f840efd026efcb94e.jpg b/.raw/MidasCivilAnalysisReference/images/page-210_56a2e78c8bf8a604e991587ca7f742c2af766663a5fd0c4f840efd026efcb94e.jpg new file mode 100644 index 00000000..aa807f0b Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-210_56a2e78c8bf8a604e991587ca7f742c2af766663a5fd0c4f840efd026efcb94e.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-211_0044341b525aa7e69d93cb84bf973066b75358246cfbb93999dff38bb903a629.jpg b/.raw/MidasCivilAnalysisReference/images/page-211_0044341b525aa7e69d93cb84bf973066b75358246cfbb93999dff38bb903a629.jpg new file mode 100644 index 00000000..2dbf3250 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-211_0044341b525aa7e69d93cb84bf973066b75358246cfbb93999dff38bb903a629.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-211_063505e5cad55165af9785eb96aac81c527cc7705ae7affe912359e8db585129.jpg b/.raw/MidasCivilAnalysisReference/images/page-211_063505e5cad55165af9785eb96aac81c527cc7705ae7affe912359e8db585129.jpg new file mode 100644 index 00000000..81cf0e7a Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-211_063505e5cad55165af9785eb96aac81c527cc7705ae7affe912359e8db585129.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-211_39bd474590ce3c7ac6c43fa561c59798d0ae0e44a23a06d8052f1eafaedd05cb.jpg b/.raw/MidasCivilAnalysisReference/images/page-211_39bd474590ce3c7ac6c43fa561c59798d0ae0e44a23a06d8052f1eafaedd05cb.jpg new file mode 100644 index 00000000..fe15df2d Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-211_39bd474590ce3c7ac6c43fa561c59798d0ae0e44a23a06d8052f1eafaedd05cb.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-211_3babb3eb5e162af37470e9c428b0252b2b416d91a52170eaddf774af39e27ad8.jpg b/.raw/MidasCivilAnalysisReference/images/page-211_3babb3eb5e162af37470e9c428b0252b2b416d91a52170eaddf774af39e27ad8.jpg new file mode 100644 index 00000000..2bcdbc54 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-211_3babb3eb5e162af37470e9c428b0252b2b416d91a52170eaddf774af39e27ad8.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-211_5b22e537175d81864ef878b513ccd03d493798233438838a3634d57d38eff799.jpg b/.raw/MidasCivilAnalysisReference/images/page-211_5b22e537175d81864ef878b513ccd03d493798233438838a3634d57d38eff799.jpg new file mode 100644 index 00000000..f04d158b Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-211_5b22e537175d81864ef878b513ccd03d493798233438838a3634d57d38eff799.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-211_ca6f1135b0187ba96c341925a61fe1c90250ec2e134478c479328c3e610c7f37.jpg b/.raw/MidasCivilAnalysisReference/images/page-211_ca6f1135b0187ba96c341925a61fe1c90250ec2e134478c479328c3e610c7f37.jpg new file mode 100644 index 00000000..54dfa073 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-211_ca6f1135b0187ba96c341925a61fe1c90250ec2e134478c479328c3e610c7f37.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-211_d2dfd19864e852c5d3c52caf00fa28ef6cae985cb75b2c18288bb5a6eb5a40b7.jpg b/.raw/MidasCivilAnalysisReference/images/page-211_d2dfd19864e852c5d3c52caf00fa28ef6cae985cb75b2c18288bb5a6eb5a40b7.jpg new file mode 100644 index 00000000..0c8168b5 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-211_d2dfd19864e852c5d3c52caf00fa28ef6cae985cb75b2c18288bb5a6eb5a40b7.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-212_d195cd0c2ab681ddfe1f5c6f11a9638d00fca1f4fe9a5c4acfa99a3ec83e3c12.jpg b/.raw/MidasCivilAnalysisReference/images/page-212_d195cd0c2ab681ddfe1f5c6f11a9638d00fca1f4fe9a5c4acfa99a3ec83e3c12.jpg new file mode 100644 index 00000000..cd180551 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-212_d195cd0c2ab681ddfe1f5c6f11a9638d00fca1f4fe9a5c4acfa99a3ec83e3c12.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-213_05208c82dd9f862b8764886c579f407a4845f1a8b98fb06caa7ec50e939d3784.jpg b/.raw/MidasCivilAnalysisReference/images/page-213_05208c82dd9f862b8764886c579f407a4845f1a8b98fb06caa7ec50e939d3784.jpg new file mode 100644 index 00000000..340ac5cf Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-213_05208c82dd9f862b8764886c579f407a4845f1a8b98fb06caa7ec50e939d3784.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-213_05d257437dba3aaeb5fc5c27415c03dc85e62b01f984f561f1a68dcd32f77dc8.jpg b/.raw/MidasCivilAnalysisReference/images/page-213_05d257437dba3aaeb5fc5c27415c03dc85e62b01f984f561f1a68dcd32f77dc8.jpg new file mode 100644 index 00000000..2d092086 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-213_05d257437dba3aaeb5fc5c27415c03dc85e62b01f984f561f1a68dcd32f77dc8.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-213_35c8bfee4b6443db03e19111509ca8c843e40296e20bc9fc5daaf12322029fb0.jpg b/.raw/MidasCivilAnalysisReference/images/page-213_35c8bfee4b6443db03e19111509ca8c843e40296e20bc9fc5daaf12322029fb0.jpg new file mode 100644 index 00000000..3251261c Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-213_35c8bfee4b6443db03e19111509ca8c843e40296e20bc9fc5daaf12322029fb0.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-213_869d595c29420715d8076667c54d217c56723af48158d564b81ce7ae4bf5b860.jpg b/.raw/MidasCivilAnalysisReference/images/page-213_869d595c29420715d8076667c54d217c56723af48158d564b81ce7ae4bf5b860.jpg new file mode 100644 index 00000000..3a8ad96e Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-213_869d595c29420715d8076667c54d217c56723af48158d564b81ce7ae4bf5b860.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-213_d42c59a6a5b9c190f3415616a37a16595dff4c92fb23d16397efbeda0e3dd282.jpg b/.raw/MidasCivilAnalysisReference/images/page-213_d42c59a6a5b9c190f3415616a37a16595dff4c92fb23d16397efbeda0e3dd282.jpg new file mode 100644 index 00000000..983127e4 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-213_d42c59a6a5b9c190f3415616a37a16595dff4c92fb23d16397efbeda0e3dd282.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-213_d71822a1622c1c47295f3a4adb35fef87999e5874b00f526c705068d5b233ac5.jpg b/.raw/MidasCivilAnalysisReference/images/page-213_d71822a1622c1c47295f3a4adb35fef87999e5874b00f526c705068d5b233ac5.jpg new file mode 100644 index 00000000..908e205e Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-213_d71822a1622c1c47295f3a4adb35fef87999e5874b00f526c705068d5b233ac5.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-213_eb92e58894fd46b56b4fd81aae18d43c2bd8ca17010170df6d2c8fd370a00616.jpg b/.raw/MidasCivilAnalysisReference/images/page-213_eb92e58894fd46b56b4fd81aae18d43c2bd8ca17010170df6d2c8fd370a00616.jpg new file mode 100644 index 00000000..9c480362 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-213_eb92e58894fd46b56b4fd81aae18d43c2bd8ca17010170df6d2c8fd370a00616.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-214_41482bea858e70719d3be12e472ffa38d062baf59d724a98ddb89e1d8a328ab9.jpg b/.raw/MidasCivilAnalysisReference/images/page-214_41482bea858e70719d3be12e472ffa38d062baf59d724a98ddb89e1d8a328ab9.jpg new file mode 100644 index 00000000..e04c47e2 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-214_41482bea858e70719d3be12e472ffa38d062baf59d724a98ddb89e1d8a328ab9.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-214_8d5c9e8d871b607bf7803a0a292f7ea2410928fbf63e6c8e230268c98ea59d93.jpg b/.raw/MidasCivilAnalysisReference/images/page-214_8d5c9e8d871b607bf7803a0a292f7ea2410928fbf63e6c8e230268c98ea59d93.jpg new file mode 100644 index 00000000..2716480c Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-214_8d5c9e8d871b607bf7803a0a292f7ea2410928fbf63e6c8e230268c98ea59d93.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-214_8ea6b3829f4937405f190815ba13faf70b67322f683f47ef1ed14b65f11638d7.jpg b/.raw/MidasCivilAnalysisReference/images/page-214_8ea6b3829f4937405f190815ba13faf70b67322f683f47ef1ed14b65f11638d7.jpg new file mode 100644 index 00000000..afe47958 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-214_8ea6b3829f4937405f190815ba13faf70b67322f683f47ef1ed14b65f11638d7.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-214_c173c61bef98a78a49968febbcf88975caa6babdecbd6631b32bb17b9eca38dc.jpg b/.raw/MidasCivilAnalysisReference/images/page-214_c173c61bef98a78a49968febbcf88975caa6babdecbd6631b32bb17b9eca38dc.jpg new file mode 100644 index 00000000..2a106c70 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-214_c173c61bef98a78a49968febbcf88975caa6babdecbd6631b32bb17b9eca38dc.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-214_dc508171cdeae8a24f2447f1034d51997166e18417c773df7e34ad1bc9e201d3.jpg b/.raw/MidasCivilAnalysisReference/images/page-214_dc508171cdeae8a24f2447f1034d51997166e18417c773df7e34ad1bc9e201d3.jpg new file mode 100644 index 00000000..72366044 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-214_dc508171cdeae8a24f2447f1034d51997166e18417c773df7e34ad1bc9e201d3.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-214_de43b4a46aea5a84838645a30854bc2bfafd5f02484c475bf4790f3861ebfb7a.jpg b/.raw/MidasCivilAnalysisReference/images/page-214_de43b4a46aea5a84838645a30854bc2bfafd5f02484c475bf4790f3861ebfb7a.jpg new file mode 100644 index 00000000..8a863a41 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-214_de43b4a46aea5a84838645a30854bc2bfafd5f02484c475bf4790f3861ebfb7a.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-214_e8c12b32e955def81a854d4a0c3732ad496ce4d8962f16281b9bc65009f9cff5.jpg b/.raw/MidasCivilAnalysisReference/images/page-214_e8c12b32e955def81a854d4a0c3732ad496ce4d8962f16281b9bc65009f9cff5.jpg new file mode 100644 index 00000000..0922af65 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-214_e8c12b32e955def81a854d4a0c3732ad496ce4d8962f16281b9bc65009f9cff5.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-215_32e5b0c382b9aeb17140018284d5ec272acc9ac4182007e698de3059d9f6ed3f.jpg b/.raw/MidasCivilAnalysisReference/images/page-215_32e5b0c382b9aeb17140018284d5ec272acc9ac4182007e698de3059d9f6ed3f.jpg new file mode 100644 index 00000000..d0d7f98c Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-215_32e5b0c382b9aeb17140018284d5ec272acc9ac4182007e698de3059d9f6ed3f.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-215_4ae3dc7ba242ce6ec793a6b13ee93b7773206bb1810404303dd73a0d646843aa.jpg b/.raw/MidasCivilAnalysisReference/images/page-215_4ae3dc7ba242ce6ec793a6b13ee93b7773206bb1810404303dd73a0d646843aa.jpg new file mode 100644 index 00000000..ffeca2d8 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-215_4ae3dc7ba242ce6ec793a6b13ee93b7773206bb1810404303dd73a0d646843aa.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-216_66a5ef815953f33a5c152f63abbba6f9386089737833cb8520980a120748494e.jpg b/.raw/MidasCivilAnalysisReference/images/page-216_66a5ef815953f33a5c152f63abbba6f9386089737833cb8520980a120748494e.jpg new file mode 100644 index 00000000..2d69d3c2 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-216_66a5ef815953f33a5c152f63abbba6f9386089737833cb8520980a120748494e.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-216_7883c96cd491b0448535da1341ab2bb201e65844a611a42894ad9a01c6b224a2.jpg b/.raw/MidasCivilAnalysisReference/images/page-216_7883c96cd491b0448535da1341ab2bb201e65844a611a42894ad9a01c6b224a2.jpg new file mode 100644 index 00000000..2517a733 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-216_7883c96cd491b0448535da1341ab2bb201e65844a611a42894ad9a01c6b224a2.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-216_82e614e33e9d0f3985b022eb427682e3c4e8fcb60acd5c1d5e05399ccded40b7.jpg b/.raw/MidasCivilAnalysisReference/images/page-216_82e614e33e9d0f3985b022eb427682e3c4e8fcb60acd5c1d5e05399ccded40b7.jpg new file mode 100644 index 00000000..da9874d2 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-216_82e614e33e9d0f3985b022eb427682e3c4e8fcb60acd5c1d5e05399ccded40b7.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-216_e58a15467ac30d4771a6425cbd85f1a34f9a937126893f7d8bb739f186a9ee29.jpg b/.raw/MidasCivilAnalysisReference/images/page-216_e58a15467ac30d4771a6425cbd85f1a34f9a937126893f7d8bb739f186a9ee29.jpg new file mode 100644 index 00000000..916ee587 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-216_e58a15467ac30d4771a6425cbd85f1a34f9a937126893f7d8bb739f186a9ee29.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-216_f29f2fe3e0926e34a226b18144e7e3fd95d830136176e5b819d225f5d2edf7f8.jpg b/.raw/MidasCivilAnalysisReference/images/page-216_f29f2fe3e0926e34a226b18144e7e3fd95d830136176e5b819d225f5d2edf7f8.jpg new file mode 100644 index 00000000..f524e225 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-216_f29f2fe3e0926e34a226b18144e7e3fd95d830136176e5b819d225f5d2edf7f8.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-217_27ee28673875d246f8430da0fb1a81fbbbc38bf7f37665af3b21fa3fc4006824.jpg b/.raw/MidasCivilAnalysisReference/images/page-217_27ee28673875d246f8430da0fb1a81fbbbc38bf7f37665af3b21fa3fc4006824.jpg new file mode 100644 index 00000000..057bf7da Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-217_27ee28673875d246f8430da0fb1a81fbbbc38bf7f37665af3b21fa3fc4006824.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-217_f7a1055324e1fbf818e090f21e01162481d7bc2470c364b402f8ad13b1000e45.jpg b/.raw/MidasCivilAnalysisReference/images/page-217_f7a1055324e1fbf818e090f21e01162481d7bc2470c364b402f8ad13b1000e45.jpg new file mode 100644 index 00000000..2e82f7ad Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-217_f7a1055324e1fbf818e090f21e01162481d7bc2470c364b402f8ad13b1000e45.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-218_1ee700887b453ece28797a7ec6b5903bad87c6515b1b3ce9e5cb47d408048d11.jpg b/.raw/MidasCivilAnalysisReference/images/page-218_1ee700887b453ece28797a7ec6b5903bad87c6515b1b3ce9e5cb47d408048d11.jpg new file mode 100644 index 00000000..33a8a395 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-218_1ee700887b453ece28797a7ec6b5903bad87c6515b1b3ce9e5cb47d408048d11.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-218_20616cb676056aaee9ec5dc5d147f25a379bc77a9360948f143cb386ce4cd00d.jpg b/.raw/MidasCivilAnalysisReference/images/page-218_20616cb676056aaee9ec5dc5d147f25a379bc77a9360948f143cb386ce4cd00d.jpg new file mode 100644 index 00000000..e8433065 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-218_20616cb676056aaee9ec5dc5d147f25a379bc77a9360948f143cb386ce4cd00d.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-218_4fe93bc6f395a07299e8587f29f4fb0138726620d6257d8086eb815f44478fbe.jpg b/.raw/MidasCivilAnalysisReference/images/page-218_4fe93bc6f395a07299e8587f29f4fb0138726620d6257d8086eb815f44478fbe.jpg new file mode 100644 index 00000000..4fe9e5ac Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-218_4fe93bc6f395a07299e8587f29f4fb0138726620d6257d8086eb815f44478fbe.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-218_6b2aef9733f1c39c9568f143dc98b096b267fcb8c798314becd5e75fadf3dbda.jpg b/.raw/MidasCivilAnalysisReference/images/page-218_6b2aef9733f1c39c9568f143dc98b096b267fcb8c798314becd5e75fadf3dbda.jpg new file mode 100644 index 00000000..00354dc4 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-218_6b2aef9733f1c39c9568f143dc98b096b267fcb8c798314becd5e75fadf3dbda.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-218_857ccd3ebd86aa5d52c32e58ed540be22e17bdc182732c55adf457d2a64be884.jpg b/.raw/MidasCivilAnalysisReference/images/page-218_857ccd3ebd86aa5d52c32e58ed540be22e17bdc182732c55adf457d2a64be884.jpg new file mode 100644 index 00000000..da624e68 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-218_857ccd3ebd86aa5d52c32e58ed540be22e17bdc182732c55adf457d2a64be884.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-218_a83a1c6cad82ef2fdf2d00b76b752f3a3d3504637e921932459f0d06ee89980f.jpg b/.raw/MidasCivilAnalysisReference/images/page-218_a83a1c6cad82ef2fdf2d00b76b752f3a3d3504637e921932459f0d06ee89980f.jpg new file mode 100644 index 00000000..496a5091 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-218_a83a1c6cad82ef2fdf2d00b76b752f3a3d3504637e921932459f0d06ee89980f.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-218_ac79b4bc77b6ca5a51e7a42342450715bd7f4d2e7241f99c00848de35a750b09.jpg b/.raw/MidasCivilAnalysisReference/images/page-218_ac79b4bc77b6ca5a51e7a42342450715bd7f4d2e7241f99c00848de35a750b09.jpg new file mode 100644 index 00000000..4415b919 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-218_ac79b4bc77b6ca5a51e7a42342450715bd7f4d2e7241f99c00848de35a750b09.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-218_ad9bc77381c7a986958302faf73b4632b801a940445a70f3cf68228a8d738a8f.jpg b/.raw/MidasCivilAnalysisReference/images/page-218_ad9bc77381c7a986958302faf73b4632b801a940445a70f3cf68228a8d738a8f.jpg new file mode 100644 index 00000000..75e06f53 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-218_ad9bc77381c7a986958302faf73b4632b801a940445a70f3cf68228a8d738a8f.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-218_b73a91ba31263f6d32e7a0c0be9f56952024f11cb4df9d19e46099c887edc9ea.jpg b/.raw/MidasCivilAnalysisReference/images/page-218_b73a91ba31263f6d32e7a0c0be9f56952024f11cb4df9d19e46099c887edc9ea.jpg new file mode 100644 index 00000000..5096973c Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-218_b73a91ba31263f6d32e7a0c0be9f56952024f11cb4df9d19e46099c887edc9ea.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-218_bb68a51b346da031ec9a61bed17629ba3e9233dc3acfb13769beec40f588c3d4.jpg b/.raw/MidasCivilAnalysisReference/images/page-218_bb68a51b346da031ec9a61bed17629ba3e9233dc3acfb13769beec40f588c3d4.jpg new file mode 100644 index 00000000..d0ba637d Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-218_bb68a51b346da031ec9a61bed17629ba3e9233dc3acfb13769beec40f588c3d4.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-218_bd11da458325b02c6586308eceaad0a591a5f58a4910320c7b4333351c1c6ebc.jpg b/.raw/MidasCivilAnalysisReference/images/page-218_bd11da458325b02c6586308eceaad0a591a5f58a4910320c7b4333351c1c6ebc.jpg new file mode 100644 index 00000000..b6ad4578 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-218_bd11da458325b02c6586308eceaad0a591a5f58a4910320c7b4333351c1c6ebc.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-218_e9740c7058f76e69ba17ca9ca430ed4e066ed3aa2e7b303183e72140175bbde1.jpg b/.raw/MidasCivilAnalysisReference/images/page-218_e9740c7058f76e69ba17ca9ca430ed4e066ed3aa2e7b303183e72140175bbde1.jpg new file mode 100644 index 00000000..a121b03b Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-218_e9740c7058f76e69ba17ca9ca430ed4e066ed3aa2e7b303183e72140175bbde1.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-219_0775ca69a83fa1e7316b0fac8df7e9138ea1a5db045c1fcc9e09cd89087e334f.jpg b/.raw/MidasCivilAnalysisReference/images/page-219_0775ca69a83fa1e7316b0fac8df7e9138ea1a5db045c1fcc9e09cd89087e334f.jpg new file mode 100644 index 00000000..56339e87 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-219_0775ca69a83fa1e7316b0fac8df7e9138ea1a5db045c1fcc9e09cd89087e334f.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-219_249d936740b802f8cfe1bbac3c6005dd23ce3f009fea574fcdaa86ca168ce79d.jpg b/.raw/MidasCivilAnalysisReference/images/page-219_249d936740b802f8cfe1bbac3c6005dd23ce3f009fea574fcdaa86ca168ce79d.jpg new file mode 100644 index 00000000..fcc1c7aa Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-219_249d936740b802f8cfe1bbac3c6005dd23ce3f009fea574fcdaa86ca168ce79d.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-219_c5623cb455d1df27e83852c657dce7693708eb6e95e4b194d700410e95672551.jpg b/.raw/MidasCivilAnalysisReference/images/page-219_c5623cb455d1df27e83852c657dce7693708eb6e95e4b194d700410e95672551.jpg new file mode 100644 index 00000000..b58c054d Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-219_c5623cb455d1df27e83852c657dce7693708eb6e95e4b194d700410e95672551.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-220_0964bdeb74dff2b84478b1a59cb8abddb427a56d0d0d75e62e12fb40b5f8d77e.jpg b/.raw/MidasCivilAnalysisReference/images/page-220_0964bdeb74dff2b84478b1a59cb8abddb427a56d0d0d75e62e12fb40b5f8d77e.jpg new file mode 100644 index 00000000..4ea2a3dc Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-220_0964bdeb74dff2b84478b1a59cb8abddb427a56d0d0d75e62e12fb40b5f8d77e.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-220_5b39f3ed64dfce145aac99ea55c2836c24bf9150e5932b2b1fd1a2ffa0c75773.jpg b/.raw/MidasCivilAnalysisReference/images/page-220_5b39f3ed64dfce145aac99ea55c2836c24bf9150e5932b2b1fd1a2ffa0c75773.jpg new file mode 100644 index 00000000..484d2a8c Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-220_5b39f3ed64dfce145aac99ea55c2836c24bf9150e5932b2b1fd1a2ffa0c75773.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-220_e45523a995376b6ea3242e1ba4659b0bb96eca3d90e557a9defacc921caf7dd2.jpg b/.raw/MidasCivilAnalysisReference/images/page-220_e45523a995376b6ea3242e1ba4659b0bb96eca3d90e557a9defacc921caf7dd2.jpg new file mode 100644 index 00000000..b71b5c9e Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-220_e45523a995376b6ea3242e1ba4659b0bb96eca3d90e557a9defacc921caf7dd2.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-221_20140864fe321b795ffd0f9b4b2d5c6f247ea81ee185e33279ffb46992832eff.jpg b/.raw/MidasCivilAnalysisReference/images/page-221_20140864fe321b795ffd0f9b4b2d5c6f247ea81ee185e33279ffb46992832eff.jpg new file mode 100644 index 00000000..16fe77e8 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-221_20140864fe321b795ffd0f9b4b2d5c6f247ea81ee185e33279ffb46992832eff.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-221_4cac6b35ee13b76a4e68c14523c30c7c7828dc9a8688a67a6ebe6a25e24aab48.jpg b/.raw/MidasCivilAnalysisReference/images/page-221_4cac6b35ee13b76a4e68c14523c30c7c7828dc9a8688a67a6ebe6a25e24aab48.jpg new file mode 100644 index 00000000..7e6b3318 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-221_4cac6b35ee13b76a4e68c14523c30c7c7828dc9a8688a67a6ebe6a25e24aab48.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-221_794e04ef3a7816bcb5867586420cdc5c4ca726fa50e9520e15acfe0214852369.jpg b/.raw/MidasCivilAnalysisReference/images/page-221_794e04ef3a7816bcb5867586420cdc5c4ca726fa50e9520e15acfe0214852369.jpg new file mode 100644 index 00000000..981765b4 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-221_794e04ef3a7816bcb5867586420cdc5c4ca726fa50e9520e15acfe0214852369.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-221_aff9bbb935e3053e29993e163479db1b6395190f4bb4334505bb33302fd4d20b.jpg b/.raw/MidasCivilAnalysisReference/images/page-221_aff9bbb935e3053e29993e163479db1b6395190f4bb4334505bb33302fd4d20b.jpg new file mode 100644 index 00000000..953154dd Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-221_aff9bbb935e3053e29993e163479db1b6395190f4bb4334505bb33302fd4d20b.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-222_1fe1302b600b55144939e16464ab079d184786591d88cd67f6df29957831b6c5.jpg b/.raw/MidasCivilAnalysisReference/images/page-222_1fe1302b600b55144939e16464ab079d184786591d88cd67f6df29957831b6c5.jpg new file mode 100644 index 00000000..6b70b910 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-222_1fe1302b600b55144939e16464ab079d184786591d88cd67f6df29957831b6c5.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-222_6c501d2fdf8247b31e26a1e5fac9eee3a7b4ab94582d1a8ea55399f70ba8806f.jpg b/.raw/MidasCivilAnalysisReference/images/page-222_6c501d2fdf8247b31e26a1e5fac9eee3a7b4ab94582d1a8ea55399f70ba8806f.jpg new file mode 100644 index 00000000..1ce1ffa3 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-222_6c501d2fdf8247b31e26a1e5fac9eee3a7b4ab94582d1a8ea55399f70ba8806f.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-222_8af4aa56c5d9b23dbe1177308ea90a7c716b352b2a95bfec99c8c135b5c779cc.jpg b/.raw/MidasCivilAnalysisReference/images/page-222_8af4aa56c5d9b23dbe1177308ea90a7c716b352b2a95bfec99c8c135b5c779cc.jpg new file mode 100644 index 00000000..2a484d3e Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-222_8af4aa56c5d9b23dbe1177308ea90a7c716b352b2a95bfec99c8c135b5c779cc.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-222_a11455763d0dc5152816ad8007c9f8480b544fc48dad8d40dcaf8024898d020f.jpg b/.raw/MidasCivilAnalysisReference/images/page-222_a11455763d0dc5152816ad8007c9f8480b544fc48dad8d40dcaf8024898d020f.jpg new file mode 100644 index 00000000..b7093289 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-222_a11455763d0dc5152816ad8007c9f8480b544fc48dad8d40dcaf8024898d020f.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-222_c3816d157ad41254f7c605da900aafc66f36104f0e61142fc16a4ce35b06ff0a.jpg b/.raw/MidasCivilAnalysisReference/images/page-222_c3816d157ad41254f7c605da900aafc66f36104f0e61142fc16a4ce35b06ff0a.jpg new file mode 100644 index 00000000..6943647d Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-222_c3816d157ad41254f7c605da900aafc66f36104f0e61142fc16a4ce35b06ff0a.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-222_d02636cadc0f4d2b0fdacd4c65be8a1582803080a4583969b8dbb00ed3d27cb5.jpg b/.raw/MidasCivilAnalysisReference/images/page-222_d02636cadc0f4d2b0fdacd4c65be8a1582803080a4583969b8dbb00ed3d27cb5.jpg new file mode 100644 index 00000000..7f2cfd3d Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-222_d02636cadc0f4d2b0fdacd4c65be8a1582803080a4583969b8dbb00ed3d27cb5.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-222_d67c92c6cccf734c09152a31d3289e27ccf768ebeff8054616fe2bfac6bbb42d.jpg b/.raw/MidasCivilAnalysisReference/images/page-222_d67c92c6cccf734c09152a31d3289e27ccf768ebeff8054616fe2bfac6bbb42d.jpg new file mode 100644 index 00000000..dee4ef65 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-222_d67c92c6cccf734c09152a31d3289e27ccf768ebeff8054616fe2bfac6bbb42d.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-222_dfa3e2824b3df83d1c1a64cec1c2c21b2324ece41631b93662481439280a7b44.jpg b/.raw/MidasCivilAnalysisReference/images/page-222_dfa3e2824b3df83d1c1a64cec1c2c21b2324ece41631b93662481439280a7b44.jpg new file mode 100644 index 00000000..621346c7 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-222_dfa3e2824b3df83d1c1a64cec1c2c21b2324ece41631b93662481439280a7b44.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-223_d6dc7c53d04e1b9b7beec3cc23dc6459fe74baace1aeeb6cbc6387f7fc395f86.jpg b/.raw/MidasCivilAnalysisReference/images/page-223_d6dc7c53d04e1b9b7beec3cc23dc6459fe74baace1aeeb6cbc6387f7fc395f86.jpg new file mode 100644 index 00000000..d49ebe56 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-223_d6dc7c53d04e1b9b7beec3cc23dc6459fe74baace1aeeb6cbc6387f7fc395f86.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-224_76b3fe55d303642b897540bd843f94b79f2595102b0f1979c20a00b781f92180.jpg b/.raw/MidasCivilAnalysisReference/images/page-224_76b3fe55d303642b897540bd843f94b79f2595102b0f1979c20a00b781f92180.jpg new file mode 100644 index 00000000..e534d7ec Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-224_76b3fe55d303642b897540bd843f94b79f2595102b0f1979c20a00b781f92180.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-225_ede9f93d9ce1ee89d847875c3917aeae4d247db3c6f8c85bb2cb75e15e020f89.jpg b/.raw/MidasCivilAnalysisReference/images/page-225_ede9f93d9ce1ee89d847875c3917aeae4d247db3c6f8c85bb2cb75e15e020f89.jpg new file mode 100644 index 00000000..ae25e509 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-225_ede9f93d9ce1ee89d847875c3917aeae4d247db3c6f8c85bb2cb75e15e020f89.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-226_f8b4133aa719f4ea2e1e8e413200c13df3e2dbb56b753d9c56a70427767a5d1a.jpg b/.raw/MidasCivilAnalysisReference/images/page-226_f8b4133aa719f4ea2e1e8e413200c13df3e2dbb56b753d9c56a70427767a5d1a.jpg new file mode 100644 index 00000000..ddf47a2c Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-226_f8b4133aa719f4ea2e1e8e413200c13df3e2dbb56b753d9c56a70427767a5d1a.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-226_fe24cd77280aa650b49c56fe0aab57295264205b510fc9039dfdb92a0e3f5ea1.jpg b/.raw/MidasCivilAnalysisReference/images/page-226_fe24cd77280aa650b49c56fe0aab57295264205b510fc9039dfdb92a0e3f5ea1.jpg new file mode 100644 index 00000000..90134213 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-226_fe24cd77280aa650b49c56fe0aab57295264205b510fc9039dfdb92a0e3f5ea1.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-227_84bfdb8ce5c5708026b22db04090716b620b2383c303dd256dc6f6e454f26fa9.jpg b/.raw/MidasCivilAnalysisReference/images/page-227_84bfdb8ce5c5708026b22db04090716b620b2383c303dd256dc6f6e454f26fa9.jpg new file mode 100644 index 00000000..5ac97cdb Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-227_84bfdb8ce5c5708026b22db04090716b620b2383c303dd256dc6f6e454f26fa9.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-227_9226db8367d5a93a33aa3186b84f5ff53e3713017f5313915d5d45d8a4aafd14.jpg b/.raw/MidasCivilAnalysisReference/images/page-227_9226db8367d5a93a33aa3186b84f5ff53e3713017f5313915d5d45d8a4aafd14.jpg new file mode 100644 index 00000000..18134c8b Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-227_9226db8367d5a93a33aa3186b84f5ff53e3713017f5313915d5d45d8a4aafd14.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-227_f56a4ffbe81868428f115f5de3f96db4f49759763bca63a0b535e21334221ca1.jpg b/.raw/MidasCivilAnalysisReference/images/page-227_f56a4ffbe81868428f115f5de3f96db4f49759763bca63a0b535e21334221ca1.jpg new file mode 100644 index 00000000..2d5e7d47 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-227_f56a4ffbe81868428f115f5de3f96db4f49759763bca63a0b535e21334221ca1.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-228_3f9e35671bbe062df1bbc5a37fb6b8a756a1a84cec37a0058ce71ac494e0f6ce.jpg b/.raw/MidasCivilAnalysisReference/images/page-228_3f9e35671bbe062df1bbc5a37fb6b8a756a1a84cec37a0058ce71ac494e0f6ce.jpg new file mode 100644 index 00000000..49f36651 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-228_3f9e35671bbe062df1bbc5a37fb6b8a756a1a84cec37a0058ce71ac494e0f6ce.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-228_4da4673b04df99cc2dfa32b38f227d1e6f68b82fa17bc04c2681d3f6f36ed5e8.jpg b/.raw/MidasCivilAnalysisReference/images/page-228_4da4673b04df99cc2dfa32b38f227d1e6f68b82fa17bc04c2681d3f6f36ed5e8.jpg new file mode 100644 index 00000000..24319749 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-228_4da4673b04df99cc2dfa32b38f227d1e6f68b82fa17bc04c2681d3f6f36ed5e8.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-228_ffc04f292728e20fa8719bdf40b8ffd684504d1f53bf227bbb4656cf916332d3.jpg b/.raw/MidasCivilAnalysisReference/images/page-228_ffc04f292728e20fa8719bdf40b8ffd684504d1f53bf227bbb4656cf916332d3.jpg new file mode 100644 index 00000000..e424a35d Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-228_ffc04f292728e20fa8719bdf40b8ffd684504d1f53bf227bbb4656cf916332d3.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-229_205ac8c88a7a1f5de1c37ef6883285c654d2effeaaeb19b9ee2bb9deec0b015a.jpg b/.raw/MidasCivilAnalysisReference/images/page-229_205ac8c88a7a1f5de1c37ef6883285c654d2effeaaeb19b9ee2bb9deec0b015a.jpg new file mode 100644 index 00000000..6c65cb6b Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-229_205ac8c88a7a1f5de1c37ef6883285c654d2effeaaeb19b9ee2bb9deec0b015a.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-229_7b4e956b6adb911170146d44052f851e1af3d549e398b75fbaa075f54c97f6ef.jpg b/.raw/MidasCivilAnalysisReference/images/page-229_7b4e956b6adb911170146d44052f851e1af3d549e398b75fbaa075f54c97f6ef.jpg new file mode 100644 index 00000000..5d7b194c Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-229_7b4e956b6adb911170146d44052f851e1af3d549e398b75fbaa075f54c97f6ef.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-229_e861f46223d1ebe9cf4c92b64157ea6576ab4bde0b0ab1c02972321164f220d8.jpg b/.raw/MidasCivilAnalysisReference/images/page-229_e861f46223d1ebe9cf4c92b64157ea6576ab4bde0b0ab1c02972321164f220d8.jpg new file mode 100644 index 00000000..1df87d5c Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-229_e861f46223d1ebe9cf4c92b64157ea6576ab4bde0b0ab1c02972321164f220d8.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-229_f521cfc13333cdd0b2dae6e783da6b9ef63c7e25fad67c346e1cf7947ee7b1b6.jpg b/.raw/MidasCivilAnalysisReference/images/page-229_f521cfc13333cdd0b2dae6e783da6b9ef63c7e25fad67c346e1cf7947ee7b1b6.jpg new file mode 100644 index 00000000..4e331094 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-229_f521cfc13333cdd0b2dae6e783da6b9ef63c7e25fad67c346e1cf7947ee7b1b6.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-230_1f2cf5b17976964b37e984103b402ff126f7e4c6241c7f35524571bd5edfbf49.jpg b/.raw/MidasCivilAnalysisReference/images/page-230_1f2cf5b17976964b37e984103b402ff126f7e4c6241c7f35524571bd5edfbf49.jpg new file mode 100644 index 00000000..a51d16cb Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-230_1f2cf5b17976964b37e984103b402ff126f7e4c6241c7f35524571bd5edfbf49.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-230_6fd4ba9fbf99630811d39aa0bd7ca6e70c202dbdf266c93219318f02cfdc2bbd.jpg b/.raw/MidasCivilAnalysisReference/images/page-230_6fd4ba9fbf99630811d39aa0bd7ca6e70c202dbdf266c93219318f02cfdc2bbd.jpg new file mode 100644 index 00000000..a4eeda42 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-230_6fd4ba9fbf99630811d39aa0bd7ca6e70c202dbdf266c93219318f02cfdc2bbd.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-231_57a627fda65e33f9aad0aa0c111a25cde7b7230c38054cc02236a327066ec46e.jpg b/.raw/MidasCivilAnalysisReference/images/page-231_57a627fda65e33f9aad0aa0c111a25cde7b7230c38054cc02236a327066ec46e.jpg new file mode 100644 index 00000000..55d346ff Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-231_57a627fda65e33f9aad0aa0c111a25cde7b7230c38054cc02236a327066ec46e.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-231_5f41bfea0b700bdc2fb3b5c14084822c25d807702f63ac03f386d339d8106d57.jpg b/.raw/MidasCivilAnalysisReference/images/page-231_5f41bfea0b700bdc2fb3b5c14084822c25d807702f63ac03f386d339d8106d57.jpg new file mode 100644 index 00000000..ef978407 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-231_5f41bfea0b700bdc2fb3b5c14084822c25d807702f63ac03f386d339d8106d57.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-231_5f4ec4bff10e9c8bbb30b94a4bfc7395e728f97d14b701f06362a7289022069c.jpg b/.raw/MidasCivilAnalysisReference/images/page-231_5f4ec4bff10e9c8bbb30b94a4bfc7395e728f97d14b701f06362a7289022069c.jpg new file mode 100644 index 00000000..e0d6ba8c Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-231_5f4ec4bff10e9c8bbb30b94a4bfc7395e728f97d14b701f06362a7289022069c.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-231_8ec34551462efde1af4f7b6a098b69910e8f140808bd090073606bc89a6a3048.jpg b/.raw/MidasCivilAnalysisReference/images/page-231_8ec34551462efde1af4f7b6a098b69910e8f140808bd090073606bc89a6a3048.jpg new file mode 100644 index 00000000..5062c680 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-231_8ec34551462efde1af4f7b6a098b69910e8f140808bd090073606bc89a6a3048.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-231_adb19e11a0640f0b5a3a0645a20b145d95ccdd2d077a838acf6b8062d3bdc15f.jpg b/.raw/MidasCivilAnalysisReference/images/page-231_adb19e11a0640f0b5a3a0645a20b145d95ccdd2d077a838acf6b8062d3bdc15f.jpg new file mode 100644 index 00000000..c38d8173 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-231_adb19e11a0640f0b5a3a0645a20b145d95ccdd2d077a838acf6b8062d3bdc15f.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-231_de7e762e7c636e8688759bbbdff26095b744cb385993188ec5e18fe772a1addb.jpg b/.raw/MidasCivilAnalysisReference/images/page-231_de7e762e7c636e8688759bbbdff26095b744cb385993188ec5e18fe772a1addb.jpg new file mode 100644 index 00000000..7702474c Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-231_de7e762e7c636e8688759bbbdff26095b744cb385993188ec5e18fe772a1addb.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-231_e58dd2b31baa8c16b71c0488663c0546890239791b6981a565df19630777a415.jpg b/.raw/MidasCivilAnalysisReference/images/page-231_e58dd2b31baa8c16b71c0488663c0546890239791b6981a565df19630777a415.jpg new file mode 100644 index 00000000..3c5a7136 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-231_e58dd2b31baa8c16b71c0488663c0546890239791b6981a565df19630777a415.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-232_34f755fedf6e979a70debdd3d9be1e37e4f5aacce7f3d8fe3579b6233bca9ce4.jpg b/.raw/MidasCivilAnalysisReference/images/page-232_34f755fedf6e979a70debdd3d9be1e37e4f5aacce7f3d8fe3579b6233bca9ce4.jpg new file mode 100644 index 00000000..b7430546 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-232_34f755fedf6e979a70debdd3d9be1e37e4f5aacce7f3d8fe3579b6233bca9ce4.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-232_552bcb30bb29774d5017ad689c6b6af74eeba5460a3b4cf53dbd5ac456aee604.jpg b/.raw/MidasCivilAnalysisReference/images/page-232_552bcb30bb29774d5017ad689c6b6af74eeba5460a3b4cf53dbd5ac456aee604.jpg new file mode 100644 index 00000000..76c28be4 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-232_552bcb30bb29774d5017ad689c6b6af74eeba5460a3b4cf53dbd5ac456aee604.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-232_8b7890d6fc6498e7171017ee7d7b4ad0cb73b28a1110d45a69bd8616599b1792.jpg b/.raw/MidasCivilAnalysisReference/images/page-232_8b7890d6fc6498e7171017ee7d7b4ad0cb73b28a1110d45a69bd8616599b1792.jpg new file mode 100644 index 00000000..f7a029ba Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-232_8b7890d6fc6498e7171017ee7d7b4ad0cb73b28a1110d45a69bd8616599b1792.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-232_c6670beca739f899b40cd7dbb3a9ddb34c3f05e68c362839ee9f5e298e02113f.jpg b/.raw/MidasCivilAnalysisReference/images/page-232_c6670beca739f899b40cd7dbb3a9ddb34c3f05e68c362839ee9f5e298e02113f.jpg new file mode 100644 index 00000000..57fbde21 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-232_c6670beca739f899b40cd7dbb3a9ddb34c3f05e68c362839ee9f5e298e02113f.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-232_eea6da0249683e35237ac270f13393b25671d1b64b5d5f38b8508db0aa2eba61.jpg b/.raw/MidasCivilAnalysisReference/images/page-232_eea6da0249683e35237ac270f13393b25671d1b64b5d5f38b8508db0aa2eba61.jpg new file mode 100644 index 00000000..a3ad233a Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-232_eea6da0249683e35237ac270f13393b25671d1b64b5d5f38b8508db0aa2eba61.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-233_ccae5dfb0ace4919c6961c4bc4b95082c53ccfeab1732d6994bc8264c630db48.jpg b/.raw/MidasCivilAnalysisReference/images/page-233_ccae5dfb0ace4919c6961c4bc4b95082c53ccfeab1732d6994bc8264c630db48.jpg new file mode 100644 index 00000000..17f8569d Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-233_ccae5dfb0ace4919c6961c4bc4b95082c53ccfeab1732d6994bc8264c630db48.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-235_7a642b1652e9367e6389c0222aa94797b5f0a175635cef64b35c93aa079dc0b5.jpg b/.raw/MidasCivilAnalysisReference/images/page-235_7a642b1652e9367e6389c0222aa94797b5f0a175635cef64b35c93aa079dc0b5.jpg new file mode 100644 index 00000000..3af9f11a Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-235_7a642b1652e9367e6389c0222aa94797b5f0a175635cef64b35c93aa079dc0b5.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-235_95c64008f8dece3ffaa858d11020c45049fc1d35cabeb2658a378363d43c1523.jpg b/.raw/MidasCivilAnalysisReference/images/page-235_95c64008f8dece3ffaa858d11020c45049fc1d35cabeb2658a378363d43c1523.jpg new file mode 100644 index 00000000..f08c48f8 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-235_95c64008f8dece3ffaa858d11020c45049fc1d35cabeb2658a378363d43c1523.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-235_b3dd0e557589996e501c1de6a27877c0e35e4a6b5a060d8d3f84d10edfdef811.jpg b/.raw/MidasCivilAnalysisReference/images/page-235_b3dd0e557589996e501c1de6a27877c0e35e4a6b5a060d8d3f84d10edfdef811.jpg new file mode 100644 index 00000000..52dc5f8d Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-235_b3dd0e557589996e501c1de6a27877c0e35e4a6b5a060d8d3f84d10edfdef811.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-236_1fc8208f793b330436b436d4d4c04ef41da1fb701925e5a4ecd7732e4a3cad62.jpg b/.raw/MidasCivilAnalysisReference/images/page-236_1fc8208f793b330436b436d4d4c04ef41da1fb701925e5a4ecd7732e4a3cad62.jpg new file mode 100644 index 00000000..e565f4e3 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-236_1fc8208f793b330436b436d4d4c04ef41da1fb701925e5a4ecd7732e4a3cad62.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-236_6a0d88d8a5dc56392c28cba94844cdc9fb2bd137736864c7cff011e6e03b555f.jpg b/.raw/MidasCivilAnalysisReference/images/page-236_6a0d88d8a5dc56392c28cba94844cdc9fb2bd137736864c7cff011e6e03b555f.jpg new file mode 100644 index 00000000..ccafcbbe Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-236_6a0d88d8a5dc56392c28cba94844cdc9fb2bd137736864c7cff011e6e03b555f.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-236_702e442e0ae9b5d3fa67ec190562a262dee09244d8ffda8331baaf7f563b95d5.jpg b/.raw/MidasCivilAnalysisReference/images/page-236_702e442e0ae9b5d3fa67ec190562a262dee09244d8ffda8331baaf7f563b95d5.jpg new file mode 100644 index 00000000..14a903c7 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-236_702e442e0ae9b5d3fa67ec190562a262dee09244d8ffda8331baaf7f563b95d5.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-236_95a0e9ab8a617be6c3690fcc826f306bc6b93869c0727990dd9c68727fc8c4c1.jpg b/.raw/MidasCivilAnalysisReference/images/page-236_95a0e9ab8a617be6c3690fcc826f306bc6b93869c0727990dd9c68727fc8c4c1.jpg new file mode 100644 index 00000000..eb036bb6 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-236_95a0e9ab8a617be6c3690fcc826f306bc6b93869c0727990dd9c68727fc8c4c1.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-236_ccd4ae99450b44362b8f98d0d57716d0b79e847d3e1d07732e99a3e19d305013.jpg b/.raw/MidasCivilAnalysisReference/images/page-236_ccd4ae99450b44362b8f98d0d57716d0b79e847d3e1d07732e99a3e19d305013.jpg new file mode 100644 index 00000000..99246cc8 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-236_ccd4ae99450b44362b8f98d0d57716d0b79e847d3e1d07732e99a3e19d305013.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-237_fe6a03e2beda6d3e28068f1dd6d0929ad060a37721839985cfcfa52c9b253c15.jpg b/.raw/MidasCivilAnalysisReference/images/page-237_fe6a03e2beda6d3e28068f1dd6d0929ad060a37721839985cfcfa52c9b253c15.jpg new file mode 100644 index 00000000..06783604 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-237_fe6a03e2beda6d3e28068f1dd6d0929ad060a37721839985cfcfa52c9b253c15.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-238_e2a22b3c49027955539bc46c8b1f80fde862c3ca7821d67787537b39d7bb8bbd.jpg b/.raw/MidasCivilAnalysisReference/images/page-238_e2a22b3c49027955539bc46c8b1f80fde862c3ca7821d67787537b39d7bb8bbd.jpg new file mode 100644 index 00000000..7755427b Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-238_e2a22b3c49027955539bc46c8b1f80fde862c3ca7821d67787537b39d7bb8bbd.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-239_bfa12fef3e8aaa3080145486a569a49a1804b968b32872de25c47c61eef5b1d8.jpg b/.raw/MidasCivilAnalysisReference/images/page-239_bfa12fef3e8aaa3080145486a569a49a1804b968b32872de25c47c61eef5b1d8.jpg new file mode 100644 index 00000000..57d68ebb Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-239_bfa12fef3e8aaa3080145486a569a49a1804b968b32872de25c47c61eef5b1d8.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-239_fb90eec126461bfe37872067b6002b296e81ac041754789d5562fd99bb0ce07f.jpg b/.raw/MidasCivilAnalysisReference/images/page-239_fb90eec126461bfe37872067b6002b296e81ac041754789d5562fd99bb0ce07f.jpg new file mode 100644 index 00000000..620aef76 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-239_fb90eec126461bfe37872067b6002b296e81ac041754789d5562fd99bb0ce07f.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-242_dc767c909d274f76b797dde06a5d47e3d1934c1f7e832b8782d4f3810fe306c2.jpg b/.raw/MidasCivilAnalysisReference/images/page-242_dc767c909d274f76b797dde06a5d47e3d1934c1f7e832b8782d4f3810fe306c2.jpg new file mode 100644 index 00000000..413f2856 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-242_dc767c909d274f76b797dde06a5d47e3d1934c1f7e832b8782d4f3810fe306c2.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-243_b40f210f79dd03ef7dc1314bb190338baa300fcf38644edff42e05232ee63e4d.jpg b/.raw/MidasCivilAnalysisReference/images/page-243_b40f210f79dd03ef7dc1314bb190338baa300fcf38644edff42e05232ee63e4d.jpg new file mode 100644 index 00000000..407e0d30 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-243_b40f210f79dd03ef7dc1314bb190338baa300fcf38644edff42e05232ee63e4d.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-244_33fef4e8dde5ebab8a95faa13eb19d37e5791131a4d668574843d2830f490987.jpg b/.raw/MidasCivilAnalysisReference/images/page-244_33fef4e8dde5ebab8a95faa13eb19d37e5791131a4d668574843d2830f490987.jpg new file mode 100644 index 00000000..8c28e8f2 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-244_33fef4e8dde5ebab8a95faa13eb19d37e5791131a4d668574843d2830f490987.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-245_2778959635e8fd1dac741a0bed8019b44714931b9af0250af8411cc1f76ee291.jpg b/.raw/MidasCivilAnalysisReference/images/page-245_2778959635e8fd1dac741a0bed8019b44714931b9af0250af8411cc1f76ee291.jpg new file mode 100644 index 00000000..1a97184d Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-245_2778959635e8fd1dac741a0bed8019b44714931b9af0250af8411cc1f76ee291.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-245_93a873259ece04b4873325ba65b48d170a17a32ac8d69963198ac3de88576eef.jpg b/.raw/MidasCivilAnalysisReference/images/page-245_93a873259ece04b4873325ba65b48d170a17a32ac8d69963198ac3de88576eef.jpg new file mode 100644 index 00000000..4bcc239f Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-245_93a873259ece04b4873325ba65b48d170a17a32ac8d69963198ac3de88576eef.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-245_fdd0eb657781e291e6121b3d7127202b98a4871abecf229eb50cb4a1f49c7c3c.jpg b/.raw/MidasCivilAnalysisReference/images/page-245_fdd0eb657781e291e6121b3d7127202b98a4871abecf229eb50cb4a1f49c7c3c.jpg new file mode 100644 index 00000000..e0ade91e Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-245_fdd0eb657781e291e6121b3d7127202b98a4871abecf229eb50cb4a1f49c7c3c.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-247_1a087d942a7f1818af4237c4b0f29a9d51fc8268cbb570125915e472ec460322.jpg b/.raw/MidasCivilAnalysisReference/images/page-247_1a087d942a7f1818af4237c4b0f29a9d51fc8268cbb570125915e472ec460322.jpg new file mode 100644 index 00000000..42df9512 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-247_1a087d942a7f1818af4237c4b0f29a9d51fc8268cbb570125915e472ec460322.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-247_398dc4ff8b15adbf04415d7276656239a0237c20993ec45e73b9f52b920d0cd8.jpg b/.raw/MidasCivilAnalysisReference/images/page-247_398dc4ff8b15adbf04415d7276656239a0237c20993ec45e73b9f52b920d0cd8.jpg new file mode 100644 index 00000000..5786e762 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-247_398dc4ff8b15adbf04415d7276656239a0237c20993ec45e73b9f52b920d0cd8.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-247_556dddc05a2b02c1b8a7ee5df432dae64fb8abd12b85a0ef29699463bf0c9100.jpg b/.raw/MidasCivilAnalysisReference/images/page-247_556dddc05a2b02c1b8a7ee5df432dae64fb8abd12b85a0ef29699463bf0c9100.jpg new file mode 100644 index 00000000..37fbef6f Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-247_556dddc05a2b02c1b8a7ee5df432dae64fb8abd12b85a0ef29699463bf0c9100.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-247_95e012dc83ab6236600e56600f4a94e3b35385712598b8b28653c0a44342dc0a.jpg b/.raw/MidasCivilAnalysisReference/images/page-247_95e012dc83ab6236600e56600f4a94e3b35385712598b8b28653c0a44342dc0a.jpg new file mode 100644 index 00000000..414b1b66 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-247_95e012dc83ab6236600e56600f4a94e3b35385712598b8b28653c0a44342dc0a.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-247_d5b61ad65334fd0df97d59baf33262e395b711c80cf8a7a2590e7b872f043b23.jpg b/.raw/MidasCivilAnalysisReference/images/page-247_d5b61ad65334fd0df97d59baf33262e395b711c80cf8a7a2590e7b872f043b23.jpg new file mode 100644 index 00000000..d78e36a2 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-247_d5b61ad65334fd0df97d59baf33262e395b711c80cf8a7a2590e7b872f043b23.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-248_21454721d3d5667168e624fc80ef41cd3d4e6dc5a98044453044e0f35d7b97ef.jpg b/.raw/MidasCivilAnalysisReference/images/page-248_21454721d3d5667168e624fc80ef41cd3d4e6dc5a98044453044e0f35d7b97ef.jpg new file mode 100644 index 00000000..d93f0eea Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-248_21454721d3d5667168e624fc80ef41cd3d4e6dc5a98044453044e0f35d7b97ef.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-248_25e40362f65ee620e6fa45924487777e91feae45030b18904dbb489ad0e8bd84.jpg b/.raw/MidasCivilAnalysisReference/images/page-248_25e40362f65ee620e6fa45924487777e91feae45030b18904dbb489ad0e8bd84.jpg new file mode 100644 index 00000000..9e87c448 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-248_25e40362f65ee620e6fa45924487777e91feae45030b18904dbb489ad0e8bd84.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-248_a3f5bce156c49651e5dcbde910206067d665da3eed8e5aadc3fb5089a56dc20f.jpg b/.raw/MidasCivilAnalysisReference/images/page-248_a3f5bce156c49651e5dcbde910206067d665da3eed8e5aadc3fb5089a56dc20f.jpg new file mode 100644 index 00000000..8de3ceb2 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-248_a3f5bce156c49651e5dcbde910206067d665da3eed8e5aadc3fb5089a56dc20f.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-248_c3b10c6a6f53a061c60e65c5cc1f3c910bcd0756be5cd62997213ba6d695d790.jpg b/.raw/MidasCivilAnalysisReference/images/page-248_c3b10c6a6f53a061c60e65c5cc1f3c910bcd0756be5cd62997213ba6d695d790.jpg new file mode 100644 index 00000000..23a56528 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-248_c3b10c6a6f53a061c60e65c5cc1f3c910bcd0756be5cd62997213ba6d695d790.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-249_b1aa300340258038b51ad2c844b8ad28395a1f2ab7409a978c41ee8d00fe669f.jpg b/.raw/MidasCivilAnalysisReference/images/page-249_b1aa300340258038b51ad2c844b8ad28395a1f2ab7409a978c41ee8d00fe669f.jpg new file mode 100644 index 00000000..fec1b9d5 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-249_b1aa300340258038b51ad2c844b8ad28395a1f2ab7409a978c41ee8d00fe669f.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-251_1bc2442505343c9fe4957d915a928e2f5172591da37a31474633a1e84afd4dcd.jpg b/.raw/MidasCivilAnalysisReference/images/page-251_1bc2442505343c9fe4957d915a928e2f5172591da37a31474633a1e84afd4dcd.jpg new file mode 100644 index 00000000..5ebf6ef6 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-251_1bc2442505343c9fe4957d915a928e2f5172591da37a31474633a1e84afd4dcd.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-251_2f3fa6a97a36bc365f71ded5832a6ba31fbbc89a648ef9bacda5103c5e039dec.jpg b/.raw/MidasCivilAnalysisReference/images/page-251_2f3fa6a97a36bc365f71ded5832a6ba31fbbc89a648ef9bacda5103c5e039dec.jpg new file mode 100644 index 00000000..987ed05e Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-251_2f3fa6a97a36bc365f71ded5832a6ba31fbbc89a648ef9bacda5103c5e039dec.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-251_2f828a627ecb70decceb3136b32384b4afb9c93408a83b6fac8fdcff9da1cfac.jpg b/.raw/MidasCivilAnalysisReference/images/page-251_2f828a627ecb70decceb3136b32384b4afb9c93408a83b6fac8fdcff9da1cfac.jpg new file mode 100644 index 00000000..9d026f7e Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-251_2f828a627ecb70decceb3136b32384b4afb9c93408a83b6fac8fdcff9da1cfac.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-251_4a283880d5f3458e3e00a0bea1639c05174bfd455779cfdaeda43a63b40f31d6.jpg b/.raw/MidasCivilAnalysisReference/images/page-251_4a283880d5f3458e3e00a0bea1639c05174bfd455779cfdaeda43a63b40f31d6.jpg new file mode 100644 index 00000000..2bc6e846 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-251_4a283880d5f3458e3e00a0bea1639c05174bfd455779cfdaeda43a63b40f31d6.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-251_530f5fce94cd5f7d3af18005b962fb7b0b82d828522c51055299256fa1897854.jpg b/.raw/MidasCivilAnalysisReference/images/page-251_530f5fce94cd5f7d3af18005b962fb7b0b82d828522c51055299256fa1897854.jpg new file mode 100644 index 00000000..8841828f Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-251_530f5fce94cd5f7d3af18005b962fb7b0b82d828522c51055299256fa1897854.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-251_dd1ae40099276684a20e38d7aae5761ef9c099ac59d1573f90bc020c4400a360.jpg b/.raw/MidasCivilAnalysisReference/images/page-251_dd1ae40099276684a20e38d7aae5761ef9c099ac59d1573f90bc020c4400a360.jpg new file mode 100644 index 00000000..41f2669f Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-251_dd1ae40099276684a20e38d7aae5761ef9c099ac59d1573f90bc020c4400a360.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-251_ea2735813773a91adc37f35ee56755d6db975b5a070fcb35a1fa10792a18b46b.jpg b/.raw/MidasCivilAnalysisReference/images/page-251_ea2735813773a91adc37f35ee56755d6db975b5a070fcb35a1fa10792a18b46b.jpg new file mode 100644 index 00000000..dd55bb21 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-251_ea2735813773a91adc37f35ee56755d6db975b5a070fcb35a1fa10792a18b46b.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-252_5c4147d811af03d954e8e067da83e663b1ffe0f3780873fd492c708b45c616c4.jpg b/.raw/MidasCivilAnalysisReference/images/page-252_5c4147d811af03d954e8e067da83e663b1ffe0f3780873fd492c708b45c616c4.jpg new file mode 100644 index 00000000..50c09320 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-252_5c4147d811af03d954e8e067da83e663b1ffe0f3780873fd492c708b45c616c4.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-252_eae3686475c9e4a6a0ce91a03b8e35aedd4b81236355a3c6102583cf14b1a7c9.jpg b/.raw/MidasCivilAnalysisReference/images/page-252_eae3686475c9e4a6a0ce91a03b8e35aedd4b81236355a3c6102583cf14b1a7c9.jpg new file mode 100644 index 00000000..e86da991 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-252_eae3686475c9e4a6a0ce91a03b8e35aedd4b81236355a3c6102583cf14b1a7c9.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-253_529d40b8f9296495db5980c59a0b45856ac48b6d33184148d77a71140c68f6cd.jpg b/.raw/MidasCivilAnalysisReference/images/page-253_529d40b8f9296495db5980c59a0b45856ac48b6d33184148d77a71140c68f6cd.jpg new file mode 100644 index 00000000..f7b8bc27 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-253_529d40b8f9296495db5980c59a0b45856ac48b6d33184148d77a71140c68f6cd.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-253_c0416b24ef0aaedb48975427cfe2e13f8c5633313b0b6c5d98862fafab779fd4.jpg b/.raw/MidasCivilAnalysisReference/images/page-253_c0416b24ef0aaedb48975427cfe2e13f8c5633313b0b6c5d98862fafab779fd4.jpg new file mode 100644 index 00000000..00c70000 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-253_c0416b24ef0aaedb48975427cfe2e13f8c5633313b0b6c5d98862fafab779fd4.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-253_dd633def22abf63249df01de10be7bf0dcf7c8788e08b70b291e1a102e0c586a.jpg b/.raw/MidasCivilAnalysisReference/images/page-253_dd633def22abf63249df01de10be7bf0dcf7c8788e08b70b291e1a102e0c586a.jpg new file mode 100644 index 00000000..cf527491 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-253_dd633def22abf63249df01de10be7bf0dcf7c8788e08b70b291e1a102e0c586a.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-254_04e47f55e2945afe2a4368a790e72f43d7b543dc8bbb969dafe6d84321afac89.jpg b/.raw/MidasCivilAnalysisReference/images/page-254_04e47f55e2945afe2a4368a790e72f43d7b543dc8bbb969dafe6d84321afac89.jpg new file mode 100644 index 00000000..142f0848 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-254_04e47f55e2945afe2a4368a790e72f43d7b543dc8bbb969dafe6d84321afac89.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-254_0894ac52f1f67e8b97f3c5599a6170200e155e1bfa965409eba1978849da83aa.jpg b/.raw/MidasCivilAnalysisReference/images/page-254_0894ac52f1f67e8b97f3c5599a6170200e155e1bfa965409eba1978849da83aa.jpg new file mode 100644 index 00000000..d4767962 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-254_0894ac52f1f67e8b97f3c5599a6170200e155e1bfa965409eba1978849da83aa.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-254_14384671ec935f6271007aa8f75d73950ace5f9c4eae30535716aabec362144d.jpg b/.raw/MidasCivilAnalysisReference/images/page-254_14384671ec935f6271007aa8f75d73950ace5f9c4eae30535716aabec362144d.jpg new file mode 100644 index 00000000..e6d267c0 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-254_14384671ec935f6271007aa8f75d73950ace5f9c4eae30535716aabec362144d.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-254_3fb78ed8e003ef8c274850e3facbb2ef421b15075a6cc497deebf24281a9c43f.jpg b/.raw/MidasCivilAnalysisReference/images/page-254_3fb78ed8e003ef8c274850e3facbb2ef421b15075a6cc497deebf24281a9c43f.jpg new file mode 100644 index 00000000..a74481df Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-254_3fb78ed8e003ef8c274850e3facbb2ef421b15075a6cc497deebf24281a9c43f.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-254_4b57a70fceb1e3d5c77be7e14d50cada329fa026cdb99176272ea950bbc4b9f3.jpg b/.raw/MidasCivilAnalysisReference/images/page-254_4b57a70fceb1e3d5c77be7e14d50cada329fa026cdb99176272ea950bbc4b9f3.jpg new file mode 100644 index 00000000..29de0b3d Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-254_4b57a70fceb1e3d5c77be7e14d50cada329fa026cdb99176272ea950bbc4b9f3.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-254_b5b8507567ad4e09fde5e3a78072cb6992cb642652b439c0f70ab3b99c80445a.jpg b/.raw/MidasCivilAnalysisReference/images/page-254_b5b8507567ad4e09fde5e3a78072cb6992cb642652b439c0f70ab3b99c80445a.jpg new file mode 100644 index 00000000..c0813e9b Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-254_b5b8507567ad4e09fde5e3a78072cb6992cb642652b439c0f70ab3b99c80445a.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-254_d857da8a2dc125d340e21f659e05157229ccc70b7dc1d01db7fbe1f30665a14e.jpg b/.raw/MidasCivilAnalysisReference/images/page-254_d857da8a2dc125d340e21f659e05157229ccc70b7dc1d01db7fbe1f30665a14e.jpg new file mode 100644 index 00000000..a1c0539b Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-254_d857da8a2dc125d340e21f659e05157229ccc70b7dc1d01db7fbe1f30665a14e.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-254_e04d86f6dbcf4412c28b8688b23cc04b496758ee2f227c592ed9a1b6379ce32e.jpg b/.raw/MidasCivilAnalysisReference/images/page-254_e04d86f6dbcf4412c28b8688b23cc04b496758ee2f227c592ed9a1b6379ce32e.jpg new file mode 100644 index 00000000..82a9a73f Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-254_e04d86f6dbcf4412c28b8688b23cc04b496758ee2f227c592ed9a1b6379ce32e.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-255_26198c933fcde7ef63d597bed9a2b18b4e57f726c6f1415c50a11868c7a23b64.jpg b/.raw/MidasCivilAnalysisReference/images/page-255_26198c933fcde7ef63d597bed9a2b18b4e57f726c6f1415c50a11868c7a23b64.jpg new file mode 100644 index 00000000..af1585e3 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-255_26198c933fcde7ef63d597bed9a2b18b4e57f726c6f1415c50a11868c7a23b64.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-255_40ef40430f7ee2f8f3de306f990a38a6f7d835815b1b760133c0394263ed3feb.jpg b/.raw/MidasCivilAnalysisReference/images/page-255_40ef40430f7ee2f8f3de306f990a38a6f7d835815b1b760133c0394263ed3feb.jpg new file mode 100644 index 00000000..237b96a7 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-255_40ef40430f7ee2f8f3de306f990a38a6f7d835815b1b760133c0394263ed3feb.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-255_834bf081bd764073f4e4d6daf1a55ec7178c4c65a3c12985c6d1de39d298a995.jpg b/.raw/MidasCivilAnalysisReference/images/page-255_834bf081bd764073f4e4d6daf1a55ec7178c4c65a3c12985c6d1de39d298a995.jpg new file mode 100644 index 00000000..3dedf5a0 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-255_834bf081bd764073f4e4d6daf1a55ec7178c4c65a3c12985c6d1de39d298a995.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-255_c01b1935ce30007f2a52bec194a6aa33842dc0c14e6f36848431120f6012d6c4.jpg b/.raw/MidasCivilAnalysisReference/images/page-255_c01b1935ce30007f2a52bec194a6aa33842dc0c14e6f36848431120f6012d6c4.jpg new file mode 100644 index 00000000..3b9d92a9 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-255_c01b1935ce30007f2a52bec194a6aa33842dc0c14e6f36848431120f6012d6c4.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-256_54c550eed5b06bffff3ba2bfe98d788d2e1314b938da2729de996d130addeda8.jpg b/.raw/MidasCivilAnalysisReference/images/page-256_54c550eed5b06bffff3ba2bfe98d788d2e1314b938da2729de996d130addeda8.jpg new file mode 100644 index 00000000..76590fb2 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-256_54c550eed5b06bffff3ba2bfe98d788d2e1314b938da2729de996d130addeda8.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-256_f0db9a5661221aa34696db45f3a79497ada26c184b215e204459d08887efa6fa.jpg b/.raw/MidasCivilAnalysisReference/images/page-256_f0db9a5661221aa34696db45f3a79497ada26c184b215e204459d08887efa6fa.jpg new file mode 100644 index 00000000..e9eabc4d Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-256_f0db9a5661221aa34696db45f3a79497ada26c184b215e204459d08887efa6fa.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-257_04febd40717b1a5c9129deb179578bf07c3d3f0a57e29935a2de23c3a2843dd3.jpg b/.raw/MidasCivilAnalysisReference/images/page-257_04febd40717b1a5c9129deb179578bf07c3d3f0a57e29935a2de23c3a2843dd3.jpg new file mode 100644 index 00000000..d7926c51 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-257_04febd40717b1a5c9129deb179578bf07c3d3f0a57e29935a2de23c3a2843dd3.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-257_b2e54033e5e20170042b43cf0f7fccdbce623e106de5b9644ab5fef00b8b4041.jpg b/.raw/MidasCivilAnalysisReference/images/page-257_b2e54033e5e20170042b43cf0f7fccdbce623e106de5b9644ab5fef00b8b4041.jpg new file mode 100644 index 00000000..2d2cd829 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-257_b2e54033e5e20170042b43cf0f7fccdbce623e106de5b9644ab5fef00b8b4041.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-257_f5044e83274f6f83ba7e91dc57daa64593d7c1ca0f437a3f898dc67fade853e2.jpg b/.raw/MidasCivilAnalysisReference/images/page-257_f5044e83274f6f83ba7e91dc57daa64593d7c1ca0f437a3f898dc67fade853e2.jpg new file mode 100644 index 00000000..44f5f66c Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-257_f5044e83274f6f83ba7e91dc57daa64593d7c1ca0f437a3f898dc67fade853e2.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-258_052415997eeebae770bea2c2d5c484dff113ee1626fc8062a7ee5f408800ffba.jpg b/.raw/MidasCivilAnalysisReference/images/page-258_052415997eeebae770bea2c2d5c484dff113ee1626fc8062a7ee5f408800ffba.jpg new file mode 100644 index 00000000..17757f79 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-258_052415997eeebae770bea2c2d5c484dff113ee1626fc8062a7ee5f408800ffba.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-258_593e63df3d6b0bdc7309f29f9d64188ea58115a3d0ff4b72f02a1abd354c40ba.jpg b/.raw/MidasCivilAnalysisReference/images/page-258_593e63df3d6b0bdc7309f29f9d64188ea58115a3d0ff4b72f02a1abd354c40ba.jpg new file mode 100644 index 00000000..34f1a029 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-258_593e63df3d6b0bdc7309f29f9d64188ea58115a3d0ff4b72f02a1abd354c40ba.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-258_d6b536c70d0d69b2ae7e3b57542a14cf821524897f159f3f36613bf1d65402d1.jpg b/.raw/MidasCivilAnalysisReference/images/page-258_d6b536c70d0d69b2ae7e3b57542a14cf821524897f159f3f36613bf1d65402d1.jpg new file mode 100644 index 00000000..3cfc81f0 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-258_d6b536c70d0d69b2ae7e3b57542a14cf821524897f159f3f36613bf1d65402d1.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-259_56ce5ea0c6a7154ffaf286030323634f5eb989133172ab5f1d39acd92978bda1.jpg b/.raw/MidasCivilAnalysisReference/images/page-259_56ce5ea0c6a7154ffaf286030323634f5eb989133172ab5f1d39acd92978bda1.jpg new file mode 100644 index 00000000..147a814f Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-259_56ce5ea0c6a7154ffaf286030323634f5eb989133172ab5f1d39acd92978bda1.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-259_93d9bf2c61f1bc088ebdc5dc0fbf39779972c26d840f77a0ef9ae86c900f1ae5.jpg b/.raw/MidasCivilAnalysisReference/images/page-259_93d9bf2c61f1bc088ebdc5dc0fbf39779972c26d840f77a0ef9ae86c900f1ae5.jpg new file mode 100644 index 00000000..5b51a58f Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-259_93d9bf2c61f1bc088ebdc5dc0fbf39779972c26d840f77a0ef9ae86c900f1ae5.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-260_36521070eec7c8ab38e71903e506b4e57c73e900efd3eb2ad5b2263bfcfa03a0.jpg b/.raw/MidasCivilAnalysisReference/images/page-260_36521070eec7c8ab38e71903e506b4e57c73e900efd3eb2ad5b2263bfcfa03a0.jpg new file mode 100644 index 00000000..da74bf5d Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-260_36521070eec7c8ab38e71903e506b4e57c73e900efd3eb2ad5b2263bfcfa03a0.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-260_5799dab032d9fa71878fb1c364b96fe78f843d76a0f3efa96153cd3c6f6cb195.jpg b/.raw/MidasCivilAnalysisReference/images/page-260_5799dab032d9fa71878fb1c364b96fe78f843d76a0f3efa96153cd3c6f6cb195.jpg new file mode 100644 index 00000000..f2877d76 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-260_5799dab032d9fa71878fb1c364b96fe78f843d76a0f3efa96153cd3c6f6cb195.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-260_5e9e715589697d6d78d7231b90d5999189faa9b58cb1bf290d3afe9f4ac17766.jpg b/.raw/MidasCivilAnalysisReference/images/page-260_5e9e715589697d6d78d7231b90d5999189faa9b58cb1bf290d3afe9f4ac17766.jpg new file mode 100644 index 00000000..a345194f Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-260_5e9e715589697d6d78d7231b90d5999189faa9b58cb1bf290d3afe9f4ac17766.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-261_0ce0dcec42247e74da6db7a2771c805d264c7f7af465d9a5d1a6507b6d77d86a.jpg b/.raw/MidasCivilAnalysisReference/images/page-261_0ce0dcec42247e74da6db7a2771c805d264c7f7af465d9a5d1a6507b6d77d86a.jpg new file mode 100644 index 00000000..69da6760 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-261_0ce0dcec42247e74da6db7a2771c805d264c7f7af465d9a5d1a6507b6d77d86a.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-261_64ac4d67b0fc6c8cf891914c068947e7aa431d52c511bb926e4543281aa0f510.jpg b/.raw/MidasCivilAnalysisReference/images/page-261_64ac4d67b0fc6c8cf891914c068947e7aa431d52c511bb926e4543281aa0f510.jpg new file mode 100644 index 00000000..ee82ea43 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-261_64ac4d67b0fc6c8cf891914c068947e7aa431d52c511bb926e4543281aa0f510.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-261_7d1042ff6872468f38a1a1a10bbba1fb75588a2454fc7eca3307a8073af87aaa.jpg b/.raw/MidasCivilAnalysisReference/images/page-261_7d1042ff6872468f38a1a1a10bbba1fb75588a2454fc7eca3307a8073af87aaa.jpg new file mode 100644 index 00000000..66f87d3e Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-261_7d1042ff6872468f38a1a1a10bbba1fb75588a2454fc7eca3307a8073af87aaa.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-261_adb97aa5e811fac84b067d0b4496d222e31d2ae92c2245a76daa58d931a8781f.jpg b/.raw/MidasCivilAnalysisReference/images/page-261_adb97aa5e811fac84b067d0b4496d222e31d2ae92c2245a76daa58d931a8781f.jpg new file mode 100644 index 00000000..134672b4 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-261_adb97aa5e811fac84b067d0b4496d222e31d2ae92c2245a76daa58d931a8781f.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-262_a4f3bc17de53f9647f91a21b7713fc33bf329b08a0b6149729cbe4d0fbb0c3e3.jpg b/.raw/MidasCivilAnalysisReference/images/page-262_a4f3bc17de53f9647f91a21b7713fc33bf329b08a0b6149729cbe4d0fbb0c3e3.jpg new file mode 100644 index 00000000..4396559e Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-262_a4f3bc17de53f9647f91a21b7713fc33bf329b08a0b6149729cbe4d0fbb0c3e3.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-262_b60f6e5b4cb105bc553d64ca6b74ac3d62a0956e00e1d0affacb6e4e44f8e984.jpg b/.raw/MidasCivilAnalysisReference/images/page-262_b60f6e5b4cb105bc553d64ca6b74ac3d62a0956e00e1d0affacb6e4e44f8e984.jpg new file mode 100644 index 00000000..8fef0967 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-262_b60f6e5b4cb105bc553d64ca6b74ac3d62a0956e00e1d0affacb6e4e44f8e984.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-262_dbfc06e459bd66133e6f527e3869129bfae511726160e0980593c97ae0c53b07.jpg b/.raw/MidasCivilAnalysisReference/images/page-262_dbfc06e459bd66133e6f527e3869129bfae511726160e0980593c97ae0c53b07.jpg new file mode 100644 index 00000000..7ea76573 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-262_dbfc06e459bd66133e6f527e3869129bfae511726160e0980593c97ae0c53b07.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-263_9f56ccc196697f04c709ae99969a372bb34bea0fcf7c19f607eb7baa5846ab11.jpg b/.raw/MidasCivilAnalysisReference/images/page-263_9f56ccc196697f04c709ae99969a372bb34bea0fcf7c19f607eb7baa5846ab11.jpg new file mode 100644 index 00000000..8d20314f Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-263_9f56ccc196697f04c709ae99969a372bb34bea0fcf7c19f607eb7baa5846ab11.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-264_9085f407eaae8ff7c18a8d5b90188738b077e56958ed7511f8a89cbcb4f6170b.jpg b/.raw/MidasCivilAnalysisReference/images/page-264_9085f407eaae8ff7c18a8d5b90188738b077e56958ed7511f8a89cbcb4f6170b.jpg new file mode 100644 index 00000000..f91f3946 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-264_9085f407eaae8ff7c18a8d5b90188738b077e56958ed7511f8a89cbcb4f6170b.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-265_0de65a56a734513b72e3696c68198b6226d1de6708e37fd3d58795776b2cc0e2.jpg b/.raw/MidasCivilAnalysisReference/images/page-265_0de65a56a734513b72e3696c68198b6226d1de6708e37fd3d58795776b2cc0e2.jpg new file mode 100644 index 00000000..0d4fc0ac Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-265_0de65a56a734513b72e3696c68198b6226d1de6708e37fd3d58795776b2cc0e2.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-265_98ac7f053363f19128333b5f379ba0dfb212140373d1e6075d4abbebbd5a9229.jpg b/.raw/MidasCivilAnalysisReference/images/page-265_98ac7f053363f19128333b5f379ba0dfb212140373d1e6075d4abbebbd5a9229.jpg new file mode 100644 index 00000000..13838053 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-265_98ac7f053363f19128333b5f379ba0dfb212140373d1e6075d4abbebbd5a9229.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-265_aacc39208ace0697fe57aa9a1044c7df321b2e049e102388579bb11d4ef5f296.jpg b/.raw/MidasCivilAnalysisReference/images/page-265_aacc39208ace0697fe57aa9a1044c7df321b2e049e102388579bb11d4ef5f296.jpg new file mode 100644 index 00000000..6e1e12a2 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-265_aacc39208ace0697fe57aa9a1044c7df321b2e049e102388579bb11d4ef5f296.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-265_fa1d5bba3287a1aa3bc0bdc9d110683b40268c6e260d5606617494691a181d30.jpg b/.raw/MidasCivilAnalysisReference/images/page-265_fa1d5bba3287a1aa3bc0bdc9d110683b40268c6e260d5606617494691a181d30.jpg new file mode 100644 index 00000000..b26e4a19 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-265_fa1d5bba3287a1aa3bc0bdc9d110683b40268c6e260d5606617494691a181d30.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-266_84ae1806bb78a44a957a7c0db135da0138a8f22fe42e80a40a8652b07fcd7213.jpg b/.raw/MidasCivilAnalysisReference/images/page-266_84ae1806bb78a44a957a7c0db135da0138a8f22fe42e80a40a8652b07fcd7213.jpg new file mode 100644 index 00000000..f99b25a5 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-266_84ae1806bb78a44a957a7c0db135da0138a8f22fe42e80a40a8652b07fcd7213.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-267_c5934d1c8f22afda8e6507d1316d467174e2e6442d3f72266475f653b0f1712d.jpg b/.raw/MidasCivilAnalysisReference/images/page-267_c5934d1c8f22afda8e6507d1316d467174e2e6442d3f72266475f653b0f1712d.jpg new file mode 100644 index 00000000..fe608407 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-267_c5934d1c8f22afda8e6507d1316d467174e2e6442d3f72266475f653b0f1712d.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-268_312a4e65d0d5739ded25eb6630f3d9496e1b5baf69d239c516e71ee5984b044e.jpg b/.raw/MidasCivilAnalysisReference/images/page-268_312a4e65d0d5739ded25eb6630f3d9496e1b5baf69d239c516e71ee5984b044e.jpg new file mode 100644 index 00000000..dc04d12b Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-268_312a4e65d0d5739ded25eb6630f3d9496e1b5baf69d239c516e71ee5984b044e.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-268_d7a02f0c8f74a5192974668585e28886d614211d3cfb06a4f5d1afee1602d9e1.jpg b/.raw/MidasCivilAnalysisReference/images/page-268_d7a02f0c8f74a5192974668585e28886d614211d3cfb06a4f5d1afee1602d9e1.jpg new file mode 100644 index 00000000..deebb78e Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-268_d7a02f0c8f74a5192974668585e28886d614211d3cfb06a4f5d1afee1602d9e1.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-269_9c01af34624cace33c209eeb45bc42a0dd61d896cd3d3ecc9acfa706a2f3a20d.jpg b/.raw/MidasCivilAnalysisReference/images/page-269_9c01af34624cace33c209eeb45bc42a0dd61d896cd3d3ecc9acfa706a2f3a20d.jpg new file mode 100644 index 00000000..4d34d661 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-269_9c01af34624cace33c209eeb45bc42a0dd61d896cd3d3ecc9acfa706a2f3a20d.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-269_9eb171f2c9016fae17d001f49c1f776fa8453aa368742cc8cfafe8ad96617775.jpg b/.raw/MidasCivilAnalysisReference/images/page-269_9eb171f2c9016fae17d001f49c1f776fa8453aa368742cc8cfafe8ad96617775.jpg new file mode 100644 index 00000000..28c5a8ea Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-269_9eb171f2c9016fae17d001f49c1f776fa8453aa368742cc8cfafe8ad96617775.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-270_d9d2d9517ddfa29cf9cda520278b910d21c200999db1f80b4b805987accd6d41.jpg b/.raw/MidasCivilAnalysisReference/images/page-270_d9d2d9517ddfa29cf9cda520278b910d21c200999db1f80b4b805987accd6d41.jpg new file mode 100644 index 00000000..981acf49 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-270_d9d2d9517ddfa29cf9cda520278b910d21c200999db1f80b4b805987accd6d41.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-271_e6a9b9efe3cbb31d266a1696d74c7f5a546ff760d19af0f5891c1ec833fab450.jpg b/.raw/MidasCivilAnalysisReference/images/page-271_e6a9b9efe3cbb31d266a1696d74c7f5a546ff760d19af0f5891c1ec833fab450.jpg new file mode 100644 index 00000000..e5583079 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-271_e6a9b9efe3cbb31d266a1696d74c7f5a546ff760d19af0f5891c1ec833fab450.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-272_01ff5d2a2e17b97d878a03f4af30a20f3542d6fd59d3c28ec5768b92cfbf28e3.jpg b/.raw/MidasCivilAnalysisReference/images/page-272_01ff5d2a2e17b97d878a03f4af30a20f3542d6fd59d3c28ec5768b92cfbf28e3.jpg new file mode 100644 index 00000000..13b6a9ab Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-272_01ff5d2a2e17b97d878a03f4af30a20f3542d6fd59d3c28ec5768b92cfbf28e3.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-272_766cf4702ae53899f1420f7c69bdfde1f5fffb2cc1653631713954d45337e412.jpg b/.raw/MidasCivilAnalysisReference/images/page-272_766cf4702ae53899f1420f7c69bdfde1f5fffb2cc1653631713954d45337e412.jpg new file mode 100644 index 00000000..45265471 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-272_766cf4702ae53899f1420f7c69bdfde1f5fffb2cc1653631713954d45337e412.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-273_9dfe176ebbcbc728e784517de92e2b994a65e22407a8379b910da750d04f30db.jpg b/.raw/MidasCivilAnalysisReference/images/page-273_9dfe176ebbcbc728e784517de92e2b994a65e22407a8379b910da750d04f30db.jpg new file mode 100644 index 00000000..006e5eea Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-273_9dfe176ebbcbc728e784517de92e2b994a65e22407a8379b910da750d04f30db.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-273_cd57387ba5cee58485d6c1f479e3d2aa18f0d78aa8fef5b075df7a9741d3a4c3.jpg b/.raw/MidasCivilAnalysisReference/images/page-273_cd57387ba5cee58485d6c1f479e3d2aa18f0d78aa8fef5b075df7a9741d3a4c3.jpg new file mode 100644 index 00000000..38662fca Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-273_cd57387ba5cee58485d6c1f479e3d2aa18f0d78aa8fef5b075df7a9741d3a4c3.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-274_d63aba2db249aa9e0ab7ed7096cb998b78faee301896ab85eb364f25fdfded88.jpg b/.raw/MidasCivilAnalysisReference/images/page-274_d63aba2db249aa9e0ab7ed7096cb998b78faee301896ab85eb364f25fdfded88.jpg new file mode 100644 index 00000000..05943783 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-274_d63aba2db249aa9e0ab7ed7096cb998b78faee301896ab85eb364f25fdfded88.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-275_3517fd673d5d613278f3438882604b0992b5c2e90d11fc8e172447c6316aeea1.jpg b/.raw/MidasCivilAnalysisReference/images/page-275_3517fd673d5d613278f3438882604b0992b5c2e90d11fc8e172447c6316aeea1.jpg new file mode 100644 index 00000000..8aefc836 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-275_3517fd673d5d613278f3438882604b0992b5c2e90d11fc8e172447c6316aeea1.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-275_46ac025acc58dd89b2e9c5148b7fe32117cc5b4ccaeb7c0ffc4c494d02244821.jpg b/.raw/MidasCivilAnalysisReference/images/page-275_46ac025acc58dd89b2e9c5148b7fe32117cc5b4ccaeb7c0ffc4c494d02244821.jpg new file mode 100644 index 00000000..80bab659 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-275_46ac025acc58dd89b2e9c5148b7fe32117cc5b4ccaeb7c0ffc4c494d02244821.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-275_9e1880f7596020191be77ac7d6df10155acada59d428887835ffda21aae76154.jpg b/.raw/MidasCivilAnalysisReference/images/page-275_9e1880f7596020191be77ac7d6df10155acada59d428887835ffda21aae76154.jpg new file mode 100644 index 00000000..163dd570 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-275_9e1880f7596020191be77ac7d6df10155acada59d428887835ffda21aae76154.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-275_b8a3df997274a12bfc51cda3fefe4370bee880f3fb3a78b3a77984184de8c1fd.jpg b/.raw/MidasCivilAnalysisReference/images/page-275_b8a3df997274a12bfc51cda3fefe4370bee880f3fb3a78b3a77984184de8c1fd.jpg new file mode 100644 index 00000000..39502df9 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-275_b8a3df997274a12bfc51cda3fefe4370bee880f3fb3a78b3a77984184de8c1fd.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-276_4196658b8f2b4ad063772837d9d2a8e982ae4f9ab2d2ffa0a51d6fce780f1de2.jpg b/.raw/MidasCivilAnalysisReference/images/page-276_4196658b8f2b4ad063772837d9d2a8e982ae4f9ab2d2ffa0a51d6fce780f1de2.jpg new file mode 100644 index 00000000..7cf9053c Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-276_4196658b8f2b4ad063772837d9d2a8e982ae4f9ab2d2ffa0a51d6fce780f1de2.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-276_e041a6e6d33679a0b55f4684ab25615d521f92bdff868ef2dc48d5e3b90a7d99.jpg b/.raw/MidasCivilAnalysisReference/images/page-276_e041a6e6d33679a0b55f4684ab25615d521f92bdff868ef2dc48d5e3b90a7d99.jpg new file mode 100644 index 00000000..4c0ab9d0 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-276_e041a6e6d33679a0b55f4684ab25615d521f92bdff868ef2dc48d5e3b90a7d99.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-277_3251c3023d3c9fb7d196530d10c555b898f6aa663c5423239a13310b44c53931.jpg b/.raw/MidasCivilAnalysisReference/images/page-277_3251c3023d3c9fb7d196530d10c555b898f6aa663c5423239a13310b44c53931.jpg new file mode 100644 index 00000000..fd07c91b Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-277_3251c3023d3c9fb7d196530d10c555b898f6aa663c5423239a13310b44c53931.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-277_845a4ea7955c088ccc8798cd9f0b7cc9c8664e5a45623433420b71bb6349680e.jpg b/.raw/MidasCivilAnalysisReference/images/page-277_845a4ea7955c088ccc8798cd9f0b7cc9c8664e5a45623433420b71bb6349680e.jpg new file mode 100644 index 00000000..d8ff3dcd Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-277_845a4ea7955c088ccc8798cd9f0b7cc9c8664e5a45623433420b71bb6349680e.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-278_7b81882e4f757bcc1e9288745495800413a006d5766f3fd081f995331655af18.jpg b/.raw/MidasCivilAnalysisReference/images/page-278_7b81882e4f757bcc1e9288745495800413a006d5766f3fd081f995331655af18.jpg new file mode 100644 index 00000000..398b04df Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-278_7b81882e4f757bcc1e9288745495800413a006d5766f3fd081f995331655af18.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-278_d7e19527779524c8c141b1c0c12fb6d48d2af6c3827f92a02fd6f93906ced96c.jpg b/.raw/MidasCivilAnalysisReference/images/page-278_d7e19527779524c8c141b1c0c12fb6d48d2af6c3827f92a02fd6f93906ced96c.jpg new file mode 100644 index 00000000..0e85b44f Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-278_d7e19527779524c8c141b1c0c12fb6d48d2af6c3827f92a02fd6f93906ced96c.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-278_e71cb393cfa7661ddd2f10b51ccd3ac93c8a6bfac25f6ea8f7c535f6d48cc23f.jpg b/.raw/MidasCivilAnalysisReference/images/page-278_e71cb393cfa7661ddd2f10b51ccd3ac93c8a6bfac25f6ea8f7c535f6d48cc23f.jpg new file mode 100644 index 00000000..c7adeb4a Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-278_e71cb393cfa7661ddd2f10b51ccd3ac93c8a6bfac25f6ea8f7c535f6d48cc23f.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-279_0f79bcdd88209740449b9aa950fb55492c7c9d8efa445ac3a7dfe719b9b0b47a.jpg b/.raw/MidasCivilAnalysisReference/images/page-279_0f79bcdd88209740449b9aa950fb55492c7c9d8efa445ac3a7dfe719b9b0b47a.jpg new file mode 100644 index 00000000..92933b5e Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-279_0f79bcdd88209740449b9aa950fb55492c7c9d8efa445ac3a7dfe719b9b0b47a.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-279_2450213fbd9be0e070e8846cafaef36bed7499d169c5c73c8827bef23efa85d1.jpg b/.raw/MidasCivilAnalysisReference/images/page-279_2450213fbd9be0e070e8846cafaef36bed7499d169c5c73c8827bef23efa85d1.jpg new file mode 100644 index 00000000..bfc55704 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-279_2450213fbd9be0e070e8846cafaef36bed7499d169c5c73c8827bef23efa85d1.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-279_91b86249485ff3fb4db0588f19d982e19f777760a86223eae89b30f4177e042a.jpg b/.raw/MidasCivilAnalysisReference/images/page-279_91b86249485ff3fb4db0588f19d982e19f777760a86223eae89b30f4177e042a.jpg new file mode 100644 index 00000000..1af1b5aa Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-279_91b86249485ff3fb4db0588f19d982e19f777760a86223eae89b30f4177e042a.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-279_ec1e3a33ad447082424f066c8679db1fd1a760bca00beb22fa65888b46a6aa9d.jpg b/.raw/MidasCivilAnalysisReference/images/page-279_ec1e3a33ad447082424f066c8679db1fd1a760bca00beb22fa65888b46a6aa9d.jpg new file mode 100644 index 00000000..27db1385 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-279_ec1e3a33ad447082424f066c8679db1fd1a760bca00beb22fa65888b46a6aa9d.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-280_a3d47c6a4d3e1df2332e5feefddff06608d5743aff48b2f20ab8b18449b8dbbe.jpg b/.raw/MidasCivilAnalysisReference/images/page-280_a3d47c6a4d3e1df2332e5feefddff06608d5743aff48b2f20ab8b18449b8dbbe.jpg new file mode 100644 index 00000000..db5a5654 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-280_a3d47c6a4d3e1df2332e5feefddff06608d5743aff48b2f20ab8b18449b8dbbe.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-280_abfc0adff701d60d535e7c59fa40716c6c2c1153e2c3c94eac884bd14d26eaff.jpg b/.raw/MidasCivilAnalysisReference/images/page-280_abfc0adff701d60d535e7c59fa40716c6c2c1153e2c3c94eac884bd14d26eaff.jpg new file mode 100644 index 00000000..19250a55 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-280_abfc0adff701d60d535e7c59fa40716c6c2c1153e2c3c94eac884bd14d26eaff.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-280_c1794aa4b76b1af9de71b2f08d8696ee8abd863e03eddbd9861dbbe74c772f03.jpg b/.raw/MidasCivilAnalysisReference/images/page-280_c1794aa4b76b1af9de71b2f08d8696ee8abd863e03eddbd9861dbbe74c772f03.jpg new file mode 100644 index 00000000..4b1f1d0f Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-280_c1794aa4b76b1af9de71b2f08d8696ee8abd863e03eddbd9861dbbe74c772f03.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-280_f2a738096bb60cd9635c522e78360479059127d3e8451fa707d08a25388ef6b3.jpg b/.raw/MidasCivilAnalysisReference/images/page-280_f2a738096bb60cd9635c522e78360479059127d3e8451fa707d08a25388ef6b3.jpg new file mode 100644 index 00000000..4bc14db5 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-280_f2a738096bb60cd9635c522e78360479059127d3e8451fa707d08a25388ef6b3.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-281_451a4e50cf2103fca70f190b78fb8b92dde882b4a38faf5d7eff3a19aa92d9ac.jpg b/.raw/MidasCivilAnalysisReference/images/page-281_451a4e50cf2103fca70f190b78fb8b92dde882b4a38faf5d7eff3a19aa92d9ac.jpg new file mode 100644 index 00000000..2e72c165 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-281_451a4e50cf2103fca70f190b78fb8b92dde882b4a38faf5d7eff3a19aa92d9ac.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-281_d2ed9b474eb73a5fa4b613380cbdf300ab06eb3208b3ca050a008edf3b3ea6b2.jpg b/.raw/MidasCivilAnalysisReference/images/page-281_d2ed9b474eb73a5fa4b613380cbdf300ab06eb3208b3ca050a008edf3b3ea6b2.jpg new file mode 100644 index 00000000..b42ebbfd Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-281_d2ed9b474eb73a5fa4b613380cbdf300ab06eb3208b3ca050a008edf3b3ea6b2.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-281_e444f40cf8e76dfc2c93292573909f7c62ee95a30d696dab49d9fd716e2b28e9.jpg b/.raw/MidasCivilAnalysisReference/images/page-281_e444f40cf8e76dfc2c93292573909f7c62ee95a30d696dab49d9fd716e2b28e9.jpg new file mode 100644 index 00000000..b97625f4 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-281_e444f40cf8e76dfc2c93292573909f7c62ee95a30d696dab49d9fd716e2b28e9.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-282_3b2aa48c9f3444727c7e075cfc4b34d5846fed78ee18c44ae14543c28dcaab4c.jpg b/.raw/MidasCivilAnalysisReference/images/page-282_3b2aa48c9f3444727c7e075cfc4b34d5846fed78ee18c44ae14543c28dcaab4c.jpg new file mode 100644 index 00000000..d86090cb Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-282_3b2aa48c9f3444727c7e075cfc4b34d5846fed78ee18c44ae14543c28dcaab4c.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-282_9da900dcbb29da649b07bca77be4b423962781d43b9289f960d266b4f97535a9.jpg b/.raw/MidasCivilAnalysisReference/images/page-282_9da900dcbb29da649b07bca77be4b423962781d43b9289f960d266b4f97535a9.jpg new file mode 100644 index 00000000..5cb14de2 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-282_9da900dcbb29da649b07bca77be4b423962781d43b9289f960d266b4f97535a9.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-282_f273ee0bdb43789fe0c4bb9d085cacf86f5c5d1d2f4cce3384df013983bd24d5.jpg b/.raw/MidasCivilAnalysisReference/images/page-282_f273ee0bdb43789fe0c4bb9d085cacf86f5c5d1d2f4cce3384df013983bd24d5.jpg new file mode 100644 index 00000000..76b0b3d7 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-282_f273ee0bdb43789fe0c4bb9d085cacf86f5c5d1d2f4cce3384df013983bd24d5.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-283_2b604a6e9078d93f3d60ffe2367c3ed9907b6a5bc539313f3daf99a482f51441.jpg b/.raw/MidasCivilAnalysisReference/images/page-283_2b604a6e9078d93f3d60ffe2367c3ed9907b6a5bc539313f3daf99a482f51441.jpg new file mode 100644 index 00000000..f3a52ea3 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-283_2b604a6e9078d93f3d60ffe2367c3ed9907b6a5bc539313f3daf99a482f51441.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-283_c4ca34abc3529b7ac653c2b9ec77448c78554af19df377339d0394539779ccab.jpg b/.raw/MidasCivilAnalysisReference/images/page-283_c4ca34abc3529b7ac653c2b9ec77448c78554af19df377339d0394539779ccab.jpg new file mode 100644 index 00000000..8b6a2722 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-283_c4ca34abc3529b7ac653c2b9ec77448c78554af19df377339d0394539779ccab.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-284_1df0d6dbd3c9c2e3ff1bec0dc1b29d29533db973a1abdf3fe536a223a70b171d.jpg b/.raw/MidasCivilAnalysisReference/images/page-284_1df0d6dbd3c9c2e3ff1bec0dc1b29d29533db973a1abdf3fe536a223a70b171d.jpg new file mode 100644 index 00000000..1bdeeb8d Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-284_1df0d6dbd3c9c2e3ff1bec0dc1b29d29533db973a1abdf3fe536a223a70b171d.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-284_2f5e4e7de78dc80dbe2d8312dac0e50a160c9fd8c2289defb7e764055e136da9.jpg b/.raw/MidasCivilAnalysisReference/images/page-284_2f5e4e7de78dc80dbe2d8312dac0e50a160c9fd8c2289defb7e764055e136da9.jpg new file mode 100644 index 00000000..c2bc2755 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-284_2f5e4e7de78dc80dbe2d8312dac0e50a160c9fd8c2289defb7e764055e136da9.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-284_bc02c003ebc202ff42db969abe7ccc9b3c48d95ceef374d6e7383a715899fd3f.jpg b/.raw/MidasCivilAnalysisReference/images/page-284_bc02c003ebc202ff42db969abe7ccc9b3c48d95ceef374d6e7383a715899fd3f.jpg new file mode 100644 index 00000000..62ea9e30 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-284_bc02c003ebc202ff42db969abe7ccc9b3c48d95ceef374d6e7383a715899fd3f.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-284_f7dbdae4dd860b3efaf44ce08c695587d9ae522b4b7c40a72ec2c0edd503063a.jpg b/.raw/MidasCivilAnalysisReference/images/page-284_f7dbdae4dd860b3efaf44ce08c695587d9ae522b4b7c40a72ec2c0edd503063a.jpg new file mode 100644 index 00000000..5ceaf91b Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-284_f7dbdae4dd860b3efaf44ce08c695587d9ae522b4b7c40a72ec2c0edd503063a.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-285_35efdf394cf96d1662710268b5a6afab5f76c86d4055eabba33e44a49b9e2a72.jpg b/.raw/MidasCivilAnalysisReference/images/page-285_35efdf394cf96d1662710268b5a6afab5f76c86d4055eabba33e44a49b9e2a72.jpg new file mode 100644 index 00000000..e2fc6ba1 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-285_35efdf394cf96d1662710268b5a6afab5f76c86d4055eabba33e44a49b9e2a72.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-285_586b1f757113a2d348dfa9bae09bf5e9c1e1458c93cf36e244b7356a545e807d.jpg b/.raw/MidasCivilAnalysisReference/images/page-285_586b1f757113a2d348dfa9bae09bf5e9c1e1458c93cf36e244b7356a545e807d.jpg new file mode 100644 index 00000000..e23b9d02 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-285_586b1f757113a2d348dfa9bae09bf5e9c1e1458c93cf36e244b7356a545e807d.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-285_8121c7a411d293dccc00e94d993799d344d254f18a5b6cd41e80d431674c365f.jpg b/.raw/MidasCivilAnalysisReference/images/page-285_8121c7a411d293dccc00e94d993799d344d254f18a5b6cd41e80d431674c365f.jpg new file mode 100644 index 00000000..7418c178 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-285_8121c7a411d293dccc00e94d993799d344d254f18a5b6cd41e80d431674c365f.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-285_f277754f829a4aeb44ead48e1c02f4629397bf416263b6d39ba09fbbfb6309c6.jpg b/.raw/MidasCivilAnalysisReference/images/page-285_f277754f829a4aeb44ead48e1c02f4629397bf416263b6d39ba09fbbfb6309c6.jpg new file mode 100644 index 00000000..04e50a4c Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-285_f277754f829a4aeb44ead48e1c02f4629397bf416263b6d39ba09fbbfb6309c6.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-286_062d3bf8f3332d9d8585a00e27c9ca6eb592034956935ab8c5423f48a52a5382.jpg b/.raw/MidasCivilAnalysisReference/images/page-286_062d3bf8f3332d9d8585a00e27c9ca6eb592034956935ab8c5423f48a52a5382.jpg new file mode 100644 index 00000000..e1fc48b4 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-286_062d3bf8f3332d9d8585a00e27c9ca6eb592034956935ab8c5423f48a52a5382.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-286_5b5e613100b1b52133811c65ed6134d1267ec21e53fe0757429f869bccfcb60b.jpg b/.raw/MidasCivilAnalysisReference/images/page-286_5b5e613100b1b52133811c65ed6134d1267ec21e53fe0757429f869bccfcb60b.jpg new file mode 100644 index 00000000..609c7133 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-286_5b5e613100b1b52133811c65ed6134d1267ec21e53fe0757429f869bccfcb60b.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-286_8adecef3d416a749dea945fa9c4f943d1b90dd1c77d5477f2869f90b37b04101.jpg b/.raw/MidasCivilAnalysisReference/images/page-286_8adecef3d416a749dea945fa9c4f943d1b90dd1c77d5477f2869f90b37b04101.jpg new file mode 100644 index 00000000..e41f97ad Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-286_8adecef3d416a749dea945fa9c4f943d1b90dd1c77d5477f2869f90b37b04101.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-286_e6066b92a97946eca1e0b82bde50853e4e200a5ff2ca01b673e17c10b41c9245.jpg b/.raw/MidasCivilAnalysisReference/images/page-286_e6066b92a97946eca1e0b82bde50853e4e200a5ff2ca01b673e17c10b41c9245.jpg new file mode 100644 index 00000000..7dd40d96 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-286_e6066b92a97946eca1e0b82bde50853e4e200a5ff2ca01b673e17c10b41c9245.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-287_432b47206621bfaf46b6a4b5ca96fa09c1a42e597518cbcd7569b60b4b81f9f3.jpg b/.raw/MidasCivilAnalysisReference/images/page-287_432b47206621bfaf46b6a4b5ca96fa09c1a42e597518cbcd7569b60b4b81f9f3.jpg new file mode 100644 index 00000000..79e0cc75 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-287_432b47206621bfaf46b6a4b5ca96fa09c1a42e597518cbcd7569b60b4b81f9f3.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-288_c6a1dea0463d2fb5a9be416c389967cfa6421e3c3241cf57c8369b7f27d47477.jpg b/.raw/MidasCivilAnalysisReference/images/page-288_c6a1dea0463d2fb5a9be416c389967cfa6421e3c3241cf57c8369b7f27d47477.jpg new file mode 100644 index 00000000..64a9900c Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-288_c6a1dea0463d2fb5a9be416c389967cfa6421e3c3241cf57c8369b7f27d47477.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-288_f317ad8320a023e498dbbb19396c9a5b44330cf3b8448523d7c5b80d03d010d5.jpg b/.raw/MidasCivilAnalysisReference/images/page-288_f317ad8320a023e498dbbb19396c9a5b44330cf3b8448523d7c5b80d03d010d5.jpg new file mode 100644 index 00000000..3f3f89ff Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-288_f317ad8320a023e498dbbb19396c9a5b44330cf3b8448523d7c5b80d03d010d5.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-289_3fd4d96049c1c924b01cebe6ee522eb9f08e1eacae7864fba6e6a6249794d996.jpg b/.raw/MidasCivilAnalysisReference/images/page-289_3fd4d96049c1c924b01cebe6ee522eb9f08e1eacae7864fba6e6a6249794d996.jpg new file mode 100644 index 00000000..9b97298f Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-289_3fd4d96049c1c924b01cebe6ee522eb9f08e1eacae7864fba6e6a6249794d996.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-289_9338cb48adf5a7b4b7d49e3c5d9db370150286dbb918af17e5bfb7f9976bc581.jpg b/.raw/MidasCivilAnalysisReference/images/page-289_9338cb48adf5a7b4b7d49e3c5d9db370150286dbb918af17e5bfb7f9976bc581.jpg new file mode 100644 index 00000000..9c52c35b Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-289_9338cb48adf5a7b4b7d49e3c5d9db370150286dbb918af17e5bfb7f9976bc581.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-289_d767cd8c7ce8c7fae5e2bd806da6ac556c39ecb72263dcfa432e068de58fecc8.jpg b/.raw/MidasCivilAnalysisReference/images/page-289_d767cd8c7ce8c7fae5e2bd806da6ac556c39ecb72263dcfa432e068de58fecc8.jpg new file mode 100644 index 00000000..260f27a2 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-289_d767cd8c7ce8c7fae5e2bd806da6ac556c39ecb72263dcfa432e068de58fecc8.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-290_187dd3c087c69337bdb72a062a780f859bac24bb512c55be69de4574bdb4cd47.jpg b/.raw/MidasCivilAnalysisReference/images/page-290_187dd3c087c69337bdb72a062a780f859bac24bb512c55be69de4574bdb4cd47.jpg new file mode 100644 index 00000000..da862ead Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-290_187dd3c087c69337bdb72a062a780f859bac24bb512c55be69de4574bdb4cd47.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-290_598e7416bf3cf54da74e6e1085682f0f26b0905c55cbe05fb5a3a59a88c70dbf.jpg b/.raw/MidasCivilAnalysisReference/images/page-290_598e7416bf3cf54da74e6e1085682f0f26b0905c55cbe05fb5a3a59a88c70dbf.jpg new file mode 100644 index 00000000..82f61cc6 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-290_598e7416bf3cf54da74e6e1085682f0f26b0905c55cbe05fb5a3a59a88c70dbf.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-290_cd9c4087ecf4f06e1cc2681dcd6c886c89e307fabd8bda4f9f398ac8e626738f.jpg b/.raw/MidasCivilAnalysisReference/images/page-290_cd9c4087ecf4f06e1cc2681dcd6c886c89e307fabd8bda4f9f398ac8e626738f.jpg new file mode 100644 index 00000000..99acc3f3 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-290_cd9c4087ecf4f06e1cc2681dcd6c886c89e307fabd8bda4f9f398ac8e626738f.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-291_addd07f3b8675a9a17caf679ab6952aa1fd4eeabff7ce7776893f63b9a2869d6.jpg b/.raw/MidasCivilAnalysisReference/images/page-291_addd07f3b8675a9a17caf679ab6952aa1fd4eeabff7ce7776893f63b9a2869d6.jpg new file mode 100644 index 00000000..4d61d5d2 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-291_addd07f3b8675a9a17caf679ab6952aa1fd4eeabff7ce7776893f63b9a2869d6.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-293_3772cb31c96362603ef7529adc7ab5c9bc73697b79813593d7bf0ad74301652b.jpg b/.raw/MidasCivilAnalysisReference/images/page-293_3772cb31c96362603ef7529adc7ab5c9bc73697b79813593d7bf0ad74301652b.jpg new file mode 100644 index 00000000..9052c100 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-293_3772cb31c96362603ef7529adc7ab5c9bc73697b79813593d7bf0ad74301652b.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-293_fdc4dbd65995e92443f742d82217908e07adc00a7bdfeccad3665c62ac3b4a96.jpg b/.raw/MidasCivilAnalysisReference/images/page-293_fdc4dbd65995e92443f742d82217908e07adc00a7bdfeccad3665c62ac3b4a96.jpg new file mode 100644 index 00000000..7088bdcf Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-293_fdc4dbd65995e92443f742d82217908e07adc00a7bdfeccad3665c62ac3b4a96.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-295_e5c92926396ee263a7e303b9a020b5379d4d95ff0e5f1fad3bc0ed39562d9110.jpg b/.raw/MidasCivilAnalysisReference/images/page-295_e5c92926396ee263a7e303b9a020b5379d4d95ff0e5f1fad3bc0ed39562d9110.jpg new file mode 100644 index 00000000..215f23ad Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-295_e5c92926396ee263a7e303b9a020b5379d4d95ff0e5f1fad3bc0ed39562d9110.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-295_f51496b581d6a7ab467285d2c7b3761bd5e55452bb388e5c0e9aa15da0be0c8f.jpg b/.raw/MidasCivilAnalysisReference/images/page-295_f51496b581d6a7ab467285d2c7b3761bd5e55452bb388e5c0e9aa15da0be0c8f.jpg new file mode 100644 index 00000000..250a0202 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-295_f51496b581d6a7ab467285d2c7b3761bd5e55452bb388e5c0e9aa15da0be0c8f.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-296_d2c16a517c85c7341011aa7f5fdedc2b4a60b922805341ce67706636876a7c22.jpg b/.raw/MidasCivilAnalysisReference/images/page-296_d2c16a517c85c7341011aa7f5fdedc2b4a60b922805341ce67706636876a7c22.jpg new file mode 100644 index 00000000..7413fb1a Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-296_d2c16a517c85c7341011aa7f5fdedc2b4a60b922805341ce67706636876a7c22.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-297_b177e9e05eb3315bd2b265d910c9c922fd3a577548ee1386b257a3a4cb01487a.jpg b/.raw/MidasCivilAnalysisReference/images/page-297_b177e9e05eb3315bd2b265d910c9c922fd3a577548ee1386b257a3a4cb01487a.jpg new file mode 100644 index 00000000..3f186e8a Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-297_b177e9e05eb3315bd2b265d910c9c922fd3a577548ee1386b257a3a4cb01487a.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-298_1790a348cf2fb39648def59434dd1dbcc883932d53d92b2c1fe2e6a981099c53.jpg b/.raw/MidasCivilAnalysisReference/images/page-298_1790a348cf2fb39648def59434dd1dbcc883932d53d92b2c1fe2e6a981099c53.jpg new file mode 100644 index 00000000..dccb751d Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-298_1790a348cf2fb39648def59434dd1dbcc883932d53d92b2c1fe2e6a981099c53.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-298_2df90371f48abf9f523e9cd89d6ee3f7cc4ba2dcfb72f1853dba450d080847c2.jpg b/.raw/MidasCivilAnalysisReference/images/page-298_2df90371f48abf9f523e9cd89d6ee3f7cc4ba2dcfb72f1853dba450d080847c2.jpg new file mode 100644 index 00000000..e3fd0baf Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-298_2df90371f48abf9f523e9cd89d6ee3f7cc4ba2dcfb72f1853dba450d080847c2.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-298_2eb06efdf66ad084c347aa65f26456cbd74f018140e1e2a076e7c0efd2b9355a.jpg b/.raw/MidasCivilAnalysisReference/images/page-298_2eb06efdf66ad084c347aa65f26456cbd74f018140e1e2a076e7c0efd2b9355a.jpg new file mode 100644 index 00000000..a5f306b1 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-298_2eb06efdf66ad084c347aa65f26456cbd74f018140e1e2a076e7c0efd2b9355a.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-298_71427447cb2dafa87c9a7695ef49b173a16cc725316369c7670af3955e6c2a56.jpg b/.raw/MidasCivilAnalysisReference/images/page-298_71427447cb2dafa87c9a7695ef49b173a16cc725316369c7670af3955e6c2a56.jpg new file mode 100644 index 00000000..fb8cba33 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-298_71427447cb2dafa87c9a7695ef49b173a16cc725316369c7670af3955e6c2a56.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-298_ab10e0778dc37c259e8f006714179736affaf6dbbbf6687e47b41b03eb5b6025.jpg b/.raw/MidasCivilAnalysisReference/images/page-298_ab10e0778dc37c259e8f006714179736affaf6dbbbf6687e47b41b03eb5b6025.jpg new file mode 100644 index 00000000..71b71370 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-298_ab10e0778dc37c259e8f006714179736affaf6dbbbf6687e47b41b03eb5b6025.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-300_1da5bb76bc9ff3e384106e36a946d48c42994ed0234abd9c949e40e9ddddc83a.jpg b/.raw/MidasCivilAnalysisReference/images/page-300_1da5bb76bc9ff3e384106e36a946d48c42994ed0234abd9c949e40e9ddddc83a.jpg new file mode 100644 index 00000000..c78c7e03 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-300_1da5bb76bc9ff3e384106e36a946d48c42994ed0234abd9c949e40e9ddddc83a.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-300_6ace698ef40b612b283a6ce793056e6a90df7103b84658bdee0d6b2096ac2353.jpg b/.raw/MidasCivilAnalysisReference/images/page-300_6ace698ef40b612b283a6ce793056e6a90df7103b84658bdee0d6b2096ac2353.jpg new file mode 100644 index 00000000..b8387944 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-300_6ace698ef40b612b283a6ce793056e6a90df7103b84658bdee0d6b2096ac2353.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-300_78967604404583438b86499114237a5375d22aa98ffd2e4c1a0d8eb56d2a5d32.jpg b/.raw/MidasCivilAnalysisReference/images/page-300_78967604404583438b86499114237a5375d22aa98ffd2e4c1a0d8eb56d2a5d32.jpg new file mode 100644 index 00000000..b0e29c48 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-300_78967604404583438b86499114237a5375d22aa98ffd2e4c1a0d8eb56d2a5d32.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-302_67072b4a1d6e90cc8b16d7a62086613d1f4bbdf88f0517f8d6eac8802e0a68af.jpg b/.raw/MidasCivilAnalysisReference/images/page-302_67072b4a1d6e90cc8b16d7a62086613d1f4bbdf88f0517f8d6eac8802e0a68af.jpg new file mode 100644 index 00000000..efb30bc0 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-302_67072b4a1d6e90cc8b16d7a62086613d1f4bbdf88f0517f8d6eac8802e0a68af.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-304_8537b5fd1acc5e650c6f74ab5058a5ba5e4aeade633d2e026b89b8e03e73ec1f.jpg b/.raw/MidasCivilAnalysisReference/images/page-304_8537b5fd1acc5e650c6f74ab5058a5ba5e4aeade633d2e026b89b8e03e73ec1f.jpg new file mode 100644 index 00000000..1f8a0cde Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-304_8537b5fd1acc5e650c6f74ab5058a5ba5e4aeade633d2e026b89b8e03e73ec1f.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-304_885f5216a77c14b160db4b0ecb005b0942d86797be652f4898ae4d856c7cb34a.jpg b/.raw/MidasCivilAnalysisReference/images/page-304_885f5216a77c14b160db4b0ecb005b0942d86797be652f4898ae4d856c7cb34a.jpg new file mode 100644 index 00000000..e0bf4e9f Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-304_885f5216a77c14b160db4b0ecb005b0942d86797be652f4898ae4d856c7cb34a.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-304_b51b46469a98225f141f77796f2c942c91b12f1cc1c0039f12d619afebc15746.jpg b/.raw/MidasCivilAnalysisReference/images/page-304_b51b46469a98225f141f77796f2c942c91b12f1cc1c0039f12d619afebc15746.jpg new file mode 100644 index 00000000..b774bc7a Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-304_b51b46469a98225f141f77796f2c942c91b12f1cc1c0039f12d619afebc15746.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-304_fe056e4c370df36f86f25fafe88f3caec1312118110c04f352d4cf2387a7185f.jpg b/.raw/MidasCivilAnalysisReference/images/page-304_fe056e4c370df36f86f25fafe88f3caec1312118110c04f352d4cf2387a7185f.jpg new file mode 100644 index 00000000..780890d2 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-304_fe056e4c370df36f86f25fafe88f3caec1312118110c04f352d4cf2387a7185f.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-305_d7fc28bd8e6c924a77aa6a6565433b2f6ff123bef775bba0f3cfffa174e2a519.jpg b/.raw/MidasCivilAnalysisReference/images/page-305_d7fc28bd8e6c924a77aa6a6565433b2f6ff123bef775bba0f3cfffa174e2a519.jpg new file mode 100644 index 00000000..e0b8b80c Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-305_d7fc28bd8e6c924a77aa6a6565433b2f6ff123bef775bba0f3cfffa174e2a519.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-305_d84a058cfdf24a1354a859fda0c275c2a0bd23ebe58d8885dcddb0e57a61ae80.jpg b/.raw/MidasCivilAnalysisReference/images/page-305_d84a058cfdf24a1354a859fda0c275c2a0bd23ebe58d8885dcddb0e57a61ae80.jpg new file mode 100644 index 00000000..268c4806 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-305_d84a058cfdf24a1354a859fda0c275c2a0bd23ebe58d8885dcddb0e57a61ae80.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-308_0795ced57e0e1a4dbf99479c7a16fef12aa001644c12b1e412615239abfe8ad6.jpg b/.raw/MidasCivilAnalysisReference/images/page-308_0795ced57e0e1a4dbf99479c7a16fef12aa001644c12b1e412615239abfe8ad6.jpg new file mode 100644 index 00000000..dadc9bff Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-308_0795ced57e0e1a4dbf99479c7a16fef12aa001644c12b1e412615239abfe8ad6.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-308_2965fa297c3cfa73aab817d0e0bdb2d35fa102d0928b73771e43197f1cf7266c.jpg b/.raw/MidasCivilAnalysisReference/images/page-308_2965fa297c3cfa73aab817d0e0bdb2d35fa102d0928b73771e43197f1cf7266c.jpg new file mode 100644 index 00000000..1eff0a51 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-308_2965fa297c3cfa73aab817d0e0bdb2d35fa102d0928b73771e43197f1cf7266c.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-308_a81813f899a275ac9f690cc285dd7b41c37195761b2fc3277d6841d7d507c036.jpg b/.raw/MidasCivilAnalysisReference/images/page-308_a81813f899a275ac9f690cc285dd7b41c37195761b2fc3277d6841d7d507c036.jpg new file mode 100644 index 00000000..886ba4ee Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-308_a81813f899a275ac9f690cc285dd7b41c37195761b2fc3277d6841d7d507c036.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-308_dcc6e1ba1e0ae1d16d82d0a65f1645531ffc9cf652cba3cda2ed509fbf1acc78.jpg b/.raw/MidasCivilAnalysisReference/images/page-308_dcc6e1ba1e0ae1d16d82d0a65f1645531ffc9cf652cba3cda2ed509fbf1acc78.jpg new file mode 100644 index 00000000..28ff4c5e Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-308_dcc6e1ba1e0ae1d16d82d0a65f1645531ffc9cf652cba3cda2ed509fbf1acc78.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-309_4dc889c31ba0af072f79e9e05bc6f1425eaf17706fbe26842e50fd5c175d0482.jpg b/.raw/MidasCivilAnalysisReference/images/page-309_4dc889c31ba0af072f79e9e05bc6f1425eaf17706fbe26842e50fd5c175d0482.jpg new file mode 100644 index 00000000..35dc7b3d Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-309_4dc889c31ba0af072f79e9e05bc6f1425eaf17706fbe26842e50fd5c175d0482.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-310_b549d216cc64ac3d189a00ce7a8d1a51f9d92853419def978e4a47114bca20c3.jpg b/.raw/MidasCivilAnalysisReference/images/page-310_b549d216cc64ac3d189a00ce7a8d1a51f9d92853419def978e4a47114bca20c3.jpg new file mode 100644 index 00000000..ad740dfb Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-310_b549d216cc64ac3d189a00ce7a8d1a51f9d92853419def978e4a47114bca20c3.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-310_c48e00c9acc2537ef6c80ad4616b399adbd98ecdc3fa9a3e96d6f3ffa9373d48.jpg b/.raw/MidasCivilAnalysisReference/images/page-310_c48e00c9acc2537ef6c80ad4616b399adbd98ecdc3fa9a3e96d6f3ffa9373d48.jpg new file mode 100644 index 00000000..545b365e Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-310_c48e00c9acc2537ef6c80ad4616b399adbd98ecdc3fa9a3e96d6f3ffa9373d48.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-310_ce53fefc2ef186ed19dfec972d4ce9e791aad66f84f3e9031ca6fb19d18f06f5.jpg b/.raw/MidasCivilAnalysisReference/images/page-310_ce53fefc2ef186ed19dfec972d4ce9e791aad66f84f3e9031ca6fb19d18f06f5.jpg new file mode 100644 index 00000000..234100e0 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-310_ce53fefc2ef186ed19dfec972d4ce9e791aad66f84f3e9031ca6fb19d18f06f5.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-311_ae15d872da7cb43ac84eba8a2d2fd893a31b8448d95b91085d73ef5583a77d41.jpg b/.raw/MidasCivilAnalysisReference/images/page-311_ae15d872da7cb43ac84eba8a2d2fd893a31b8448d95b91085d73ef5583a77d41.jpg new file mode 100644 index 00000000..d0202f2c Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-311_ae15d872da7cb43ac84eba8a2d2fd893a31b8448d95b91085d73ef5583a77d41.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-311_b4b24354b1299dd43767931cede790bb3c76de30df4e4644c0b4ceb75e7dd7ab.jpg b/.raw/MidasCivilAnalysisReference/images/page-311_b4b24354b1299dd43767931cede790bb3c76de30df4e4644c0b4ceb75e7dd7ab.jpg new file mode 100644 index 00000000..8c73c7b8 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-311_b4b24354b1299dd43767931cede790bb3c76de30df4e4644c0b4ceb75e7dd7ab.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-311_c355c29748d4b2f6dbefae0451ff6b75864b56e30fe980b54186db412a491715.jpg b/.raw/MidasCivilAnalysisReference/images/page-311_c355c29748d4b2f6dbefae0451ff6b75864b56e30fe980b54186db412a491715.jpg new file mode 100644 index 00000000..820b59f7 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-311_c355c29748d4b2f6dbefae0451ff6b75864b56e30fe980b54186db412a491715.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-312_363508893f8795637fdee7c85cee019206283e90ba2629073a7757f7237901ce.jpg b/.raw/MidasCivilAnalysisReference/images/page-312_363508893f8795637fdee7c85cee019206283e90ba2629073a7757f7237901ce.jpg new file mode 100644 index 00000000..1d5e3773 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-312_363508893f8795637fdee7c85cee019206283e90ba2629073a7757f7237901ce.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-312_e3f608c0c8cbbcb1ae10ca59b011504ae1549358f9ca7f5bb07c3e35f1530bdd.jpg b/.raw/MidasCivilAnalysisReference/images/page-312_e3f608c0c8cbbcb1ae10ca59b011504ae1549358f9ca7f5bb07c3e35f1530bdd.jpg new file mode 100644 index 00000000..059e581b Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-312_e3f608c0c8cbbcb1ae10ca59b011504ae1549358f9ca7f5bb07c3e35f1530bdd.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-313_326e372130529006abbec35dcdb32f39f8cc287a3ea329f87728b129344e9093.jpg b/.raw/MidasCivilAnalysisReference/images/page-313_326e372130529006abbec35dcdb32f39f8cc287a3ea329f87728b129344e9093.jpg new file mode 100644 index 00000000..d45a4eb7 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-313_326e372130529006abbec35dcdb32f39f8cc287a3ea329f87728b129344e9093.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-313_9da1658717e66dc27ca446e0ee4ec1d923f64970f000cc75b4346b3125c7ea9b.jpg b/.raw/MidasCivilAnalysisReference/images/page-313_9da1658717e66dc27ca446e0ee4ec1d923f64970f000cc75b4346b3125c7ea9b.jpg new file mode 100644 index 00000000..e3815420 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-313_9da1658717e66dc27ca446e0ee4ec1d923f64970f000cc75b4346b3125c7ea9b.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-314_13355f53d905ea40be1d322e959fbf4f3fdc89a8dce50a582cb50ce8e96d45b1.jpg b/.raw/MidasCivilAnalysisReference/images/page-314_13355f53d905ea40be1d322e959fbf4f3fdc89a8dce50a582cb50ce8e96d45b1.jpg new file mode 100644 index 00000000..f627a5c5 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-314_13355f53d905ea40be1d322e959fbf4f3fdc89a8dce50a582cb50ce8e96d45b1.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-315_6aa6b3d95a7d3a3432e875cd0621b67890bf28b57830fb1138aafe985860fa1e.jpg b/.raw/MidasCivilAnalysisReference/images/page-315_6aa6b3d95a7d3a3432e875cd0621b67890bf28b57830fb1138aafe985860fa1e.jpg new file mode 100644 index 00000000..32c627ab Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-315_6aa6b3d95a7d3a3432e875cd0621b67890bf28b57830fb1138aafe985860fa1e.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-315_78d0d1ab72dce00a0e73eb77b92e7c3d77bf527cf3a4e8420bd96a475e0fd669.jpg b/.raw/MidasCivilAnalysisReference/images/page-315_78d0d1ab72dce00a0e73eb77b92e7c3d77bf527cf3a4e8420bd96a475e0fd669.jpg new file mode 100644 index 00000000..01a17cb5 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-315_78d0d1ab72dce00a0e73eb77b92e7c3d77bf527cf3a4e8420bd96a475e0fd669.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-315_ee29deb5e94d8154a4ac54da6c18979f81da74deaf2090b6b7612c5e530d6ce2.jpg b/.raw/MidasCivilAnalysisReference/images/page-315_ee29deb5e94d8154a4ac54da6c18979f81da74deaf2090b6b7612c5e530d6ce2.jpg new file mode 100644 index 00000000..b6f157a2 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-315_ee29deb5e94d8154a4ac54da6c18979f81da74deaf2090b6b7612c5e530d6ce2.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-317_18a93f8c8968fcd7c1908dd79d8dcc10f712d72f6e2fb7d71f2190460107268a.jpg b/.raw/MidasCivilAnalysisReference/images/page-317_18a93f8c8968fcd7c1908dd79d8dcc10f712d72f6e2fb7d71f2190460107268a.jpg new file mode 100644 index 00000000..979e2817 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-317_18a93f8c8968fcd7c1908dd79d8dcc10f712d72f6e2fb7d71f2190460107268a.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-317_1c9d90b80741f5ccc6edb020fd3c1341e5a4510f5958067a7df034e3e2ff8449.jpg b/.raw/MidasCivilAnalysisReference/images/page-317_1c9d90b80741f5ccc6edb020fd3c1341e5a4510f5958067a7df034e3e2ff8449.jpg new file mode 100644 index 00000000..eedfd3bf Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-317_1c9d90b80741f5ccc6edb020fd3c1341e5a4510f5958067a7df034e3e2ff8449.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-317_2f37b821314bd0ac494b58d98b2527d2dff04757bbb5707138cf920e23835334.jpg b/.raw/MidasCivilAnalysisReference/images/page-317_2f37b821314bd0ac494b58d98b2527d2dff04757bbb5707138cf920e23835334.jpg new file mode 100644 index 00000000..c808d37c Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-317_2f37b821314bd0ac494b58d98b2527d2dff04757bbb5707138cf920e23835334.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-318_cd6554daa7b0ea86689cbc34b4bcf3210efbb399716e9f1d14b8040ebe2fc622.jpg b/.raw/MidasCivilAnalysisReference/images/page-318_cd6554daa7b0ea86689cbc34b4bcf3210efbb399716e9f1d14b8040ebe2fc622.jpg new file mode 100644 index 00000000..b12ff564 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-318_cd6554daa7b0ea86689cbc34b4bcf3210efbb399716e9f1d14b8040ebe2fc622.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-319_199db7af27bef2e019b25a0ecaf1d308c017ad138fa25826087513fd4653d921.jpg b/.raw/MidasCivilAnalysisReference/images/page-319_199db7af27bef2e019b25a0ecaf1d308c017ad138fa25826087513fd4653d921.jpg new file mode 100644 index 00000000..2abf640c Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-319_199db7af27bef2e019b25a0ecaf1d308c017ad138fa25826087513fd4653d921.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-319_233a33f0dba984c48992b787d2667f892eb1c890b55a1107db3201142609d8b3.jpg b/.raw/MidasCivilAnalysisReference/images/page-319_233a33f0dba984c48992b787d2667f892eb1c890b55a1107db3201142609d8b3.jpg new file mode 100644 index 00000000..d579adf5 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-319_233a33f0dba984c48992b787d2667f892eb1c890b55a1107db3201142609d8b3.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-319_40c64b5d6f8b0c6d7f0368015bc2fde9cbb397de66deb179c8e2fc4934a321ce.jpg b/.raw/MidasCivilAnalysisReference/images/page-319_40c64b5d6f8b0c6d7f0368015bc2fde9cbb397de66deb179c8e2fc4934a321ce.jpg new file mode 100644 index 00000000..078f3274 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-319_40c64b5d6f8b0c6d7f0368015bc2fde9cbb397de66deb179c8e2fc4934a321ce.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-319_c849832009c6c7d1258b11bfd72fbbf39857d1ba5122d40dfceb68c1ebbaf12e.jpg b/.raw/MidasCivilAnalysisReference/images/page-319_c849832009c6c7d1258b11bfd72fbbf39857d1ba5122d40dfceb68c1ebbaf12e.jpg new file mode 100644 index 00000000..c0675220 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-319_c849832009c6c7d1258b11bfd72fbbf39857d1ba5122d40dfceb68c1ebbaf12e.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-319_ec3706a1463c31772572d7dc9ba3e82489a2fa10f40cc2998f2ff5df8f2c3b2b.jpg b/.raw/MidasCivilAnalysisReference/images/page-319_ec3706a1463c31772572d7dc9ba3e82489a2fa10f40cc2998f2ff5df8f2c3b2b.jpg new file mode 100644 index 00000000..b2339100 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-319_ec3706a1463c31772572d7dc9ba3e82489a2fa10f40cc2998f2ff5df8f2c3b2b.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-321_0aecaf8cc3eb382646cf202b6e6c56258e711e75e286de4e8ac0419f8e5d1e09.jpg b/.raw/MidasCivilAnalysisReference/images/page-321_0aecaf8cc3eb382646cf202b6e6c56258e711e75e286de4e8ac0419f8e5d1e09.jpg new file mode 100644 index 00000000..8a2f7b23 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-321_0aecaf8cc3eb382646cf202b6e6c56258e711e75e286de4e8ac0419f8e5d1e09.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-321_225af5b757a426fd8aee91dabb72b092c74e8181e1ae02bbdf5c093187431c88.jpg b/.raw/MidasCivilAnalysisReference/images/page-321_225af5b757a426fd8aee91dabb72b092c74e8181e1ae02bbdf5c093187431c88.jpg new file mode 100644 index 00000000..1e2bf379 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-321_225af5b757a426fd8aee91dabb72b092c74e8181e1ae02bbdf5c093187431c88.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-321_74fcb2a5c255e872a5bb4efc41fd37b57a6ef75bf79e664d5ec06cb21adb8a7a.jpg b/.raw/MidasCivilAnalysisReference/images/page-321_74fcb2a5c255e872a5bb4efc41fd37b57a6ef75bf79e664d5ec06cb21adb8a7a.jpg new file mode 100644 index 00000000..e36e7fc8 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-321_74fcb2a5c255e872a5bb4efc41fd37b57a6ef75bf79e664d5ec06cb21adb8a7a.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-321_edacae29983b94af78df9ed5ef63b182a5eb8fb2a107e4461a20fe2a7388dd3f.jpg b/.raw/MidasCivilAnalysisReference/images/page-321_edacae29983b94af78df9ed5ef63b182a5eb8fb2a107e4461a20fe2a7388dd3f.jpg new file mode 100644 index 00000000..7082f861 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-321_edacae29983b94af78df9ed5ef63b182a5eb8fb2a107e4461a20fe2a7388dd3f.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-321_fb4e039b97607d39fdfe461329cf5dde93ecc012515881eb6bbfcfe525ab4cae.jpg b/.raw/MidasCivilAnalysisReference/images/page-321_fb4e039b97607d39fdfe461329cf5dde93ecc012515881eb6bbfcfe525ab4cae.jpg new file mode 100644 index 00000000..3b4e1f47 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-321_fb4e039b97607d39fdfe461329cf5dde93ecc012515881eb6bbfcfe525ab4cae.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-322_247aef3774bbe88998328a073998503fa9bc2ee85990f4dade1561a3f606b2e4.jpg b/.raw/MidasCivilAnalysisReference/images/page-322_247aef3774bbe88998328a073998503fa9bc2ee85990f4dade1561a3f606b2e4.jpg new file mode 100644 index 00000000..7bc864f2 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-322_247aef3774bbe88998328a073998503fa9bc2ee85990f4dade1561a3f606b2e4.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-323_40114157c0849e18e179eb23064959b61c87d601041e382166401d2f31e35445.jpg b/.raw/MidasCivilAnalysisReference/images/page-323_40114157c0849e18e179eb23064959b61c87d601041e382166401d2f31e35445.jpg new file mode 100644 index 00000000..d81aeda4 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-323_40114157c0849e18e179eb23064959b61c87d601041e382166401d2f31e35445.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-323_4329829aadb73d8071432f420fb644fe5512f4caf06ae7889d1715d6741d2d02.jpg b/.raw/MidasCivilAnalysisReference/images/page-323_4329829aadb73d8071432f420fb644fe5512f4caf06ae7889d1715d6741d2d02.jpg new file mode 100644 index 00000000..d69ebd81 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-323_4329829aadb73d8071432f420fb644fe5512f4caf06ae7889d1715d6741d2d02.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-323_56d7cb8eaeefbad581349a485ccda69ab3831e43186fb2a13b5dbf2ebc5afc6b.jpg b/.raw/MidasCivilAnalysisReference/images/page-323_56d7cb8eaeefbad581349a485ccda69ab3831e43186fb2a13b5dbf2ebc5afc6b.jpg new file mode 100644 index 00000000..c293363e Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-323_56d7cb8eaeefbad581349a485ccda69ab3831e43186fb2a13b5dbf2ebc5afc6b.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-323_843e7ca153852879c38ae31b704714de3b682db2f5e78963d9de630e2a21e0ef.jpg b/.raw/MidasCivilAnalysisReference/images/page-323_843e7ca153852879c38ae31b704714de3b682db2f5e78963d9de630e2a21e0ef.jpg new file mode 100644 index 00000000..88c8f085 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-323_843e7ca153852879c38ae31b704714de3b682db2f5e78963d9de630e2a21e0ef.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-323_921884861d90e0490921bd88dbd8f50684fa051ac180abe750cc58b497b8e5eb.jpg b/.raw/MidasCivilAnalysisReference/images/page-323_921884861d90e0490921bd88dbd8f50684fa051ac180abe750cc58b497b8e5eb.jpg new file mode 100644 index 00000000..89136252 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-323_921884861d90e0490921bd88dbd8f50684fa051ac180abe750cc58b497b8e5eb.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-323_a5ff0b1d8283c196d4948e081382eb3444a80da5a9d716175274a0327d99ae2e.jpg b/.raw/MidasCivilAnalysisReference/images/page-323_a5ff0b1d8283c196d4948e081382eb3444a80da5a9d716175274a0327d99ae2e.jpg new file mode 100644 index 00000000..eb7de88d Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-323_a5ff0b1d8283c196d4948e081382eb3444a80da5a9d716175274a0327d99ae2e.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-323_e7e809a103202cbb1fe2b78feded186b926a1a2586b2516a320234685a164cc2.jpg b/.raw/MidasCivilAnalysisReference/images/page-323_e7e809a103202cbb1fe2b78feded186b926a1a2586b2516a320234685a164cc2.jpg new file mode 100644 index 00000000..f54b3690 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-323_e7e809a103202cbb1fe2b78feded186b926a1a2586b2516a320234685a164cc2.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-324_128b5e637a7f9d9e34ff2a21c6b270d228aae03ad0a98c4d0c2d5e8f67273162.jpg b/.raw/MidasCivilAnalysisReference/images/page-324_128b5e637a7f9d9e34ff2a21c6b270d228aae03ad0a98c4d0c2d5e8f67273162.jpg new file mode 100644 index 00000000..50419ac9 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-324_128b5e637a7f9d9e34ff2a21c6b270d228aae03ad0a98c4d0c2d5e8f67273162.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-324_25100ef08a8fea24137c21c614a830881100ef3958c1bd3b783b01ba3361dec0.jpg b/.raw/MidasCivilAnalysisReference/images/page-324_25100ef08a8fea24137c21c614a830881100ef3958c1bd3b783b01ba3361dec0.jpg new file mode 100644 index 00000000..eb119de6 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-324_25100ef08a8fea24137c21c614a830881100ef3958c1bd3b783b01ba3361dec0.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-324_45c97b61f359430700d39d9b4da9360e75e67eaf243657d02e58bb6ef05d2b50.jpg b/.raw/MidasCivilAnalysisReference/images/page-324_45c97b61f359430700d39d9b4da9360e75e67eaf243657d02e58bb6ef05d2b50.jpg new file mode 100644 index 00000000..45b6aa31 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-324_45c97b61f359430700d39d9b4da9360e75e67eaf243657d02e58bb6ef05d2b50.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-324_6355d5d3d9b66fcfa94622c6bbadced5847a97cc7cec5589cf1e7f688032f5e1.jpg b/.raw/MidasCivilAnalysisReference/images/page-324_6355d5d3d9b66fcfa94622c6bbadced5847a97cc7cec5589cf1e7f688032f5e1.jpg new file mode 100644 index 00000000..43295a64 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-324_6355d5d3d9b66fcfa94622c6bbadced5847a97cc7cec5589cf1e7f688032f5e1.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-324_75676a861d346798eeea3c2bf419f5b5c67fda9459a96078ee5aedf306c93425.jpg b/.raw/MidasCivilAnalysisReference/images/page-324_75676a861d346798eeea3c2bf419f5b5c67fda9459a96078ee5aedf306c93425.jpg new file mode 100644 index 00000000..c2e99d37 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-324_75676a861d346798eeea3c2bf419f5b5c67fda9459a96078ee5aedf306c93425.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-324_7974ea9ca81aa51d987d45914e98e04862a72edf533c7e487e9ebe7182e5e160.jpg b/.raw/MidasCivilAnalysisReference/images/page-324_7974ea9ca81aa51d987d45914e98e04862a72edf533c7e487e9ebe7182e5e160.jpg new file mode 100644 index 00000000..751065ec Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-324_7974ea9ca81aa51d987d45914e98e04862a72edf533c7e487e9ebe7182e5e160.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-324_eb83e8ba0ab0a1fe328cdbb1d3245e9101fd32762453834a3394414a89ff2c0c.jpg b/.raw/MidasCivilAnalysisReference/images/page-324_eb83e8ba0ab0a1fe328cdbb1d3245e9101fd32762453834a3394414a89ff2c0c.jpg new file mode 100644 index 00000000..dbb70b55 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-324_eb83e8ba0ab0a1fe328cdbb1d3245e9101fd32762453834a3394414a89ff2c0c.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-324_efa0f7d57369208986b55acf36b6224daaf987129fda9c9338060c529c350228.jpg b/.raw/MidasCivilAnalysisReference/images/page-324_efa0f7d57369208986b55acf36b6224daaf987129fda9c9338060c529c350228.jpg new file mode 100644 index 00000000..c7c14250 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-324_efa0f7d57369208986b55acf36b6224daaf987129fda9c9338060c529c350228.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-325_3c5ea2dd9fbe10d5ff6aefb2c090247f5a8d6054281b4ebe7d0588b7b1a19cd6.jpg b/.raw/MidasCivilAnalysisReference/images/page-325_3c5ea2dd9fbe10d5ff6aefb2c090247f5a8d6054281b4ebe7d0588b7b1a19cd6.jpg new file mode 100644 index 00000000..e42da5b1 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-325_3c5ea2dd9fbe10d5ff6aefb2c090247f5a8d6054281b4ebe7d0588b7b1a19cd6.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-325_e43d50b4dda3036285d4c322393965934efe05b26ec0f6dea346a7b96154bcbb.jpg b/.raw/MidasCivilAnalysisReference/images/page-325_e43d50b4dda3036285d4c322393965934efe05b26ec0f6dea346a7b96154bcbb.jpg new file mode 100644 index 00000000..2e049269 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-325_e43d50b4dda3036285d4c322393965934efe05b26ec0f6dea346a7b96154bcbb.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-325_f8b164d7139b633219850aeab8da9477ec772a138f7211f8c375c957a5bb13cd.jpg b/.raw/MidasCivilAnalysisReference/images/page-325_f8b164d7139b633219850aeab8da9477ec772a138f7211f8c375c957a5bb13cd.jpg new file mode 100644 index 00000000..1ec8f2bc Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-325_f8b164d7139b633219850aeab8da9477ec772a138f7211f8c375c957a5bb13cd.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-326_cde249a31c9bf78b053d224bb3705392b23b2d9a9ba70316f41d718ec7cb65f8.jpg b/.raw/MidasCivilAnalysisReference/images/page-326_cde249a31c9bf78b053d224bb3705392b23b2d9a9ba70316f41d718ec7cb65f8.jpg new file mode 100644 index 00000000..aa016834 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-326_cde249a31c9bf78b053d224bb3705392b23b2d9a9ba70316f41d718ec7cb65f8.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-326_e5ad41fbdfe71911f0398969f5b49dfa7963371a73e5553e04135a5be67f089b.jpg b/.raw/MidasCivilAnalysisReference/images/page-326_e5ad41fbdfe71911f0398969f5b49dfa7963371a73e5553e04135a5be67f089b.jpg new file mode 100644 index 00000000..9460d7c6 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-326_e5ad41fbdfe71911f0398969f5b49dfa7963371a73e5553e04135a5be67f089b.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-327_f8ca172fdcd79742e5529ab3f36664f027180d6778b5e0ee14486178edc207d4.jpg b/.raw/MidasCivilAnalysisReference/images/page-327_f8ca172fdcd79742e5529ab3f36664f027180d6778b5e0ee14486178edc207d4.jpg new file mode 100644 index 00000000..191111dd Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-327_f8ca172fdcd79742e5529ab3f36664f027180d6778b5e0ee14486178edc207d4.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-328_56eb149817df5aed1a9c463e86fa3050a257b29671416947809d319507d55c1d.jpg b/.raw/MidasCivilAnalysisReference/images/page-328_56eb149817df5aed1a9c463e86fa3050a257b29671416947809d319507d55c1d.jpg new file mode 100644 index 00000000..a5ec7deb Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-328_56eb149817df5aed1a9c463e86fa3050a257b29671416947809d319507d55c1d.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-328_78a7cf1d29880ab7e8d2c6954830329e5a4a543829b7eaf108b3b2f4ecad5095.jpg b/.raw/MidasCivilAnalysisReference/images/page-328_78a7cf1d29880ab7e8d2c6954830329e5a4a543829b7eaf108b3b2f4ecad5095.jpg new file mode 100644 index 00000000..d4a33c21 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-328_78a7cf1d29880ab7e8d2c6954830329e5a4a543829b7eaf108b3b2f4ecad5095.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-328_ee1ffdccaa83a49bce90c3ca9197b64ed1f9501cf4b523d4cb0cad12d73d6b09.jpg b/.raw/MidasCivilAnalysisReference/images/page-328_ee1ffdccaa83a49bce90c3ca9197b64ed1f9501cf4b523d4cb0cad12d73d6b09.jpg new file mode 100644 index 00000000..94bd7f60 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-328_ee1ffdccaa83a49bce90c3ca9197b64ed1f9501cf4b523d4cb0cad12d73d6b09.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-329_015f5eb9e1adc86462f2d4958b7d636406d6b0115bcefe3ffb7882a6a982b85a.jpg b/.raw/MidasCivilAnalysisReference/images/page-329_015f5eb9e1adc86462f2d4958b7d636406d6b0115bcefe3ffb7882a6a982b85a.jpg new file mode 100644 index 00000000..77c2d9aa Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-329_015f5eb9e1adc86462f2d4958b7d636406d6b0115bcefe3ffb7882a6a982b85a.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-329_13b643d643afe9d10608020624d3456e333b3766eee397837e04c599e0538994.jpg b/.raw/MidasCivilAnalysisReference/images/page-329_13b643d643afe9d10608020624d3456e333b3766eee397837e04c599e0538994.jpg new file mode 100644 index 00000000..f5ef70df Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-329_13b643d643afe9d10608020624d3456e333b3766eee397837e04c599e0538994.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-329_15812ac93a9eec6006d491b41d1a18ca1142cfc73b17e1c5b1a9f6808e9ccc5a.jpg b/.raw/MidasCivilAnalysisReference/images/page-329_15812ac93a9eec6006d491b41d1a18ca1142cfc73b17e1c5b1a9f6808e9ccc5a.jpg new file mode 100644 index 00000000..8199c14f Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-329_15812ac93a9eec6006d491b41d1a18ca1142cfc73b17e1c5b1a9f6808e9ccc5a.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-329_1c45f4e5cf8def4d07563ba9d02c9ea06cf81a5c21a83d34fcd11e3f56d9fdcc.jpg b/.raw/MidasCivilAnalysisReference/images/page-329_1c45f4e5cf8def4d07563ba9d02c9ea06cf81a5c21a83d34fcd11e3f56d9fdcc.jpg new file mode 100644 index 00000000..7371644e Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-329_1c45f4e5cf8def4d07563ba9d02c9ea06cf81a5c21a83d34fcd11e3f56d9fdcc.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-329_48cb68d76a4c620e20b9c1d57bac4747cd865da36d45151440edcf71ff97f6b4.jpg b/.raw/MidasCivilAnalysisReference/images/page-329_48cb68d76a4c620e20b9c1d57bac4747cd865da36d45151440edcf71ff97f6b4.jpg new file mode 100644 index 00000000..1ea5d31c Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-329_48cb68d76a4c620e20b9c1d57bac4747cd865da36d45151440edcf71ff97f6b4.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-329_74ad84a974ee8ffd11e4d398308d854fa5a0b3188779d7ec7760f152dbe1f224.jpg b/.raw/MidasCivilAnalysisReference/images/page-329_74ad84a974ee8ffd11e4d398308d854fa5a0b3188779d7ec7760f152dbe1f224.jpg new file mode 100644 index 00000000..00c9e10e Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-329_74ad84a974ee8ffd11e4d398308d854fa5a0b3188779d7ec7760f152dbe1f224.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-329_88e5c1c6ab988bf38a17cc394cb54f44e109c3a8d810dc548d1b3f8e396d7d85.jpg b/.raw/MidasCivilAnalysisReference/images/page-329_88e5c1c6ab988bf38a17cc394cb54f44e109c3a8d810dc548d1b3f8e396d7d85.jpg new file mode 100644 index 00000000..25fcffdc Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-329_88e5c1c6ab988bf38a17cc394cb54f44e109c3a8d810dc548d1b3f8e396d7d85.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-329_a7409a7adffafbd611de18d9f94f8dfbae8cf6cf5a03d42e82b3ebd749c564e4.jpg b/.raw/MidasCivilAnalysisReference/images/page-329_a7409a7adffafbd611de18d9f94f8dfbae8cf6cf5a03d42e82b3ebd749c564e4.jpg new file mode 100644 index 00000000..2f86ebb3 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-329_a7409a7adffafbd611de18d9f94f8dfbae8cf6cf5a03d42e82b3ebd749c564e4.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-329_bd18fbfff23b1867b677c2652850efcc8db06bca55e9eca290b4b870dc913eb5.jpg b/.raw/MidasCivilAnalysisReference/images/page-329_bd18fbfff23b1867b677c2652850efcc8db06bca55e9eca290b4b870dc913eb5.jpg new file mode 100644 index 00000000..cea6e345 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-329_bd18fbfff23b1867b677c2652850efcc8db06bca55e9eca290b4b870dc913eb5.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-329_c4d0d819697c553aad1e9dfb104fb1a302fdd4742abe1e655a709220d275b1c8.jpg b/.raw/MidasCivilAnalysisReference/images/page-329_c4d0d819697c553aad1e9dfb104fb1a302fdd4742abe1e655a709220d275b1c8.jpg new file mode 100644 index 00000000..7fc4b604 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-329_c4d0d819697c553aad1e9dfb104fb1a302fdd4742abe1e655a709220d275b1c8.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-329_c87c9da9c7613642e4e131b5b982944d5a93c9ea78ddbfc7cd60460625235e27.jpg b/.raw/MidasCivilAnalysisReference/images/page-329_c87c9da9c7613642e4e131b5b982944d5a93c9ea78ddbfc7cd60460625235e27.jpg new file mode 100644 index 00000000..ebc99852 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-329_c87c9da9c7613642e4e131b5b982944d5a93c9ea78ddbfc7cd60460625235e27.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-329_e7bf0381c386e9d56d16feda23845ac2cc46d7ed90354086f770b4255f99d959.jpg b/.raw/MidasCivilAnalysisReference/images/page-329_e7bf0381c386e9d56d16feda23845ac2cc46d7ed90354086f770b4255f99d959.jpg new file mode 100644 index 00000000..843545a9 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-329_e7bf0381c386e9d56d16feda23845ac2cc46d7ed90354086f770b4255f99d959.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-330_2a3793b56c0b420fd5903aed02108d688469ce8d8fc2627554ce69231cd6b905.jpg b/.raw/MidasCivilAnalysisReference/images/page-330_2a3793b56c0b420fd5903aed02108d688469ce8d8fc2627554ce69231cd6b905.jpg new file mode 100644 index 00000000..154f6a6e Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-330_2a3793b56c0b420fd5903aed02108d688469ce8d8fc2627554ce69231cd6b905.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-330_801a0502a47a8e6b450662a25a91bdcdd773a9ac9b98cae3b7fbc024b8c04995.jpg b/.raw/MidasCivilAnalysisReference/images/page-330_801a0502a47a8e6b450662a25a91bdcdd773a9ac9b98cae3b7fbc024b8c04995.jpg new file mode 100644 index 00000000..4c275a94 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-330_801a0502a47a8e6b450662a25a91bdcdd773a9ac9b98cae3b7fbc024b8c04995.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-330_ab9a9979c474c19882a300e2b037a9b043a671a70f2c3829afd62a1c65aba63b.jpg b/.raw/MidasCivilAnalysisReference/images/page-330_ab9a9979c474c19882a300e2b037a9b043a671a70f2c3829afd62a1c65aba63b.jpg new file mode 100644 index 00000000..a381e1ef Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-330_ab9a9979c474c19882a300e2b037a9b043a671a70f2c3829afd62a1c65aba63b.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-331_28630419f8b02ca45cd542ff788071918160c6b3ac0a7ed8aea4bb57244628b1.jpg b/.raw/MidasCivilAnalysisReference/images/page-331_28630419f8b02ca45cd542ff788071918160c6b3ac0a7ed8aea4bb57244628b1.jpg new file mode 100644 index 00000000..36f7ebb3 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-331_28630419f8b02ca45cd542ff788071918160c6b3ac0a7ed8aea4bb57244628b1.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-331_418a6e3195d2d12112131a26620f81256cc1ebc678c8144c9d328d17e657f036.jpg b/.raw/MidasCivilAnalysisReference/images/page-331_418a6e3195d2d12112131a26620f81256cc1ebc678c8144c9d328d17e657f036.jpg new file mode 100644 index 00000000..a4cae11b Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-331_418a6e3195d2d12112131a26620f81256cc1ebc678c8144c9d328d17e657f036.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-331_567547de9b048a476f16bf5f96790f67a0c91efb86f195d599f2091ff56b4241.jpg b/.raw/MidasCivilAnalysisReference/images/page-331_567547de9b048a476f16bf5f96790f67a0c91efb86f195d599f2091ff56b4241.jpg new file mode 100644 index 00000000..2e49a14a Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-331_567547de9b048a476f16bf5f96790f67a0c91efb86f195d599f2091ff56b4241.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-331_59db54673b0dad019edfeeb446cb0cc75cfe28110551bd11fcd7015d0c04f328.jpg b/.raw/MidasCivilAnalysisReference/images/page-331_59db54673b0dad019edfeeb446cb0cc75cfe28110551bd11fcd7015d0c04f328.jpg new file mode 100644 index 00000000..05c1941f Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-331_59db54673b0dad019edfeeb446cb0cc75cfe28110551bd11fcd7015d0c04f328.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-331_68fc0747639983ca0c7acbd00829bc2345e60c0ef2794b0fefc22d5873bf54bb.jpg b/.raw/MidasCivilAnalysisReference/images/page-331_68fc0747639983ca0c7acbd00829bc2345e60c0ef2794b0fefc22d5873bf54bb.jpg new file mode 100644 index 00000000..9b81a5f2 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-331_68fc0747639983ca0c7acbd00829bc2345e60c0ef2794b0fefc22d5873bf54bb.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-331_8ace83d7949e82d4d6a6f94347db3c34247f2f5ab288cf1b4eb602a95344dd09.jpg b/.raw/MidasCivilAnalysisReference/images/page-331_8ace83d7949e82d4d6a6f94347db3c34247f2f5ab288cf1b4eb602a95344dd09.jpg new file mode 100644 index 00000000..f010b6d7 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-331_8ace83d7949e82d4d6a6f94347db3c34247f2f5ab288cf1b4eb602a95344dd09.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-331_d5bf7618cd16d785f361cc23b6d7b0a6e21787a8416bb81bde167d936c1649d9.jpg b/.raw/MidasCivilAnalysisReference/images/page-331_d5bf7618cd16d785f361cc23b6d7b0a6e21787a8416bb81bde167d936c1649d9.jpg new file mode 100644 index 00000000..8336d40b Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-331_d5bf7618cd16d785f361cc23b6d7b0a6e21787a8416bb81bde167d936c1649d9.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-332_118b0bead422065cc7a08f6730538f6229d373d4b6e1de99ab234b913d4e6255.jpg b/.raw/MidasCivilAnalysisReference/images/page-332_118b0bead422065cc7a08f6730538f6229d373d4b6e1de99ab234b913d4e6255.jpg new file mode 100644 index 00000000..4de1e5ed Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-332_118b0bead422065cc7a08f6730538f6229d373d4b6e1de99ab234b913d4e6255.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-332_23583675ddbc63ef422e9a068262485019e90b20caae4a9c6d1b79e38da334c0.jpg b/.raw/MidasCivilAnalysisReference/images/page-332_23583675ddbc63ef422e9a068262485019e90b20caae4a9c6d1b79e38da334c0.jpg new file mode 100644 index 00000000..263f1397 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-332_23583675ddbc63ef422e9a068262485019e90b20caae4a9c6d1b79e38da334c0.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-332_24f1afdb546af72be16fbcfe6022274d311dde9e40a41d00d8ce634343f1a721.jpg b/.raw/MidasCivilAnalysisReference/images/page-332_24f1afdb546af72be16fbcfe6022274d311dde9e40a41d00d8ce634343f1a721.jpg new file mode 100644 index 00000000..e71f3ddc Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-332_24f1afdb546af72be16fbcfe6022274d311dde9e40a41d00d8ce634343f1a721.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-332_662b954ec4cfb4a2f7cbf7d9e299818225e65da7c1104c81f9942dbc565af586.jpg b/.raw/MidasCivilAnalysisReference/images/page-332_662b954ec4cfb4a2f7cbf7d9e299818225e65da7c1104c81f9942dbc565af586.jpg new file mode 100644 index 00000000..d6cbc8e1 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-332_662b954ec4cfb4a2f7cbf7d9e299818225e65da7c1104c81f9942dbc565af586.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-332_a15d0b16d11189a9c1f07c77dacc464a2976ed65c064116618418563f00c485d.jpg b/.raw/MidasCivilAnalysisReference/images/page-332_a15d0b16d11189a9c1f07c77dacc464a2976ed65c064116618418563f00c485d.jpg new file mode 100644 index 00000000..a13e6ed5 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-332_a15d0b16d11189a9c1f07c77dacc464a2976ed65c064116618418563f00c485d.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-332_cd98a35027a1e1a3693fe21fca3d726f5989e05cfd22eb9ccd5dc6214ede65b5.jpg b/.raw/MidasCivilAnalysisReference/images/page-332_cd98a35027a1e1a3693fe21fca3d726f5989e05cfd22eb9ccd5dc6214ede65b5.jpg new file mode 100644 index 00000000..ac3f4a6e Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-332_cd98a35027a1e1a3693fe21fca3d726f5989e05cfd22eb9ccd5dc6214ede65b5.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-332_f0cb2ab270320ebcd3bd74e870981919b5df135493c2fd2950f237813dc040cb.jpg b/.raw/MidasCivilAnalysisReference/images/page-332_f0cb2ab270320ebcd3bd74e870981919b5df135493c2fd2950f237813dc040cb.jpg new file mode 100644 index 00000000..d880a95f Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-332_f0cb2ab270320ebcd3bd74e870981919b5df135493c2fd2950f237813dc040cb.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-334_2c69040e796e7dcb08b9a60479049d40b0acc861a53f069b51cc42f9543f0568.jpg b/.raw/MidasCivilAnalysisReference/images/page-334_2c69040e796e7dcb08b9a60479049d40b0acc861a53f069b51cc42f9543f0568.jpg new file mode 100644 index 00000000..8c824daf Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-334_2c69040e796e7dcb08b9a60479049d40b0acc861a53f069b51cc42f9543f0568.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-334_40115e54d4ea6cf2f1a4bdcc96fba9b7182a359565dd6141c2f13c00b68fcc32.jpg b/.raw/MidasCivilAnalysisReference/images/page-334_40115e54d4ea6cf2f1a4bdcc96fba9b7182a359565dd6141c2f13c00b68fcc32.jpg new file mode 100644 index 00000000..ed1e18c0 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-334_40115e54d4ea6cf2f1a4bdcc96fba9b7182a359565dd6141c2f13c00b68fcc32.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-334_7a7dc0f1702ab98fb4692fa08ced4aaace161c47a483e6387e52270bf5811dc4.jpg b/.raw/MidasCivilAnalysisReference/images/page-334_7a7dc0f1702ab98fb4692fa08ced4aaace161c47a483e6387e52270bf5811dc4.jpg new file mode 100644 index 00000000..ce4e4646 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-334_7a7dc0f1702ab98fb4692fa08ced4aaace161c47a483e6387e52270bf5811dc4.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-334_a6c1badcee55eb65550394f54d30ae4a8eb6e1c4a905c8744649792eb8360e58.jpg b/.raw/MidasCivilAnalysisReference/images/page-334_a6c1badcee55eb65550394f54d30ae4a8eb6e1c4a905c8744649792eb8360e58.jpg new file mode 100644 index 00000000..6503fcc3 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-334_a6c1badcee55eb65550394f54d30ae4a8eb6e1c4a905c8744649792eb8360e58.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-334_cc5fedae066b263fe032f429c870ec92bbfcbe6954ccacd9fdb11fc9c9b15274.jpg b/.raw/MidasCivilAnalysisReference/images/page-334_cc5fedae066b263fe032f429c870ec92bbfcbe6954ccacd9fdb11fc9c9b15274.jpg new file mode 100644 index 00000000..efbf37a0 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-334_cc5fedae066b263fe032f429c870ec92bbfcbe6954ccacd9fdb11fc9c9b15274.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-335_1138197651b03467b24791c6c00111d7952f1d75757d12a7418ca0f512c1333e.jpg b/.raw/MidasCivilAnalysisReference/images/page-335_1138197651b03467b24791c6c00111d7952f1d75757d12a7418ca0f512c1333e.jpg new file mode 100644 index 00000000..0abba02c Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-335_1138197651b03467b24791c6c00111d7952f1d75757d12a7418ca0f512c1333e.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-335_2258b984c86de1a77b29603ab8c67e052fddd32bdc95c58d40ec5c39fa87da11.jpg b/.raw/MidasCivilAnalysisReference/images/page-335_2258b984c86de1a77b29603ab8c67e052fddd32bdc95c58d40ec5c39fa87da11.jpg new file mode 100644 index 00000000..07c0f91d Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-335_2258b984c86de1a77b29603ab8c67e052fddd32bdc95c58d40ec5c39fa87da11.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-335_5cae28c2e6120778c42389c840edb48b5488d89bc829391824fad814ea3d6cbb.jpg b/.raw/MidasCivilAnalysisReference/images/page-335_5cae28c2e6120778c42389c840edb48b5488d89bc829391824fad814ea3d6cbb.jpg new file mode 100644 index 00000000..2104f7c9 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-335_5cae28c2e6120778c42389c840edb48b5488d89bc829391824fad814ea3d6cbb.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-336_2634a94a77e957dad36284805b57074d96d45a0d1e9e0543d3fc77a4db09d02c.jpg b/.raw/MidasCivilAnalysisReference/images/page-336_2634a94a77e957dad36284805b57074d96d45a0d1e9e0543d3fc77a4db09d02c.jpg new file mode 100644 index 00000000..ad57f6c1 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-336_2634a94a77e957dad36284805b57074d96d45a0d1e9e0543d3fc77a4db09d02c.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-336_6534888e900757398c463ec334a688e03732b6038711be3b56833fa8b318d36b.jpg b/.raw/MidasCivilAnalysisReference/images/page-336_6534888e900757398c463ec334a688e03732b6038711be3b56833fa8b318d36b.jpg new file mode 100644 index 00000000..be40c8d6 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-336_6534888e900757398c463ec334a688e03732b6038711be3b56833fa8b318d36b.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-336_8617409f02a6a649e33ea2cec5615648f9901537b39010854f1557c84b5f5782.jpg b/.raw/MidasCivilAnalysisReference/images/page-336_8617409f02a6a649e33ea2cec5615648f9901537b39010854f1557c84b5f5782.jpg new file mode 100644 index 00000000..c9a308b6 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-336_8617409f02a6a649e33ea2cec5615648f9901537b39010854f1557c84b5f5782.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-336_d6a33c71c8c361f4be076ecc0e077197ce323c1f80a2a556f552d81314f938bf.jpg b/.raw/MidasCivilAnalysisReference/images/page-336_d6a33c71c8c361f4be076ecc0e077197ce323c1f80a2a556f552d81314f938bf.jpg new file mode 100644 index 00000000..22f43fe6 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-336_d6a33c71c8c361f4be076ecc0e077197ce323c1f80a2a556f552d81314f938bf.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-336_ffb7c8acb0aadcadf223b7fefec68ad8fae320fbb7ae72ee690ed470fe2026fa.jpg b/.raw/MidasCivilAnalysisReference/images/page-336_ffb7c8acb0aadcadf223b7fefec68ad8fae320fbb7ae72ee690ed470fe2026fa.jpg new file mode 100644 index 00000000..367ee9f3 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-336_ffb7c8acb0aadcadf223b7fefec68ad8fae320fbb7ae72ee690ed470fe2026fa.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-337_2e7cb5cf83d38f8e58803863fda728b5f716b3122b9750919e6b9f9b19f1ed3b.jpg b/.raw/MidasCivilAnalysisReference/images/page-337_2e7cb5cf83d38f8e58803863fda728b5f716b3122b9750919e6b9f9b19f1ed3b.jpg new file mode 100644 index 00000000..8a9168f2 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-337_2e7cb5cf83d38f8e58803863fda728b5f716b3122b9750919e6b9f9b19f1ed3b.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-337_42cdde59a8307f3e3817256742dfbf0c94ce6fe885a7c9088b8637d4d67d5c97.jpg b/.raw/MidasCivilAnalysisReference/images/page-337_42cdde59a8307f3e3817256742dfbf0c94ce6fe885a7c9088b8637d4d67d5c97.jpg new file mode 100644 index 00000000..fed58f64 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-337_42cdde59a8307f3e3817256742dfbf0c94ce6fe885a7c9088b8637d4d67d5c97.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-337_ff707941e4b5122bccd33f9f0b7065d53bbd3562b941be3157eed95c76d3b4a4.jpg b/.raw/MidasCivilAnalysisReference/images/page-337_ff707941e4b5122bccd33f9f0b7065d53bbd3562b941be3157eed95c76d3b4a4.jpg new file mode 100644 index 00000000..a0889ee0 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-337_ff707941e4b5122bccd33f9f0b7065d53bbd3562b941be3157eed95c76d3b4a4.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-339_890041ccc85cc9bfb8cb4b4d2f753e1c0c91603a4976f46af07e10dc8df39c3a.jpg b/.raw/MidasCivilAnalysisReference/images/page-339_890041ccc85cc9bfb8cb4b4d2f753e1c0c91603a4976f46af07e10dc8df39c3a.jpg new file mode 100644 index 00000000..f4e500be Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-339_890041ccc85cc9bfb8cb4b4d2f753e1c0c91603a4976f46af07e10dc8df39c3a.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-339_c27a71a11696860db98107052ca1462c995a245eb2272fd6d0056dd5ea6254a1.jpg b/.raw/MidasCivilAnalysisReference/images/page-339_c27a71a11696860db98107052ca1462c995a245eb2272fd6d0056dd5ea6254a1.jpg new file mode 100644 index 00000000..634d1b6d Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-339_c27a71a11696860db98107052ca1462c995a245eb2272fd6d0056dd5ea6254a1.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-340_c64fb6e1f013c4ebc05616d9149dc2006fd678a8d58d38cb1dad409e2738d30d.jpg b/.raw/MidasCivilAnalysisReference/images/page-340_c64fb6e1f013c4ebc05616d9149dc2006fd678a8d58d38cb1dad409e2738d30d.jpg new file mode 100644 index 00000000..71af7d67 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-340_c64fb6e1f013c4ebc05616d9149dc2006fd678a8d58d38cb1dad409e2738d30d.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-341_368f195d8a7a4f1a730def9f463be15004f6a2c491dd2116142030657a316c7f.jpg b/.raw/MidasCivilAnalysisReference/images/page-341_368f195d8a7a4f1a730def9f463be15004f6a2c491dd2116142030657a316c7f.jpg new file mode 100644 index 00000000..3dee6357 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-341_368f195d8a7a4f1a730def9f463be15004f6a2c491dd2116142030657a316c7f.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-341_6c5b4d82cdd0212a3e873bb02be3368bc79cb3fd40622de6a944eec580a9199b.jpg b/.raw/MidasCivilAnalysisReference/images/page-341_6c5b4d82cdd0212a3e873bb02be3368bc79cb3fd40622de6a944eec580a9199b.jpg new file mode 100644 index 00000000..4c0810c5 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-341_6c5b4d82cdd0212a3e873bb02be3368bc79cb3fd40622de6a944eec580a9199b.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-341_b882878fa1b605a129a5c942ba40bc698c63bde543eab3e0168ec3e5adbc7162.jpg b/.raw/MidasCivilAnalysisReference/images/page-341_b882878fa1b605a129a5c942ba40bc698c63bde543eab3e0168ec3e5adbc7162.jpg new file mode 100644 index 00000000..6c34c08e Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-341_b882878fa1b605a129a5c942ba40bc698c63bde543eab3e0168ec3e5adbc7162.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-341_f322886ce4af9abd5e4c13fcdf8e7414eba3fa443018183fce82cf68ea2c3539.jpg b/.raw/MidasCivilAnalysisReference/images/page-341_f322886ce4af9abd5e4c13fcdf8e7414eba3fa443018183fce82cf68ea2c3539.jpg new file mode 100644 index 00000000..941a2dd3 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-341_f322886ce4af9abd5e4c13fcdf8e7414eba3fa443018183fce82cf68ea2c3539.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-342_39e9d469c2a33e1d5189c6359d80bd711004e026fa9c5aca1fbbd980d7d50967.jpg b/.raw/MidasCivilAnalysisReference/images/page-342_39e9d469c2a33e1d5189c6359d80bd711004e026fa9c5aca1fbbd980d7d50967.jpg new file mode 100644 index 00000000..67f719be Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-342_39e9d469c2a33e1d5189c6359d80bd711004e026fa9c5aca1fbbd980d7d50967.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-342_68ce721a72f936e3eaa00504931679187cfe15cff078e464a90451bd57801bcc.jpg b/.raw/MidasCivilAnalysisReference/images/page-342_68ce721a72f936e3eaa00504931679187cfe15cff078e464a90451bd57801bcc.jpg new file mode 100644 index 00000000..39f01183 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-342_68ce721a72f936e3eaa00504931679187cfe15cff078e464a90451bd57801bcc.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-342_9d54df8515717f13c61fdb11840277d31ae806f4ea6d52d016d370f7df362a16.jpg b/.raw/MidasCivilAnalysisReference/images/page-342_9d54df8515717f13c61fdb11840277d31ae806f4ea6d52d016d370f7df362a16.jpg new file mode 100644 index 00000000..fafbac6d Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-342_9d54df8515717f13c61fdb11840277d31ae806f4ea6d52d016d370f7df362a16.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-342_dee87500b593a08f83000b2dd080d91d57dc9a278ff52bdbf41597bc68bfb3c3.jpg b/.raw/MidasCivilAnalysisReference/images/page-342_dee87500b593a08f83000b2dd080d91d57dc9a278ff52bdbf41597bc68bfb3c3.jpg new file mode 100644 index 00000000..04111a3b Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-342_dee87500b593a08f83000b2dd080d91d57dc9a278ff52bdbf41597bc68bfb3c3.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-343_1a83fc7c6cc1b9244b90343adf6a61eab07f950e2883e8f757dc5b74b537d954.jpg b/.raw/MidasCivilAnalysisReference/images/page-343_1a83fc7c6cc1b9244b90343adf6a61eab07f950e2883e8f757dc5b74b537d954.jpg new file mode 100644 index 00000000..c5ad0fba Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-343_1a83fc7c6cc1b9244b90343adf6a61eab07f950e2883e8f757dc5b74b537d954.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-344_96ee1cadfb440f195747e76a11165d89bfe2147d7699ccca651aa727448b3f11.jpg b/.raw/MidasCivilAnalysisReference/images/page-344_96ee1cadfb440f195747e76a11165d89bfe2147d7699ccca651aa727448b3f11.jpg new file mode 100644 index 00000000..107236b0 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-344_96ee1cadfb440f195747e76a11165d89bfe2147d7699ccca651aa727448b3f11.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-344_f58decd9bbd0614e1a281787050211fa11e9d49d41cec5e629b358ca0adeb08e.jpg b/.raw/MidasCivilAnalysisReference/images/page-344_f58decd9bbd0614e1a281787050211fa11e9d49d41cec5e629b358ca0adeb08e.jpg new file mode 100644 index 00000000..b6e936dd Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-344_f58decd9bbd0614e1a281787050211fa11e9d49d41cec5e629b358ca0adeb08e.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-345_06af4d7d9754a0a452466aec2b4c6afaedc392eae500ff547bacbc425bf3c1f0.jpg b/.raw/MidasCivilAnalysisReference/images/page-345_06af4d7d9754a0a452466aec2b4c6afaedc392eae500ff547bacbc425bf3c1f0.jpg new file mode 100644 index 00000000..4c5df7bc Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-345_06af4d7d9754a0a452466aec2b4c6afaedc392eae500ff547bacbc425bf3c1f0.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-345_508937f7007a7c643cc61e7ecd530a546830e2ee287b5c3556c93b939062ac2b.jpg b/.raw/MidasCivilAnalysisReference/images/page-345_508937f7007a7c643cc61e7ecd530a546830e2ee287b5c3556c93b939062ac2b.jpg new file mode 100644 index 00000000..bb48ceb3 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-345_508937f7007a7c643cc61e7ecd530a546830e2ee287b5c3556c93b939062ac2b.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-345_6f43f704bb18fb30f213c3214ce2a30d15380a26cb7016f6f4b6d836a8c58719.jpg b/.raw/MidasCivilAnalysisReference/images/page-345_6f43f704bb18fb30f213c3214ce2a30d15380a26cb7016f6f4b6d836a8c58719.jpg new file mode 100644 index 00000000..c79c6fed Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-345_6f43f704bb18fb30f213c3214ce2a30d15380a26cb7016f6f4b6d836a8c58719.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-345_786b1ee4922a1a11854e330fed14c77fd054f01bff33cb17630035cbea470eb4.jpg b/.raw/MidasCivilAnalysisReference/images/page-345_786b1ee4922a1a11854e330fed14c77fd054f01bff33cb17630035cbea470eb4.jpg new file mode 100644 index 00000000..03e06a1a Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-345_786b1ee4922a1a11854e330fed14c77fd054f01bff33cb17630035cbea470eb4.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-345_df40cc447771c067fd39ca20c63b32fd21e57e08ec667e4357f9261f7951126e.jpg b/.raw/MidasCivilAnalysisReference/images/page-345_df40cc447771c067fd39ca20c63b32fd21e57e08ec667e4357f9261f7951126e.jpg new file mode 100644 index 00000000..2798bc04 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-345_df40cc447771c067fd39ca20c63b32fd21e57e08ec667e4357f9261f7951126e.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-346_120965fa32f77e21cf071915dded34666fe790e8d37ebe34b403ae48bb70411d.jpg b/.raw/MidasCivilAnalysisReference/images/page-346_120965fa32f77e21cf071915dded34666fe790e8d37ebe34b403ae48bb70411d.jpg new file mode 100644 index 00000000..8f3350bf Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-346_120965fa32f77e21cf071915dded34666fe790e8d37ebe34b403ae48bb70411d.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-346_54259e20c6d949b9e1daa9a9ab9970c9a6f0efb216e5a507f38666da6366e66e.jpg b/.raw/MidasCivilAnalysisReference/images/page-346_54259e20c6d949b9e1daa9a9ab9970c9a6f0efb216e5a507f38666da6366e66e.jpg new file mode 100644 index 00000000..ff2ac849 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-346_54259e20c6d949b9e1daa9a9ab9970c9a6f0efb216e5a507f38666da6366e66e.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-346_8a020665156e9ff0e16e2f511f5ee00d1f2e6cb87f4d74c3f4609b041f23cb6c.jpg b/.raw/MidasCivilAnalysisReference/images/page-346_8a020665156e9ff0e16e2f511f5ee00d1f2e6cb87f4d74c3f4609b041f23cb6c.jpg new file mode 100644 index 00000000..e40952f0 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-346_8a020665156e9ff0e16e2f511f5ee00d1f2e6cb87f4d74c3f4609b041f23cb6c.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-347_6a31363448c27f3ddb7eed7f71f94de016554758ed62aa28191cf6270f97d185.jpg b/.raw/MidasCivilAnalysisReference/images/page-347_6a31363448c27f3ddb7eed7f71f94de016554758ed62aa28191cf6270f97d185.jpg new file mode 100644 index 00000000..4d828dd6 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-347_6a31363448c27f3ddb7eed7f71f94de016554758ed62aa28191cf6270f97d185.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-347_71d7e8cc03b1c3998fb667ce21c4fe77f82cd1dfe38c0356702d768ee6ce1ff9.jpg b/.raw/MidasCivilAnalysisReference/images/page-347_71d7e8cc03b1c3998fb667ce21c4fe77f82cd1dfe38c0356702d768ee6ce1ff9.jpg new file mode 100644 index 00000000..4b9325a3 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-347_71d7e8cc03b1c3998fb667ce21c4fe77f82cd1dfe38c0356702d768ee6ce1ff9.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-347_7c2181091af234573abfe9c6b353544d6844ff8f59e5044af91b61461180cb43.jpg b/.raw/MidasCivilAnalysisReference/images/page-347_7c2181091af234573abfe9c6b353544d6844ff8f59e5044af91b61461180cb43.jpg new file mode 100644 index 00000000..aa938d66 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-347_7c2181091af234573abfe9c6b353544d6844ff8f59e5044af91b61461180cb43.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-348_0627f26099d0a2fc829dc9e538813fd7135bb931fa6d6b6e1837edebbc42aabb.jpg b/.raw/MidasCivilAnalysisReference/images/page-348_0627f26099d0a2fc829dc9e538813fd7135bb931fa6d6b6e1837edebbc42aabb.jpg new file mode 100644 index 00000000..9582b204 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-348_0627f26099d0a2fc829dc9e538813fd7135bb931fa6d6b6e1837edebbc42aabb.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-348_55688fa358c66a37178e7ff09810e52f9f09e9330639a84d882a07a29fd5e63d.jpg b/.raw/MidasCivilAnalysisReference/images/page-348_55688fa358c66a37178e7ff09810e52f9f09e9330639a84d882a07a29fd5e63d.jpg new file mode 100644 index 00000000..94465ea4 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-348_55688fa358c66a37178e7ff09810e52f9f09e9330639a84d882a07a29fd5e63d.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-348_9dbec6f3069908b7781fc60e8230cfe13b5fadf75daea5cc9125bd049a4a72a0.jpg b/.raw/MidasCivilAnalysisReference/images/page-348_9dbec6f3069908b7781fc60e8230cfe13b5fadf75daea5cc9125bd049a4a72a0.jpg new file mode 100644 index 00000000..60284094 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-348_9dbec6f3069908b7781fc60e8230cfe13b5fadf75daea5cc9125bd049a4a72a0.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-349_93c653607324510ff3ac6deca6e7931c548d23e35f44189a342f06e91f35337d.jpg b/.raw/MidasCivilAnalysisReference/images/page-349_93c653607324510ff3ac6deca6e7931c548d23e35f44189a342f06e91f35337d.jpg new file mode 100644 index 00000000..773d542e Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-349_93c653607324510ff3ac6deca6e7931c548d23e35f44189a342f06e91f35337d.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-350_123f8dad8d7e09953a74e939a8feccea96cdfd935c74df53303985685531fe0b.jpg b/.raw/MidasCivilAnalysisReference/images/page-350_123f8dad8d7e09953a74e939a8feccea96cdfd935c74df53303985685531fe0b.jpg new file mode 100644 index 00000000..f6e97ba0 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-350_123f8dad8d7e09953a74e939a8feccea96cdfd935c74df53303985685531fe0b.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-351_4c2edbbdeb4b582adccb5ef90bb13ef6e4ea773b53d0d3f7bbd76eb2ecb27022.jpg b/.raw/MidasCivilAnalysisReference/images/page-351_4c2edbbdeb4b582adccb5ef90bb13ef6e4ea773b53d0d3f7bbd76eb2ecb27022.jpg new file mode 100644 index 00000000..aff4f71a Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-351_4c2edbbdeb4b582adccb5ef90bb13ef6e4ea773b53d0d3f7bbd76eb2ecb27022.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-352_862bcbd8b76498530b24adb208919e5d18f8221e798bdf38bc5ed2d23b36321c.jpg b/.raw/MidasCivilAnalysisReference/images/page-352_862bcbd8b76498530b24adb208919e5d18f8221e798bdf38bc5ed2d23b36321c.jpg new file mode 100644 index 00000000..b6f3947c Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-352_862bcbd8b76498530b24adb208919e5d18f8221e798bdf38bc5ed2d23b36321c.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-352_8d1503bf21eb926be97a980dbead5ae790bfc6b53796528a970a76c331493a19.jpg b/.raw/MidasCivilAnalysisReference/images/page-352_8d1503bf21eb926be97a980dbead5ae790bfc6b53796528a970a76c331493a19.jpg new file mode 100644 index 00000000..392e7af4 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-352_8d1503bf21eb926be97a980dbead5ae790bfc6b53796528a970a76c331493a19.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-353_0560d03c37dc3e30fafb225efbafa7b29e0d23da878f2bb084958fb1cec9f313.jpg b/.raw/MidasCivilAnalysisReference/images/page-353_0560d03c37dc3e30fafb225efbafa7b29e0d23da878f2bb084958fb1cec9f313.jpg new file mode 100644 index 00000000..ba1c73ba Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-353_0560d03c37dc3e30fafb225efbafa7b29e0d23da878f2bb084958fb1cec9f313.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-355_74c5bdf0c387b15f8ba010b7d2f5527caa2c9c86493f0d74aabd0d95fb16819a.jpg b/.raw/MidasCivilAnalysisReference/images/page-355_74c5bdf0c387b15f8ba010b7d2f5527caa2c9c86493f0d74aabd0d95fb16819a.jpg new file mode 100644 index 00000000..b37c8601 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-355_74c5bdf0c387b15f8ba010b7d2f5527caa2c9c86493f0d74aabd0d95fb16819a.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-356_0e5350727787ffc508c2b20b620c27d6603cb810f007ed74adaa7941b2577c98.jpg b/.raw/MidasCivilAnalysisReference/images/page-356_0e5350727787ffc508c2b20b620c27d6603cb810f007ed74adaa7941b2577c98.jpg new file mode 100644 index 00000000..ecd96090 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-356_0e5350727787ffc508c2b20b620c27d6603cb810f007ed74adaa7941b2577c98.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-356_385b21b35ce12ac55d2c2bfadedefdc68377816f018fb350d954cfb939c3d1be.jpg b/.raw/MidasCivilAnalysisReference/images/page-356_385b21b35ce12ac55d2c2bfadedefdc68377816f018fb350d954cfb939c3d1be.jpg new file mode 100644 index 00000000..e4110cab Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-356_385b21b35ce12ac55d2c2bfadedefdc68377816f018fb350d954cfb939c3d1be.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-356_7e2738b627f6bafc9a7a9fa9c747746b7224f0bd0bd50421737c1bf46b8c13f2.jpg b/.raw/MidasCivilAnalysisReference/images/page-356_7e2738b627f6bafc9a7a9fa9c747746b7224f0bd0bd50421737c1bf46b8c13f2.jpg new file mode 100644 index 00000000..b8f70251 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-356_7e2738b627f6bafc9a7a9fa9c747746b7224f0bd0bd50421737c1bf46b8c13f2.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-356_8bbfa584a837ec06c83db1450900acdac2ba169ae0a1333c6903b6699768e3cf.jpg b/.raw/MidasCivilAnalysisReference/images/page-356_8bbfa584a837ec06c83db1450900acdac2ba169ae0a1333c6903b6699768e3cf.jpg new file mode 100644 index 00000000..710007f3 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-356_8bbfa584a837ec06c83db1450900acdac2ba169ae0a1333c6903b6699768e3cf.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-356_a174b49f1ebdce589faa7fed7545f8f9faa3c250b77f71c5b3f1d9812a34210f.jpg b/.raw/MidasCivilAnalysisReference/images/page-356_a174b49f1ebdce589faa7fed7545f8f9faa3c250b77f71c5b3f1d9812a34210f.jpg new file mode 100644 index 00000000..a48962d6 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-356_a174b49f1ebdce589faa7fed7545f8f9faa3c250b77f71c5b3f1d9812a34210f.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-356_b922a3f9af0aa0dbb981975af0205f69d0f22eb22cab8652f48cb70e381cdff3.jpg b/.raw/MidasCivilAnalysisReference/images/page-356_b922a3f9af0aa0dbb981975af0205f69d0f22eb22cab8652f48cb70e381cdff3.jpg new file mode 100644 index 00000000..741894bb Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-356_b922a3f9af0aa0dbb981975af0205f69d0f22eb22cab8652f48cb70e381cdff3.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-357_048d750979f6b10bf46d350e5421d70fd4753c17f3510517253dffcb3b210bb3.jpg b/.raw/MidasCivilAnalysisReference/images/page-357_048d750979f6b10bf46d350e5421d70fd4753c17f3510517253dffcb3b210bb3.jpg new file mode 100644 index 00000000..fd77bb74 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-357_048d750979f6b10bf46d350e5421d70fd4753c17f3510517253dffcb3b210bb3.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-357_2046b542ec535d52ac53cb983bfa8f84dd26209b88080d0d6399b151da901f69.jpg b/.raw/MidasCivilAnalysisReference/images/page-357_2046b542ec535d52ac53cb983bfa8f84dd26209b88080d0d6399b151da901f69.jpg new file mode 100644 index 00000000..80e2487b Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-357_2046b542ec535d52ac53cb983bfa8f84dd26209b88080d0d6399b151da901f69.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-357_467ce7515421207d25fcc4ecd308c1fbbbe64a4f49fefcb3580fcae2bcd36c09.jpg b/.raw/MidasCivilAnalysisReference/images/page-357_467ce7515421207d25fcc4ecd308c1fbbbe64a4f49fefcb3580fcae2bcd36c09.jpg new file mode 100644 index 00000000..78c08922 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-357_467ce7515421207d25fcc4ecd308c1fbbbe64a4f49fefcb3580fcae2bcd36c09.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-357_5567f435d0fd199356ae9af836c36babe7761f7aedcc6a3126043ef2e1efdd47.jpg b/.raw/MidasCivilAnalysisReference/images/page-357_5567f435d0fd199356ae9af836c36babe7761f7aedcc6a3126043ef2e1efdd47.jpg new file mode 100644 index 00000000..b566b227 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-357_5567f435d0fd199356ae9af836c36babe7761f7aedcc6a3126043ef2e1efdd47.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-357_76aefd10656ab53158f4a3899fb5260a4206b5077c4cf7d7c947001cfbc6abb0.jpg b/.raw/MidasCivilAnalysisReference/images/page-357_76aefd10656ab53158f4a3899fb5260a4206b5077c4cf7d7c947001cfbc6abb0.jpg new file mode 100644 index 00000000..fae99fee Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-357_76aefd10656ab53158f4a3899fb5260a4206b5077c4cf7d7c947001cfbc6abb0.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-357_91975d14104dff31eaefa9fdd5c3db71b0e09cd72a41137ae29ac293c7412fd1.jpg b/.raw/MidasCivilAnalysisReference/images/page-357_91975d14104dff31eaefa9fdd5c3db71b0e09cd72a41137ae29ac293c7412fd1.jpg new file mode 100644 index 00000000..dd89a503 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-357_91975d14104dff31eaefa9fdd5c3db71b0e09cd72a41137ae29ac293c7412fd1.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-357_a5db7e07e2afe11db0e3f6e0175455a71f3c8adb3286138cb6404ce9b2d58407.jpg b/.raw/MidasCivilAnalysisReference/images/page-357_a5db7e07e2afe11db0e3f6e0175455a71f3c8adb3286138cb6404ce9b2d58407.jpg new file mode 100644 index 00000000..023d68c0 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-357_a5db7e07e2afe11db0e3f6e0175455a71f3c8adb3286138cb6404ce9b2d58407.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-357_a7f090aa8d5b81ce8c3d2a136f14ff79a1757dfab1ba4221b21f4309c47caa54.jpg b/.raw/MidasCivilAnalysisReference/images/page-357_a7f090aa8d5b81ce8c3d2a136f14ff79a1757dfab1ba4221b21f4309c47caa54.jpg new file mode 100644 index 00000000..286ea65a Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-357_a7f090aa8d5b81ce8c3d2a136f14ff79a1757dfab1ba4221b21f4309c47caa54.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-358_2abd09639074efab887c20e30188c65b6d23afc6d27e78225e3aac3eab49db64.jpg b/.raw/MidasCivilAnalysisReference/images/page-358_2abd09639074efab887c20e30188c65b6d23afc6d27e78225e3aac3eab49db64.jpg new file mode 100644 index 00000000..dc5ab29b Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-358_2abd09639074efab887c20e30188c65b6d23afc6d27e78225e3aac3eab49db64.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-361_7cd2062b703c42e7758a70b6d8efc8f53e090a1d31c924ec2526ad3eb8827be1.jpg b/.raw/MidasCivilAnalysisReference/images/page-361_7cd2062b703c42e7758a70b6d8efc8f53e090a1d31c924ec2526ad3eb8827be1.jpg new file mode 100644 index 00000000..2f1c61df Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-361_7cd2062b703c42e7758a70b6d8efc8f53e090a1d31c924ec2526ad3eb8827be1.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-361_bc0e2b1c272fb7ca3c2c17f482bf7be71c548cdbf9fff46b2bb0bb9c68676305.jpg b/.raw/MidasCivilAnalysisReference/images/page-361_bc0e2b1c272fb7ca3c2c17f482bf7be71c548cdbf9fff46b2bb0bb9c68676305.jpg new file mode 100644 index 00000000..959102ab Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-361_bc0e2b1c272fb7ca3c2c17f482bf7be71c548cdbf9fff46b2bb0bb9c68676305.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-363_518d7f804d38d01796f2257450ea8db054c0cf8f5d3329a2c24f08802042bc7e.jpg b/.raw/MidasCivilAnalysisReference/images/page-363_518d7f804d38d01796f2257450ea8db054c0cf8f5d3329a2c24f08802042bc7e.jpg new file mode 100644 index 00000000..cf65ba37 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-363_518d7f804d38d01796f2257450ea8db054c0cf8f5d3329a2c24f08802042bc7e.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-363_647ef068c641dfd5a29a9862dee5b81fcc0dee9f8431a3d575cf7e7316f9f015.jpg b/.raw/MidasCivilAnalysisReference/images/page-363_647ef068c641dfd5a29a9862dee5b81fcc0dee9f8431a3d575cf7e7316f9f015.jpg new file mode 100644 index 00000000..48cf9dad Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-363_647ef068c641dfd5a29a9862dee5b81fcc0dee9f8431a3d575cf7e7316f9f015.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-365_8b9d0b907fd6f66a50825f5ee725c6201f04089c9d713ee9cb5657663048bb5f.jpg b/.raw/MidasCivilAnalysisReference/images/page-365_8b9d0b907fd6f66a50825f5ee725c6201f04089c9d713ee9cb5657663048bb5f.jpg new file mode 100644 index 00000000..c5c4ff2c Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-365_8b9d0b907fd6f66a50825f5ee725c6201f04089c9d713ee9cb5657663048bb5f.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-365_aa6cbb09ec4473fbf0a2ad11dad52589a207f81ac462f06c91a4dedcbf7c601f.jpg b/.raw/MidasCivilAnalysisReference/images/page-365_aa6cbb09ec4473fbf0a2ad11dad52589a207f81ac462f06c91a4dedcbf7c601f.jpg new file mode 100644 index 00000000..8c964795 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-365_aa6cbb09ec4473fbf0a2ad11dad52589a207f81ac462f06c91a4dedcbf7c601f.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-366_472028e5ba011df5fef31e5f577dd484f17c2fec662b092554f3af4ed3c68838.jpg b/.raw/MidasCivilAnalysisReference/images/page-366_472028e5ba011df5fef31e5f577dd484f17c2fec662b092554f3af4ed3c68838.jpg new file mode 100644 index 00000000..52cf9941 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-366_472028e5ba011df5fef31e5f577dd484f17c2fec662b092554f3af4ed3c68838.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-366_8732fab353fdbddc72a082084b3f107a63a95399f778dc972e83c6a17d5df75e.jpg b/.raw/MidasCivilAnalysisReference/images/page-366_8732fab353fdbddc72a082084b3f107a63a95399f778dc972e83c6a17d5df75e.jpg new file mode 100644 index 00000000..888e7375 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-366_8732fab353fdbddc72a082084b3f107a63a95399f778dc972e83c6a17d5df75e.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-366_93cca4265a31d298cdbea24a3d0d930117cd8028a8aa57909966476ce82183d5.jpg b/.raw/MidasCivilAnalysisReference/images/page-366_93cca4265a31d298cdbea24a3d0d930117cd8028a8aa57909966476ce82183d5.jpg new file mode 100644 index 00000000..35342035 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-366_93cca4265a31d298cdbea24a3d0d930117cd8028a8aa57909966476ce82183d5.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-366_dd45a43c5dd8f1334b8fee7177c185608e288c332d35faa65243fb411d2b5c32.jpg b/.raw/MidasCivilAnalysisReference/images/page-366_dd45a43c5dd8f1334b8fee7177c185608e288c332d35faa65243fb411d2b5c32.jpg new file mode 100644 index 00000000..9fcd35ba Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-366_dd45a43c5dd8f1334b8fee7177c185608e288c332d35faa65243fb411d2b5c32.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-367_11e225479e583c0ee1f06714da66438d0b44195003dc09d2c56d2792b9cc4b95.jpg b/.raw/MidasCivilAnalysisReference/images/page-367_11e225479e583c0ee1f06714da66438d0b44195003dc09d2c56d2792b9cc4b95.jpg new file mode 100644 index 00000000..1f495158 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-367_11e225479e583c0ee1f06714da66438d0b44195003dc09d2c56d2792b9cc4b95.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-367_2c44cfe617c5ebbc5ebdf306c5c0421f2234229a8cacc083547be9a20c97d659.jpg b/.raw/MidasCivilAnalysisReference/images/page-367_2c44cfe617c5ebbc5ebdf306c5c0421f2234229a8cacc083547be9a20c97d659.jpg new file mode 100644 index 00000000..5055f862 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-367_2c44cfe617c5ebbc5ebdf306c5c0421f2234229a8cacc083547be9a20c97d659.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-367_56a57fbf992280c7d782221e002a58ac8ab3b0c3d538f1404ede7299df0c1d95.jpg b/.raw/MidasCivilAnalysisReference/images/page-367_56a57fbf992280c7d782221e002a58ac8ab3b0c3d538f1404ede7299df0c1d95.jpg new file mode 100644 index 00000000..f34905be Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-367_56a57fbf992280c7d782221e002a58ac8ab3b0c3d538f1404ede7299df0c1d95.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-367_6f186fede7e2dd4457144d02100469320b0d3c511dcd5a3c4cac51ef3f4af0bf.jpg b/.raw/MidasCivilAnalysisReference/images/page-367_6f186fede7e2dd4457144d02100469320b0d3c511dcd5a3c4cac51ef3f4af0bf.jpg new file mode 100644 index 00000000..c7377e8c Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-367_6f186fede7e2dd4457144d02100469320b0d3c511dcd5a3c4cac51ef3f4af0bf.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-367_75e249025738d78c0f6c4d691c496675f0c0e494764e2e2a39a64a98d6a2ca96.jpg b/.raw/MidasCivilAnalysisReference/images/page-367_75e249025738d78c0f6c4d691c496675f0c0e494764e2e2a39a64a98d6a2ca96.jpg new file mode 100644 index 00000000..86a9c52d Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-367_75e249025738d78c0f6c4d691c496675f0c0e494764e2e2a39a64a98d6a2ca96.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-367_cc8dff9a1f222ae7e50e439a8a6c47a402ee3858e3b6bd018c55f1be3ac7fb96.jpg b/.raw/MidasCivilAnalysisReference/images/page-367_cc8dff9a1f222ae7e50e439a8a6c47a402ee3858e3b6bd018c55f1be3ac7fb96.jpg new file mode 100644 index 00000000..8cdd9c7f Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-367_cc8dff9a1f222ae7e50e439a8a6c47a402ee3858e3b6bd018c55f1be3ac7fb96.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-367_fdca8b81a0017bb91167f870fdf4daaff73bbdb46e77664ad6341f50c6b5c7af.jpg b/.raw/MidasCivilAnalysisReference/images/page-367_fdca8b81a0017bb91167f870fdf4daaff73bbdb46e77664ad6341f50c6b5c7af.jpg new file mode 100644 index 00000000..abb71744 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-367_fdca8b81a0017bb91167f870fdf4daaff73bbdb46e77664ad6341f50c6b5c7af.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-368_371a88122d2c7562f37338f87588737dc7ddbf0c56f040bc3e346230d205b083.jpg b/.raw/MidasCivilAnalysisReference/images/page-368_371a88122d2c7562f37338f87588737dc7ddbf0c56f040bc3e346230d205b083.jpg new file mode 100644 index 00000000..67a50c48 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-368_371a88122d2c7562f37338f87588737dc7ddbf0c56f040bc3e346230d205b083.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-368_50e5625b3248622ae4cd5114d0642d160850472b674c98099b1e3c8e63b0591f.jpg b/.raw/MidasCivilAnalysisReference/images/page-368_50e5625b3248622ae4cd5114d0642d160850472b674c98099b1e3c8e63b0591f.jpg new file mode 100644 index 00000000..f8648bac Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-368_50e5625b3248622ae4cd5114d0642d160850472b674c98099b1e3c8e63b0591f.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-368_baa03d7e75e510c7ee25acc5638c43e4c877000fb479014d4bf5c0e3e38e0de3.jpg b/.raw/MidasCivilAnalysisReference/images/page-368_baa03d7e75e510c7ee25acc5638c43e4c877000fb479014d4bf5c0e3e38e0de3.jpg new file mode 100644 index 00000000..cf1e0701 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-368_baa03d7e75e510c7ee25acc5638c43e4c877000fb479014d4bf5c0e3e38e0de3.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-368_dc85712f0506efac07d2dad398409497f7fb8e464c48718088c69afbec0fddee.jpg b/.raw/MidasCivilAnalysisReference/images/page-368_dc85712f0506efac07d2dad398409497f7fb8e464c48718088c69afbec0fddee.jpg new file mode 100644 index 00000000..31ed6f51 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-368_dc85712f0506efac07d2dad398409497f7fb8e464c48718088c69afbec0fddee.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-369_0448568b82c9b042fe7a5204df02cdb007af2ad33c492617d17db05d8f700afe.jpg b/.raw/MidasCivilAnalysisReference/images/page-369_0448568b82c9b042fe7a5204df02cdb007af2ad33c492617d17db05d8f700afe.jpg new file mode 100644 index 00000000..cc9214e6 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-369_0448568b82c9b042fe7a5204df02cdb007af2ad33c492617d17db05d8f700afe.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-369_32e80ba261accd3f4c0ba9c62e132d63e74015b81a2bcc1735e8e75cd46ac4cb.jpg b/.raw/MidasCivilAnalysisReference/images/page-369_32e80ba261accd3f4c0ba9c62e132d63e74015b81a2bcc1735e8e75cd46ac4cb.jpg new file mode 100644 index 00000000..2ca677c5 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-369_32e80ba261accd3f4c0ba9c62e132d63e74015b81a2bcc1735e8e75cd46ac4cb.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-369_6324cb3d3a941cf71850d9ac19932082385a04fd938575cdab3832d869c605c0.jpg b/.raw/MidasCivilAnalysisReference/images/page-369_6324cb3d3a941cf71850d9ac19932082385a04fd938575cdab3832d869c605c0.jpg new file mode 100644 index 00000000..d27cd27c Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-369_6324cb3d3a941cf71850d9ac19932082385a04fd938575cdab3832d869c605c0.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-369_7930399a78395558553f9f6e4d7b26c1c85478e12e0d26853f10e6756da9bba5.jpg b/.raw/MidasCivilAnalysisReference/images/page-369_7930399a78395558553f9f6e4d7b26c1c85478e12e0d26853f10e6756da9bba5.jpg new file mode 100644 index 00000000..7be4e7b2 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-369_7930399a78395558553f9f6e4d7b26c1c85478e12e0d26853f10e6756da9bba5.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-369_9b9736261657c0a3b0542f923f47e0cee2426ac996cf838ba593c7a966f21dfc.jpg b/.raw/MidasCivilAnalysisReference/images/page-369_9b9736261657c0a3b0542f923f47e0cee2426ac996cf838ba593c7a966f21dfc.jpg new file mode 100644 index 00000000..5cd00cd1 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-369_9b9736261657c0a3b0542f923f47e0cee2426ac996cf838ba593c7a966f21dfc.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-369_e251ab6171aea9d5e071d392acc04ecb70251a37d2356f666fb7c3043021b624.jpg b/.raw/MidasCivilAnalysisReference/images/page-369_e251ab6171aea9d5e071d392acc04ecb70251a37d2356f666fb7c3043021b624.jpg new file mode 100644 index 00000000..a4eb2273 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-369_e251ab6171aea9d5e071d392acc04ecb70251a37d2356f666fb7c3043021b624.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-370_090251038416f0939c8df35179b22dc24e35fb90f2acadbad5c2e3311e48648d.jpg b/.raw/MidasCivilAnalysisReference/images/page-370_090251038416f0939c8df35179b22dc24e35fb90f2acadbad5c2e3311e48648d.jpg new file mode 100644 index 00000000..a2ce8477 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-370_090251038416f0939c8df35179b22dc24e35fb90f2acadbad5c2e3311e48648d.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-371_5e11b54fdbe8a6453b4d16f22ef1ce9ff4a14b02d0529c035eeb28d0b5a1fc40.jpg b/.raw/MidasCivilAnalysisReference/images/page-371_5e11b54fdbe8a6453b4d16f22ef1ce9ff4a14b02d0529c035eeb28d0b5a1fc40.jpg new file mode 100644 index 00000000..b0a22ed5 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-371_5e11b54fdbe8a6453b4d16f22ef1ce9ff4a14b02d0529c035eeb28d0b5a1fc40.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-373_b4eeb570aaa3a3d440fbb049ba17c42867b9d581f136b10d176016fab8c487ba.jpg b/.raw/MidasCivilAnalysisReference/images/page-373_b4eeb570aaa3a3d440fbb049ba17c42867b9d581f136b10d176016fab8c487ba.jpg new file mode 100644 index 00000000..bf34b2a9 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-373_b4eeb570aaa3a3d440fbb049ba17c42867b9d581f136b10d176016fab8c487ba.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-374_0a7f1809fadbd2b31032311285bf33f4e52857b48430cbe5ab3eff5db59ba5e5.jpg b/.raw/MidasCivilAnalysisReference/images/page-374_0a7f1809fadbd2b31032311285bf33f4e52857b48430cbe5ab3eff5db59ba5e5.jpg new file mode 100644 index 00000000..45302e91 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-374_0a7f1809fadbd2b31032311285bf33f4e52857b48430cbe5ab3eff5db59ba5e5.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-375_24fe7c52c7424c0d28ff74dba9fb435290b528629d4c2c4c7761b4a3c64b6bf8.jpg b/.raw/MidasCivilAnalysisReference/images/page-375_24fe7c52c7424c0d28ff74dba9fb435290b528629d4c2c4c7761b4a3c64b6bf8.jpg new file mode 100644 index 00000000..50c3af58 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-375_24fe7c52c7424c0d28ff74dba9fb435290b528629d4c2c4c7761b4a3c64b6bf8.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-375_bdf95c05da8737027806889712bbb0b2bad1f9002264485f44db765095cb5b82.jpg b/.raw/MidasCivilAnalysisReference/images/page-375_bdf95c05da8737027806889712bbb0b2bad1f9002264485f44db765095cb5b82.jpg new file mode 100644 index 00000000..9e48b702 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-375_bdf95c05da8737027806889712bbb0b2bad1f9002264485f44db765095cb5b82.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-375_c5595499821201c292edcc8fc990b98bec0127e232959a82ce3c79852a7d63a4.jpg b/.raw/MidasCivilAnalysisReference/images/page-375_c5595499821201c292edcc8fc990b98bec0127e232959a82ce3c79852a7d63a4.jpg new file mode 100644 index 00000000..e462f370 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-375_c5595499821201c292edcc8fc990b98bec0127e232959a82ce3c79852a7d63a4.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-376_23b4167be163e18a662e2daf48f40fea6cc58dbd150e7019ae88c7bc40280887.jpg b/.raw/MidasCivilAnalysisReference/images/page-376_23b4167be163e18a662e2daf48f40fea6cc58dbd150e7019ae88c7bc40280887.jpg new file mode 100644 index 00000000..b4747fd6 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-376_23b4167be163e18a662e2daf48f40fea6cc58dbd150e7019ae88c7bc40280887.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-376_904dde8f84a13fea7750efc74feb0e9089ce4057f98eb2a6fe75fde812365bc0.jpg b/.raw/MidasCivilAnalysisReference/images/page-376_904dde8f84a13fea7750efc74feb0e9089ce4057f98eb2a6fe75fde812365bc0.jpg new file mode 100644 index 00000000..7bb5888d Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-376_904dde8f84a13fea7750efc74feb0e9089ce4057f98eb2a6fe75fde812365bc0.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-376_ff68a26750483aa24169cc8f5d4399de69d27fc8e9558aa1cdb4998903f36d9e.jpg b/.raw/MidasCivilAnalysisReference/images/page-376_ff68a26750483aa24169cc8f5d4399de69d27fc8e9558aa1cdb4998903f36d9e.jpg new file mode 100644 index 00000000..a8157b38 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-376_ff68a26750483aa24169cc8f5d4399de69d27fc8e9558aa1cdb4998903f36d9e.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-377_365892140fb9f184892320692666ecf9aa8bca98a4c2d93a02ca1388f72f1c2b.jpg b/.raw/MidasCivilAnalysisReference/images/page-377_365892140fb9f184892320692666ecf9aa8bca98a4c2d93a02ca1388f72f1c2b.jpg new file mode 100644 index 00000000..816ce845 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-377_365892140fb9f184892320692666ecf9aa8bca98a4c2d93a02ca1388f72f1c2b.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-378_0f45b39d8e62a16c554724fa2f91b1488686833a126cdfc6e6dd81c77da39177.jpg b/.raw/MidasCivilAnalysisReference/images/page-378_0f45b39d8e62a16c554724fa2f91b1488686833a126cdfc6e6dd81c77da39177.jpg new file mode 100644 index 00000000..62b199bd Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-378_0f45b39d8e62a16c554724fa2f91b1488686833a126cdfc6e6dd81c77da39177.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-378_4e0dabb8b45650b0ca730826bf54ce8980c170fc62b4b69ab412294f713bc83f.jpg b/.raw/MidasCivilAnalysisReference/images/page-378_4e0dabb8b45650b0ca730826bf54ce8980c170fc62b4b69ab412294f713bc83f.jpg new file mode 100644 index 00000000..59f3935d Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-378_4e0dabb8b45650b0ca730826bf54ce8980c170fc62b4b69ab412294f713bc83f.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-378_a7d8a98a54eebe2c4b8768b3e89d3462268b51e4065c43b745d806ba61fe956d.jpg b/.raw/MidasCivilAnalysisReference/images/page-378_a7d8a98a54eebe2c4b8768b3e89d3462268b51e4065c43b745d806ba61fe956d.jpg new file mode 100644 index 00000000..36a6650e Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-378_a7d8a98a54eebe2c4b8768b3e89d3462268b51e4065c43b745d806ba61fe956d.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-378_ad1c68b9c453b1b16fa6782c88985240762d49434cdb3d9e419097128b3ce519.jpg b/.raw/MidasCivilAnalysisReference/images/page-378_ad1c68b9c453b1b16fa6782c88985240762d49434cdb3d9e419097128b3ce519.jpg new file mode 100644 index 00000000..71d37ad5 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-378_ad1c68b9c453b1b16fa6782c88985240762d49434cdb3d9e419097128b3ce519.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-378_cb014fed2d3020ddfb645a52de977c9742f07ee120fdebb4e2e44afbe4d98d6a.jpg b/.raw/MidasCivilAnalysisReference/images/page-378_cb014fed2d3020ddfb645a52de977c9742f07ee120fdebb4e2e44afbe4d98d6a.jpg new file mode 100644 index 00000000..5097f014 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-378_cb014fed2d3020ddfb645a52de977c9742f07ee120fdebb4e2e44afbe4d98d6a.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-378_cdccc11e161300440164d06784743aa5f77402c00762039db476108ff2f8f121.jpg b/.raw/MidasCivilAnalysisReference/images/page-378_cdccc11e161300440164d06784743aa5f77402c00762039db476108ff2f8f121.jpg new file mode 100644 index 00000000..9fd6fdeb Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-378_cdccc11e161300440164d06784743aa5f77402c00762039db476108ff2f8f121.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-379_c16d2dc6f0dd444510305cbfacea09b22cd0f8641f76b322c30553a9f016ada0.jpg b/.raw/MidasCivilAnalysisReference/images/page-379_c16d2dc6f0dd444510305cbfacea09b22cd0f8641f76b322c30553a9f016ada0.jpg new file mode 100644 index 00000000..13ecfa30 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-379_c16d2dc6f0dd444510305cbfacea09b22cd0f8641f76b322c30553a9f016ada0.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-380_40413c518e817e73410f7c0f1c768be5b62ff88e64a8c521f594b8f752a17e13.jpg b/.raw/MidasCivilAnalysisReference/images/page-380_40413c518e817e73410f7c0f1c768be5b62ff88e64a8c521f594b8f752a17e13.jpg new file mode 100644 index 00000000..cb2d4dfc Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-380_40413c518e817e73410f7c0f1c768be5b62ff88e64a8c521f594b8f752a17e13.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-380_480c9a387230b6a975790b3a207266f45a0cfe094e961a2221e8020d386ad0e0.jpg b/.raw/MidasCivilAnalysisReference/images/page-380_480c9a387230b6a975790b3a207266f45a0cfe094e961a2221e8020d386ad0e0.jpg new file mode 100644 index 00000000..ec5864a7 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-380_480c9a387230b6a975790b3a207266f45a0cfe094e961a2221e8020d386ad0e0.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-380_4ec66ed975b32ecf5f2ef4786da928032e458105b6a35acc6439d78c8ef55feb.jpg b/.raw/MidasCivilAnalysisReference/images/page-380_4ec66ed975b32ecf5f2ef4786da928032e458105b6a35acc6439d78c8ef55feb.jpg new file mode 100644 index 00000000..8001ef56 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-380_4ec66ed975b32ecf5f2ef4786da928032e458105b6a35acc6439d78c8ef55feb.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-380_677f64e988b7c27763c385dfd7633ab1bc19ff1c931d9b9c730fe3e1693da223.jpg b/.raw/MidasCivilAnalysisReference/images/page-380_677f64e988b7c27763c385dfd7633ab1bc19ff1c931d9b9c730fe3e1693da223.jpg new file mode 100644 index 00000000..f6f9e9f1 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-380_677f64e988b7c27763c385dfd7633ab1bc19ff1c931d9b9c730fe3e1693da223.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-380_732a7c31e36cdb035c017591775a537d4c7c40c8cd0bdc466425a90fd455a5ac.jpg b/.raw/MidasCivilAnalysisReference/images/page-380_732a7c31e36cdb035c017591775a537d4c7c40c8cd0bdc466425a90fd455a5ac.jpg new file mode 100644 index 00000000..7d1321ed Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-380_732a7c31e36cdb035c017591775a537d4c7c40c8cd0bdc466425a90fd455a5ac.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-380_73f0f1deb260dceef0650a6bf236d8ef79011d8a0ab266614e697e47099741d0.jpg b/.raw/MidasCivilAnalysisReference/images/page-380_73f0f1deb260dceef0650a6bf236d8ef79011d8a0ab266614e697e47099741d0.jpg new file mode 100644 index 00000000..03e74b0a Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-380_73f0f1deb260dceef0650a6bf236d8ef79011d8a0ab266614e697e47099741d0.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-380_75fc8297a16fc6ba46b91b09fd039a490b84b608d912dafb1f92f269f67369e8.jpg b/.raw/MidasCivilAnalysisReference/images/page-380_75fc8297a16fc6ba46b91b09fd039a490b84b608d912dafb1f92f269f67369e8.jpg new file mode 100644 index 00000000..3baeea10 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-380_75fc8297a16fc6ba46b91b09fd039a490b84b608d912dafb1f92f269f67369e8.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-380_9a741a80e0c2460deaea5e060ae920f0cd414de4c9d6f285954537f81973a04d.jpg b/.raw/MidasCivilAnalysisReference/images/page-380_9a741a80e0c2460deaea5e060ae920f0cd414de4c9d6f285954537f81973a04d.jpg new file mode 100644 index 00000000..cb245e7e Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-380_9a741a80e0c2460deaea5e060ae920f0cd414de4c9d6f285954537f81973a04d.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-380_ab0efee062289caec5990d186213e5b16202dd2bdd72c14d76b2af4a428f6fa5.jpg b/.raw/MidasCivilAnalysisReference/images/page-380_ab0efee062289caec5990d186213e5b16202dd2bdd72c14d76b2af4a428f6fa5.jpg new file mode 100644 index 00000000..ed7ec099 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-380_ab0efee062289caec5990d186213e5b16202dd2bdd72c14d76b2af4a428f6fa5.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-380_ce8048c93a227ca2fae6d78d952dfc13c134fab5e0d09dfe79e5b96a3b29a379.jpg b/.raw/MidasCivilAnalysisReference/images/page-380_ce8048c93a227ca2fae6d78d952dfc13c134fab5e0d09dfe79e5b96a3b29a379.jpg new file mode 100644 index 00000000..a9c5d5c5 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-380_ce8048c93a227ca2fae6d78d952dfc13c134fab5e0d09dfe79e5b96a3b29a379.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-380_fc06a152c9a577ad9a32963746f4f9f705cd44b109e61c678e58fe1dc625193c.jpg b/.raw/MidasCivilAnalysisReference/images/page-380_fc06a152c9a577ad9a32963746f4f9f705cd44b109e61c678e58fe1dc625193c.jpg new file mode 100644 index 00000000..4c498591 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-380_fc06a152c9a577ad9a32963746f4f9f705cd44b109e61c678e58fe1dc625193c.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-381_65cc56bae09626319df92415f6a773ca21b9e017cd28d2733ff077f5cb27ca48.jpg b/.raw/MidasCivilAnalysisReference/images/page-381_65cc56bae09626319df92415f6a773ca21b9e017cd28d2733ff077f5cb27ca48.jpg new file mode 100644 index 00000000..a6737cfa Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-381_65cc56bae09626319df92415f6a773ca21b9e017cd28d2733ff077f5cb27ca48.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-382_4d5687d59d78c1cf54509174cc1a134855e1aec8c24b0423fc61999fe7ba4ca2.jpg b/.raw/MidasCivilAnalysisReference/images/page-382_4d5687d59d78c1cf54509174cc1a134855e1aec8c24b0423fc61999fe7ba4ca2.jpg new file mode 100644 index 00000000..9301fa54 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-382_4d5687d59d78c1cf54509174cc1a134855e1aec8c24b0423fc61999fe7ba4ca2.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-382_5bf57c5ceb1c08cc1959fb1e3a3b1a56121fed98d918ded1a680ed2fc1b8b02e.jpg b/.raw/MidasCivilAnalysisReference/images/page-382_5bf57c5ceb1c08cc1959fb1e3a3b1a56121fed98d918ded1a680ed2fc1b8b02e.jpg new file mode 100644 index 00000000..9c156c9e Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-382_5bf57c5ceb1c08cc1959fb1e3a3b1a56121fed98d918ded1a680ed2fc1b8b02e.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-382_728ee6b60c950b90fdf93321743468be044d49c4483e4318865cc78be22ade05.jpg b/.raw/MidasCivilAnalysisReference/images/page-382_728ee6b60c950b90fdf93321743468be044d49c4483e4318865cc78be22ade05.jpg new file mode 100644 index 00000000..47365868 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-382_728ee6b60c950b90fdf93321743468be044d49c4483e4318865cc78be22ade05.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-382_77b170aca7d3797e83d2f59899ca617f620c722fbb62c11e0613cb33e8dc6e1f.jpg b/.raw/MidasCivilAnalysisReference/images/page-382_77b170aca7d3797e83d2f59899ca617f620c722fbb62c11e0613cb33e8dc6e1f.jpg new file mode 100644 index 00000000..46c3ca82 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-382_77b170aca7d3797e83d2f59899ca617f620c722fbb62c11e0613cb33e8dc6e1f.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-382_965ca4322ad7189ea17a3c47581f6d47ff9e4bd0577e6e6666fe268111677952.jpg b/.raw/MidasCivilAnalysisReference/images/page-382_965ca4322ad7189ea17a3c47581f6d47ff9e4bd0577e6e6666fe268111677952.jpg new file mode 100644 index 00000000..d442c928 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-382_965ca4322ad7189ea17a3c47581f6d47ff9e4bd0577e6e6666fe268111677952.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-382_9990225ae1560db1b4c992549017b639b00e20343f8ad9bd611961182e6b1e04.jpg b/.raw/MidasCivilAnalysisReference/images/page-382_9990225ae1560db1b4c992549017b639b00e20343f8ad9bd611961182e6b1e04.jpg new file mode 100644 index 00000000..098021e0 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-382_9990225ae1560db1b4c992549017b639b00e20343f8ad9bd611961182e6b1e04.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-382_9ce7df9217f6215e99d56e34ad4ace2f697a7a3b0d58e8930b5854beddbde00a.jpg b/.raw/MidasCivilAnalysisReference/images/page-382_9ce7df9217f6215e99d56e34ad4ace2f697a7a3b0d58e8930b5854beddbde00a.jpg new file mode 100644 index 00000000..dbffc5a5 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-382_9ce7df9217f6215e99d56e34ad4ace2f697a7a3b0d58e8930b5854beddbde00a.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-382_b581a643588fa0c3880da121df321d37d7dabe352c9410fe2aa4c0bcb7ce1f35.jpg b/.raw/MidasCivilAnalysisReference/images/page-382_b581a643588fa0c3880da121df321d37d7dabe352c9410fe2aa4c0bcb7ce1f35.jpg new file mode 100644 index 00000000..aeba69bd Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-382_b581a643588fa0c3880da121df321d37d7dabe352c9410fe2aa4c0bcb7ce1f35.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-382_c624103d61de2f72bca48a2b13eebf9c87544848ffc9ec3dff074973f80b847c.jpg b/.raw/MidasCivilAnalysisReference/images/page-382_c624103d61de2f72bca48a2b13eebf9c87544848ffc9ec3dff074973f80b847c.jpg new file mode 100644 index 00000000..0e7da0a6 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-382_c624103d61de2f72bca48a2b13eebf9c87544848ffc9ec3dff074973f80b847c.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-382_caf13d73cbfd6d2a8873ef760dd26facbd8619e1820561bc17685ecb666e2652.jpg b/.raw/MidasCivilAnalysisReference/images/page-382_caf13d73cbfd6d2a8873ef760dd26facbd8619e1820561bc17685ecb666e2652.jpg new file mode 100644 index 00000000..aa2529e4 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-382_caf13d73cbfd6d2a8873ef760dd26facbd8619e1820561bc17685ecb666e2652.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-382_f8e9f542d76d30f0b6a0092bf257e1455902685c49924e168e923905e960c260.jpg b/.raw/MidasCivilAnalysisReference/images/page-382_f8e9f542d76d30f0b6a0092bf257e1455902685c49924e168e923905e960c260.jpg new file mode 100644 index 00000000..a735ed04 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-382_f8e9f542d76d30f0b6a0092bf257e1455902685c49924e168e923905e960c260.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-383_81dae41f1b58baf19424fda92e59fb9b7473bb0d0cbfb640535c60ab4422fdb5.jpg b/.raw/MidasCivilAnalysisReference/images/page-383_81dae41f1b58baf19424fda92e59fb9b7473bb0d0cbfb640535c60ab4422fdb5.jpg new file mode 100644 index 00000000..d25f389f Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-383_81dae41f1b58baf19424fda92e59fb9b7473bb0d0cbfb640535c60ab4422fdb5.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-384_25ad3cc2405f059c9cbbeb40a84de7dcd04c3c05e46bd877cf0be51577e6c34e.jpg b/.raw/MidasCivilAnalysisReference/images/page-384_25ad3cc2405f059c9cbbeb40a84de7dcd04c3c05e46bd877cf0be51577e6c34e.jpg new file mode 100644 index 00000000..d16e504a Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-384_25ad3cc2405f059c9cbbeb40a84de7dcd04c3c05e46bd877cf0be51577e6c34e.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-384_269a2a0c8c8e301c3c3bd4e05383d2bda15fd83d83d96610b1c4e402133bd57e.jpg b/.raw/MidasCivilAnalysisReference/images/page-384_269a2a0c8c8e301c3c3bd4e05383d2bda15fd83d83d96610b1c4e402133bd57e.jpg new file mode 100644 index 00000000..d130ccf2 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-384_269a2a0c8c8e301c3c3bd4e05383d2bda15fd83d83d96610b1c4e402133bd57e.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-384_30a70ded576b769aefe19663fd58aec518718f39358e862e573638d48795e34d.jpg b/.raw/MidasCivilAnalysisReference/images/page-384_30a70ded576b769aefe19663fd58aec518718f39358e862e573638d48795e34d.jpg new file mode 100644 index 00000000..9d6e8b57 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-384_30a70ded576b769aefe19663fd58aec518718f39358e862e573638d48795e34d.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-384_5e1d766d28c0e0ac397741e64990ada2552a3c833af3f08cf789a3950dc6173a.jpg b/.raw/MidasCivilAnalysisReference/images/page-384_5e1d766d28c0e0ac397741e64990ada2552a3c833af3f08cf789a3950dc6173a.jpg new file mode 100644 index 00000000..7833b496 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-384_5e1d766d28c0e0ac397741e64990ada2552a3c833af3f08cf789a3950dc6173a.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-384_97eb32462945c9d44ca1ffbd2e07fab08a6054d8ce612bb92e090fde0aa1b64c.jpg b/.raw/MidasCivilAnalysisReference/images/page-384_97eb32462945c9d44ca1ffbd2e07fab08a6054d8ce612bb92e090fde0aa1b64c.jpg new file mode 100644 index 00000000..409699eb Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-384_97eb32462945c9d44ca1ffbd2e07fab08a6054d8ce612bb92e090fde0aa1b64c.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-384_9efd1a7f1bad3b6b04018ad38a7a370e7aff54cc7223d88e47dc4954c4521ea9.jpg b/.raw/MidasCivilAnalysisReference/images/page-384_9efd1a7f1bad3b6b04018ad38a7a370e7aff54cc7223d88e47dc4954c4521ea9.jpg new file mode 100644 index 00000000..7b372974 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-384_9efd1a7f1bad3b6b04018ad38a7a370e7aff54cc7223d88e47dc4954c4521ea9.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-384_bf140edcb1e45bc46773bcbe21ce22cb2553c4adc141ae99aa42465e391faa00.jpg b/.raw/MidasCivilAnalysisReference/images/page-384_bf140edcb1e45bc46773bcbe21ce22cb2553c4adc141ae99aa42465e391faa00.jpg new file mode 100644 index 00000000..ea367043 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-384_bf140edcb1e45bc46773bcbe21ce22cb2553c4adc141ae99aa42465e391faa00.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-384_c1a3611acffbcb79cb284672ba0b28f8aa8f306db1f3f99cd5981543ff16f081.jpg b/.raw/MidasCivilAnalysisReference/images/page-384_c1a3611acffbcb79cb284672ba0b28f8aa8f306db1f3f99cd5981543ff16f081.jpg new file mode 100644 index 00000000..356c08ec Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-384_c1a3611acffbcb79cb284672ba0b28f8aa8f306db1f3f99cd5981543ff16f081.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-384_e36df3f2ea640d2062eed194062c77ea8a10fa11e2109b0266063b31617896a5.jpg b/.raw/MidasCivilAnalysisReference/images/page-384_e36df3f2ea640d2062eed194062c77ea8a10fa11e2109b0266063b31617896a5.jpg new file mode 100644 index 00000000..f875ac65 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-384_e36df3f2ea640d2062eed194062c77ea8a10fa11e2109b0266063b31617896a5.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-384_e73300bd6881401e266849c391380a99a3c334e3db75f70559fde318bf053bb9.jpg b/.raw/MidasCivilAnalysisReference/images/page-384_e73300bd6881401e266849c391380a99a3c334e3db75f70559fde318bf053bb9.jpg new file mode 100644 index 00000000..10e2a0b4 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-384_e73300bd6881401e266849c391380a99a3c334e3db75f70559fde318bf053bb9.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-384_f53b11cc3840a1afe79ed9e36bc468c116d495baf308ed31f15535e9a78215d9.jpg b/.raw/MidasCivilAnalysisReference/images/page-384_f53b11cc3840a1afe79ed9e36bc468c116d495baf308ed31f15535e9a78215d9.jpg new file mode 100644 index 00000000..9c4b4046 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-384_f53b11cc3840a1afe79ed9e36bc468c116d495baf308ed31f15535e9a78215d9.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-385_28fded88a53c02201ef88d888160cdf9be344b86db0c9fcef07e378bde5de801.jpg b/.raw/MidasCivilAnalysisReference/images/page-385_28fded88a53c02201ef88d888160cdf9be344b86db0c9fcef07e378bde5de801.jpg new file mode 100644 index 00000000..a844f4ed Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-385_28fded88a53c02201ef88d888160cdf9be344b86db0c9fcef07e378bde5de801.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-385_373abbcf16c664c0ead2506f24cd2930cdfd035b3caa0f154688d704b0668cd4.jpg b/.raw/MidasCivilAnalysisReference/images/page-385_373abbcf16c664c0ead2506f24cd2930cdfd035b3caa0f154688d704b0668cd4.jpg new file mode 100644 index 00000000..57463900 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-385_373abbcf16c664c0ead2506f24cd2930cdfd035b3caa0f154688d704b0668cd4.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-386_174232234284881312fd3a77bd47bee5659b9cedacbca2456af1b757403e710a.jpg b/.raw/MidasCivilAnalysisReference/images/page-386_174232234284881312fd3a77bd47bee5659b9cedacbca2456af1b757403e710a.jpg new file mode 100644 index 00000000..fe77fe62 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-386_174232234284881312fd3a77bd47bee5659b9cedacbca2456af1b757403e710a.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-386_30b981f3b108b4cb83e102d0b0e0c6816532ee49a8554ec4bb8bffffbde50c49.jpg b/.raw/MidasCivilAnalysisReference/images/page-386_30b981f3b108b4cb83e102d0b0e0c6816532ee49a8554ec4bb8bffffbde50c49.jpg new file mode 100644 index 00000000..b6005f60 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-386_30b981f3b108b4cb83e102d0b0e0c6816532ee49a8554ec4bb8bffffbde50c49.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-386_41f4a3b78c031cfd94aabd79e66573521b2e72a30233fd8448c729dc49986daf.jpg b/.raw/MidasCivilAnalysisReference/images/page-386_41f4a3b78c031cfd94aabd79e66573521b2e72a30233fd8448c729dc49986daf.jpg new file mode 100644 index 00000000..f3978324 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-386_41f4a3b78c031cfd94aabd79e66573521b2e72a30233fd8448c729dc49986daf.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-386_83d68cc988a5dafc5249eca3b87bfc76f847f3437e79adb8f9997bd5c7158389.jpg b/.raw/MidasCivilAnalysisReference/images/page-386_83d68cc988a5dafc5249eca3b87bfc76f847f3437e79adb8f9997bd5c7158389.jpg new file mode 100644 index 00000000..df79264b Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-386_83d68cc988a5dafc5249eca3b87bfc76f847f3437e79adb8f9997bd5c7158389.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-386_89d6114d1cb0538013109f62a891941e734c000ab355c8b96ae054ddb0f2d049.jpg b/.raw/MidasCivilAnalysisReference/images/page-386_89d6114d1cb0538013109f62a891941e734c000ab355c8b96ae054ddb0f2d049.jpg new file mode 100644 index 00000000..cfd8db5d Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-386_89d6114d1cb0538013109f62a891941e734c000ab355c8b96ae054ddb0f2d049.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-386_9a52443a374a286615620f9687ff47c24eacf8b4b01407a7ecf2e0cebe094d31.jpg b/.raw/MidasCivilAnalysisReference/images/page-386_9a52443a374a286615620f9687ff47c24eacf8b4b01407a7ecf2e0cebe094d31.jpg new file mode 100644 index 00000000..1f5c7632 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-386_9a52443a374a286615620f9687ff47c24eacf8b4b01407a7ecf2e0cebe094d31.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-386_b10e1094c829fae0c6c6d4e93252000176421b49b46c1f47a15e16e1caaca17a.jpg b/.raw/MidasCivilAnalysisReference/images/page-386_b10e1094c829fae0c6c6d4e93252000176421b49b46c1f47a15e16e1caaca17a.jpg new file mode 100644 index 00000000..3f74aeee Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-386_b10e1094c829fae0c6c6d4e93252000176421b49b46c1f47a15e16e1caaca17a.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-386_b1d2ad5b23eef9c6250261ddf35310582eba31c365042b946f9cadfcbfec2ce4.jpg b/.raw/MidasCivilAnalysisReference/images/page-386_b1d2ad5b23eef9c6250261ddf35310582eba31c365042b946f9cadfcbfec2ce4.jpg new file mode 100644 index 00000000..5394dc24 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-386_b1d2ad5b23eef9c6250261ddf35310582eba31c365042b946f9cadfcbfec2ce4.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-386_cb674441001cca6becd69983ef599437139570495166a34bfbb9910bc7629821.jpg b/.raw/MidasCivilAnalysisReference/images/page-386_cb674441001cca6becd69983ef599437139570495166a34bfbb9910bc7629821.jpg new file mode 100644 index 00000000..24fb1310 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-386_cb674441001cca6becd69983ef599437139570495166a34bfbb9910bc7629821.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-386_e33a9cc5d4530c3facdde7723e836a8d7cc2eaa41fdb141cb66c97d750498506.jpg b/.raw/MidasCivilAnalysisReference/images/page-386_e33a9cc5d4530c3facdde7723e836a8d7cc2eaa41fdb141cb66c97d750498506.jpg new file mode 100644 index 00000000..3a801f17 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-386_e33a9cc5d4530c3facdde7723e836a8d7cc2eaa41fdb141cb66c97d750498506.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-387_2889d453f57de3a557d464566e35efd190fc09a419758574a8dc76e61fb9ea0f.jpg b/.raw/MidasCivilAnalysisReference/images/page-387_2889d453f57de3a557d464566e35efd190fc09a419758574a8dc76e61fb9ea0f.jpg new file mode 100644 index 00000000..9ae9c5f1 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-387_2889d453f57de3a557d464566e35efd190fc09a419758574a8dc76e61fb9ea0f.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-388_18ebaf868b185e263558d3c6169b6bde223181e2ecaaf05a44f253c9c9bb4758.jpg b/.raw/MidasCivilAnalysisReference/images/page-388_18ebaf868b185e263558d3c6169b6bde223181e2ecaaf05a44f253c9c9bb4758.jpg new file mode 100644 index 00000000..99cfb45d Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-388_18ebaf868b185e263558d3c6169b6bde223181e2ecaaf05a44f253c9c9bb4758.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-388_558fa9a98fc70aa22eb8732399faf7c2ec3c88e1f2857435be984f03dc3d0fef.jpg b/.raw/MidasCivilAnalysisReference/images/page-388_558fa9a98fc70aa22eb8732399faf7c2ec3c88e1f2857435be984f03dc3d0fef.jpg new file mode 100644 index 00000000..7c043392 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-388_558fa9a98fc70aa22eb8732399faf7c2ec3c88e1f2857435be984f03dc3d0fef.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-388_91411705a5001407626f974ddb54f51fa4014c3274f01335fd4cbf66216261af.jpg b/.raw/MidasCivilAnalysisReference/images/page-388_91411705a5001407626f974ddb54f51fa4014c3274f01335fd4cbf66216261af.jpg new file mode 100644 index 00000000..28f3cdc6 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-388_91411705a5001407626f974ddb54f51fa4014c3274f01335fd4cbf66216261af.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-388_aac0f8595f9d1814773261643551738baa02407563abd6603539021f27c3b2b1.jpg b/.raw/MidasCivilAnalysisReference/images/page-388_aac0f8595f9d1814773261643551738baa02407563abd6603539021f27c3b2b1.jpg new file mode 100644 index 00000000..522c43d0 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-388_aac0f8595f9d1814773261643551738baa02407563abd6603539021f27c3b2b1.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-388_f2d281a51bad7325553e75d4958d00bcdea55eb61cc3aa63b250e01aa35f40cc.jpg b/.raw/MidasCivilAnalysisReference/images/page-388_f2d281a51bad7325553e75d4958d00bcdea55eb61cc3aa63b250e01aa35f40cc.jpg new file mode 100644 index 00000000..e22d7c32 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-388_f2d281a51bad7325553e75d4958d00bcdea55eb61cc3aa63b250e01aa35f40cc.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-389_08ab4c4801261d9e7fc17a47c7a58529ff315b51c12e24e440f9c7690bdb1827.jpg b/.raw/MidasCivilAnalysisReference/images/page-389_08ab4c4801261d9e7fc17a47c7a58529ff315b51c12e24e440f9c7690bdb1827.jpg new file mode 100644 index 00000000..67f7649a Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-389_08ab4c4801261d9e7fc17a47c7a58529ff315b51c12e24e440f9c7690bdb1827.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-390_158d47dd53a7506363019f0d21bc4392d5adacd85d342b964e86c2a738ec95fe.jpg b/.raw/MidasCivilAnalysisReference/images/page-390_158d47dd53a7506363019f0d21bc4392d5adacd85d342b964e86c2a738ec95fe.jpg new file mode 100644 index 00000000..9c1fc14d Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-390_158d47dd53a7506363019f0d21bc4392d5adacd85d342b964e86c2a738ec95fe.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-390_2401c1a8686d43a1313b7902c49013d0637fd76321ee3f47393773430053c351.jpg b/.raw/MidasCivilAnalysisReference/images/page-390_2401c1a8686d43a1313b7902c49013d0637fd76321ee3f47393773430053c351.jpg new file mode 100644 index 00000000..18b5dbf8 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-390_2401c1a8686d43a1313b7902c49013d0637fd76321ee3f47393773430053c351.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-390_24ffe895142563254f322e54b4cb75ef01cd0f7eb59ee986a31e90e981acc8f5.jpg b/.raw/MidasCivilAnalysisReference/images/page-390_24ffe895142563254f322e54b4cb75ef01cd0f7eb59ee986a31e90e981acc8f5.jpg new file mode 100644 index 00000000..7ea6e6dc Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-390_24ffe895142563254f322e54b4cb75ef01cd0f7eb59ee986a31e90e981acc8f5.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-390_6f1319c786975ef1b185256e37d0bbc6d9c3f3c50dc61a7a544b8fd1b352055d.jpg b/.raw/MidasCivilAnalysisReference/images/page-390_6f1319c786975ef1b185256e37d0bbc6d9c3f3c50dc61a7a544b8fd1b352055d.jpg new file mode 100644 index 00000000..2468088a Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-390_6f1319c786975ef1b185256e37d0bbc6d9c3f3c50dc61a7a544b8fd1b352055d.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-390_7163284492f899fa163c00ff82d2f55beae34e301ea942b16f423dfecfbe80bb.jpg b/.raw/MidasCivilAnalysisReference/images/page-390_7163284492f899fa163c00ff82d2f55beae34e301ea942b16f423dfecfbe80bb.jpg new file mode 100644 index 00000000..efc20cad Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-390_7163284492f899fa163c00ff82d2f55beae34e301ea942b16f423dfecfbe80bb.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-390_7432a89a186cd854259570b97444a453f80a1ce185b87c1deee07773befc7c6f.jpg b/.raw/MidasCivilAnalysisReference/images/page-390_7432a89a186cd854259570b97444a453f80a1ce185b87c1deee07773befc7c6f.jpg new file mode 100644 index 00000000..b6fc4c16 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-390_7432a89a186cd854259570b97444a453f80a1ce185b87c1deee07773befc7c6f.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-390_7ebf78ca750f683458c1a535e5bb90c8b1dc8303f51ff5f31a95582e0b7b0593.jpg b/.raw/MidasCivilAnalysisReference/images/page-390_7ebf78ca750f683458c1a535e5bb90c8b1dc8303f51ff5f31a95582e0b7b0593.jpg new file mode 100644 index 00000000..07759546 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-390_7ebf78ca750f683458c1a535e5bb90c8b1dc8303f51ff5f31a95582e0b7b0593.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-390_994f41dee45747fc650e937c96b1bfd3115493c8dc93dbe2ca06ef27a8a9e85c.jpg b/.raw/MidasCivilAnalysisReference/images/page-390_994f41dee45747fc650e937c96b1bfd3115493c8dc93dbe2ca06ef27a8a9e85c.jpg new file mode 100644 index 00000000..a201fc3e Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-390_994f41dee45747fc650e937c96b1bfd3115493c8dc93dbe2ca06ef27a8a9e85c.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-390_9f2c2c7a701adb13853b737137358ed9c54dd0432626f0cbf11c7439ca374e72.jpg b/.raw/MidasCivilAnalysisReference/images/page-390_9f2c2c7a701adb13853b737137358ed9c54dd0432626f0cbf11c7439ca374e72.jpg new file mode 100644 index 00000000..ad8c2948 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-390_9f2c2c7a701adb13853b737137358ed9c54dd0432626f0cbf11c7439ca374e72.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-390_d50bf9ebfea65c368553248853115a42fed833f867dcaddd8ac2e51943e86b6b.jpg b/.raw/MidasCivilAnalysisReference/images/page-390_d50bf9ebfea65c368553248853115a42fed833f867dcaddd8ac2e51943e86b6b.jpg new file mode 100644 index 00000000..139ea73f Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-390_d50bf9ebfea65c368553248853115a42fed833f867dcaddd8ac2e51943e86b6b.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-390_e55552430d848eb7aa0efed45247fcb3a8df3018185cc34fc1198ba721173c36.jpg b/.raw/MidasCivilAnalysisReference/images/page-390_e55552430d848eb7aa0efed45247fcb3a8df3018185cc34fc1198ba721173c36.jpg new file mode 100644 index 00000000..c3705d80 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-390_e55552430d848eb7aa0efed45247fcb3a8df3018185cc34fc1198ba721173c36.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-390_e6cfa4ab8eefc16519d90ee1d23c2bc6d66d7352b270b8ff97865a525bf0e392.jpg b/.raw/MidasCivilAnalysisReference/images/page-390_e6cfa4ab8eefc16519d90ee1d23c2bc6d66d7352b270b8ff97865a525bf0e392.jpg new file mode 100644 index 00000000..eea29c70 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-390_e6cfa4ab8eefc16519d90ee1d23c2bc6d66d7352b270b8ff97865a525bf0e392.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-390_eaa8a35e47b83b899d3db24dcdee975916cf81caa3fa8f9f555fcd050130625f.jpg b/.raw/MidasCivilAnalysisReference/images/page-390_eaa8a35e47b83b899d3db24dcdee975916cf81caa3fa8f9f555fcd050130625f.jpg new file mode 100644 index 00000000..294ca0ce Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-390_eaa8a35e47b83b899d3db24dcdee975916cf81caa3fa8f9f555fcd050130625f.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-391_649217cb40ef2ca2f2972ac04f5ed293ad893bc009cdb4b67aadd72ae92c35f2.jpg b/.raw/MidasCivilAnalysisReference/images/page-391_649217cb40ef2ca2f2972ac04f5ed293ad893bc009cdb4b67aadd72ae92c35f2.jpg new file mode 100644 index 00000000..524805b8 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-391_649217cb40ef2ca2f2972ac04f5ed293ad893bc009cdb4b67aadd72ae92c35f2.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-391_e747c080b64d85effdd0912b9e415bc0b845fb3386e08c8042c4e8d8ce8bd9ad.jpg b/.raw/MidasCivilAnalysisReference/images/page-391_e747c080b64d85effdd0912b9e415bc0b845fb3386e08c8042c4e8d8ce8bd9ad.jpg new file mode 100644 index 00000000..b443580c Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-391_e747c080b64d85effdd0912b9e415bc0b845fb3386e08c8042c4e8d8ce8bd9ad.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-392_a6681ed2251efd4642b1ce2faa957835e85b6ca62c3a756dd127b2b7951bf37b.jpg b/.raw/MidasCivilAnalysisReference/images/page-392_a6681ed2251efd4642b1ce2faa957835e85b6ca62c3a756dd127b2b7951bf37b.jpg new file mode 100644 index 00000000..897cd30a Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-392_a6681ed2251efd4642b1ce2faa957835e85b6ca62c3a756dd127b2b7951bf37b.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-392_bebd06e822494fb2cd8d18edce7b25c2d6ea8b4f1acf6714faca3d1b5399767f.jpg b/.raw/MidasCivilAnalysisReference/images/page-392_bebd06e822494fb2cd8d18edce7b25c2d6ea8b4f1acf6714faca3d1b5399767f.jpg new file mode 100644 index 00000000..36bc6421 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-392_bebd06e822494fb2cd8d18edce7b25c2d6ea8b4f1acf6714faca3d1b5399767f.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-393_5d1ec2269a443ca7931c773186feb6acf953bdb0d2d2b27a83b5a09b4c0fd13d.jpg b/.raw/MidasCivilAnalysisReference/images/page-393_5d1ec2269a443ca7931c773186feb6acf953bdb0d2d2b27a83b5a09b4c0fd13d.jpg new file mode 100644 index 00000000..3d2d0641 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-393_5d1ec2269a443ca7931c773186feb6acf953bdb0d2d2b27a83b5a09b4c0fd13d.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-394_01bd3b5f58d705b0f03ce63cf1ca912ea37aadd145a2f2c4af4675a60c19ea7e.jpg b/.raw/MidasCivilAnalysisReference/images/page-394_01bd3b5f58d705b0f03ce63cf1ca912ea37aadd145a2f2c4af4675a60c19ea7e.jpg new file mode 100644 index 00000000..89983172 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-394_01bd3b5f58d705b0f03ce63cf1ca912ea37aadd145a2f2c4af4675a60c19ea7e.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-394_0fb53ebc916b126b64579bf57296e0785800f9bf65c4d0b9d47419d1d1a7a92a.jpg b/.raw/MidasCivilAnalysisReference/images/page-394_0fb53ebc916b126b64579bf57296e0785800f9bf65c4d0b9d47419d1d1a7a92a.jpg new file mode 100644 index 00000000..a830e0e3 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-394_0fb53ebc916b126b64579bf57296e0785800f9bf65c4d0b9d47419d1d1a7a92a.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-394_2076df45623dbf5532385760412ae9b7d676a8af5218b1e2a7d9c85ac20c35a0.jpg b/.raw/MidasCivilAnalysisReference/images/page-394_2076df45623dbf5532385760412ae9b7d676a8af5218b1e2a7d9c85ac20c35a0.jpg new file mode 100644 index 00000000..dc39f2e9 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-394_2076df45623dbf5532385760412ae9b7d676a8af5218b1e2a7d9c85ac20c35a0.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-394_20f55d7a5467d326ac3c0cba166349cf4d71bb5fc211019a1f444d2abfa17ada.jpg b/.raw/MidasCivilAnalysisReference/images/page-394_20f55d7a5467d326ac3c0cba166349cf4d71bb5fc211019a1f444d2abfa17ada.jpg new file mode 100644 index 00000000..fb53abff Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-394_20f55d7a5467d326ac3c0cba166349cf4d71bb5fc211019a1f444d2abfa17ada.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-394_3386f2b34fdd3c556d7a463303babe8b92e6486d539682187903552c5ffab59e.jpg b/.raw/MidasCivilAnalysisReference/images/page-394_3386f2b34fdd3c556d7a463303babe8b92e6486d539682187903552c5ffab59e.jpg new file mode 100644 index 00000000..27b57592 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-394_3386f2b34fdd3c556d7a463303babe8b92e6486d539682187903552c5ffab59e.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-394_34f1107e3d2bcb1b985813a57d521d0739b20ac7fbf13a585c70ad18481f503e.jpg b/.raw/MidasCivilAnalysisReference/images/page-394_34f1107e3d2bcb1b985813a57d521d0739b20ac7fbf13a585c70ad18481f503e.jpg new file mode 100644 index 00000000..52d81c9e Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-394_34f1107e3d2bcb1b985813a57d521d0739b20ac7fbf13a585c70ad18481f503e.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-394_4b3d9dc938c98423c97a877d766d3bfd3fc49e61388cbfe11656c4d9d2d12a74.jpg b/.raw/MidasCivilAnalysisReference/images/page-394_4b3d9dc938c98423c97a877d766d3bfd3fc49e61388cbfe11656c4d9d2d12a74.jpg new file mode 100644 index 00000000..32a50796 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-394_4b3d9dc938c98423c97a877d766d3bfd3fc49e61388cbfe11656c4d9d2d12a74.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-394_7563d9cacdba1aa7b861863518451772260024ab874363460064e97c96ce32cd.jpg b/.raw/MidasCivilAnalysisReference/images/page-394_7563d9cacdba1aa7b861863518451772260024ab874363460064e97c96ce32cd.jpg new file mode 100644 index 00000000..28d802d1 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-394_7563d9cacdba1aa7b861863518451772260024ab874363460064e97c96ce32cd.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-394_7d29daf289f58bb3a9597c000f3c3a3d5e3cc16e0f18dd0868774d126b53a415.jpg b/.raw/MidasCivilAnalysisReference/images/page-394_7d29daf289f58bb3a9597c000f3c3a3d5e3cc16e0f18dd0868774d126b53a415.jpg new file mode 100644 index 00000000..0fd248d1 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-394_7d29daf289f58bb3a9597c000f3c3a3d5e3cc16e0f18dd0868774d126b53a415.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-394_7e094ff038865a8b63341e8e9f10172b362a4ab68b52ba67365233f5ce2e2e58.jpg b/.raw/MidasCivilAnalysisReference/images/page-394_7e094ff038865a8b63341e8e9f10172b362a4ab68b52ba67365233f5ce2e2e58.jpg new file mode 100644 index 00000000..7b1f20f0 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-394_7e094ff038865a8b63341e8e9f10172b362a4ab68b52ba67365233f5ce2e2e58.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-394_830c38794488fa0b8ada6c7c76b9d4bb169d4c23e6704a0b75ab1fe337f1c434.jpg b/.raw/MidasCivilAnalysisReference/images/page-394_830c38794488fa0b8ada6c7c76b9d4bb169d4c23e6704a0b75ab1fe337f1c434.jpg new file mode 100644 index 00000000..0bdbff5b Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-394_830c38794488fa0b8ada6c7c76b9d4bb169d4c23e6704a0b75ab1fe337f1c434.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-394_93403fc41497f6c1dffc9423040ce24eb412624c8a9c0679b3f99e6315dfb08f.jpg b/.raw/MidasCivilAnalysisReference/images/page-394_93403fc41497f6c1dffc9423040ce24eb412624c8a9c0679b3f99e6315dfb08f.jpg new file mode 100644 index 00000000..f4bb926a Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-394_93403fc41497f6c1dffc9423040ce24eb412624c8a9c0679b3f99e6315dfb08f.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-394_b9ba7eff971c16de584ebc7198ef4c929b28c2688f5c846a0156d3f4b173fff5.jpg b/.raw/MidasCivilAnalysisReference/images/page-394_b9ba7eff971c16de584ebc7198ef4c929b28c2688f5c846a0156d3f4b173fff5.jpg new file mode 100644 index 00000000..858caf39 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-394_b9ba7eff971c16de584ebc7198ef4c929b28c2688f5c846a0156d3f4b173fff5.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-394_bd841eadce86ca3f73da4afbe6a38d08f16f7a7940ca593fe1eb4d28754a8f78.jpg b/.raw/MidasCivilAnalysisReference/images/page-394_bd841eadce86ca3f73da4afbe6a38d08f16f7a7940ca593fe1eb4d28754a8f78.jpg new file mode 100644 index 00000000..318731db Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-394_bd841eadce86ca3f73da4afbe6a38d08f16f7a7940ca593fe1eb4d28754a8f78.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-394_d81a8d4dbeaac39836171e54a88fa0d2691286ed505ebc70d46e113f9329cf76.jpg b/.raw/MidasCivilAnalysisReference/images/page-394_d81a8d4dbeaac39836171e54a88fa0d2691286ed505ebc70d46e113f9329cf76.jpg new file mode 100644 index 00000000..52171f96 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-394_d81a8d4dbeaac39836171e54a88fa0d2691286ed505ebc70d46e113f9329cf76.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-394_e9212fbb264ed815772751cecc9398c4262a60b134466f806d8bc0dd1e393f78.jpg b/.raw/MidasCivilAnalysisReference/images/page-394_e9212fbb264ed815772751cecc9398c4262a60b134466f806d8bc0dd1e393f78.jpg new file mode 100644 index 00000000..59b34c06 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-394_e9212fbb264ed815772751cecc9398c4262a60b134466f806d8bc0dd1e393f78.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-395_dbeeac1295ed5a0eaaa6c774d524ffad04caf0cdcc1d2e961b3bbcb50f34df3f.jpg b/.raw/MidasCivilAnalysisReference/images/page-395_dbeeac1295ed5a0eaaa6c774d524ffad04caf0cdcc1d2e961b3bbcb50f34df3f.jpg new file mode 100644 index 00000000..4f120249 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-395_dbeeac1295ed5a0eaaa6c774d524ffad04caf0cdcc1d2e961b3bbcb50f34df3f.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-396_19c414cb3a81173ffc2c2a7a68cfdbddb5f4ddc8d596116a0b8ac6dcee256b51.jpg b/.raw/MidasCivilAnalysisReference/images/page-396_19c414cb3a81173ffc2c2a7a68cfdbddb5f4ddc8d596116a0b8ac6dcee256b51.jpg new file mode 100644 index 00000000..720904b8 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-396_19c414cb3a81173ffc2c2a7a68cfdbddb5f4ddc8d596116a0b8ac6dcee256b51.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-396_28cc5feba7540abbf730531c26afe1a64439526fdc6beef0980f840bae93968d.jpg b/.raw/MidasCivilAnalysisReference/images/page-396_28cc5feba7540abbf730531c26afe1a64439526fdc6beef0980f840bae93968d.jpg new file mode 100644 index 00000000..47fb3972 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-396_28cc5feba7540abbf730531c26afe1a64439526fdc6beef0980f840bae93968d.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-396_40f642836c704739a513f28dc591fb790361b534e737f27bd69b4435d6d55631.jpg b/.raw/MidasCivilAnalysisReference/images/page-396_40f642836c704739a513f28dc591fb790361b534e737f27bd69b4435d6d55631.jpg new file mode 100644 index 00000000..db36c100 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-396_40f642836c704739a513f28dc591fb790361b534e737f27bd69b4435d6d55631.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-396_4cac5c38a6a4776c05edff7183836cdf096acc6021bdc4041cfb513cdc72153e.jpg b/.raw/MidasCivilAnalysisReference/images/page-396_4cac5c38a6a4776c05edff7183836cdf096acc6021bdc4041cfb513cdc72153e.jpg new file mode 100644 index 00000000..560da0e1 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-396_4cac5c38a6a4776c05edff7183836cdf096acc6021bdc4041cfb513cdc72153e.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-396_51d4468da07aa20aea52361741a9d45d46ed5d1256e467b818ea61abdc223dfa.jpg b/.raw/MidasCivilAnalysisReference/images/page-396_51d4468da07aa20aea52361741a9d45d46ed5d1256e467b818ea61abdc223dfa.jpg new file mode 100644 index 00000000..304eb98b Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-396_51d4468da07aa20aea52361741a9d45d46ed5d1256e467b818ea61abdc223dfa.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-396_746d20b7bcc38662b5e7b8ce35b474e7d9e6941ab17d5dfc1ecebcf9e4a0c108.jpg b/.raw/MidasCivilAnalysisReference/images/page-396_746d20b7bcc38662b5e7b8ce35b474e7d9e6941ab17d5dfc1ecebcf9e4a0c108.jpg new file mode 100644 index 00000000..9689fdcf Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-396_746d20b7bcc38662b5e7b8ce35b474e7d9e6941ab17d5dfc1ecebcf9e4a0c108.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-396_77784b02cbf90d8b5292ddf35caf85894cfcb22534e2fbb60aed0bf5eba1ca60.jpg b/.raw/MidasCivilAnalysisReference/images/page-396_77784b02cbf90d8b5292ddf35caf85894cfcb22534e2fbb60aed0bf5eba1ca60.jpg new file mode 100644 index 00000000..b4b40de0 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-396_77784b02cbf90d8b5292ddf35caf85894cfcb22534e2fbb60aed0bf5eba1ca60.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-396_800f9deb61fa6a46c1ab1556a520ccb76e953a6a7cffa1bb39de680025ce2f5d.jpg b/.raw/MidasCivilAnalysisReference/images/page-396_800f9deb61fa6a46c1ab1556a520ccb76e953a6a7cffa1bb39de680025ce2f5d.jpg new file mode 100644 index 00000000..feb85a13 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-396_800f9deb61fa6a46c1ab1556a520ccb76e953a6a7cffa1bb39de680025ce2f5d.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-396_a8fc12d2a9e589c6e2a252695264081ff31d56607c5498adba6887c11d710ae8.jpg b/.raw/MidasCivilAnalysisReference/images/page-396_a8fc12d2a9e589c6e2a252695264081ff31d56607c5498adba6887c11d710ae8.jpg new file mode 100644 index 00000000..2d458610 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-396_a8fc12d2a9e589c6e2a252695264081ff31d56607c5498adba6887c11d710ae8.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-396_d089ba0e6a0bebf7f0a626b85b6051b712f566a1068446a4c96d8f057db01ac1.jpg b/.raw/MidasCivilAnalysisReference/images/page-396_d089ba0e6a0bebf7f0a626b85b6051b712f566a1068446a4c96d8f057db01ac1.jpg new file mode 100644 index 00000000..04c226eb Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-396_d089ba0e6a0bebf7f0a626b85b6051b712f566a1068446a4c96d8f057db01ac1.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-396_d6f0828dde9744d0ac441a0ceefe9b441d349d66859a65dc60e02cda7ca8d87d.jpg b/.raw/MidasCivilAnalysisReference/images/page-396_d6f0828dde9744d0ac441a0ceefe9b441d349d66859a65dc60e02cda7ca8d87d.jpg new file mode 100644 index 00000000..0a7f4ca1 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-396_d6f0828dde9744d0ac441a0ceefe9b441d349d66859a65dc60e02cda7ca8d87d.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-396_e7aac2bc9d5f7555776ecf8644558faeb0313f88f580e0fbf90a1a442349b76d.jpg b/.raw/MidasCivilAnalysisReference/images/page-396_e7aac2bc9d5f7555776ecf8644558faeb0313f88f580e0fbf90a1a442349b76d.jpg new file mode 100644 index 00000000..1b51cfa2 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-396_e7aac2bc9d5f7555776ecf8644558faeb0313f88f580e0fbf90a1a442349b76d.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-396_fde3fa2c6658acc347bc9aa959bcaf0ccb69825d96ab056186a6c27ed1eb3015.jpg b/.raw/MidasCivilAnalysisReference/images/page-396_fde3fa2c6658acc347bc9aa959bcaf0ccb69825d96ab056186a6c27ed1eb3015.jpg new file mode 100644 index 00000000..641dee4a Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-396_fde3fa2c6658acc347bc9aa959bcaf0ccb69825d96ab056186a6c27ed1eb3015.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-397_05b65f9526cbacf484384a313b6b9ae86bc84e48a018eb95ae29cafbfa948c6f.jpg b/.raw/MidasCivilAnalysisReference/images/page-397_05b65f9526cbacf484384a313b6b9ae86bc84e48a018eb95ae29cafbfa948c6f.jpg new file mode 100644 index 00000000..300f8bdd Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-397_05b65f9526cbacf484384a313b6b9ae86bc84e48a018eb95ae29cafbfa948c6f.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-397_6c281edccc0fe6e3d2c21f4583872b3f20a747a5832473d78ebb03f132eb7226.jpg b/.raw/MidasCivilAnalysisReference/images/page-397_6c281edccc0fe6e3d2c21f4583872b3f20a747a5832473d78ebb03f132eb7226.jpg new file mode 100644 index 00000000..410adbc5 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-397_6c281edccc0fe6e3d2c21f4583872b3f20a747a5832473d78ebb03f132eb7226.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-398_77631913f7131708c900f65032f522429973840f883b3ddebe8e24dd91d08368.jpg b/.raw/MidasCivilAnalysisReference/images/page-398_77631913f7131708c900f65032f522429973840f883b3ddebe8e24dd91d08368.jpg new file mode 100644 index 00000000..2af4c6d3 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-398_77631913f7131708c900f65032f522429973840f883b3ddebe8e24dd91d08368.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-398_e3592a6509fd98e01d53a2eb835dc7e5677b170e6d8d730d417cb0a2ec084571.jpg b/.raw/MidasCivilAnalysisReference/images/page-398_e3592a6509fd98e01d53a2eb835dc7e5677b170e6d8d730d417cb0a2ec084571.jpg new file mode 100644 index 00000000..0383ffe5 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-398_e3592a6509fd98e01d53a2eb835dc7e5677b170e6d8d730d417cb0a2ec084571.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-399_b80a975183c986e71e2967375f1a0a2ed9ffa2ff1c89d16e764ad4fd1d16a2dc.jpg b/.raw/MidasCivilAnalysisReference/images/page-399_b80a975183c986e71e2967375f1a0a2ed9ffa2ff1c89d16e764ad4fd1d16a2dc.jpg new file mode 100644 index 00000000..6b6bd190 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-399_b80a975183c986e71e2967375f1a0a2ed9ffa2ff1c89d16e764ad4fd1d16a2dc.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-401_9fa6fcc2433a5b697a4b2424158dc53a340a89dc2f55d18246007bbf9a00d44b.jpg b/.raw/MidasCivilAnalysisReference/images/page-401_9fa6fcc2433a5b697a4b2424158dc53a340a89dc2f55d18246007bbf9a00d44b.jpg new file mode 100644 index 00000000..c3062932 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-401_9fa6fcc2433a5b697a4b2424158dc53a340a89dc2f55d18246007bbf9a00d44b.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-402_067530b614853f8c4be3b6fee8a06d851c028359e9be9b43a08776764e511768.jpg b/.raw/MidasCivilAnalysisReference/images/page-402_067530b614853f8c4be3b6fee8a06d851c028359e9be9b43a08776764e511768.jpg new file mode 100644 index 00000000..6d8b4342 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-402_067530b614853f8c4be3b6fee8a06d851c028359e9be9b43a08776764e511768.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-402_2933745ea68beb9f2c776100b47da2c4727dc339849ab44b62345f8ea8123b1d.jpg b/.raw/MidasCivilAnalysisReference/images/page-402_2933745ea68beb9f2c776100b47da2c4727dc339849ab44b62345f8ea8123b1d.jpg new file mode 100644 index 00000000..7960cb97 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-402_2933745ea68beb9f2c776100b47da2c4727dc339849ab44b62345f8ea8123b1d.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-402_88f3b6bcd9931ce391c00511f2899a56774d122fef03bf6d0d315b205b13a76c.jpg b/.raw/MidasCivilAnalysisReference/images/page-402_88f3b6bcd9931ce391c00511f2899a56774d122fef03bf6d0d315b205b13a76c.jpg new file mode 100644 index 00000000..3baeea10 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-402_88f3b6bcd9931ce391c00511f2899a56774d122fef03bf6d0d315b205b13a76c.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-402_a0a3d77014376277468951ed0733f6d4e528e0afdc2811161c06ba5de754dca3.jpg b/.raw/MidasCivilAnalysisReference/images/page-402_a0a3d77014376277468951ed0733f6d4e528e0afdc2811161c06ba5de754dca3.jpg new file mode 100644 index 00000000..699b71ce Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-402_a0a3d77014376277468951ed0733f6d4e528e0afdc2811161c06ba5de754dca3.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-402_b64a7038f99089427634bc06feaf5b72191fa295f10d47412c2e9e96ce1cb9e6.jpg b/.raw/MidasCivilAnalysisReference/images/page-402_b64a7038f99089427634bc06feaf5b72191fa295f10d47412c2e9e96ce1cb9e6.jpg new file mode 100644 index 00000000..b7c64c58 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-402_b64a7038f99089427634bc06feaf5b72191fa295f10d47412c2e9e96ce1cb9e6.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-402_cf29f8c68301ad5818efb644d7aa180b6f356b4c3ba9ff1c11546490c91d0803.jpg b/.raw/MidasCivilAnalysisReference/images/page-402_cf29f8c68301ad5818efb644d7aa180b6f356b4c3ba9ff1c11546490c91d0803.jpg new file mode 100644 index 00000000..fab96681 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-402_cf29f8c68301ad5818efb644d7aa180b6f356b4c3ba9ff1c11546490c91d0803.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-403_ba4b1b8b380da722adc86139af2b8643020347c72f5a813b402e75745d14ed70.jpg b/.raw/MidasCivilAnalysisReference/images/page-403_ba4b1b8b380da722adc86139af2b8643020347c72f5a813b402e75745d14ed70.jpg new file mode 100644 index 00000000..edb99e15 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-403_ba4b1b8b380da722adc86139af2b8643020347c72f5a813b402e75745d14ed70.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-404_1e7aad84dae64f76dc8715a627e2f489f250c30213b15620adcf73c29e694343.jpg b/.raw/MidasCivilAnalysisReference/images/page-404_1e7aad84dae64f76dc8715a627e2f489f250c30213b15620adcf73c29e694343.jpg new file mode 100644 index 00000000..b3971a83 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-404_1e7aad84dae64f76dc8715a627e2f489f250c30213b15620adcf73c29e694343.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-404_233aa10f06888ac575006aaa4044f0057f4e9d15868cea8033a147d52b365ddc.jpg b/.raw/MidasCivilAnalysisReference/images/page-404_233aa10f06888ac575006aaa4044f0057f4e9d15868cea8033a147d52b365ddc.jpg new file mode 100644 index 00000000..65dcacc1 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-404_233aa10f06888ac575006aaa4044f0057f4e9d15868cea8033a147d52b365ddc.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-404_245f17a0a7e7595d5e8b6c08c0e838ff4830441360c631e177ec67628e083440.jpg b/.raw/MidasCivilAnalysisReference/images/page-404_245f17a0a7e7595d5e8b6c08c0e838ff4830441360c631e177ec67628e083440.jpg new file mode 100644 index 00000000..8c5f79d4 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-404_245f17a0a7e7595d5e8b6c08c0e838ff4830441360c631e177ec67628e083440.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-404_27d8358beb4dbfce0475e7aa5bcca11f5f69192de146f087707e53cc9d3ea3b9.jpg b/.raw/MidasCivilAnalysisReference/images/page-404_27d8358beb4dbfce0475e7aa5bcca11f5f69192de146f087707e53cc9d3ea3b9.jpg new file mode 100644 index 00000000..9b9a11e0 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-404_27d8358beb4dbfce0475e7aa5bcca11f5f69192de146f087707e53cc9d3ea3b9.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-404_487b6c89b2a41cb17f5ac012ed9e3f403e409a975904fdf36042b01f630ad845.jpg b/.raw/MidasCivilAnalysisReference/images/page-404_487b6c89b2a41cb17f5ac012ed9e3f403e409a975904fdf36042b01f630ad845.jpg new file mode 100644 index 00000000..7a592bf8 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-404_487b6c89b2a41cb17f5ac012ed9e3f403e409a975904fdf36042b01f630ad845.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-404_48d40b4dd76ac46bb2b5c353eefe9d87fdd82c906b90de675abb88d2e9c19c3e.jpg b/.raw/MidasCivilAnalysisReference/images/page-404_48d40b4dd76ac46bb2b5c353eefe9d87fdd82c906b90de675abb88d2e9c19c3e.jpg new file mode 100644 index 00000000..632bb3cf Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-404_48d40b4dd76ac46bb2b5c353eefe9d87fdd82c906b90de675abb88d2e9c19c3e.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-404_84d93ab474621d68dfcd03ac17a6ad44f931db8366a85281d23f4015624135cf.jpg b/.raw/MidasCivilAnalysisReference/images/page-404_84d93ab474621d68dfcd03ac17a6ad44f931db8366a85281d23f4015624135cf.jpg new file mode 100644 index 00000000..b0495220 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-404_84d93ab474621d68dfcd03ac17a6ad44f931db8366a85281d23f4015624135cf.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-404_8bf446b35044b9ae4fb66debfb8452644f73f340b6fd67b1754a9e1bad64423f.jpg b/.raw/MidasCivilAnalysisReference/images/page-404_8bf446b35044b9ae4fb66debfb8452644f73f340b6fd67b1754a9e1bad64423f.jpg new file mode 100644 index 00000000..8bf21247 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-404_8bf446b35044b9ae4fb66debfb8452644f73f340b6fd67b1754a9e1bad64423f.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-404_9263fc4b31360e7793703c11dcb8f1c990334cad1fbe1c1555561945f396bc29.jpg b/.raw/MidasCivilAnalysisReference/images/page-404_9263fc4b31360e7793703c11dcb8f1c990334cad1fbe1c1555561945f396bc29.jpg new file mode 100644 index 00000000..1fd6d37b Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-404_9263fc4b31360e7793703c11dcb8f1c990334cad1fbe1c1555561945f396bc29.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-404_b5313ff93c3810632be5a8d932d8d98906c7e77c8ba4bc1602385589dffb602c.jpg b/.raw/MidasCivilAnalysisReference/images/page-404_b5313ff93c3810632be5a8d932d8d98906c7e77c8ba4bc1602385589dffb602c.jpg new file mode 100644 index 00000000..4761925b Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-404_b5313ff93c3810632be5a8d932d8d98906c7e77c8ba4bc1602385589dffb602c.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-404_b7fbadfc995cec2b67690f858d17e9106704cd3f0f69e29a62707ed501242376.jpg b/.raw/MidasCivilAnalysisReference/images/page-404_b7fbadfc995cec2b67690f858d17e9106704cd3f0f69e29a62707ed501242376.jpg new file mode 100644 index 00000000..6c90c8c4 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-404_b7fbadfc995cec2b67690f858d17e9106704cd3f0f69e29a62707ed501242376.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-405_047dfcf9ddf4a9a19d071395e60b6815c2215ed74a851ed49e215a95c62021ce.jpg b/.raw/MidasCivilAnalysisReference/images/page-405_047dfcf9ddf4a9a19d071395e60b6815c2215ed74a851ed49e215a95c62021ce.jpg new file mode 100644 index 00000000..475719a5 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-405_047dfcf9ddf4a9a19d071395e60b6815c2215ed74a851ed49e215a95c62021ce.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-405_0ff8136378ef0fc4a6a81803cf26d2246e4b139b8327964877f854a0d99de0b0.jpg b/.raw/MidasCivilAnalysisReference/images/page-405_0ff8136378ef0fc4a6a81803cf26d2246e4b139b8327964877f854a0d99de0b0.jpg new file mode 100644 index 00000000..592448b3 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-405_0ff8136378ef0fc4a6a81803cf26d2246e4b139b8327964877f854a0d99de0b0.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-405_3ee5b90e4899797bdd21a9cc62fe579e2fc8c4f21c0d624ad131199c01f6dbfd.jpg b/.raw/MidasCivilAnalysisReference/images/page-405_3ee5b90e4899797bdd21a9cc62fe579e2fc8c4f21c0d624ad131199c01f6dbfd.jpg new file mode 100644 index 00000000..db36c100 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-405_3ee5b90e4899797bdd21a9cc62fe579e2fc8c4f21c0d624ad131199c01f6dbfd.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-405_65ec8a90c6f5bebac8201199a9502c36f61312ae05c4b29dfb6c8b066fe6b8c5.jpg b/.raw/MidasCivilAnalysisReference/images/page-405_65ec8a90c6f5bebac8201199a9502c36f61312ae05c4b29dfb6c8b066fe6b8c5.jpg new file mode 100644 index 00000000..d17aaa29 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-405_65ec8a90c6f5bebac8201199a9502c36f61312ae05c4b29dfb6c8b066fe6b8c5.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-405_938c1a1479e5e8a9ad58623b273af4e08826ddf7b3a57e11eea8bfd18b347e40.jpg b/.raw/MidasCivilAnalysisReference/images/page-405_938c1a1479e5e8a9ad58623b273af4e08826ddf7b3a57e11eea8bfd18b347e40.jpg new file mode 100644 index 00000000..bbf0d2ed Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-405_938c1a1479e5e8a9ad58623b273af4e08826ddf7b3a57e11eea8bfd18b347e40.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-405_977a678a5bfa4179480195c2aabbb12b2329c690e0ce360937c65a78441021e2.jpg b/.raw/MidasCivilAnalysisReference/images/page-405_977a678a5bfa4179480195c2aabbb12b2329c690e0ce360937c65a78441021e2.jpg new file mode 100644 index 00000000..7b407557 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-405_977a678a5bfa4179480195c2aabbb12b2329c690e0ce360937c65a78441021e2.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-405_eeb9b3db2bdb7f06614641bf355c43113fbce42f5967033acd90bc47a9fce4b8.jpg b/.raw/MidasCivilAnalysisReference/images/page-405_eeb9b3db2bdb7f06614641bf355c43113fbce42f5967033acd90bc47a9fce4b8.jpg new file mode 100644 index 00000000..feb85a13 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-405_eeb9b3db2bdb7f06614641bf355c43113fbce42f5967033acd90bc47a9fce4b8.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-406_2747f53a00a89414290e14069fc825ba869c00880d74d11ce59e8f7d64ba587e.jpg b/.raw/MidasCivilAnalysisReference/images/page-406_2747f53a00a89414290e14069fc825ba869c00880d74d11ce59e8f7d64ba587e.jpg new file mode 100644 index 00000000..13ef9e3a Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-406_2747f53a00a89414290e14069fc825ba869c00880d74d11ce59e8f7d64ba587e.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-406_f1c948786c6a3f44d805894f34c0c717bc0df75adad6d66ff6df903272b55d52.jpg b/.raw/MidasCivilAnalysisReference/images/page-406_f1c948786c6a3f44d805894f34c0c717bc0df75adad6d66ff6df903272b55d52.jpg new file mode 100644 index 00000000..9205958c Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-406_f1c948786c6a3f44d805894f34c0c717bc0df75adad6d66ff6df903272b55d52.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-407_514e3e3905cc5769ed73ec61854255e81d57a868fe761a5533fe8b6c2e15bfcd.jpg b/.raw/MidasCivilAnalysisReference/images/page-407_514e3e3905cc5769ed73ec61854255e81d57a868fe761a5533fe8b6c2e15bfcd.jpg new file mode 100644 index 00000000..133f2d8d Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-407_514e3e3905cc5769ed73ec61854255e81d57a868fe761a5533fe8b6c2e15bfcd.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-407_65b88686472d5484514413869e408df12d2283316e517ac13c5277fd3eb71acb.jpg b/.raw/MidasCivilAnalysisReference/images/page-407_65b88686472d5484514413869e408df12d2283316e517ac13c5277fd3eb71acb.jpg new file mode 100644 index 00000000..136b2889 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-407_65b88686472d5484514413869e408df12d2283316e517ac13c5277fd3eb71acb.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-407_efbe64d9004f6340bf2550847ab456ee44b40ab1c6cc80ff94f8ba607e31f847.jpg b/.raw/MidasCivilAnalysisReference/images/page-407_efbe64d9004f6340bf2550847ab456ee44b40ab1c6cc80ff94f8ba607e31f847.jpg new file mode 100644 index 00000000..4f4e63ba Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-407_efbe64d9004f6340bf2550847ab456ee44b40ab1c6cc80ff94f8ba607e31f847.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-408_3d0bc51d381c399775e47095369f9ba85f40d1527b2b07818aaa9d719552ccac.jpg b/.raw/MidasCivilAnalysisReference/images/page-408_3d0bc51d381c399775e47095369f9ba85f40d1527b2b07818aaa9d719552ccac.jpg new file mode 100644 index 00000000..d4d8382e Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-408_3d0bc51d381c399775e47095369f9ba85f40d1527b2b07818aaa9d719552ccac.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-408_4cc1ec1e4ab5cd080f23e16a9d4025180ef21387353d7e162e919af30433cce1.jpg b/.raw/MidasCivilAnalysisReference/images/page-408_4cc1ec1e4ab5cd080f23e16a9d4025180ef21387353d7e162e919af30433cce1.jpg new file mode 100644 index 00000000..d52b50f9 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-408_4cc1ec1e4ab5cd080f23e16a9d4025180ef21387353d7e162e919af30433cce1.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-408_58d83fbc2d4fd63584899f1058039e57a5e612c68fd7cb5455196ebe4452afa9.jpg b/.raw/MidasCivilAnalysisReference/images/page-408_58d83fbc2d4fd63584899f1058039e57a5e612c68fd7cb5455196ebe4452afa9.jpg new file mode 100644 index 00000000..f15c1cb6 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-408_58d83fbc2d4fd63584899f1058039e57a5e612c68fd7cb5455196ebe4452afa9.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-408_6725bb0cc5986283558f31bb85bb3bd242b46a58c3330d9790a3da8ded72b192.jpg b/.raw/MidasCivilAnalysisReference/images/page-408_6725bb0cc5986283558f31bb85bb3bd242b46a58c3330d9790a3da8ded72b192.jpg new file mode 100644 index 00000000..fb02c55c Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-408_6725bb0cc5986283558f31bb85bb3bd242b46a58c3330d9790a3da8ded72b192.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-408_b21093e877a2245be6f4b85afeb4d36b7b809516aa0e64b447b4e38b49b1f397.jpg b/.raw/MidasCivilAnalysisReference/images/page-408_b21093e877a2245be6f4b85afeb4d36b7b809516aa0e64b447b4e38b49b1f397.jpg new file mode 100644 index 00000000..16035bbc Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-408_b21093e877a2245be6f4b85afeb4d36b7b809516aa0e64b447b4e38b49b1f397.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-408_cbb65ceba1adc6639af01bf6f15800c09cb60f4930bff996752e3fd20a2a56a3.jpg b/.raw/MidasCivilAnalysisReference/images/page-408_cbb65ceba1adc6639af01bf6f15800c09cb60f4930bff996752e3fd20a2a56a3.jpg new file mode 100644 index 00000000..e2cbcc34 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-408_cbb65ceba1adc6639af01bf6f15800c09cb60f4930bff996752e3fd20a2a56a3.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-408_d56382885bf863c6559ad697785e73ba671db68e5547e9a478e5840fcd755960.jpg b/.raw/MidasCivilAnalysisReference/images/page-408_d56382885bf863c6559ad697785e73ba671db68e5547e9a478e5840fcd755960.jpg new file mode 100644 index 00000000..641dee4a Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-408_d56382885bf863c6559ad697785e73ba671db68e5547e9a478e5840fcd755960.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-409_62e19b56370d3c369c0a5df056e8e70d420ff3110dc1a252f7320a65b237cc58.jpg b/.raw/MidasCivilAnalysisReference/images/page-409_62e19b56370d3c369c0a5df056e8e70d420ff3110dc1a252f7320a65b237cc58.jpg new file mode 100644 index 00000000..2cc27080 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-409_62e19b56370d3c369c0a5df056e8e70d420ff3110dc1a252f7320a65b237cc58.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-409_8a932360a494d6ebc63a5d9a2d522b31e764a117819c5395759f516a5387a223.jpg b/.raw/MidasCivilAnalysisReference/images/page-409_8a932360a494d6ebc63a5d9a2d522b31e764a117819c5395759f516a5387a223.jpg new file mode 100644 index 00000000..14082f95 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-409_8a932360a494d6ebc63a5d9a2d522b31e764a117819c5395759f516a5387a223.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-409_fcb97fbb8c139b6a27779e0145442e048f47079e03e26d3623d0102090cdba95.jpg b/.raw/MidasCivilAnalysisReference/images/page-409_fcb97fbb8c139b6a27779e0145442e048f47079e03e26d3623d0102090cdba95.jpg new file mode 100644 index 00000000..37a1037f Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-409_fcb97fbb8c139b6a27779e0145442e048f47079e03e26d3623d0102090cdba95.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-410_5b9c0325840fc975e777825001f98d65921cc6a645a66ec7a787b3a6ae5b1929.jpg b/.raw/MidasCivilAnalysisReference/images/page-410_5b9c0325840fc975e777825001f98d65921cc6a645a66ec7a787b3a6ae5b1929.jpg new file mode 100644 index 00000000..3d9c0ef7 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-410_5b9c0325840fc975e777825001f98d65921cc6a645a66ec7a787b3a6ae5b1929.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-411_88589e7d7f0ed773079aa0eddc1778518824ec7bd324c1b84e350469648ca68e.jpg b/.raw/MidasCivilAnalysisReference/images/page-411_88589e7d7f0ed773079aa0eddc1778518824ec7bd324c1b84e350469648ca68e.jpg new file mode 100644 index 00000000..b8ec6b88 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-411_88589e7d7f0ed773079aa0eddc1778518824ec7bd324c1b84e350469648ca68e.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-411_a029fea715a5ced1183d9373ceee3474c4a6da0e5545684238dc54738a6b8a91.jpg b/.raw/MidasCivilAnalysisReference/images/page-411_a029fea715a5ced1183d9373ceee3474c4a6da0e5545684238dc54738a6b8a91.jpg new file mode 100644 index 00000000..e70fb6c3 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-411_a029fea715a5ced1183d9373ceee3474c4a6da0e5545684238dc54738a6b8a91.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-411_e22becbe0ea446ab69dd6acd828fa9135d266c596c519975dcf1a9dcd584356e.jpg b/.raw/MidasCivilAnalysisReference/images/page-411_e22becbe0ea446ab69dd6acd828fa9135d266c596c519975dcf1a9dcd584356e.jpg new file mode 100644 index 00000000..eb03bcc5 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-411_e22becbe0ea446ab69dd6acd828fa9135d266c596c519975dcf1a9dcd584356e.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-412_2c5dd56d1eaba848f30963ff5c8fdadf9c54b99c38884e04f4609379b2f0829a.jpg b/.raw/MidasCivilAnalysisReference/images/page-412_2c5dd56d1eaba848f30963ff5c8fdadf9c54b99c38884e04f4609379b2f0829a.jpg new file mode 100644 index 00000000..81a9416e Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-412_2c5dd56d1eaba848f30963ff5c8fdadf9c54b99c38884e04f4609379b2f0829a.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-412_f28dc016521443848cb904d338d9a792e6933041f4d39a8c1f9505a502208393.jpg b/.raw/MidasCivilAnalysisReference/images/page-412_f28dc016521443848cb904d338d9a792e6933041f4d39a8c1f9505a502208393.jpg new file mode 100644 index 00000000..136ffa9d Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-412_f28dc016521443848cb904d338d9a792e6933041f4d39a8c1f9505a502208393.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-413_1d56e8c73544ae86fafcc6140d71efe26dc66e0223ec8dc2833a1491c58314bc.jpg b/.raw/MidasCivilAnalysisReference/images/page-413_1d56e8c73544ae86fafcc6140d71efe26dc66e0223ec8dc2833a1491c58314bc.jpg new file mode 100644 index 00000000..d38dec11 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-413_1d56e8c73544ae86fafcc6140d71efe26dc66e0223ec8dc2833a1491c58314bc.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-413_4e25ec14c53f56981205858186fbbb8eb0220965d1fac882b0e0ce0a76140dfe.jpg b/.raw/MidasCivilAnalysisReference/images/page-413_4e25ec14c53f56981205858186fbbb8eb0220965d1fac882b0e0ce0a76140dfe.jpg new file mode 100644 index 00000000..4ed60820 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-413_4e25ec14c53f56981205858186fbbb8eb0220965d1fac882b0e0ce0a76140dfe.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-413_e6b2c6c9cb3be3f86e600da9346850d8dabf9b79ecc70a7a20b82d4d7a448592.jpg b/.raw/MidasCivilAnalysisReference/images/page-413_e6b2c6c9cb3be3f86e600da9346850d8dabf9b79ecc70a7a20b82d4d7a448592.jpg new file mode 100644 index 00000000..16b2342b Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-413_e6b2c6c9cb3be3f86e600da9346850d8dabf9b79ecc70a7a20b82d4d7a448592.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-414_401506e1b77a524c4640815e96a2db6751866c48722d33a9d49c6490c78af4e6.jpg b/.raw/MidasCivilAnalysisReference/images/page-414_401506e1b77a524c4640815e96a2db6751866c48722d33a9d49c6490c78af4e6.jpg new file mode 100644 index 00000000..432fc2b4 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-414_401506e1b77a524c4640815e96a2db6751866c48722d33a9d49c6490c78af4e6.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-416_1ba8030fe8b37c82cf69f0ef6bd6778c72f545a0f79b4700e06105da427a407b.jpg b/.raw/MidasCivilAnalysisReference/images/page-416_1ba8030fe8b37c82cf69f0ef6bd6778c72f545a0f79b4700e06105da427a407b.jpg new file mode 100644 index 00000000..7cfba99c Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-416_1ba8030fe8b37c82cf69f0ef6bd6778c72f545a0f79b4700e06105da427a407b.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-416_33564e8cc63cfa321069a38022233ef06264df914d9caaae8b022554bd8b6648.jpg b/.raw/MidasCivilAnalysisReference/images/page-416_33564e8cc63cfa321069a38022233ef06264df914d9caaae8b022554bd8b6648.jpg new file mode 100644 index 00000000..fabefcc2 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-416_33564e8cc63cfa321069a38022233ef06264df914d9caaae8b022554bd8b6648.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-416_45df0ca71a71c7872b1b943753183443af297bbd8c75c70936083c6d7d30a178.jpg b/.raw/MidasCivilAnalysisReference/images/page-416_45df0ca71a71c7872b1b943753183443af297bbd8c75c70936083c6d7d30a178.jpg new file mode 100644 index 00000000..e3e8f663 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-416_45df0ca71a71c7872b1b943753183443af297bbd8c75c70936083c6d7d30a178.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-416_49e5adfffa00e7a2ac14ae9449179e395ecf9b223d8968b856b4ab271b4448ce.jpg b/.raw/MidasCivilAnalysisReference/images/page-416_49e5adfffa00e7a2ac14ae9449179e395ecf9b223d8968b856b4ab271b4448ce.jpg new file mode 100644 index 00000000..cb719921 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-416_49e5adfffa00e7a2ac14ae9449179e395ecf9b223d8968b856b4ab271b4448ce.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-416_7439c37c0240e77e41486dc8ffc36eeede7486d97551cb749e41f034dbd84178.jpg b/.raw/MidasCivilAnalysisReference/images/page-416_7439c37c0240e77e41486dc8ffc36eeede7486d97551cb749e41f034dbd84178.jpg new file mode 100644 index 00000000..60e3267d Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-416_7439c37c0240e77e41486dc8ffc36eeede7486d97551cb749e41f034dbd84178.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-416_9bda33487b66bb64eb43c618b2ec2ccaa390d00fa83efa9a2737d0118dca016e.jpg b/.raw/MidasCivilAnalysisReference/images/page-416_9bda33487b66bb64eb43c618b2ec2ccaa390d00fa83efa9a2737d0118dca016e.jpg new file mode 100644 index 00000000..abb64f2a Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-416_9bda33487b66bb64eb43c618b2ec2ccaa390d00fa83efa9a2737d0118dca016e.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-416_aab9ce68548bd360f3ea6301296f062526f3ec398907d023301fd0b2a1096056.jpg b/.raw/MidasCivilAnalysisReference/images/page-416_aab9ce68548bd360f3ea6301296f062526f3ec398907d023301fd0b2a1096056.jpg new file mode 100644 index 00000000..e6a8391d Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-416_aab9ce68548bd360f3ea6301296f062526f3ec398907d023301fd0b2a1096056.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-416_af1d36a9d2b82054b1e3a6f22d7a46245c438463b83263167fe07d6e47cd732b.jpg b/.raw/MidasCivilAnalysisReference/images/page-416_af1d36a9d2b82054b1e3a6f22d7a46245c438463b83263167fe07d6e47cd732b.jpg new file mode 100644 index 00000000..ee0fc56b Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-416_af1d36a9d2b82054b1e3a6f22d7a46245c438463b83263167fe07d6e47cd732b.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-416_c91877dd34d6591960efdf1031ee74894c58d4420c5db8b48c44a941cf8779e2.jpg b/.raw/MidasCivilAnalysisReference/images/page-416_c91877dd34d6591960efdf1031ee74894c58d4420c5db8b48c44a941cf8779e2.jpg new file mode 100644 index 00000000..4e4fa720 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-416_c91877dd34d6591960efdf1031ee74894c58d4420c5db8b48c44a941cf8779e2.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-417_b4251b75ba1b5be6bcf87a64ac4e0d3e0eac392f5e41c0d6bdad53657d04cc1c.jpg b/.raw/MidasCivilAnalysisReference/images/page-417_b4251b75ba1b5be6bcf87a64ac4e0d3e0eac392f5e41c0d6bdad53657d04cc1c.jpg new file mode 100644 index 00000000..9a00a44b Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-417_b4251b75ba1b5be6bcf87a64ac4e0d3e0eac392f5e41c0d6bdad53657d04cc1c.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-417_b72e2ba5e735561c63f62d923a1f78c796b964ae0e788e4988d1de36a38ef29b.jpg b/.raw/MidasCivilAnalysisReference/images/page-417_b72e2ba5e735561c63f62d923a1f78c796b964ae0e788e4988d1de36a38ef29b.jpg new file mode 100644 index 00000000..612ca099 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-417_b72e2ba5e735561c63f62d923a1f78c796b964ae0e788e4988d1de36a38ef29b.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-417_d5735f7d39c558c010423cd45f2e3c96a6e6f5a3f234b8a6be8660a31c34c109.jpg b/.raw/MidasCivilAnalysisReference/images/page-417_d5735f7d39c558c010423cd45f2e3c96a6e6f5a3f234b8a6be8660a31c34c109.jpg new file mode 100644 index 00000000..0f5d1e6b Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-417_d5735f7d39c558c010423cd45f2e3c96a6e6f5a3f234b8a6be8660a31c34c109.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-417_f3d59ff9a8e9397c72ef7dda624166ba19e5c0df196db06956204b417c6d36ce.jpg b/.raw/MidasCivilAnalysisReference/images/page-417_f3d59ff9a8e9397c72ef7dda624166ba19e5c0df196db06956204b417c6d36ce.jpg new file mode 100644 index 00000000..06b962f5 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-417_f3d59ff9a8e9397c72ef7dda624166ba19e5c0df196db06956204b417c6d36ce.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-418_0b76eaeca839274fae52730fe7c8ff47ff67b16ffbedb7e473e2564a4600862c.jpg b/.raw/MidasCivilAnalysisReference/images/page-418_0b76eaeca839274fae52730fe7c8ff47ff67b16ffbedb7e473e2564a4600862c.jpg new file mode 100644 index 00000000..a746c14b Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-418_0b76eaeca839274fae52730fe7c8ff47ff67b16ffbedb7e473e2564a4600862c.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-418_54e39d77cad01f1c260c7ac1b486ec9a77a7e47892064b34f14c9286421dcb24.jpg b/.raw/MidasCivilAnalysisReference/images/page-418_54e39d77cad01f1c260c7ac1b486ec9a77a7e47892064b34f14c9286421dcb24.jpg new file mode 100644 index 00000000..9cf7fe05 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-418_54e39d77cad01f1c260c7ac1b486ec9a77a7e47892064b34f14c9286421dcb24.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-418_ef8992fce7c20639e8ef0b7b829003629cfe4d91252168b370aa47b347e21737.jpg b/.raw/MidasCivilAnalysisReference/images/page-418_ef8992fce7c20639e8ef0b7b829003629cfe4d91252168b370aa47b347e21737.jpg new file mode 100644 index 00000000..95e856e6 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-418_ef8992fce7c20639e8ef0b7b829003629cfe4d91252168b370aa47b347e21737.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-419_0042f6b4da274ec67e39d6798568b553f3ac7651877c8129c3569d42bbd10af4.jpg b/.raw/MidasCivilAnalysisReference/images/page-419_0042f6b4da274ec67e39d6798568b553f3ac7651877c8129c3569d42bbd10af4.jpg new file mode 100644 index 00000000..f29a80af Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-419_0042f6b4da274ec67e39d6798568b553f3ac7651877c8129c3569d42bbd10af4.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-419_1611f5a14ea69749f10c3aad0d65aa03a7ba68fd3ef1b181f58fd860cd4e1299.jpg b/.raw/MidasCivilAnalysisReference/images/page-419_1611f5a14ea69749f10c3aad0d65aa03a7ba68fd3ef1b181f58fd860cd4e1299.jpg new file mode 100644 index 00000000..c36f6f19 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-419_1611f5a14ea69749f10c3aad0d65aa03a7ba68fd3ef1b181f58fd860cd4e1299.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-420_82b98e05f75259ddc012f7c16c2649af883023fe115d17241f50a9d6fb07602f.jpg b/.raw/MidasCivilAnalysisReference/images/page-420_82b98e05f75259ddc012f7c16c2649af883023fe115d17241f50a9d6fb07602f.jpg new file mode 100644 index 00000000..87bebd15 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-420_82b98e05f75259ddc012f7c16c2649af883023fe115d17241f50a9d6fb07602f.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-420_e4b89897838bff6843ad90fa43995ca8a9557a8a8820735eaa7a3bc9bb87a719.jpg b/.raw/MidasCivilAnalysisReference/images/page-420_e4b89897838bff6843ad90fa43995ca8a9557a8a8820735eaa7a3bc9bb87a719.jpg new file mode 100644 index 00000000..92529712 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-420_e4b89897838bff6843ad90fa43995ca8a9557a8a8820735eaa7a3bc9bb87a719.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-421_7117ee7255a45884c80139e5fb1ba565648d12603caa5038ddd92ff05f28688a.jpg b/.raw/MidasCivilAnalysisReference/images/page-421_7117ee7255a45884c80139e5fb1ba565648d12603caa5038ddd92ff05f28688a.jpg new file mode 100644 index 00000000..819ed360 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-421_7117ee7255a45884c80139e5fb1ba565648d12603caa5038ddd92ff05f28688a.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-421_c018a6d4d34189f331a0eac50edcd20cfb648dd4952a14758d52d71b56073e33.jpg b/.raw/MidasCivilAnalysisReference/images/page-421_c018a6d4d34189f331a0eac50edcd20cfb648dd4952a14758d52d71b56073e33.jpg new file mode 100644 index 00000000..681ce072 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-421_c018a6d4d34189f331a0eac50edcd20cfb648dd4952a14758d52d71b56073e33.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-421_c96a9404191f98e8c5d23e89ae599c4e66b04fcbdb36e6aa68eb11c209b10498.jpg b/.raw/MidasCivilAnalysisReference/images/page-421_c96a9404191f98e8c5d23e89ae599c4e66b04fcbdb36e6aa68eb11c209b10498.jpg new file mode 100644 index 00000000..296edbe0 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-421_c96a9404191f98e8c5d23e89ae599c4e66b04fcbdb36e6aa68eb11c209b10498.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-422_fe45cb06c9581de7ef91acb992d8d49f26fc7bec88b8f9212f368bec16a72807.jpg b/.raw/MidasCivilAnalysisReference/images/page-422_fe45cb06c9581de7ef91acb992d8d49f26fc7bec88b8f9212f368bec16a72807.jpg new file mode 100644 index 00000000..d2ccac9e Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-422_fe45cb06c9581de7ef91acb992d8d49f26fc7bec88b8f9212f368bec16a72807.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-423_432a3763d337a5001f0d009536cfcaffaca271112fb7f8aa4b7409bb76033e73.jpg b/.raw/MidasCivilAnalysisReference/images/page-423_432a3763d337a5001f0d009536cfcaffaca271112fb7f8aa4b7409bb76033e73.jpg new file mode 100644 index 00000000..0f3e36a7 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-423_432a3763d337a5001f0d009536cfcaffaca271112fb7f8aa4b7409bb76033e73.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-423_f71dbfe42940721932157e429bea83d04b2745f3e85622ccd1799fd6d8989ce3.jpg b/.raw/MidasCivilAnalysisReference/images/page-423_f71dbfe42940721932157e429bea83d04b2745f3e85622ccd1799fd6d8989ce3.jpg new file mode 100644 index 00000000..739a72e0 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-423_f71dbfe42940721932157e429bea83d04b2745f3e85622ccd1799fd6d8989ce3.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-425_3cdb5496a13568befde4065ddd3cd21333d8fd3011821c729e2b320d8f13af56.jpg b/.raw/MidasCivilAnalysisReference/images/page-425_3cdb5496a13568befde4065ddd3cd21333d8fd3011821c729e2b320d8f13af56.jpg new file mode 100644 index 00000000..0201e41b Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-425_3cdb5496a13568befde4065ddd3cd21333d8fd3011821c729e2b320d8f13af56.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-425_6b8f33518aa913068ac4013ca347c56bf015d8cb550de1764e4ac24e483eeddc.jpg b/.raw/MidasCivilAnalysisReference/images/page-425_6b8f33518aa913068ac4013ca347c56bf015d8cb550de1764e4ac24e483eeddc.jpg new file mode 100644 index 00000000..8cb8b807 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-425_6b8f33518aa913068ac4013ca347c56bf015d8cb550de1764e4ac24e483eeddc.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-425_73a22a1388bfa136a40b0e7c1642ded8b12a07ab8a468dc099ca910b50a35e8b.jpg b/.raw/MidasCivilAnalysisReference/images/page-425_73a22a1388bfa136a40b0e7c1642ded8b12a07ab8a468dc099ca910b50a35e8b.jpg new file mode 100644 index 00000000..2d15b312 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-425_73a22a1388bfa136a40b0e7c1642ded8b12a07ab8a468dc099ca910b50a35e8b.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-425_ab691234dfe5ab7e47d23f643486b4c7f5b5fce053fec574484e0d0f6963db7a.jpg b/.raw/MidasCivilAnalysisReference/images/page-425_ab691234dfe5ab7e47d23f643486b4c7f5b5fce053fec574484e0d0f6963db7a.jpg new file mode 100644 index 00000000..9dacb59e Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-425_ab691234dfe5ab7e47d23f643486b4c7f5b5fce053fec574484e0d0f6963db7a.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-426_6515f51f98131995ec8b187820772adcd500fe5528de5bbe96e6ea7bb33eebbe.jpg b/.raw/MidasCivilAnalysisReference/images/page-426_6515f51f98131995ec8b187820772adcd500fe5528de5bbe96e6ea7bb33eebbe.jpg new file mode 100644 index 00000000..d80f7d8c Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-426_6515f51f98131995ec8b187820772adcd500fe5528de5bbe96e6ea7bb33eebbe.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-426_8cd5be57f6deebe52515a123290f19255416f155464c061396070b782f1e157a.jpg b/.raw/MidasCivilAnalysisReference/images/page-426_8cd5be57f6deebe52515a123290f19255416f155464c061396070b782f1e157a.jpg new file mode 100644 index 00000000..50e1f0c9 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-426_8cd5be57f6deebe52515a123290f19255416f155464c061396070b782f1e157a.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-427_27697a57933274a36a97e6cba1c553a510aa342a490e13d8c3d5f6ad1ef695da.jpg b/.raw/MidasCivilAnalysisReference/images/page-427_27697a57933274a36a97e6cba1c553a510aa342a490e13d8c3d5f6ad1ef695da.jpg new file mode 100644 index 00000000..f79c8d9d Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-427_27697a57933274a36a97e6cba1c553a510aa342a490e13d8c3d5f6ad1ef695da.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-427_6b401ed1bfef641ce6fc590360d3416824b8cfa5735c37d834be4b3c1084b917.jpg b/.raw/MidasCivilAnalysisReference/images/page-427_6b401ed1bfef641ce6fc590360d3416824b8cfa5735c37d834be4b3c1084b917.jpg new file mode 100644 index 00000000..23e6246d Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-427_6b401ed1bfef641ce6fc590360d3416824b8cfa5735c37d834be4b3c1084b917.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-427_a623600ccc8b9e74577bf196689d0ade5ba825aa32c630bcb6533c2d13349227.jpg b/.raw/MidasCivilAnalysisReference/images/page-427_a623600ccc8b9e74577bf196689d0ade5ba825aa32c630bcb6533c2d13349227.jpg new file mode 100644 index 00000000..7727b6d3 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-427_a623600ccc8b9e74577bf196689d0ade5ba825aa32c630bcb6533c2d13349227.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-428_0a72e4400f2ec55ea7cb6c9ef24d073d28d2c42e0ef4e447883953e58dba567d.jpg b/.raw/MidasCivilAnalysisReference/images/page-428_0a72e4400f2ec55ea7cb6c9ef24d073d28d2c42e0ef4e447883953e58dba567d.jpg new file mode 100644 index 00000000..d1324931 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-428_0a72e4400f2ec55ea7cb6c9ef24d073d28d2c42e0ef4e447883953e58dba567d.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-429_206a0e82d267fefe12c530129ebc772909ded02e4306aadac27cd33b1fd1cbba.jpg b/.raw/MidasCivilAnalysisReference/images/page-429_206a0e82d267fefe12c530129ebc772909ded02e4306aadac27cd33b1fd1cbba.jpg new file mode 100644 index 00000000..267e01c5 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-429_206a0e82d267fefe12c530129ebc772909ded02e4306aadac27cd33b1fd1cbba.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-430_b6d2665a18f254158734df756897e74e19c5a775fe69ada8442cfd67cdd3b1ee.jpg b/.raw/MidasCivilAnalysisReference/images/page-430_b6d2665a18f254158734df756897e74e19c5a775fe69ada8442cfd67cdd3b1ee.jpg new file mode 100644 index 00000000..ecd3b148 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-430_b6d2665a18f254158734df756897e74e19c5a775fe69ada8442cfd67cdd3b1ee.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-431_25a2b65c9ae3399e073b852e09eec0b7c3e85941bc2f097540387694a87e1fa8.jpg b/.raw/MidasCivilAnalysisReference/images/page-431_25a2b65c9ae3399e073b852e09eec0b7c3e85941bc2f097540387694a87e1fa8.jpg new file mode 100644 index 00000000..83115729 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-431_25a2b65c9ae3399e073b852e09eec0b7c3e85941bc2f097540387694a87e1fa8.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-431_caf1dbd5e7daa70c8fc58f32b226da3d7e175fe483ce5bd5cee8ebaf8b6f71a0.jpg b/.raw/MidasCivilAnalysisReference/images/page-431_caf1dbd5e7daa70c8fc58f32b226da3d7e175fe483ce5bd5cee8ebaf8b6f71a0.jpg new file mode 100644 index 00000000..8e7d8044 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-431_caf1dbd5e7daa70c8fc58f32b226da3d7e175fe483ce5bd5cee8ebaf8b6f71a0.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-432_3a541733bbbd0961be6827ebe87c480a98841156613dc8268c16a9e6c4731462.jpg b/.raw/MidasCivilAnalysisReference/images/page-432_3a541733bbbd0961be6827ebe87c480a98841156613dc8268c16a9e6c4731462.jpg new file mode 100644 index 00000000..f5bb42a2 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-432_3a541733bbbd0961be6827ebe87c480a98841156613dc8268c16a9e6c4731462.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-432_4122c09cd565b40a2d98da152a377e8e5cfb4ca13d374cbc5b37ab962baabab2.jpg b/.raw/MidasCivilAnalysisReference/images/page-432_4122c09cd565b40a2d98da152a377e8e5cfb4ca13d374cbc5b37ab962baabab2.jpg new file mode 100644 index 00000000..ca30c835 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-432_4122c09cd565b40a2d98da152a377e8e5cfb4ca13d374cbc5b37ab962baabab2.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-432_765bacaf3be2e4e86ec490e2ef7daaf5facce104a1d2467e94aa7613a01b43f1.jpg b/.raw/MidasCivilAnalysisReference/images/page-432_765bacaf3be2e4e86ec490e2ef7daaf5facce104a1d2467e94aa7613a01b43f1.jpg new file mode 100644 index 00000000..4a72658a Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-432_765bacaf3be2e4e86ec490e2ef7daaf5facce104a1d2467e94aa7613a01b43f1.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-432_ce022742634563f3c660d33692404126be37bf7bb356173090627f86f230b389.jpg b/.raw/MidasCivilAnalysisReference/images/page-432_ce022742634563f3c660d33692404126be37bf7bb356173090627f86f230b389.jpg new file mode 100644 index 00000000..749771b9 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-432_ce022742634563f3c660d33692404126be37bf7bb356173090627f86f230b389.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-433_20cc589df011c9451b75895eaefc1e22f6245c8bb5dcdd186fe50c7624dd4584.jpg b/.raw/MidasCivilAnalysisReference/images/page-433_20cc589df011c9451b75895eaefc1e22f6245c8bb5dcdd186fe50c7624dd4584.jpg new file mode 100644 index 00000000..9efcf3ff Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-433_20cc589df011c9451b75895eaefc1e22f6245c8bb5dcdd186fe50c7624dd4584.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-433_5cb546d15ec19f35091d3617fc559683dca9235dd6dcd4a0a53d638ebd3dbafb.jpg b/.raw/MidasCivilAnalysisReference/images/page-433_5cb546d15ec19f35091d3617fc559683dca9235dd6dcd4a0a53d638ebd3dbafb.jpg new file mode 100644 index 00000000..63c384a2 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-433_5cb546d15ec19f35091d3617fc559683dca9235dd6dcd4a0a53d638ebd3dbafb.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-433_73038d497fa01eed7a71778b156b6089bd082a11ed7f23070f9470947b4051e2.jpg b/.raw/MidasCivilAnalysisReference/images/page-433_73038d497fa01eed7a71778b156b6089bd082a11ed7f23070f9470947b4051e2.jpg new file mode 100644 index 00000000..2ecbba7b Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-433_73038d497fa01eed7a71778b156b6089bd082a11ed7f23070f9470947b4051e2.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-433_7b4fe04e863ad35ddd0e65e4062f2c7ad0a9b30c1dc0bb23a6fbc73b0be66dc1.jpg b/.raw/MidasCivilAnalysisReference/images/page-433_7b4fe04e863ad35ddd0e65e4062f2c7ad0a9b30c1dc0bb23a6fbc73b0be66dc1.jpg new file mode 100644 index 00000000..2afc80dd Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-433_7b4fe04e863ad35ddd0e65e4062f2c7ad0a9b30c1dc0bb23a6fbc73b0be66dc1.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-433_97691a6290a3d27b690811a6eed86ac5dc6f15312d7e380bafd8863d7a6aa316.jpg b/.raw/MidasCivilAnalysisReference/images/page-433_97691a6290a3d27b690811a6eed86ac5dc6f15312d7e380bafd8863d7a6aa316.jpg new file mode 100644 index 00000000..0cfe2e2d Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-433_97691a6290a3d27b690811a6eed86ac5dc6f15312d7e380bafd8863d7a6aa316.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-433_d3c9e673e01b0193cfe3fd04215c7ccf2c200b7d24d8104b551f4bcac60b3a9e.jpg b/.raw/MidasCivilAnalysisReference/images/page-433_d3c9e673e01b0193cfe3fd04215c7ccf2c200b7d24d8104b551f4bcac60b3a9e.jpg new file mode 100644 index 00000000..a0afcd08 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-433_d3c9e673e01b0193cfe3fd04215c7ccf2c200b7d24d8104b551f4bcac60b3a9e.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-433_d4d0c46eaebb21a2d9e8f15491a63d72c3a941a214dadeffc3ae0bbe1c83ff43.jpg b/.raw/MidasCivilAnalysisReference/images/page-433_d4d0c46eaebb21a2d9e8f15491a63d72c3a941a214dadeffc3ae0bbe1c83ff43.jpg new file mode 100644 index 00000000..6cefac7d Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-433_d4d0c46eaebb21a2d9e8f15491a63d72c3a941a214dadeffc3ae0bbe1c83ff43.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-433_f39feb16828b665ccd3c47b29be856bd951202d74cbdf92cc3ec589f2f7e6bad.jpg b/.raw/MidasCivilAnalysisReference/images/page-433_f39feb16828b665ccd3c47b29be856bd951202d74cbdf92cc3ec589f2f7e6bad.jpg new file mode 100644 index 00000000..6838dee0 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-433_f39feb16828b665ccd3c47b29be856bd951202d74cbdf92cc3ec589f2f7e6bad.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-433_f3c2fb7ec881eb3aa3ba616969c221aec3b1141fe6b63ad38353aae61850ac74.jpg b/.raw/MidasCivilAnalysisReference/images/page-433_f3c2fb7ec881eb3aa3ba616969c221aec3b1141fe6b63ad38353aae61850ac74.jpg new file mode 100644 index 00000000..270e7f41 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-433_f3c2fb7ec881eb3aa3ba616969c221aec3b1141fe6b63ad38353aae61850ac74.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-434_800b493f5aa3af5abece36f3090c5bf62f57badfd6ff049ca6408b9dc6ab5aa3.jpg b/.raw/MidasCivilAnalysisReference/images/page-434_800b493f5aa3af5abece36f3090c5bf62f57badfd6ff049ca6408b9dc6ab5aa3.jpg new file mode 100644 index 00000000..70191003 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-434_800b493f5aa3af5abece36f3090c5bf62f57badfd6ff049ca6408b9dc6ab5aa3.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-434_9c798272b72b4a159fa482b6db3b9ee5555e53d08abde13c061a916239c863ea.jpg b/.raw/MidasCivilAnalysisReference/images/page-434_9c798272b72b4a159fa482b6db3b9ee5555e53d08abde13c061a916239c863ea.jpg new file mode 100644 index 00000000..fa1411b6 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-434_9c798272b72b4a159fa482b6db3b9ee5555e53d08abde13c061a916239c863ea.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-435_a94fcd0140aff2565f91d6521a97230391889645e2e61631f4b11bbb020e8542.jpg b/.raw/MidasCivilAnalysisReference/images/page-435_a94fcd0140aff2565f91d6521a97230391889645e2e61631f4b11bbb020e8542.jpg new file mode 100644 index 00000000..574d9bb3 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-435_a94fcd0140aff2565f91d6521a97230391889645e2e61631f4b11bbb020e8542.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-436_1400cbca8dd63fcc9b8472693cd13e03a77de570590de91ea6590e07d424e9fd.jpg b/.raw/MidasCivilAnalysisReference/images/page-436_1400cbca8dd63fcc9b8472693cd13e03a77de570590de91ea6590e07d424e9fd.jpg new file mode 100644 index 00000000..12af63b0 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-436_1400cbca8dd63fcc9b8472693cd13e03a77de570590de91ea6590e07d424e9fd.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-436_2241988a36ba143bbaeb473f4994b628e11a6ae1c6cf262f8d6c53349fcaadb6.jpg b/.raw/MidasCivilAnalysisReference/images/page-436_2241988a36ba143bbaeb473f4994b628e11a6ae1c6cf262f8d6c53349fcaadb6.jpg new file mode 100644 index 00000000..b8f2c299 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-436_2241988a36ba143bbaeb473f4994b628e11a6ae1c6cf262f8d6c53349fcaadb6.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-436_6e6679ca070ba4d7d0ff0e77a0e7275a9e439aa757dab00af831cfea8d1fc3ac.jpg b/.raw/MidasCivilAnalysisReference/images/page-436_6e6679ca070ba4d7d0ff0e77a0e7275a9e439aa757dab00af831cfea8d1fc3ac.jpg new file mode 100644 index 00000000..9ecb14c8 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-436_6e6679ca070ba4d7d0ff0e77a0e7275a9e439aa757dab00af831cfea8d1fc3ac.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-436_bbdd95bd5b2c835fbbc38a4cb2fdbaac5dec6df0ff330dd4e5d257e376ffb069.jpg b/.raw/MidasCivilAnalysisReference/images/page-436_bbdd95bd5b2c835fbbc38a4cb2fdbaac5dec6df0ff330dd4e5d257e376ffb069.jpg new file mode 100644 index 00000000..389a6b58 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-436_bbdd95bd5b2c835fbbc38a4cb2fdbaac5dec6df0ff330dd4e5d257e376ffb069.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-437_1f7c2030dbfc1c7fea0e56db037f3187f0066867329fb51c143a55538793c2dc.jpg b/.raw/MidasCivilAnalysisReference/images/page-437_1f7c2030dbfc1c7fea0e56db037f3187f0066867329fb51c143a55538793c2dc.jpg new file mode 100644 index 00000000..fef7c7d8 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-437_1f7c2030dbfc1c7fea0e56db037f3187f0066867329fb51c143a55538793c2dc.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-437_ad770045d11969f2f8989b2e1a606e1335b1a65f2705a2e4468c9f348dbc187d.jpg b/.raw/MidasCivilAnalysisReference/images/page-437_ad770045d11969f2f8989b2e1a606e1335b1a65f2705a2e4468c9f348dbc187d.jpg new file mode 100644 index 00000000..be3956c4 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-437_ad770045d11969f2f8989b2e1a606e1335b1a65f2705a2e4468c9f348dbc187d.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-437_eeea2e5d9e52ab386a275736e95d19475b0c5929abf8c2fe674d51160ed8acf3.jpg b/.raw/MidasCivilAnalysisReference/images/page-437_eeea2e5d9e52ab386a275736e95d19475b0c5929abf8c2fe674d51160ed8acf3.jpg new file mode 100644 index 00000000..ac22fed3 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-437_eeea2e5d9e52ab386a275736e95d19475b0c5929abf8c2fe674d51160ed8acf3.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-438_1b3564b01798434ba65da99114866e861ef860ec02bdea52e67c78adc1299b92.jpg b/.raw/MidasCivilAnalysisReference/images/page-438_1b3564b01798434ba65da99114866e861ef860ec02bdea52e67c78adc1299b92.jpg new file mode 100644 index 00000000..773bcbc0 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-438_1b3564b01798434ba65da99114866e861ef860ec02bdea52e67c78adc1299b92.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-438_5310617bb17e79ce4f9d158fb2fd104b186f5addc345e78ce263efd03941dfe8.jpg b/.raw/MidasCivilAnalysisReference/images/page-438_5310617bb17e79ce4f9d158fb2fd104b186f5addc345e78ce263efd03941dfe8.jpg new file mode 100644 index 00000000..8c3c0831 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-438_5310617bb17e79ce4f9d158fb2fd104b186f5addc345e78ce263efd03941dfe8.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-438_b5a385f23be6f1ea328ae521d0f98fed3c88dbeb40929ff56d4d38e735cda394.jpg b/.raw/MidasCivilAnalysisReference/images/page-438_b5a385f23be6f1ea328ae521d0f98fed3c88dbeb40929ff56d4d38e735cda394.jpg new file mode 100644 index 00000000..5fb7b1f4 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-438_b5a385f23be6f1ea328ae521d0f98fed3c88dbeb40929ff56d4d38e735cda394.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-438_da06aa1ec218a62345ddf805e89890e17b9ffd754a7a07121cd407ff53ba479d.jpg b/.raw/MidasCivilAnalysisReference/images/page-438_da06aa1ec218a62345ddf805e89890e17b9ffd754a7a07121cd407ff53ba479d.jpg new file mode 100644 index 00000000..4eb1caa7 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-438_da06aa1ec218a62345ddf805e89890e17b9ffd754a7a07121cd407ff53ba479d.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-438_f30242af5bc78b98f199648e417787a36ed3799dc4505d67b113ba401660dbac.jpg b/.raw/MidasCivilAnalysisReference/images/page-438_f30242af5bc78b98f199648e417787a36ed3799dc4505d67b113ba401660dbac.jpg new file mode 100644 index 00000000..c5a2c942 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-438_f30242af5bc78b98f199648e417787a36ed3799dc4505d67b113ba401660dbac.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-439_756ded40c41c65ce46a70583436eefa2e2c47bbb76e431b9479a880bf1493276.jpg b/.raw/MidasCivilAnalysisReference/images/page-439_756ded40c41c65ce46a70583436eefa2e2c47bbb76e431b9479a880bf1493276.jpg new file mode 100644 index 00000000..375bc035 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-439_756ded40c41c65ce46a70583436eefa2e2c47bbb76e431b9479a880bf1493276.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-439_e8b048d30e3169d1b9c58f29664611018ad415ac22578c37e10529b66e92af60.jpg b/.raw/MidasCivilAnalysisReference/images/page-439_e8b048d30e3169d1b9c58f29664611018ad415ac22578c37e10529b66e92af60.jpg new file mode 100644 index 00000000..a3ac0027 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-439_e8b048d30e3169d1b9c58f29664611018ad415ac22578c37e10529b66e92af60.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-439_fb4ded7af2530e7a3d64f9355b01a614af70b8ff9ead94cc532c106db80aba18.jpg b/.raw/MidasCivilAnalysisReference/images/page-439_fb4ded7af2530e7a3d64f9355b01a614af70b8ff9ead94cc532c106db80aba18.jpg new file mode 100644 index 00000000..56f62404 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-439_fb4ded7af2530e7a3d64f9355b01a614af70b8ff9ead94cc532c106db80aba18.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-440_2d7a4b0365dd1f034cd522aaaf502745bdc26384af650b2a2da4fc5a1aa42aae.jpg b/.raw/MidasCivilAnalysisReference/images/page-440_2d7a4b0365dd1f034cd522aaaf502745bdc26384af650b2a2da4fc5a1aa42aae.jpg new file mode 100644 index 00000000..7be9e5aa Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-440_2d7a4b0365dd1f034cd522aaaf502745bdc26384af650b2a2da4fc5a1aa42aae.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-440_3056be7fb39851a84566c234d970eba9ce137bc8740a3a98de0319e5fc9b3a1a.jpg b/.raw/MidasCivilAnalysisReference/images/page-440_3056be7fb39851a84566c234d970eba9ce137bc8740a3a98de0319e5fc9b3a1a.jpg new file mode 100644 index 00000000..74dff33f Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-440_3056be7fb39851a84566c234d970eba9ce137bc8740a3a98de0319e5fc9b3a1a.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-440_8409c1a3f63bcb5bead01f3da637af2b017d6e191d8faa92c882323da7c44ce0.jpg b/.raw/MidasCivilAnalysisReference/images/page-440_8409c1a3f63bcb5bead01f3da637af2b017d6e191d8faa92c882323da7c44ce0.jpg new file mode 100644 index 00000000..eba4fae6 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-440_8409c1a3f63bcb5bead01f3da637af2b017d6e191d8faa92c882323da7c44ce0.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-440_9b7bbb5fbeb3b72000edd40d6fbc939b9ae5c832a357e70cd32a5999e9adf7e4.jpg b/.raw/MidasCivilAnalysisReference/images/page-440_9b7bbb5fbeb3b72000edd40d6fbc939b9ae5c832a357e70cd32a5999e9adf7e4.jpg new file mode 100644 index 00000000..49a7ba58 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-440_9b7bbb5fbeb3b72000edd40d6fbc939b9ae5c832a357e70cd32a5999e9adf7e4.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-440_9bb2586730a59dc3a8ee27781c1f9b725d5f293a2d2660ee5f8816ddb534857f.jpg b/.raw/MidasCivilAnalysisReference/images/page-440_9bb2586730a59dc3a8ee27781c1f9b725d5f293a2d2660ee5f8816ddb534857f.jpg new file mode 100644 index 00000000..15da1d08 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-440_9bb2586730a59dc3a8ee27781c1f9b725d5f293a2d2660ee5f8816ddb534857f.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-441_1e016bcf41cc7e987e4c999966b0457b8dd9dc95ff7ff2d00a21c8a31b742a2a.jpg b/.raw/MidasCivilAnalysisReference/images/page-441_1e016bcf41cc7e987e4c999966b0457b8dd9dc95ff7ff2d00a21c8a31b742a2a.jpg new file mode 100644 index 00000000..cbf4c1fd Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-441_1e016bcf41cc7e987e4c999966b0457b8dd9dc95ff7ff2d00a21c8a31b742a2a.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-441_f32989a276ffe9dd35bf17835c42078c19f3bf6ecc7e39ad41e7a3d47e3396f8.jpg b/.raw/MidasCivilAnalysisReference/images/page-441_f32989a276ffe9dd35bf17835c42078c19f3bf6ecc7e39ad41e7a3d47e3396f8.jpg new file mode 100644 index 00000000..f13f3f09 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-441_f32989a276ffe9dd35bf17835c42078c19f3bf6ecc7e39ad41e7a3d47e3396f8.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-442_8597c967279e912c22b7acdab7bdf417c65cb7ef2000767b38029acbfdd989d4.jpg b/.raw/MidasCivilAnalysisReference/images/page-442_8597c967279e912c22b7acdab7bdf417c65cb7ef2000767b38029acbfdd989d4.jpg new file mode 100644 index 00000000..789741ab Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-442_8597c967279e912c22b7acdab7bdf417c65cb7ef2000767b38029acbfdd989d4.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-442_be9e66cbdf8a2d367de6e745f7c062300534bee9fe89cabf814329033931abe0.jpg b/.raw/MidasCivilAnalysisReference/images/page-442_be9e66cbdf8a2d367de6e745f7c062300534bee9fe89cabf814329033931abe0.jpg new file mode 100644 index 00000000..1f2a0bd3 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-442_be9e66cbdf8a2d367de6e745f7c062300534bee9fe89cabf814329033931abe0.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-443_37c0674287e4f7e189149a68078f31205d4b22911b73be843f1887097a0da85c.jpg b/.raw/MidasCivilAnalysisReference/images/page-443_37c0674287e4f7e189149a68078f31205d4b22911b73be843f1887097a0da85c.jpg new file mode 100644 index 00000000..cd6e1042 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-443_37c0674287e4f7e189149a68078f31205d4b22911b73be843f1887097a0da85c.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-443_5c40d6df7215a99930507169408870a30203e95d890c7a04dadcb8be69cbcaed.jpg b/.raw/MidasCivilAnalysisReference/images/page-443_5c40d6df7215a99930507169408870a30203e95d890c7a04dadcb8be69cbcaed.jpg new file mode 100644 index 00000000..0e5adb3b Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-443_5c40d6df7215a99930507169408870a30203e95d890c7a04dadcb8be69cbcaed.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-443_60290934ca38b3cba30c6e69e107dd2ff9407d08363aac8edd3e6be5017b4698.jpg b/.raw/MidasCivilAnalysisReference/images/page-443_60290934ca38b3cba30c6e69e107dd2ff9407d08363aac8edd3e6be5017b4698.jpg new file mode 100644 index 00000000..83af15c6 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-443_60290934ca38b3cba30c6e69e107dd2ff9407d08363aac8edd3e6be5017b4698.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-443_7ead3537bef95a595493f4e82c5fdb14c85ef197b4feb555a43b00035ccc2f32.jpg b/.raw/MidasCivilAnalysisReference/images/page-443_7ead3537bef95a595493f4e82c5fdb14c85ef197b4feb555a43b00035ccc2f32.jpg new file mode 100644 index 00000000..cc603074 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-443_7ead3537bef95a595493f4e82c5fdb14c85ef197b4feb555a43b00035ccc2f32.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-443_9aaf73bc7bf6921b20e7d2531152bce6faeabe9c15858972fd9f0fa3218c14c5.jpg b/.raw/MidasCivilAnalysisReference/images/page-443_9aaf73bc7bf6921b20e7d2531152bce6faeabe9c15858972fd9f0fa3218c14c5.jpg new file mode 100644 index 00000000..e797df2f Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-443_9aaf73bc7bf6921b20e7d2531152bce6faeabe9c15858972fd9f0fa3218c14c5.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-443_daa5569f8f5e1ccc8039ddf2e53483bc4f55b660840f77359fde04296634ae88.jpg b/.raw/MidasCivilAnalysisReference/images/page-443_daa5569f8f5e1ccc8039ddf2e53483bc4f55b660840f77359fde04296634ae88.jpg new file mode 100644 index 00000000..f8134bdf Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-443_daa5569f8f5e1ccc8039ddf2e53483bc4f55b660840f77359fde04296634ae88.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-444_07d88d32ecd8ba444da376f78cc52fa5b42d295cf30451b7dc65ee5d43551676.jpg b/.raw/MidasCivilAnalysisReference/images/page-444_07d88d32ecd8ba444da376f78cc52fa5b42d295cf30451b7dc65ee5d43551676.jpg new file mode 100644 index 00000000..e15d6fb0 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-444_07d88d32ecd8ba444da376f78cc52fa5b42d295cf30451b7dc65ee5d43551676.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-444_0e31370c522b0ac7826885773fd2c492716a2bbf7dba05e4e48593a8d5372573.jpg b/.raw/MidasCivilAnalysisReference/images/page-444_0e31370c522b0ac7826885773fd2c492716a2bbf7dba05e4e48593a8d5372573.jpg new file mode 100644 index 00000000..8c325774 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-444_0e31370c522b0ac7826885773fd2c492716a2bbf7dba05e4e48593a8d5372573.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-444_4ccb8f4efced6b9cd76063083763f032dd323c488396743794b8c73140a530db.jpg b/.raw/MidasCivilAnalysisReference/images/page-444_4ccb8f4efced6b9cd76063083763f032dd323c488396743794b8c73140a530db.jpg new file mode 100644 index 00000000..35e6fab4 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-444_4ccb8f4efced6b9cd76063083763f032dd323c488396743794b8c73140a530db.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-444_ac4f4df83a9417994af4db9f197d1bd50ebd977236dc7478f8c47b542f35ca3b.jpg b/.raw/MidasCivilAnalysisReference/images/page-444_ac4f4df83a9417994af4db9f197d1bd50ebd977236dc7478f8c47b542f35ca3b.jpg new file mode 100644 index 00000000..876d13f4 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-444_ac4f4df83a9417994af4db9f197d1bd50ebd977236dc7478f8c47b542f35ca3b.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-445_1a352d088e4509d0b6b11c8f1f23267e279a4402fe7e47b5de4517ba34093dd8.jpg b/.raw/MidasCivilAnalysisReference/images/page-445_1a352d088e4509d0b6b11c8f1f23267e279a4402fe7e47b5de4517ba34093dd8.jpg new file mode 100644 index 00000000..9379b398 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-445_1a352d088e4509d0b6b11c8f1f23267e279a4402fe7e47b5de4517ba34093dd8.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-445_31fac22ce4e80059b3e5908b7397e868722a8d4293787f8be58b3350019df304.jpg b/.raw/MidasCivilAnalysisReference/images/page-445_31fac22ce4e80059b3e5908b7397e868722a8d4293787f8be58b3350019df304.jpg new file mode 100644 index 00000000..e2319b51 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-445_31fac22ce4e80059b3e5908b7397e868722a8d4293787f8be58b3350019df304.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-445_6d2cc5a3e2003a7eeb334c08ac6861fd76bc8f30707a706aaf388d9323b305e0.jpg b/.raw/MidasCivilAnalysisReference/images/page-445_6d2cc5a3e2003a7eeb334c08ac6861fd76bc8f30707a706aaf388d9323b305e0.jpg new file mode 100644 index 00000000..70c23fa4 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-445_6d2cc5a3e2003a7eeb334c08ac6861fd76bc8f30707a706aaf388d9323b305e0.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-445_7bce0e1b90d17f3b876e83f80c4f89da7ae5d76133e0ab3e4f8f3e0514b99c75.jpg b/.raw/MidasCivilAnalysisReference/images/page-445_7bce0e1b90d17f3b876e83f80c4f89da7ae5d76133e0ab3e4f8f3e0514b99c75.jpg new file mode 100644 index 00000000..76c570a6 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-445_7bce0e1b90d17f3b876e83f80c4f89da7ae5d76133e0ab3e4f8f3e0514b99c75.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-445_ca1f6db44428ac462bd6ccfa58c6f6422ed2a80ba2d01175944c7e3c61b86068.jpg b/.raw/MidasCivilAnalysisReference/images/page-445_ca1f6db44428ac462bd6ccfa58c6f6422ed2a80ba2d01175944c7e3c61b86068.jpg new file mode 100644 index 00000000..48b6ba0e Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-445_ca1f6db44428ac462bd6ccfa58c6f6422ed2a80ba2d01175944c7e3c61b86068.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-445_fdb050af125114ac588a7468d838085a53feaf0065f0ea7fd98a15dd9c1b244f.jpg b/.raw/MidasCivilAnalysisReference/images/page-445_fdb050af125114ac588a7468d838085a53feaf0065f0ea7fd98a15dd9c1b244f.jpg new file mode 100644 index 00000000..d75d006b Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-445_fdb050af125114ac588a7468d838085a53feaf0065f0ea7fd98a15dd9c1b244f.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-446_7cc5960f125030a02dd054282f70056d8d10f2929b6f370cd8d8e35195ec5b17.jpg b/.raw/MidasCivilAnalysisReference/images/page-446_7cc5960f125030a02dd054282f70056d8d10f2929b6f370cd8d8e35195ec5b17.jpg new file mode 100644 index 00000000..e3fb6d61 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-446_7cc5960f125030a02dd054282f70056d8d10f2929b6f370cd8d8e35195ec5b17.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-446_edcfde15abafc3b58e273ab8f301ad27576cbd674d413ab647bda5ed90ae5377.jpg b/.raw/MidasCivilAnalysisReference/images/page-446_edcfde15abafc3b58e273ab8f301ad27576cbd674d413ab647bda5ed90ae5377.jpg new file mode 100644 index 00000000..ca80951b Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-446_edcfde15abafc3b58e273ab8f301ad27576cbd674d413ab647bda5ed90ae5377.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-447_29148467356dd4078f742c5f89239f98a224762ab30beff35643e7798b344284.jpg b/.raw/MidasCivilAnalysisReference/images/page-447_29148467356dd4078f742c5f89239f98a224762ab30beff35643e7798b344284.jpg new file mode 100644 index 00000000..d029c8a9 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-447_29148467356dd4078f742c5f89239f98a224762ab30beff35643e7798b344284.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-447_44940cbc7a4a0f1c2fa4152fd898a0051524607a227855995d2511528ac45755.jpg b/.raw/MidasCivilAnalysisReference/images/page-447_44940cbc7a4a0f1c2fa4152fd898a0051524607a227855995d2511528ac45755.jpg new file mode 100644 index 00000000..cf580eac Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-447_44940cbc7a4a0f1c2fa4152fd898a0051524607a227855995d2511528ac45755.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-447_85409f79959e00ec03720ae5360c3785303b4da24db95892515e6901cb26cf9a.jpg b/.raw/MidasCivilAnalysisReference/images/page-447_85409f79959e00ec03720ae5360c3785303b4da24db95892515e6901cb26cf9a.jpg new file mode 100644 index 00000000..8beba2a8 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-447_85409f79959e00ec03720ae5360c3785303b4da24db95892515e6901cb26cf9a.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-447_c9520ed29dc713329c2edc92d7299d67d5904f5e6ae5338427b2c8aa9d24e46e.jpg b/.raw/MidasCivilAnalysisReference/images/page-447_c9520ed29dc713329c2edc92d7299d67d5904f5e6ae5338427b2c8aa9d24e46e.jpg new file mode 100644 index 00000000..f11f25bf Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-447_c9520ed29dc713329c2edc92d7299d67d5904f5e6ae5338427b2c8aa9d24e46e.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-447_e7f7dbfb6f6d516d17c9466f4e22b3a9d121a92914a3cd50f571c0888e4f1163.jpg b/.raw/MidasCivilAnalysisReference/images/page-447_e7f7dbfb6f6d516d17c9466f4e22b3a9d121a92914a3cd50f571c0888e4f1163.jpg new file mode 100644 index 00000000..5d549446 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-447_e7f7dbfb6f6d516d17c9466f4e22b3a9d121a92914a3cd50f571c0888e4f1163.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-450_863d0be89da1abf6f5997a3a6c5461838ed5945a1b7db2907b5e1cc64d0f6ab4.jpg b/.raw/MidasCivilAnalysisReference/images/page-450_863d0be89da1abf6f5997a3a6c5461838ed5945a1b7db2907b5e1cc64d0f6ab4.jpg new file mode 100644 index 00000000..44614f8a Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-450_863d0be89da1abf6f5997a3a6c5461838ed5945a1b7db2907b5e1cc64d0f6ab4.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-451_36849830adcbbd4883a82e1fba1e6d3652c0b6c17dddc8b0af09a582c049b414.jpg b/.raw/MidasCivilAnalysisReference/images/page-451_36849830adcbbd4883a82e1fba1e6d3652c0b6c17dddc8b0af09a582c049b414.jpg new file mode 100644 index 00000000..fd612d28 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-451_36849830adcbbd4883a82e1fba1e6d3652c0b6c17dddc8b0af09a582c049b414.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-451_c22fe356fac49e25b5c97359ddb13ac590a35a79b5ff156109c5fcfbd4b0000f.jpg b/.raw/MidasCivilAnalysisReference/images/page-451_c22fe356fac49e25b5c97359ddb13ac590a35a79b5ff156109c5fcfbd4b0000f.jpg new file mode 100644 index 00000000..fea59049 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-451_c22fe356fac49e25b5c97359ddb13ac590a35a79b5ff156109c5fcfbd4b0000f.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-451_eb6e770168920d4192a4b429170f3285c7fe20d36689a7ed236c17855d341bdc.jpg b/.raw/MidasCivilAnalysisReference/images/page-451_eb6e770168920d4192a4b429170f3285c7fe20d36689a7ed236c17855d341bdc.jpg new file mode 100644 index 00000000..344012f5 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-451_eb6e770168920d4192a4b429170f3285c7fe20d36689a7ed236c17855d341bdc.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-452_3e6fda8f887fe4047ef8d3bafa701675a98154070cb37e37b9c6d29a7eec9d1d.jpg b/.raw/MidasCivilAnalysisReference/images/page-452_3e6fda8f887fe4047ef8d3bafa701675a98154070cb37e37b9c6d29a7eec9d1d.jpg new file mode 100644 index 00000000..86444d0a Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-452_3e6fda8f887fe4047ef8d3bafa701675a98154070cb37e37b9c6d29a7eec9d1d.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-452_932b9ec5a841e3799f512226f9af6b212751698d86aa863c7db5866da9893007.jpg b/.raw/MidasCivilAnalysisReference/images/page-452_932b9ec5a841e3799f512226f9af6b212751698d86aa863c7db5866da9893007.jpg new file mode 100644 index 00000000..fdd01ed0 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-452_932b9ec5a841e3799f512226f9af6b212751698d86aa863c7db5866da9893007.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-452_d33d86f23d433010d0b43cebcf45ed09b93caafb594b33d8cfec6f76c2d871da.jpg b/.raw/MidasCivilAnalysisReference/images/page-452_d33d86f23d433010d0b43cebcf45ed09b93caafb594b33d8cfec6f76c2d871da.jpg new file mode 100644 index 00000000..31746481 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-452_d33d86f23d433010d0b43cebcf45ed09b93caafb594b33d8cfec6f76c2d871da.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-452_eccbeb68e136d489ceb0bfbe6dc4e6ad4f0b640d51a3e47f9c77e7f778aa4541.jpg b/.raw/MidasCivilAnalysisReference/images/page-452_eccbeb68e136d489ceb0bfbe6dc4e6ad4f0b640d51a3e47f9c77e7f778aa4541.jpg new file mode 100644 index 00000000..ef03668c Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-452_eccbeb68e136d489ceb0bfbe6dc4e6ad4f0b640d51a3e47f9c77e7f778aa4541.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-453_82d47a46ab9bde1ea5822df42c206b2d631ab3b4396334d9b70576d055e2433d.jpg b/.raw/MidasCivilAnalysisReference/images/page-453_82d47a46ab9bde1ea5822df42c206b2d631ab3b4396334d9b70576d055e2433d.jpg new file mode 100644 index 00000000..1f9ab5cc Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-453_82d47a46ab9bde1ea5822df42c206b2d631ab3b4396334d9b70576d055e2433d.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-454_01aae777e9462a31a31d81758b04f309be3dae6ff281577f632d9ba8addb5a5a.jpg b/.raw/MidasCivilAnalysisReference/images/page-454_01aae777e9462a31a31d81758b04f309be3dae6ff281577f632d9ba8addb5a5a.jpg new file mode 100644 index 00000000..a2950647 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-454_01aae777e9462a31a31d81758b04f309be3dae6ff281577f632d9ba8addb5a5a.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-454_0e743a1a66b066112457e2026d33002ba933158aee91e9f8445ce2b15f6e4917.jpg b/.raw/MidasCivilAnalysisReference/images/page-454_0e743a1a66b066112457e2026d33002ba933158aee91e9f8445ce2b15f6e4917.jpg new file mode 100644 index 00000000..b721eea2 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-454_0e743a1a66b066112457e2026d33002ba933158aee91e9f8445ce2b15f6e4917.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-454_b3e17b11411c92ab7961fb765c6e5bd43fd9b9b5988c8fceeba3c35d6a9d156f.jpg b/.raw/MidasCivilAnalysisReference/images/page-454_b3e17b11411c92ab7961fb765c6e5bd43fd9b9b5988c8fceeba3c35d6a9d156f.jpg new file mode 100644 index 00000000..232fc7c2 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-454_b3e17b11411c92ab7961fb765c6e5bd43fd9b9b5988c8fceeba3c35d6a9d156f.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-456_2a5451062bdd5af99378279dc46f38d3c55df054a05acc2bfdf805787acfcc84.jpg b/.raw/MidasCivilAnalysisReference/images/page-456_2a5451062bdd5af99378279dc46f38d3c55df054a05acc2bfdf805787acfcc84.jpg new file mode 100644 index 00000000..29541ddc Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-456_2a5451062bdd5af99378279dc46f38d3c55df054a05acc2bfdf805787acfcc84.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-456_41c334d0bd60aefbacf45f2db29f28b2343bc503c2e094a675c096454126b8b2.jpg b/.raw/MidasCivilAnalysisReference/images/page-456_41c334d0bd60aefbacf45f2db29f28b2343bc503c2e094a675c096454126b8b2.jpg new file mode 100644 index 00000000..53a38bd0 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-456_41c334d0bd60aefbacf45f2db29f28b2343bc503c2e094a675c096454126b8b2.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-456_62b722de24aa48505f12cdb7091d77a55a0177cfd5552c64ffc88737e19ed53e.jpg b/.raw/MidasCivilAnalysisReference/images/page-456_62b722de24aa48505f12cdb7091d77a55a0177cfd5552c64ffc88737e19ed53e.jpg new file mode 100644 index 00000000..21e62b90 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-456_62b722de24aa48505f12cdb7091d77a55a0177cfd5552c64ffc88737e19ed53e.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-456_7ec811bdc6ed38f699d9de6daab0df0571b58efdd936975e03c980c5bbc39230.jpg b/.raw/MidasCivilAnalysisReference/images/page-456_7ec811bdc6ed38f699d9de6daab0df0571b58efdd936975e03c980c5bbc39230.jpg new file mode 100644 index 00000000..122c2653 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-456_7ec811bdc6ed38f699d9de6daab0df0571b58efdd936975e03c980c5bbc39230.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-456_e4a5fd63f2084adf405bf37fc3680cad94513eaae1b4cf9f9b09fd2f57c47d47.jpg b/.raw/MidasCivilAnalysisReference/images/page-456_e4a5fd63f2084adf405bf37fc3680cad94513eaae1b4cf9f9b09fd2f57c47d47.jpg new file mode 100644 index 00000000..712d9cb3 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-456_e4a5fd63f2084adf405bf37fc3680cad94513eaae1b4cf9f9b09fd2f57c47d47.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-456_e7983d2c52b80da4f1bedb253bb9e1e3dd09a9440aceaa416c22776c86ac2fd0.jpg b/.raw/MidasCivilAnalysisReference/images/page-456_e7983d2c52b80da4f1bedb253bb9e1e3dd09a9440aceaa416c22776c86ac2fd0.jpg new file mode 100644 index 00000000..9be2a12e Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-456_e7983d2c52b80da4f1bedb253bb9e1e3dd09a9440aceaa416c22776c86ac2fd0.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-456_f054c1e583a1885a341e7c2c7fabf6422ab92da23c3e891c43fdb83ff41790a3.jpg b/.raw/MidasCivilAnalysisReference/images/page-456_f054c1e583a1885a341e7c2c7fabf6422ab92da23c3e891c43fdb83ff41790a3.jpg new file mode 100644 index 00000000..c2fa2fee Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-456_f054c1e583a1885a341e7c2c7fabf6422ab92da23c3e891c43fdb83ff41790a3.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-457_09f7bb3c72326ab65c41e9755409b7dfed47682273564141a120077e8ff3da01.jpg b/.raw/MidasCivilAnalysisReference/images/page-457_09f7bb3c72326ab65c41e9755409b7dfed47682273564141a120077e8ff3da01.jpg new file mode 100644 index 00000000..c462858d Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-457_09f7bb3c72326ab65c41e9755409b7dfed47682273564141a120077e8ff3da01.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-457_1643b94ee1bbc21498b80a32d803e642a029e4c14aa33e7aa358d4ed0d8fa3ca.jpg b/.raw/MidasCivilAnalysisReference/images/page-457_1643b94ee1bbc21498b80a32d803e642a029e4c14aa33e7aa358d4ed0d8fa3ca.jpg new file mode 100644 index 00000000..6399153f Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-457_1643b94ee1bbc21498b80a32d803e642a029e4c14aa33e7aa358d4ed0d8fa3ca.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-457_171950453b5cb9f7adaa97b81c89348b1e1b62acc2036904661495f0f73d762c.jpg b/.raw/MidasCivilAnalysisReference/images/page-457_171950453b5cb9f7adaa97b81c89348b1e1b62acc2036904661495f0f73d762c.jpg new file mode 100644 index 00000000..6e874c46 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-457_171950453b5cb9f7adaa97b81c89348b1e1b62acc2036904661495f0f73d762c.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-457_455e39aaff6ae4554afc37ee58f229b88f404bf6266d6cd7fcbcd6e28cedbc4f.jpg b/.raw/MidasCivilAnalysisReference/images/page-457_455e39aaff6ae4554afc37ee58f229b88f404bf6266d6cd7fcbcd6e28cedbc4f.jpg new file mode 100644 index 00000000..0d2488b4 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-457_455e39aaff6ae4554afc37ee58f229b88f404bf6266d6cd7fcbcd6e28cedbc4f.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-457_9edf5aae66d2b6badef947171912f90b463fc71e15b4c51301d06b9a1e9e6c13.jpg b/.raw/MidasCivilAnalysisReference/images/page-457_9edf5aae66d2b6badef947171912f90b463fc71e15b4c51301d06b9a1e9e6c13.jpg new file mode 100644 index 00000000..5488fbc7 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-457_9edf5aae66d2b6badef947171912f90b463fc71e15b4c51301d06b9a1e9e6c13.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-457_a3efc6bf930eec65b0b5544c2f457ff6158f54ca6beb5c5ec55be61eabdce554.jpg b/.raw/MidasCivilAnalysisReference/images/page-457_a3efc6bf930eec65b0b5544c2f457ff6158f54ca6beb5c5ec55be61eabdce554.jpg new file mode 100644 index 00000000..635947cd Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-457_a3efc6bf930eec65b0b5544c2f457ff6158f54ca6beb5c5ec55be61eabdce554.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-458_2db1f1c1db0d2d20052f40a1dad1fe3d8826d07d097e53da7a13f26863545131.jpg b/.raw/MidasCivilAnalysisReference/images/page-458_2db1f1c1db0d2d20052f40a1dad1fe3d8826d07d097e53da7a13f26863545131.jpg new file mode 100644 index 00000000..b18c5338 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-458_2db1f1c1db0d2d20052f40a1dad1fe3d8826d07d097e53da7a13f26863545131.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-459_2bde5a9ad2c16ab9a98f0e41ac085becef2f3ca6f9d7aaee9cd5de58490e51c9.jpg b/.raw/MidasCivilAnalysisReference/images/page-459_2bde5a9ad2c16ab9a98f0e41ac085becef2f3ca6f9d7aaee9cd5de58490e51c9.jpg new file mode 100644 index 00000000..fda51910 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-459_2bde5a9ad2c16ab9a98f0e41ac085becef2f3ca6f9d7aaee9cd5de58490e51c9.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-459_3667641241cd334897bd266a151bff4a230a6013dca7e90147f4688d0cb1b8da.jpg b/.raw/MidasCivilAnalysisReference/images/page-459_3667641241cd334897bd266a151bff4a230a6013dca7e90147f4688d0cb1b8da.jpg new file mode 100644 index 00000000..e7d711b5 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-459_3667641241cd334897bd266a151bff4a230a6013dca7e90147f4688d0cb1b8da.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-459_882dfeadd07f75c1cdd870499bea596ff8f88b4ca61b7983d594ae5feccf9fea.jpg b/.raw/MidasCivilAnalysisReference/images/page-459_882dfeadd07f75c1cdd870499bea596ff8f88b4ca61b7983d594ae5feccf9fea.jpg new file mode 100644 index 00000000..307c8cbf Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-459_882dfeadd07f75c1cdd870499bea596ff8f88b4ca61b7983d594ae5feccf9fea.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-459_e1e91d3a0590c2d98d4568cf634e04ff4589d095ded5d6ef88f55b0087a8a699.jpg b/.raw/MidasCivilAnalysisReference/images/page-459_e1e91d3a0590c2d98d4568cf634e04ff4589d095ded5d6ef88f55b0087a8a699.jpg new file mode 100644 index 00000000..ff009a94 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-459_e1e91d3a0590c2d98d4568cf634e04ff4589d095ded5d6ef88f55b0087a8a699.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-460_1325d1a95f85471dd253820e2faf3114461efbf7c416e85dbb44d53b5341e1af.jpg b/.raw/MidasCivilAnalysisReference/images/page-460_1325d1a95f85471dd253820e2faf3114461efbf7c416e85dbb44d53b5341e1af.jpg new file mode 100644 index 00000000..228ba0c5 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-460_1325d1a95f85471dd253820e2faf3114461efbf7c416e85dbb44d53b5341e1af.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-460_915b39bb341883cced17fd6e22bd5292146248823bd4cf3be322061d184ab394.jpg b/.raw/MidasCivilAnalysisReference/images/page-460_915b39bb341883cced17fd6e22bd5292146248823bd4cf3be322061d184ab394.jpg new file mode 100644 index 00000000..98fae21e Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-460_915b39bb341883cced17fd6e22bd5292146248823bd4cf3be322061d184ab394.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-460_d1251111ac193d6f53ba69ccfe1c6bd79de8c3162b20004cad422800b4daddd5.jpg b/.raw/MidasCivilAnalysisReference/images/page-460_d1251111ac193d6f53ba69ccfe1c6bd79de8c3162b20004cad422800b4daddd5.jpg new file mode 100644 index 00000000..d09b3498 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-460_d1251111ac193d6f53ba69ccfe1c6bd79de8c3162b20004cad422800b4daddd5.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-462_dc8e601bebe63c0676d9b1aef33b84c2b83b266b537b36308edf8fb7f5a49df1.jpg b/.raw/MidasCivilAnalysisReference/images/page-462_dc8e601bebe63c0676d9b1aef33b84c2b83b266b537b36308edf8fb7f5a49df1.jpg new file mode 100644 index 00000000..a6c24943 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-462_dc8e601bebe63c0676d9b1aef33b84c2b83b266b537b36308edf8fb7f5a49df1.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-465_6040a28ef5ad24a3ca04f76438e068f072b12914fc43a18e23deec23ea9654ab.jpg b/.raw/MidasCivilAnalysisReference/images/page-465_6040a28ef5ad24a3ca04f76438e068f072b12914fc43a18e23deec23ea9654ab.jpg new file mode 100644 index 00000000..739cca4d Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-465_6040a28ef5ad24a3ca04f76438e068f072b12914fc43a18e23deec23ea9654ab.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-468_33d12f996cca59896eb8c246a7eba3ffe9f733b806213ae2459f1725e203b25c.jpg b/.raw/MidasCivilAnalysisReference/images/page-468_33d12f996cca59896eb8c246a7eba3ffe9f733b806213ae2459f1725e203b25c.jpg new file mode 100644 index 00000000..fc98a522 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-468_33d12f996cca59896eb8c246a7eba3ffe9f733b806213ae2459f1725e203b25c.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-470_8c25894a0c4d9b5fdb1c276fb4730338be7af19dd4a6483241f965d22f4fc220.jpg b/.raw/MidasCivilAnalysisReference/images/page-470_8c25894a0c4d9b5fdb1c276fb4730338be7af19dd4a6483241f965d22f4fc220.jpg new file mode 100644 index 00000000..0d470bb8 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-470_8c25894a0c4d9b5fdb1c276fb4730338be7af19dd4a6483241f965d22f4fc220.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-470_92464f61a59b05216201b28c7fac57bcbc55d223d290ae18cb12de8a317c8aad.jpg b/.raw/MidasCivilAnalysisReference/images/page-470_92464f61a59b05216201b28c7fac57bcbc55d223d290ae18cb12de8a317c8aad.jpg new file mode 100644 index 00000000..a71db37b Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-470_92464f61a59b05216201b28c7fac57bcbc55d223d290ae18cb12de8a317c8aad.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-470_f6edb52665a0a1e4d914ca710d210c02543d7e59ef769173c86022562de346f9.jpg b/.raw/MidasCivilAnalysisReference/images/page-470_f6edb52665a0a1e4d914ca710d210c02543d7e59ef769173c86022562de346f9.jpg new file mode 100644 index 00000000..5b628fe7 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-470_f6edb52665a0a1e4d914ca710d210c02543d7e59ef769173c86022562de346f9.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-471_0f7814e1ef1455bc1b482a9737ceae9689733a64affc8719d73f028a61d54fb4.jpg b/.raw/MidasCivilAnalysisReference/images/page-471_0f7814e1ef1455bc1b482a9737ceae9689733a64affc8719d73f028a61d54fb4.jpg new file mode 100644 index 00000000..992f0ca2 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-471_0f7814e1ef1455bc1b482a9737ceae9689733a64affc8719d73f028a61d54fb4.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-471_7442136adcb5608339af357e214640edc01a97321b0fc01f6a4ac8d567b854ac.jpg b/.raw/MidasCivilAnalysisReference/images/page-471_7442136adcb5608339af357e214640edc01a97321b0fc01f6a4ac8d567b854ac.jpg new file mode 100644 index 00000000..5fadcbfe Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-471_7442136adcb5608339af357e214640edc01a97321b0fc01f6a4ac8d567b854ac.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-471_9c56974a69f10045e30b2045dd3f9d92b95c11ea4243a3bf39f7db93c43c6bbf.jpg b/.raw/MidasCivilAnalysisReference/images/page-471_9c56974a69f10045e30b2045dd3f9d92b95c11ea4243a3bf39f7db93c43c6bbf.jpg new file mode 100644 index 00000000..5d026924 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-471_9c56974a69f10045e30b2045dd3f9d92b95c11ea4243a3bf39f7db93c43c6bbf.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-471_aa9f38fbc30ac730ae33b48e9aaf8616c708a5a340e966115c17ed33d8a96189.jpg b/.raw/MidasCivilAnalysisReference/images/page-471_aa9f38fbc30ac730ae33b48e9aaf8616c708a5a340e966115c17ed33d8a96189.jpg new file mode 100644 index 00000000..5d334c49 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-471_aa9f38fbc30ac730ae33b48e9aaf8616c708a5a340e966115c17ed33d8a96189.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-471_aac1b67db0f9c65e4a553303ad0373ce8bda72e5ed924cb651b099e09a764193.jpg b/.raw/MidasCivilAnalysisReference/images/page-471_aac1b67db0f9c65e4a553303ad0373ce8bda72e5ed924cb651b099e09a764193.jpg new file mode 100644 index 00000000..7b3729b4 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-471_aac1b67db0f9c65e4a553303ad0373ce8bda72e5ed924cb651b099e09a764193.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-471_ee282104c417ef1fe9fc241f2aa3fd28618d11aa5819c8711e7db57ab2f81a22.jpg b/.raw/MidasCivilAnalysisReference/images/page-471_ee282104c417ef1fe9fc241f2aa3fd28618d11aa5819c8711e7db57ab2f81a22.jpg new file mode 100644 index 00000000..26bfb6b8 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-471_ee282104c417ef1fe9fc241f2aa3fd28618d11aa5819c8711e7db57ab2f81a22.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-472_60c6b38b2e9cc23f422405947f1c0427aa11af0ad4ab411efbdd746694f7e869.jpg b/.raw/MidasCivilAnalysisReference/images/page-472_60c6b38b2e9cc23f422405947f1c0427aa11af0ad4ab411efbdd746694f7e869.jpg new file mode 100644 index 00000000..d34ef05a Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-472_60c6b38b2e9cc23f422405947f1c0427aa11af0ad4ab411efbdd746694f7e869.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-472_7142613b75359ccb691b454a16dbde56fed248a7a2770463cf227506588d5c52.jpg b/.raw/MidasCivilAnalysisReference/images/page-472_7142613b75359ccb691b454a16dbde56fed248a7a2770463cf227506588d5c52.jpg new file mode 100644 index 00000000..68ec6390 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-472_7142613b75359ccb691b454a16dbde56fed248a7a2770463cf227506588d5c52.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-472_ddbb1abc4fff4e820e53d0f2d613bea35f07c2aa6c216e663cc6d70fad7d094f.jpg b/.raw/MidasCivilAnalysisReference/images/page-472_ddbb1abc4fff4e820e53d0f2d613bea35f07c2aa6c216e663cc6d70fad7d094f.jpg new file mode 100644 index 00000000..e4a6ffa9 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-472_ddbb1abc4fff4e820e53d0f2d613bea35f07c2aa6c216e663cc6d70fad7d094f.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-473_7b50f060898bfde6d8f248faebdf7e8c4e2410f41cde47ae0459003dffaa32e8.jpg b/.raw/MidasCivilAnalysisReference/images/page-473_7b50f060898bfde6d8f248faebdf7e8c4e2410f41cde47ae0459003dffaa32e8.jpg new file mode 100644 index 00000000..4d12dfe6 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-473_7b50f060898bfde6d8f248faebdf7e8c4e2410f41cde47ae0459003dffaa32e8.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-473_dd3e2b33da6f3cec20f88db917a3b157461dd9709f17ef2fa07bc2dc61e39d62.jpg b/.raw/MidasCivilAnalysisReference/images/page-473_dd3e2b33da6f3cec20f88db917a3b157461dd9709f17ef2fa07bc2dc61e39d62.jpg new file mode 100644 index 00000000..84509c4e Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-473_dd3e2b33da6f3cec20f88db917a3b157461dd9709f17ef2fa07bc2dc61e39d62.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-474_db6aa1a9404c1ae42fd30185ac38f9bad500110c5b8835758a3ae88d26f2a02d.jpg b/.raw/MidasCivilAnalysisReference/images/page-474_db6aa1a9404c1ae42fd30185ac38f9bad500110c5b8835758a3ae88d26f2a02d.jpg new file mode 100644 index 00000000..188874a0 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-474_db6aa1a9404c1ae42fd30185ac38f9bad500110c5b8835758a3ae88d26f2a02d.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-476_67a7f74b8738bc4aab8f4e9bca19999ccf7a4f96e5cd9da0efdf0b4b26f15240.jpg b/.raw/MidasCivilAnalysisReference/images/page-476_67a7f74b8738bc4aab8f4e9bca19999ccf7a4f96e5cd9da0efdf0b4b26f15240.jpg new file mode 100644 index 00000000..5c81eda4 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-476_67a7f74b8738bc4aab8f4e9bca19999ccf7a4f96e5cd9da0efdf0b4b26f15240.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-477_3dd05969ca8e3fa08e86c4b444a9eb6d13151aec16f2365c7c5861c897646088.jpg b/.raw/MidasCivilAnalysisReference/images/page-477_3dd05969ca8e3fa08e86c4b444a9eb6d13151aec16f2365c7c5861c897646088.jpg new file mode 100644 index 00000000..f9936726 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-477_3dd05969ca8e3fa08e86c4b444a9eb6d13151aec16f2365c7c5861c897646088.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-477_4afe5c7e6f5b9a97469c4ff0388c7490ea8c76b0d3398145479a164264eaaffb.jpg b/.raw/MidasCivilAnalysisReference/images/page-477_4afe5c7e6f5b9a97469c4ff0388c7490ea8c76b0d3398145479a164264eaaffb.jpg new file mode 100644 index 00000000..1eeffe62 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-477_4afe5c7e6f5b9a97469c4ff0388c7490ea8c76b0d3398145479a164264eaaffb.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-477_67fcc8cb01ead02dd34405be2db03eef6fd1c8d2ac6b421c8d6a2dce8dae93ec.jpg b/.raw/MidasCivilAnalysisReference/images/page-477_67fcc8cb01ead02dd34405be2db03eef6fd1c8d2ac6b421c8d6a2dce8dae93ec.jpg new file mode 100644 index 00000000..9a13f966 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-477_67fcc8cb01ead02dd34405be2db03eef6fd1c8d2ac6b421c8d6a2dce8dae93ec.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-477_86d17c1d40aeb76e7c1d1b0a6a59eafb025272f44522bfdb689adc8d8ae5991f.jpg b/.raw/MidasCivilAnalysisReference/images/page-477_86d17c1d40aeb76e7c1d1b0a6a59eafb025272f44522bfdb689adc8d8ae5991f.jpg new file mode 100644 index 00000000..f1c6892c Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-477_86d17c1d40aeb76e7c1d1b0a6a59eafb025272f44522bfdb689adc8d8ae5991f.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-478_866191372813deb111a42de213bb8d9b2ffb2db704f5c46a3444f41413ba53bc.jpg b/.raw/MidasCivilAnalysisReference/images/page-478_866191372813deb111a42de213bb8d9b2ffb2db704f5c46a3444f41413ba53bc.jpg new file mode 100644 index 00000000..3c68b470 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-478_866191372813deb111a42de213bb8d9b2ffb2db704f5c46a3444f41413ba53bc.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-479_05de733ae0052d2cb191ceaee1393d3e6c7b5e082065263c0182667a47d4871c.jpg b/.raw/MidasCivilAnalysisReference/images/page-479_05de733ae0052d2cb191ceaee1393d3e6c7b5e082065263c0182667a47d4871c.jpg new file mode 100644 index 00000000..df777432 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-479_05de733ae0052d2cb191ceaee1393d3e6c7b5e082065263c0182667a47d4871c.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-479_51b07ef1fc6708ac40f6825d5d1459b5d3eb9c0125a23f6800c7d385e4d57c38.jpg b/.raw/MidasCivilAnalysisReference/images/page-479_51b07ef1fc6708ac40f6825d5d1459b5d3eb9c0125a23f6800c7d385e4d57c38.jpg new file mode 100644 index 00000000..054e3c22 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-479_51b07ef1fc6708ac40f6825d5d1459b5d3eb9c0125a23f6800c7d385e4d57c38.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-479_55aefa4ca75f8da3fe1d5d841b5273d9e309257c11868779e14c2e56dfff01a1.jpg b/.raw/MidasCivilAnalysisReference/images/page-479_55aefa4ca75f8da3fe1d5d841b5273d9e309257c11868779e14c2e56dfff01a1.jpg new file mode 100644 index 00000000..a0ff57d6 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-479_55aefa4ca75f8da3fe1d5d841b5273d9e309257c11868779e14c2e56dfff01a1.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-479_59f15d3c6ebc6b142652dc06e5e06c533aa36ad2e119f1b6e1c13b1485d70e64.jpg b/.raw/MidasCivilAnalysisReference/images/page-479_59f15d3c6ebc6b142652dc06e5e06c533aa36ad2e119f1b6e1c13b1485d70e64.jpg new file mode 100644 index 00000000..ac26db5c Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-479_59f15d3c6ebc6b142652dc06e5e06c533aa36ad2e119f1b6e1c13b1485d70e64.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-479_b5ae40f801f8404c99a4812a0b3c291df3a9ed5aae8e9e39a1bf2e5236745fa0.jpg b/.raw/MidasCivilAnalysisReference/images/page-479_b5ae40f801f8404c99a4812a0b3c291df3a9ed5aae8e9e39a1bf2e5236745fa0.jpg new file mode 100644 index 00000000..a19cb64c Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-479_b5ae40f801f8404c99a4812a0b3c291df3a9ed5aae8e9e39a1bf2e5236745fa0.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-479_ee8882c744acbe19ad66acc535095cbd2a6252ea7737478e0576493b4d4fb024.jpg b/.raw/MidasCivilAnalysisReference/images/page-479_ee8882c744acbe19ad66acc535095cbd2a6252ea7737478e0576493b4d4fb024.jpg new file mode 100644 index 00000000..80c8fbea Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-479_ee8882c744acbe19ad66acc535095cbd2a6252ea7737478e0576493b4d4fb024.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-479_f27ce54934d39fab3e379f9d575a0b3ff24b6f201c197764fab7327b4a2fdf72.jpg b/.raw/MidasCivilAnalysisReference/images/page-479_f27ce54934d39fab3e379f9d575a0b3ff24b6f201c197764fab7327b4a2fdf72.jpg new file mode 100644 index 00000000..11324c5b Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-479_f27ce54934d39fab3e379f9d575a0b3ff24b6f201c197764fab7327b4a2fdf72.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-481_04582d25d3111050b342aceab6f3c97252744754aaac6ffe2c05dc27747e8058.jpg b/.raw/MidasCivilAnalysisReference/images/page-481_04582d25d3111050b342aceab6f3c97252744754aaac6ffe2c05dc27747e8058.jpg new file mode 100644 index 00000000..58a27f00 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-481_04582d25d3111050b342aceab6f3c97252744754aaac6ffe2c05dc27747e8058.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-481_24415e8fb1388d39abd935c23f2f8c2799a01d77964e9ad1e4f6bc7e805b7204.jpg b/.raw/MidasCivilAnalysisReference/images/page-481_24415e8fb1388d39abd935c23f2f8c2799a01d77964e9ad1e4f6bc7e805b7204.jpg new file mode 100644 index 00000000..edc89a38 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-481_24415e8fb1388d39abd935c23f2f8c2799a01d77964e9ad1e4f6bc7e805b7204.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-481_d29dd76b0904ca596f6a447578c3251253079261cd7f1bb6ec3980059bc07f93.jpg b/.raw/MidasCivilAnalysisReference/images/page-481_d29dd76b0904ca596f6a447578c3251253079261cd7f1bb6ec3980059bc07f93.jpg new file mode 100644 index 00000000..cc71328d Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-481_d29dd76b0904ca596f6a447578c3251253079261cd7f1bb6ec3980059bc07f93.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-481_d31c6767220b043e04663f91ecc2edcf013c707296099b0f6487d575f9f791a3.jpg b/.raw/MidasCivilAnalysisReference/images/page-481_d31c6767220b043e04663f91ecc2edcf013c707296099b0f6487d575f9f791a3.jpg new file mode 100644 index 00000000..fb1c76cc Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-481_d31c6767220b043e04663f91ecc2edcf013c707296099b0f6487d575f9f791a3.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-482_01cdab8d9aa70f10f72c577c386560e77f920b8ba18793b5c7d8091cae598624.jpg b/.raw/MidasCivilAnalysisReference/images/page-482_01cdab8d9aa70f10f72c577c386560e77f920b8ba18793b5c7d8091cae598624.jpg new file mode 100644 index 00000000..07246a11 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-482_01cdab8d9aa70f10f72c577c386560e77f920b8ba18793b5c7d8091cae598624.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-482_a19a1358dac3588411b5af52eff995469e7f3e70bc9fa247748f401e533daddc.jpg b/.raw/MidasCivilAnalysisReference/images/page-482_a19a1358dac3588411b5af52eff995469e7f3e70bc9fa247748f401e533daddc.jpg new file mode 100644 index 00000000..a3a523ff Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-482_a19a1358dac3588411b5af52eff995469e7f3e70bc9fa247748f401e533daddc.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-482_cd1c8fa0c4baed20fa919fdb59de65ad679591a0d3366e1622b3e3c1a978385f.jpg b/.raw/MidasCivilAnalysisReference/images/page-482_cd1c8fa0c4baed20fa919fdb59de65ad679591a0d3366e1622b3e3c1a978385f.jpg new file mode 100644 index 00000000..8588f805 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-482_cd1c8fa0c4baed20fa919fdb59de65ad679591a0d3366e1622b3e3c1a978385f.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-483_5883cdf8859ded77e177f7ecfede7ffc334205e5f631967fe741115f6493ebc8.jpg b/.raw/MidasCivilAnalysisReference/images/page-483_5883cdf8859ded77e177f7ecfede7ffc334205e5f631967fe741115f6493ebc8.jpg new file mode 100644 index 00000000..b5177cf1 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-483_5883cdf8859ded77e177f7ecfede7ffc334205e5f631967fe741115f6493ebc8.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-483_ecf35b431534674128bf371f8bf6203590ec0af392d2a497c6b6711c1f7af7b9.jpg b/.raw/MidasCivilAnalysisReference/images/page-483_ecf35b431534674128bf371f8bf6203590ec0af392d2a497c6b6711c1f7af7b9.jpg new file mode 100644 index 00000000..ac7470da Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-483_ecf35b431534674128bf371f8bf6203590ec0af392d2a497c6b6711c1f7af7b9.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-484_9dff79a6f83766f9e43eaa9364a3902b0116ee8c907689945f4fcc2af1f3ef29.jpg b/.raw/MidasCivilAnalysisReference/images/page-484_9dff79a6f83766f9e43eaa9364a3902b0116ee8c907689945f4fcc2af1f3ef29.jpg new file mode 100644 index 00000000..219ff5d4 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-484_9dff79a6f83766f9e43eaa9364a3902b0116ee8c907689945f4fcc2af1f3ef29.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-485_33c246c198aa4782a97ac33b4e2bbd1df209b60828cb4ffe2de8aa242566edc9.jpg b/.raw/MidasCivilAnalysisReference/images/page-485_33c246c198aa4782a97ac33b4e2bbd1df209b60828cb4ffe2de8aa242566edc9.jpg new file mode 100644 index 00000000..0327bac5 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-485_33c246c198aa4782a97ac33b4e2bbd1df209b60828cb4ffe2de8aa242566edc9.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-485_a8a5742e002ce05dcd843ae8d80a8b70512ba159d2bb44b4cfc63d310fa93aa0.jpg b/.raw/MidasCivilAnalysisReference/images/page-485_a8a5742e002ce05dcd843ae8d80a8b70512ba159d2bb44b4cfc63d310fa93aa0.jpg new file mode 100644 index 00000000..e5a8f845 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-485_a8a5742e002ce05dcd843ae8d80a8b70512ba159d2bb44b4cfc63d310fa93aa0.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-485_cd341db889c05521ba2a6ec48874c2a073bcf2a1e744bc2b1aeb878d7b413f03.jpg b/.raw/MidasCivilAnalysisReference/images/page-485_cd341db889c05521ba2a6ec48874c2a073bcf2a1e744bc2b1aeb878d7b413f03.jpg new file mode 100644 index 00000000..d5dcceb1 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-485_cd341db889c05521ba2a6ec48874c2a073bcf2a1e744bc2b1aeb878d7b413f03.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-486_217a4e7772ef9ba86953189d6b4cde46e68f55fff7b0543d8a7ec93330cbfc43.jpg b/.raw/MidasCivilAnalysisReference/images/page-486_217a4e7772ef9ba86953189d6b4cde46e68f55fff7b0543d8a7ec93330cbfc43.jpg new file mode 100644 index 00000000..69f52ea4 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-486_217a4e7772ef9ba86953189d6b4cde46e68f55fff7b0543d8a7ec93330cbfc43.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-486_514de689747d1880646f2c33dd401138c349d1519ea8af532c67b90c8bc208b2.jpg b/.raw/MidasCivilAnalysisReference/images/page-486_514de689747d1880646f2c33dd401138c349d1519ea8af532c67b90c8bc208b2.jpg new file mode 100644 index 00000000..4a4d72ad Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-486_514de689747d1880646f2c33dd401138c349d1519ea8af532c67b90c8bc208b2.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-486_51f560048d1603a7108fe4caed40c917b5c1b634058901bcc272eaef2e5f3d89.jpg b/.raw/MidasCivilAnalysisReference/images/page-486_51f560048d1603a7108fe4caed40c917b5c1b634058901bcc272eaef2e5f3d89.jpg new file mode 100644 index 00000000..7e8b296f Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-486_51f560048d1603a7108fe4caed40c917b5c1b634058901bcc272eaef2e5f3d89.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-486_7dfcdcaa60d66b6b1544964a92223530dbc8225f7ab2fb4320aea884fc985a53.jpg b/.raw/MidasCivilAnalysisReference/images/page-486_7dfcdcaa60d66b6b1544964a92223530dbc8225f7ab2fb4320aea884fc985a53.jpg new file mode 100644 index 00000000..f19aa2e3 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-486_7dfcdcaa60d66b6b1544964a92223530dbc8225f7ab2fb4320aea884fc985a53.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-486_8cd6bb2589cb70991c474a2621a550a1ae53d67b00f9ba0eb379f706995db3f8.jpg b/.raw/MidasCivilAnalysisReference/images/page-486_8cd6bb2589cb70991c474a2621a550a1ae53d67b00f9ba0eb379f706995db3f8.jpg new file mode 100644 index 00000000..0d985296 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-486_8cd6bb2589cb70991c474a2621a550a1ae53d67b00f9ba0eb379f706995db3f8.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-486_d60bc51813070b373ff9a18a4ac3fd7c37a5584170274db1183c4970665fb35f.jpg b/.raw/MidasCivilAnalysisReference/images/page-486_d60bc51813070b373ff9a18a4ac3fd7c37a5584170274db1183c4970665fb35f.jpg new file mode 100644 index 00000000..f0c4ebd3 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-486_d60bc51813070b373ff9a18a4ac3fd7c37a5584170274db1183c4970665fb35f.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-489_00b9cc76ceacb719ee9cfc5ea49cdf25f77137c1b9f306d83a0e81ca825f5820.jpg b/.raw/MidasCivilAnalysisReference/images/page-489_00b9cc76ceacb719ee9cfc5ea49cdf25f77137c1b9f306d83a0e81ca825f5820.jpg new file mode 100644 index 00000000..0508ae41 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-489_00b9cc76ceacb719ee9cfc5ea49cdf25f77137c1b9f306d83a0e81ca825f5820.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-490_0d79d1142be7d7b4e494dba02d5444f5272247b83a70b3c9c22b842cb10ef256.jpg b/.raw/MidasCivilAnalysisReference/images/page-490_0d79d1142be7d7b4e494dba02d5444f5272247b83a70b3c9c22b842cb10ef256.jpg new file mode 100644 index 00000000..072e6374 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-490_0d79d1142be7d7b4e494dba02d5444f5272247b83a70b3c9c22b842cb10ef256.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-490_db2a222a95bafc9cc85bf40a65ff9bf7724b397098c6768ff77c12db7de9487d.jpg b/.raw/MidasCivilAnalysisReference/images/page-490_db2a222a95bafc9cc85bf40a65ff9bf7724b397098c6768ff77c12db7de9487d.jpg new file mode 100644 index 00000000..7a5626ff Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-490_db2a222a95bafc9cc85bf40a65ff9bf7724b397098c6768ff77c12db7de9487d.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-490_f464b63460646e83205042bb80f737168773c82417169da0e7d2abcbbb7bd953.jpg b/.raw/MidasCivilAnalysisReference/images/page-490_f464b63460646e83205042bb80f737168773c82417169da0e7d2abcbbb7bd953.jpg new file mode 100644 index 00000000..0d7c282c Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-490_f464b63460646e83205042bb80f737168773c82417169da0e7d2abcbbb7bd953.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-491_3f601dbcfb2b35fae27b12863c15bd5c8cfd12003cf22e4859cd49ac9921d7b0.jpg b/.raw/MidasCivilAnalysisReference/images/page-491_3f601dbcfb2b35fae27b12863c15bd5c8cfd12003cf22e4859cd49ac9921d7b0.jpg new file mode 100644 index 00000000..6e8ec025 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-491_3f601dbcfb2b35fae27b12863c15bd5c8cfd12003cf22e4859cd49ac9921d7b0.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-491_8ad6b4ed8a2995034179c88c1642bd2aea7617c16774ec0aceec455df60f844c.jpg b/.raw/MidasCivilAnalysisReference/images/page-491_8ad6b4ed8a2995034179c88c1642bd2aea7617c16774ec0aceec455df60f844c.jpg new file mode 100644 index 00000000..9a4597bc Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-491_8ad6b4ed8a2995034179c88c1642bd2aea7617c16774ec0aceec455df60f844c.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-492_00b72cfb3d171da09073cd7375ab73af46e18e09d3e2252540b4477f0f24467a.jpg b/.raw/MidasCivilAnalysisReference/images/page-492_00b72cfb3d171da09073cd7375ab73af46e18e09d3e2252540b4477f0f24467a.jpg new file mode 100644 index 00000000..a77b657a Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-492_00b72cfb3d171da09073cd7375ab73af46e18e09d3e2252540b4477f0f24467a.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-492_69c664fe699e7b5e45614f58884897765277fd0a48fc67303eda792008912d7f.jpg b/.raw/MidasCivilAnalysisReference/images/page-492_69c664fe699e7b5e45614f58884897765277fd0a48fc67303eda792008912d7f.jpg new file mode 100644 index 00000000..4f76e6bf Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-492_69c664fe699e7b5e45614f58884897765277fd0a48fc67303eda792008912d7f.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-493_aa83be6100e226d44525f4c85734f2f79dcf1877240903b84d2875673e2f2af5.jpg b/.raw/MidasCivilAnalysisReference/images/page-493_aa83be6100e226d44525f4c85734f2f79dcf1877240903b84d2875673e2f2af5.jpg new file mode 100644 index 00000000..faea0544 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-493_aa83be6100e226d44525f4c85734f2f79dcf1877240903b84d2875673e2f2af5.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-494_015fe6f7d334ce96bcf0ba61e9a690371d0964dcdf7acd1780d8ad250b790c00.jpg b/.raw/MidasCivilAnalysisReference/images/page-494_015fe6f7d334ce96bcf0ba61e9a690371d0964dcdf7acd1780d8ad250b790c00.jpg new file mode 100644 index 00000000..4528b1a6 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-494_015fe6f7d334ce96bcf0ba61e9a690371d0964dcdf7acd1780d8ad250b790c00.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-494_a16a6b79d183636ea20554e99bbc985e47a4acafc9974124770d0cfd77b21547.jpg b/.raw/MidasCivilAnalysisReference/images/page-494_a16a6b79d183636ea20554e99bbc985e47a4acafc9974124770d0cfd77b21547.jpg new file mode 100644 index 00000000..9180cee9 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-494_a16a6b79d183636ea20554e99bbc985e47a4acafc9974124770d0cfd77b21547.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-494_a39ec0300c1dfba5ccb6c3dc7ce16bdf0f1fa480f40223e52b78242cd7554be8.jpg b/.raw/MidasCivilAnalysisReference/images/page-494_a39ec0300c1dfba5ccb6c3dc7ce16bdf0f1fa480f40223e52b78242cd7554be8.jpg new file mode 100644 index 00000000..199459f9 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-494_a39ec0300c1dfba5ccb6c3dc7ce16bdf0f1fa480f40223e52b78242cd7554be8.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-495_0f8ba747e99eb196ec1345c64b9d8d7b8d3114b20f883cbf328e2396cb982e91.jpg b/.raw/MidasCivilAnalysisReference/images/page-495_0f8ba747e99eb196ec1345c64b9d8d7b8d3114b20f883cbf328e2396cb982e91.jpg new file mode 100644 index 00000000..4ecffb0d Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-495_0f8ba747e99eb196ec1345c64b9d8d7b8d3114b20f883cbf328e2396cb982e91.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-495_30fe39eab6a7801af02d899c2f0856708733efcbfe800313b61017316cd7f209.jpg b/.raw/MidasCivilAnalysisReference/images/page-495_30fe39eab6a7801af02d899c2f0856708733efcbfe800313b61017316cd7f209.jpg new file mode 100644 index 00000000..38db68e3 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-495_30fe39eab6a7801af02d899c2f0856708733efcbfe800313b61017316cd7f209.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-495_4dbc4956d7da2d50adeadc147cabc697a79e7e27ae786052c9326a726056a8d5.jpg b/.raw/MidasCivilAnalysisReference/images/page-495_4dbc4956d7da2d50adeadc147cabc697a79e7e27ae786052c9326a726056a8d5.jpg new file mode 100644 index 00000000..87952106 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-495_4dbc4956d7da2d50adeadc147cabc697a79e7e27ae786052c9326a726056a8d5.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-495_697166432479e4e49e0f50960547f6894ddbd7db05fbedef224d1cb9355177be.jpg b/.raw/MidasCivilAnalysisReference/images/page-495_697166432479e4e49e0f50960547f6894ddbd7db05fbedef224d1cb9355177be.jpg new file mode 100644 index 00000000..4b07c5b1 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-495_697166432479e4e49e0f50960547f6894ddbd7db05fbedef224d1cb9355177be.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-495_9b79197bfcceb51e5d39e6f341dd0bd25b4b9d5a09575eded8da56b9bc4eded5.jpg b/.raw/MidasCivilAnalysisReference/images/page-495_9b79197bfcceb51e5d39e6f341dd0bd25b4b9d5a09575eded8da56b9bc4eded5.jpg new file mode 100644 index 00000000..dc4397c6 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-495_9b79197bfcceb51e5d39e6f341dd0bd25b4b9d5a09575eded8da56b9bc4eded5.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-495_a1ee6636cc580924d2ebea02d8bf5049c7e6c94eaaf89f01b6983fda6ecae8b7.jpg b/.raw/MidasCivilAnalysisReference/images/page-495_a1ee6636cc580924d2ebea02d8bf5049c7e6c94eaaf89f01b6983fda6ecae8b7.jpg new file mode 100644 index 00000000..7c34781a Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-495_a1ee6636cc580924d2ebea02d8bf5049c7e6c94eaaf89f01b6983fda6ecae8b7.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-495_b8e12f5a2ad791a377548812e9d6a7575b986aafa2ef89f91583732c206f4893.jpg b/.raw/MidasCivilAnalysisReference/images/page-495_b8e12f5a2ad791a377548812e9d6a7575b986aafa2ef89f91583732c206f4893.jpg new file mode 100644 index 00000000..7d8d71ff Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-495_b8e12f5a2ad791a377548812e9d6a7575b986aafa2ef89f91583732c206f4893.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-495_ef5c7f32b1028d35c91e8c53410bf2b1aea28299d8b4a088b221f3308378847f.jpg b/.raw/MidasCivilAnalysisReference/images/page-495_ef5c7f32b1028d35c91e8c53410bf2b1aea28299d8b4a088b221f3308378847f.jpg new file mode 100644 index 00000000..5148407a Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-495_ef5c7f32b1028d35c91e8c53410bf2b1aea28299d8b4a088b221f3308378847f.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-496_010b8c66eec83f7131f67d8b90065c64b697fa82f15380b78ead018cc71f3b8e.jpg b/.raw/MidasCivilAnalysisReference/images/page-496_010b8c66eec83f7131f67d8b90065c64b697fa82f15380b78ead018cc71f3b8e.jpg new file mode 100644 index 00000000..6fc7914d Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-496_010b8c66eec83f7131f67d8b90065c64b697fa82f15380b78ead018cc71f3b8e.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-496_3fbc6141f9bd8e742909e06a4c96e826257c45b341bd09c6287ccaf3079d4d01.jpg b/.raw/MidasCivilAnalysisReference/images/page-496_3fbc6141f9bd8e742909e06a4c96e826257c45b341bd09c6287ccaf3079d4d01.jpg new file mode 100644 index 00000000..17984678 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-496_3fbc6141f9bd8e742909e06a4c96e826257c45b341bd09c6287ccaf3079d4d01.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-496_9fec669a4e2396c9c158f88a8bd1a953dd008624e5c0132742e0c95b26c9de5c.jpg b/.raw/MidasCivilAnalysisReference/images/page-496_9fec669a4e2396c9c158f88a8bd1a953dd008624e5c0132742e0c95b26c9de5c.jpg new file mode 100644 index 00000000..9d3744f0 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-496_9fec669a4e2396c9c158f88a8bd1a953dd008624e5c0132742e0c95b26c9de5c.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-497_209f1b07a911e34497ae8b1b2c56f1e6c806d40b419f75fe003d0ea5673f75cb.jpg b/.raw/MidasCivilAnalysisReference/images/page-497_209f1b07a911e34497ae8b1b2c56f1e6c806d40b419f75fe003d0ea5673f75cb.jpg new file mode 100644 index 00000000..160f6fd9 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-497_209f1b07a911e34497ae8b1b2c56f1e6c806d40b419f75fe003d0ea5673f75cb.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-497_2e6c3c74883f805438a03c1d345b4fe5dff9f9eda01bc5839f5be9fa0efd5303.jpg b/.raw/MidasCivilAnalysisReference/images/page-497_2e6c3c74883f805438a03c1d345b4fe5dff9f9eda01bc5839f5be9fa0efd5303.jpg new file mode 100644 index 00000000..75589801 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-497_2e6c3c74883f805438a03c1d345b4fe5dff9f9eda01bc5839f5be9fa0efd5303.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-497_490940f5b3fa0d5720de6b790a58942d8302f13e9fb04876e1e0e5793fd1a285.jpg b/.raw/MidasCivilAnalysisReference/images/page-497_490940f5b3fa0d5720de6b790a58942d8302f13e9fb04876e1e0e5793fd1a285.jpg new file mode 100644 index 00000000..fd8891ca Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-497_490940f5b3fa0d5720de6b790a58942d8302f13e9fb04876e1e0e5793fd1a285.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-497_6350dcdde7e54bb6ed2914bac062ad10895581ca4861d5631fc5e9e02a2022fe.jpg b/.raw/MidasCivilAnalysisReference/images/page-497_6350dcdde7e54bb6ed2914bac062ad10895581ca4861d5631fc5e9e02a2022fe.jpg new file mode 100644 index 00000000..384a0a56 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-497_6350dcdde7e54bb6ed2914bac062ad10895581ca4861d5631fc5e9e02a2022fe.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-497_6b4bee6ac2af1f181a2cd3058743fd4894a1662816b57ed9cb39455514120c5a.jpg b/.raw/MidasCivilAnalysisReference/images/page-497_6b4bee6ac2af1f181a2cd3058743fd4894a1662816b57ed9cb39455514120c5a.jpg new file mode 100644 index 00000000..a188d9bf Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-497_6b4bee6ac2af1f181a2cd3058743fd4894a1662816b57ed9cb39455514120c5a.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-497_7a869c4bf8bd8a760541895e341d41ec7b7f7f132bcac144f9f1b85b8336e2bf.jpg b/.raw/MidasCivilAnalysisReference/images/page-497_7a869c4bf8bd8a760541895e341d41ec7b7f7f132bcac144f9f1b85b8336e2bf.jpg new file mode 100644 index 00000000..790151a9 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-497_7a869c4bf8bd8a760541895e341d41ec7b7f7f132bcac144f9f1b85b8336e2bf.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-497_7d494659c443ce01575025024e86ba52a116f0a63a674bdd41ce60ab8b8e86d0.jpg b/.raw/MidasCivilAnalysisReference/images/page-497_7d494659c443ce01575025024e86ba52a116f0a63a674bdd41ce60ab8b8e86d0.jpg new file mode 100644 index 00000000..38856101 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-497_7d494659c443ce01575025024e86ba52a116f0a63a674bdd41ce60ab8b8e86d0.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-497_8404f414bb8d967312156b94842668ade903bea0f623536c6d4d55e75108b6ba.jpg b/.raw/MidasCivilAnalysisReference/images/page-497_8404f414bb8d967312156b94842668ade903bea0f623536c6d4d55e75108b6ba.jpg new file mode 100644 index 00000000..adb30a71 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-497_8404f414bb8d967312156b94842668ade903bea0f623536c6d4d55e75108b6ba.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-497_8b6cc0ee86cdb504479d84ef6cbed00099e37aea47a8bd33226373fc48c111cc.jpg b/.raw/MidasCivilAnalysisReference/images/page-497_8b6cc0ee86cdb504479d84ef6cbed00099e37aea47a8bd33226373fc48c111cc.jpg new file mode 100644 index 00000000..7f4ce3c1 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-497_8b6cc0ee86cdb504479d84ef6cbed00099e37aea47a8bd33226373fc48c111cc.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-497_b9296bcc5fdb62ac506b119d3294d77e2b8d48e8aa0be705889af5505bcf3a3a.jpg b/.raw/MidasCivilAnalysisReference/images/page-497_b9296bcc5fdb62ac506b119d3294d77e2b8d48e8aa0be705889af5505bcf3a3a.jpg new file mode 100644 index 00000000..0f00ec44 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-497_b9296bcc5fdb62ac506b119d3294d77e2b8d48e8aa0be705889af5505bcf3a3a.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-497_ddf88e741e0a8da978258bf1d37b39564ae8cbd6ba410491abd3a2bda424d784.jpg b/.raw/MidasCivilAnalysisReference/images/page-497_ddf88e741e0a8da978258bf1d37b39564ae8cbd6ba410491abd3a2bda424d784.jpg new file mode 100644 index 00000000..e938097b Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-497_ddf88e741e0a8da978258bf1d37b39564ae8cbd6ba410491abd3a2bda424d784.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-497_dfa489fd4aa73841e958c721c98122639250faefaa4ba010f20acde97a8ba062.jpg b/.raw/MidasCivilAnalysisReference/images/page-497_dfa489fd4aa73841e958c721c98122639250faefaa4ba010f20acde97a8ba062.jpg new file mode 100644 index 00000000..b28387cb Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-497_dfa489fd4aa73841e958c721c98122639250faefaa4ba010f20acde97a8ba062.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-499_034a48f12bc91f3112c3f5c432bf15123dc3345e3a1aa6425b95734eb09f1ec1.jpg b/.raw/MidasCivilAnalysisReference/images/page-499_034a48f12bc91f3112c3f5c432bf15123dc3345e3a1aa6425b95734eb09f1ec1.jpg new file mode 100644 index 00000000..a26c61bd Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-499_034a48f12bc91f3112c3f5c432bf15123dc3345e3a1aa6425b95734eb09f1ec1.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-503_c0308b244fd3abd2e16365d997519de04d274ec0ba30aeb92ec11224f35e8cac.jpg b/.raw/MidasCivilAnalysisReference/images/page-503_c0308b244fd3abd2e16365d997519de04d274ec0ba30aeb92ec11224f35e8cac.jpg new file mode 100644 index 00000000..1aafcc94 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-503_c0308b244fd3abd2e16365d997519de04d274ec0ba30aeb92ec11224f35e8cac.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-503_d78540bb1c1ea41403f5ffb1cc7851ab0fcb333118b5d2bbfcb198f1062068fe.jpg b/.raw/MidasCivilAnalysisReference/images/page-503_d78540bb1c1ea41403f5ffb1cc7851ab0fcb333118b5d2bbfcb198f1062068fe.jpg new file mode 100644 index 00000000..063770f5 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-503_d78540bb1c1ea41403f5ffb1cc7851ab0fcb333118b5d2bbfcb198f1062068fe.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-505_f8f984e985cafdd40245402b2213289dc9b8cdeda39fc24e42bcb75fef7316b3.jpg b/.raw/MidasCivilAnalysisReference/images/page-505_f8f984e985cafdd40245402b2213289dc9b8cdeda39fc24e42bcb75fef7316b3.jpg new file mode 100644 index 00000000..db7a14aa Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-505_f8f984e985cafdd40245402b2213289dc9b8cdeda39fc24e42bcb75fef7316b3.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-506_386c3c65b0095ecddc41a823445750b298908fb3ec294dfdc79769bb6cf7c45c.jpg b/.raw/MidasCivilAnalysisReference/images/page-506_386c3c65b0095ecddc41a823445750b298908fb3ec294dfdc79769bb6cf7c45c.jpg new file mode 100644 index 00000000..4041bae5 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-506_386c3c65b0095ecddc41a823445750b298908fb3ec294dfdc79769bb6cf7c45c.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-506_404d9b230d6dc8f9d4ddcb9f4f3688d85465fa77d214952b041d0d39313a2439.jpg b/.raw/MidasCivilAnalysisReference/images/page-506_404d9b230d6dc8f9d4ddcb9f4f3688d85465fa77d214952b041d0d39313a2439.jpg new file mode 100644 index 00000000..a0f16b61 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-506_404d9b230d6dc8f9d4ddcb9f4f3688d85465fa77d214952b041d0d39313a2439.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-506_487c71e7c7083b01b456a1559606da467e0949d226ba8364e4bbc424fbb57f03.jpg b/.raw/MidasCivilAnalysisReference/images/page-506_487c71e7c7083b01b456a1559606da467e0949d226ba8364e4bbc424fbb57f03.jpg new file mode 100644 index 00000000..195613ba Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-506_487c71e7c7083b01b456a1559606da467e0949d226ba8364e4bbc424fbb57f03.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-506_a603c5c65c1f2c83ea1a7ee2afb551315f3effe3be30cfd25682e30b5e3d6116.jpg b/.raw/MidasCivilAnalysisReference/images/page-506_a603c5c65c1f2c83ea1a7ee2afb551315f3effe3be30cfd25682e30b5e3d6116.jpg new file mode 100644 index 00000000..2dacf7e4 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-506_a603c5c65c1f2c83ea1a7ee2afb551315f3effe3be30cfd25682e30b5e3d6116.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-507_266242620fc8ed998e4e1679dea8f4195a41be0b438c5de16a289a4a7af5a164.jpg b/.raw/MidasCivilAnalysisReference/images/page-507_266242620fc8ed998e4e1679dea8f4195a41be0b438c5de16a289a4a7af5a164.jpg new file mode 100644 index 00000000..b2d3a87c Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-507_266242620fc8ed998e4e1679dea8f4195a41be0b438c5de16a289a4a7af5a164.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-507_3c2befe4deb591904f6e62dc3d02a5e92396a0880b6f99f8d4a5afab448648d7.jpg b/.raw/MidasCivilAnalysisReference/images/page-507_3c2befe4deb591904f6e62dc3d02a5e92396a0880b6f99f8d4a5afab448648d7.jpg new file mode 100644 index 00000000..82a5266a Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-507_3c2befe4deb591904f6e62dc3d02a5e92396a0880b6f99f8d4a5afab448648d7.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-508_5c501454e7759bea4b6a65c22f82ac6d370d221388799eef22d4baee5d100c1e.jpg b/.raw/MidasCivilAnalysisReference/images/page-508_5c501454e7759bea4b6a65c22f82ac6d370d221388799eef22d4baee5d100c1e.jpg new file mode 100644 index 00000000..94902a00 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-508_5c501454e7759bea4b6a65c22f82ac6d370d221388799eef22d4baee5d100c1e.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-508_addab9d1b8e537afab12d008849b350e7a131ab2b0d5ed59b055214f1acceb82.jpg b/.raw/MidasCivilAnalysisReference/images/page-508_addab9d1b8e537afab12d008849b350e7a131ab2b0d5ed59b055214f1acceb82.jpg new file mode 100644 index 00000000..d4352866 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-508_addab9d1b8e537afab12d008849b350e7a131ab2b0d5ed59b055214f1acceb82.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-511_645c03489c570ab209346ffa5b84305c3df18b323221ac2cfa3f70e0cc2e344c.jpg b/.raw/MidasCivilAnalysisReference/images/page-511_645c03489c570ab209346ffa5b84305c3df18b323221ac2cfa3f70e0cc2e344c.jpg new file mode 100644 index 00000000..29a4c6ef Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-511_645c03489c570ab209346ffa5b84305c3df18b323221ac2cfa3f70e0cc2e344c.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-512_60ae8e67ff4cb417d2b1ac252bd839a11b67aa4cea3ec69016d6bc158a92e8df.jpg b/.raw/MidasCivilAnalysisReference/images/page-512_60ae8e67ff4cb417d2b1ac252bd839a11b67aa4cea3ec69016d6bc158a92e8df.jpg new file mode 100644 index 00000000..15030d31 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-512_60ae8e67ff4cb417d2b1ac252bd839a11b67aa4cea3ec69016d6bc158a92e8df.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-512_90d7c7833e9a52a345b7404cc361c6ce4fb45d5197a668e0258e389051951366.jpg b/.raw/MidasCivilAnalysisReference/images/page-512_90d7c7833e9a52a345b7404cc361c6ce4fb45d5197a668e0258e389051951366.jpg new file mode 100644 index 00000000..fb73d90c Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-512_90d7c7833e9a52a345b7404cc361c6ce4fb45d5197a668e0258e389051951366.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-512_c9bd69a17bfad337002f51e01ba2316fb0f93b457e2b5fc6fae0d5b77b4cfd28.jpg b/.raw/MidasCivilAnalysisReference/images/page-512_c9bd69a17bfad337002f51e01ba2316fb0f93b457e2b5fc6fae0d5b77b4cfd28.jpg new file mode 100644 index 00000000..8b37fc8e Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-512_c9bd69a17bfad337002f51e01ba2316fb0f93b457e2b5fc6fae0d5b77b4cfd28.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-513_5803887d5ef3739b19aabc1cddb32bc7b62d52a54e6a23bf9ec0195e6a46407b.jpg b/.raw/MidasCivilAnalysisReference/images/page-513_5803887d5ef3739b19aabc1cddb32bc7b62d52a54e6a23bf9ec0195e6a46407b.jpg new file mode 100644 index 00000000..c5c533ab Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-513_5803887d5ef3739b19aabc1cddb32bc7b62d52a54e6a23bf9ec0195e6a46407b.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-513_5c5cfcf123fff939859352ae37ae07390b9846fc41d27006a847f62dcb168f76.jpg b/.raw/MidasCivilAnalysisReference/images/page-513_5c5cfcf123fff939859352ae37ae07390b9846fc41d27006a847f62dcb168f76.jpg new file mode 100644 index 00000000..667c5382 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-513_5c5cfcf123fff939859352ae37ae07390b9846fc41d27006a847f62dcb168f76.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-513_e8d303f414b5b9422fc1d1049b8b5fc30aaf51974b6d9e1bd782ef4adf215369.jpg b/.raw/MidasCivilAnalysisReference/images/page-513_e8d303f414b5b9422fc1d1049b8b5fc30aaf51974b6d9e1bd782ef4adf215369.jpg new file mode 100644 index 00000000..8f7304ba Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-513_e8d303f414b5b9422fc1d1049b8b5fc30aaf51974b6d9e1bd782ef4adf215369.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-513_fcd32cc77926ec1c96dbd55a9e423bb57d5ac49cac79ccef32f7e81a6b78641d.jpg b/.raw/MidasCivilAnalysisReference/images/page-513_fcd32cc77926ec1c96dbd55a9e423bb57d5ac49cac79ccef32f7e81a6b78641d.jpg new file mode 100644 index 00000000..87025972 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-513_fcd32cc77926ec1c96dbd55a9e423bb57d5ac49cac79ccef32f7e81a6b78641d.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-514_0188e4811cd80640b5daa503b09fe5efaf6035d98f90d89912cc5fb3614eb004.jpg b/.raw/MidasCivilAnalysisReference/images/page-514_0188e4811cd80640b5daa503b09fe5efaf6035d98f90d89912cc5fb3614eb004.jpg new file mode 100644 index 00000000..13dfedfb Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-514_0188e4811cd80640b5daa503b09fe5efaf6035d98f90d89912cc5fb3614eb004.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-514_384ceff6a242bc1caf29a42a120cb1a745d6afd36bc2bf9bbfae73d36479332f.jpg b/.raw/MidasCivilAnalysisReference/images/page-514_384ceff6a242bc1caf29a42a120cb1a745d6afd36bc2bf9bbfae73d36479332f.jpg new file mode 100644 index 00000000..18d2310e Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-514_384ceff6a242bc1caf29a42a120cb1a745d6afd36bc2bf9bbfae73d36479332f.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-514_5d2dead4609ac9ce991cdb2fe6df2e9778b57afe36f4aa486e7516a866eea5c4.jpg b/.raw/MidasCivilAnalysisReference/images/page-514_5d2dead4609ac9ce991cdb2fe6df2e9778b57afe36f4aa486e7516a866eea5c4.jpg new file mode 100644 index 00000000..64bf5dcc Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-514_5d2dead4609ac9ce991cdb2fe6df2e9778b57afe36f4aa486e7516a866eea5c4.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-514_af837d259e4d1e87c3e55746d60d7f503d31e5ae70ac7969f1b0e5d7aaa70458.jpg b/.raw/MidasCivilAnalysisReference/images/page-514_af837d259e4d1e87c3e55746d60d7f503d31e5ae70ac7969f1b0e5d7aaa70458.jpg new file mode 100644 index 00000000..4f7e93bc Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-514_af837d259e4d1e87c3e55746d60d7f503d31e5ae70ac7969f1b0e5d7aaa70458.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-514_da68e7e902d18b950704d5ee75f85274cdc312fe7d6a4ce400ef8a1e284cd977.jpg b/.raw/MidasCivilAnalysisReference/images/page-514_da68e7e902d18b950704d5ee75f85274cdc312fe7d6a4ce400ef8a1e284cd977.jpg new file mode 100644 index 00000000..0b169327 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-514_da68e7e902d18b950704d5ee75f85274cdc312fe7d6a4ce400ef8a1e284cd977.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-515_1b098c08bf338b2d4c01a04411f0651a0757f45c09ba7c9c6dc27ceb4c584697.jpg b/.raw/MidasCivilAnalysisReference/images/page-515_1b098c08bf338b2d4c01a04411f0651a0757f45c09ba7c9c6dc27ceb4c584697.jpg new file mode 100644 index 00000000..6826dfe9 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-515_1b098c08bf338b2d4c01a04411f0651a0757f45c09ba7c9c6dc27ceb4c584697.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-515_1d76ece60f2f0452cca3687c0871d6cb1c73528595c836dfd64fc76ef13ec171.jpg b/.raw/MidasCivilAnalysisReference/images/page-515_1d76ece60f2f0452cca3687c0871d6cb1c73528595c836dfd64fc76ef13ec171.jpg new file mode 100644 index 00000000..d581ef0c Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-515_1d76ece60f2f0452cca3687c0871d6cb1c73528595c836dfd64fc76ef13ec171.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-515_1f8e7848ec0ce22c3809705515aac673b2556ccb7f2a4c2f115f69a253c28021.jpg b/.raw/MidasCivilAnalysisReference/images/page-515_1f8e7848ec0ce22c3809705515aac673b2556ccb7f2a4c2f115f69a253c28021.jpg new file mode 100644 index 00000000..ddd01029 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-515_1f8e7848ec0ce22c3809705515aac673b2556ccb7f2a4c2f115f69a253c28021.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-515_37c9f3eba5ccdfae191443220c743a0a4549b2b798793fcc4cacdf3c81a9c70b.jpg b/.raw/MidasCivilAnalysisReference/images/page-515_37c9f3eba5ccdfae191443220c743a0a4549b2b798793fcc4cacdf3c81a9c70b.jpg new file mode 100644 index 00000000..837cc82b Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-515_37c9f3eba5ccdfae191443220c743a0a4549b2b798793fcc4cacdf3c81a9c70b.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-515_4cb7ea59794de3e52b579d25c9819e62c9e8230ccd4f09e412bc253bca4cec32.jpg b/.raw/MidasCivilAnalysisReference/images/page-515_4cb7ea59794de3e52b579d25c9819e62c9e8230ccd4f09e412bc253bca4cec32.jpg new file mode 100644 index 00000000..b3d8776f Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-515_4cb7ea59794de3e52b579d25c9819e62c9e8230ccd4f09e412bc253bca4cec32.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-515_6622901604e009a512ff18bbd13b72b065f9970e9cfa5a13d6756e29450908fc.jpg b/.raw/MidasCivilAnalysisReference/images/page-515_6622901604e009a512ff18bbd13b72b065f9970e9cfa5a13d6756e29450908fc.jpg new file mode 100644 index 00000000..c51f6f72 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-515_6622901604e009a512ff18bbd13b72b065f9970e9cfa5a13d6756e29450908fc.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-516_2040fb55cfc0383eb4dc2765205574c15a05cd27f5354080fc6040f9ad0e1ca1.jpg b/.raw/MidasCivilAnalysisReference/images/page-516_2040fb55cfc0383eb4dc2765205574c15a05cd27f5354080fc6040f9ad0e1ca1.jpg new file mode 100644 index 00000000..7406b258 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-516_2040fb55cfc0383eb4dc2765205574c15a05cd27f5354080fc6040f9ad0e1ca1.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-516_afcc47e72f4de47e0de5217661e4e28deedbcbfaaa51c3e4da25be3ff8cd4a70.jpg b/.raw/MidasCivilAnalysisReference/images/page-516_afcc47e72f4de47e0de5217661e4e28deedbcbfaaa51c3e4da25be3ff8cd4a70.jpg new file mode 100644 index 00000000..e2c7daa8 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-516_afcc47e72f4de47e0de5217661e4e28deedbcbfaaa51c3e4da25be3ff8cd4a70.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-516_c5c67e6b9a53e632ba8d4898534db39689c06398dbff31d260537130c12507ca.jpg b/.raw/MidasCivilAnalysisReference/images/page-516_c5c67e6b9a53e632ba8d4898534db39689c06398dbff31d260537130c12507ca.jpg new file mode 100644 index 00000000..ffc21b91 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-516_c5c67e6b9a53e632ba8d4898534db39689c06398dbff31d260537130c12507ca.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-516_d77feef39ffe48f305b396013147a6100be7c29b842b3d186682486211b1a7dc.jpg b/.raw/MidasCivilAnalysisReference/images/page-516_d77feef39ffe48f305b396013147a6100be7c29b842b3d186682486211b1a7dc.jpg new file mode 100644 index 00000000..36805188 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-516_d77feef39ffe48f305b396013147a6100be7c29b842b3d186682486211b1a7dc.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-517_8abd5ba9a46d70c3d40dc9dad8ce28182791fb290554f4ca61c72a3e36b8e681.jpg b/.raw/MidasCivilAnalysisReference/images/page-517_8abd5ba9a46d70c3d40dc9dad8ce28182791fb290554f4ca61c72a3e36b8e681.jpg new file mode 100644 index 00000000..369e3490 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-517_8abd5ba9a46d70c3d40dc9dad8ce28182791fb290554f4ca61c72a3e36b8e681.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-517_d97af68c055f4f7da75ba0c021cd2e793658c7b0a5e42899c10325352ba7912b.jpg b/.raw/MidasCivilAnalysisReference/images/page-517_d97af68c055f4f7da75ba0c021cd2e793658c7b0a5e42899c10325352ba7912b.jpg new file mode 100644 index 00000000..23899e1f Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-517_d97af68c055f4f7da75ba0c021cd2e793658c7b0a5e42899c10325352ba7912b.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-518_1ee6c91a54effd00c0083695210eeb98f82135c4cdb29f36c6719a642ef23e85.jpg b/.raw/MidasCivilAnalysisReference/images/page-518_1ee6c91a54effd00c0083695210eeb98f82135c4cdb29f36c6719a642ef23e85.jpg new file mode 100644 index 00000000..e3156f6f Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-518_1ee6c91a54effd00c0083695210eeb98f82135c4cdb29f36c6719a642ef23e85.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-518_3aced3673c9bd7273490213f18d39465a4606b47166d6a15aed744a157cf1b22.jpg b/.raw/MidasCivilAnalysisReference/images/page-518_3aced3673c9bd7273490213f18d39465a4606b47166d6a15aed744a157cf1b22.jpg new file mode 100644 index 00000000..14cd70bf Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-518_3aced3673c9bd7273490213f18d39465a4606b47166d6a15aed744a157cf1b22.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-519_b51369b5602db937ef99cd03f6558ae32157462ca18ff292d0367040d8b5c9f2.jpg b/.raw/MidasCivilAnalysisReference/images/page-519_b51369b5602db937ef99cd03f6558ae32157462ca18ff292d0367040d8b5c9f2.jpg new file mode 100644 index 00000000..75163719 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-519_b51369b5602db937ef99cd03f6558ae32157462ca18ff292d0367040d8b5c9f2.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-520_30bab4381a9a878787a152890579fa56173d1974777bcaaa9ca2f2ed9e7e9729.jpg b/.raw/MidasCivilAnalysisReference/images/page-520_30bab4381a9a878787a152890579fa56173d1974777bcaaa9ca2f2ed9e7e9729.jpg new file mode 100644 index 00000000..26b8edf5 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-520_30bab4381a9a878787a152890579fa56173d1974777bcaaa9ca2f2ed9e7e9729.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-522_82ab7b3c6633a96b5afd48a86050a709c03e487901bbc072625129299fc11d0a.jpg b/.raw/MidasCivilAnalysisReference/images/page-522_82ab7b3c6633a96b5afd48a86050a709c03e487901bbc072625129299fc11d0a.jpg new file mode 100644 index 00000000..b4b9257c Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-522_82ab7b3c6633a96b5afd48a86050a709c03e487901bbc072625129299fc11d0a.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-523_03eab31deae6e6cecca2c7c0cb0d9277e98171896a59dd72a58ac59177fe67c5.jpg b/.raw/MidasCivilAnalysisReference/images/page-523_03eab31deae6e6cecca2c7c0cb0d9277e98171896a59dd72a58ac59177fe67c5.jpg new file mode 100644 index 00000000..7411d2cf Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-523_03eab31deae6e6cecca2c7c0cb0d9277e98171896a59dd72a58ac59177fe67c5.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-523_2ec73a95965663b5b84a81c2d96f4606d8bcb7f5d578308c08a372bc895be4ff.jpg b/.raw/MidasCivilAnalysisReference/images/page-523_2ec73a95965663b5b84a81c2d96f4606d8bcb7f5d578308c08a372bc895be4ff.jpg new file mode 100644 index 00000000..4570f508 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-523_2ec73a95965663b5b84a81c2d96f4606d8bcb7f5d578308c08a372bc895be4ff.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-523_334b7561c38c6a9e7cc7f25e9a8546f22654ba494f95ed48e68e23f2c71a165d.jpg b/.raw/MidasCivilAnalysisReference/images/page-523_334b7561c38c6a9e7cc7f25e9a8546f22654ba494f95ed48e68e23f2c71a165d.jpg new file mode 100644 index 00000000..5d8630ee Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-523_334b7561c38c6a9e7cc7f25e9a8546f22654ba494f95ed48e68e23f2c71a165d.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-523_b294e61e4e7d99703e60265b5dc15a81ad02452fcfddcbe291a03e045d085d5a.jpg b/.raw/MidasCivilAnalysisReference/images/page-523_b294e61e4e7d99703e60265b5dc15a81ad02452fcfddcbe291a03e045d085d5a.jpg new file mode 100644 index 00000000..110100b6 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-523_b294e61e4e7d99703e60265b5dc15a81ad02452fcfddcbe291a03e045d085d5a.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-524_3d17a0d8368525e9b16b48315f9b503b30db3663926e8f45b4a979060ae0c80d.jpg b/.raw/MidasCivilAnalysisReference/images/page-524_3d17a0d8368525e9b16b48315f9b503b30db3663926e8f45b4a979060ae0c80d.jpg new file mode 100644 index 00000000..4daffb31 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-524_3d17a0d8368525e9b16b48315f9b503b30db3663926e8f45b4a979060ae0c80d.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-524_4c5400fd0bad0db30ea670d31ab6da0c7b9a86ecf5bd2abe641d15a88bca33cc.jpg b/.raw/MidasCivilAnalysisReference/images/page-524_4c5400fd0bad0db30ea670d31ab6da0c7b9a86ecf5bd2abe641d15a88bca33cc.jpg new file mode 100644 index 00000000..adc317e7 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-524_4c5400fd0bad0db30ea670d31ab6da0c7b9a86ecf5bd2abe641d15a88bca33cc.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-524_9dcdda0f9268673c1541f54bab022e55ebe49e488169b99beafff86e66da2389.jpg b/.raw/MidasCivilAnalysisReference/images/page-524_9dcdda0f9268673c1541f54bab022e55ebe49e488169b99beafff86e66da2389.jpg new file mode 100644 index 00000000..caad09ff Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-524_9dcdda0f9268673c1541f54bab022e55ebe49e488169b99beafff86e66da2389.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-524_a0f142e0469b83220c2c50e4ecb2bf2cbbbb98f30a44b051cf2e54386b2e5163.jpg b/.raw/MidasCivilAnalysisReference/images/page-524_a0f142e0469b83220c2c50e4ecb2bf2cbbbb98f30a44b051cf2e54386b2e5163.jpg new file mode 100644 index 00000000..af6bc4f6 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-524_a0f142e0469b83220c2c50e4ecb2bf2cbbbb98f30a44b051cf2e54386b2e5163.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-524_fba0c531beadd221d29da72107673cda0b4902ddcd7dcd72c13c4521f4a9d5d5.jpg b/.raw/MidasCivilAnalysisReference/images/page-524_fba0c531beadd221d29da72107673cda0b4902ddcd7dcd72c13c4521f4a9d5d5.jpg new file mode 100644 index 00000000..fd5421e2 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-524_fba0c531beadd221d29da72107673cda0b4902ddcd7dcd72c13c4521f4a9d5d5.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-525_29184c0dc0d63001d5dac56884f5dac1b663daf58863a0917a99f6ef376ac3cb.jpg b/.raw/MidasCivilAnalysisReference/images/page-525_29184c0dc0d63001d5dac56884f5dac1b663daf58863a0917a99f6ef376ac3cb.jpg new file mode 100644 index 00000000..6f812d7a Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-525_29184c0dc0d63001d5dac56884f5dac1b663daf58863a0917a99f6ef376ac3cb.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-525_e9bc5ae4c44a4f05973163e06e301ec1c95a9f77e53b310ed7b99540cf545d7c.jpg b/.raw/MidasCivilAnalysisReference/images/page-525_e9bc5ae4c44a4f05973163e06e301ec1c95a9f77e53b310ed7b99540cf545d7c.jpg new file mode 100644 index 00000000..dcac9bf3 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-525_e9bc5ae4c44a4f05973163e06e301ec1c95a9f77e53b310ed7b99540cf545d7c.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-526_255d0b75069e59480a79cca6b78276fb263538341a4e4ba6cf438a4a9dd0db29.jpg b/.raw/MidasCivilAnalysisReference/images/page-526_255d0b75069e59480a79cca6b78276fb263538341a4e4ba6cf438a4a9dd0db29.jpg new file mode 100644 index 00000000..40909351 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-526_255d0b75069e59480a79cca6b78276fb263538341a4e4ba6cf438a4a9dd0db29.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-526_f6303c718fe4a989a1118feb4b9bb327ffa500280c41284abbbbd4bd80e07733.jpg b/.raw/MidasCivilAnalysisReference/images/page-526_f6303c718fe4a989a1118feb4b9bb327ffa500280c41284abbbbd4bd80e07733.jpg new file mode 100644 index 00000000..5e812898 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-526_f6303c718fe4a989a1118feb4b9bb327ffa500280c41284abbbbd4bd80e07733.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-527_476e8fa51a73371d88b0e19444454ed4133b9da4f8925f81d32b313d3ab0ca1f.jpg b/.raw/MidasCivilAnalysisReference/images/page-527_476e8fa51a73371d88b0e19444454ed4133b9da4f8925f81d32b313d3ab0ca1f.jpg new file mode 100644 index 00000000..3fd22750 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-527_476e8fa51a73371d88b0e19444454ed4133b9da4f8925f81d32b313d3ab0ca1f.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-527_8acbdcd99eb7427238c47ccc7962a950d5e5a900f5634a050f865c4eca75050b.jpg b/.raw/MidasCivilAnalysisReference/images/page-527_8acbdcd99eb7427238c47ccc7962a950d5e5a900f5634a050f865c4eca75050b.jpg new file mode 100644 index 00000000..26eb8735 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-527_8acbdcd99eb7427238c47ccc7962a950d5e5a900f5634a050f865c4eca75050b.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-527_ba73b008509de3411cc421e0c29db252399db2979fb6c7f1e339c61a61960b78.jpg b/.raw/MidasCivilAnalysisReference/images/page-527_ba73b008509de3411cc421e0c29db252399db2979fb6c7f1e339c61a61960b78.jpg new file mode 100644 index 00000000..b363ba3b Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-527_ba73b008509de3411cc421e0c29db252399db2979fb6c7f1e339c61a61960b78.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-527_ffa1515fbd6ba2fab088f3e9c369a7414fb99689889d4809b48082b5fa63ff1b.jpg b/.raw/MidasCivilAnalysisReference/images/page-527_ffa1515fbd6ba2fab088f3e9c369a7414fb99689889d4809b48082b5fa63ff1b.jpg new file mode 100644 index 00000000..eb5510c9 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-527_ffa1515fbd6ba2fab088f3e9c369a7414fb99689889d4809b48082b5fa63ff1b.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-528_bd87833eb841ef1b3b42f63596d2be25cb3e1c575f992b37e362dbb93612f536.jpg b/.raw/MidasCivilAnalysisReference/images/page-528_bd87833eb841ef1b3b42f63596d2be25cb3e1c575f992b37e362dbb93612f536.jpg new file mode 100644 index 00000000..c982e754 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-528_bd87833eb841ef1b3b42f63596d2be25cb3e1c575f992b37e362dbb93612f536.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-529_46024cdf078f6289e37bf47008aadd695632820b151e575c7942691d3c19875c.jpg b/.raw/MidasCivilAnalysisReference/images/page-529_46024cdf078f6289e37bf47008aadd695632820b151e575c7942691d3c19875c.jpg new file mode 100644 index 00000000..ed47143a Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-529_46024cdf078f6289e37bf47008aadd695632820b151e575c7942691d3c19875c.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-529_a127e81d376ff7818a769d8ecc1aa41657dec7259b34f5f9ec50f0c609e6ceb3.jpg b/.raw/MidasCivilAnalysisReference/images/page-529_a127e81d376ff7818a769d8ecc1aa41657dec7259b34f5f9ec50f0c609e6ceb3.jpg new file mode 100644 index 00000000..036d4697 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-529_a127e81d376ff7818a769d8ecc1aa41657dec7259b34f5f9ec50f0c609e6ceb3.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-530_fedc0fcd4e64e69b25ff3565d0e0fa529e8c95880802aa3b21865e9f45c486b6.jpg b/.raw/MidasCivilAnalysisReference/images/page-530_fedc0fcd4e64e69b25ff3565d0e0fa529e8c95880802aa3b21865e9f45c486b6.jpg new file mode 100644 index 00000000..6f78c1db Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-530_fedc0fcd4e64e69b25ff3565d0e0fa529e8c95880802aa3b21865e9f45c486b6.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-531_04852ae456f3ef09685b532ae4c55fddc61581d690153a432a3f960558001af3.jpg b/.raw/MidasCivilAnalysisReference/images/page-531_04852ae456f3ef09685b532ae4c55fddc61581d690153a432a3f960558001af3.jpg new file mode 100644 index 00000000..bbcb3453 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-531_04852ae456f3ef09685b532ae4c55fddc61581d690153a432a3f960558001af3.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-531_0ac6e54a4797a0678d30d424e7b071764ceb73c0d82da2c9137eef3c4fbe2d81.jpg b/.raw/MidasCivilAnalysisReference/images/page-531_0ac6e54a4797a0678d30d424e7b071764ceb73c0d82da2c9137eef3c4fbe2d81.jpg new file mode 100644 index 00000000..b90a11bf Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-531_0ac6e54a4797a0678d30d424e7b071764ceb73c0d82da2c9137eef3c4fbe2d81.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-531_8aef28cd7a2088783adc92f884cf0bb9f397c0c484fe995c80110541d98705de.jpg b/.raw/MidasCivilAnalysisReference/images/page-531_8aef28cd7a2088783adc92f884cf0bb9f397c0c484fe995c80110541d98705de.jpg new file mode 100644 index 00000000..1a58f3e8 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-531_8aef28cd7a2088783adc92f884cf0bb9f397c0c484fe995c80110541d98705de.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-531_f9882313ee1ede9df471d0d957f79fbd4b32f80a9599238aae81a2929f11391a.jpg b/.raw/MidasCivilAnalysisReference/images/page-531_f9882313ee1ede9df471d0d957f79fbd4b32f80a9599238aae81a2929f11391a.jpg new file mode 100644 index 00000000..d02b9caa Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-531_f9882313ee1ede9df471d0d957f79fbd4b32f80a9599238aae81a2929f11391a.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-533_26e17d4b0f502f40ad3d8167686bdda054731e6af61b624f6c74bfac6eb790d2.jpg b/.raw/MidasCivilAnalysisReference/images/page-533_26e17d4b0f502f40ad3d8167686bdda054731e6af61b624f6c74bfac6eb790d2.jpg new file mode 100644 index 00000000..47211b89 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-533_26e17d4b0f502f40ad3d8167686bdda054731e6af61b624f6c74bfac6eb790d2.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-534_b4da31bee1018cad1d1980020d607d8d0f76f093e3406163eac2434be2c5811a.jpg b/.raw/MidasCivilAnalysisReference/images/page-534_b4da31bee1018cad1d1980020d607d8d0f76f093e3406163eac2434be2c5811a.jpg new file mode 100644 index 00000000..7541789e Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-534_b4da31bee1018cad1d1980020d607d8d0f76f093e3406163eac2434be2c5811a.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-538_41f82d11769a053136f87890669982dfa43a595498cfc9381e2693483097530b.jpg b/.raw/MidasCivilAnalysisReference/images/page-538_41f82d11769a053136f87890669982dfa43a595498cfc9381e2693483097530b.jpg new file mode 100644 index 00000000..fd49e4c7 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-538_41f82d11769a053136f87890669982dfa43a595498cfc9381e2693483097530b.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-538_46b7ef42ed1a6028d83d9be4aed8f054d32f309544cb60244fc79ba787172431.jpg b/.raw/MidasCivilAnalysisReference/images/page-538_46b7ef42ed1a6028d83d9be4aed8f054d32f309544cb60244fc79ba787172431.jpg new file mode 100644 index 00000000..bc8b4335 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-538_46b7ef42ed1a6028d83d9be4aed8f054d32f309544cb60244fc79ba787172431.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-538_6d837c9b9d65b78b4875d46796cbc909a0903afc0d8527825c608c623daf4393.jpg b/.raw/MidasCivilAnalysisReference/images/page-538_6d837c9b9d65b78b4875d46796cbc909a0903afc0d8527825c608c623daf4393.jpg new file mode 100644 index 00000000..f301f917 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-538_6d837c9b9d65b78b4875d46796cbc909a0903afc0d8527825c608c623daf4393.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-539_2726fafbc1f0987a2b601a0d4eff536cf15f89aa9e8f325d5cb05d3426e20405.jpg b/.raw/MidasCivilAnalysisReference/images/page-539_2726fafbc1f0987a2b601a0d4eff536cf15f89aa9e8f325d5cb05d3426e20405.jpg new file mode 100644 index 00000000..c9d1d50d Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-539_2726fafbc1f0987a2b601a0d4eff536cf15f89aa9e8f325d5cb05d3426e20405.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-539_6753d57dc4d2b274a6076d3ee47bf1d6d2d7786d9616f71d3117c76d33e250d7.jpg b/.raw/MidasCivilAnalysisReference/images/page-539_6753d57dc4d2b274a6076d3ee47bf1d6d2d7786d9616f71d3117c76d33e250d7.jpg new file mode 100644 index 00000000..54b0a54a Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-539_6753d57dc4d2b274a6076d3ee47bf1d6d2d7786d9616f71d3117c76d33e250d7.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-539_df3a8141c1a67421bf8156b972b4165661d7284f4a2a7abf808e76941302b74d.jpg b/.raw/MidasCivilAnalysisReference/images/page-539_df3a8141c1a67421bf8156b972b4165661d7284f4a2a7abf808e76941302b74d.jpg new file mode 100644 index 00000000..7465c159 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-539_df3a8141c1a67421bf8156b972b4165661d7284f4a2a7abf808e76941302b74d.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-539_f3bd6bd1accc03c3cecf6f95958358a59e7d259d513a70f678dbb2d6ff9326ca.jpg b/.raw/MidasCivilAnalysisReference/images/page-539_f3bd6bd1accc03c3cecf6f95958358a59e7d259d513a70f678dbb2d6ff9326ca.jpg new file mode 100644 index 00000000..dcd98b55 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-539_f3bd6bd1accc03c3cecf6f95958358a59e7d259d513a70f678dbb2d6ff9326ca.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-540_435165be0d057c1d5b1d40c4526f5148791531b335351e1ddbdabc3eea828de2.jpg b/.raw/MidasCivilAnalysisReference/images/page-540_435165be0d057c1d5b1d40c4526f5148791531b335351e1ddbdabc3eea828de2.jpg new file mode 100644 index 00000000..d7c77a7c Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-540_435165be0d057c1d5b1d40c4526f5148791531b335351e1ddbdabc3eea828de2.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-540_c0e0d3299781c8f20d3e4e2b9134df61e0d1376d817843a02bdebc95a717ac40.jpg b/.raw/MidasCivilAnalysisReference/images/page-540_c0e0d3299781c8f20d3e4e2b9134df61e0d1376d817843a02bdebc95a717ac40.jpg new file mode 100644 index 00000000..068d63b2 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-540_c0e0d3299781c8f20d3e4e2b9134df61e0d1376d817843a02bdebc95a717ac40.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-541_09a5f2bf14afdde4d8505e549f5ddc940eff37230596dc5d285a6ea0d2953f84.jpg b/.raw/MidasCivilAnalysisReference/images/page-541_09a5f2bf14afdde4d8505e549f5ddc940eff37230596dc5d285a6ea0d2953f84.jpg new file mode 100644 index 00000000..d1765e3c Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-541_09a5f2bf14afdde4d8505e549f5ddc940eff37230596dc5d285a6ea0d2953f84.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-541_2579d60217e8647c8ab2a1b445b295b1781839708481a113efa0e05e0a8a3834.jpg b/.raw/MidasCivilAnalysisReference/images/page-541_2579d60217e8647c8ab2a1b445b295b1781839708481a113efa0e05e0a8a3834.jpg new file mode 100644 index 00000000..bb50b11a Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-541_2579d60217e8647c8ab2a1b445b295b1781839708481a113efa0e05e0a8a3834.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-541_33a0c123552d88c5d33f3e6ad615e9a643eaed2e720f0db86c6bef2c0eaf7ba9.jpg b/.raw/MidasCivilAnalysisReference/images/page-541_33a0c123552d88c5d33f3e6ad615e9a643eaed2e720f0db86c6bef2c0eaf7ba9.jpg new file mode 100644 index 00000000..d3ac125a Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-541_33a0c123552d88c5d33f3e6ad615e9a643eaed2e720f0db86c6bef2c0eaf7ba9.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-541_3a011434d482d9c46c5089ce3277e23bc7276db4cef8741e225d60f6c6e7fcf7.jpg b/.raw/MidasCivilAnalysisReference/images/page-541_3a011434d482d9c46c5089ce3277e23bc7276db4cef8741e225d60f6c6e7fcf7.jpg new file mode 100644 index 00000000..60bd6479 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-541_3a011434d482d9c46c5089ce3277e23bc7276db4cef8741e225d60f6c6e7fcf7.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-541_3edab63c71bcbee57b3ada4a98cf1a051a2cfb69834e4fff464a19da8f390e84.jpg b/.raw/MidasCivilAnalysisReference/images/page-541_3edab63c71bcbee57b3ada4a98cf1a051a2cfb69834e4fff464a19da8f390e84.jpg new file mode 100644 index 00000000..0245190a Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-541_3edab63c71bcbee57b3ada4a98cf1a051a2cfb69834e4fff464a19da8f390e84.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-542_77eabcc1694e9acc08555cddf44f6c8114f73d19c8903b70cdd9b3dfe72e9007.jpg b/.raw/MidasCivilAnalysisReference/images/page-542_77eabcc1694e9acc08555cddf44f6c8114f73d19c8903b70cdd9b3dfe72e9007.jpg new file mode 100644 index 00000000..d574579d Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-542_77eabcc1694e9acc08555cddf44f6c8114f73d19c8903b70cdd9b3dfe72e9007.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-542_f70b9b5bda8349c859e38615f4fd0be05b0ebf74359ffb4d783cf81e9c516a60.jpg b/.raw/MidasCivilAnalysisReference/images/page-542_f70b9b5bda8349c859e38615f4fd0be05b0ebf74359ffb4d783cf81e9c516a60.jpg new file mode 100644 index 00000000..3ee00560 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-542_f70b9b5bda8349c859e38615f4fd0be05b0ebf74359ffb4d783cf81e9c516a60.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-543_21c8eb624d68fbcde4fb5d1224d9497c38f3e0aa15d02e91c89d4edc80ba6c8f.jpg b/.raw/MidasCivilAnalysisReference/images/page-543_21c8eb624d68fbcde4fb5d1224d9497c38f3e0aa15d02e91c89d4edc80ba6c8f.jpg new file mode 100644 index 00000000..2d280a9e Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-543_21c8eb624d68fbcde4fb5d1224d9497c38f3e0aa15d02e91c89d4edc80ba6c8f.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-543_73735a93a347acc1b7a4bffeba2e645c21c030e03e7b6b7c7ab2268fd05328dd.jpg b/.raw/MidasCivilAnalysisReference/images/page-543_73735a93a347acc1b7a4bffeba2e645c21c030e03e7b6b7c7ab2268fd05328dd.jpg new file mode 100644 index 00000000..bd190bdb Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-543_73735a93a347acc1b7a4bffeba2e645c21c030e03e7b6b7c7ab2268fd05328dd.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-543_820b4c6f7f71c178554ebc50da29f995054ff8056c956162bdab09da010b2b55.jpg b/.raw/MidasCivilAnalysisReference/images/page-543_820b4c6f7f71c178554ebc50da29f995054ff8056c956162bdab09da010b2b55.jpg new file mode 100644 index 00000000..4b59e41a Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-543_820b4c6f7f71c178554ebc50da29f995054ff8056c956162bdab09da010b2b55.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-543_897ad32a9d8721bfa1966073f4fe52ef6ba3df8b1deb96f53ed5250eafc4c620.jpg b/.raw/MidasCivilAnalysisReference/images/page-543_897ad32a9d8721bfa1966073f4fe52ef6ba3df8b1deb96f53ed5250eafc4c620.jpg new file mode 100644 index 00000000..79af39b2 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-543_897ad32a9d8721bfa1966073f4fe52ef6ba3df8b1deb96f53ed5250eafc4c620.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-543_8a4a51f8d64a2e54ef2d749ee944f98590211a76def750201e99bc15194f80f6.jpg b/.raw/MidasCivilAnalysisReference/images/page-543_8a4a51f8d64a2e54ef2d749ee944f98590211a76def750201e99bc15194f80f6.jpg new file mode 100644 index 00000000..ef680816 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-543_8a4a51f8d64a2e54ef2d749ee944f98590211a76def750201e99bc15194f80f6.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-543_bd5e330820e6cd828c7514b7d9810b72f47233006587709fe45caed4d6276c39.jpg b/.raw/MidasCivilAnalysisReference/images/page-543_bd5e330820e6cd828c7514b7d9810b72f47233006587709fe45caed4d6276c39.jpg new file mode 100644 index 00000000..d4b6e4e1 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-543_bd5e330820e6cd828c7514b7d9810b72f47233006587709fe45caed4d6276c39.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-543_caf139b480b9ec7497dad1d6d21cae4aadd4a429904e6cf8e50b2030d7158fad.jpg b/.raw/MidasCivilAnalysisReference/images/page-543_caf139b480b9ec7497dad1d6d21cae4aadd4a429904e6cf8e50b2030d7158fad.jpg new file mode 100644 index 00000000..b9be7c6f Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-543_caf139b480b9ec7497dad1d6d21cae4aadd4a429904e6cf8e50b2030d7158fad.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-543_cfe66a9b0b3ea17cb8e2a4d87da8882fb3dd5d88ca388c9c4ce3dce070d1972b.jpg b/.raw/MidasCivilAnalysisReference/images/page-543_cfe66a9b0b3ea17cb8e2a4d87da8882fb3dd5d88ca388c9c4ce3dce070d1972b.jpg new file mode 100644 index 00000000..5a758f86 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-543_cfe66a9b0b3ea17cb8e2a4d87da8882fb3dd5d88ca388c9c4ce3dce070d1972b.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-544_181666451bfe73a41b72ebf564c7eb03dd4d51bf2ae90ebc3179bb382967b581.jpg b/.raw/MidasCivilAnalysisReference/images/page-544_181666451bfe73a41b72ebf564c7eb03dd4d51bf2ae90ebc3179bb382967b581.jpg new file mode 100644 index 00000000..51f3b982 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-544_181666451bfe73a41b72ebf564c7eb03dd4d51bf2ae90ebc3179bb382967b581.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-544_1c0511296293a312ba77076a46f66e53d71379eea0a5b3999a1feb3a39e30104.jpg b/.raw/MidasCivilAnalysisReference/images/page-544_1c0511296293a312ba77076a46f66e53d71379eea0a5b3999a1feb3a39e30104.jpg new file mode 100644 index 00000000..e695739c Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-544_1c0511296293a312ba77076a46f66e53d71379eea0a5b3999a1feb3a39e30104.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-544_69ced4e8359ffacb9283edb42a1b985c39a59b064076b622a9d4a95030cdae70.jpg b/.raw/MidasCivilAnalysisReference/images/page-544_69ced4e8359ffacb9283edb42a1b985c39a59b064076b622a9d4a95030cdae70.jpg new file mode 100644 index 00000000..5cfb10ba Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-544_69ced4e8359ffacb9283edb42a1b985c39a59b064076b622a9d4a95030cdae70.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-544_6ff8b38030edc8c348a69b3b0a2e2ede9ba25312f300577931cd77c55dc5efa9.jpg b/.raw/MidasCivilAnalysisReference/images/page-544_6ff8b38030edc8c348a69b3b0a2e2ede9ba25312f300577931cd77c55dc5efa9.jpg new file mode 100644 index 00000000..4c3e0646 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-544_6ff8b38030edc8c348a69b3b0a2e2ede9ba25312f300577931cd77c55dc5efa9.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-544_85744bfea33a0830385cf86a9a3abd89e1dbf574d1c23416373b18c8159bbf08.jpg b/.raw/MidasCivilAnalysisReference/images/page-544_85744bfea33a0830385cf86a9a3abd89e1dbf574d1c23416373b18c8159bbf08.jpg new file mode 100644 index 00000000..b4ac59b1 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-544_85744bfea33a0830385cf86a9a3abd89e1dbf574d1c23416373b18c8159bbf08.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-544_a0d613354100cf69224befae2b8fd63043c73d7d25a7ace0350805393061195f.jpg b/.raw/MidasCivilAnalysisReference/images/page-544_a0d613354100cf69224befae2b8fd63043c73d7d25a7ace0350805393061195f.jpg new file mode 100644 index 00000000..7e13c3f4 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-544_a0d613354100cf69224befae2b8fd63043c73d7d25a7ace0350805393061195f.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-544_da0d565b01f1de2188e8d5f50da960264ec409a303a5393b3bc7341a6216d476.jpg b/.raw/MidasCivilAnalysisReference/images/page-544_da0d565b01f1de2188e8d5f50da960264ec409a303a5393b3bc7341a6216d476.jpg new file mode 100644 index 00000000..b264411f Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-544_da0d565b01f1de2188e8d5f50da960264ec409a303a5393b3bc7341a6216d476.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-545_08e2586e98ae671e9a3c776032e1f86ed76760a928157b155344609988947217.jpg b/.raw/MidasCivilAnalysisReference/images/page-545_08e2586e98ae671e9a3c776032e1f86ed76760a928157b155344609988947217.jpg new file mode 100644 index 00000000..c90922ca Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-545_08e2586e98ae671e9a3c776032e1f86ed76760a928157b155344609988947217.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-545_26bf5d23d651026eeaa3dc017d3b067dadcde788aeabaf72dea69e91b3124b85.jpg b/.raw/MidasCivilAnalysisReference/images/page-545_26bf5d23d651026eeaa3dc017d3b067dadcde788aeabaf72dea69e91b3124b85.jpg new file mode 100644 index 00000000..eee5beb7 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-545_26bf5d23d651026eeaa3dc017d3b067dadcde788aeabaf72dea69e91b3124b85.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-545_ec90fefe7c1d1c656949ad989ebb3000021aa40a677a5ac257acb1459a69b48a.jpg b/.raw/MidasCivilAnalysisReference/images/page-545_ec90fefe7c1d1c656949ad989ebb3000021aa40a677a5ac257acb1459a69b48a.jpg new file mode 100644 index 00000000..5fdab752 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-545_ec90fefe7c1d1c656949ad989ebb3000021aa40a677a5ac257acb1459a69b48a.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-546_29ad0ec7d6fb06329d67f52b82b6844c986f696a0e1c75e10b9696973293e15a.jpg b/.raw/MidasCivilAnalysisReference/images/page-546_29ad0ec7d6fb06329d67f52b82b6844c986f696a0e1c75e10b9696973293e15a.jpg new file mode 100644 index 00000000..51f01a2d Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-546_29ad0ec7d6fb06329d67f52b82b6844c986f696a0e1c75e10b9696973293e15a.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-547_58db8fd09375f143ba26f0051b90f17b054c3d7ac29cae137faea18b630636eb.jpg b/.raw/MidasCivilAnalysisReference/images/page-547_58db8fd09375f143ba26f0051b90f17b054c3d7ac29cae137faea18b630636eb.jpg new file mode 100644 index 00000000..161c5a41 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-547_58db8fd09375f143ba26f0051b90f17b054c3d7ac29cae137faea18b630636eb.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-547_bc901780c7180cc23b4e2cba529643dee07ce14ef59da56cd1779b42d30350dd.jpg b/.raw/MidasCivilAnalysisReference/images/page-547_bc901780c7180cc23b4e2cba529643dee07ce14ef59da56cd1779b42d30350dd.jpg new file mode 100644 index 00000000..a1acf711 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-547_bc901780c7180cc23b4e2cba529643dee07ce14ef59da56cd1779b42d30350dd.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-548_85f495337ba66929e19d2a1ffca3fd46e5413b6e6a60b29deeb305d636aa2568.jpg b/.raw/MidasCivilAnalysisReference/images/page-548_85f495337ba66929e19d2a1ffca3fd46e5413b6e6a60b29deeb305d636aa2568.jpg new file mode 100644 index 00000000..8652b342 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-548_85f495337ba66929e19d2a1ffca3fd46e5413b6e6a60b29deeb305d636aa2568.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-549_3f470b220293f6e1b4461f04e57a7d745597ba75c020c06bc3da2ceba0c8ea56.jpg b/.raw/MidasCivilAnalysisReference/images/page-549_3f470b220293f6e1b4461f04e57a7d745597ba75c020c06bc3da2ceba0c8ea56.jpg new file mode 100644 index 00000000..5a387475 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-549_3f470b220293f6e1b4461f04e57a7d745597ba75c020c06bc3da2ceba0c8ea56.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-549_50558aed1f709a6f635540c62836d09e0a851b3ec369a7eabcd44d110c18776f.jpg b/.raw/MidasCivilAnalysisReference/images/page-549_50558aed1f709a6f635540c62836d09e0a851b3ec369a7eabcd44d110c18776f.jpg new file mode 100644 index 00000000..a8401f44 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-549_50558aed1f709a6f635540c62836d09e0a851b3ec369a7eabcd44d110c18776f.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-550_1d57d0c787baf0527b2e3482afc8f209be00ac156c210573421251cea4abb7cf.jpg b/.raw/MidasCivilAnalysisReference/images/page-550_1d57d0c787baf0527b2e3482afc8f209be00ac156c210573421251cea4abb7cf.jpg new file mode 100644 index 00000000..27e26fa0 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-550_1d57d0c787baf0527b2e3482afc8f209be00ac156c210573421251cea4abb7cf.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-550_2cb3761714232795919fb6b635632d0cdc1706df2db956f917dce01d66bf93ff.jpg b/.raw/MidasCivilAnalysisReference/images/page-550_2cb3761714232795919fb6b635632d0cdc1706df2db956f917dce01d66bf93ff.jpg new file mode 100644 index 00000000..c09d36fc Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-550_2cb3761714232795919fb6b635632d0cdc1706df2db956f917dce01d66bf93ff.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-550_83cf9dd24188da2e4642af0ca9ba3bc8d2a902caa579c8406dd5959d003e8a27.jpg b/.raw/MidasCivilAnalysisReference/images/page-550_83cf9dd24188da2e4642af0ca9ba3bc8d2a902caa579c8406dd5959d003e8a27.jpg new file mode 100644 index 00000000..29f5d2ae Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-550_83cf9dd24188da2e4642af0ca9ba3bc8d2a902caa579c8406dd5959d003e8a27.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-550_9b959dbb7044bce6e0f2956ff3fa0d71fc05e9e54fe1c15ed2161c811773abf8.jpg b/.raw/MidasCivilAnalysisReference/images/page-550_9b959dbb7044bce6e0f2956ff3fa0d71fc05e9e54fe1c15ed2161c811773abf8.jpg new file mode 100644 index 00000000..037ab256 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-550_9b959dbb7044bce6e0f2956ff3fa0d71fc05e9e54fe1c15ed2161c811773abf8.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-550_aaf6f07eaab61fc3812fd3ea6ea18f036c407632814ff3b1e4844b8b61ddcd11.jpg b/.raw/MidasCivilAnalysisReference/images/page-550_aaf6f07eaab61fc3812fd3ea6ea18f036c407632814ff3b1e4844b8b61ddcd11.jpg new file mode 100644 index 00000000..781600e7 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-550_aaf6f07eaab61fc3812fd3ea6ea18f036c407632814ff3b1e4844b8b61ddcd11.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-550_cb2dd510e73ff582d97d4b6e37f25afa4c909ad1d5bb172c5f96fad339530cb1.jpg b/.raw/MidasCivilAnalysisReference/images/page-550_cb2dd510e73ff582d97d4b6e37f25afa4c909ad1d5bb172c5f96fad339530cb1.jpg new file mode 100644 index 00000000..06f39d48 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-550_cb2dd510e73ff582d97d4b6e37f25afa4c909ad1d5bb172c5f96fad339530cb1.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-550_d40d2757c8527ba67fe7faba6d025c205d468902802ee723df020938ae8fe04e.jpg b/.raw/MidasCivilAnalysisReference/images/page-550_d40d2757c8527ba67fe7faba6d025c205d468902802ee723df020938ae8fe04e.jpg new file mode 100644 index 00000000..59a3a9a2 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-550_d40d2757c8527ba67fe7faba6d025c205d468902802ee723df020938ae8fe04e.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-550_d6ebd57ecb0efdcc554734671868abb083d07f64acd8cc1dfff9777372ae3788.jpg b/.raw/MidasCivilAnalysisReference/images/page-550_d6ebd57ecb0efdcc554734671868abb083d07f64acd8cc1dfff9777372ae3788.jpg new file mode 100644 index 00000000..9575d6c6 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-550_d6ebd57ecb0efdcc554734671868abb083d07f64acd8cc1dfff9777372ae3788.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-550_eda0bd879e75509f993c26a50e9b6281431b4afc041a8ccfc92661f819926e8f.jpg b/.raw/MidasCivilAnalysisReference/images/page-550_eda0bd879e75509f993c26a50e9b6281431b4afc041a8ccfc92661f819926e8f.jpg new file mode 100644 index 00000000..7f001e05 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-550_eda0bd879e75509f993c26a50e9b6281431b4afc041a8ccfc92661f819926e8f.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-550_eeab95dcf937f4d683202a6796e7d10e398ae58fa3ef3c07ed7a6e86bbd8d463.jpg b/.raw/MidasCivilAnalysisReference/images/page-550_eeab95dcf937f4d683202a6796e7d10e398ae58fa3ef3c07ed7a6e86bbd8d463.jpg new file mode 100644 index 00000000..cc6527a2 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-550_eeab95dcf937f4d683202a6796e7d10e398ae58fa3ef3c07ed7a6e86bbd8d463.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-551_40876c1069521a0f2641bcd568cdbac63e8236988d86a9c20778bd8ce5165c0a.jpg b/.raw/MidasCivilAnalysisReference/images/page-551_40876c1069521a0f2641bcd568cdbac63e8236988d86a9c20778bd8ce5165c0a.jpg new file mode 100644 index 00000000..8047c9b8 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-551_40876c1069521a0f2641bcd568cdbac63e8236988d86a9c20778bd8ce5165c0a.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-551_64d3cbd28e888f31cb1c5d23f0d2195091d531ad8ca4a4836104c29d0035ad86.jpg b/.raw/MidasCivilAnalysisReference/images/page-551_64d3cbd28e888f31cb1c5d23f0d2195091d531ad8ca4a4836104c29d0035ad86.jpg new file mode 100644 index 00000000..da866baa Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-551_64d3cbd28e888f31cb1c5d23f0d2195091d531ad8ca4a4836104c29d0035ad86.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-551_d34ef324672fce3f40a378ce3bb2f651a4846e5104044e2a3577eeef23bbb499.jpg b/.raw/MidasCivilAnalysisReference/images/page-551_d34ef324672fce3f40a378ce3bb2f651a4846e5104044e2a3577eeef23bbb499.jpg new file mode 100644 index 00000000..f04dfcd8 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-551_d34ef324672fce3f40a378ce3bb2f651a4846e5104044e2a3577eeef23bbb499.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-551_e46437254c6e9ac7ebb1950f8cd08bf28745f64a7a094d4c65b11c303bc68c30.jpg b/.raw/MidasCivilAnalysisReference/images/page-551_e46437254c6e9ac7ebb1950f8cd08bf28745f64a7a094d4c65b11c303bc68c30.jpg new file mode 100644 index 00000000..2cc40180 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-551_e46437254c6e9ac7ebb1950f8cd08bf28745f64a7a094d4c65b11c303bc68c30.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-552_2d3ace239d988d0cf6cf79f2585a2bb3f51d9d8ab71a02f8fd79ccc37305ef8d.jpg b/.raw/MidasCivilAnalysisReference/images/page-552_2d3ace239d988d0cf6cf79f2585a2bb3f51d9d8ab71a02f8fd79ccc37305ef8d.jpg new file mode 100644 index 00000000..691b1720 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-552_2d3ace239d988d0cf6cf79f2585a2bb3f51d9d8ab71a02f8fd79ccc37305ef8d.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-553_3600a4c744c352aa356b0e40350acdfbd5d7f77bf794e7f700de4ab2c9c7f534.jpg b/.raw/MidasCivilAnalysisReference/images/page-553_3600a4c744c352aa356b0e40350acdfbd5d7f77bf794e7f700de4ab2c9c7f534.jpg new file mode 100644 index 00000000..d749917f Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-553_3600a4c744c352aa356b0e40350acdfbd5d7f77bf794e7f700de4ab2c9c7f534.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-554_19c38a730b7daec7823449bdd4cd276f089e50e3a32fd7003a3c46bacde4dc47.jpg b/.raw/MidasCivilAnalysisReference/images/page-554_19c38a730b7daec7823449bdd4cd276f089e50e3a32fd7003a3c46bacde4dc47.jpg new file mode 100644 index 00000000..54b5949a Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-554_19c38a730b7daec7823449bdd4cd276f089e50e3a32fd7003a3c46bacde4dc47.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-554_d3d36589650cbe82966b9981eb2f532f7e0f732ddff80bf2662cdb520ac6243d.jpg b/.raw/MidasCivilAnalysisReference/images/page-554_d3d36589650cbe82966b9981eb2f532f7e0f732ddff80bf2662cdb520ac6243d.jpg new file mode 100644 index 00000000..9d1ce876 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-554_d3d36589650cbe82966b9981eb2f532f7e0f732ddff80bf2662cdb520ac6243d.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-554_f587be1a928d827fa8b27aa71802155fadf24be7619775a81fae2dcf4c0b5745.jpg b/.raw/MidasCivilAnalysisReference/images/page-554_f587be1a928d827fa8b27aa71802155fadf24be7619775a81fae2dcf4c0b5745.jpg new file mode 100644 index 00000000..25727498 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-554_f587be1a928d827fa8b27aa71802155fadf24be7619775a81fae2dcf4c0b5745.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-555_7cf31361ced0bdda7f26f4e81e4f727516688ed8733cc922353d773a62a1b7c3.jpg b/.raw/MidasCivilAnalysisReference/images/page-555_7cf31361ced0bdda7f26f4e81e4f727516688ed8733cc922353d773a62a1b7c3.jpg new file mode 100644 index 00000000..04a50bff Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-555_7cf31361ced0bdda7f26f4e81e4f727516688ed8733cc922353d773a62a1b7c3.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-556_b7934afe3ec4ba3d9ec6b5c49f0ba96490769b42c23feec330b40cf6e7a3ee53.jpg b/.raw/MidasCivilAnalysisReference/images/page-556_b7934afe3ec4ba3d9ec6b5c49f0ba96490769b42c23feec330b40cf6e7a3ee53.jpg new file mode 100644 index 00000000..937c0899 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-556_b7934afe3ec4ba3d9ec6b5c49f0ba96490769b42c23feec330b40cf6e7a3ee53.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-557_07eb9940c0e99ad98b4aee047369e51f3a069b98d01516596acc11b6acf89be9.jpg b/.raw/MidasCivilAnalysisReference/images/page-557_07eb9940c0e99ad98b4aee047369e51f3a069b98d01516596acc11b6acf89be9.jpg new file mode 100644 index 00000000..fe1b45d8 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-557_07eb9940c0e99ad98b4aee047369e51f3a069b98d01516596acc11b6acf89be9.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-557_3bf9d5f8c972f1e7d5f98c818b2f4e95821b544213cde94487ba75c9899cf0f8.jpg b/.raw/MidasCivilAnalysisReference/images/page-557_3bf9d5f8c972f1e7d5f98c818b2f4e95821b544213cde94487ba75c9899cf0f8.jpg new file mode 100644 index 00000000..1a3ffa65 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-557_3bf9d5f8c972f1e7d5f98c818b2f4e95821b544213cde94487ba75c9899cf0f8.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-557_4b907ed87d06ce35a0ad380061f5ac361e3f006adc29922d3b112fd0dc86c149.jpg b/.raw/MidasCivilAnalysisReference/images/page-557_4b907ed87d06ce35a0ad380061f5ac361e3f006adc29922d3b112fd0dc86c149.jpg new file mode 100644 index 00000000..763819a6 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-557_4b907ed87d06ce35a0ad380061f5ac361e3f006adc29922d3b112fd0dc86c149.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-557_925fb4ca42f79b077a680c924842130f4df4c876395d3f64fd061a9324538fac.jpg b/.raw/MidasCivilAnalysisReference/images/page-557_925fb4ca42f79b077a680c924842130f4df4c876395d3f64fd061a9324538fac.jpg new file mode 100644 index 00000000..a81a3e9e Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-557_925fb4ca42f79b077a680c924842130f4df4c876395d3f64fd061a9324538fac.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-558_9db1a145d7ed0902693f8bee523d2ad64117af400597e2277b22916a60009804.jpg b/.raw/MidasCivilAnalysisReference/images/page-558_9db1a145d7ed0902693f8bee523d2ad64117af400597e2277b22916a60009804.jpg new file mode 100644 index 00000000..0c945700 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-558_9db1a145d7ed0902693f8bee523d2ad64117af400597e2277b22916a60009804.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-558_f4ddfdc3a07b06db60505db17ce64db0d9284bac8d781dcfb292ed27c5f78518.jpg b/.raw/MidasCivilAnalysisReference/images/page-558_f4ddfdc3a07b06db60505db17ce64db0d9284bac8d781dcfb292ed27c5f78518.jpg new file mode 100644 index 00000000..c2f36ebe Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-558_f4ddfdc3a07b06db60505db17ce64db0d9284bac8d781dcfb292ed27c5f78518.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-559_0c9befb52c9e496ed2d73e8b489432fe97d0b5ecd828bf1265ff8c5e1cd9d2cb.jpg b/.raw/MidasCivilAnalysisReference/images/page-559_0c9befb52c9e496ed2d73e8b489432fe97d0b5ecd828bf1265ff8c5e1cd9d2cb.jpg new file mode 100644 index 00000000..49f1e401 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-559_0c9befb52c9e496ed2d73e8b489432fe97d0b5ecd828bf1265ff8c5e1cd9d2cb.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-559_48d92dbbd264b979dec2d2c47c57ed9e92996ddd794a96207354eeaa539f8bde.jpg b/.raw/MidasCivilAnalysisReference/images/page-559_48d92dbbd264b979dec2d2c47c57ed9e92996ddd794a96207354eeaa539f8bde.jpg new file mode 100644 index 00000000..9c816867 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-559_48d92dbbd264b979dec2d2c47c57ed9e92996ddd794a96207354eeaa539f8bde.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-559_4fdf263f111bfa4343ff4c11c65a8a14c072436540d1c73fd169356a1077d280.jpg b/.raw/MidasCivilAnalysisReference/images/page-559_4fdf263f111bfa4343ff4c11c65a8a14c072436540d1c73fd169356a1077d280.jpg new file mode 100644 index 00000000..7f7b60ac Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-559_4fdf263f111bfa4343ff4c11c65a8a14c072436540d1c73fd169356a1077d280.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-559_582876159e21fa385c7dc9fe7f1ac35750bb7c385eb6596cebd0701038b4edcb.jpg b/.raw/MidasCivilAnalysisReference/images/page-559_582876159e21fa385c7dc9fe7f1ac35750bb7c385eb6596cebd0701038b4edcb.jpg new file mode 100644 index 00000000..af94f971 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-559_582876159e21fa385c7dc9fe7f1ac35750bb7c385eb6596cebd0701038b4edcb.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-559_7395c72dd438c245291821685ca6d74498c68ab9914bcdecb1f57e793240cc16.jpg b/.raw/MidasCivilAnalysisReference/images/page-559_7395c72dd438c245291821685ca6d74498c68ab9914bcdecb1f57e793240cc16.jpg new file mode 100644 index 00000000..c6a41a20 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-559_7395c72dd438c245291821685ca6d74498c68ab9914bcdecb1f57e793240cc16.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-559_89653757a20192dcb359d89543aacac76658d30a66bc9833f0d821275e35b330.jpg b/.raw/MidasCivilAnalysisReference/images/page-559_89653757a20192dcb359d89543aacac76658d30a66bc9833f0d821275e35b330.jpg new file mode 100644 index 00000000..d0190477 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-559_89653757a20192dcb359d89543aacac76658d30a66bc9833f0d821275e35b330.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-559_ad41272e2d87261b346a0d06768959fd88e5cf70b53ce437104e71d8dd32668a.jpg b/.raw/MidasCivilAnalysisReference/images/page-559_ad41272e2d87261b346a0d06768959fd88e5cf70b53ce437104e71d8dd32668a.jpg new file mode 100644 index 00000000..0c55ddc3 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-559_ad41272e2d87261b346a0d06768959fd88e5cf70b53ce437104e71d8dd32668a.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-559_d7142c77a6b9a58c848a4fda480f728221bedd1a367aebf827ba3f32c2ed53df.jpg b/.raw/MidasCivilAnalysisReference/images/page-559_d7142c77a6b9a58c848a4fda480f728221bedd1a367aebf827ba3f32c2ed53df.jpg new file mode 100644 index 00000000..e94a6f31 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-559_d7142c77a6b9a58c848a4fda480f728221bedd1a367aebf827ba3f32c2ed53df.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-559_ec09125067db6ed06e0b1aa9b5d1e54713ddff81f3ec67b68fb86d5e6b4ee73a.jpg b/.raw/MidasCivilAnalysisReference/images/page-559_ec09125067db6ed06e0b1aa9b5d1e54713ddff81f3ec67b68fb86d5e6b4ee73a.jpg new file mode 100644 index 00000000..d947f49f Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-559_ec09125067db6ed06e0b1aa9b5d1e54713ddff81f3ec67b68fb86d5e6b4ee73a.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-559_eedc14b98a8fe9784eb462ab7b506a9bed5c9d1e5fc122eea183d330c29c3760.jpg b/.raw/MidasCivilAnalysisReference/images/page-559_eedc14b98a8fe9784eb462ab7b506a9bed5c9d1e5fc122eea183d330c29c3760.jpg new file mode 100644 index 00000000..342a6682 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-559_eedc14b98a8fe9784eb462ab7b506a9bed5c9d1e5fc122eea183d330c29c3760.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-561_4a993c84f5060d635ab16dc86b0ad43aebc539ebff33422bfcbd480e05d3e0fc.jpg b/.raw/MidasCivilAnalysisReference/images/page-561_4a993c84f5060d635ab16dc86b0ad43aebc539ebff33422bfcbd480e05d3e0fc.jpg new file mode 100644 index 00000000..f271e140 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-561_4a993c84f5060d635ab16dc86b0ad43aebc539ebff33422bfcbd480e05d3e0fc.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-561_9c3811bf14a64bf9562a2cfde48eea0d5fe8b470302bdccf991d3576cc1bef47.jpg b/.raw/MidasCivilAnalysisReference/images/page-561_9c3811bf14a64bf9562a2cfde48eea0d5fe8b470302bdccf991d3576cc1bef47.jpg new file mode 100644 index 00000000..aee74fcf Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-561_9c3811bf14a64bf9562a2cfde48eea0d5fe8b470302bdccf991d3576cc1bef47.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-561_ad280d6c8c53431740cca9fc25376f1fbecdda8e2f158780d3504e26d3251549.jpg b/.raw/MidasCivilAnalysisReference/images/page-561_ad280d6c8c53431740cca9fc25376f1fbecdda8e2f158780d3504e26d3251549.jpg new file mode 100644 index 00000000..193f7709 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-561_ad280d6c8c53431740cca9fc25376f1fbecdda8e2f158780d3504e26d3251549.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-562_30f5a5ab9cc0278c4a242bcab8e877a1d987f54782afdace1223cc48c8bd996a.jpg b/.raw/MidasCivilAnalysisReference/images/page-562_30f5a5ab9cc0278c4a242bcab8e877a1d987f54782afdace1223cc48c8bd996a.jpg new file mode 100644 index 00000000..e246dce2 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-562_30f5a5ab9cc0278c4a242bcab8e877a1d987f54782afdace1223cc48c8bd996a.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-562_a68ec97481e3ade03d1ea5ecaa47b3bdb58c31c75dc7f0b6b36450634d4c3f01.jpg b/.raw/MidasCivilAnalysisReference/images/page-562_a68ec97481e3ade03d1ea5ecaa47b3bdb58c31c75dc7f0b6b36450634d4c3f01.jpg new file mode 100644 index 00000000..07146a61 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-562_a68ec97481e3ade03d1ea5ecaa47b3bdb58c31c75dc7f0b6b36450634d4c3f01.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-563_055c01d0b349b3bdc01f9d9c230e623d47055e42736bc655bb442fd25e147a5c.jpg b/.raw/MidasCivilAnalysisReference/images/page-563_055c01d0b349b3bdc01f9d9c230e623d47055e42736bc655bb442fd25e147a5c.jpg new file mode 100644 index 00000000..c14d093a Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-563_055c01d0b349b3bdc01f9d9c230e623d47055e42736bc655bb442fd25e147a5c.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-563_1467508306cb359a3362c12f7719208f0e0466125d2765ef4bf9147f6c99386e.jpg b/.raw/MidasCivilAnalysisReference/images/page-563_1467508306cb359a3362c12f7719208f0e0466125d2765ef4bf9147f6c99386e.jpg new file mode 100644 index 00000000..cba8ea2a Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-563_1467508306cb359a3362c12f7719208f0e0466125d2765ef4bf9147f6c99386e.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-563_294817ca1877ef47411f6a1613949e11c718f9802ed68c7cd34a5066d3b8f7e8.jpg b/.raw/MidasCivilAnalysisReference/images/page-563_294817ca1877ef47411f6a1613949e11c718f9802ed68c7cd34a5066d3b8f7e8.jpg new file mode 100644 index 00000000..2de0d8a0 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-563_294817ca1877ef47411f6a1613949e11c718f9802ed68c7cd34a5066d3b8f7e8.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-563_8bb449e53089556354d4bbc3b4828c003894af7ad706dd02dc3c6fa279d798a1.jpg b/.raw/MidasCivilAnalysisReference/images/page-563_8bb449e53089556354d4bbc3b4828c003894af7ad706dd02dc3c6fa279d798a1.jpg new file mode 100644 index 00000000..5a8488d1 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-563_8bb449e53089556354d4bbc3b4828c003894af7ad706dd02dc3c6fa279d798a1.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-563_d8a3c5d198278299432d8cd4c752ca0d22439574c3d90d6d9770203fd6cbe2da.jpg b/.raw/MidasCivilAnalysisReference/images/page-563_d8a3c5d198278299432d8cd4c752ca0d22439574c3d90d6d9770203fd6cbe2da.jpg new file mode 100644 index 00000000..0ff4f21b Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-563_d8a3c5d198278299432d8cd4c752ca0d22439574c3d90d6d9770203fd6cbe2da.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-564_7fc5f93f1ecc02f83e2abd9604e2955586f4fafb348f0203456cfcd61f40dda7.jpg b/.raw/MidasCivilAnalysisReference/images/page-564_7fc5f93f1ecc02f83e2abd9604e2955586f4fafb348f0203456cfcd61f40dda7.jpg new file mode 100644 index 00000000..bbedceaf Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-564_7fc5f93f1ecc02f83e2abd9604e2955586f4fafb348f0203456cfcd61f40dda7.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-565_cf1d09efdf4859fa353866a22e38f5679fd906f9ef5224d90801323aa4f35067.jpg b/.raw/MidasCivilAnalysisReference/images/page-565_cf1d09efdf4859fa353866a22e38f5679fd906f9ef5224d90801323aa4f35067.jpg new file mode 100644 index 00000000..56f17a4a Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-565_cf1d09efdf4859fa353866a22e38f5679fd906f9ef5224d90801323aa4f35067.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-565_d3a885364da98400d626ad4fa2aad6cca4a12d7431845d3ef9c786af40feed4c.jpg b/.raw/MidasCivilAnalysisReference/images/page-565_d3a885364da98400d626ad4fa2aad6cca4a12d7431845d3ef9c786af40feed4c.jpg new file mode 100644 index 00000000..46e29cc3 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-565_d3a885364da98400d626ad4fa2aad6cca4a12d7431845d3ef9c786af40feed4c.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-567_2f4f94b38a82d765bbb7ae6d2b569a3e47969e287f36582e4f3f44ca0110bde5.jpg b/.raw/MidasCivilAnalysisReference/images/page-567_2f4f94b38a82d765bbb7ae6d2b569a3e47969e287f36582e4f3f44ca0110bde5.jpg new file mode 100644 index 00000000..2c9a6b77 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-567_2f4f94b38a82d765bbb7ae6d2b569a3e47969e287f36582e4f3f44ca0110bde5.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-567_f817c02a69e10d34802ea87fc548285365a364ede8e30ad23b1f0145d54b4117.jpg b/.raw/MidasCivilAnalysisReference/images/page-567_f817c02a69e10d34802ea87fc548285365a364ede8e30ad23b1f0145d54b4117.jpg new file mode 100644 index 00000000..216ab207 Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-567_f817c02a69e10d34802ea87fc548285365a364ede8e30ad23b1f0145d54b4117.jpg differ diff --git a/.raw/MidasCivilAnalysisReference/images/page-568_68c2db8b26e4413ac8fab8241320e7ede496bac1656fdf7a6d25c2fe30355ce7.jpg b/.raw/MidasCivilAnalysisReference/images/page-568_68c2db8b26e4413ac8fab8241320e7ede496bac1656fdf7a6d25c2fe30355ce7.jpg new file mode 100644 index 00000000..17d11bbd Binary files /dev/null and b/.raw/MidasCivilAnalysisReference/images/page-568_68c2db8b26e4413ac8fab8241320e7ede496bac1656fdf7a6d25c2fe30355ce7.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_001.md b/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_001.md new file mode 100644 index 00000000..3102250b --- /dev/null +++ b/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_001.md @@ -0,0 +1,212 @@ + + +MIDAS Family Program은 + +주식회사 마이다스아이티에서 개발한 + +구조해석 및 설계, 지반 및 터널해석, 가시설전용 해석 및 설계용 소프트웨어 패키지입니다. + +MIDAS Family Program과 관련 책자는 + +컴퓨터 프로그램 보호법과 저작권법에 의하여 보호를 받고 있습니다. + +프로그램이나 관련 자료에 대한 문의는 아래 연락처를 참조 바랍니다. + +![](images/page-001_a124ac6f2e757115af2060d5e8a38ad7a55fc95bb93f1e2251c73b8d9878b134.jpg) + +# OOO + +# MIDAS Information Technology Co.,Ltd. + +경기도 성남시 중원구 상대원1동 190-1 SKn 테크노파크 테크센타 15층 + +Phone : 031-789-2000 + +Fax : 031-789-2100 + +E-mail : midas@midasit.com + +http://www.midasuser.com + +Modeling, Integrated Design & Analysis Software + +본 사용자 지침서의 작성에 인용된 상표(trademark) 및 등록상표(registered trademark)는다음과 같습니다. + +ADINA is a trademark of ADINA R&D, Inc. + +AutoCAD is a registered trademark of Autodesk, Inc. + +SAP2000 is registered trademark of Computer and Structure, Inc. + +Excel is a trademark of Microsoft Corporation. + +IBM is a registered trademarks of International Business Machines Corporation. + +Intel 386, 486, and Pentium are trademark of Intel Corporation. + +MIDAS is a trademark of MIDAS Information Technology Corporation. + +MSC/NASTRAN is a registered trademark of the National Aeronautics and Space Administration(NASA). + +NISAⅡis a trademark of Engineering Mechanics Research Corporation. + +ScreenCam is a trademark of Lotus Development Corporation. + +Sentinel is a trademark of Rainbow Technologies, Inc. + +Windows is a trademark of Microsoft Corporation. + +Internet Explorer is a trademark of Microsoft Corporation. + + + +MIDAS Family Program은 개발단계에서 수천종의 예제 문제를 통하여 이론치 그리고 타 S/W와의 비교검증을 마친바 있으며 최신의 이론을 내장하여 우수한 해석결과를 산출합니다. + +그리고 1989년 개발이후 관공서를 포함해 국내외 5,000여 프로젝트에 적용하여 정확성과 효용성이 입증되었습니다. + +MIDAS Family Program은 사단법인 한국전산구조공학회와 사단법인 한국터널공학회의 엄격한 검증과정을 거친 프로그램입니다. + +그러나 방대한 양의 이론과 설계지식이 집적되는 구조해석 및 설계 프로그램의 특성상, MIDAS FamilyProgram을 사용함으로써 발생될 수 있는 어떠한 이익과 손실에 대해서도 MIDAS Family Program의 개발 후원자와 개발자 그리고, 검증참여기관에게는 권리와 책임이 없습니다. + +따라서 프로그램을 사용하기 전에 사용자지침서에 대한 충분한 이해과정이 필요하며 프로그램의 수행결과에 대해서도 사용자의 검증이 반드시 필요합니다. + +# DISCLAIMER + +Developers and sponsors assume no responsibility for the use of MIDAS Family Program (midas FEA, midas Civil, midas FX+, midas Abutment, midas Pier, midas Deck, midas GTS, midas GeoX, midas Gen, midas ADS, midas SDS, midas Set ; hereinafter referred to as “MIDAS package”) or for the accuracy or validity of any results obtained from the MIDAS package. + +Developers and sponsors shall not be liable for loss of profit, loss of business, or financial loss which may be caused directly or indirectly by the MIDAS package, when used for any purpose or use, due to any defect or deficiency therein. + + + +# 감사의 글 + +MIDAS Family Program은 포스코그룹의 창사 이래 엔지니어링과 건설분야에서 축적한 구조설계 및 설계 기술을 집적하고 지반, 터널 및 가시설분야로 영역을 확대하여 국내의 여러 교수님들과 기술자 여러분들의 도움으로 만들어진 것입니다. + +본 프로그램의 개발과 사용자지침서의 작성에 도움을 주신 학계 교수님, 그리고 관련 분야의 기술자여러분께 감사드립니다. + +그리고 본 프로그램의 개발을 위해 지원을 아끼지 않으신 대한토목학회, 한국강구조공학회, 한국전산구조공학회의 여러분께도 본 지면을 빌어 감사의 말씀을 올립니다. + +폐사는 이러한 헌신적인 기여에 보답하기 위해서 최선을 다하여 프로그램 개발에 전념할 것입니다.본 프로그램이 우리나라 구조해석 및 설계 분야의 기술 신장과 대외 기술 경쟁력 확보에 다소나마 기여할 수 있기를 바랍니다. + +주식회사 마이다스아이티 + + + +# 머리말 + +midas FEA는 다양하고 복잡한 토목 구조물의 형상을 손쉽게 모델링하고 비선형 상세 해석을 빠른 시간내에 완성할 수 있도록 개발된 “비선형 상세 전용 해석 프로그램”입니다.. + +# midas FEA와 MIDAS Family Program에 대하여 + +midas FEA는 2005년부터 개발되기 시작한 MIDAS Family Program 중 하나입니다. MIDAS FamilyProgram은 구조물 해석 및 설계에 수반되는 단위설계 업무의 전 과정을 자동화하기 위한 목적으로 개발된 package software로서 다음과 같이 구성되어 있습니다. + +
토목분야midas FEAAdvanced Nonlinear and Detail Analysis Program교량 및 토목구조물 전용 비선형 상세 프로그램
midas CivilIntegrated Solution System for Bridge and Civil Structures교량 및 토목구조물 전용 구조해석 통합 시스템
midas FX+General Pre & Post-processors for Finite Element Analysis유한요소해석 범용 전/후처리 프로그램
토목설계자동화분야midas AbutmentAbutment Automatic Design System교대설계(계산서, 도면, 수량) 자동화 시스템
midas PierPier Automatic Design System교각설계(계산서, 도면, 수량) 자동화 시스템
midas DeckDeck Automatic Design System콘크리트 바닥판 설계(계산서, 도면, 수량) 자동화 시스템
지반 및 터널분야midas GTSGeotechnical and Tunnel analysis System지반 및 터널구조물 전용 해석 시스템
가시설분야midas GeoXTemporary Works Analysis and Design System가시설전용 해석 및 설계 프로그램
건축분야midas GenIntegrated Design System for Building and General Structures건물 및 일반구조물 구조설계 통합 시스템
midas ADSShear wall type Apartment Design System전단벽식 아파트 구조해석 및 최적설계 시스템
midas SDSSlab & basemat Design System바닥판 · 기초판 구조해석 및 최적설계 시스템
midas SetStructural Engineer’s Tools단위구조설계 도움 프로그램
+ + + +midas FEA는 토목 구조 분야 학계 교수님들과 관련 업계의 실무자 여러분들 및 TNO DIANA와의 조력 및 협력으로 개발된 프로그램이며 윈도우 기반의 객체 지향적 특성, 첨단의 컴퓨터 그래픽과 해석기술을 집적하여 만들어진 새로운 개념의 비선형 상세 해석 전용 프로그램입니다. 사용자 중심의 환경과 Visual C++로 개발 되었기 때문에 빠르고 쉽게 익혀서 실무에 적용할 수 있습니다. + +midas FEA는 토목 구조 분야의 비선형 상세 해석에 필요한 제반기능을 집적하여 개발된 최적의 솔루션입니다. 기존의 프로그램과는 전혀 다른 직관적이고 다양한 모델링 기능, 엄격한 품질보증 및 관리시스템을 통해 검증된 강력한 해석기능, 최신의 해석 솔버를 탑재한 최고 성능의 해석속도, 탁월한 그래픽 표현 그리고 기술자를 위한 실용적인 해석결과 및 설계 등을 제공할 수 있도록 개발되었습니다. + +또한 midas FEA는 개발 과정에서 이론적 접근이 가능한 검증용 문제를 통하여 모든 기능에 대해 이론치 및 타 범용 프로그램과의 비교 검토를 마쳤으며, 다양한 실무 프로젝트에 적용되어 신뢰성과 효용성에 대한 검증이 이루어졌습니다. + +이들 중에서 대표적인 검증 예제를 발췌하여 작성한 Verification을 홈페이지(http://kor.midasuser.com/fea)에 등재하였습니다. 해석 결과의 정확도를 결정하는 유한요소의 알고리즘 측면에서도 최신의 이론을적용하였기 때문에 타 유사 프로그램에 비해 우수한 결과를 산출합니다. + +# 끝으로 + +midas FEA는 국내외의 수많은 기술자들과 학계에 계신 교수님들의 노력과 협조로 탄생된 것입니다.이제부터 midas FEA를 사용하실 여러분의 성공적인 성과를 기대하며, 사용상 불편한 점이나 개선사항이 있으면 연락 바랍니다. + +끝으로 midas FEA를 개발하는 동안 참여해 주신 여러분께 감사 드리며, 특히 헌신적인 희생을 감내하여 주신 개발자와 가족 여러분께 지면을 빌어 감사의 말을 전합니다. + + + +midas FEA의 사용자지침서는 다음과 같이 2권의 책자와 On-line Manual로 구성되어 있습니다. + +제 1권, Getting Started & Tutorials + +프로그램의 개요와 기본 예제 따라하기 + +제 2권, Analysis & Algorithm + +수치해석모델과 요소 및 해석기능에 대한 해설 + +On-line Manual + +프로그램에 내장되어 있으며 각 기능에 대한 자세한 사용법과 각 입력항목에 대한 설명 + +midas FEA의 특성과 기능을 효과적으로 이해하고 습득하기 위해서는 다음과 같은 순서에 따라 지침서에 수록된 내용을 먼저 이해한 후에 프로그램을 사용하는 것이 바람직합니다. + +1. midas FEA의 해석 기능에 대한 해설 내용을 포함하고 있는 제 2권을 보시기 바랍니다. + +제 2권에 기술된 내용들은 midas FEA를 이용하여 유한요소해석과 재료 및 기하 비선형 해석을 수행하는데 필요한 기초적 고려사항과 해석과정에서 기본적으로 숙지해야 하는 내용들을 포함하고 있습니다. 실제로 구조해석 이론과 사용 프로그램에 대한 이해가 부족한 상태에서 구조해석을 수행할 경우,오류가 포함될 확률이 90%를 상회하는 것으로 외국 저널 등에 심각하게 보고되고 있습니다. + +2. 제 1권의 “설치하기”부분을 보시고 안내된 절차에 따라 midas FEA를 설치하십시오. + +그리고 midas FEA를 사용하는데 필요한 기본개념을 포함하고 있는 제 1권의 나머지 부분들을 읽어보시기 바랍니다. + +제 1권에는 midas FEA의 전체적인 개념도 및 효과적인 운용을 위해 반드시 숙지해야 할 GUI 환경에대한 사용법과 “모델링하기”, “해석”, “결과분석” 등 실제 해석 작업에 반드시 필요한 기능들에 대한설명이 포함되어 있습니다. + +참고로 각 기능에 대한 자세한 사용법과 각 입력항목에 대한 설명은 midas FEA의 Help메뉴에 내장된On-line Manual의 “midas FEA의 기능” 부분에 기술되어 있습니다. + + + +midas FEA의 홈페이지(http://kor.midasuser.com/fea)에는 midas FEA의 주요 해석기능에 대하여 다양한예제를 통하여 타 구조해석 프로그램의 결과와 비교, 검증한 내용을 담은 Verification Examples이 등재되어 있습니다. 검증예제들은 대부분 관련분야 교육 과정에서 다루어지거나 기타 상용 프로그램과 비교 가능한 실무 예제를 중심으로 구성되어 있습니다. 그러므로 가시설분야에 입문하는 초보 사용자들과 실무자들에게 수치해석과 설계에 관한 개념을 파악하고 이해하는 자료로 활용될 수 있습니다. + +이외에도 midas FEA의 홈페이지에는 “프로그램 소개”, “프로그램 특징”, “프로그램 기능 설명”, “실무적용 사례 기술자료” 그리고 “Q&A” 등의 코너를 운용하고 있습니다. 홈페이지에서는 midas FEA의 사용법은 물론 실무를 수행하는데 필요한 관련 기술을 전파하고 기타 유용한 자료를 실시간으로 제공하여,항상 사용자 여러분의 옆에 존재하는 살아있는 매뉴얼의 역할을 수행하고자 최선을 다하겠습니다. + + + + + +# CONTENTS + +# Part 1 Element Library + +# Chapter 1. Structural Elements 03 + +1-1 개요 / 03 +1-2 트러스(Truss) 요소 / 11 +1-3 보(Beam) 요소 / 16 +1-4 평면응력(Plane Stress) 요소 / 32 +1-5 판(Plate) 요소 / 43 +1-6 평면변형(Plane Strain) 요소 / 70 +1-7 축대칭(Axisymmetric) 요소 / 77 +1-8 입체요소(Solid) 요소 / 87 +1-9 스프링(Spring) / 100 +1-10 강체연결(Rigid Link) / 103 + +# Chaper 2. Embedded Reinforcements ·········· 109 + +2-1 개요 / 109 +2-2 철근의 형태 / 110 +2-3 유한요소 정식화 / 116 +2-4 평면요소에 삽입되는 철근 / 121 +2-5 축대칭요소에 삽입되는 철근 / 126 +2-6 평면응력요소에 삽입되는 철근 / 131 +2-7 입체요소에 삽입되는 철근 / 134 +2-8 판요소에 삽입되는 철근 / 141 +2-9 철근의 프리스트레스 / 146 + + + +# CONTENTS + +# Chapter 3. Interface Elements 153 + +3-1 개요 / 153 +3-2 좌표계 및 상대변위 / 155 +3-3 점계면 요소 / 157 +3-4 선계면 요소 / 159 +3-5 면계면 요소 / 161 +3-6 유한요소 정식화 / 165 +3-7 계면요소 해석결과 / 166 + +# Chapter 4. Geometric Nonlinearity 167 + +4-1 개요 / 167 +4-2 트러스(Truss) 요소 / 172 +4-3 평면응력요소(Plane Stress) 요소 / 175 +4-4 판(Plate) 요소 / 178 +4-5 평면변형률(Plane Strain) 요소 / 183 +4-6 축대칭(Axisymmetric) 요소 / 184 +4-7 입체요소(Solid) 요소 / 187 diff --git a/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_002.md b/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_002.md new file mode 100644 index 00000000..4b5a35b3 --- /dev/null +++ b/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_002.md @@ -0,0 +1,277 @@ + + +# CONTENTS + +# Part 2 Material Library + +# Chapter 1. Material Models 193 + +1-1 개요 / 193 + +1-2 항복기준 / 205 + +# Chapter 2. Total Strain Crack 225 + +2-1 개요 / 225 + +2-2 기본특성 / 229 + +2-3 재하 및 제하 / 232 + +2-4 균열변형률 변화 / 234 + +2-5 강성 행렬 / 236 + +2-6 압축 모델 / 242 + +2-7 인장 모델 / 248 + +2-8 전단 모델 / 258 + +2-9 횡방향 영향 / 261 + +# Chapter 3. Interface Nonlinearities 269 + +3-1 개요 / 269 + +3-2 이산균열 / 271 + +3-3 균열팽창 / 277 + +3-4 부착슬립 / 287 + +3-5 쿨롱마찰 / 290 + +3-6 복합파괴 모델 / 294 + + + +# CONTENTS + +# Part 3 General Algorithms + +# Chapter 1. Load and Boundary ··· 305 + +1-1 자유도 구속조건 / 305 + +1-2 경사 지지조건 / 308 + +1-3 구속 방정식 / 309 + +1-4 절점하중 / 312 + +1-5 요소압력하중 / 313 + +1-6 체적력 / 315 + +1-7 강제변위 / 316 + +# Chapter 2. Equation Solver 317 + +2-1 개요 / 317 + +2-2 직접해법 / 318 + +2-3 반복해법 / 322 + +2-4 특징 / 324 + +# Chapter 3. Iteration Methods 325 + +3-1 개요 / 325 + +3-2 초기강성법 / 328 + +3-3 뉴튼 랩슨법 / 329 + +3-4 수정 뉴튼 랩슨법 / 330 + +3-5 호장법 / 331 + +3-6 수렴조건 / 335 + +3-7 자동전환기능 / 336 + + + +# CONTENTS + +# Part 4. Linear Analysis + +# Chapter 1. Linear Static Analysis 339 + +1-1 개요 / 339 +1-2 비선형요소의 선형정적해석/ 340 + +# Chapter 2. Modal Analysis · 341 + +2-1 개요 / 341 +2-2 Lanczos 반복법 / 345 +2-3 Subspace 반복법 / 347 +2-4 관련기능 / 349 + +# Chapter 3. Time History Analysis · 353 + +3-1 개요 / 353 +3-2 모드중첩법 / 354 +3-3 직접적분법 / 356 +3-4 감쇠 / 359 +3-5 주의사항 / 361 + +# Chapter 4. Response Spectrum Analysis ······ 363 + +4-1 개요 / 363 +4-2 스펙트럼 함수 / 369 + +# Chapter 5. Linear Buckling Analysis ············ 371 + +5-1 개요 / 371 +5-2 기하강성 / 374 +5-3 임계하중계수 추출방법 / 381 +5-4 관련 기능 / 382 + + + +# CONTENTS + +# Part 5. Construction Stage Analysis + +Chapter 1. Construction Stage Analysis········· 385 + +1-1 개요 / 385 +1-2 시공단계의 구성 / 387 +1-3 요소의 생성 및 제거 / 391 +1-4 하중의 재하 및 제거와 적용 / 392 +1-5 경계조건의 추가 및 제거 / 394 + +# Part 6. Potential Flow Analysis + +Chapter 1. General Heat Transfer Analysis ··· 397 + +1-1 개요 / 397 +1-2 열전달 방정식 / 398 +1-3 요소 / 403 +1-4 하중과 경계조건 / 406 +1-5 단계별 열전달 해석과 결과 / 410 + +Chapter 2. Heat of Hydration Analysis ········· 413 + +2-1 개요 / 413 +2-2 열전달 해석 / 414 +2-3 열응력 해석 / 416 +2-4 시공단계를 고려한 수화열 해석 / 420 +2-5 시간의존적 재질특성 / 421 +2-6 수화열 해석 결과 / 444 + + + +# CONTENTS + +# Part 7. Contact Analysis + +Chapter 1. Static Contact Analysis · 447 + +1-1 개요 / 447 +1-2 접촉 검색 / 448 +1-3 기능 및 결과 / 456 + +# Part 8. Fatigue Analysis + +Chapter 1. Fatigue Analysis 461 + +1-1 개요 / 461 +1-2 반복하중 / 462 +1-3 평균응력의 영향 / 465 +1-4 수정계수 / 467 +1-5 레인플로 집계 / 474 +1-6 피로해석의 단계 / 477 + + + +# CONTENTS + +# Part 9. CFD(Computational Fluid Dynamic) + +# Analysis + +Chapter 1. CFD Analysis 481 + +1-1 개요 / 481 +1-2 RANS 방정식과 난류모델 / 482 +1-3 공간이산화 / 485 +1-4 정상유동 / 487 +1-5 비정상유동 / 488 +1-6 수치적 안정성 / 490 +1-7 전산유체 해석결과 /491 + + + +# Analysis and Algorithm Manual + +# Part 1 Element Library + +Chapter 1. Structural Elements + +Chapter 2. Embedded Reinforcements + +Chapter 3. Interface Elements + +Chapter 4. Geometric Nonlinearity + + + + + +# Chapter 1. Structural Elements + +# 1-1 개요 + +구조물의 모델링과 유한요소를 이해하는데 필요한 공간상의 좌표계는 다음과 같다. + +- 전체좌표계 (GCS : Global Coordinate System) +구조물 전체를 동일한 하나의 기준으로 표현하는 좌표계이다. 오른손 법칙에 따라 X,Y,Z (대문자) 축으로 표현한다. +- 요소좌표계 (ECS: Element Coordinate System) +각각의 요소를 기준으로 독립적으로 설정되는 좌표계이다. 각 요소의 x,y,z (소문자) 축으로 표현한다. + +\- 기타 좌표계 + +절점좌표계(NCS: Nodal Coordinate System), 출력좌표계(OCS: Output Coordinate System), 재료좌표계(MCS: Material Coordinate System) 등이 있다. + +![](images/page-019_258ab7ca72d297f5ce253777162e857f3475aff76812b71fd099fb8c77f13647.jpg) + +
+text_image + +MCS +2 +3 +1 +z +y +ECS +x +z +y +GCS +X +
+ +그림 1.1.1 여러가지 좌표계 + + + +midas FEA에서 사용 가능한 유한요소의 종류는 다음과 같다. + +- 트러스요소(truss element) +• 보요소(beam element) +• 평면응력요소(plane stress element) +• 판요소(plate element) +• 평면변형요소(plane strain element) +• 축대칭요소(axisymmetric element) +- 입체요소(solid element) + +\- 탄성연결(elastic link)/ 절점스프링(point spring)/ 절점감쇄(point damping) / 행렬스프링(matrix spring)/ 강체연결(rigid link) 요소 + +유한요소는 기본적으로 요소 종류와 연결 절점번호로 정의하며, 절점의 연결 순서에 따라 요소좌표계가 결정된다. 각각의 요소에는 모양, 크기, 재질(material) 등을 규정하는 데이터가 할당되며, 요소 종류별로 필요한 데이터는 다음과 같다. + +
요소필요 데이터
트러스요소단면적, 재질
보요소단면성질, 재질
평면응력요소두께, 재질
판요소두께, 재질, 재료좌표계
평면변형요소두께, 재질
축대칭요소재질
입체요소재질, 재료좌표계
diff --git a/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_003.md b/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_003.md new file mode 100644 index 00000000..5d0a2a98 --- /dev/null +++ b/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_003.md @@ -0,0 +1,211 @@ + + +각 요소별로 사용 가능한 해석 종류는 다음과 같다. + +'✓' 표시는 사용 가능함을 의미한다. + +
요소점류요소차수선형해석비선형해석포텐설유동해석
1차2차선형정적해석고유치해석선형좌점해석시간이력해석지시응답스페트럼해석지시재료비진형해석지시기하비신형해석지수화열해석지수일반열진달해석지점촉해석지점리로해석
트러스√√√√√√√√√√
보√√√√√√√
평면응력√√√√√√√√√√√√
판√√√√√√√√√√√√
평면변형√√√√√√√√√√
축대칭√√√√√√√
입체√√√√√√√√√√√√√
+ +\* 전산유체해석에서는 별도의 전용 판요소 사용. + + + +각 요소별로 사용 가능한 하중조건은 다음과 같다. + +
요소종류정적 하중
체적력cancel압력하중cancel모서리하중cancel보요소하중프리스트레스cancel온도하중cancel온도구배하중
트러스요소√√√
보요소√√√√√
평면응력요소√√√√√
판요소√√√√√√
평면변형요소√√√√√
축대칭요소√√√√
입체요소√√√√
+ + + +각 재료별로 가능한 해석 종류는 다음과 같다. + +
량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량선형해석비선형해석
직업전용량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량Load량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량 Load량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량总额고유치해석직업전용량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량직업전용량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량Load량량량량Load량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량직업전용력량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량력량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량직업전용賦량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량額량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량량
Elastic√√√√√√√
Multi-Elastic√√
Rankine√√
Tresca√√
Von Mises√√
Drucker Prager√√
Mohr Clulomb√√
Total Strain Crack√√
User Supplied√√
Creep/Shrinkage $\checkmark^*$
+ +(\* 선형시공단계해석, 수화열해석) + + + +각 재료별로 사용 가능한 요소종류는 다음과 같다. + +
지료종류요소종류
트러스요소보요소평면응력요소판요소평면변형요소축대칭요소임체요소
Elastic√√√√√√√
Multi-Elastic√
Rankine√√√√√√
Tresca√√√√√√
Von Mises√√√√√√
Drucker Prager√√√√√√
Mohr Clulomb√√√√√√
Total Strain Crack√√√√√√
User Supplied√√√√√√
Creep/Shrinkage√√√√√√√
+ + + +각 요소별 정적 해석 결과로 출력되는 데이터는 다음과 같다. + +'✓' 표시는 데이터가 출력됨을 의미한다. + +
요소종류요소결과
응력변형률요소내력기본출력좌표계(좌표계 변환)
트러스요소√√ECS (no)
보요소√ECS (no)
평면응력요소√√ECS (yes)
판요소√√√ECS (yes)
평면변형요소√√GCS (yes)
축대칭요소√√GCS (yes)
입체요소√√GCS (yes)
+ +일반적으로 트러스/보 요소는 요소 크기의 영향을 크게 받지 않지만, 평면 또는 입체 요소의 경우에는 요소의 크기와 분포에 따라 결과값에 큰 영향을 받게 된다. 일반적으로 요소의 세분화가 필요한 부위는 다음과 같다. + +- 기하학적 불연속 부위 또는 개구부 주위 +- 하중의 변화가 심한 부위 또는 집중 하중이 작용하는 부위 +- 단면적/두께 또는 재료의 성질이 불연속적인 부위 +• 정밀한 응력/내력 결과가 필요한 부위 + +효과적인 수치해석 모델을 만들기 위해서는 요소의 크기뿐만 아니라 요소의 형상 및 연결 상태와 함께 다음과 같은 사항을 고려하는 것이 좋다. + +- 인접 요소간의 크기 차이가 1/2 이하가 되도록 한다. +• 응력을 구하고자 할 경우에는 4절점 평면요소와 8절점 입체요소를 사용한 + + + +다. 3절점 평면요소와 4절점 입체요소의 경우에는 고차요소를 사용한다. + +- 평면요소의 경우 정다각형 형태를 유지하도록 한다. 사각형의 경우 모서리 각도는 45° \~ 135°를 유지하며, 삼각형의 경우에는 30° \~ 150°가 되도록 한다. +4절점 평면 요소의 경우 절점이 동일 평면상에 존재하도록 한다. +- 회전자유도에 대한 강성을 가지고 있지 않은 요소(트러스/평면응력/입체요소)들이 절점을 공유하는 경우에는 특이성 오류(singular error)가 발생할 수 있다. midas FEA에서는 이러한 경우 해당절점의 회전자유도를 자동으로 구속시키는 기능이 있다. + +유한요소법에서 개별 요소의 특성은 요소좌표계에서의 요소강성 $K^{e}$ 로 표현되며, 선형해석에 사용되는 강성행렬 $K^{e}$ 는 다음과 같은 형태로 표현된다 + +$$ +\mathbf {K} ^ {e} = \int_ {V _ {e}} \mathbf {B} ^ {T} \mathbf {D} \mathbf {B} d V \tag {1.1.1} +$$ + +여기서, 행렬 B 는 형상함수 N 의 미분값으로 이루지며, D 는 응력과 변형률 관계를 나타내는 행렬이다. N 은 절점별 형상함수 $N_{i}$ 로 구성되고, 무차원 특성좌표계 (natural coordinate) $\xi - \eta - \zeta$ 에서 정의한다. + + + +# 1-2 트러스요소 + +# 1-2-1 개요 + +트러스요소는 2개의 절점에 의해 정의되는 “Uniaxial Tension-Compression 3DLine Element”이다. 트러스요소는 일반적으로 공간트러스(space truss) 또는 대각부재(diagonal brace) 등을 모델링하는데 사용되며, 정적(선형/비선형) 해석 및 동적해석에 모두 사용할 수 있다. 트러스요소는 축방향(axial) 힘만을 전달하며, 일정한초기간격(gap/hook distance)을 가진 인장전담(tension-only) 또는 압축전담(compression-only) 특성을 부여할 수 있다. 트러스요소의 변형을 정의하는 응력과변형률은 다음과 같다. + +$$ +\boldsymbol {\sigma} = \left\{\sigma_ {x x} \right\}, \quad \boldsymbol {\varepsilon} = \left\{\varepsilon_ {x x} \right\} +$$ + +(축방향 응력과 변형률) + +그림 1.2.1과 같이 트러스요소는 축방향에 대한 자유도만을 갖기 때문에 요소좌표계의 x 축 방향만 의미를 가지고 있다. 요소좌표계 x 축 방향은 절점 1에서 절점 2 방향을 향한다. + +![](images/page-027_d00fcee6db61deae3e3e3f55ba32035767fa0b3f03ba804280661190b41f585b.jpg) + +
+text_image + +ECS x-axis +σₓₓ, εₓₓ +2 +1 +σₓₓ, εₓₓ +
+ +그림 1.2.1 트러스요소의 좌표계와 응력/변형률 + + + +# 1-2-2. 유한요소 정식화 + +트러스요소의 단면적은 전체길이에서 동일하다고 가정한다. 트러스요소는 요소좌표계에서 x 방향 이동변위(translation) u 만을 갖는다. + +$$ +\mathbf {u} _ {i} = \left\{u _ {i} \right\} \tag {1.2.1} +$$ + +임의의 좌표 x 와 이동변위 u 는 형상함수를 이용하여 다음과 같이 나타낸다. + +$$ +x = \sum_ {i = 1} ^ {2} N _ {i} x _ {i}, u = \sum_ {i = 1} ^ {2} N _ {i} u _ {i} \tag {1.2.2} +$$ + +$$ +N _ {1} = \frac {1 - \xi}{2}, \quad N _ {2} = \frac {1 + \xi}{2} \quad (- 1 \leq \xi \leq 1) \tag {1.2.3} +$$ + +절점 변위와 변형률의 관계는 Bi 에 의하여 식 (1.2.4)와 같이 나타낼 수 있다. + +$$ +\boldsymbol {\varepsilon} = \sum_ {i = 1} ^ {2} \mathbf {B} _ {i} \mathbf {u} _ {i} \tag {1.2.4} +$$ + +행렬 Bi 는 형상함수의 미분값으로 다음과 같이 표현된다. + +$$ +\mathbf {B} _ {i} = \left\{\frac {\partial N _ {i}}{\partial x} \right\} \tag {1.2.5} +$$ + +행렬 Bi 를 이용하여 축방향 변형에 관계된 요소강성행렬을 표현하면 다음과 같다. + +$$ +\mathbf {K} _ {i j} = \int_ {L _ {e}} \mathbf {B} _ {i} ^ {T} \mathbf {D} \mathbf {B} _ {j} d L, \quad \mathbf {D} = A \{E \} \tag {1.2.6} +$$ + +여기서, + +$$ +\begin{array}{l} A: \text { 단면적 } \\ L _ {e} \quad : \text { 요소 길이 } \\ \end{array} +$$ + +식 (1.2.6)을 정리하여 트러스요소의 강성행렬을 다음과 같이 계산할 수 있다. + +$$ +\mathbf {K} = \frac {E A}{L _ {e}} \left[ \begin{array}{l l} 1 & - 1 \\ - 1 & 1 \end{array} \right] \tag {1.2.7} +$$ + + + +# 1-2-3. 하중과 질량 + +트러스요소에 적용되는 하중은 체적력(body force), 프리스트레스하중(prestress), 온도하중(thermal) 등이 있다. 체적력은 요소의 자중이나 관성력을 표현하고자 하는 하중이다. 프리스트레스 하중은 트러스요소에 내력으로 인장력을 주고자 하는 경우에 사용한다. 온도하중은 절점온도, 요소온도 하중에 의한 열변형을 반영하고자 할 때 사용한다. 프리스트레스와 온도하중에 의한 요소좌표계의 하중벡터는 아래와 같다. + +\- 체적력 + +$$ +\mathbf {F} _ {i} = A \int_ {L _ {e}} N _ {i} \left\{ \begin{array}{l} \omega_ {x} \\ \omega_ {y} \\ \omega_ {z} \end{array} \right\} d L \tag {1.2.8} +$$ + +여기서, + +$$ +\omega_ {x}, \omega_ {y}, \omega_ {z} \quad : \text { 단위 체적당 자중(방향별) } +$$ + +\- 프리스트레스 + +$$ +\mathbf {F} _ {i} = - \int_ {L _ {e}} \mathbf {B} _ {i} ^ {T} P d L \tag {1.2.9} +$$ + +여기서, + +$$ +P: \text { 축방향 프리스트레스 (힘) } +$$ + +\- 온도 하중 + +$$ +\mathbf {F} _ {i} = \int_ {L _ {e}} \mathbf {B} _ {i} ^ {T} E A \alpha \Delta T d L \tag {1.2.10} +$$ + +여기서, + +$$ +\alpha : \text { 열팽창계수 } +$$ + +$$ +\Delta T \quad : \text { 온도변화 } +$$ + + + +트러스요소의 집중질량(lumped mass)과 분포질량(consistent mass) 행렬은 아래와 같다. 트러스요소의 질량은 x,y,z 의 이동변위에 대해서만 구성된다. + +\- 집중질량 + +$$ +\mathbf {M} = \frac {\rho A L _ {e}}{2} \left[ \begin{array}{c c c c c c} 1 & & & & & \\ 0 & 1 & & & \text { symm. } & \\ 0 & 0 & 1 & & & \\ 0 & 0 & 0 & 1 & & \\ 0 & 0 & 0 & 0 & 1 & \\ 0 & 0 & 0 & 0 & 0 & 1 \end{array} \right] \tag {1.2.11} +$$ + +여기서, + +ρ : 밀도 + +\- 분포질량 + +$$ +\mathbf {M} = \frac {\rho A L _ {e}}{6} \left[ \begin{array}{c c c c c c} 2 & & & & & \\ 0 & 2 & & & \text {symm.} & \\ 0 & 0 & 2 & & & \\ 1 & 0 & 0 & 2 & & \\ 0 & 1 & 0 & 0 & 2 & \\ 0 & 0 & 1 & 0 & 0 & 2 \end{array} \right] \tag {1.2.12} +$$ diff --git a/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_004.md b/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_004.md new file mode 100644 index 00000000..0f4658c5 --- /dev/null +++ b/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_004.md @@ -0,0 +1,365 @@ + + +# 1-2-4. 요소결과 + +트러스요소의 해석결과로는 절점 1, 2 위치에서 요소내력(element force)과 응력을출력하고, 방향은 요소좌표계를 따른다. 출력되는 요소내력은 축방향의 힘( ) N A x xx= σ 이고, 그림 1.2.2와 같이 인장응력이 작용할 경우에 ‘+’ 부호를 갖는다.일반적으로 양 끝단에서의 요소내력은 동일하지만 자중이 입력된 경우에는 달라질수도 있다. + +![](images/page-031_44d5a7c712ab2d54adf66903d276612ea70420d5297f4cf2a132a2f27796c3a7.jpg) + +
+text_image + +ECS x-axis +Nₓ,σₓ +J - End +I - End +Nₓ,σₓₓ +
+ +그림 1.2.2 트러스요소의 결과 출력위치 및 성분 + + + +# 1-3 보요소 + +# 1-3-1 개요 + +보요소는 2개의 절점에 의해 정의되는 선요소이며, 단면의 치수에 비하여 길이가 긴골조부재의 모델링에 주로 이용된다. 보요소는 정적(선형/비선형) 해석 및 동적 해석에 모두 이용할 수 있으며, 축방향(axial) 변형, 휨(bending), 비틀림(torsion), 전단(shear) 변형 등을 고려할 수 있다. 보요소에서 변형을 정의하는 응력, 변형률, 요소내력(element force)은 다음과 같다. + +$$ +\boldsymbol {\sigma} = \left\{\sigma_ {x x} \right\}, \boldsymbol {\varepsilon} = \left\{\varepsilon_ {x x} \right\} +$$ + +(축방향 응력과 변형률) + +$$ +\mathbf {M} = \left\{ \begin{array}{l} M _ {y} \\ M _ {z} \end{array} \right\}, \quad \mathbf {K} = \left\{ \begin{array}{l} \kappa_ {y} \\ \kappa_ {z} \end{array} \right\} +$$ + +(휨모멘트와 곡률) + +$$ +\mathbf {T} = \left\{M _ {x} \right\}, \quad \boldsymbol {\varphi} = \left\{\phi_ {x} \right\} +$$ + +(비틀림모멘트와 비틀림) + +$$ +\mathbf {Q} = \left\{ \begin{array}{l} Q _ {y} \\ Q _ {z} \end{array} \right\}, \quad \boldsymbol {\gamma} = \left\{ \begin{array}{l} \gamma_ {x y} \\ \gamma_ {z x} \end{array} \right\} +$$ + +(전단력과 전단변형률) + +요소내력과 응력에 대한 부호규약은 그림 1.3.1과 같고, 화살표 방향이 ‘+’ 부호를의미한다. 길이에 대한 단면의 폭 또는 높이비가 대략 1/5 보다 커질 경우에는 전단변형에 의한 영향이 커지게 되므로, 보요소를 사용하지 않고 판요소나 입체요소를 사용하는 것이 바람직하다. + + + +![](images/page-033_c0f6316ee0c9c54b5cb7b5c41133803694a69a98abf266eaafea6d58a0357301.jpg) + +
+text_image + +ECS x-axis +ECS y-axis +ECS z-axis +1 +2 +Qz,γzx +Mz,κz +σxx,εxx +My,κy +My,φx +My,φx +Qy,γyz +Qy,γyz +My,κy +My,φx +σxx,εxx +Mx,φx +Mz,κz +Qz,γzx +
+ +그림 1.3.1 보요소의 좌표계와 응력/변형률/내력 + +보요소 좌표계는 요소내력 또는 응력의 출력 기준이 되고, 특히 보요소의 전단강성과휨강성의 방향을 정하는 기준이 된다. 요소좌표계의 x 축은 절점 1에서 절점 2로 진행하는 방향이 되고, y z, 축의 방향은 그림 1.3.1과 같고, 그림 1.3.2 \~ 1.3.4와 같이3가지 방법을 사용하여 지정한다. 요소좌표계 z 축을 정하면 y 축은 오른손법칙에따라 자동적으로 정해진다. + +그림 1.3.2는 요소좌표계 z 축과 전체좌표계와의 각도인 “Beta Angle” β를 입력하여 z 축을 입력하는 방법이다. 요소좌표계 x 축이 전체좌표계 Z 축과 평행하면β이 전체좌표계 Z 축과 평행하지 않으면 β 는 전체좌표계 Z 축과 요소좌표계 x − z평면이 이루는 수직각도가 된다. + +그림 1.3.3은 요소좌표계 x− z 평면상의 임의의 절점을 입력하여 요소좌표계를 지정하는 방법이다. 입력되는 임의의 절점은 요소좌표계 x 축 선상에 있지 않아야 한다.그림 1.3.4는 요소좌표계 x − z 평면상에 임의의 벡터를 입력하여 요소좌표계를 지정하는 방법이다. 입력되는 임의의 벡터는 요소좌표계 x 축과 평행하지 않아야 한다. + + + +![](images/page-034_0819fd808d1cf870d3db1f248e1606efd35846377c0a93173b46824aec1cc20a.jpg) + +
+text_image + +Y' +z +x +β +Z' +y +X' +
+ +X′: axis passing through node N1 and parallel with the global X-axis +Y′: axis passing through node N1 and parallel with the global Y-axis +Z′: axis passing through node N1 and parallel with the global Z-axis + +(a) 수직부재인 경우 (요소좌표계 x축이 전체좌표계 Z축과 평행할 경우) +![](images/page-034_cba36d96decf719618eb408c5b8a3dcc1dd86fd07a6dcf5e0d0c91442077a281.jpg) + +
+text_image + +z +β +y +x +Y +z' +
+ +![](images/page-034_a67da1b53a0521590c75a2886841420254feb276b34b60df3c781b7a69cdfe27.jpg) + +
+text_image + +Z' +z +x +X' +y +Y' +β +
+ +![](images/page-034_4a284522372b8626a91421f7dbb7cd5116e82cfb6af163700acfb707f21f2839.jpg) +(b) 수평 또는 대각부재인 경우 (요소좌표계 x축이 전체좌표계 Z축과 평행하지 않을 경우) +그림 1.3.2 Beta Angle을 이용한 요소좌표계 정의 + + + +![](images/page-035_3c453cd700412e1d03a95d9af34dfd4a08947a5accac9df225c0a5f52e298ffa.jpg) + +
+text_image + +Local z +x-z plane +Vz +3 +2 +Local x +Local y +1 +Z +Y +X +GCS +
+ +그림 1.3.3 Node를 사용한 요소좌표계 정의 + +![](images/page-035_b1c0ae61f3073d01b141f726258867b07ff542ec4934d5151acd21845008ca3c.jpg) + +
+text_image + +Local z +x-z plane +V(α, β, γ) +Local x +Local y +1 +2 +Z +Y +X +GCS +
+ +그림 1.3.4 K-Vector 를 사용한 요소좌표계의 정의 + + + +# 1-3-2 유한요소 정식화 + +보요소의 단면 형상과 크기는 전체 길이에서 균일하다고 가정한다. 보요소는 요소좌표계에서 3개의 이동변위(translation)와 3개의 회전변위(rotation)를 모두 갖는다. 축방향 변형에 대한 강성은 트러스요소와 동일한 방법으로 계산하며, 비틀림 강성 또한유사한 과정으로 계산할 수 있다. 그리고 휨과 전단강성에 대해서는 Timoshenko 보이론 또는 Euler 보이론을 사용한다 + +축방향 변형에 대해서는 트러스요소와 동일한 방법을 적용한다. + +$$ +\mathbf {u} _ {i} = \left\{u _ {i} \right\} \tag {1.3.1} +$$ + +$$ +\mathbf {K} _ {\text { axial }} = \frac {E A}{L _ {e}} \left[ \begin{array}{l l} 1 & - 1 \\ - 1 & 1 \end{array} \right] \tag {1.3.2} +$$ + +여기서, + +$$ +\begin{array}{l} A \quad : \text { 단면적 } \\ L _ {e} \quad : \text { 요소 길이 } \\ \end{array} +$$ + +비틀림에 대해서도 유사한 과정을 거쳐 다음과 같은 강성행렬을 얻을 수 있다. + +$$ +\mathbf {u} _ {i} = \left\{\theta_ {x i} \right\} \tag {1.3.3} +$$ + +$$ +\mathbf {K} _ {\text { torsional }} = \frac {G I _ {x}}{L _ {e}} \left[ \begin{array}{l l} 1 & - 1 \\ - 1 & 1 \end{array} \right] \tag {1.3.4} +$$ + +여기서, + +$$ +\begin{array}{l} I _ {x} \quad : \text { 비틀림강성 (torsional resistance) } \\ L _ {e} \quad : \text { 요소 길이 } \\ \end{array} +$$ + +전단과 휨 강성은 서로 연관되어 있으므로 연관강성을 구성한다. 먼저 Timoshenko보이론에 기반하여 전단변형을 고려한 강성에 대하여 설명한다. 요소좌표계의 x − z평면 상에서 전단변형과 휨을 표현하는 변위는 z 방향 이동변위 w 와 y 축에 대한회전이다. + + + +$$ +\mathbf {u} _ {i} = \left\{w _ {i} \quad \theta_ {y i} \right\} ^ {T} \tag {1.3.5} +$$ + +요소의 축방향으로 임의의 좌표 x와 이동변위 w는 다음과 같이 나타낼 수 있다. + +$$ +x = \sum_ {i = 1} ^ {2} N _ {i} x _ {i}, \quad w = \sum_ {i = 1} ^ {2} N _ {i} w _ {i} \tag {1.3.6} +$$ + +요소좌표계의 y 축에 대한 회전은 다음과 같이 2차로 표현된다. + +$$ +\theta_ {y} = \sum_ {i = 1} ^ {2} N _ {i} \theta_ {y i} + P _ {3} \Delta \theta_ {y 3} \tag {1.3.7} +$$ + +$$ +N _ {1} = \frac {1 - \xi}{2}, \quad N _ {2} = \frac {1 + \xi}{2}, \quad \mathrm{P} _ {3} = 1 - \xi^ {2} \quad (- 1 \leq \xi \leq 1) \tag {1.3.8} +$$ + +요소 중앙의 가상 절점 회전각 $\Delta\theta_{y3}$ 를 구하기 위하여 다음과 같은 가정을 이용한다. + +\- 전단력과 힘모멘트의 평형식을 만족한다. + +$$ +Q _ {z} = - \frac {\partial M _ {y}}{\partial x} \tag {1.3.9} +$$ + +\- 전단력과 휩모멘트의 평형식에 의해 계산되는 평균 전단변형률 $\overline{\gamma}_{zx}$ 는 형상함수로부터 계산되는 전단변형률 $\gamma_{zx}$ 와 다음의 관계를 만족한다. + +$$ +\int_ {L _ {e}} \left(\gamma_ {z x} - \overline {{\gamma}} _ {z x}\right) d L = 0 \tag {1.3.10} +$$ + +위의 가정을 이용하면 중앙 가상 절점의 회전각 $\Delta\theta_{y3}$ 를 다음처럼 표현할 수 있다. + +$$ +\Delta \theta_ {y 3} = \frac {3}{2 L _ {e} (1 + \phi_ {3})} \left[ \begin{array}{l l l l} 1 & - \frac {L _ {e}}{2} & - 1 & - \frac {L _ {e}}{2} \end{array} \right] \left\{ \begin{array}{l} w _ {1} \\ \theta_ {y 1} \\ w _ {2} \\ \theta_ {y 2} \end{array} \right\}, \quad \phi_ {3} = \frac {1 2 E I _ {y}}{G A _ {s z} L _ {e} ^ {2}} \tag {1.3.11} +$$ + +$A_{sz}$ : 유효전단면적 (effective shear area) + +$I_{y}$ : 단면2차 모멘트 (area moment of inertia) + + + +절점 변위와 곡률 κ y 의 관계는 Bbi에 의해 다음과 같이 표현된다. + +$$ +\kappa_ {y} = \sum_ {i = 1} ^ {2} \mathbf {B} _ {b i} \mathbf {u} _ {i} \tag {1.3.12} +$$ + +$$ +\mathbf {B} _ {b i} = \left\{\frac {3 a _ {i}}{2 L _ {e} (1 + \phi_ {3})} \frac {\partial P _ {3}}{\partial x} \quad \frac {3}{4 (1 + \phi_ {3})} \frac {\partial P _ {3}}{\partial x} - \frac {\partial N _ {i}}{\partial x} \right\}, \quad a _ {1} = - 1, \quad a _ {2} = 1 \tag {1.3.13} +$$ + +전단변형은 zxγ 를 이용하며, zxγ 와 절점 변위와의 관계는 Bsi 에 의해 다음과 같이나타낼 수 있다. + +$$ +\overline {{\gamma}} _ {z x} = \sum_ {i = 1} ^ {2} \mathbf {B} _ {s i} \mathbf {u} _ {i} \tag {1.3.14} +$$ + +$$ +\mathbf {B} _ {s i} = \left\{\frac {a _ {i}}{L _ {e} (1 + \phi_ {3})} \quad \frac {L _ {e}}{2 (1 + \phi_ {3})} \right\}, \quad a _ {1} = - 1, \quad a _ {2} = 1 \tag {1.3.15} +$$ + +따라서 전단과 휨 강성은 다음과 같이 계산할 수 있다. + +$$ +\mathbf {K} _ {i j} = \int_ {L _ {e}} \left(\mathbf {B} _ {b i} ^ {T} \mathbf {D} _ {b} \mathbf {B} _ {b j} + \mathbf {B} _ {s i} ^ {T} \mathbf {D} _ {s} \mathbf {B} _ {s j}\right) d L \tag {1.3.16} +$$ + +$$ +\mathbf {D} _ {b} = I _ {y} \left\{E \right\}, \quad \mathbf {D} _ {s} = A _ {s y} \left\{G \right\} \tag {1.3.17} +$$ + +식 (1.3.16)을 적분하여 정리하면, x − z 평면의 휨과 전단에 대한 강성을 계산할 수있고, 행렬식으로 표현하면 다음과 같다. + +$$ +\mathbf {K} _ {\text {bending}} + \mathbf {K} _ {\text {shear}} = \frac {E I _ {y}}{1 + \phi_ {3}} \left[ \begin{array}{c c c c} \frac {1 2}{L _ {e} ^ {3}} & - \frac {6}{L _ {e} ^ {2}} & - \frac {1 2}{L _ {e} ^ {3}} & - \frac {6}{L _ {e} ^ {2}} \\ & \frac {4}{L _ {e}} \left(1 + \frac {\phi_ {3}}{4}\right) & \frac {6}{L _ {e} ^ {2}} & \frac {2}{L _ {e}} \left(1 - \frac {\phi_ {3}}{2}\right) \\ & & \frac {1 2}{L _ {e} ^ {3}} & \frac {6}{L _ {e} ^ {2}} \\ \text {Symm.} & & & \frac {4}{L} \left(1 + \frac {\phi_ {3}}{4}\right) \end{array} \right] \tag {1.3.18} +$$ + +요소좌표계의 x − y 평면에 대한 휨과 전단 강성은 동일한 방법으로 계산할 수 있다.전단변형을 고려하지 않는 Euler 보이론에 의한 강성을 계산하려면, 전단변형률의 가 + + + +정인 식 (1.3.10) 대신 다음의 식을 이용하여 정식화한다. + +$$ +\int_ {L _ {e}} \gamma_ {z x} d L = 0 \tag {1.3.19} +$$ + +따라서, 식 (1.3.18)에서 $\phi_{3}=0$ 으로 두면 힘 강성을 얻을 수 있다. + +# 1-3-3. 하중과 질량 + +보요소에 적용되는 하중은 보요소 하중(beam load), 체적력(body force), 프리스트레스하중(prestress load), 온도하중(thermal load) 등이 있다. 체적력은 요소의 자중이나 관성력을 표현하는 하중이고, 보요소 하중은 보요소의 길이에 따라 작용하는 집중 또는 분포하중이다. 보요소 하중에는 집중하중과 분포하중이 있고, 요소좌표축과 전체좌표축 방향으로 입력할 수 있다. 프리스트레스 하중은 보요소에 내력으로 인정 변형을 주고자 하는 경우에 사용한다. 프리스트레스 하중은 요소 좌표계의 길이 방향으로 작용하며, 요소의 두 절점이 구속된 경우에 입력한 프리스트레스 하중이 요소내력과 같도록 하는 하중을 도입한다. 보요소의 온도하중에는 길이방향 열 변형하중과 온도구배(temperature gradient)와 같은 휩 하중이 있다. + +# • 보요소 하중 + +보요소의 두 절점사이에 입력되는 집중 또는 분포하중이다. 하중 재하 구간은 사용자가 임의로 입력할 수 있고, 해석 시에는 등가 절점하중으로 변환하여 사용한다. 보요소 하중은 요소내력 계산시에 반영하며, 요소내력은 부재 내부의 지정된 위치(I-end, 1/4, 1/2, 3/4, J-end)에서 정확하게 계산된다. 그림 1.3.5는 보요소 하중의 입력 예를 나타내고 있다. + + + +Part 1 Element Library +![](images/page-040_01569af719e74e1b00af59dd8da04aadfb71796ceacff756fe3ec2d187de1636.jpg) + +
+text_image + +p₁ p₂ +m₁ m₂ +I J +P M +I J +
+ +(a) 보요소 하중의 종류(분포하중, 집중하중) + +![](images/page-040_4a6b85ff7490b350acfeefc6c726157e13eac77bac415abadf4e51a2e43f4cb6.jpg) + +
+text_image + +W +L +θ +Z +X +W +L +θ +Z +X +
+ +(b) 전체좌표축에 대한 투영(Projection) 여부에 따른 분포하중의 계산 +그림 1.3.5 보요소 하중의 입력 예 diff --git a/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_005.md b/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_005.md new file mode 100644 index 00000000..ac216fd2 --- /dev/null +++ b/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_005.md @@ -0,0 +1,333 @@ + + +\- 체적력 + +$$ +\mathbf {F} _ {i} = A \int_ {L _ {e}} N _ {i} \left\{ \begin{array}{l} \omega_ {x} \\ \omega_ {y} \\ \omega_ {z} \end{array} \right\} d L \tag {1.3,20} +$$ + +여기서, + +$$ +\omega_ {x}, \omega_ {y}, \omega_ {z} \quad : \text { 단위 체적당 자중(방향별) } +$$ + +\- 프리스트레스 + +$$ +\mathbf {F} _ {i} = - \int_ {L _ {e}} \mathbf {B} _ {i} ^ {T} P d L \quad \left(\mathbf {B} _ {i}: \text {트러스요소와 동일}\right) \tag {1.3.21} +$$ + +여기서, + +$$ +P \quad : \text { 축방향 프리스트레스 (힘) } +$$ + +\- 온도하중 + +$$ +\mathbf {F} _ {i} = \int_ {L _ {e}} \mathbf {B} _ {i} ^ {T} E A \alpha \Delta T d L \tag {1.3.22} +$$ + +여기서, + +$$ +\alpha \quad : \text { 열팽창계수 } +$$ + +$$ +\Delta T \quad : \text { 온도변화 } +$$ + + + +보요소의 질량은 집중질량(lumped mass)과 분포질량(consistent mass)을 반영할 수 있다. 집중질량은 x,y,z 의 이동변위에 대해서 구성되고, 트러스요소와 같이 크기 $\left(\rho AL_{e}\right)/2$ 로 방향에 무관하게 동일한 값을 갖는다. 보요소의 분포질량은 이동변위와 회전변위에 대하여 모두 반영된다. + +# - 분포질량 + +$$ +\mathbf {M} = \frac {\rho A L _ {e}}{4 2 0} \left[ \begin{array}{c c c c c c c c c c c c} 1 4 0 & & & & & & & & & & \\ 0 & 1 5 6 & & & & & & & & & \text {symm.} \\ 0 & 0 & 1 5 6 & & & & & & & & \\ 0 & 0 & 0 & 1 4 0 \frac {J}{A} & & & & & & & \\ 0 & 0 & - 2 2 L _ {e} & 0 & 4 L _ {e} ^ {2} & & & & & & \\ 0 & 2 2 L _ {e} & 0 & 0 & 0 & 4 L _ {e} ^ {2} & & & & & \\ 7 0 & 0 & 0 & 0 & 0 & 0 & 1 4 0 & & & & \\ 0 & 5 4 & 0 & 0 & 0 & 1 3 L _ {e} & 0 & 1 5 6 & & & \\ 0 & 0 & 5 4 & 0 & - 1 3 L _ {e} & 0 & 0 & 0 & 1 5 6 & & \\ 0 & 0 & 0 & 7 0 \frac {J}{A} & 0 & 0 & 0 & 0 & 0 & 1 4 0 \frac {J}{A} & & \\ 0 & 0 & 1 3 L _ {e} & 0 & - 3 L _ {e} ^ {2} & 0 & 0 & 0 & 2 2 L _ {e} & 0 & 4 L _ {e} ^ {2} \\ 0 & - 1 3 L _ {e} & 0 & 0 & 0 & - 3 L _ {e} ^ {2} & 0 & - 2 2 L _ {e} & 0 & 0 & 0 & 4 L _ {e} ^ {2} \end{array} \right] \tag {1.3.23} +$$ + +여기서, + +J : 극관성 모멘트(polar moment of inertia) + + + +# 1-3-4 경계조건 + +보요소의 내부 경계조건으로는 단부 해제조건(end release)과 옵셋(offset)이 있다.부재의 양단부가 핀접합 또는 슬롯홀(slot hole) 등에 의해 연결될 경우에 그림 1.3.6과 같은 단부 해제조건을 사용한다. 단부 해제조건은 요소를 구성하는 모든 자유도에대하여 입력이 가능하다. 단부 해제가 수행되는 방향은 요소좌표계를 따르므로, 전체좌표계에 대한 강성의 연결해제를 입력할 경우에는 요소좌표계와의 관계에 주의하여야 한다. 또한 요소의 단부 해제에 따른 강성의 변화가 특이성오류를 발생시킬 수 있으므로, 전체구조물에 대한 충분한 고려가 필요하다. + +![](images/page-043_4d9cf0badba6b44d133c2756dbf402863c17e98dfabc6cf78f991fe2f56013de.jpg) + +
+text_image + +rotational d.o.f. released +girder +beam +
+ +(a) 핀접합의 경우 + +![](images/page-043_b268c15508015868b130dae396fa2ad32205bed28cde91a8c1be5e0fd31589e4.jpg) + +
+text_image + +girder +axial direction +d.o.f. released +slot hole +column +
+ +(b) 슬롯홀접합의 경우 + +![](images/page-043_79745a40c065cb80b824051bada35d5e8a94d3c763e8841906f0a82e93a4cfbe.jpg) + +
+text_image + +rotational d.o.f. released +rigid connection +
+ +(c) 여러 개의 보 요소가 한 절점에 핀접합으로 연결된 경우 + + + +![](images/page-044_fa9720b0eb3b79785a236e28dd68746799aff02baefbfbd565cb7158b52a3b52.jpg) + +
+text_image + +beam +wall +rigid connection +beam element +rigid beam element +for connectivity +all rotational degree +of freedom and +vertical displacement +degree of freedom +released +plane stress or plate element +
+ +(d) 절점자유도가 서로 다른 요소끼리 연결된 경우 +그림 1.3.6 보요소의 단부해제 조건 + +보요소의 중립축이 절점과 격리되어 있는 경우 또는 연결되는 2개의 요소의 중립축이 일치하지 않는 경우에 별도 절점을 생성하지 않고 옵셋(offset)을 사용하여 모델링 할 수 있다. 옵셋은 Local Offset과 Global Offset으로 분류할 수 있다. + +Global Offset은 전체좌표계를 기준으로 보요소의 양 절점에서 옵셋거리를 정의한다.그림 1.3.7의 경우가 이에 해당되며, 강성과 요소하중의 변환은 강체연결과 같은 변환 관계를 갖는다. 부재의 분포하중, 체적력, 질량 등은 옵셋된 절점을 기준으로 계산되고 요소내력 출력도 옵셋된 요소좌표계에 따른다. + + + +![](images/page-045_83c253f888f6e4b32b1f2ebe8485c8e00d3da9383c3129d95a7eacd90547a76d.jpg) +그림 1.3.7 Global Offset + +Local Offset은 그림 1.3.8과 같이 요소의 길이 방향으로 옵셋을 정의한다. 요소의강성을 계산할 때, 축방향 강성과 비틀림 강성에 대해서는 양 절점 사이의 길이가 사용되고, 전단 강성과 휨 강성을 계산할 때는 옵셋을 제외한 길이를 사용한다. + +![](images/page-045_44a21e3fc9f4676d0bc38d7135b8fa9f5dbccc803585e7a7c0a9d6f2dc01c09d.jpg) + +
+text_image + +column member +beam member +column member +Rigid Zone +Rigid Zone +B +A +B +clear length of beam +A +length between nodes (L) +
+ +그림 1.3.8 Local Offset (A, B: Local Offset Distance) + + + +Local Offset을 사용한 요소의 분포하중에 대한 계산은 그림 1.3.9와 같다. 옵셋 구간에 재하되는 하중은 절점상에 전단력만 고려하고, 나머지 구간에 재하된 분포하중은 전단력과 모멘트로 치환하여 고려한다. 체적력은 분포하중과 같은 방법으로 계산한다. 요소내력의 출력위치는 옵셋된 위치에서 출력한다. + +![](images/page-046_b44a70a0f00874ad1967653242fa8ba840bed9d1b91ed11483cec21ff69cb95b.jpg) + +
+text_image + +rigid end offset location at i-th node +rigid end offset location at j-th node +distributed load on beam element +i-th node +L₁ +L₁ (length for shear/bending stiffness calculation) +Lⱼ +j-th node +Story Level +L +zone in which load is converted into +shear force only at i-th node +zone in which load is converted into both +shear and moment +zone in which load is converted into +shear force only at j-th node +V₃ +V₁ +V₂ +V₄ +M₁ +M₂ +locations for member force output(▼) +
+ +L=1.0×R "Panel Zone"is selected for the locations of member force output Li=Z× Ri “Offset Position"is selected for the locations of member force output Li= 1.0 × Ri“Panel Zone"is selected for the locations of member force output Li=ZF×Ri “Offset Position"is selected for the locations of member force output R rigid end offset distance at i-th node R rigid end offset distance at j-th node Z rigid end Offset Factor Vi.V2 shear forces due to distributed load between the offset ends Mi,M2 moments due to distributed load between the offset ends V3.V4 shear forces due to distributed load between the offset ends and the nodal points + +그림 1.3.9 보요소의 분포하중 계산 방법 + + + +# 1-3-5 요소결과 + +보요소의 해석결과로는 요소당 5개 위치에서의 요소내력을 출력하고, 모든 내력은 요소좌표계를 따른다. 요소의 축방향으로 출력위치는 I-End, 1/4, 2/4, 3/4, J-End이며, 부호는 그림 1.3.10에서 화살표 방향을 향할 때 ‘+’ 부호로 한다. 출력되는 요소내력의 종류는 다음과 같다. + +- 축방향 내력 $N_{x}$ +- 전단력 $Q_{y}, Q_{z}$ +• 비틀림모멘트 $M_{x}$ +• 힘모멘트 $M_{y}, M_{z}$ + +![](images/page-047_157f9a49327b73ac9fda7556dbf8fb59c8229a039b0a0d263da7e49b6b530f0b.jpg) + +
+text_image + +ECS x-axis +ECS y-axis +ECS z-axis +1-End +1/4 pt +1/2 pt +3/4 pt +J-End +Mx +Mz +Mx +My +Qy +Qz +Nx +Qz +
+ +그림 1.3.10 보요소의 결과 출력위치 및 성분 + + + +# 1-4 평면응력요소 + +# 1-4-1 개요 + +평면응력요소는 동일 평면상에 위치한 3, 4, 6, 8개의 절점에 의해 정의되는 삼각형혹은 사각형 요소이며, 두께가 균일한 박판(membrane)을 모델링하는데 주로 사용된다. 요소의 두께방향 응력성분은 존재하지 않으며, 두께방향의 변형률은 포아송(Poisson) 효과에 의해 존재하는 것으로 가정한다. 평면응력요소는 면내변형(in-plane deformation)만을 고려할 수 있으며, 정적(선형/비선형) 해석 및 동적 해석에모두 사용할 수 있다. 평면응력요소에서 변형을 정의하는 응력과 변형률은 다음과 같다. + +$$ +\boldsymbol {\sigma} = \left\{ \begin{array}{l} \sigma_ {x x} \\ \sigma_ {y y} \\ \tau_ {x y} \end{array} \right\}, \quad \boldsymbol {\varepsilon} = \left\{ \begin{array}{l} \varepsilon_ {x x} \\ \varepsilon_ {y y} \\ \gamma_ {x y} \end{array} \right\} +$$ + +(면내방향 응력과 변형률) + +응력과 변형률에 대한 부호규약은 그림 1.4.1과 같고, 화살표 방향이 ‘+’를 의미한다. + +![](images/page-048_e971f5a81d5a43516d32e0cbd7895bfd15bf8a68ea0c0041ea8bb90e97fb7801.jpg) + +
+text_image + +z +y +ECS +x +τxy, γxy +σxx, εxx +τxy, γxy +σyy, εyy +
+ +그림 1.4.1 평면응력요소의 응력/변형률 + + + +요소좌표계는 오른손법칙에 준한 x, , y z 축의 직교좌표계를 따르며, 방향은 그림1.4.2와 같이 설정된다. 사각형 요소는 절점 1과 절점 4의 중점에서 절점 2와 절점 3의 중점을 향하는 방향을 x 축 방향으로 하며, 삼각형 요소는 절점 1에서 2를 향하는 방향을 x 축 방향으로 설정한다. + +![](images/page-049_7345ce977e36acbd3d559fed2bb3a0be000d27442a4483a39e473bf679216dfb.jpg) + +
+text_image + +ECS z-axis +3 +ECS y-axis +1 +ECS x-axis +(1→2 direction) +2 +
+ +![](images/page-049_0b5e32feb99400ec9229ead7b0947bab995c575ea0700360d7bcb589612abc85.jpg) + +
+text_image + +ECS z-axis +4 +ECS y-axis +1 +3 +ECS x-axis +2 +
+ + + +![](images/page-050_0107250c2922f939936a0525167b4fcd82a83c328d5c8abc89353277d142f74f.jpg) + +
+text_image + +ECS z-axis +ECS y-axis +ECS x-axis +(1 → 2 direction) +1 +2 +3 +4 +5 +6 +
+ +![](images/page-050_073c8bab400442948daed95fe5e4a8a522520353a139af888e6392cb029d192f.jpg) + +
+text_image + +ECS z-axis +4 +8 +7 +ECS y-axis +1 +3 +6 +ECS x-axis +5 +2 +
+ +그림 1.4.2 평면응력요소의 좌표계 + +평면응력요소의 종류는 연결된 절점 수에 따라 두 가지로 구분할 수 있다. 3절점 삼각형 요소와 4절점 사각형 요소는 1차(linear) 요소이며, 6절점 삼각형 요소와 8절점사각형 요소는 2차(quadratic) 요소이다. 1차 요소의 경우 4절점 사각형 요소는 변위및 응력값의 정확도가 높지만, 3절점 삼각형 요소는 변위에 비해 응력의 정확도가낮은 경향이 있다. 따라서 정밀한 해석결과가 필요한 부위에서는 3절점 삼각형 요소의 사용을 피하는 것이 바람직하다. diff --git a/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_006.md b/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_006.md new file mode 100644 index 00000000..126847cc --- /dev/null +++ b/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_006.md @@ -0,0 +1,340 @@ + + +# 1-4-2 유한요소 정식화 + +평면응력요소는 전체 면적에 대해 두께가 일정하다고 가정한다. 평면응력요소는 등매 개변수(isoparametric) 요소로 구성되어 있으며, 4절점 사각형 요소의 경우에는 비적합(incompatible) 모드를 이용한다. 그리고 평면응력요소는 요소좌표계에서 x,y 방향의 이동변위(translation) u, v 만을 가진다. + +$$ +\mathbf {u} _ {i} = \left\{u _ {i} \quad v _ {i} \right\} ^ {T} \tag {1.4.1} +$$ + +평면응력요소는 4절점 사각형 요소의 비적합 모드를 제외하면 절점 개수에 관계 없이 유사한 과정으로 강성을 계산할 수 있다. 따라서 절점 수 N 개를 가지는 요소에 대하여 일괄적으로 설명한다. + +요소 내 임의의 좌표 x,y 와 이동변위 u,v 는 다음과 같이 나타낼 수 있다. + +$$ +x = \sum_ {i = 1} ^ {N} N _ {i} x _ {i}, \quad y = \sum_ {i = 1} ^ {N} N _ {i} y _ {i}, \quad u = \sum_ {i = 1} ^ {N} N _ {i} u _ {i}, \quad v = \sum_ {i = 1} ^ {N} N _ {i} v _ {i} \tag {1.4.2} +$$ + +• 3절점 삼각형 + +$$ +N _ {1} = 1 - \xi - \eta , N _ {2} = \xi , N _ {3} = \eta \tag {1.4.3} +$$ + +4절점 사각형 + +$$ +N _ {1} = \frac {1}{4} (1 - \xi) (1 - \eta), N _ {2} = \frac {1}{4} (1 + \xi) (1 - \eta) +$$ + +$$ +N _ {3} = \frac {1}{4} (1 + \xi) (1 + \eta), N _ {4} = \frac {1}{4} (1 - \xi) (1 + \eta) \tag {1.4.4} +$$ + +6절점 삼각형 + +$$ +N _ {1} = (1 - \xi - \eta) (1 - 2 \xi - 2 \eta), \quad N _ {2} = \xi (2 \xi - 1), \quad N _ {3} = \eta (2 \eta - 1) +$$ + +$$ +N _ {4} = 4 \xi (1 - \xi - \eta), N _ {5} = 4 \xi \eta , N _ {6} = 4 \eta (1 - \xi - \eta) \tag {1.4.5} +$$ + + + +• 8절점 사각형 + +$$ +N _ {1} = \frac {1}{4} (1 - \xi) (1 - \eta) - \frac {1}{2} N _ {5} - \frac {1}{2} N _ {8}, \quad N _ {2} = \frac {1}{4} (1 + \xi) (1 - \eta) - \frac {1}{2} N _ {5} - \frac {1}{2} N _ {6} +$$ + +$$ +N _ {3} = \frac {1}{4} (1 + \xi) (1 + \eta) - \frac {1}{2} N _ {6} - \frac {1}{2} N _ {7}, \quad N _ {4} = \frac {1}{4} (1 - \xi) (1 + \eta) - \frac {1}{2} N _ {7} - \frac {1}{2} N _ {8} +$$ + +$$ +N _ {5} = \frac {1}{2} (1 - \xi^ {2}) (1 - \eta), N _ {6} = \frac {1}{2} (1 + \xi) (1 - \eta^ {2}), N _ {7} = \frac {1}{2} (1 - \xi^ {2}) (1 + \eta) +$$ + +$$ +N _ {8} = \frac {1}{2} (1 - \xi) (1 - \eta^ {2}) \tag {1.4.6} +$$ + +절점 변위 u와 변형률 ε의 관계는 $B_{i}$ 에 의하여 식(1.4.7)과 같이 나타낼 수 있다. + +$$ +\boldsymbol {\varepsilon} = \sum_ {i = 1} ^ {N} \mathbf {B} _ {i} \mathbf {u} _ {i} \tag {1.4.7} +$$ + +행렬 $B_{i}$ 는 형상함수의 미분값으로 다음과 같이 표현된다. + +$$ +\mathbf {B} _ {i} = \left[ \begin{array}{c c} \frac {\partial N _ {i}}{\partial x} & 0 \\ 0 & \frac {\partial N _ {i}}{\partial y} \\ \frac {\partial N _ {i}}{\partial y} & \frac {\partial N _ {i}}{\partial x} \end{array} \right] \tag {1.4.8} +$$ + +행렬 $B_{i}$ 를 이용하여 면내변형에 관계된 요소강성 행렬을 표현하면 다음과 같다. + +$$ +\mathbf {K} _ {i j} = t \int_ {A _ {e}} \mathbf {B} _ {i} ^ {T} \mathbf {D} \mathbf {B} _ {j} d A \tag {1.4.9} +$$ + +여기서, + +$$ +\begin{array}{l l} t & : \text {두께} \\ A _ {e} & : \text {면적} \end{array} +$$ + +등방성(isotropic) 재료의 경우 응력과 변형률의 관계를 나타내는 행렬 D 는 다음과 같다. + + + +$$ +\mathbf {D} = \frac {E}{1 - \nu^ {2}} \left[ \begin{array}{c c c} 1 & \nu & 0 \\ \nu & 1 & 0 \\ 0 & 0 & \frac {1 - \nu}{2} \end{array} \right] \tag {1.4.10} +$$ + +선형 해석 시 4절점 사각형 요소는 비적합 모드를 포함하여 계산한다. 비적합 모드를 포함한 경우에는 절점변위 이외에 다음과 같은 추가적인 자유도를 가지게 된다. + +$$ +\mathbf {u} _ {a} = \left\{a _ {1} \quad b _ {1} \quad a _ {2} \quad b _ {2} \right\} ^ {T} \tag {1.4.11} +$$ + +좌표 x, y 와 이동변위 u v, 는 다음과 같이 나타낸다. + +$$ +x = \sum_ {i = 1} ^ {4} N _ {i} x _ {i}, y = \sum_ {i = 1} ^ {4} N _ {i} y _ {i}, u = \sum_ {i = 1} ^ {4} N _ {i} u _ {i} + a _ {1} P _ {1} + a _ {2} P _ {2}, v = \sum_ {i = 1} ^ {4} N _ {i} v _ {i} + b _ {1} P _ {1} + b _ {2} P _ {2} \tag {1.4.12} +$$ + +비적합 모드를 의미하는 형상함수는 다음과 같다. + +$$ +P _ {1} = 1 - \xi^ {2}, P _ {2} = 1 - \eta^ {2} \tag {1.4.13} +$$ + +변형률 ε 는 절점변위와 비적합 모드를 동시에 고려하여 다음과 같이 표현된다. + +$$ +\boldsymbol {\varepsilon} = \sum_ {i = 1} ^ {4} \mathbf {B} _ {i} \mathbf {u} _ {i} + \mathbf {B} _ {a} \mathbf {u} _ {a} \tag {1.4.14} +$$ + +행렬 B 는 식 (1.4.15)과 같고, 비적합 모드에 관계된 B 는 다음과 같다. + +$$ +\mathbf {B} _ {a} = \left[ \begin{array}{c c c c} \frac {\partial P _ {1}}{\partial x} & 0 & \frac {\partial P _ {2}}{\partial x} & 0 \\ 0 & \frac {\partial P _ {1}}{\partial y} & 0 & \frac {\partial P _ {2}}{\partial y} \\ \frac {\partial P _ {1}}{\partial y} & \frac {\partial P _ {1}}{\partial x} & \frac {\partial P _ {2}}{\partial y} & \frac {\partial P _ {2}}{\partial x} \end{array} \right] \tag {1.4.15} +$$ + +행렬 B 와 B 를 이용하여 면내변형에 관계된 요소강성 행렬을 계산하면, 다음과같이 4개의 행렬을 얻을 수 있다. + + + +$$ +\mathbf {K} _ {i j} = t \int_ {A _ {e}} \mathbf {B} _ {i} ^ {T} \mathbf {D} \mathbf {B} _ {j} d A, \mathbf {K} _ {i a} = t \int_ {A _ {e}} \mathbf {B} _ {i} ^ {T} \mathbf {D} \mathbf {B} _ {a} d A +$$ + +$$ +\mathbf {K} _ {a i} = t \int_ {A _ {e}} \mathbf {B} _ {a} ^ {T} \mathbf {D} \mathbf {B} _ {i} d A, \quad \mathbf {K} _ {a a} = t \int_ {A _ {e}} \mathbf {B} _ {a} ^ {T} \mathbf {D} \mathbf {B} _ {a} d A \tag {1.4.16} +$$ + +식 (1.4.16)의 4가지 강성행렬은 다음과 같은 관계를 가진다. + +$$ +\left[ \begin{array}{l l} \left[ \mathbf {K} _ {i j} \right] & \left[ \mathbf {K} _ {i a} \right] \\ \left[ \mathbf {K} _ {a j} \right] & \mathbf {K} _ {a a} \end{array} \right] \left\{ \begin{array}{l} \left\{\mathbf {u} _ {j} \right\} \\ \mathbf {u} _ {a} \end{array} \right\} = \left\{ \begin{array}{l} \left\{\mathbf {F} _ {i} \right\} \\ \mathbf {0} \end{array} \right\} \tag {1.4.17} +$$ + +비적합 $\Xi \subseteq 0 \|$ 대한 강성은 정적 축약(static condensation)에 의해 다음과 같이 소거한다. + +$$ +\left[ \mathbf {K} _ {i j} \right] = \left[ \mathbf {K} _ {i j} \right] - \left[ \mathbf {K} _ {i a} \right] \mathbf {K} _ {a a} ^ {- 1} \left[ \mathbf {K} _ {a j} \right] \tag {1.4.18} +$$ + +비적합 모드에 관한 행렬 $\mathbf { B } _ { a }$ 의 구성에 필요한 형상함수를 미분할 때는 무차원 자연좌표계(natural coordinate)의 원점( $\xi = \eta = 0$ )에서 계산된 Jacobian을 이용한다. 비적합 모드는 그림 1.4.3과 같은 굽힘 형태의 변위를 모사할 수 있기 때문에 요소의성능 향상에 기여하게 된다. + +![](images/page-054_8d4976e26ad410b08df08e9e1cb0e75faba91b07d100984b131907bf145b4d08.jpg) + +
+natural_image + +Simple geometric diagram of a rectangle with dashed curved lines on its sides (no text or symbols) +
+ +![](images/page-054_89db5cfa3809db5521f4f76f4174c426adb36b15fa6846907311ef8a61c798e0.jpg) + +
+natural_image + +Simple geometric diagram of a rectangle with dashed lines indicating hidden edges (no text or symbols) +
+ +그림 1.4.3 비적합 모드의 형상 (굽힘) + + + +# 1-4-3 하중과 질량 + +평면응력요소에 적용되는 하중은 체적력(body force), 압력하중(pressure load), 모서리하중(edge load), 온도하중(thermal load), 프리스트레스하중(prestress load) 등이 있다. 체적력은 요소의 자중이나 관성력을 표현하고자 하는 하중이고, 압력하중은 요소의 면에 가해지는 분포하중이다. 모서리하중은 요소의 변에 가해지는 분포하중이이며, 온도하중에는 절점온도, 요소온도 하중과 같은 면내방향 열 변형하중이다. + +\- 체적력 + +$$ +\mathbf {F} _ {i} = t \int_ {A _ {e}} N _ {i} \left\{ \begin{array}{l} \omega_ {x} \\ \omega_ {y} \\ \omega_ {z} \end{array} \right\} d A \tag {1.4.19} +$$ + +여기서, + +$$ +\omega_ {x}, \omega_ {y}, \omega_ {z} \quad : \text { 단위 체적당 자중(방향별) } +$$ + +\- 압력하중 + +$$ +\mathbf {F} _ {i} = \int_ {A _ {e}} N _ {i} \left\{ \begin{array}{l} P _ {x} \\ P _ {y} \\ 0 \end{array} \right\} d A \tag {1.4.20} +$$ + +여기서, + +$$ +P _ {x}, P _ {y} \quad : \text { 단위면적당 하중(방향별) } +$$ + +\- 모서리하중 + +$$ +\mathbf {F} _ {i} = \int_ {L} N _ {i} \left\{ \begin{array}{l} P _ {x} \\ P _ {y} \\ 0 \end{array} \right\} d s \tag {1.4.21} +$$ + +여기서, + +$$ +P _ {x}, P _ {y} \quad : \text { 단위길이당 하중(방향별) } +$$ + + + +\- 온도하중 + +$$ +\mathbf {F} _ {i} = t \int_ {A _ {e}} \mathbf {B} _ {i} ^ {T} \mathbf {D} \left\{ \begin{array}{l} \alpha_ {x} \\ \alpha_ {y} \\ 0 \end{array} \right\} \Delta T d A \tag {1.4.22} +$$ + +여기서, + +$$ +\alpha_ {x}, \alpha_ {y} \quad : \text { 열팽창계수(방향별) } +$$ + +$$ +\Delta T: \text { 온도변화 } +$$ + +평면응력요소의 질량은 집중질량(lumped mass)과 분포질량(consistent mass)을 사용할 수 있으며, x,y,z 방향의 이동변위만을 반영한다. + +\- 분포질량 + +$$ +\mathbf {M} _ {i j} = \rho t \int_ {A _ {e}} N _ {i} N _ {j} d A \tag {1.4.23} +$$ + +\- 집중질량 + +집중질량은 요소 전체질량( $\rho tA_{e}$ )을 분포질량의 대각 항 비율로 분배하여 사용한다. + + + +# 1-4-4 요소결과 + +평면응력요소의 해석 결과로는 절점에서의 응력과 변형률이 있으며, 부호와 방향은 요소좌표계를 따른다. 요소좌표계를 기준으로 출력된 결과는 전체좌표계 또는 출력좌표계로 변환하여 볼 수 있다. 평면응력요소에서 출력되는 응력과 변형률의 종류는 다음과 같다. 주변형률 성분 중 $E_{3}$ 은 선형해석의 경우 0으로 간주하였음에 주의해야 한다. + +• 응력 성분 $\sigma_{xx}, \sigma_{yy}, \tau_{xy}$ +- Von-Mises 응력 $\sqrt{\left(P_{1}^{2}+P_{2}^{2}-P_{1}P_{2}\right)}$ +• 최대 전단응력 $\sqrt{\left(\frac{\sigma_{xx}-\sigma_{yy}}{2}\right)^{2}+\tau_{xy}^{2}}$ +• 주응력 $P_{1}, P_{2}$ + +$$ +P _ {i} = \frac {\sigma_ {x x} + \sigma_ {y y}}{2} \pm \sqrt {\left(\frac {\sigma_ {x x} - \sigma_ {y y}}{2}\right) ^ {2} + \tau_ {x y} ^ {2}} +$$ + +• 변형률 성분 $\varepsilon_{xx}, \varepsilon_{yy}, \gamma_{xy}$ +- Von-Mises 변형률 $\frac{2}{3}\sqrt{\left(E_{1}^{2}+E_{2}^{2}-E_{1}E_{2}\right)}$ +• 체적(volumetric) 변형률 $E_{1} + E_{2}$ +• 주변형률 $E_{1}, E_{2}$ + + + +$$ +E _ {i} = \frac {\varepsilon_ {x x} + \varepsilon_ {y y}}{2} \pm \sqrt {\left(\frac {\varepsilon_ {x x} - \varepsilon_ {y y}}{2}\right) ^ {2} + \frac {\gamma_ {x y} ^ {2}}{4}} +$$ + +절점에서의 응력과 변형률은 요소 내의 적분점에서 계산된 결과를 이용하여 외삽법 (extrapolation)에 의해 산출된다. 평면응력요소의 적분점은 다음과 같다. + +• 3절점 삼각형 : 1 점 가우스 적분 +• 4절점 사각형 : 4 점 가우스 적분 +- 6절점 삼각형 : 3 점 가우스 적분 +• 8절점 사각형 : 9 점 가우스 적분 + +응력과 변형률에 대한 부호규약은 그림 1.4.1과 같다. + + + +# 1-5 판요소 + +# 1-5-1 개요 + +판요소는 3, 4, 6, 8개의 절점에 의해 정의되는 삼각형 또는 사각형 요소이며, 압력용기, 토류벽, 교량 상판 등의 모델링에 사용할 수 있다. 판요소는 평면응력(planestress) 상태의 면내변형(in-plane deformation), 휨(bending). 전단(shear)으로 이루어진 면외변형(out-of-plane deformation)을 고려할 수 있으며, 정적(선형/비선형)해석 및 동적 해석에 모두 사용할 수 있다. 판요소의 변형을 정의하는 응력과 변형률은 다음과 같다. + +$$ +\boldsymbol {\sigma} = \left\{ \begin{array}{l} \sigma_ {x x} \\ \sigma_ {y y} \\ \sigma_ {x y} \end{array} \right\}, \quad \boldsymbol {\varepsilon} = \left\{ \begin{array}{l} \varepsilon_ {x x} \\ \varepsilon_ {y y} \\ \gamma_ {x y} \end{array} \right\} +$$ + +(면내방향 응력과 변형률) + +$$ +\mathbf {M} = \left\{ \begin{array}{l} M _ {x x} \\ M _ {y y} \\ M _ {x y} \end{array} \right\}, \quad \mathbf {K} = \left\{ \begin{array}{l} \kappa_ {x x} \\ \kappa_ {y y} \\ \kappa_ {x y} \end{array} \right\} +$$ + +(휨모멘트와 곡률) + +$$ +\mathbf {Q} = \left\{ \begin{array}{l} Q _ {z x} \\ Q _ {y z} \end{array} \right\}, \quad \boldsymbol {\gamma} = \left\{ \begin{array}{l} \gamma_ {z x} \\ \gamma_ {y z} \end{array} \right\} +$$ + +(전단력과 전단변형률) + +면내방향 응력과 변형률의 부호 규약은 평면응력요소와 동일하며, 휨모멘트와 전단력의 방향은 그림 1.5.1과 같이 정의한다. 그림에서 화살표 방향이 ‘+’를 의미한다. + + + +![](images/page-060_061df440ce12761890a986cf325e5373c3efb0790dfbbc9bed4959348bbfd8b6.jpg) + +
+text_image + +z +y +Qzx,γzx +ECS +Mxx,κxx +Mxy,κxy +Mxy,κxy +Qyz,γyz +Myy,κyy +
+ +그림 1.5.1 판요소의 내력 + +요소좌표계는 오른손법칙에 준한 x, , y z 축의 직교좌표계를 따르며, 방향은 그림1.5.2와 같이 설정된다. 사각형 요소는 절점 1과 절점 4의 중점에서 절점 2와 절점 3의 중점을 향하는 방향을 x 축 방향으로 하며, 삼각형 요소는 절점 1에서 2를 향하는 방향을 x 축 방향으로 설정한다. + +![](images/page-060_4d174980d7f471bde2e24e3dcde866ca81490491613f81c1da991a07bedffa30.jpg) + +
+text_image + +ECS z-axis +3 +ECS y-axis +1 +ECS x-axis +(1 → 2 direction) +2 +
diff --git a/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_007.md b/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_007.md new file mode 100644 index 00000000..d29999e6 --- /dev/null +++ b/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_007.md @@ -0,0 +1,321 @@ + + +![](images/page-061_563d61f9f0d5c24b8c76c56c96a203fe69213a864a10c1752d7c4be8353ea1c2.jpg) +그림 1.5.2 판요소의 좌표계 + + + +판요소는 연결된 절점 수에 따라 두 가지로 구분할 수 있다. 3 또는 4절점 판요소는절점들이 하나의 평면 위에 위치한다고 가정하여 평면판(flat plate)이라 한다. 6절점삼각형 요소와 8절점 사각형 요소는 절점들이 곡면 상에 위치할 수 있기 때문에 곡면판(curved plate)이라 한다. 4절점 판요소는 변위 및 응력 값의 정확도가 높지만,3절점 판요소는 변위에 비해 응력의 정확도가 낮은 경향이 있다. 따라서 정밀 해석결과가 필요한 부위에서는 3절점 판요소의 사용을 피하는 것이 바람직하다. + + + +# 1-5-2 유한요소 정식화 + +판소는 요소좌표계에서 3개의 이동변위(translation)와 x, y 축에 대한 회전변위(rotation)를 갖는다. 평면판은 면내변형과 면외변형 강성을 독립적으로 고려하는 반면, 곡면판은 3차원 탄성이론을 기반으로 한 “연속체 셀이론(continuum shell approach)”을 이용한다. + +# (1) 평면판 + +평면판요소에서 변형성분 별로 사용할 수 있는 강성의 종류는 다음과 같다. + +# - 면내변형 + +3절점 요소 +등매개변수 요소(평면응력요소와 동일), z 축에 대한 회전 자유도를 고려한 요소 $^{1}$ +4절점 요소 +등매개변수 요소(평면응력요소와 동일), z 축에 대한 회전 자유도를 고려한 요소 + +# - 면외변형 + +• 3절점 요소 +DKT $^{2}$ (Discrete Kirchhoff Triangle), DKMT $^{3}$ (Discrete Kirchhoff Mindlin Triangle) + +4절점 요소 + +DKQ $^{4}$ (Discrete Kirchhoff Quadrilateral), DKMQ $^{5}$ (Discrete Kirchhoff + +$^{1}$ D.J. Allman, “A Compatible Triangular Element Including Vertex Rotations for Plane Elasticity Analysis,” Comput. Struct., Vol. 19, 1-8, 1984 +$^{2}$ J.L. Batoz, K.J. Bathe and L.W. Ho, “A Study of Three-Node Triangular Plate Bending Elements,” International Journal for Numerical Methods in Engineering, Vol. 15, 1771-1812, 1980 +$^{3}$ I. Katili, “A New Discrete Kirchhoff-Mindlin Element Based on Mindlin-Reissner Plate Theory and Assumed Shear Strain Fields Part I: An Extended DKT Element for Thick-Plate Bending Analysis,” International Journal for Numerical Methods in Engineering, Vol. 36, 1859-1883, 1993 +$^{4}$ J.L. Batoz and M. Ben Tahar, “Evaluation of a New Thin Plate Quadrilateral Element,” International Journal for Numerical Methods in Engineering, Vol. 18, 1655-1678, 1982 +$^{5}$ I. Katili, “A New Discrete Kirchhoff-Mindlin Element Based on Mindlin-Reissner Plate Theory and Assumed Shear Strain Fields-Part II: An Extended DKQ Element for Thick-Plate Bending Analysis,” International Journal for Numerical Methods in Engineering, Vol. 36, 1885-1908, 1993 + + + +# Mindlin Quadrilateral) + +면내변형 강성으로 사용할 수 있는 방법 중 등매개변수 요소는 평면응력요소와 동일하므로 “1.4 평면응력요소”에서 설명한다. z 축에 대한 회전 자유도가 고려 가능한 요소는 요소좌표계에서의 x, y 방향의 이동변위(translation) u, v 와 z 축에 대한 회전변위(rotation) $\theta_{z}$ 의 영향을 고려한다. + +$$ +\mathbf {u} _ {i} = \left\{u _ {i}, v _ {i}, \theta_ {z i} \right\} ^ {T} \tag {1.5.1} +$$ + +절점 수 N 개를 가지는 요소 내 임의의 좌표 x, y 와 이동변위 u, v 는 다음과 같이 나타낼 수 있다. + +$$ +x = \sum_ {i = 1} ^ {N} N _ {i} x _ {i}, y = \sum_ {i = 1} ^ {N} N _ {i} y _ {i} \tag {1.5.2} +$$ + +$$ +u = \sum_ {i = 1} ^ {N} N _ {i} u _ {i} + \frac {1}{8} \sum_ {i = 1} ^ {N} P _ {i} \left(y _ {j} - y _ {i}\right) \left(\theta_ {z j} - \theta_ {z i}\right), v = \sum_ {i = 1} ^ {N} N _ {i} v _ {i} - \frac {1}{8} \sum_ {i = 1} ^ {N} P _ {i} \left(x _ {j} - x _ {i}\right) \left(\theta_ {z j} - \theta_ {z i}\right) +$$ + +$$ +i = 1, 2,.., N - 1, N \quad j = 2, 3,.., N, 1 \tag {1.5.3} +$$ + +여기서, $\theta_{zi}$ 는 절점에서의 회전자유도이고, 형상함수는 다음과 같다. + +3절점 요소 + +$$ +N _ {1} = 1 - \xi - \eta , N _ {2} = \xi , N _ {3} = \eta \tag {1.5.4} +$$ + +$$ +P _ {1} = 4 \xi (1 - \xi - \eta), \quad P _ {2} = 4 \xi \eta , \quad P _ {3} = 4 \eta (1 - \xi - \eta) \tag {1.5.5} +$$ + +4절점 요소 + +$$ +N _ {1} = \frac {1}{4} (1 - \xi) (1 - \eta), N _ {2} = \frac {1}{4} (1 + \xi) (1 - \eta), N _ {3} = \frac {1}{4} (1 + \xi) (1 + \eta), +$$ + +$$ +N _ {4} = \frac {1}{4} (1 - \xi) (1 + \eta) \tag {1.5.6} +$$ + + + +$$ +P _ {1} = \frac {1}{2} (1 - \xi^ {2}) (1 - \eta), P _ {2} = \frac {1}{2} (1 + \xi) (1 - \eta^ {2}), P _ {3} = \frac {1}{2} (1 - \xi^ {2}) (1 + \eta), +$$ + +$$ +P _ {4} = \frac {1}{2} (1 - \xi) (1 - \eta^ {2}) \tag {1.5.7} +$$ + +절점변위 u 와 면내 변형률 ε 의 관계는 B 에 의하여 식(1.5.8)과 같이 나타낼 수있다. + +$$ +\boldsymbol {\varepsilon} = \sum_ {i = 1} ^ {N} \mathbf {B} _ {i} \mathbf {u} _ {i} \tag {1.5.8} +$$ + +행렬 B 는 형상함수의 미분값으로 다음과 같이 표현된다. + +$$ +\mathbf {B} _ {i} = \left[ \begin{array}{c c c c} \frac {\partial N _ {i}}{\partial x} & 0 & \frac {\left(y _ {i} - y _ {k}\right)}{8} \frac {\partial P _ {k}}{\partial x} - \frac {\left(y _ {j} - y _ {i}\right)}{8} \frac {\partial P _ {i}}{\partial x} \\ 0 & \frac {\partial N _ {i}}{\partial y} & \frac {\left(x _ {k} - x _ {i}\right)}{8} \frac {\partial P _ {k}}{\partial y} - \frac {\left(x _ {i} - x _ {j}\right)}{8} \frac {\partial P _ {i}}{\partial y} \\ \frac {\partial N _ {i}}{\partial y} & \frac {\partial N _ {i}}{\partial x} & \frac {\left(y _ {i} - y _ {k}\right)}{8} \frac {\partial P _ {k}}{\partial y} - \frac {\left(y _ {j} - y _ {i}\right)}{8} \frac {\partial P _ {i}}{\partial y} + \frac {\left(x _ {k} - x _ {i}\right)}{8} \frac {\partial P _ {k}}{\partial x} - \frac {\left(x _ {i} - x _ {j}\right)}{8} \frac {\partial P _ {i}}{\partial x} \end{array} \right] +$$ + +$$ +i = 1, 2,.., N - 1, N, \quad j = 2, 3,.., N, 1, \quad k = N, 1,.., N - 2, N - 1 \tag {1.5.9} +$$ + +행렬 Bi 를 이용하여 면내변형에 관계된 요소강성 행렬을 표현하면 다음과 같다. + +$$ +\mathbf {K} _ {i j} ^ {(I)} = \int_ {A _ {e}} t \mathbf {B} _ {i} ^ {T} \mathbf {D} \mathbf {B} _ {j} d A \tag {1.5.10} +$$ + +여기서, + +t : 두께 + +Ae : 면적 + +등방성(isotropic) 재료의 경우 응력과 변형률의 관계를 나타내는 행렬 D 는 다음과같다. + + + +$$ +\mathbf {D} = \frac {E}{1 - \nu^ {2}} \left[ \begin{array}{c c c} 1 & \nu & 0 \\ \nu & 1 & 0 \\ 0 & 0 & \frac {1 - \nu}{2} \end{array} \right] \tag {1.5.11} +$$ + +면내변형에 대하여 z 축에 대한 회전 자유도를 고려한 요소를 사용하게 되면 요소의변에 수직한 이동변위를 2차로 보간하게 된다. 회전 자유도와 이동변위의 관계는 그림 1.5.3과 같은 휨 형태의 변형에서 꼭지점 부분에는 전단변형이 존재하지 않는 사실에 착안한 것이다. + +![](images/page-066_15dbe7897d4be3d1c59f931ebc7cc648599f1452eff128b9e7e6e52a03c9b379.jpg) + +
+flowchart +```mermaid +graph TD + A["Start"] --> B["Loop"] + B --> C["End"] + C -->|Feedback| B + B -->|Feedback| A +``` +
+ +그림 1.5.3 굽힘 변형과 회전자유도 관계 + +면외변형에 대한 강성으로 사용할 수 있는 방법 중 DKMT(3절점)와 DKMQ(4절점) 요소는 전단변형을 고려하며, 전단변형률 가정법을 이용한다. 절점에서의 자유도는 요소좌표계에서 z 방향의 이동변위 w 와 x, y 축에 대한 회전변위 , θ x θ y 를 고려한다. + +$$ +\mathbf {u} _ {i} = \left\{w _ {i} \quad \theta_ {x i} \quad \theta_ {y i} \right\} ^ {T} \tag {1.5.12} +$$ + +요소 내 임의의 좌표 x, y 는 식 (1.5.2)과 같이 계산하고, 회전변위 $\theta _ { x } , \ \theta _ { y } \in \ \mathsf { L } \mathsf { I } \frac { \circ } { \mathsf { I } }$ 과 같이 2차로 표현한다. + +$$ +\theta_ {x} = \sum_ {i = 1} ^ {N} N _ {i} \theta_ {x i} + \sum_ {i = 1} ^ {N} P _ {i} S _ {i j} \Delta \theta_ {n i}, \quad \theta_ {y} = \sum_ {i = 1} ^ {N} N _ {i} \theta_ {y i} - \sum_ {i = 1} ^ {N} P _ {i} C _ {i j} \Delta \theta_ {n i} \tag {1.5.13} +$$ + +$$ +C _ {i j} = - x _ {i j} / L _ {i j}, S _ {i j} = - y _ {i j} / L _ {i j}, x _ {i j} = x _ {i} - x _ {j}, y _ {i j} = y _ {i} - y _ {j}, L _ {i j} ^ {2} = x _ {i j} ^ {2} + y _ {i j} ^ {2} +$$ + + + +$$ +i = 1, 2,.., N - 1, N \quad j = 2, 3,.., N, 1 +$$ + +여기서, 형상함수 $N_{i}$ , $P_{i}$ 는 식 (1.5.4)\~(1.5.7)과 같다. 요소 변 중앙에서의 가상 회전각 $\Delta\theta_{ni}$ 를 구하기 위해 다음과 같은 가정을 이용한다. + +\- N 개의 변을 따라 전단력과 휩모멘트의 평형식을 만족한다. + +$$ +Q _ {s} = - M _ {s, s} - M _ {n s, n} \tag {1.5.14} +$$ + +\- 변에 수직한 축에 대한 회전변위는 변을 따라 2차이고, 접선방향 축에 대한 회전변위는 1차이다. + +$$ +\theta_ {n} = \left(1 - \frac {S}{L _ {i j}}\right) \theta_ {n i} + \frac {S}{L _ {i j}} \theta_ {n j} + 4 \frac {S}{L _ {i j}} \left(1 - \frac {S}{L _ {i j}}\right) \Delta \theta_ {n i}, \quad \theta_ {s} = \left(1 - \frac {S}{L _ {i j}}\right) \theta_ {s i} + \frac {S}{L _ {i j}} \theta_ {s j} +$$ + +$$ +i = 1, 2,.., N - 1, N \quad j = 2, 3,.., N, 1 \tag {1.5.15} +$$ + +\- 식 (1.5.14)를 통해 계산되는 전단변형률 $\overline{\gamma}_{sz}$ 는 형상함수로부터 직접 계산되는 전단변형률 $\gamma_{sz}$ 와 다음 관계를 만족한다. + +$$ +\int_ {0} ^ {L _ {i j}} \left(\gamma_ {s z} - \overline {{\gamma}} _ {s z}\right) d s = 0 \tag {1.5.16} +$$ + +위 가정을 통하여 구한 $\Delta\theta_{ni}$ 를 식 (1.5.13)에 대입하면, 다음과 같이 회전변위 $\theta_{x}, \theta_{y}$ 를 $u_{i}$ 로 표현할 수 있다. + +$$ +\theta_ {x} = \sum_ {i = 1} ^ {N} \mathbf {H} _ {x i} ^ {T} \mathbf {u} _ {i}, \theta_ {y} = \sum_ {i = 1} ^ {N} \mathbf {H} _ {y i} ^ {T} \mathbf {u} _ {i} \tag {1.5.17} +$$ + +여기서, $H_{xi}, H_{yi}$ 는 다음과 같다. + + + +$$ +\mathbf {H} _ {x i} = \left\{ \begin{array}{l} 0 \\ N _ {i} \\ 0 \end{array} \right\} + \left\{ \begin{array}{l} \frac {3 P _ {k} S _ {k i}}{2 L _ {k i} \left(1 + \phi_ {k i}\right)} - \frac {3 P _ {i} S _ {i j}}{2 L _ {i j} \left(1 + \phi_ {i j}\right)} \\ \frac {3 P _ {k} S _ {k i} y _ {k i}}{4 L _ {k i} \left(1 + \phi_ {k i}\right)} + \frac {3 P _ {i} S _ {i j} y _ {i j}}{4 L _ {i j} \left(1 + \phi_ {i j}\right)} \\ - \frac {3 P _ {k} S _ {k i} x _ {k i}}{4 L _ {k i} \left(1 + \phi_ {k i}\right)} - \frac {3 P _ {i} S _ {i j} x _ {i j}}{4 L _ {i j} \left(1 + \phi_ {i j}\right)} \end{array} \right\} \tag {1.5.18} +$$ + +$$ +\mathbf {H} _ {y, i} = \left\{ \begin{array}{l} 0 \\ 0 \\ N _ {i} \end{array} \right\} + \left\{ \begin{array}{l} - \frac {3 P _ {k} C _ {k i}}{2 L _ {k i} \left(1 + \phi_ {k i}\right)} + \frac {3 P _ {i} C _ {i j}}{2 L _ {i j} \left(1 + \phi_ {i j}\right)} \\ - \frac {3 P _ {k} C _ {k i} y _ {k i}}{4 L _ {k i} \left(1 + \phi_ {k i}\right)} - \frac {3 P _ {i} C _ {i j} y _ {i j}}{4 L _ {i j} \left(1 + \phi_ {i j}\right)} \\ \frac {3 P _ {k} C _ {k i} x _ {k i}}{4 L _ {k i} \left(1 + \phi_ {k i}\right)} + \frac {3 P _ {i} C _ {i j} x _ {i j}}{4 L _ {i j} \left(1 + \phi_ {i j}\right)} \end{array} \right\} \tag {1.5.19} +$$ + +$$ +\phi_ {i j} = \frac {2}{\kappa (1 - \nu)} (\frac {t ^ {2}}{L _ {i j} ^ {2}}) \quad \text {(등방성 재료의 경우)} +$$ + +$$ +i = 1, 2,.., N - 1, N \quad j = 2, 3,.., N, 1 \quad k = N, 1,.., N - 2, N - 1 +$$ + +절점변위와 곡률 κ 의 관계는 $\mathbf { B } _ { b i }$ 에 의해 다음과 같이 표현된다. + +$$ +\mathbf {k} = \sum_ {i = 1} ^ {N} \mathbf {B} _ {b i} \mathbf {u} _ {i} \tag {1.5.20} +$$ + +$$ +\mathbf {B} _ {b i} = \left[ \begin{array}{c} - \frac {\partial \mathbf {H} _ {y i} ^ {T}}{\partial x} \\ \frac {\partial \mathbf {H} _ {x i} ^ {T}}{\partial y} \\ \frac {\partial \mathbf {H} _ {x i} ^ {T}}{\partial x} - \frac {\partial \mathbf {H} _ {y i} ^ {T}}{\partial y} \end{array} \right] \tag {1.5.21} +$$ + +전단변형 γ 의 계산에는 식 (1.5.16)으로부터 계산되는 $\overline { { \gamma } } _ { s z }$ 를 이용하며, 절점변위와의 관계를 정의하는 행렬 $\mathbf { B } _ { s i }$ 는 다음과 같다. + +$$ +\boldsymbol {\gamma} = \sum_ {i = 1} ^ {N} \mathbf {B} _ {s i} \mathbf {u} _ {i} \tag {1.5.22} +$$ + + + +3절점 요소 + +$$ +\mathbf {B} _ {s i} = \left[ \begin{array}{c c} \left(\frac {S _ {j k}}{A _ {j}} N _ {j} - \frac {S _ {k i}}{A _ {i}} N _ {i}\right) \frac {\phi_ {i j}}{L _ {i j} \left(1 + \phi_ {i j}\right)} & \left(\frac {S _ {i j}}{A _ {i}} N _ {i} - \frac {S _ {j k}}{A _ {k}} N _ {k}\right) \frac {\phi_ {k i}}{L _ {k i} \left(1 + \phi_ {k i}\right)} \\ \left(\frac {C _ {k i}}{A _ {i}} N _ {i} - \frac {S _ {j k}}{A _ {j}} N _ {j}\right) \frac {\phi_ {i j}}{L _ {i j} \left(1 + \phi_ {i j}\right)} & \left(\frac {S}{A _ {k}} N _ {k} - \frac {S}{A _ {i}} N _ {i}\right) \frac {\phi_ {k i}}{L _ {k i} \left(1 + \phi_ {k i}\right)} \end{array} \right] \left[ \begin{array}{c c c} 1 & \frac {- y _ {i j}}{2} & \frac {x _ {i j}}{2} \\ - 1 & \frac {- y _ {k i}}{2} & \frac {x _ {k i}}{2} \end{array} \right] +$$ + +$$ +i = 1, 2, 3 \quad j = 2, 3, 1 \quad k = 3, 1, 2, A _ {i} = C _ {i j} S _ {k i} - C _ {k i} S _ {i j} \tag {1.5.23} +$$ + +4절점 요소 + +$$ +\mathbf {B} _ {s i} = \left[ \begin{array}{l l} \frac {\partial N _ {i}}{\partial \lambda} \frac {\partial \lambda}{\partial x} \frac {\phi_ {i j}}{\left(1 + \phi_ {i j}\right)} & \frac {\partial N _ {k}}{\partial \lambda} \frac {\partial \lambda}{\partial x} \frac {\phi_ {k i}}{\left(1 + \phi_ {k i}\right)} \\ \frac {\partial N _ {i}}{\partial \lambda} \frac {\partial \lambda}{\partial y} \frac {\phi_ {i j}}{\left(1 + \phi_ {i j}\right)} & \frac {\partial N _ {k}}{\partial \lambda} \frac {\partial \lambda}{\partial y} \frac {\phi_ {k i}}{\left(1 + \phi_ {k i}\right)} \end{array} \right] \left[ \begin{array}{c c c} 1 & \frac {- y _ {i j}}{2} & \frac {x _ {i j}}{2} \\ - 1 & \frac {- y _ {k i}}{2} & \frac {x _ {k i}}{2} \end{array} \right] \tag {1.5.24} +$$ + +$$ +i = 1, 2, 3, 4 \quad j = 2, 3, 4, 1 \quad k = 4, 1, 2, 3 +$$ + +$$ +\frac {\partial N _ {i}}{\partial \lambda} \frac {\partial \lambda}{\partial x} = \left\{ \begin{array}{l l} \frac {\partial N _ {i}}{\partial \xi} \frac {\partial \xi}{\partial x} & i = 1, 3 \\ \frac {\partial N _ {i}}{\partial \eta} \frac {\partial \eta}{\partial x} & i = 2, 4 \end{array} , \quad \frac {\partial N _ {i}}{\partial \lambda} \frac {\partial \lambda}{\partial y} = \left\{ \begin{array}{l l} \frac {\partial N _ {i}}{\partial \xi} \frac {\partial \xi}{\partial y} & i = 1, 3 \\ \frac {\partial N _ {i}}{\partial \eta} \frac {\partial \eta}{\partial y} & i = 2, 4 \end{array} \right. \right. \tag {1.5.25} +$$ + +따라서 힘과 전단변형에 관계된 요소 강성은 다음과 같다. + +$$ +\mathbf {K} _ {i j} ^ {(O)} = \int_ {A _ {e}} \left(\mathbf {B} _ {b i} ^ {T} \mathbf {D} \mathbf {B} _ {b j} \frac {t ^ {3}}{1 2} + \mathbf {B} _ {s i} ^ {T} \mathbf {D} \mathbf {B} _ {s j} t\right) d A \tag {1.5.26} +$$ + +면외변형에 대한 강성으로 사용할 수 있는 방법 중 DKT(3절점) 요소와 DKQ(4절점) 요소는 전단변형을 고려하지 않는다. 이들 두 요소는 Kirchhoff–Love 가정의 이산화 (discretization)를 이용한다. 절점에서의 자유도는 식 (1.5.12)와 같이 요소좌표계에서 z 방향의 이동변위 w와 x, y 축에 대한 회전변위 $\theta_{x}$ , $\theta_{y}$ 를 고려한다. 요소 내 임의의 좌표 x, y 는 식 (1.5.2)와 같이 계산하고, 회전변위 $\theta_{x}$ , $\theta_{y}$ 는 다음과 같이 2차로 표현한다. + +$$ +\theta_ {x} = \sum_ {i = 1} ^ {N} N _ {i} \theta_ {x i} + \sum_ {i = 1} ^ {N} N _ {i + N} \Delta \theta_ {x i}, \quad \theta_ {y} = \sum_ {i = 1} ^ {N} N _ {i} \theta_ {y i} + \sum_ {i = 1} ^ {N} N _ {i + N} \Delta \theta_ {y i} \tag {1.5.27} +$$ + + + +3절점 요소 + +$$ +N _ {1} = (1 - \xi - \eta) (1 - 2 \xi - 2 \eta), N _ {2} = \xi (2 \xi - 1), N _ {3} = \eta (2 \eta - 1) +$$ + +$$ +N _ {4} = 4 \xi (1 - \xi - \eta), N _ {5} = 4 \xi \eta , N _ {6} = 4 \eta (1 - \xi - \eta) \tag {1.5.28} +$$ + +4절점 요소 + +$$ +N _ {1} = \frac {1}{4} (1 - \xi) (1 - \eta) - \frac {1}{2} N _ {5} - \frac {1}{2} N _ {8}, N _ {2} = \frac {1}{4} (1 + \xi) (1 - \eta) - \frac {1}{2} N _ {5} - \frac {1}{2} N _ {6} +$$ + +$$ +N _ {3} = \frac {1}{4} (1 + \xi) (1 + \eta) - \frac {1}{2} N _ {6} - \frac {1}{2} N _ {7}, \quad N _ {4} = \frac {1}{4} (1 - \xi) (1 + \eta) - \frac {1}{2} N _ {7} - \frac {1}{2} N _ {8} +$$ + +$$ +N _ {5} = \frac {1}{2} (1 - \xi^ {2}) (1 - \eta), N _ {6} = \frac {1}{2} (1 + \xi) (1 - \eta^ {2}) +$$ + +$$ +N _ {7} = \frac {1}{2} (1 - \xi^ {2}) (1 + \eta), N _ {8} = \frac {1}{2} (1 - \xi) (1 - \eta^ {2}) \tag {1.5.29} +$$ + +요소 변 중앙에서의 가상 회전각 $\Delta\theta_{xi}$ , $\Delta\theta_{yi}$ 를 구하기 위해 다음과 같은 가정을 이용한다. + +\- Kirchhoff-Love의 가정을 각 절점과 변의 중점에서 적용한다. + +절점: $-\theta_{x}+\frac{\partial w}{\partial y}=0$ , $\theta_{y}+\frac{\partial w}{\partial x}=0$ , 변의 중점: $-\theta_{n}+\frac{\partial w}{\partial s}=0$ (1.5.30) + +\- 면외방향 이동변위는 변을 따라 3차이고, 변의 접선방향을 향하는 축에 대한 회전변위는 1차이다. + +$$ +\frac {\partial w (L _ {i j} / 2)}{\partial s} = - \frac {3}{2 L _ {i j}} w _ {i} - \frac {1}{4} \frac {\partial w (0)}{\partial s} + \frac {3}{2 L _ {i j}} w _ {j} + \frac {1}{4} \frac {\partial w (L _ {i j})}{\partial s} \tag {1.5.31} +$$ + +$$ +\Delta \theta_ {s i} = \frac {1}{2} (\theta_ {s i} + \theta_ {s j}) , i = 1, 2,..., N - 1, N \quad j = 2, 3,..., N, 1 \tag {1.5.32} +$$ diff --git a/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_008.md b/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_008.md new file mode 100644 index 00000000..8cbc6502 --- /dev/null +++ b/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_008.md @@ -0,0 +1,350 @@ + + +위의 가정을 통하여 구한 $\Delta \theta _ { x i } , \ \Delta \theta _ { y i }$ 를 식 (1.5.27)에 대입하면, 다음과 같이 회전변위 , θ x θ y 를 $\mathbf { u } _ { i }$ 로 표현할 수 있다. + +$$ +\theta_ {x} = \sum_ {i = 1} ^ {N} \mathbf {H} _ {x i} ^ {T} \mathbf {u} _ {i}, \theta_ {y} = \sum_ {i = 1} ^ {N} \mathbf {H} _ {y i} ^ {T} \mathbf {u} _ {i} \tag {1.5.33} +$$ + +여기서, $\mathbf { H } _ { x i } , ~ \mathbf { H } _ { y i } \\\Breve { \mathbf { } } ~ \mathsf { L } \mathsf { F } \Breve { \mathbf { } } \Breve { \mathbf { \updownarrow } } ~ \Breve { \mathbf { \mu } } \Breve { \mathbf { \mu } } ,$ . + +$$ +\mathbf {H} _ {x} = \left\{ \begin{array}{c} - \frac {3}{2} \left(d _ {i j} N _ {i + N} - d _ {k i} N _ {k + N}\right) \\ N _ {i} - e _ {i j} N _ {i + N} - e _ {k i} N _ {k + N} \\ b _ {i j} N _ {i + N} + b _ {k i} N _ {k + N} \end{array} \right\}, \quad \mathbf {H} _ {y i} = \left\{ \begin{array}{c} \frac {3}{2} \left(a _ {i j} N _ {i + N} - a _ {k i} N _ {k + N}\right) \\ b _ {i j} N _ {i + N} + b _ {k i} N _ {k + N} \\ N _ {i} - c _ {i j} N _ {i + N} - c _ {k i} N _ {k + N} \end{array} \right\} \tag {1.5.34} +$$ + +$$ +i = 1, 2,.., N - 1, N \quad j = 2, 3,.., N, 1 \quad k = N, 1,.., N - 2, N - 1 +$$ + +$a _ { i j } , \ b _ { i j } , \ c _ { i j } , \ d _ { i j } , \ e _ { i j }$ 는 요소의 기하학적 형상에 의해 결정되는 값이며, 다음과 같이정의된다. + +$$ +a _ {i j} = - x _ {i j} / L _ {i j} ^ {2}, b _ {i j} = \frac {3}{4} x _ {i j} y _ {i j} / L _ {i j} ^ {2}, c _ {i j} = \left(\frac {1}{4} x _ {i j} ^ {2} - \frac {1}{2} y _ {i j} ^ {2}\right) / L _ {i j} ^ {2} +$$ + +$$ +d _ {i j} = - y _ {i j} ^ {2} / L _ {i j} ^ {2}, e _ {i j} = \left(\frac {1}{4} y _ {i j} ^ {2} - \frac {1}{2} x _ {i j} ^ {2}\right) / L _ {i j} ^ {2} \tag {1.5.35} +$$ + +절점변위와 횡방향 곡률 κ 의 관계는 식 (1.5.20)과 같고, $\mathbf { B } _ { b i } \triangleq$ 식 (1.5.21)과 같다.DKT와 DKQ 요소의 경우에는 전단변형을 고려하지 $\Omega \underline { { \underline { { \circ } } } } \underline { { \underline { { \circ } } } } \underline { { \underline { { \circ } } } } $ 로, 면외변형에 관계된 요소 강성은 다음과 같다. + +$$ +\mathbf {K} _ {i j} ^ {(O)} = \int_ {A _ {e}} \mathbf {B} _ {b i} ^ {T} \mathbf {D} \mathbf {B} _ {b j} \frac {t ^ {3}}{1 2} d A \tag {1.5.36} +$$ + +4절점 평면판요소는 절점의 좌표가 하나의 평면 위에 존재하지 않는 경우가 있다.평면에 존재하지 않는 요소를 위와 같은 정식화 과정으로 계산하게 되면 요소의 기하학적 형상을 정확하게 고려하지 못한다. 따라서 요소좌표계에서 정의된 절점 변위가 왜곡될 수 있다. 이러한 문제점을 해결하기 위하여 ${ \mathsf { M a c N e a l } } ^ { 6 } 0 { \mathsf { 1 } }$ 제안한 강성 보정방법을 이용한다. 그림 1.5.4와 같이 A − B C D − − 평면에서 계산된 강성행렬 $\mathbf { K } _ { P }$ + + + +를 실제 절점 위치 1 2 3 4 − − − 에 대한 강성 K 로 변환하기 위해 변환 행렬 S 를이용한다. + +$$ +\mathbf {K} = \mathbf {S} ^ {T} \mathbf {K} _ {P} \mathbf {S} \tag {1.5.37} +$$ + +![](images/page-072_42bbab7e0d99a31a134cfc544b9ec26a45c78c1a510665a281e7ef046fb860ad.jpg) + +
+text_image + +4 +h* +D +z +y +x +A +h* +1 +2 +3 +F32 +C +B +F23 +ΔFz2 +
+ +그림 1.5.4 4절점 판요소가 곡면 상에 존재하는 경우 + +변환행렬 S 는 평면 위의 점( A − B C D − − )에서의 힘 $\mathbf { F } _ { P }$ 를 절점( 1 2 3 4 − − − ) 위치에서의 힘 F 로 변환하는 행렬이다. + +$$ +\mathbf {F} = \mathbf {S} ^ {T} \mathbf {F} _ {P} \tag {1.5.38} +$$ + +$$ +\mathbf {F} _ {P} = \left\{\mathbf {F} _ {P 1} ^ {T}, \mathbf {F} _ {P 2} ^ {T}, \mathbf {F} _ {P 3} ^ {T}, \mathbf {F} _ {P 4} ^ {T} \right\} ^ {T} \tag {1.5.39} +$$ + +$$ +\mathbf {F} _ {P i} = \left\{F _ {x}, F _ {y}, F _ {z}, M _ {x}, M _ {y}, M _ {z} \right\} _ {P i} ^ {T} \tag {1.5.40} +$$ + + + +힘의 변환행렬 계산시 고려하는 성분은 절점( 1 2 3 4 − − − )을 연결한 요소의 변이 평면( A − − − B C D )와 이루는 각도로 인해 발생하는 면외방향 힘과 모멘트이다. + +$$ +- \Delta F _ {z 3} = \Delta F _ {z 2} = h ^ {*} (\frac {F _ {3 2}}{L _ {2 3}} - \frac {F _ {2 3}}{L _ {2 3}}) \tag {1.5.41} +$$ + +$$ +- \Delta M _ {z 3} = \Delta M _ {z 2} = h ^ {*} (\frac {M _ {3 2}}{L _ {2 3}} - \frac {M _ {2 3}}{L _ {2 3}}) \tag {1.5.42} +$$ + +면외방향으로 발생하는 모멘트 ∆Mzi 는 변환행렬 S 의 구성에 직접 이용하지 않고등가의 힘으로 치환하여 적용한다. + +# (2) 곡면판 + +곡면판요소의 정식화는 “Continuum Shell Approach”를 이용하므로, 면내변형과면외변형에 대한 강성을 동시에 계산할 수 있다. 곡면판요소는 요소좌표계에서x, y, z 방향의 이동변위 u, v, w 와 벡터 $\mathbf { V } _ { 1 i }$ , $\mathbf { V } _ { 2 i }$ 에 대한 회전변위 $\theta _ { 1 } ,$ $\theta _ { 2 }$ 의 영향을 고려한다. + +$$ +\mathbf {u} _ {i} = \left\{u _ {i} \quad v _ {i} \quad w _ {i} \quad \theta_ {1 i} \quad \theta_ {2 i} \right\} ^ {T} \tag {1.5.43} +$$ + +각 절점마다 정의된 3개의 벡터 V 는 그림 1.5.5와 같다. $\mathbf { V } _ { 1 }$ 은 요소좌표계의 $x \equiv$ 을 곡면에 투영하여 계산하고 이를 이용하여 $\mathbf { V } _ { 2 }$ , V 를 얻을 수 있다. + +$$ +\mathbf {V} _ {1 i} = \left\{l _ {1 i}, m _ {1 i}, n _ {1 i} \right\} ^ {T}, \quad \mathbf {V} _ {2 i} = \left\{l _ {2 i}, m _ {2 i}, n _ {2 i} \right\} ^ {T}, \quad \mathbf {V} _ {3 i} = \left\{l _ {3 i}, m _ {3 i}, n _ {3 i} \right\} ^ {T} \tag {1.5.44} +$$ + + + +![](images/page-074_f709954d94828cc45a677d89d0d8fa26676916fbf97ec454e741f569f35852ab.jpg) + +
+text_image + +V3 +V2 +V1 +projection +ECS +z +y +x +
+ +그림 1.5.5 요소좌표계의 투영에 의해 생성되는 절점좌표계 + +절점수 N 개를 가지는 요소 내 임의의 좌표 x, y, z 와 이동변위 u, v w , 는 다음과 같이 나타낼 수 있다. + +$$ +x = \sum_ {i = 1} ^ {N} N _ {i} (x _ {i} + \zeta \frac {t _ {i}}{2} l _ {3 i}) , y = \sum_ {i = 1} ^ {N} N _ {i} (y _ {i} + \zeta \frac {t _ {i}}{2} m _ {3 i}) , z = \sum_ {i = 1} ^ {N} N _ {i} (z _ {i} + \zeta \frac {t _ {i}}{2} n _ {3 i}) \tag {1.5.45} +$$ + +$$ +u = \sum_ {i = 1} ^ {N} N _ {i} \left\{u _ {i} + \zeta \frac {t _ {i}}{2} \left(\mu_ {1 1 i} \theta_ {1 i} + \mu_ {1 2 i} \theta_ {2 i}\right) \right\} +$$ + +$$ +v = \sum_ {i = 1} ^ {N} N _ {i} \left\{v _ {i} + \zeta \frac {t _ {i}}{2} \left(\mu_ {2 1 i} \theta_ {1 i} + \mu_ {2 2 i} \theta_ {2 i}\right) \right\} \tag {1.5.46} +$$ + +$$ +w = \sum_ {i = 1} ^ {N} N _ {i} \left\{w _ {i} + \zeta \frac {t _ {i}}{2} \left(\mu_ {3 1 i} \theta_ {1 i} + \mu_ {3 2 i} \theta_ {2 i}\right) \right\} +$$ + +여기서, + +it : 절점에서의 두께 + +µi : 회전행렬 + + + +6절점 요소 + +$$ +N _ {1} = (1 - \xi - \eta) (1 - 2 \xi - 2 \eta), N _ {2} = \xi (2 \xi - 1), N _ {3} = \eta (2 \eta - 1) +$$ + +$$ +N _ {4} = 4 \xi (1 - \xi - \eta), N _ {5} = 4 \xi \eta , N _ {6} = 4 \eta (1 - \xi - \eta) \tag {1.5.47} +$$ + +8절점 요소 + +$$ +N _ {1} = \frac {1}{4} (1 - \xi) (1 - \eta) - \frac {1}{2} N _ {5} - \frac {1}{2} N _ {8}, N _ {2} = \frac {1}{4} (1 + \xi) (1 - \eta) - \frac {1}{2} N _ {5} - \frac {1}{2} N _ {6} +$$ + +$$ +N _ {3} = \frac {1}{4} (1 + \xi) (1 + \eta) - \frac {1}{2} N _ {6} - \frac {1}{2} N _ {7}, \quad N _ {4} = \frac {1}{4} (1 - \xi) (1 + \eta) - \frac {1}{2} N _ {7} - \frac {1}{2} N _ {8} +$$ + +$$ +N _ {5} = \frac {1}{2} (1 - \xi^ {2}) (1 - \eta), N _ {6} = \frac {1}{2} (1 + \xi) (1 - \eta^ {2}) +$$ + +$$ +N _ {7} = \frac {1}{2} (1 - \xi^ {2}) (1 + \eta), N _ {8} = \frac {1}{2} (1 - \xi) (1 - \eta^ {2}) \tag {1.5.48} +$$ + +절점변위와 변형률 $\varepsilon_{G}$ 의 관계는 $B_{i}$ 에 의해 다음과 같이 표현할 수 있다. + +$$ +\boldsymbol {\varepsilon} _ {G} = \sum_ {i = 1} ^ {N} \mathbf {B} _ {i} \mathbf {u} _ {i} \tag {1.5.49} +$$ + +여기서, 변형률 $\varepsilon_{G}$ 는 3차원 응력 텐서의 모든 항을 포함한다. + +$$ +\boldsymbol {\varepsilon} _ {G} = \left\{\varepsilon_ {x x}, \varepsilon_ {y y}, \varepsilon_ {z z}, \gamma_ {x y}, \gamma_ {y z}, \gamma_ {z x} \right\} ^ {T} \tag {1.5.50} +$$ + +행렬 $B_{i}$ 는 식 (1.5.51)과 같다. + + + +$$ +\mathbf {B} _ {i} = \mathbf {H} \left[ \begin{array}{c c c} \mathbf {J} ^ {- 1} & \mathbf {0} & \mathbf {0} \\ \mathbf {0} & \mathbf {J} ^ {- 1} & \mathbf {0} \\ \mathbf {0} & \mathbf {0} & \mathbf {J} ^ {- 1} \end{array} \right] \left[ \begin{array}{c c c c c} \frac {\partial N _ {i}}{\partial \xi} & 0 & 0 & - \zeta \frac {t _ {i}}{2} \frac {\partial N _ {i}}{\partial \xi} l _ {2, i} & \zeta \frac {t _ {i}}{2} \frac {\partial N _ {i}}{\partial \xi} l _ {1, i} \\ \frac {\partial N _ {i}}{\partial \eta} & 0 & 0 & - \zeta \frac {t _ {i}}{2} \frac {\partial N _ {i}}{\partial \eta} l _ {2, i} & \zeta \frac {t _ {i}}{2} \frac {\partial N _ {i}}{\partial \eta} l _ {1, i} \\ 0 & 0 & 0 & - \frac {t _ {i}}{2} N _ {i} l _ {2, i} & \frac {t _ {i}}{2} N _ {i} l _ {1, i} \\ 0 & \frac {\partial N _ {i}}{\partial \xi} & 0 & - \zeta \frac {t _ {i}}{2} \frac {\partial N _ {i}}{\partial \xi} m _ {2, i} & \zeta \frac {t _ {i}}{2} \frac {\partial N _ {i}}{\partial \xi} m _ {1, i} \\ \vdots & \vdots & \vdots & \vdots & \vdots \\ 0 & 0 & 0 & - \frac {t _ {i}}{2} N _ {i} n _ {2, i} & \frac {t _ {i}}{2} N _ {i} n _ {1, i} \end{array} \right] \tag {1.5.51} +$$ + +여기서, + +H : 불리언(Boolean) 행렬 + +J : 자코비안(Jacobian) 행렬 + +행렬 $B_{i}$ 에 의해 계산되는 변형률 $\varepsilon_{G}$ 를 곡면에 투영하여, 다음과 같이 강성 행렬을 계산한다. + +$$ +\mathbf {K} _ {i j} = \int_ {V _ {e}} \mathbf {B} _ {i} ^ {T} \mathbf {T} ^ {T} \mathbf {D} \mathbf {T} \mathbf {B} _ {j} d V \tag {1.5.52} +$$ + +z 축에 대한 회전자유도를 고려하고자 하는 경우에는 Zienkiewicz 와 Taylor $^{7}$ 에서 제시한 가상의 회전 강성을 이용한다. 이 방법은 회전자유도에 의한 변형 에너지를 다음과 같이 가정하여, 일정량의 강성을 발생시키는 방법이다. + +$$ +\Pi^ {e} = \int_ {\Omega_ {e}} f (\mathbf {D} _ {M}, t ^ {n}, (\theta_ {z} - \overline {{\theta_ {z}}}) ^ {2}) d \Omega \tag {1.5.53} +$$ + +여기서, + +$D_{M}$ : 응력과 변형률의 관계를 타나내는 행렬 + + + +# 1-5-3 하중과 질량 + +판요소에 적용되는 하중은 체적력(body force), 압력하중(pressure load), 모서리하중(edge load), 온도하중(thermal load), 프리스트레스하중(prestress load) 등이 있다. 체적력은 요소의 자중이나 관성력을 표현하고자 하는 하중이고, 압력하중은 요소의 면에 가해지는 분포하중이다. 모서리하중은 요소의 변에 가해지는 분포하중이며, 온도하중에는 절점온도, 요소온도 하중과 같은 면내방향 열 변형하중과 온도구배(temperature gradient)와 같이 힐을 유발하는 하중이 있다. + +\- 체적력 + +$$ +\mathbf {F} _ {i} = \int_ {A _ {e}} t N _ {i} \left\{ \begin{array}{l} \omega_ {x} \\ \omega_ {y} \\ \omega_ {z} \end{array} \right\} d A \tag {1.5.54} +$$ + +여기서, + +$$ +\omega_ {x}, \omega_ {y}, \omega_ {z} \quad : \text { 단위 체적당 자중(방향별) } +$$ + +\- 압력하중 + +$$ +\mathbf {F} _ {i} = \int_ {A _ {e}} N _ {i} \left\{ \begin{array}{l} P _ {x} \\ P _ {y} \\ P _ {z} \end{array} \right\} d A \tag {1.5.55} +$$ + +여기서, + +$$ +P _ {x}, P _ {y}, P _ {z} \quad : \text { 단위 면적당 하중(방향별) } +$$ + + + +\- 모서리하중 + +$$ +\mathbf {F} _ {i} = \int_ {L} N _ {i} \left\{ \begin{array}{l} P _ {x} \\ P _ {y} \\ P _ {z} \end{array} \right\} d s \tag {1.5.56} +$$ + +여기서, + +$$ +P _ {x}, P _ {y}, P _ {z} \quad : \text { 단위 길이당 하중(방향별) } +$$ + +\- 온도하중 + +$$ +\mathbf {F} _ {i} = \int_ {A _ {e}} t \mathbf {B} _ {i} ^ {T} \mathbf {D} \left\{ \begin{array}{l} \alpha_ {x} \\ \alpha_ {y} \\ 0 \end{array} \right\} \Delta T d A \tag {1.5.57} +$$ + +$$ +\alpha_ {x}, \alpha_ {y} \quad : \text { 열팽창 계수(방향별) } +$$ + +$$ +\Delta T: \text {온도변화} +$$ + +\- 온도구배하중 + +$$ +\mathbf {F} _ {i} = \int_ {A _ {e}} \frac {t ^ {3}}{1 2} \mathbf {B} _ {b i} ^ {T} \mathbf {D} \left\{ \begin{array}{l} \alpha_ {x} \\ \alpha_ {y} \\ 0 \end{array} \right\} \frac {\Delta T _ {z}}{2 H _ {z}} d A \tag {1.5.58} +$$ + +$$ +\Delta T _ {z} / H _ {z} \quad : \text { 온도 구배 } +$$ + +판요소의 질량은 집중질량(lumped mass)과 분포질량(consistent mass)을 사용할 수 있으며, x,y,z 방향의 이동변위만을 반영한다. + + + +\- 분포질량 + +$$ +\mathbf {M} _ {i j} = \rho t \int_ {A _ {e}} N _ {i} N _ {j} d A \left(N _ {i}: \text {평면응력요소와 동일}\right) \tag {1.5.59} +$$ + +\- 집중질량 + +집중질량은 요소 전체질량( $\rho tA_{e}$ )을 분포질량의 대각 항 비율로 분배하여 사용한다. + +# 1-5-4 판요소의 두께/재료방향/옵셋 + +3 또는 4절점 평면판요소의 절점 별 두께는 그림 1.5.6과 같이 설정할 수 있으며, 6 또는 8절점 곡면판요소의 경우에는 꼭지점의 두께만을 설정할 수 있다. 판요소는 거동을 방향 별로 구분하여 면내거동 두께, 휩 두께, 전단변형 두께로 분류할 수 있으며, 면내거동 두께를 t 라 할 때 다음과 같은 값을 설정할 수 있다. + +• $12I/t^{3}$ +실제 힜 강성 I와 면내거동 두께로 계산한 힜 강성의 비율 (기본값=1.0) +• $t_{s}/t$ +실제 전단변형 두께 $t_{s}$ 와 면내거동 두께 t 의 비율 (기본값=0.83333) + + + +![](images/page-080_9c3a494f677699357d79205efab1fe8ed1fd5985c0597c74abf9fbdebdfaa592.jpg) + +
+text_image + +t₁ +1 +3 +t₃ +t₂ +2 +
+ +![](images/page-080_79c13c5a67a1812d2d61a6d80ca76b55772cc3181569fcf6d894b211a7d43788.jpg) + +
+text_image + +t₁ +1 +2 +t₂ +3 +4 +t₃ +t₄ +
+ +그림 1.5.6 판요소의 절점별 두께 + +판요소에서는 재료의 성질을 변형성분별로 입력할 수 있다. 이 때, 질량행렬은 면내거동에 대한 두께와 재료를 사용하여 계산한다. 이방성 재료의 경우 1축을 임의의 방향으로 설정할 수 있도록 재료좌표계(MCS)를 설정할 수 있다. 재료 좌표계가 판의곡면과 평행하지 않은 경우에는 그림 1.5.7과 같이 재료좌표계의 x 축을 판에 투영하여 계산한다. diff --git a/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_009.md b/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_009.md new file mode 100644 index 00000000..0fa2210c --- /dev/null +++ b/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_009.md @@ -0,0 +1,247 @@ + + +![](images/page-081_c5405785c6503e0aad93d4da08e8a10f5a11bf9556292b0a4528fd966620bfae.jpg) + +
+text_image + +ECS z-axis +MCS y-axis +ECS y-axis +MCS x-axis +ECS x-axis +MCS +projection +z +y +x +
+ +그림 1.5.7 재료좌표계가 요소로 투영된 경우 + +판요소는 그림 1.5.8과 같이 절점으로부터 옵셋(offset)을 부여할 수 있다. 옵셋은 절점으로부터 실제로 구조물이 존재하는 위치까지의 거리를 의미하며 요소좌표계의 z방향을 ‘+’ 방향으로 정의한다. + + + +Part 1 Element Library + +![](images/page-082_640ed361918410e4d3ddc398e2daeca5c9d2476b3bb3f2b1ababa5291d29ff1a.jpg) + +
+text_image + +Z₁ +1 +Z₃ +3 +Z₂ +2 +
+ +![](images/page-082_9e661c721585385feeb5e0fb2255d6f46cb61ab93ff312cb6d8cdbbfe4c2a25f.jpg) + +
+text_image + +Z₁ +1 +Z₂ +2 +3 +4 +Z₃ +Z₄ +
+ +그림 1.5.8 판요소에서 옵셋의 정의 + + + +# 1-5-5 요소결과 + +판요소의 해석 결과로는 절점에서의 응력/변형률, 요소내력을 출력하며, 부호와 방향은 요소좌표계를 따른다. 요소좌표계를 기준으로 출력된 결과는 전체좌표계 또는 출력좌표계로 변환하여 볼 수 있다. 응력과 변형률은 z 축을 따라 두 지점에서 계산하며, 면내거동에 대한 두께를 기준으로 상단(z = t/2)과 하단(z = -t/2)이 기본 위치이다. 주변형률 성분 중 $E_{3}$ 은 선형해석의 경우 0으로 간주하였음에 주의해야 한다. 판요소에서 출력되는 응력/변형률, 요소내력의 종류는 다음과 같다. + +• 응력 성분 $\sigma_{xx}, \sigma_{yy}, \tau_{xy}$ +- Von-Mises 응력 $\sqrt{\left(P_{1}^{2}+P_{2}^{2}-P_{1}P_{2}\right)}$ +• 최대 전단응력 $\sqrt{\left(\frac{\sigma_{xx}-\sigma_{yy}}{2}\right)^{2}+\tau_{xy}^{2}}$ +• 주응력 $P_{1}, P_{2}$ + +$$ +P _ {i} = \frac {\sigma_ {x x} + \sigma_ {y y}}{2} \pm \sqrt {\left(\frac {\sigma_ {x x} - \sigma_ {y y}}{2}\right) ^ {2} + \tau_ {x y} ^ {2}} +$$ + +- 변형률 성분 $\varepsilon_{xx}, \varepsilon_{yy}, \gamma_{xy}$ +- Von-Mises 변형률 $\frac{2}{3}\sqrt{\left(E_{1}^{2}+E_{2}^{2}-E_{1}E_{2}\right)}$ +• 체적(volumetric) 변형률 $E_{1} + E_{2}$ +• 주변형률 $E_{1}, E_{2}$ + + + +$$ +E _ {i} = \frac {\varepsilon_ {x x} + \varepsilon_ {y y}}{2} \pm \sqrt {\left(\frac {\varepsilon_ {x x} - \varepsilon_ {y y}}{2}\right) ^ {2} + \frac {\gamma_ {x y} ^ {2}}{4}} +$$ + +• 면내방향 내력 $N_{xx}, N_{yy}, N_{xy}$ +- 힘모멘트 $M_{xx}, M_{yy}, M_{xy}$ +- 전단력 $Q_{yz}, Q_{zx}$ + +절점에서의 응력/변형률 및 요소내력은 적분점에서 계산된 결과를 이용하여 외삽법 (extrapolation)에 의해 산출된다. 판요소의 적분점은 다음과 같다. + +• 3절점 삼각형 요소: 3 점 가우스 적분 +• 4절점 사각형 요소: 4 점 가우스 적분 +• 6절점 삼각형 요소: 3 점 가우스 적분 +• 8절점 사각형 요소: 4 점 가우스 적분 + + + +응력과 변형률에 대한 부호규약은 평면응력요소와 동일하며, 휨모멘트와 전단력의 방향은 그림 1.5.1과 같다. 그림 1.5.9는 요소내력 중 면내방향 성분의 방향을 나타내며,화살표 방향이 ‘+’ 부호를 의미한다. + +![](images/page-085_1c542c5ba66aaddb35586bd2c2e3ea566a208fa93f4d819e8edd4f0effa35436.jpg) + +
+text_image + +z +y +x +Nxy +ECS +Nx +Nx +Nxy +Nyy +
+ +그림 1.5.9 판요소의 결과 방향과 성분(면내 성분) + + + +# 1-6 평면변형요소 + +# 1-6-1 개요 + +평면변형요소는 동일 평면상에 위치한 3, 4, 6, 8개의 절점에 의해 정의되는 삼각형또는 사각형 요소가 있으며, 주로 댐(dam) 또는 터널(tunnel) 등과 같이 일정한 단면을 유지하면서 길이가 긴 구조물의 해석에 사용된다. 요소의 두께방향 변형률 성분은존재하지 않으며, 두께방향 응력은 포아송(Poisson) 효과에 의해 존재한다. 평면변형요소는 면내응력만을 고려할 수 있으며, 정적(선형/비선형) 해석 및 동적 해석에 모두 사용할 수 있다. 평면변형요소에서 변형을 정의하는 응력과 변형률은 다음과 같다. + +$$ +\boldsymbol {\sigma} = \left\{ \begin{array}{l} \sigma_ {x x} \\ \sigma_ {y y} \\ \tau_ {x y} \end{array} \right\}, \quad \boldsymbol {\varepsilon} = \left\{ \begin{array}{l} \varepsilon_ {x x} \\ \varepsilon_ {y y} \\ \gamma_ {x y} \end{array} \right\} +$$ + +(면내방향 응력과 변형률) + +응력과 변형률에 대한 부호규약은 그림 1.6.1과 같고, 화살표 방향이 ‘+’를 의미한다. + +![](images/page-086_189b1de82d9e422efe7188cf01ba909d8b2bedc0d939890352e3eb95ca748957.jpg) + +
+text_image + +y +x +τxy,γxy +σxx,εxx +τxy,γxy +σyy,εyy +
+ +그림 1.6.1 평면변형 요소의 응력/변형률 + + + +요소좌표계는 오른손법칙에 준한 x, , y z 축의 직교좌표계를 따르며, 방향은 그림1.6.2와 같이 설정된다. 사각형 요소는 절점 1과 절점 4의 중점에서 절점 2와 절점 3의 중점을 향하는 방향을 x 축 방향으로 하며, 삼각형 요소는 절점 1에서 2를 향하는 방향을 x 축 방향으로 설정한다. + +![](images/page-087_22fcd872f3cec1dd8a1227d4ef07726bc24f80a2cfc98806cf5b73ceff95535e.jpg) + +
+text_image + +3 ECS y-axis +1 ECS x-axis +(1→2 direction) +2 +
+ +![](images/page-087_b79146df34f049bebf78dea3822b2f479351b8e92f4d078cb2514d0f24f5f555.jpg) + +
+text_image + +ECS y-axis +3 +4 +ECS x-axis +1 +2 +
+ + + +![](images/page-088_c479e4c229e79da8dd020d71c236f874da04b39f61b014c34697ccce9092d472.jpg) + +
+text_image + +ECS y-axis +ECS x-axis +(1→2 direction) +1 +2 +3 +4 +5 +6 +
+ +![](images/page-088_3ca372b976f00c6674a9f429560a6fc79a286c2e122585d54824e4d65027b8d0.jpg) + +
+text_image + +ECS y-axis +3 +7 +4 +8 +6 +ECS x-axis +1 +5 +2 +
+ +그림 1.6.2 평면변형요소의 좌표계 + +평면변형요소의 종류는 연결된 절점 수에 따라 두 가지로 구분할 수 있다. 3절점 삼각형 요소와 4절점 사각형 요소는 1차(linear) 요소이며, 6절점 삼각형 요소와 8절점사각형 요소는 2차(quadratic) 요소이다. 1차 요소의 경우 4절점 사각형 요소는 변위및 응력값의 정확도가 높지만, 3절점 삼각형 요소는 변위에 비해 응력의 정확도가낮은 경향이 있다. 따라서 정밀한 해석결과가 필요한 부위에서는 3절점 삼각형 요소의 사용을 피하는 것이 바람직하다. 평면변형요소는 면외방향으로 변형률이 없지만,응력 $\sigma _ { z z }$ 는 존재함에 주의하여야 한다. + + + +# 1-6-2 유한요소 정식화 + +평면변형요소는 등매개변수(isoparametric) 요소로 구성되어 있으며, 4절점 사각형요소의 경우에는 비적합(incompatible) 모드를 이용한다. 그러나, 요소의 정식화 과정은 평면응력요소와 동일하므로 동일하므로 “1.4 평면응력요소”에서 설명했다.등방성(isotropic) 재료인 경우 평면변형요소에서 사용하는 응력과 변형률의 관계는다음과 같다. + +$$ +\mathbf {D} = \frac {E (1 - \nu)}{(1 + \nu) (1 - 2 \nu)} \left[ \begin{array}{c c c} 1 & \frac {\nu}{1 - \nu} & 0 \\ \frac {\nu}{1 - \nu} & 1 & 0 \\ 0 & 0 & \frac {1 - 2 \nu}{2 (1 - \nu)} \end{array} \right] \tag {1.6.1} +$$ + +강성과 변형을 계산하는데 사용되는 변위는 모두 면내에서 정의하지만, 두께 방향의응력이 존재한다. 두께 방향 응력은 등방성 재료인 경우 다음과 같이 계산할 수 있다. + +$$ +\sigma_ {z z} = \nu \left(\sigma_ {x x} + \sigma_ {y y}\right) \tag {1.6.2} +$$ + + + +# 1-6-3. 하중과 질량 + +평면변형요소에 적용되는 하중은 체적력(body force), 모서리하중 (edge load), 온도하중(thermal load), 프리스트레스하중(prestress load) 등이 있다. 체적력은 요소의 자중이나 관성력을 표현하고자 하는 하중이고, 모서리하중은 요소의 변에 가해지는 분포하중이다. 온도하중에는 절점온도, 요소온도 하중과 같은 면내방향 열 변형하중이 있다. 체적력과 모서리하중, 온도하중은 평면응력요소와 유사한 방법으로 계산한다. 온도하중의 경우 포아송 효과에 의한 면내 변형을 고려한다. + +평면변형요소의 질량은 집중질량(lumped mass)과 분포질량(consistent mass)을 사용할 수 있으며, x,y 방향의 이동변위만을 반영한다. + +\- 분포질량 + +$$ +\mathbf {M} _ {i j} = \rho t \int_ {A _ {e}} N _ {i} N _ {j} d A \tag {1.6.3} +$$ + +\- 집중질량 + +집중질량은 요소 전체질량( $\rho tA_{e}$ )을 분포질량의 대각 항 비율로 분배하여 사용한다 diff --git a/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_010.md b/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_010.md new file mode 100644 index 00000000..5cce89a5 --- /dev/null +++ b/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_010.md @@ -0,0 +1,316 @@ + + +# 1-6-4 요소결과 + +평면변형요소의 해석 결과로는 절점에서의 응력과 변형률을 출력한다. 평면변형요소를 이용한 해석은 전체좌표계의 X-Y, Y-Z, X-Z 중 하나의 평면에서 수행하므로, 응력과 변형률의 결과를 전체좌표계에서 출력한다. 전체좌표계를 기준으로 출력된 결과는 요소좌표계 또는 출력좌표계로 변환하여 볼 수 있다. + +다음은 X-Y 평면에서 해석하였을 경우, 요소에서 출력되는 응력과 변형률의 종류이다. + +• 응력 성분 $\sigma_{XX}, \sigma_{YY}, \sigma_{ZZ}, \tau_{XY}$ +- Von-Mises 응력 $\sqrt{\left(P_{1}^{2}+P_{2}^{2}+P_{3}^{2}-P_{1}P_{2}-P_{2}P_{3}-P_{3}P_{1}\right)}$ +• 최대 전단응력 $\frac{\max(|P_{1}-P_{2}|,|P_{2}-P_{3}|,|P_{3}-P_{1}|)}{2}$ +• 주응력 $P_{1}, P_{2}, P_{3}$ + +$$ +P _ {i} = \frac {\sigma_ {X X} + \sigma_ {Y Y}}{2} \pm \sqrt {\left(\frac {\sigma_ {X X} - \sigma_ {Y Y}}{2}\right) ^ {2} + \tau_ {X Y} ^ {2}} \quad \text {과} \quad \sigma_ {Z Z} \quad \text {중 큰 값부터} \quad P _ {1}, P _ {2}, P _ {3} \text {이} +$$ + +다. + +• 변형률 성분 $\varepsilon_{XX}, \varepsilon_{YY}, \gamma_{XY}$ +- Von-Mises 변형률 $\frac{2}{3}\sqrt{\left(E_{1}^{2}+E_{2}^{2}-E_{1}E_{2}\right)}$ +• 체적(volumetric) 변형률 $E_{1} + E_{2}$ + + + +\- 주변형률 + +$$ +E _ {1}, E _ {2} +$$ + +$$ +E _ {i} = \frac {\varepsilon_ {X X} + \varepsilon_ {Y Y}}{2} \pm \sqrt {\left(\frac {\varepsilon_ {X X} - \varepsilon_ {Y Y}}{2}\right) ^ {2} + \frac {\gamma_ {X Y} ^ {2}}{4}} +$$ + +절점에서의 응력/변형률 및 요소내력은 적분점에서 계산된 결과를 이용하여 외삽법 (extrapolation)에 의해 산출된다. 평면변형요소의 적분점은 다음과 같다. + +• 3절점 삼각형 요소: 1 점 가우스 적분 +• 4절점 사각형 요소: 4 점 가우스 적분 +• 6절점 삼각형 요소: 3 점 가우스 적분 +• 8절점 사각형 요소: 9 점 가우스 적분 + +응력과 변형률에 대한 부호규약은 그림 1.6.3과 같고, 화살표 방향이 ‘+’ 부호를 의미한다. + +![](images/page-092_679fb66e1b84ddf81a4d7194982db4ce0d777a078a179449b5f54dd4cf571374.jpg) + +
+text_image + +σ_zz +(ε_zz = 0) +τ_xy, γ_xy +σ_xx, ε_xx +τ_xy, γ_xy +σ_yy, ε_yy +GCS +X +Y +Z +
+ +그림 1.6.3 평면변형요소의 결과 방향과 성분 + + + +# 1-7 축대칭요소 + +# 1-7-1 개요 + +축대칭요소는 형상, 재질, 하중 조건 등이 임의의 축에 대해 회전대칭 조건을 만족하는 구조물(deep well, circular foundation 및 circular tunnel 등)의 해석에 사용한다. 축대칭요소는 다른 종류의 요소들과 혼용할 수 없으며, 정적(선형/비선형) 해석에사용 가능하다. 축대칭요소는 구조물의 축대칭적 특성을 근거로 하기 때문에 원주방향 전단변형을 고려하지 않는다. 축대칭요소에서 변형을 정의하는 응력과 변형률은다음과 같다. + +$$ +\boldsymbol {\sigma} = \left\{ \begin{array}{l} \sigma_ {x x} \\ \sigma_ {y y} \\ \sigma_ {z z} \\ \tau_ {x y} \end{array} \right\}, \quad \boldsymbol {\varepsilon} = \left\{ \begin{array}{l} \varepsilon_ {x x} \\ \varepsilon_ {y y} \\ \varepsilon_ {z z} \\ \gamma_ {x y} \end{array} \right\} (\text {단면내방향 응력/변형률과 원주방향 수직응력/변형률}) +$$ + +응력과 변형률에 대한 부호규약은 그림 1.7.1과 같고, 화살표 방향이‘+’를 의미한다. + +![](images/page-093_d68827407c4f99fa7b9c95376e290c604fbb61f36be0231c718be04bc1b56c6e.jpg) + +
+text_image + +GCS → X +z +y +ECS +x +z +σxx, εxx +τxy, γxy +σyy, εyy +τxy, γxy +
+ +그림 1.7.1 축대칭요소의 응력/변형률 + + + +요소좌표계는 오른손법칙에 준한 x, y, z 축의 직교좌표계를 따르며, 방향은 그림1.7.2와 같이 설정된다. 사각형 요소는 절점 1과 절점 4의 중점에서 절점 2와 절점 3의 중점 방향을 x 축을 향하는 방향으로 하며, 삼각형 요소는 절점 1에서 2를 향하는 방향을 x 축 방향으로 설정한다. + +![](images/page-094_81d3133e679e480f36a468028fd089e314d80d0323145911740002f77f08c7e0.jpg) + +
+text_image + +3 ECS y-axis +1 ECS x-axis +(1 → 2 direction) +2 +
+ +![](images/page-094_96d4c33e4e5c9126414bc7f041d1d24ef7d75583a72f1110acfdf15a867fccb0.jpg) + +
+text_image + +ECS y-axis +4 +3 +ECS x-axis +1 +2 +
+ + + +![](images/page-095_60ed38affcef1c40bfff01db1e3ceb0d55fb67014191cc6241973b800fafa9f9.jpg) + +
+text_image + +ECS y-axis +ECS x-axis +(1→2 direction) +1 +2 +3 +4 +5 +6 +
+ +![](images/page-095_ebfbdcf0393b0e913a3a651010279973f7d48d96e59c125cef8e41d9267285f4.jpg) + +
+text_image + +ECS y-axis +3 +7 +4 +8 +6 +ECS x-axis +1 +5 +2 +
+ +그림 1.7.2 축대칭요소의 좌표계 + + + +축대칭요소는 전체좌표계에서 Z 축에 대한 대칭 구조만을 모사하므로, X Z − 평면에존재하여야 한다. 요소의 두께는 그림 1.7.3과 같이 단위 폭( 1.0radian )만큼 자동으로 고려된다. + +![](images/page-096_c04a003fd833954ba3a5814a68340bac6f2bb7a494e01e72e9c4cc0d9f3cac6c.jpg) + +
+text_image + +Z(axis of rotation) +1.0 radian (unit width) +an axisymmetric element +X +Y +
+ +그림 1.7.3 축대칭요소의 강성 계산영역 + +축대칭요소의 종류는 연결된 절점 수에 따라 두 가지로 구분할 수 있다. 3절점 삼각형 요소와 4절점 사각형 요소는 1차(linear) 요소이며, 6절점 삼각형 요소와 8절점사각형 요소는 2차(quadratic) 요소이다. 축대칭요소에는 원주방향으로 수직응력과변형률이 존재함에 주의해야 한다. + + + +# 1-7-2 유한요소 정식화 + +축대칭요소는 등매개변수(isoparametric) 요소로 구성되어 있으며, 비적합 (incompatible) 모드는 사용하지 않는다. 그리고 요소좌표계에서 x, y 방향의 이동 변위(translation) u, v 만을 가진다. + +$$ +\mathbf {u} _ {i} = \left\{u _ {i} \quad v _ {i} \right\} ^ {T} \tag {1.7.1} +$$ + +축대칭요소는 절점 개수에 관계 없이 유사한 과정으로 강성을 계산할 수 있기 때문에, 절점 수 N 개를 가지는 요소에 대하여 일괄적으로 설명한다. + +요소 내 임의의 좌표 x,y 와 이동변위 u,v 는 다음과 같이 나타낼 수 있다. + +$$ +x = \sum_ {i = 1} ^ {N} N _ {i} x _ {i}, y = \sum_ {i = 1} ^ {N} N _ {i} y _ {i}, u = \sum_ {i = 1} ^ {N} N _ {i} u _ {i}, v = \sum_ {i = 1} ^ {N} N _ {i} v _ {i} \tag {1.7.2} +$$ + +• 3절점 삼각형 + +$$ +N _ {1} = 1 - \xi - \eta , N _ {2} = \xi , N _ {3} = \eta \tag {1.7.3} +$$ + +4절점 사각형 + +$$ +N _ {1} = \frac {1}{4} (1 - \xi) (1 - \eta), N _ {2} = \frac {1}{4} (1 + \xi) (1 - \eta), +$$ + +$$ +N _ {3} = \frac {1}{4} (1 + \xi) (1 + \eta), N _ {4} = \frac {1}{4} (1 - \xi) (1 + \eta) \tag {1.7.4} +$$ + +6절점 삼각형 + +$$ +N _ {1} = (1 - \xi - \eta) (1 - 2 \xi - 2 \eta), N _ {2} = \xi (2 \xi - 1), N _ {3} = \eta (2 \eta - 1) +$$ + +$$ +N _ {4} = 4 \xi (1 - \xi - \eta), N _ {5} = 4 \xi \eta , N _ {6} = 4 \eta (1 - \xi - \eta) \tag {1.7.5} +$$ + + + +• 8절점 사각형 + +$$ +N _ {1} = \frac {1}{4} (1 - \xi) (1 - \eta) - \frac {1}{2} N _ {5} - \frac {1}{2} N _ {8}, \quad N _ {2} = \frac {1}{4} (1 + \xi) (1 - \eta) - \frac {1}{2} N _ {5} - \frac {1}{2} N _ {6} +$$ + +$$ +N _ {3} = \frac {1}{4} (1 + \xi) (1 + \eta) - \frac {1}{2} N _ {6} - \frac {1}{2} N _ {7}, \quad N _ {4} = \frac {1}{4} (1 - \xi) (1 + \eta) - \frac {1}{2} N _ {7} - \frac {1}{2} N _ {8} +$$ + +$$ +N _ {5} = \frac {1}{2} (1 - \xi^ {2}) (1 - \eta), N _ {6} = \frac {1}{2} (1 + \xi) (1 - \eta^ {2}), N _ {7} = \frac {1}{2} (1 - \xi^ {2}) (1 + \eta) +$$ + +$$ +N _ {8} = \frac {1}{2} (1 - \xi) (1 - \eta^ {2}) \tag {1.7.6} +$$ + +절점 변위와 변형률 ε 의 관계는 $B_{i}$ 에 의하여 다음과 같이 나타낼 수 있다. + +$$ +\boldsymbol {\varepsilon} = \sum_ {i = 1} ^ {N} \mathbf {B} _ {i} \mathbf {u} _ {i} \tag {1.7.7} +$$ + +행렬 $B_{i}$ 는 형상함수의 미분값으로 다음과 같이 표현된다. + +$$ +\mathbf {B} _ {i} = \left[ \begin{array}{c c} \frac {\partial N _ {i}}{\partial x} & 0 \\ 0 & \frac {\partial N _ {i}}{\partial y} \\ \alpha \frac {N _ {i}}{X} & \beta \frac {N _ {i}}{X} \\ \frac {\partial N _ {i}}{\partial y} & \frac {\partial N _ {i}}{\partial x} \end{array} \right] \tag {1.7.8} +$$ + +$$ +\alpha : \vec {x} \bullet \overrightarrow {X} +$$ + +$$ +\beta \quad : \vec {y} \bullet \vec {X} +$$ + +요소좌표계 x,y 에서의 반지름으로 전체좌표계의 X 값을 사용하였다. + +행렬 $B_{i}$ 를 이용하여 면내변형에 관계된 요소강성 행렬을 표현하면 다음과 같다. + +$$ +\mathbf {K} _ {i j} = \int_ {A _ {e}} X \mathbf {B} _ {i} ^ {T} \mathbf {D} \mathbf {B} _ {j} d A \tag {1.7.9} +$$ + + + +위 식은 1.0radian 에 대한 강성이 된다. 등방성(isotropic) 재료인 경우 축대칭요소에서 사용하는 응력과 변형률의 관계는 다음과 같다. + +$$ +\mathbf {D} = \frac {E (1 - \nu)}{(1 + \nu) (1 - 2 \nu)} \left[ \begin{array}{c c c c} 1 & \frac {\nu}{1 - \nu} & \frac {\nu}{1 - \nu} & 0 \\ \frac {\nu}{1 - \nu} & 1 & \frac {\nu}{1 - \nu} & 0 \\ \frac {\nu}{1 - \nu} & \frac {\nu}{1 - \nu} & 1 & 0 \\ 0 & 0 & 0 & \frac {1 - 2 \nu}{2 (1 - \nu)} \end{array} \right] \tag {1.7.10} +$$ + + + +# 1-7-3 하중 + +축대칭요소에 적용되는 하중은 체적력(body force), 모서리하중(edge load), 온도하중(thermal load), 프리스트레스하중(prestress load) 등이 있다. 체적력은 요소의 자중이나 관성력을 표현하고자 하는 하중이고, 모서리하중은 요소의 변에 가해지는 분포하중이다. 축대칭요소에서 사용하는 모서리하중은 단위 면적당 힘을 의미하며, 절점에 입력된 하중은 원주(2πr)에 대해 적분된 값으로 간주하므로 입력 시에 주의해야 한다. + +\- 체적력 + +$$ +\mathbf {F} _ {i} = \int_ {A _ {e}} X N _ {i} \left\{ \begin{array}{l} \omega_ {x} \\ \omega_ {y} \\ \omega_ {z} \end{array} \right\} d A \tag {1.7.11} +$$ + +여기서, + +$$ +\omega_ {x}, \omega_ {y}, \omega_ {z} \quad : \text { 단위 체적당 자중(방향별) } +$$ + +\- 모서리하중 + +$$ +\mathbf {F} _ {i} = \int_ {L} X N _ {i} \left\{ \begin{array}{l} P _ {x} \\ P _ {y} \\ 0 \end{array} \right\} d s \tag {1.7.12} +$$ + +여기서, + +$$ +P _ {x}, P _ {y} \quad : \text { 단위 면적당 하중(방향별) } +$$ + +\- 온도하중 + +$$ +\mathbf {F} _ {i} = \int_ {A _ {e}} X \mathbf {B} _ {i} ^ {T} \mathbf {D} \left\{ \begin{array}{l} \alpha_ {x} \\ \alpha_ {y} \\ \alpha_ {z} \\ 0 \end{array} \right\} \Delta T d A \tag {1.7.13} +$$ diff --git a/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_011.md b/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_011.md new file mode 100644 index 00000000..99333032 --- /dev/null +++ b/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_011.md @@ -0,0 +1,338 @@ + + +여기서, + +$$ +\alpha_ {x}, \alpha_ {y}, \alpha_ {z} \quad : \text { 열팽창 계수(방향별) } +$$ + +# 1-7-4 요소결과 + +축대칭요소의 해석 결과로는 절점에서의 응력과 변형률을 출력한다. 축대칭요소를 이용한 해석은 전체좌표계의 Z 축에 대한 대칭 구조물에 사용하므로, X Z − 평면에서해석을 수행한다. 따라서 응력과 변형률의 결과를 전체좌표계에서 출력한다. 전체좌표계를 기준으로 출력된 결과는 요소좌표계 또는 출력좌표계로 변환하여 볼 수 있다.다음은 요소에서 출력되는 응력과 변형률의 종류이다. + +응력 성분 $\sigma _ { \scriptscriptstyle { X X } } , \sigma _ { \scriptscriptstyle { Y Y } } , \sigma _ { \scriptscriptstyle { Z Z } } , \tau _ { \scriptscriptstyle { Z X } }$ + +Von-Mises 응력 $\sqrt { \left( P _ { 1 } ^ { 2 } + P _ { 2 } ^ { 2 } + P _ { 3 } ^ { 2 } - P _ { 1 } P _ { 2 } - P _ { 2 } P _ { 3 } - P _ { 3 } P _ { 1 } \right) }$ + +최대 전단응력 $\frac { \operatorname* { m a x } ( \big | P _ { 1 } - P _ { 2 } \big | , \big | P _ { 2 } - P _ { 3 } \big | , \big | P _ { 3 } - P _ { 1 } \big | ) } { 2 }$ + +주응력 $P _ { 1 } , P _ { 2 } , P _ { 3 }$ + +$$ +P _ {i} = \frac {\sigma_ {X X} + \sigma_ {Z Z}}{2} \pm \sqrt {\left(\frac {\sigma_ {X X} - \sigma_ {Z Z}}{2}\right) ^ {2} + \tau_ {Z X} ^ {2}} \text {와} \sigma_ {Y Y} \text {종 큰 값부터} P _ {1}, P _ {2}, P _ {3} \text {이다.} +$$ + +변형률 성분 $\varepsilon _ { X X } , \varepsilon _ { Y Y } , \varepsilon _ { Z Z } , \gamma _ { Z X }$ + +Von-Mises 변형률 $\frac { 2 } { 3 } \sqrt { \left( E _ { 1 } ^ { 2 } + E _ { 2 } ^ { 2 } + E _ { 3 } ^ { 2 } - E _ { 1 } E _ { 2 } - E _ { 2 } E _ { 3 } - E _ { 3 } E _ { 1 } \right) }$ + +$r _ { 1 } ( \ r _ { 1 } , \ r _ { 1 } ) ( \ r _ { 2 } , \ r _ { 1 } , \ r _ { 2 } ) , t _ { 1 } = 0 , t _ { 2 } = 1$ $E _ { 1 } + E _ { 2 } + E _ { 3 }$ + + + +주변형률 + +$$ +E _ {1}, E _ {2}, E _ {3} +$$ + +$$ +E _ {i} = \frac {\varepsilon_ {X X} + \varepsilon_ {Z Z}}{2} \pm \sqrt {\left(\frac {\varepsilon_ {X X} - \varepsilon_ {Z Z}}{2}\right) ^ {2} + \frac {\gamma_ {Z X} ^ {2}}{4}} \text {와} \varepsilon_ {Y Y} \text {중 큰 값부터} E _ {1}, E _ {2}, E _ {3} \text {이다.} +$$ + +절점에서의 응력/변형률은 요소 내의 적분점에서 계산된 결과를 이용하여 외삽법 (extrapolation)에 의해 산출된다. 축대칭요소의 적분점은 다음과 같다. + +• 3절점 삼각형 : 1 점 가우스 적분 +• 4절점 사각형 : 4 점 가우스 적분 +• 6절점 삼각형 : 3 점 가우스 적분 +• 8절점 사각형 : 9 점 가우스 적분 + +응력과 변형률에 대한 부호규약은 그림 1.7.4와 같고, 화살표 방향이 ‘+’ 부호를 의미한다. + +![](images/page-102_c622a48f238209186e98998b23e9ed90707d6b77ab5cc667facb63bd9733d696.jpg) + +
+text_image + +Z +GCS → X +τzx, γzx +σyy, εyy +σxx, εxx +τzx, γzx +σzz, εzz +
+ +그림 1.7.4 축대칭요소의 결과 방향과 성분 + + + +# 1-8 입체요소 + +# 1-8-1 개요 + +입체요소는 주로 콘크리트 기초, 자동차 엔진, 두꺼운 벽, 고무 등과 같이 부피가 있는 구조물의 모델링에 주로 이용된다. midas FEA에서 사용할 수 있는 입체요소로는8면체(hexahedron), 4면체(tetrahedron), 5면체(pentahedron) 요소가 있으며, 정적(선형/비선형) 해석 및 동적 해석에 모두 사용할 수 있다. 입체요소에서 변형을 정의하는 응력과 변형률은 다음과 같다. + +$$ +\boldsymbol {\sigma} = \left\{ \begin{array}{l} \sigma_ {x x} \\ \sigma_ {y y} \\ \sigma_ {z z} \\ \tau_ {x y} \\ \tau_ {y z} \\ \tau_ {z x} \end{array} \right\}, \quad \boldsymbol {\varepsilon} = \left\{ \begin{array}{l} \varepsilon_ {x x} \\ \varepsilon_ {y y} \\ \varepsilon_ {z z} \\ \gamma_ {x y} \\ \gamma_ {y z} \\ \gamma_ {z x} \end{array} \right\} \quad (\text {응력/변형률}) +$$ + +응력과 변형률에 대한 부호규약은 그림 1.8.1과 같고, 화살표 방향이 ‘+’를 의미한다. + +![](images/page-103_b927e7fc78481ecd1cc18b07444bfa773c6cec727e65b2b799a1fa43964fc1d3.jpg) + +
+text_image + +σ_zz, ε_z +τ_yz, γ_yz +τ_zx, γ_zx +z +y +τ_zx, γ_zx +τ_xy, γ_xy +ECS +x +σ_xx, ε_xx +σ_yy, ε_yy +τ_yz, γ_yz +
+ +그림 1.8.1 입체요소의 응력/변형률 + + + +요소좌표계는 오른손법칙에 준한 x, , y z 축의 직교좌표계를 따르며, 방향은 그림1.8.2와 같이 설정한다. 8면체요소는 절점 1과 절점 4의 중점에서 절점 2와 절점 3의중점 방향을 x 축을 향하는 방향으로 하며, 4면체와 5면체 요소는 절점 1에서 2를향하는 방향을 x 축 방향으로 설정한다. + + + +![](images/page-105_77405d30d36b083170e2747acb3fa8cea4f589094e31efaba874dbb79c32ac88.jpg) +그림 1.8.2. 입체요소의 좌표계 + +입체요소 에서 사용하는 형상 함수는 최대 2차 함수로 되어 있다. 8절점 6면체 요소,6절점 5면체 요소, 4절점 4면체 요소는 1차(linear) 요소이며, 20절점 6면체 요소,15절점 5면체 요소, 10절점 4면체 요소는 2차(quadratic) 요소이다. 일반적으로 4면체, 5면체 요소 보다 6면체 요소를 사용했을 때 보다 정확한 해를 구할 수 있으므로,정밀한 해석 결과가 필요한 부분에서는 6면체 요소를 사용하는 것이 바람직하다. + + + +# 1-8-2 유한요소 정식화 + +입체요소는 등매개변수(isoparametric) 요소로 구성되어 있으며, 8절점 6면체 요소와 6절점 5면체 요소의경우에는 비적합(incompatible) 모드를 이용한다. 입체요소는 요소좌표계에서 x, y, z 방향의 이동변위(translation) u, v, w 만을 가진다. + +$$ +\mathbf {u} _ {i} = \left\{u _ {i} \quad v _ {i} \quad w _ {i} \right\} ^ {T} \tag {1.8.1} +$$ + +비적합 모드를 제외하면 절점 개수에 관계 없이 모든 입체요소의 강성을 유사한 과정으로 계산할 수 있기 때문에, 절점 수 N 개를 가지는 요소에 대하여 일괄적으로 설명한다. + +요소 내 임의의 좌표 x, y, z와 이동변위 u, v, w는 다음과 같이 나타낼 수 있다 + +$$ +x = \sum_ {i = 1} ^ {N} N _ {i} x _ {i}, y = \sum_ {i = 1} ^ {N} N _ {i} y _ {i}, z = \sum_ {i = 1} ^ {N} N _ {i} z _ {i} +$$ + +$$ +u = \sum_ {i = 1} ^ {N} N _ {i} u _ {i}, v = \sum_ {i = 1} ^ {N} N _ {i} v _ {i}, w = \sum_ {i = 1} ^ {N} N _ {i} w _ {i} \tag {1.8.2} +$$ + +4절점 4면체 + +$$ +N _ {1} = 1 - \xi - \eta - \zeta , N _ {2} = \xi , N _ {3} = \eta , N _ {4} = \zeta \tag {1.8.3} +$$ + +6절점 5면체 + +$$ +N _ {1} = \frac {\lambda}{2} (1 - \zeta), N _ {2} = \frac {\xi}{2} (1 - \zeta), N _ {3} = \frac {\eta}{2} (1 - \zeta), N _ {4} = \frac {\lambda}{2} (1 + \zeta) +$$ + +$$ +N _ {5} = \frac {\xi}{2} (1 + \zeta), N _ {6} = \frac {\eta}{2} (1 + \zeta) \tag {1.8.4} +$$ + +$$ +\lambda = 1 - \xi - \eta +$$ + +8절점 6면체 + +$$ +N _ {1} = \frac {1}{8} (1 - \xi) (1 - \eta) (1 - \zeta), N _ {2} = \frac {1}{8} (1 + \xi) (1 - \eta) (1 - \zeta) +$$ + + + +$$ +N _ {3} = \frac {1}{8} (1 + \xi) (1 + \eta) (1 - \zeta), N _ {4} = \frac {1}{8} (1 - \xi) (1 + \eta) (1 - \zeta) +$$ + +$$ +N _ {5} = \frac {1}{8} (1 - \xi) (1 - \eta) (1 + \zeta), N _ {6} = \frac {1}{8} (1 + \xi) (1 - \eta) (1 + \zeta) +$$ + +$$ +N _ {7} = \frac {1}{8} (1 + \xi) (1 + \eta) (1 + \zeta), N _ {8} = \frac {1}{8} (1 - \xi) (1 + \eta) (1 + \zeta) \tag {1.8.5} +$$ + +• 10절점 4면체 + +$$ +N _ {1} = - \lambda (1 - 2 \lambda), N _ {2} = - \xi (1 - 2 \xi), N _ {3} = - \eta (1 - 2 \eta), N _ {4} = - \zeta (1 - 2 \zeta) +$$ + +$$ +N _ {5} = 4 \xi \lambda , N _ {6} = 4 \xi \eta , N _ {7} = 4 \eta \lambda , N _ {8} = 4 \zeta \lambda , N _ {9} = 4 \xi \zeta , N _ {1 0} = 4 \eta \zeta +$$ + +$$ +\lambda = 1 - \xi - \eta - \zeta \tag {1.8.6} +$$ + +• 15절점 5면체 + +$$ +N _ {1} = \frac {\lambda}{2} (1 - 2 \lambda) \zeta (1 - \zeta), N _ {2} = \frac {\xi}{2} (1 - 2 \xi) \zeta (1 - \zeta), N _ {3} = \frac {\eta}{2} (1 - 2 \eta) \zeta (1 - \zeta) +$$ + +$$ +N _ {4} = - \frac {\lambda}{2} (1 - 2 \lambda) \zeta (1 + \zeta), N _ {5} = - \frac {\xi}{2} (1 - 2 \xi) \zeta (1 + \zeta) +$$ + +$$ +N _ {6} = - \frac {\eta}{2} (1 - 2 \eta) \zeta (1 + \zeta) +$$ + +$$ +N _ {7} = - 2 \xi \lambda \zeta (1 - \zeta), N _ {8} = - 2 \xi \eta \zeta (1 - \zeta) +$$ + +$$ +N _ {9} = - 2 \eta \lambda \zeta (1 - \zeta), N _ {1 0} = 2 \xi \lambda \zeta (1 + \zeta) +$$ + +$$ +N _ {1 1} = 2 \xi \eta \zeta (1 + \zeta), N _ {1 2} = 2 \eta \lambda \zeta (1 + \zeta), N _ {1 3} = (1 - \xi - \eta) (1 - \zeta^ {2}) +$$ + + + +$$ +N _ {1 4} = \xi \left(1 - \zeta^ {2}\right), N _ {1 5} = \eta \left(1 - \zeta^ {2}\right) +$$ + +$$ +\lambda = 1 - \xi - \eta \tag {1.8.7} +$$ + +20절점 6면체 + +$$ +N _ {1} = \frac {1}{8} (1 - \xi) (1 - \eta) (1 - \zeta) (- 2 - \xi - \eta - \zeta) +$$ + +$$ +N _ {2} = \frac {1}{8} (1 + \xi) (1 - \eta) (1 - \zeta) (- 2 + \xi - \eta - \zeta) +$$ + +$$ +N _ {3} = \frac {1}{8} (1 + \xi) (1 + \eta) (1 - \zeta) (- 2 + \xi + \eta - \zeta) +$$ + +$$ +N _ {4} = \frac {1}{8} (1 - \xi) (1 + \eta) (1 - \zeta) (- 2 - \xi + \eta - \zeta) +$$ + +$$ +N _ {5} = \frac {1}{8} (1 - \xi) (1 - \eta) (1 + \zeta) (- 2 - \xi - \eta + \zeta) +$$ + +$$ +N _ {6} = \frac {1}{8} (1 + \xi) (1 - \eta) (1 + \zeta) (- 2 + \xi - \eta + \zeta) +$$ + +$$ +N _ {7} = \frac {1}{8} (1 + \xi) (1 + \eta) (1 + \zeta) (- 2 + \xi + \eta + \zeta) +$$ + +$$ +N _ {8} = \frac {1}{8} (1 - \xi) (1 + \eta) (1 + \zeta) (- 2 - \xi + \eta + \zeta) +$$ + +$$ +N _ {9} = \frac {1}{4} (1 - \eta) (1 - \zeta) (1 - \xi^ {2}), \quad N _ {1 0} = \frac {1}{4} (1 + \xi) (1 - \zeta) (1 - \eta^ {2}) +$$ + +$$ +N _ {1 1} = \frac {1}{4} (1 + \eta) (1 - \zeta) (1 - \xi^ {2}) +$$ + +$$ +N _ {1 2} = \frac {1}{4} (1 - \xi) (1 - \zeta) (1 - \eta^ {2}), N _ {1 3} = \frac {1}{4} (1 - \eta) (1 + \zeta) (1 - \xi^ {2}) +$$ + + + +$$ +N _ {1 4} = \frac {1}{4} (1 + \xi) (1 + \zeta) (1 - \eta^ {2}) +$$ + +$$ +N _ {1 5} = \frac {1}{4} (1 - \eta) (1 + \zeta) (1 - \xi^ {2}), N _ {1 6} = \frac {1}{4} (1 + \xi) (1 - \zeta) (1 - \eta^ {2}) +$$ + +$$ +N _ {1 7} = \frac {1}{4} (1 - \xi) (1 - \eta) (1 - \zeta^ {2}) +$$ + +$$ +N _ {1 8} = \frac {1}{4} (1 + \xi) (1 - \eta) (1 - \zeta^ {2}), N _ {1 9} = \frac {1}{4} (1 + \xi) (1 + \eta) (1 - \zeta^ {2}) +$$ + +$$ +N _ {2 0} = \frac {1}{4} (1 - \xi) (1 + \eta) (1 - \zeta^ {2}) \tag {1.8.8} +$$ + +절점 변위 u 와 변형률 ε 의 관계는 Bi 에 의하여 다음과 같이 나타낼 수 있다. + +$$ +\boldsymbol {\varepsilon} = \sum_ {i = 1} ^ {N} \mathbf {B} _ {i} \mathbf {u} _ {i} \tag {1.8.9} +$$ + +행렬 Bi 는 형상함수의 미분값으로 다음과 같이 표현된다. + +$$ +\mathbf {B} _ {i} = \left[ \begin{array}{c c c} \frac {\partial N _ {i}}{\partial x} & 0 & 0 \\ 0 & \frac {\partial N _ {i}}{\partial y} & 0 \\ 0 & 0 & \frac {\partial N _ {i}}{\partial z} \\ \frac {\partial N _ {i}}{\partial y} & \frac {\partial N _ {i}}{\partial x} & 0 \\ 0 & \frac {\partial N _ {i}}{\partial z} & \frac {\partial N _ {i}}{\partial y} \\ \frac {\partial N _ {i}}{\partial z} & 0 & \frac {\partial N _ {i}}{\partial x} \end{array} \right] \tag {1.8.10} +$$ + +행렬 B 를 이용하여 면내변형에 관계된 요소강성 행렬을 표현하면 다음과 같다. + +$$ +\mathbf {K} _ {i j} = \int_ {V _ {e}} \mathbf {B} _ {i} ^ {T} \mathbf {D} \mathbf {B} _ {j} d V \tag {1.8.11} +$$ + +Ve : 요소의 체적 + + + +여기서 D 는 응력과 변형률 관계를 나타내는 행렬이며, 등방성(isotropic) 재료의 경우 다음과 같다. + +$$ +\mathbf {D} = \frac {E}{(1 + \nu) (1 - 2 \nu)} \left[ \begin{array}{c c c c c c} 1 - \nu & \nu & \nu & 0 & 0 & 0 \\ \nu & 1 - \nu & \nu & 0 & 0 & 0 \\ \nu & \nu & 1 - \nu & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac {1 - 2 \nu}{2} & 0 & 0 \\ 0 & 0 & 0 & 0 & \frac {1 - 2 \nu}{2} & 0 \\ 0 & 0 & 0 & 0 & 0 & \frac {1 - 2 \nu}{2} \end{array} \right] \tag {1.8.12} +$$ + +입체요소는 재료좌표계를 설정할 수 있으며, 설정된 재료좌표계에 따라 행렬 D 를 좌표변환 하여 사용한다. + +선형 해석 시 6절점 5면체 요소와 8절점 6면체 요소는 비적합 모드를 포함하여 계산한다. 비적합 모드를 포함한 경우에는 절점변위 이외에 추가적인 자유도를 가지게 된다. 비적합 모드를 이용하는 자세한 방법은 평면응력요소를 참조하고, 본 절에서는 비적합 모드의 형상과 주요 행렬만을 나열한다. + +6절점 5면체 + +$$ +\mathbf {u} _ {a} = \left\{a _ {1} \quad b _ {1} \quad c _ {1} \right\} ^ {T} \tag {1.8.13} +$$ + +$$ +u = \sum_ {i = 1} ^ {6} N _ {i} u _ {i} + a _ {1} P _ {1}, \quad v = \sum_ {i = 1} ^ {6} N _ {i} v _ {i} + b _ {1} P _ {1}, \quad w = \sum_ {i = 1} ^ {6} N _ {i} w _ {i} + c _ {1} P _ {1} \tag {1.8.14} +$$ + +$$ +P _ {1} = 1 - \zeta^ {2} \tag {1.8.15} +$$ diff --git a/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_012.md b/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_012.md new file mode 100644 index 00000000..f5a31414 --- /dev/null +++ b/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_012.md @@ -0,0 +1,311 @@ + + +$$ +\mathbf {B} _ {a} = \left[ \begin{array}{c c c} \frac {\partial P _ {1}}{\partial x} & 0 & 0 \\ 0 & \frac {\partial P _ {1}}{\partial y} & 0 \\ 0 & 0 & \frac {\partial P _ {1}}{\partial z} \\ \frac {\partial P _ {1}}{\partial y} & \frac {\partial P _ {1}}{\partial x} & 0 \\ 0 & \frac {\partial P _ {1}}{\partial z} & \frac {\partial P _ {1}}{\partial y} \\ \frac {\partial P _ {1}}{\partial z} & 0 & \frac {\partial P _ {1}}{\partial x} \end{array} \right] \tag {1.8.16} +$$ + +8절점 6면체 + +$$ +\mathbf {u} _ {a} = \left\{a _ {1} \quad b _ {1} \quad c _ {1} \quad a _ {2} \quad b _ {2} \quad c _ {2} \quad a _ {3} \quad b _ {3} \quad c _ {3} \right\} ^ {T} \tag {1.8.17} +$$ + +$$ +u = \sum_ {i = 1} ^ {8} N _ {i} u _ {i} + a _ {1} P _ {1} + a _ {2} P _ {2} + a _ {3} P _ {3} \quad v = \sum_ {i = 1} ^ {8} N _ {i} v _ {i} + b _ {1} P _ {1} + b _ {2} P _ {2} + b _ {3} P _ {3}, +$$ + +$$ +w = \sum_ {i = 1} ^ {8} N _ {i} w _ {i} + c _ {1} P _ {1} + c _ {2} P _ {2} + c _ {3} P _ {3} \tag {1.8.18} +$$ + +$$ +P _ {1} = 1 - \xi^ {2}, P _ {2} = 1 - \eta^ {2}, P _ {3} = 1 - \zeta^ {2} \tag {1.8.19} +$$ + +$$ +\mathbf {B} _ {a} = \left[ \begin{array}{c c c c c c c c c} \frac {\partial P _ {1}}{\partial x} & 0 & 0 & \frac {\partial P _ {2}}{\partial x} & 0 & 0 & \frac {\partial P _ {3}}{\partial x} & 0 & 0 \\ 0 & \frac {\partial P _ {1}}{\partial y} & 0 & 0 & \frac {\partial P _ {2}}{\partial y} & 0 & 0 & \frac {\partial P _ {3}}{\partial y} & 0 \\ 0 & 0 & \frac {\partial P _ {1}}{\partial z} & 0 & 0 & \frac {\partial P _ {2}}{\partial z} & 0 & 0 & \frac {\partial P _ {3}}{\partial z} \\ \frac {\partial P _ {1}}{\partial y} & \frac {\partial P _ {1}}{\partial x} & 0 & \frac {\partial P _ {2}}{\partial y} & \frac {\partial P _ {2}}{\partial x} & 0 & \frac {\partial P _ {3}}{\partial y} & \frac {\partial P _ {3}}{\partial x} & 0 \\ 0 & \frac {\partial P _ {1}}{\partial z} & \frac {\partial P _ {1}}{\partial y} & 0 & \frac {\partial P _ {2}}{\partial z} & \frac {\partial P _ {2}}{\partial y} & 0 & \frac {\partial P _ {3}}{\partial z} & \frac {\partial P _ {3}}{\partial y} \\ \frac {\partial P _ {1}}{\partial z} & 0 & \frac {\partial P _ {1}}{\partial x} & \frac {\partial P _ {2}}{\partial z} & 0 & \frac {\partial P _ {2}}{\partial x} & \frac {\partial P _ {3}}{\partial z} & 0 & \frac {\partial P _ {3}}{\partial x} \end{array} \right] \tag {1.8.20} +$$ + + + +# 1-8-3 하중과 질량 + +입체요소에 적용되는 하중은 체적력(body force), 압력하중(pressure load), 온도하중(thermal load), 프리스트레스하중(prestress load) 등이 있다. 체적력은 요소의 자중이나 관성력을 표현하고자 하는 하중이고, 압력하중은 요소의 면에 가해지는 분포하중이다. 온도하중에는 절점온도, 요소온도 하중이 있다. + +\- 체적력 + +$$ +\mathbf {F} _ {i} = \int_ {V _ {e}} N _ {i} \left\{ \begin{array}{l} \omega_ {x} \\ \omega_ {y} \\ \omega_ {z} \end{array} \right\} d V \tag {1.8.21} +$$ + +여기서, + +$$ +\omega_ {x}, \omega_ {y}, \omega_ {z} \quad : \text { 단위 체적당 자중(방향별) } +$$ + +\- 압력하중 + +$$ +\mathbf {F} _ {i} = \int_ {A _ {e}} N _ {i} \left\{ \begin{array}{l} P _ {x} \\ P _ {y} \\ P _ {z} \end{array} \right\} d A \tag {1.8,22} +$$ + +여기서, + +$$ +P _ {x}, P _ {y}, P _ {z} \quad : \text { 단위 면적당 하중(방향별) } +$$ + +\- 온도하중 + +$$ +\mathbf {F} _ {i} = \int_ {V _ {e}} \mathbf {B} _ {i} ^ {T} \mathbf {D} \left\{ \begin{array}{l} \alpha_ {x} \\ \alpha_ {y} \\ \alpha_ {z} \end{array} \right\} \Delta T d V \tag {1.8,23} +$$ + +여기서, + +$$ +\alpha_ {x}, \alpha_ {y}, \alpha_ {z} \quad : \text { 열팽창 계수(방향별) } +$$ + + + +입체요소의 질량은 집중질량(lumped mass)과 분포질량(consistent mass)을 반영할 수 있으며, x,y,z 방향의 이동변위만을 반영한다. + +\- 분포질량 + +$$ +\mathbf {M} _ {i j} = \rho \int_ {V _ {e}} N _ {i} N _ {j} d V \tag {1.8.24} +$$ + +\- 집중질량 + +집중질량은 요소 전체질량( $\rho V_{e}$ )을 분포질량의 대각 항 비율로 분배하여 사용한다. + + + +# 1-8-4 요소결과 + +입체요소의 해석 결과로는 절점에서의 응력과 변형률을 출력하며, 부호와 방향은 전체좌표계를 따른다. 전체좌표계를 기준으로 출력된 결과는 요소좌표계 또는 출력좌표계로 변환하여 볼 수 있다. 입체요소에서 출력되는 응력과 변형률의 종류는 다음과 같다. + +• 응력 성분 $\sigma_{XX}, \sigma_{YY}, \sigma_{ZZ}, \tau_{XY}, \tau_{YZ}, \tau_{ZX}$ + +\- Von-Mises 응력 $\sqrt{\left(P_{1}^{2}+P_{2}^{2}+P_{3}^{2}-P_{1}P_{2}-P_{2}P_{3}-P_{3}P_{1}\right)}$ + +• 최대 전단응력 $\frac{\max(\left|P_{1}-P_{2}\right|,\left|P_{2}-P_{3}\right|,\left|P_{3}-P_{1}\right|)}{2}$ + +• 주응력 $P_{1}, P_{2}, P_{3}$ + +$\operatorname{det}\left[\begin{array}{ccc}\sigma_{XX}-P_{i}&\tau_{XY}&\tau_{ZX}\\ \tau_{XY}&\sigma_{YY}-P_{i}&\tau_{YZ}\\ \tau_{ZX}&\tau_{YZ}&\sigma_{ZZ}-P_{i}\end{array}\right]=0$ 의 해 중 큰 값부터 $P_{1},P_{2},P_{3}$ 이다. + +\- 변형률 성분 $\varepsilon_{XX}, \varepsilon_{YY}, \varepsilon_{ZZ}, \gamma_{XY}, \gamma_{YZ}, \gamma_{ZX}$ + +\- Von-Mises 변형률 $\frac{2}{3}\sqrt{\left(E_{1}^{2}+E_{2}^{2}+E_{3}^{2}-E_{1}E_{2}-E_{2}E_{3}-E_{3}E_{1}\right)}$ + +• 체적(volumetric) 변형률 $E_{1} + E_{2} + E_{3}$ + +\- 주변형률 $E_{1}, E_{2}, E_{3}$ + +$\operatorname{det}\left[\begin{array}{ccc}\varepsilon_{XX}-E_{i}&\gamma_{XY}/2&\gamma_{ZX}/2\\\gamma_{XY}/2&\varepsilon_{YY}-E_{i}&\gamma_{YZ}/2\\\gamma_{ZX}/2&\gamma_{YZ}/2&\varepsilon_{ZZ}-E_{i}\end{array}\right]=0$ 의 해 중 큰 값부터 $E_{1},E_{2},E_{3}$ 이다. + + + +절점에서의 응력과 변형률은 요소 내의 적분점에서 계산된 결과를 이용하여 외삽법 (extrapolation)에 의해 산출된다. 입체요소의 적분점은 다음과 같다. + +• 4절점 4면체 : 1 점 가우스 적분 +• 6절점 5면체 : 6 점 가우스 적분 +• 8절점 6면체 : 8 점 가우스 적분 +• 10절점 4면체 : 4 점 가우스 적분 +• 15절점 5면체 : 9 점 가우스 적분 +• 20절점 6면체 : 27 점 가우스 적분 + +응력과 변형률에 대한 부호규약은 그림 1.8.3과 같고, 화살표 방향이 ‘+’ 부호를 의미한다. + +![](images/page-115_3b9e8e9d286e997ab5594e659d9e828aa0a33ac0033b6501532859a3058ca669.jpg) + +
+text_image + +σ_zz, ε_z +τ_yz, γ_yz +τ_zx, γ_zx +τ_zx, γ_zx +τ_xy, γ_xy +σ_xx, ε_xx +σ_yy, ε_yy +τ_yz, γ_yz +Z +Y +GCS +X +
+ +그림 1.8.3. 입체요소의 결과 성분과 방향 + + + +# 1-9 스프링 + +# 1-9-1 탄성연결요소(elastic link) + +탄성연결요소는 두 절점을 사용자가 입력한 강성으로 연결하는 기능이며, 강성을 제외한 구조적 특성은 가지고 있지 않다. 탄성연결요소의 좌표계 방향은 그림 1.9.1과같다. 탄성연결요소는 인장전담(tension-only)이나 압축전담(compression-only) 특성을 부여할 수 있는데, 이러한 경우에는 요소좌표계 x 방향으로만 강성을 입력할수 있다. 탄성연결요소의 입력은 각각 3방향의 이동(translation) 및 회전(rotation)강성으로 구성되어 있다. 탄성연결요소의 강성 크기는 이동 방향은 단위길이당 힘,회전방향은 단위각도(radian)당 모멘트로 입력한다. + +탄성연결요소는 교량구조물의 상부와 하부 교각부를 연결해주는 탄성받침이나, 압축전담 특성을 갖는 지반 경계조건에 적절하게 사용할 수 있다. 또한 강체 연결기능을선택하면 rigid link와 같이 두 절점을 강체로 연결할 수도 있다. + +![](images/page-116_e3708cb4950bd2d947740f8cd6ece82d4b12544d95ab8254d2bf7ea6419366aa.jpg) + +
+text_image + +z +Ref. +x +y +1 +2 +
+ +그림 1.9.1 탄성연결요소의 좌표계 + + + +# 1-9-2 절점 스프링(spring)/감쇄(damping) + +절점스프링은 모델의 경계부분에 위치한 인접구조물 또는 지반경계조건 등의 탄성강성을 고려할 때, 혹은 자유도가 부족한 요소(트러스, 평면응력요소 등)가 상호 접합될 경우에 발생할 수 있는 특이성 오류(singular error)를 방지하고자 할 때 사용된다. + +절점스프링은 절점당 전체좌표계 기준의 6개 자유도(이동방향 3개 성분, 회전방향 3개 성분)에 대해 입력이 가능하다. 이동방향 강성은 단위길이당 힘으로 입력하고, 회전방향 강성은 단위각도(radian)당 모멘트로 입력한다. 지반을 모델링할 때는 지반반력계수(modulus of subgrade reaction)에 해당절점의 유효면적(effective area)을곱한 강성값을 사용한다. + +절점감쇄는 절점의 감쇠 스프링을 입력하는데 사용된다. 지반의 점성 경계조건 모델링에 많이 사용되며, 절점당 6개의 자유도(이동방향 3개 성분, 회전방향 3개 성분)에대해 입력이 가능하다. 절점 감쇠는 특성상 일반 정적해석에는 반영되지 않고 동해석의 경우에만 적용된다. + +절점 스프링과 절점 감쇠의 좌표축은 기본적으로는 전체좌표계를 따르지만 절점좌표계가 선언되어 있는 경우에는 절점좌표계를 기준으로 한다. + +![](images/page-117_a5f53cdd96697fe7286946e553eac893eddff5434c6e40cecbe885a9e0be55e1.jpg) + +
+text_image + +SRY +SY +SX +Nodal Point +SZ +SRX +SRZ +Z +Y +X +
+ +그림 1.9.2 절점 스프링/감쇄의 좌표계 + + + +# 1-9-3 행렬스프링(matrix spring) + +행렬스프링은 자유도 방향으로만 강성이 입력되는 절점스프링을 보완하기 위해 사용된다. 탄성 경계조건을 정밀하게 모델링 하고자 할 경우에는 자유도별 강성뿐만 아니라 자유도사이의 연관(coupled)되는 강성까지 고려하는 것이 필요하다. 즉 이동변위가 발생할 때 동시에 발생되는 회전변위 등을 고려하기 위해서는 연관된 강성의 사용이 필요하다. 행렬스프링은 이와 같은 탄성 경계조건을 모델링하기에 적합한 요소이다. + +예를 들어 구조물의 기초에 사용되는 파일(pile)을 경계스프링으로 모델링하고자 할경우에 각 방향별 강성 이외에 자유도 상호간의 연관된 강성을 입력하여 보다 정밀한 해석을 수행할 수 있다. 행렬스프링 값은 사용자가 임의 값을 입력할 수는 있지만,전체 행렬이 양의 정부호(positive definite) 이어야 한다. 그렇지 않을 경우에는 적절한 해를 구할 수 없게 된다. 행렬스프링의 값은 대칭이 되도록 상부 삼각행렬만 입력하도록 되어 있다. + +행렬스프링의 좌표축은 기본적으로는 전체좌표계를 따르지만, 절점좌표계가 선언되어있는 경우에는 절점좌표계를 기준으로 한다. + + + +# 1-10 강체연결(rigid link) + +강체연결요소는 구조물의 기하학적(geometric) 상대거동을 상호 구속하는 기능이다.기하학적 상대거동의 구속은 임의 절점의 자유도에 한 개 또는 그 이상의 절점의 자유도를 종속시킴으로써 이루어진다. 여기서 임의 절점을 주절점(master node)이라하고 자유도가 종속되는 절점을 종속절점(slave node)이라 한다. 강체연결요소에 의한 주절점과 종속절점의 상호구속방정식은 식 (1.10.1)과 같다. + +$$ +U _ {X s} = U _ {X m} + R _ {Y m} \Delta Z - R _ {Z m} \Delta Y +$$ + +$$ +U _ {Y s} = U _ {Y m} + R _ {Z m} \Delta X - R _ {X m} \Delta Z +$$ + +$$ +U _ {Z s} = U _ {Z m} + R _ {X m} \Delta Y - R _ {Y m} \Delta X \tag {1.10.1} +$$ + +$$ +R _ {X s} = R _ {X m} +$$ + +$$ +R _ {Y s} = R _ {Y m} +$$ + +$$ +R _ {Z s} = R _ {Z m} +$$ + +$$ +\Delta X = X _ {m} - X _ {s} +$$ + +$$ +\Delta Y = Y _ {m} - Y _ {s} +$$ + +$$ +\Delta Z = Z _ {m} - Z _ {s} +$$ + +$U _ { _ { X s } } , U _ { _ { Y s } } , U _ { _ { Z s } }$ : 종속절점의 전체좌표계 X , , Y Z 방향 이동변위 + +$U _ { \chi m } \ , \ U _ { Y m } \ , \ U _ { Z m }$ : 주절점의 전체좌표계 X , , Y Z 방향 이동변위 + +$R _ { \chi _ { s } } ~ , ~ R _ { \gamma _ { s } } ~ , ~ R _ { Z s }$ : 종속절점의 전체좌표계 X , , Y Z 방향 회전변위 + +$R _ { \chi _ { s } } ~ , ~ R _ { \gamma _ { s } } ~ , ~ R _ { Z s }$ : 주절점의 전체좌표계 X , , Y Z 방향 회전변위 + +$X _ { s } ~ , ~ Y _ { s } ~ , ~ Z _ { s }$ : 종속절점의 전체좌표계 X , , Y Z 방향 좌표 + +$X _ { m } \ , \ Y _ { m } \ , \ Z _ { m }$ : 주절점의 전체좌표계 X , , Y Z 방향 좌표 + +강체연결요소는 강성이 타 구조부재보다 훨씬 커서 변형효과를 무시할 수 있는 부재의 모델링이나, 보강판에서의 판과 보강재을 상호 연결하는데 활용할 수 있다. + + + +강체연결요소는 6개 자유도에서 사용자가 임의대로 선택하여 입력할 수 있고, 입력한 자유도에 대해서만 식 (1.10.1)을 사용하여 상호구속 방정식을 구성한다. 예를 들어 전체좌표계 특정 평면에 대한 강체연결요소의 상호구속 방정식 고려해 보자. 주절점과 종속절점이 X Y− 평면상에서 평면강체(rigid plane connection)로 상호거동이구속되는 경우의 상호구속 방정식은 다음과 같다. + +$$ +U _ {X s} = U _ {X m} - R _ {Z m} \Delta Y +$$ + +$$ +U _ {Y s} = U _ {Y m} + R _ {Z m} \Delta X \tag {1.10.2} +$$ + +$$ +R _ {Z s} = R _ {Z m} +$$ + +그림 1.10.1은 구조물 바닥판(floor)의 모델에 평면강체 기능을 적용한 예를 나타낸것이다. 일반적으로 구조물이 수평력을 받을 때 바닥판 내의 모든 위치에서의 수평방향 상대변위는 다른 구조부재(수직부재, 대각부재)의 상대변위에 비해 거의 무시할만큼 작다. 이와 같은 바닥판의 강막작용(rigid diaphragm action)은 바닥판 내의 모든 면내거동을 상호 구속함으로써 고려될 수 있다. 이때 면내거동은 바닥판의 면내이동변위 2개의 성분과 면의 수직방향에 대한 회전변위 성분을 의미한다. + +![](images/page-120_d6afb926aeb94be394324dd675cdbd82ca75fd07c2c7ebcefe61024918810710.jpg) + +
+text_image + +floor diaphragm +torsional moment +Y +φ4 +4 +3 +φ3 +φ1 +1 +2 +X +φ2 +Z +1 +2 +Y +X +
+ +그림 1.10.1 바닥판이 있는 구조물이 수직축에 대해 비틀림 거동을 하는 경우 + +그림 1.10.1의 예에서 바닥판이 있는 구조물이 비틀림 거동을 할때, 바닥판의 면내강 diff --git a/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_013.md b/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_013.md new file mode 100644 index 00000000..c4a9c26a --- /dev/null +++ b/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_013.md @@ -0,0 +1,303 @@ + + +성이 수직기둥부재의 강성에 비해 무한히 클 경우, 바닥판 전체가 φ 만큼 회전하게되고 φ1 2 3 4= = = = φ φ φ φ 가 된다. 따라서 4개의 자유도를 1개의 자유도로 축약시킬수 있다. + +그림 1.10.2는 각막작용을 고려하여 절점당 6개씩의 자유도(6×4), 총 24개의 자유도를 15개의 자유도로 축약하는 과정을 나타낸 것이다. + + + +![](images/page-122_e48a342a9a15da177444c3442daf8b9b9be764f44cd6d2484d4a6674957beb71.jpg) + +
+flowchart + +```mermaid +graph TD + A["master node"] --> B["slave node"] + B --> C["floor diaphragm"] + C --> D["X"] + A --> E["UxUyUzRxRyRz"] + B --> F["UxUyUzRxRyRz"] + C --> G["UxUyUzRxRyRz"] + E --> H["↑ Y"] + F --> I["↑ X"] + G --> J["↑ X"] +``` +
+ +![](images/page-122_009ec7b516cd554fcb8146596056e2bb38cc58e0972aaa4dbb7b03b75c5e1342.jpg) + +
+text_image + +floor diaphragm +UzRxRy +UzRxRy +UzRxRy +UzRxRy +UzRxRy +master node +slave node +Y +X +Z +1 +2 +3 +4 +
+ +UX :displacement degree of freedom in the X-direction at the corresponding node +UY : displacement degree of freedom in the Y-direction at the corresponding node +UZ :displacement degree of freedom in the Z-direction at the corresponding node +RX :rotational degree of freedom about the X-axis at the corresponding node +RY :rotational degree of freedom about the Y-axis at the corresponding node +RZ : rotational degree of freedom about the Z-axis at the corresponding node + +그림 1.10.2 면내 무한강성을 갖는 바닥판의 자유도 축약 개념도 + + + +그림 1.10.3은 다양한 강체연결 기능을 이용한 예를 나타낸 것이다. 그림 1.10.3의(a)는 사각튜브의 구조적 거동을 정밀해석하기 위해, 정밀검토가 필요한 부분에 대해서는 판요소로 세분화하고, 나머지 부분은 보요소로 모델링한 것이다. 두 모델 사이는 모든 자유도를 강체연결 하였다. 그림 1.10.3의 (b)는 2차원 평면상에 있는 두 개의 기둥이 편심되어 만나는 경우에, 절점에서의 편심효과를 고려하기 위해 평면강체기능을 이용한 예이다. 이와 같이 임의 평면내에 강체연결기능을 사용하고자 할 때에는 반드시 평면내의 두 개의 이동변위 성분과 수직방향에 대한 회전 변위성분에 대해 구속조건을 부여해야 한다. 마찬가지로 그림 1.10.3의 (a)와 같이 모든 방향성분에대해 강체연결을 할 경우에는 6개 자유도 전부에 대해 구속조건을 부여해야 한다. + + + +![](images/page-124_359c7d3bbc0a8d0427af0397297d639ae2fb685ca5a056f25674310152542efa.jpg) + +
+text_image + +rectangular tube modeled with plate elements +Rigid Link +rectangular tube modeled as a beam element +master node +Z +Y +X +: slave nodes (12 nodes) +* all 6 degrees of freedom of +the slave nodes are linked +to the master node. +
+ +(a) 한 개의 튜브를 보요소와 판요소로 부위별로 모델링하여 상호연결한 경우 (Rigid Body Connection) +![](images/page-124_f16f876e2fefbe4b62d78fc44055c832ee54d5a8a525319f956cc9766f941b60.jpg) + +
+text_image + +P +P +Z +Y → X +slave node +master node +eccentricity +eccentricity +Y +Z → X +* all slave node's d.o.f in the X-Z plane are linked to the master node (translational displacement d.o.f in the X and Z-directions and rotational d.o.f about the Y-axis +
+ +(b) 두개의 기둥이 편심되어 만나는 경우 (Rigid Plane Connection) +그림 1.10.3 강체연결요소를 사용한 예 + + + +# Chapter 2. Embedded Reinforcements + +# 2-1 개요 + +midas FEA에서는 별도의 절점을 가진 요소로 사용되지 않고, 철근(reinforcement)을 포함하는 모재요소(mother element)에 철근의 요소강성이 더해지는 내재요소(embedded element)형태로 구성된다. 철근요소를 사용하는데 있어서 유의해야 할 사항은 다음과 같다. + +■ 사용자가 철근을 정의하고 철근을 모재요소에 맞게 분할하는 과정이 필요한 경우 프로그램에서 모재요소와 철근의 교차점을 계산한다. +▪ 철근은 별도의 자유도를 가지지 않는다. +▪ 철근과 모재요소는 완전부착(perfectbond)된 것으로 가정한다 +▪ 철근의 변형률(strain)은 모재요소의 변위로부터 구하여진다. + +모재요소에 삽입 되는 철근에는 바(bar)와 격자(grid) 두 가지 형태가 있다. 철근을 정의하기 위한 입력정보는 위치정보, 형상정보, 물성치 등으로 구성 된다. 위치정보는 모재요소에 삽입되는 철근의 위치절점(location point) 정보로 구성된다. 그리고 철근의 형상정보를 정의하는 방법에는 두 가지가 있다. 철근절점으로 정의되는 섹션(section)을 생성하여 전처리 과정에서 모재요소의 특성에 따라 철근을 세그먼트(segment)로 분할하는 전체입력 방식이 있다. 이 때 철근섹션은 세그먼트들로 구성되게 되며 철근세그먼트는 삽입되는 모재요소의 경계를 가로지를 수 없다. 또 다른 방법은 모재요소에 맞춰 철근섹션을 모재요소에 맞추어 직접 입력하여 1개의 섹션이 1개의 세그먼트로 바로 사용되도록 하는 개별입력 방식이다. 형상정보는 철근의 형태 및 모재요소의 형태에 따라 각각 다른 정보를 필요로 하므로, 각각의 경우에 대한 자세한 설명은 해당 장에 서술하였다. + +구조물에서 철근과 모재요소는 서로 상호작용하며 미끄러짐이 일어날 수 있다. 이런 경우 서로 완전부착 가정은 부적합 할 수 있다. 이 때는 자신의 절점을 가지는 트러스요소(truss element)를 사용하고 모재요소와 트러스요소를 연계시켜주는 경계면요소(interface element)를 사용하여 트러스요소를 모재요소와 연결시켜줄 수 있다. + + + +# 2-2 철근의 형태 + +# 2-2-1 철근바 + +철근바(bar reinforcement)에는 선(line) 형상의 철근바와 점(point) 형상의 철근바가 있다. 각각의 철근바를 삽입이 가능한 모재요소의 종류 에 따라 분류하면다음과 같다. + +- 선 : 입체요소, 판요소, 평면응력요소 +- 점 : 평면변형요소, 축대칭요소 + +각 요소의 철근바에 대한 자세한 설명은 해당 장에 서술하였다. + +midas FEA 에서 철근바를 정의하기 위해서는 위치정보를 사용하여 임의의 선이나 점을 정의하고, 섹션을 생성한다. 입력해야 하는 물성치는 다음과 같다. + +- 재료물성치(material property) +- 단면적 +- 텐던물성치(tendon property) + +텐던은 프리스트레스가 가해지는 철근을 의미한다. + +그림 2.2.1 에서는 바섹션의 종류를 보이고 있다. 바섹션은 2절점 저차 바섹션과 3절점 고차 바섹션으로 정의할 수 있다. 바섹션은 철근절점에 의해 정의된다. + +![](images/page-126_c4caa76cf4107f88c92abcba06309f56f1403682b2bb241f1f9ab6feabef8708.jpg) + +
+text_image + +1 +2 +1 +3 +2 +
+ +그림 2.2.1 바섹션과 철근절점 + + + +전처리 과정에서 바섹션은 철근이 삽입되는 모재요소에 맞게 자동적으로 철근바의 세그먼트로 분할된다. 분할된 철근바세그먼트는 삽입되는 요소 내에 존재하며, 위치절점을 사용하여 정의된다. 철근바세그먼트는 분할되기 전 바섹션이 사용하던 물성치를 동일하게 사용한다. + +모재 요소에 삽입 되는 철근바세그먼트 차수는 섹션의 차수와 동일하다. 바섹션이 2절점 저차 섹션일 경우 세그먼트도 2개의 위치절점을 사용한 저차 세그먼트로 분할 되며, 바섹션이 3절점 고차 섹션일 경우 세그먼트도 3개의 위치절점을사용한 고차 세그먼트로 분할된다. 분할된 철근바세그먼트의 형상은 후처리 과정에서 확인할 수 있다. + +전처리 과정에 의한 분할을 원하지 않고 요소에 직접 입력하고자 하는 경우 바섹션을 각각의 모재 요소에 삽입되는 형태로 사용자가 직접 입력하면 입력한 바섹션을 별도의 분할과정 없이 세그먼트로 사용한다. + +분할된 철근바세그먼트 강성은 모재요소의 강성에 더해지게 된다. 철근바세그먼트의 적분점은 기본적으로 선 형상에 대해서는 2점 적분을 사용하고 있으며,점 형상에 대해서는 1점 적분을 수행한다. 적분점의 위치는 해석과정에서 요소의등매개 좌표계(isoparametric coordinate)를 사용하여 자동적으로 계산된다. 그림 2.2.2 왼쪽 그림은 철근바의 세그먼트가 위치점을 사용하여 정의되는 것을보여주며 오른쪽 그림은 철근바의 세그먼트가 가지는 응력 성분을 보여준다. 철근바의 응력은 축방향 성분만 존재하며, 적분점에서 철근바의 변형률 xxε 와 응력 σ xx 는 적분점에서의 접선방향으로 구해진다. 적분점에서 구해진 변형률과 응력은 외삽법(extrapolation)으로 처리되어 후처리에서 위치절점에서의 결과로 출력된다. + + + +location point +△ + +![](images/page-128_516397b8c26b05c5b74095a3967c987e6295120d0e5e3e447b1e0b70d93e563d.jpg) + +
+text_image + +integration point +segment +segment +
+ +![](images/page-128_423d7d3aabbeb5e8c24b46d2922da963f6f284b90cda0ff97f4dc8015c8f02f3.jpg) +그림 2.2.2 철근바세그먼트 + +# 2-1-2 격자철근 + +철근격자 (grid reinforcement)에는 평면(plane) 형상의 격자와 선 형상의 격자가 있다. 평면형상의 격자는 다시 삼각형 형상과 사각형 형상의 두 종류로 나뉜다. 철근격자를 삽입 가능한 모재요소종류 별로 구분하여 보면 다음과 같다. + +- 평면 : 입체요소, 판요소, 평면응력요소 +- 선 : 평면변형요소, 축대칭요소 + +모재 요소에 삽입된 철근격자에 대한 자세한 설명은 각 장에 서술하였다. + +midas FEA 에서는 철근격자의 형상을 위치정보로 정의한 후, 철근격자 섹션으로 분할한다. 격자섹션을 입력하는 과정에서 철근격자의 물성치를 격자섹션에적용하게 된다. 입력하여야 하는 철근격자의 물성치는 다음과 같다. + +- 재료물성치 (등방성(isotropic)/이방성(orthotropic)) +- 철근요소축(reinforcement axis) +- 등가두께(equivalent thickness) (등방성/이방성) +- 철근세부유형 (철근이나 텐던) + +철근격자섹션은 4가지 종류가 있다. 삼각형 격자섹션은 그림 2.2.3과 같이 3절 + + + +점 혹은 6절점을 사용하여 정의되며, 사각형 격자섹션은 그림 2.2.4와 같이 4절점 혹은 8절점을 사용하여 정의된다. + +![](images/page-129_489186395938e3019b2ba63a9dfcfeb4122f4c97f25fa2ecf747e06b8c9b705e.jpg) + +
+flowchart + +```mermaid +graph TD + A["1"] --> B["2"] + B --> C["3"] + C --> D["4"] + D --> A +``` +
+ +![](images/page-129_2b809ab68a5727f36e282223be9e14682ea926b074a013d0fd4e961016db5be5.jpg) + +
+flowchart + +```mermaid +graph TD + 1 --> 2 + 2 --> 3 + 3 --> 4 + 4 --> 5 + 5 --> 6 + 6 --> 7 + 7 --> 8 + 8 --> 1 +``` +
+ +그림 2.2.3 사각형 철근격자와 철근절점 + +![](images/page-129_3fc6eeb3ec90e5c881baa2859e559a48c226621e45e1355fcda4a033596b3f5f.jpg) + +
+text_image + +1 +2 +3 +
+ +![](images/page-129_6418e1bbe2e0c1d92b27a0a715e5b003e5be26a37cf2a4b51bcd08e26797476f.jpg) + +
+text_image + +1 +6 +3 +5 +4 +2 +
+ +그림 2.2.4 삼각형 철근격자와 철근절점 + +생성된 격자섹션은 전처리 과정에서 격자섹션이 삽입되는 모재요소를 찾고, 삽입되는 모재요소의 형상에 맞게 철근격자의 세그먼트로 분할한다. 격자섹션을세그먼트로 분할할 때 삼각형 세그먼트들로 분할한다. + +전처리 과정에 의한 분할을 원하지 않고 요소에 직접 입력하고자 하는 개별입력방식의 경우 격자섹션을 모재요소에 맞게 사용자가 직접 입력하여 별도의 분할 + + + +과정을 거치지 않고 입력한 격자섹션을 격자세그먼트로 사용할 수 있다. 평면형상의 철근격자의 경우 전처리 과정에 의한 분할을 거치게 되면 세그먼트가 삼각형 형상을 가지게 되는데 비하여, 직접 입력하게 되면 철근격자를 사용자가원하는 형상으로 입력이 가능하다. + +![](images/page-130_889bb0a0dea589c451d1c4c0f4c8d28c11f657b2d9cf6a3263c32e79ee41258c.jpg) + +
+text_image + +y +x +
+ +![](images/page-130_5a3e60f7da6b137d7de7bc0438d09a8b72d8b0a82650772b418492116ed8ce52.jpg) + +
+text_image + +y +x +
+ +![](images/page-130_84f703cf28f23e82e3186b7e0d0fa1cc2202c324b6e0b83228e5d0bc0b15b58a.jpg) +element node +location point +integration point + +그림 2.2.5 철근격자 + +철근격자의 적분은 사각평면 형상에 대해서는 4점 적분을 사용하며 삼각평면 형상에 대해서는 3절점이 사용되는 저차 삼각형 격자에는 1점 적분을 사용하며 6절점이 사용되는 고차 삼각형 격자에는 3점 적분을 사용한다. 선형상에 대해서는 2점 적분을 한다. 그림 2.2.5 왼쪽 그림은 위치절점을 사용하여 철근격자가정의되는 것을 보여준다. 철근격자에 사용되는 적분점 위치는 해석과정에서 자동적으로 계산된다. + +철근격자는 격자의 요소좌표 xˆ 축을 사용자가 정의할 수 있다. 요소좌표축 xˆ , yˆ 는서로 직교하고, 적분점에서의 변형률 ε , ε 와 응력 σ , σ 는 각각 xˆ , yˆ 방향성분이다. diff --git a/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_014.md b/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_014.md new file mode 100644 index 00000000..76ef6ec1 --- /dev/null +++ b/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_014.md @@ -0,0 +1,329 @@ + + +격자는 xˆ , yˆ 방향 요소축이 있고, 각 요소축 방향으로만 강성이 존재하므로 전단력에 대해서는 저항하지 못한다. 이 때 직교성 재료를 사용하고 있는 격자의 경우 각요소축 방향에 서로 다른 두께(thickness)를 사용할 수 있다. 그리고 하나의 요소축에 입력되는 두께를 등가두께라고 한다. + + + +# 2-3 유한요소 정식화 + +철근의 공간상 위치는 철근을 구성하는 세그먼트들의 위치절점들로 정의된다. 철근세그먼트가 n개의 위치절점으로 구성되어 있고, 각각의 점은 $X_{re}^{i}, Y_{re}^{i}, Z_{re}^{i}$ 로 정의되어 있다고 하면, 철근세그먼트의 위치점을 식 (2.3.1) 과 같은 행렬로 정리할 수 있다. 여기서 위 첨자는 철근세그먼트의 위치점 번호를 나타낸다. + +$$ +\mathbf {X} _ {r e} = \left\{ \begin{array}{c c c c} X _ {r e} ^ {1} & X _ {r e} ^ {2} & \dots & X _ {r e} ^ {n} \\ Y _ {r e} ^ {1} & Y _ {r e} ^ {2} & \dots & Y _ {r e} ^ {n} \\ Z _ {r e} ^ {1} & Z _ {r e} ^ {2} & \dots & Z _ {r e} ^ {n} \end{array} \right\} \tag {2.3.1} +$$ + +철근세그먼트의 형상함수는 식 (2.3.2) 와 같이 나타낼 수 있다. + +$$ +\mathbf {N} = \left\{N _ {1} \quad N _ {2} \quad \dots \quad N _ {n} \right\} \tag {2.3.2} +$$ + +여기서 형상함수의 구성성분 $N_{i}(i=1,2,\ldots,n)$ 는 철근의 형태에 따라 다르며, 각 형상함수는 다음과 같다. + +2절점 선 형상의 철근바세그먼트의 형상함수 + +$$ +N _ {1} (\xi) = \frac {1}{2} (1 - \xi), N _ {2} (\xi) = \frac {1}{2} (1 + \xi) \tag {2.3.3} +$$ + +3절점 선 형상의 철근바세그먼트의 형상함수 + +$$ +N _ {1} (\xi) = - \frac {1}{2} (1 - \xi) \xi , N _ {2} (\xi) = \frac {1}{2} (1 + \xi) \xi , N _ {3} (\xi) = \left(1 - \xi^ {2}\right) \tag {2.3.4} +$$ + +4절점 형상의 철근격자세그먼트의 형상함수 + +$$ +N _ {1} (\xi , \eta) = \frac {1}{4} (1 - \xi) (1 - \eta), N _ {2} (\xi , \eta) = \frac {1}{4} (1 + \xi) (1 - \eta) +$$ + +$$ +N _ {3} (\xi , \eta) = \frac {1}{4} (1 + \xi) (1 + \eta), N _ {4} (\xi , \eta) = \frac {1}{4} (1 - \xi) (1 + \eta) \tag {2.3.5} +$$ + + + +8절점 형상의 철근격자세그먼트의 형상함수 + +$$ +\begin{array}{l} N _ {1} (\xi , \eta) = \frac {1}{4} (1 - \xi) (1 - \eta) - \frac {1}{2} N _ {5} - \frac {1}{2} N _ {8} \\ N _ {2} (\xi , \eta) = \frac {1}{4} (1 + \xi) (1 - \eta) - \frac {1}{2} N _ {5} - \frac {1}{2} N _ {6} \tag {2.3.6} \\ N _ {3} (\xi , \eta) = \frac {1}{4} (1 + \xi) (1 + \eta) - \frac {1}{2} N _ {6} - \frac {1}{2} N _ {7} \\ N _ {4} (\xi , \eta) = \frac {1}{4} (1 - \xi) (1 + \eta) - \frac {1}{2} N _ {7} - \frac {1}{2} N _ {8} \\ N _ {5} (\xi , \eta) = \frac {1}{2} (1 - \xi^ {2}) (1 - \eta) \\ N _ {6} (\xi , \eta) = \frac {1}{2} (1 + \xi) (1 - \eta^ {2}) \\ N _ {7} (\xi , \eta) = \frac {1}{2} (1 - \xi^ {2}) (1 + \eta) \\ N _ {8} (\xi , \eta) = \frac {1}{2} (1 - \xi) (1 - \eta^ {2}) \\ \end{array} +$$ + +3절점 삼각형 형상의 철근격자세그먼트의 형상함수 + +$$ +N _ {1} (\xi , \eta) = 1 - \xi - \eta +$$ + +$$ +N _ {2} (\xi , \eta) = \xi +$$ + +$$ +N _ {3} (\xi , \eta) = \eta \tag {2.3.7} +$$ + +6절점 삼각형 형상의 철근격자세그먼트의 형상함수 + +$$ +N _ {1} = (1 - \xi - \eta) (1 - 2 \xi - 2 \eta), N _ {2} = \xi (2 \xi - 1), N _ {3} = \eta (2 \eta - 1) +$$ + + + +$$ +N _ {4} = 4 \xi (1 - \xi - \eta), N _ {5} = 4 \xi \eta , N _ {6} = 4 \eta (1 - \xi - \eta) \tag {2.3.8} +$$ + +철근세그먼트의 j 번째 수치 적분점에서 + +$$ +\mathbf {N} (j) \mathbf {X} _ {r e} \tag {2.3.9} +$$ + +는 각각 철근세그먼트의 j 번째 적분점의 위치를 의미한다. 철근의 세그먼트는 모재요소의 내부에 삽입되어 있기 때문에 철근세그먼트의 적분점도 모재요소의 내부에 존재한다. + +midas FEA 에서 사용하는 각 요소들은 등매개 맵핑(isoparametric mapping) 기법을 사용하고 있다. 이 때 앞서 구한, 모재요소의 내부에 존재하는 철근의 적분점도 모재요소가 맵핑된 등매개 좌표 상에서의 위치를 계산할 수 있다. + +모재요소가 3차원 요소일 경우 모재요소의 등매개 좌표 상에서 철근세그먼트의 j 번째 수치 적분점은 다음과 같다. + +$$ +\mathbf {G} _ {j} (\xi , \eta , \varsigma) \tag {2.3.10} +$$ + +2차원 모재요소에 삽입되는 철근세그먼트의 j 번째 수치 적분점은 다음과 같다. + +$$ +\mathbf {G} _ {j} (\xi , \eta) \tag {2.3.11} +$$ + +철근세그먼트의 j 번째 적분점에서 철근세그먼트의 변형률과 모재요소 변위와의 관계를 나타낸 변형률-변위 행렬은 다음과 같다. + +$$ +\widehat {\mathbf {B}} _ {\text {Re in..}} ^ {j} = \mathbf {B} _ {\text {mother}} \left(\mathbf {G} _ {j}\right) \tag {2.3.12} +$$ + +이와 같이 구해진 철근세그먼트의 변형률-변위 행렬은 모재요소의 요소좌표계를 사용한다. 철근세그먼트의 강성은 철근의 요소좌표계에서의 변형률 $\varepsilon_{xx}$ , $\varepsilon_{yy}$ 를 사용하여 나타내어 진다. 따라서 앞서 구한 $\hat{B}_{rein}$ 은 모재요소의 요소좌표계에서 철근세그먼트의 요소좌표계로 회전시켜주어야 한다. + + + +모재요소가 3차원 요소라면 다음과 같이 모재요소의 요소좌표계에서의 변형률을 철근세그먼트의 요소좌표계에서의 변형률로 변환 할 수 있다. + +$$ +\mathbf {T} \left\{ \begin{array}{c} \varepsilon_ {x x} \\ \varepsilon_ {y y} \\ \vdots \\ \varepsilon_ {z z} \end{array} \right\} = \left\{ \begin{array}{c} \varepsilon_ {\widehat {x x}} \\ \varepsilon_ {\widehat {y y}} \end{array} \right\} \tag {2.3.13} +$$ + +여기서, + +T : 모재요소의 등매개좌표계와 철근세그먼트의 요소좌표계 +사이의 회전변환 행렬 + +같은 변환이 B 행렬의 변환에도 사용될 수 있다. + +$$ +\mathbf {B} _ {\text { Re in. }} ^ {j} = \mathbf {T} \widehat {\mathbf {B}} _ {\text { Re in. }} ^ {j} \tag {2.3.14} +$$ + +철근세그먼트의 구성행렬(constitutive matrix)은 다음과 같이 나타낼 수 있다. + +철근바 일 경우 + +$$ +\mathbf {D} = \mathbf {E} _ {\tilde {x} \tilde {x}} A _ {e} \tag {2.3.15} +$$ + +여기서, + +$A_{e}$ : 철근바의 단면적 + +등방성 재료를 사용하는 철근격자일 경우 + +$$ +\mathbf {D} = \left[ \begin{array}{c c} \mathbf {E} _ {\hat {x} \hat {x}} t _ {\hat {x} \hat {x}} & 0 \\ 0 & \mathbf {E} _ {\hat {x} \hat {x}} t _ {\hat {x} \hat {x}} \end{array} \right] \tag {2.3.16} +$$ + +이방성 재료를 사용하는 철근격자일 경우 + +$$ +\mathbf {D} = \left[ \begin{array}{c c} \mathbf {E} _ {\hat {x} \hat {x}} t _ {\hat {x} \hat {x}} & 0 \\ 0 & \mathbf {E} _ {\hat {y} \hat {y}} t _ {\hat {y} \hat {y}} \end{array} \right] \tag {2.3.17} +$$ + +여기서, + + + +$$ +\begin{array}{l} \mathbf {E} _ {\hat {x} \hat {x}} \quad : \hat {x} \text { 축 영 계수(Young's moduls) } \\ \mathbf {E} _ {\hat {y} \hat {y}} \quad : \hat {y} \text { 축영계수 } \\ t _ {\hat {x} \hat {x}} \quad : \hat {x} \text { 축 두께 } \\ t _ {\hat {y} \hat {y}} \quad : \hat {y} \text { 축 두께 } \\ \end{array} +$$ + +철근세그먼트의 요소좌표계에서의 응력은 다음과 같이 구할 수 있다. + +$$ +\left\{ \begin{array}{l} \sigma_ {x x} \\ \sigma_ {y y} \end{array} \right\} = \mathbf {D} \left\{ \begin{array}{l} \varepsilon_ {x x} \\ \varepsilon_ {y y} \end{array} \right\} \tag {2.3.18} +$$ + +철근세그먼트에 의한 강성행렬 식은 다음과 같다. + +$$ +\mathbf {K} _ {\text { rein }} ^ {e} = \int_ {V} \mathbf {B} _ {\text { Rein }} ^ {T} \mathbf {D} \mathbf {B} _ {\text { Rein }} d V \tag {2.3.19} +$$ + +그리고 철근세그먼트에 의한 내력(internal force)식은 다음과 같이 구해진다. + +$$ +\int_ {V} \mathbf {B} _ {\text { Rein }} ^ {T} \boldsymbol {\sigma} d V = \mathbf {F} _ {\text { rein }} \tag {2.3.20} +$$ + +여기서, + +σ : 철근세그먼트의 응력 + +식 (2.3.19)와 식 (2.3.20) 을 수치적분 식으로 다시 전개하면, 철근세그먼트의 강성행렬과 내력행렬의 수치적분 식을 다음과 같이 나타낼 수 있다. + +$$ +\sum_ {j = 1} ^ {N i p} \mathbf {B} _ {R r e i n} ^ {j} {} ^ {T} \mathbf {D} \mathbf {B} _ {R e i n} ^ {j} \det \mathbf {J} ^ {j} \tag {2.3.21} +$$ + +$$ +\sum_ {j = 1} ^ {N i p} \mathbf {B} _ {R r e i n} ^ {j} {} ^ {T} \boldsymbol {\sigma} \det \mathbf {J} ^ {j} \tag {2.3.22} +$$ + +여기서, + +Nip : 철근세그먼트의 적분점 수 + + + +# 2-4 평면변형요소에 삽입되는 철근 + +2-4-1 철근바 + +평면변형요소에 삽입되는 철근요소는 그림 2.4.1과 같이 점 형상을 가진다. + +![](images/page-137_9a86d538055ffbd21ec22bb7f32c77f78c78732f05652eba74c3d122ed02d722.jpg) + +
+text_image + +Y +t +X +Z +X +
+ +![](images/page-137_6637d4ae9698b315e96bc70aca34b0954bf566228461c9fbd51b7481f226cae9.jpg) + +
+text_image + +σₓₓ +
+ +element node +location point + +그림 2.4.1 평면변형요소에 삽입되는 철근바 + +평면변형요소에 삽입되는 철근바는 점 형상을 가지므로 철근바를 섹션으로 분할하는 별도의 분할이 필요하지 않다. + + + +그림 2.4.2의 왼쪽 그림은 고차 평면변형요소에 삽입된 철근바를 보이고 있으며오른쪽 그림은 여러 개의 철근바가 평면변형요소 1, 2, 3, 4 에 삽입이 된 모델을 보이고 있다. + +![](images/page-138_8d7822207cc88b5d28bf78e3bbf3efbfc0404f77d572031e9ab4c1e44af626ea.jpg) + +
+text_image + +1 +
+ +![](images/page-138_27a348d72c3b2008f40675eb0b4f402adbf21e6028b828909068e7926a88eb69.jpg) + +
+text_image + +2 +26 +4 +25 +1 +3 +24 +
+ +![](images/page-138_75d3910f0b3ca769f2cd40cde556c172e1a3938c31639db2d686fe06b5f328b5.jpg) + +element node + +bar location + +그림 2.4.2 평면변형요소에 삽입되는 철근바와 위치절점 + +평면변형요소에서의 철근바는 축방향 변형률 성분 $\varepsilon _ { x x }$ 와 응력성분 $\sigma _ { x x }$ 를 가지게 된다. + +# 2-4-2 철근격자 + +평면변형요소에 삽입되는 철근격자는 2점 혹은 3점으로 정의되는 선 형상을 가지고 있다. 평면변형요소에 삽입되는 철근격자는 평면응력요소에 삽입되는 철근격자와 다르게 정의된다. 평면변형요소에 삽입되는 철근격자는 평변변형요소의평면에 수직으로 삽입된다. + + + +![](images/page-139_754023eb9df91d3ac56ab7856a020fad6b61f1257620ec83f208fb8eacc825c0.jpg) + +
+text_image + +Y +Y +t +x +Z +X +element node +location point +integration point +
+ +그림 2.4.3 평면변형요소에 삽입되는 철근격자 + +평면변형요소와 삽입되는 철근격자가 정의된 선을 따라 수치적분을 수행하게 된다.평면변형요소에 삽입되는 철근격자는 요소축방향을 따라 변형률 성분 $\varepsilon _ { x x }$ 와 $\varepsilon _ { y y }$ 및응력성분 $\sigma _ { x x }$ 와 $\sigma _ { y y }$ 를 가진다. 평면변형요소에 삽입되는 철근격자의 x축은 철근격자와 평면변형요소가 접하는 선의 접선방향으로 자동적으로 설정된다. 따라서 다른철근격자와는 달리 별도의 요소축을 입력할 필요는 없다. 평면변형요소에 삽입되는철근격자는 선의 접선 방향이 보강재의 x축으로 자동적으로 설정되며 각각의 축방향으로 서로 다른 영 계수나 두께를 정의할 수 있다. + +평면변형요소에 삽입되는 철근격자는 선으로 정의한 후에 자동메쉬(auto-mesh)기능을 사용하여 철근섹션으로 분할한다. 그리고 철근섹션은 평면변형요소에 따라 철근세그먼트로 분할된다. 분할된 철근섹션은 철근절점을 사용하여 정의되며,선 형상을 가지고 있다. midas FEA 에서 입력 가능한 평면변형요소의 격자섹션은 그림 2.4.4 와 같다. 생선의 뼈와 같은 형태는 철근격자를 이루고 있는 선이평면변형요소의 두께방향으로 연장되어 삽입되어 있는 것을 나타낸다. + + + +![](images/page-140_c5cc155b05408e27e0b5449f05ac429718134b94caa8a85c5d5d34d11adb6433.jpg) + +
+text_image + +1 +2 +1 +2 +3 +Y +Z +X +
+ +그림 2.4.4 철근격자섹션과 철근절점 + +그림 2.4.5는 철근격자가 섹션 1과 섹션 2로 구성된 철근격자섹션으로 분할되는방식을 보이고 있다. 섹션 1은 고차 섹션이고, 섹션 2는 저차 섹션이다. 각각의섹션이 이루는 선과 평면변형요소의 테두리가 교차하는 점이 계산되고 섹션은세그먼트로 분할된다. 각각의 세그먼트들은 하나의 모체요소에 완전히 삽입되어있으며 위치절점을 사용하여 정의된다. 고차 세그먼트의 섹션에서 분할된 세그먼트는 중간 위치절점을 사용하여 정의된다. + +![](images/page-140_028fa209d65a3e8f13d4f175812e3aa388d87291e216810b6b5cd2579052ad2d.jpg) + +
+line + +| x | y | +|----|----| +| 24 | 24 | +| 25 | 25 | +| 26 | 26 | +| 27 | 27 | +
+ +그림 2.4.5 평면변형요소에 삽입되는 철근격자섹션 diff --git a/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_015.md b/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_015.md new file mode 100644 index 00000000..976591d9 --- /dev/null +++ b/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_015.md @@ -0,0 +1,336 @@ + + +철근이 삽입되는 섹션이 요소의 경계선을 가로지르지 않도록 세그먼트와 같이입력하면 별도의 분할 없이 그대로 세그먼트로 구성된다. + +철근세그먼트는 그림 2.4.6과 같다. 세그먼트의 차수는 섹션의 차수와 같으며 모재요소의 차수와는 관련이 없다. 고차 섹션을 사용하면 고차 세그먼트로 분할된다. + +![](images/page-141_103997b1d2a01a457d504f41379ea708027b143a04d6014dc850ba20916c8ea8.jpg) + +
+text_image + +1 +2 +1 +2 +3 +Y +Z +X +
+ +그림 2.4.6 평면변형요소에 삽입되는 철근바와 위치절점 + +그림 2.4.7은 섹션 1과 섹션 2로 구성된 섹션이 철근세그먼트로 분할되는 방식을 보이고 있다. + +![](images/page-141_ec802385f8d979c53e034463f89de7681e4461930463ed397d93cdd27d98c5fb.jpg) + +
+line + +| Segment | Value | +| ------- | ----- | +| segment1 | 21 | +| segment2 | 28 | +| segment3 | 36 | +| segment4 | 41 | +| segment5 | 25 | +| segment6 | 45 | +
+ +그림 2.4.7 평면변형요소에 삽입되는 철근바세그먼트 + + + +# 2-5 축대칭요소에 삽입되는 철근 + +# 2-5-1 철근바 + +철근바는 모든 축대칭요소에 삽입이 가능하다. 그림 2.5.1은 축대칭요소에 삽입되는 철근바는 점 형상을 가지며, 점으로 정의 되는 것을 보여준다. + +![](images/page-142_1142977eb345240aa570e71e4ebc86e674c07ebaf97d39ac1edbb740ad2cb4d8.jpg) + +
+text_image + +Z +X +Y +X +σₓₓ +
+ +그림 2.5.1 축대칭요소에 삽입되는 철근바 + +축대칭요소에 삽입된 철근바는 요소의 회전축을 중심으로 한 회전방향으로 일정한 단면적을 가지며 섹션분할이 필요하지 않다. + +그림 2.5.2의 왼쪽에는 고차 축대칭요소에 삽입된 철근바를 보이고 있으며 오른쪽에는 여러 개의 철근바가 축대칭 요소 1, 2, 3, 4 에 삽입이 된 모델을 보이고있다. + + + +![](images/page-143_1720486db79ee364458873f6c891cda351c27baff4d4f8d6a508a46bf1ee1759.jpg) + +element node + +bar location + +그림 2.5.2 축대칭요소에 삽입되는 철근바와 위치절점 + +축대칭요소에 삽입되는 철근바는 축방향으로 변형률 성분 xxε 와 응력성분 $\sigma _ { x x } \frac { \mathbf { \sigma } _ { \mathbf { \overline { { { a } } } } } } { \mathbf { \sigma } _ { \mathbf { { a } } } }$ 가지게 된다. 철근바의 변형과 응력은 둘러싸고 있는 모재요소의 변위와 연관관계를가진다. + +# 2-5-2 철근격자 + +철근격자는 모든 축대칭요소에 삽입이 가능하며 2점 혹은 3점으로 정의될 수 있다. 축대칭요소에 철근격자가 정의되는 방식은 평면변형요소에 삽입된 철근격자와 유사하다. 축대칭요소는 전체 좌표계의 Z축을 중심으로 축대칭이다. 그림2.5.3은 축대칭요소에 삽입되는 철근격자를 보이고 있다. 철근격자는 축대칭요소와 같이 축을 중심으로 회전된다. + + + +![](images/page-144_05c4cae1abe2480774d00850c7c66722d81e309eaf1826cfaa2f4c762a89c5b8.jpg) + +
+text_image + +Z +y +X +Y +X +
+ +element node +location point +integration point + +그림 2.5.3 축대칭요소에 삽입되는 철근격자 + +축대칭 요소와 삽입되는 철근격자가 정의된 선을 따라 수치적분을 수행하게 된다. 축대칭요소에 삽입되는 철근격자는 요소축방향을 따라 변형률 성분 $\varepsilon _ { x x }$ 와 $\varepsilon _ { y y }$ 및 응력성분 $\sigma _ { x x }$ 와 $\sigma _ { y y }$ 를 가진다. midas FEA에서 축대칭요소에 삽입되는 철근격자는 전체좌표계의 XZ 평면 상에서 입력이 되며 접선방향으로 자동으로 요소의 x-축 벡터가 설정된다. 따라서 사용자가 임의의 x-축 벡터를 입력할 필요가 없다. + +축대칭요소에 삽입되는 철근격자는 선을 정의한 후에 정의된 선을 자동메쉬 기능을 사용하여 철근섹션으로 분할한다. 분할된 철근섹션은 철근노드를 사용하여정의되며, 입력 가능한 축대칭요소의 철근격자섹션은 그림 2.5.4와 같다. + +![](images/page-144_2380a75fd636e0eb14f2396c6b86c891d38c977c5a5ce1c89ecadfc4f6b05c74.jpg) + +
+text_image + +1 +2 +Z +Y +X +
+ +![](images/page-144_db602e6dea32d635502fbb92d6df573b037f0886e16c81338318f0945ac357c0.jpg) +그림 2.5.4 축대칭요소에 삽입되는 철근격자섹션 + + + +그림 2.5.5에서 철근격자가 섹션 1과 섹션 2로 구성된 철근격자섹션으로 분할되는 방식을 보이고 있다. 철근격자가 2개의 섹션으로 분리되어 있다. 섹션 1은 고차 섹션이고, 섹션 2는 저차 섹션이다. 각각의 섹션이 축대칭요소의 경계선과 교차하는 점이 계산되며 섹션은 요소에 맞게 세그먼트로 분할된다. 세그먼트를 구성하는 위치절점은 교차점에서 정의되며 고차 세그먼트의 경우 중간 위치절점을추가로 사용하여 정의된다. + +![](images/page-145_1c281e67bbf6fa77dfa4eeea0474b0c1295664d3e284b00ebf0c303e73507c42.jpg) + +
+line + +| x | y | +|----|----| +| 24 | 24 | +| 25 | 25 | +| 26 | 26 | +| 27 | 27 | +
+ +그림 2.5.5 축대칭 요소에 삽입되는 철근격자섹션 + +개별입력을 하여 철근섹션이 요소에 포함되도록 정의되면 별도의 분할 없이 요소에 삽입되는 철근세그먼트로 사용된다. 이 때 입력된 철근섹션은 요소의 경계를 가로지르지 않아야 한다. 분할에 사용되는 철근세그먼트의 종류는 그림 2.5.6과 같다. + +![](images/page-145_fbf83d1d5d2d2c537dcdc2fcb47128cb8754f42e3ad396870e40394d89174795.jpg) + +
+text_image + +1 +2 +1 +2 +3 +Z +Y +X +
+ +그림 2.5.6 축대칭요소에 삽입되는 철근격자세그먼트와 위치절점 + +그림 2.5.7은 섹션 1과 섹션 2로 구성된 철근격자섹션이 위치절점으로 정의되는 + + + +세그먼트 1\~6으로 분할되어 재구성되어 있는 것을 보이고 있다. + +![](images/page-146_3219bd7fe9cd4b975104f5a17eee4b420427bbe59bfda8756bd682c33a9bffae.jpg) + +
+line + +| Segment | Value | +|---------|-------| +| segment 1 | 21 | +| segment 2 | 35 | +| segment 3 | 28 | +| segment 4 | 36 | +| segment 5 | 41 | +| segment 6 | 45 | +
+ +그림 2.5.7 축대칭 요소에 삽입되는 철근격자세그먼트와 위치절점 + + + +# 2-6 평면응력요소에 삽입되는 철근 + +# 2-6-1 철근바 + +평면응력요소에 삽입되는 철근섹션은 2점 혹은 3점으로 정의되는 바섹션으로 구성된다. 그림 2.6.1은 한 개의 철근이 두 개의 섹션 1, 섹션 2로 구성되고, 다시모재요소에 따라 철근세그먼트로 분할되는 예를 보여준다. 철근세그먼트는 모재요소의 경계를 가로지를 수 없다. 섹션 1은 3점 고차 섹션이고, 섹션 2는 2점 저차 섹션 이다. 고차 섹션이 세그먼트로 분리되면 3점의 고차 세그먼트들로 분할된다. 같은 방식으로 저차 섹션이 세그먼트로 분리되면 2점의 저차 세그먼트들로 분할된다. 그리고 각각의 세그먼트들은 위치절점으로 정의되고 있다. + +![](images/page-147_4661ae664f5aaebeebd3d679452bdbdb3640aab36cd491e465ddc9a0afb492bc.jpg) + +
+line + +| x | y | +|----|----| +| 24 | 24 | +| 25 | 25 | +| 26 | 26 | +| 27 | 27 | +
+ +reinforcement node + +location point + +그림 2.6.1 평면응력요소에 삽입되는 철근바섹션 + +그림 2.6.2는 그림 2.6.1과 같은 일련의 과정을 거쳐 최종적으로 분할된 철근세그먼트들과 모재 요소들을 보여준다. 위치절점 28, 36, 22로 정의된 세그먼트 2는 요소번호 2인 평면응력요소를 지나가고 있다. 세그먼트 2가 포함되어 있던섹션이 고차이기 때문에 세그먼트도 고차 세그먼트로 분할되어 있다. 위치절점30, 27로 정의되는 세그먼트 5는 저차 섹션으로부터 분할되었기 때문에 저차 세그먼트의 형태를 가지고 있다. + + + +![](images/page-148_d77da82402c57ff70b7e15a38b6a0b2ff8e4c803ca493859e8cef7e721e7878d.jpg) + +
+line + +| Segment | Value | +|---------|-------| +| segment1 | 21 | +| segment2 | 35 | +| segment3 | 28 | +| segment4 | 36 | +| segment5 | 41 | +| segment6 | 45 | +
+ +그림 2.6.2 평면응력요소에 삽입되는 철근바 세그먼트 + +그림 2.6.3은 8절점 사각형 평면응력요소에 삽입되어 있는 철근세그먼트를 보이고 있다. 각각의 바 세그먼트는 2개의 적분점을 가진다. + +![](images/page-148_548382ea7b012cbe4a3a5d199e3b1deb6790190081ee8abb3cce7642edbda42e.jpg) + +
+flowchart + +```mermaid +graph TD + A["•"] --> B["•"] + B --> C["•"] + C --> D["•"] + D --> E["•"] + E --> F["•"] + F --> G["•"] + G --> H["•"] + H --> I["•"] + I --> J["•"] + J --> K["•"] + K --> L["•"] + L --> M["•"] + M --> N["•"] + N --> O["•"] + O --> P["•"] + P --> Q["•"] + Q --> R["•"] + R --> S["•"] + S --> T["•"] + T --> U["•"] + U --> V["•"] + V --> W["•"] + W --> X["•"] + X --> Y["•"] + Y --> Z["•"] + Z --> A + style A fill:#f9f,stroke:#333 + style B fill:#f9f,stroke:#333 + style C fill:#f9f,stroke:#333 + style D fill:#f9f,stroke:#333 + style E fill:#f9f,stroke:#333 + style F fill:#f9f,stroke:#333 + style G fill:#f9f,stroke:#333 + style H fill:#f9f,stroke:#333 + style I fill:#f9f,stroke:#333 + style J fill:#f9f,stroke:#333 + style K fill:#f9f,stroke:#333 + style L fill:#f9f,stroke:#333 + style M fill:#f9f,stroke:#333 + style N fill:#f9f,stroke:#333 + style O fill:#f9f,stroke:#333 + style P fill:#f9f,stroke:#333 + style Q fill:#f9f,stroke:#333 + style R fill:#f9f,stroke:#333 + style S fill:#f9f,stroke:#333 + style T fill:#f9f,stroke:#333 + style U fill:#f9f,stroke:#333 + style V fill:#f9f,stroke:#333 + style W fill:#f9f,stroke:#333 + style X fill:#f9f,stroke:#333 + style Y fill:#f9f,stroke:#333 + style Z fill:#f9f,stroke:#333 +``` +
+ +element node +location point +integration point + +그림 2.6.3 평면응력요소에 삽입되는 철근바세그먼트와 위치절점 + +평면응력요소에 사용되는 철근바를 정의하기 위한 재료 물성은 단면적과 영 계수로 구성된다. + +# 2.6.2 철근격자 + +철근격자는 midas FEA 의 모든 평면응력요소에 삽입이 가능하며, 3점, 4점, 6점, 혹은 8점으로 정의될 수 있다. 평면응력요소에 삽입되는 철근격자를 생성하는 방법은 철근바의 경우와 동일하다. 평면응력요소에 삽입되는 철근격자는 요 + + + +소의 표면을 완전히 덮어야 한다. 철근격자섹션이 완전히 덮고 있는 평면응력요소에만 철근격자가 삽입되게 된다. + +![](images/page-149_db38bacd7fd6208c8f279498ba96a9213bfcc04981f85f44bcaab1b33ff7662c.jpg) + +
+text_image + +27 +24 +26 +25 +
+ +그림 2.6.4 평면응력요소에 삽입되는 철근격자섹션 + +그림 2.6.4에서와 같이 24, 25, 26, 27번 철근노드로 구성된 철근격자섹션이 격자모양으로 구성된 평면응력요소 메쉬 위에 정의되어 있다. X 표시를 한 평면응력요소에만 철근격자가 삽입된다. + + + +# 2-7 입체요소에 삽입되는 철근 + +# 2-7-1 철근바 + +철근바는 모든 고체요소에 삽입이 가능하며, 2점 혹은 3점으로 정의될 수 있다.철근바는 철근섹션과 철근세그먼트로 구성된다. 사용자가 철근섹션을 정의하면,전처리 과정에 의해 철근섹션과 모재요소 경계선의 교차점이 계산된다. 각각의철근섹션은 모재요소 내부에 포함되는 철근세그먼트로 분할된다. midas FEA 에서는 3차원 곡선을 정의하여 철근을 모델링 할 수 있다. 정의된 곡선을 자동메쉬 기능을 사용하여 철근섹션으로 분할한다. 철근 섹션은 2개나 3개의 철근 절점으로 정의된다. 철근세그먼트는 위치절점으로 정의된다. 철근의 위치절점의 좌표는 모재요소에 추가되는 철근의 강성과 내력의 계산에 사용된다. + +그림 2.7.1의 철근섹션은 2개의 섹션으로 구성되어 있다. 섹션 1은 철근절점 26,27, 28로 구성되어 있고 섹션 2는 철근절점 28, 29를 사용하여 구성되어 있다.첫 번째 섹션은 고차 섹션이며 두 번째 섹션은 저차 섹션이다. 사용자는 자동메쉬 기능을 사용하여 섹션을 정의할 수 있다. diff --git a/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_016.md b/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_016.md new file mode 100644 index 00000000..e56c2d65 --- /dev/null +++ b/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_016.md @@ -0,0 +1,471 @@ + + +![](images/page-151_446d4e8bf361ff3f3aa4aeb64d5ea1cbefdfd2dfd2dbef4e175bf50638fa92c6.jpg) + +
+line + +| Point | Value | +|-------|-------| +| 1 | 21 | +| 2 | 26 | +| 3 | 35 | +| 4 | 45 | +
+ +그림 2.7.1 입체요소에 삽입되는 철근바섹션 + +그림 2.7.2는 그림 2.7.1과 동일한 입체요소의 메쉬와 철근바로 구성되어 있다.여기서 철근은 세그먼트로 분할되어 있다. 입체요소 2, 4번을 가로지르고 있는섹션 1은 두 세그먼트로 나누어져 있으며 4, 8, 12번을 가로지르는 섹션 2는 3조각의 세그먼트로 분할되어 있다. 고차 섹션인 섹션 1에 속해있는 세그먼트들은3개의 위치절점으로 정의되어 있으며 저차 섹션인 섹션 2에 속해있는 세그먼트들은 2개의 위치절점으로 정의되어 있다. + + + +![](images/page-152_a5f301cb5cfedd20b88ebf4a25d806c9ce3a4e4ff1daa4b54ea08004197f3c39.jpg) + +
+line + +| Point | X | Y | +|-------|----|----| +| 1 | 21 | 21 | +| 2 | 28 | 28 | +| 3 | 35 | 35 | +| 4 | 45 | 45 | +
+ +그림 2.7.2 입체요소에 삽입되는 철근바의 위치절점 + +그림 2.7.3은 20절점 고체요소에 삽입된 철근바세그먼트를 보이고 있다. 바세그먼트는 위치절점의 개수와 상관없이 2개의 적분점을 가지고 있다. 적분점에서의 요소좌표계 x축 방향의 응력과 변형률 ε 는 철근바의 접선방향과 같다. + +![](images/page-152_55e947a94cc5809dd79f8e2daabdd83aa3cde90b478d712b70566128b6ff83a6.jpg) + +
+flowchart + +```mermaid +graph TD + A["•"] --> B["•"] + B --> C["•"] + C --> D["•"] + D --> E["•"] + E --> F["•"] + F --> G["•"] + G --> H["•"] + H --> I["•"] + I --> J["•"] + J --> K["•"] + K --> L["•"] + L --> M["•"] + M --> N["•"] + N --> O["•"] + O --> P["•"] + P --> Q["•"] + Q --> R["•"] + R --> S["•"] + S --> T["•"] + T --> U["•"] + U --> V["•"] + V --> W["•"] + W --> X["•"] + X --> Y["•"] + Y --> Z["•"] + Z --> A["•"] + style A fill:#fff,stroke:#000 + style B fill:#fff,stroke:#000 + style C fill:#fff,stroke:#000 + style D fill:#fff,stroke:#000 + style E fill:#fff,stroke:#000 + style F fill:#fff,stroke:#000 + style G fill:#fff,stroke:#000 + style H fill:#fff,stroke:#000 + style I fill:#fff,stroke:#000 + style J fill:#fff,stroke:#000 + style K fill:#fff,stroke:#000 + style L fill:#fff,stroke:#000 + style M fill:#fff,stroke:#000 + style N fill:#fff,stroke:#000 + style O fill:#fff,stroke:#000 + style P fill:#fff,stroke:#000 + style Q fill:#fff,stroke:#000 + style R fill:#fff,stroke:#000 + style S fill:#fff,stroke:#000 + style T fill:#fff,stroke:#000 + style U fill:#fff,stroke:#000 + style V fill:#fff,stroke:#000 + style W fill:#fff,stroke:#000 + style X fill:#fff,stroke:#000 + style Y fill:#fff,stroke:#000 + style Z fill:#fff,stroke:#000 +``` +
+ +element node +location point +integration point + +그림 2.7.3 입체요소에 삽입되는 철근바세그먼트와 위치절점 + + + +# 2-7-2 철근격자 + +철근격자는 midas FEA의 모든 고체요소에 삽입이 가능하며, 3점 혹은 6점 삼각형 철근격자와 4점 혹은 8점 사각형 철근격자를 입력할 수 있다. 철근격자를 정의하는 과정은 철근바와 유사하다. 평면을 정의하고, 자동메쉬기능을 사용하여철근섹션으로 분할 한 후 전처리 과정에서 철근격자세그먼트로 분할한다. 이 때철근격자세그먼트는 모재요소 내부에 포함된다. + +![](images/page-153_7e77db69dbaa64ad72453cbbc2927cfed2c59551d2ce642b5bf1c9564291021e.jpg) + +
+surface_3d + +| Point | X | Y | +|-------|----|----| +| 1 | 30 | 30 | +| 2 | 31 | 31 | +| 3 | 32 | 32 | +| 4 | 33 | 33 | +| 5 | 34 | 34 | +| 6 | 35 | 35 | +| 7 | 36 | 36 | +| 8 | 37 | 37 | +| 9 | 38 | 38 | +| 10 | 39 | 39 | +| 11 | 40 | 40 | +| 12 | 41 | 41 | +| 13 | 42 | 42 | +| 14 | 43 | 43 | +| 15 | 44 | 44 | +| 16 | 45 | 45 | +| 17 | 46 | 46 | +| 18 | 47 | 47 | +| 19 | 48 | 48 | +| 20 | 49 | 49 | +| 21 | 50 | 50 | +| 22 | 51 | 51 | +| 23 | 52 | 52 | +| 24 | 53 | 53 | +| 25 | 54 | 54 | +| 26 | 55 | 55 | +| 27 | 56 | 56 | +| 28 | 57 | 57 | +| 29 | 58 | 58 | +| 30 | 59 | 59 | +| 31 | 60 | 60 | +| 32 | 61 | 61 | +| 33 | 62 | 62 | +| 34 | 63 | 63 | +| 35 | 64 | 64 | +| 36 | 65 | 65 | +| 37 | 66 | 66 | +| 38 | 67 | 67 | +| 39 | 68 | 68 | +| 40 | 69 | 69 | +| 41 | 70 | 70 | +| 42 | 71 | 71 | +| 43 | 72 | 72 | +| 44 | 73 | 73 | +| 45 | 74 | 74 | +| 46 | 75 | 75 | +| 47 | 76 | 76 | +| 48 | 77 | 77 | +| 49 | 78 | 78 | +| 50 | 79 | 79 | +| 51 | 80 | 80 | +| 52 | 81 | 81 | +| 53 | 82 | 82 | +| 54 | 83 | 83 | +| 55 | 84 | 84 | +| 56 | 85 | 85 | +| 57 | 86 | 86 | +| 58 | 87 | 87 | +| 59 | 88 | 88 | +| 60 | 89 | 89 | +| 61 | 90 | 90 | +| 62 | 91 | 91 | +| 63 | 92 | 92 | +| 64 | 93 | 93 | +| 65 | 94 | 94 | +| 66 | 95 | 95 | +| 67 | 96 | 96 | +| 68 | 97 | 97 | +| 69 | 98 | 98 | +| 70 | 99 | 99 | +| 71 | 100| 100 | +The chart displays a 3D surface plot with 3D coordinates (X, Y) and the label 'reinforce node' and 'location point' at the top and bottom of the plot. The points are labeled with numbers 30 through 37. +
+ + + +![](images/page-154_8761e001c4eed19e4f5b77e2d6e9e8d4449caccbae647db8193246f70f468ac0.jpg) + +
+natural_image + +Isometric wireframe diagram of a 3D geometric structure with interconnected nodes and lines (no text or symbols) +
+ +그림 2.7.4 입체요소에 삽입되는 철근격자섹션 + +그림 2.7.4는 철근절점 30, 31, 32, 33, 34, 35, 36, 37을 사용하여 정의된 사각형 철근섹션과 입체요소를 보이고 있다. 전처리 과정을 통해서 그림 2.7.4와 같이 삼각형 격자세그먼트들로 분할된다. + +사용자가 전처리에 의한 분할과정을 거치지 않고 원하는 철근세그먼트를 직접입력하는 것도 가능하다. 요소에 직접 입력하기 위해서는 사용자가 입력하고자하는 요소에 완전히 포함되도록 철근격자섹션을 정의하면, 전처리에서 분할과정없이 사용자가 입력한 철근섹션을 세그먼트로 사용하게 된다. + + + +![](images/page-155_0b20193b4040214b672fc6f1272d574230ca32766451aff5202883f414bb2a07.jpg) + +
+flowchart + +```mermaid +graph TD + 1 --> 8 + 8 --> 4 + 4 --> 7 + 7 --> 3 + 3 --> 6 + 6 --> 2 + 2 --> 5 + 5 --> 8 + 8 -.-> 5 + 5 -.-> 6 + 6 -.-> 7 + 7 -.-> 8 + 8 -.-> 9 + 9 -.-> 1 +``` +
+ +![](images/page-155_e4708195cfc04f4b4671db7bb47f4f74b224cece3d83ba1d6b9cf0068b136ae1.jpg) + +
+natural_image + +3D geometric diagram of a cube with numbered vertices (1, 2, 3, 4) and solid/dashed lines indicating hidden edges (no text or symbols) +
+ +![](images/page-155_6a02193a05c4cdce35a21c66626227140e2785421594679456bcfc4c422d1558.jpg) + +
+text_image + +1 +2 +3 +4 +
+ +![](images/page-155_2119c324a5a96acd081b51a9506cdb9271e3ddaeffc51439bab36b3055a1aefa.jpg) + +
+flowchart + +```mermaid +graph TD + 1 --> 2 + 1 --> 3 + 1 --> 4 + 1 --> 5 + 1 --> 6 + 2 --> 3 + 2 --> 4 + 2 --> 5 + 2 --> 6 + 3 --> 4 + 3 --> 5 + 3 --> 6 + 4 --> 5 + 4 --> 6 + 5 --> 6 + 6 --> 1 + 6 --> 2 + 6 --> 3 + 6 --> 4 + 6 --> 5 +``` +
+ +![](images/page-155_f04e246fe3e10dd644969a1f1040b247e88ed7bf28057ba2ec205dc713ba6f34.jpg) + +
+natural_image + +Geometric diagram of a polyhedron with labeled vertices (no text or symbols) +
+ +![](images/page-155_d94cc7d19821aa03d8c1269b02ecd327b861ef8e4f65b3861e4f94758923b952.jpg) + +
+text_image + +1 +2 +3 +
+ +그림 2.7.5 입체요소에 삽입되는 철근격자세그먼트의 종류 + +그림 2.7.5는 요소에 직접 입력하는 방식을 사용하여 모재요소에 삽입될 수 있는 모든 철근세그먼트들과 입체요소의 조합을 보이고 있다. 전처리를 사용하여세그먼트를 생성하게 되면 세그먼트는 삼각형만으로 분할되지만, 직접 입력하게되면 사용자가 원하는 모양의 세그먼트를 입력할 수 있다. + +그림 2.7.6은 고체요소에 삽입된 철근격자세그먼트를 보여준다. 그림에서 x와 y는 철근격자세그먼트의 요소좌표축 방향이다. + + + +Part 1 Element Library +![](images/page-156_950e1acfe5a1a07dc94caac44e09def47bea22dd6521f08e75b0057667ea05bf.jpg) + +
+text_image + +Diagram showing a 3D surface with coordinate axes x and y, and marked points and triangles, likely illustrating a mathematical or geometric concept. +
+ +element node +location point +integration point + +그림 2.7.6 입체요소에 삽입되는 철근격자세그먼트와 위치절점 + + + +# 2-8 판요소에 삽입되는 철근 + +# 2-8-1 철근바 + +곡면판요소나 평면판요소에 삽입되는 철근바섹션은 2점 혹은 3점으로 정의될 수있다. 철근바섹션의 위치는 철근절점으로 정의된다. 전처리과정에서 철근섹션은철근세그먼트로 분할된다. 이 때 철근세그먼트는 하나의 판요소에 포함된다. 철근세그먼트의 위치는 위치절점으로 정의된다. + +![](images/page-157_5588852d25a832b64a9e856f0b5f842aec570658de79eaf8fbffbbe558931d77.jpg) + +
+text_image + +Diagram showing a curved surface with labeled points and a vector ξ, featuring geometric shapes and connecting lines. +
+ +그림 2.8.1 판요소에 삽입되는 철근바 + +element node +location point +integration point + +그림 2.8.1은 곡면판요소에 삽입된 철근바를 나타내고 있다. 철근바세그먼트는 3개의위치절점을 이용하여 정의되어 있으며 2개의 적분점을 가진다. 적분점은 위치절점을이용하여 자동적으로 계산된다. + + + +![](images/page-158_604f70dd7e366526d6ded0cd17634aca20611a9b54465e1733dcc9adc6216162.jpg) + +
+radar + +| Point | Value | +|-------|-------| +| 1 | 24 | +| 2 | 25 | +| 3 | 26 | +| 4 | 27 | +| 5 | 28 | +
+ +element node +location point +reinforcement node + +그림 2.8.2 곡면판에 삽입되는 철근바섹션 + +철근바섹션은 사용자가 3차원 공간에서 선을 정의하여 입력할 수 있다. 그림2.8.2에서 2개의 철근바섹션을 보이고 있다. 첫 번째 섹션은 24, 25번 철근절점으로 이루어진 직선 섹션이고 두 번째 섹션은 26, 27, 28번 철근절점으로 이루어진 곡선 고차 섹션이다. 각각의 섹션은 판요소 메쉬를 통과해 지나가므로 판요소의 경계선과 섹션의 교차점을 계산하여 위치절점을 생성한다. 각각의 세그먼트는 판요소 내부에 삽입되도록 섹션에서 분할된다. 곡선 세그먼트들은 섹션에서 분리되면서 중간 위치절점을 생성하여 섹션이 가지고 있는 곡선 형상을 각각의 세그먼트도 유지한다. + +![](images/page-158_7477ebb3e8a09064c7d4d84ea0251e42271fa04e45180eded9b50bab448bf05a.jpg) + +
+flowchart + +```mermaid +graph TD + z1["1"] --> z2["2"] + z1 --> z1 + z2 --> z1 + style z1 fill:#f9f,stroke:#333 + style z2 fill:#f9f,stroke:#333 +``` +
+ +![](images/page-158_1ba1c647dbd66e7bb719cb5c51db6e045670693a1b4e85bd4b0e2156abeb17e5.jpg) + +
+flowchart + +```mermaid +graph TD + A["1"] -->|z1| B["2"] + B -->|z2| C["3"] + C -->|z3| A +``` +
+ +그림 2.8.3 곡면판요소에 삽입되는 철근바세그먼트의 편심벡터 + + + +판요소에 삽입되는 철근세그먼트를 정의하는 각각의 위치절점은 판요소의 중립면에위치하며 요소좌표계의 z축방향으로 편심벡터를 가진다. 그림 2.8.3은 위치 절점의편심벡터들을 보이고 있다. 편심벡터는 판요소의 중립면에 위치하는 철근의 위치절점에서 철근이 판요소 내부의 실제 위치로의 벡터이다. 편심벡터는 전처리에서 섹션을세그먼트로 분리하는 과정에서 자동적으로 계산된다. + +# 2.8.2 철근격자 + +곡면판요소나 평면판요소에 삽입되는 철근격자는 3점 혹은 6점 삼각형 철근격자나 4점 혹은 8점 사각형 철근격자로 입력할 수 있다. + +![](images/page-159_67c315d38f23cd1dc613dd7f4c2504409e7b685caaffad4d2f6c8efbe8d41b1e.jpg) + +
+text_image + +element node +location point +Z +Y +X +MCS +y +x +z +t +z +O +
+ +그림 2.8.4 곡면판요소에 삽입되는 철근격자세그먼트와 위치절점에서의 편심 + +그림 2.8.4는 철근격자의 세그먼트를 나타내고 있다. 재료 모델은 등방성 선형탄성모델과 이방성 선형탄성모델 두 재료모델을 사용할 수 있다. 그림 2.8.4에서와 같이 철근격자는 요소축을 정의하여 철근격자의 방향성을 나타낼 수 있다. 이 때 x와 y는항상 직교한다. 이방성 선형탄성 모델을 적용할 경우 철근격자의 x와 y 방향에 대하여 서로 다른 격자 두께와 탄성계수를 적용할 수 있다. + + + +![](images/page-160_86896bfb5c248914da04b68d529214287640df2071f30789016ae4c6a9f9e57e.jpg) + +
+text_image + +24 +25 +26 +27 +
+ +reinforcement node + +그림 2.8.5 GRID section in curved plate elements + +그림2.8.5에서는 철근절점 24, 25, 26, 27으로 정의된 격자섹션을 보이고 있다. 격자섹션은 3점 혹은 6점 삼각형, 4점 혹은 8점 사각형으로 정의가 가능하다. 그림2.8.5에서와 같이 격자섹션 중 요소 전체를 완전히 덮고 있는 부분만이 철근격자세그먼트로 사용되고 있다. 음영처리가 된 부분이 철근격자가 삽입된 판요소이다. + +![](images/page-160_4e54fd4dcdcee1b9241a7439202d5e789ecf4f78a1a8ec2a5113f8342984458b.jpg) + +
+radar + +| Point | Value | +|-------|-------| +| 1 | 0.2 | +| 2 | 0.2 | +| 3 | 0.2 | +| 4 | 0.2 | +| 5 | 0.2 | +| 6 | 0.2 | +
+ +그림 2.8.6 Eccentricities for GRID in curved plate element + +철근격자세그먼트는 판요소의 중립면에 위치한 위치절점으로 정의된다. 각각의위치절점은 요소좌표계의 z방향으로 편심거리가 계산된다. 편심거리는 앞서 철 diff --git a/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_017.md b/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_017.md new file mode 100644 index 00000000..21361556 --- /dev/null +++ b/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_017.md @@ -0,0 +1,285 @@ + + +근바의 편심거리와 같은 의미를 가진다. 그림 2.8.6에서 2, 3번 판요소에 삽입된철근격자세그먼트의 위치절점은 같은 편심거리를 가진다. 1번 판요소에 삽입된철근격자세그먼트의 위치절점들은 각각 0.2에서 0.6사이의 편심거리를 가지고있다. + + + +# 2-9 철근의 프리스트레스 + +# 2-9-1 개요 + +철근바의 세부타입으로 철근과 텐던(tendon)이 있다. 철근은 구조물에 사용되는철근을 표현하기 위한 것이고, 텐던은 프리스트레스(prestress)가 가해진 텐던을표현하기 위한 것이다. 적용할 수 있는 프리스트레스의 형태는 두 가지이다. 하나는 일정한 긴프리스트레스가 텐던 전체에 가해지는 균일 프리스트레스(uniform prestress)형태이고, 다른 하나는 곡률(curvature)에 의한 마찰(friction)과 파상(wobble)에 의한 마찰손실과 정착장치(anchorage device)의 이동(penetration)에 의한 손실(prestress loss)을 고려할 수 있는 포스트텐션(posttension) 형태이다. + +# 2-9-2 철근의 균일 프리스트레스 + +균일 프리스트레스는 일정한 프리스트레스를 텐던타입의 철근에 작용시키는 것이다. 이때 철근격자는 2개의 축방향으로 각각 다른 프리스트레스를 적용할 수있고, 철근바는 한 개의 축방향으로 일정 크기의 프리스트레스를 적용할 수 있다. + +# 2-9-3 철근의 포스트텐션 프리스트레스 + +# (1) 프리스트레스의 손실 + +일반적으로 텐던의 프리스트레스는 텐던의 양단에 가해지는 초기 긴장력(jacking force)에 의해 발생된다. 그러나 텐던에 가해지는 긴장력에 의한 프리스트레스는 프리스트레스의 손실로 인해서 감소하게 된다. 이러한 프리스트레스의 손실은 몇 가지 요인에 의해 발생된다. + + + +곡선형태로 배치된 텐던은 텐던의 곡률에 의한 영향으로 프리스트레스는 손실되게 되며, 이러한 프리스트레스의 손실을 곡률에 의한 마찰 손실이라 한다. 그리고 곡률에 의한 손실 외에도 파상효과에 의한 손실이 있다. 또한 텐던의 긴장이완료되고 마찰에 의한 손실이 발생한 후에도 정착장치의 이동에 의해서 손실이발생할 수 있다. + +midas FEA 에서는 이러한 손실의 예측은 콘크리트와 텐던의 모델링을 통해서구할 수 있다. + +(2) 마찰손실 +![](images/page-163_ada546f086cc3fec1fe6817b9f27fac912433da89d3d0984120e6bb5256fd222.jpg) + +
+text_image + +1/k +P +p +P +
+ +그림 2.9.1 텐던의 곡률에 의한 압력 + +프리스트레스의 손실은 텐던과 이를 둘러 싼 재료와의 마찰에 의해서 발생하게된다. 이 마찰은 텐던의 곡률에 의해서 생기는 곡률마찰(curvature friction)과텐던의 길이에 의해서 생기는 파상마찰(wobble friction)로 구성된다. midasFEA에서는 두 가지 CEB-FIP 과 Korean Standard(KS) 두 가지의 설계기준을사용할 수 있다. + +곡면에 의한 마찰은 다음과 같이 구할 수 있다. 곡선 형태를 가지는 텐던의 형상을텐던의 축방향 좌표축 r 에 대해서 x(r) 로 나타낼 수 있다. 이때 외력 P 가 텐던의양단에 가해질 때 다음과 같은 압력 p 가 곡률에 의해서 생기게 된다. + + + +$$ +p = \kappa P \tag {2.9.1} +$$ + +곡률 κ 는 다음과 같이 산정된다. + +$$ +\kappa = \left| \frac {\partial^ {2} x}{\partial r ^ {2}} \right| \tag {2.9.2} +$$ + +압력 p 에 의한 단위 길이당 프리스트레스의 감소는 마찰계수 µ (/radian)을 사용하여 구할 수 있다. + +$$ +\frac {\partial P}{\partial r} = \mu p \tag {2.9.3} +$$ + +부분적인 불규칙성에 의해서 마찰이 생기는 현상을 나타내기 위해 파상효과(wobble effect)를 고려하여 해석을 수행한다. 파상효과를 고려하기 위해 CEB-FIP와 KS는 서로 다른 식을 사용한다. + +# < CEB-FIP > + +가상의 곡률 φ1 을 도입하고, 이 보조변수를 파상계수(wobble parameter)라고 한다. + +$$ +\frac {\partial P}{\partial r} = - \mu \phi_ {1} P \tag {2.9.4} +$$ + +특정위치에서의 프리스트레스의 손실은 프리스트레스가 가해지는 정착구로부터시작하여 축방향으로 적분을 하여 얻어 진다. + +$$ +\Delta P = \int \frac {\partial P}{\partial r} d r \tag {2.9.5} +$$ + +텐던이 일정한 곡률 κ 를 가지고 있고 정착구 에서 가해지는 힘을 P0 라고 하면,∆r 길이 만큼 시작점에서 떨어진 위치에서의 프리스트레스는 다음과 같이 얻어진다. + +$$ +P (\Delta r) = P _ {0} e ^ {- \mu (\kappa + \phi_ {1}) \Delta r} \tag {2.9.6} +$$ + +그리고 곡률과 파상에 의한 손실이 고려된 식 (2.9.6)을 다음과 같이 쓸 수 있다. + +$$ +P (\Delta r) = P _ {0} e ^ {- \mu \Delta \phi} \tag {2.9.7} +$$ + + + +여기서 ∆φ 는 시작 점에서의 축방향 벡터와 ∆r 떨어진 위치에서의 접선 벡터의 회전각을 의미한다. + +![](images/page-165_9395a14600022b810b40ea31c392a8d5507a48bd636b85403796d32b867fa04f.jpg) + +
+text_image + +Δr +∂x/∂r(r) +∂x/∂r(r + Δr) +
+ +그림 2.9.2 텐던의 회전각 변화 + +식 (2.9.7)에서 사용된 ∆φ 는 다음과 같다. + +$$ +\Delta \phi = \Delta r \left(\phi_ {1} + \kappa\right) = \Delta r \phi_ {1} + \left| \Delta \frac {\partial x}{\partial r} \right| \tag {2.9.8} +$$ + +# < Korean Standard > + +파상마찰계수(wobble friction coefficient) K 를 도입하여 파상효과에 의한 단위길이당 파상에 의한 손실을 나타내면 다음과 같다. + +$$ +\frac {\partial P}{\partial r} = - K P \tag {2.9.9} +$$ + +CEB-FIP와 같이 일정한 곡률 κ 를 가지고 있는 텐던에서 정착구에 가해지는 힘을P0 라고 하면, 시작점에서 ∆r 길이 만큼 떨어진 위치에서의 곡률과 파상에 의한 손실이 고려된 프리스트레스는 다음과 같다. + +$$ +P (\Delta r) = P _ {0} e ^ {- (\mu \kappa + K) \Delta r} \tag {2.9.10} +$$ + +여기서 ∆φ는 시작 점에서의 축방향 벡터와 ∆r 떨어진 위치에서의 접선 벡터의 회전각을 의미하며 ∆φ는 다음과 같다. + +$$ +\Delta \phi = \Delta r \kappa = \left| \Delta \frac {\partial x}{\partial r} \right| \tag {2.9.11} +$$ + +midas FEA 에서는 정착구 위치에서의 힘 P0 와 텐던의 물성치를 이용하여 각 기준에따른 손실을 고려한 프리스트레스를 각 철근세그먼트의 적분점에서 찾는다. + + + +# (3) 정착장치의 이동 + +텐던은 단부 정착구 에서의 쐐기에 의해서 정착이 된다. 그러므로 프리스트레스에 의해서 쐐기는 정착한 위치에서 안쪽으로 이동이 발생하게 된다. 이렇게 이동하여 들어간 길이를 ∆l 이라고 할 때, 정착구가 밀려들어가면서 텐던의 프리스트레스에 영향을주게 된다. 쐐기의 이동에 의해 텐던의 프리스트레스가 영향을 받는 길이를 ∆x 라고한다면, 다음과 같은 힘 평형식을 유도할 수 있다. + +$$ +\int_ {\Delta x} \Delta P (r) d r = \Delta l E A \tag {2.9.12} +$$ + +![](images/page-166_6e34f33208ee7c4a7c715d27169919c12c4031634961fc5c8361227d93d2bfa7.jpg) + +
+text_image + +P +before penetration +ΔP +ΔI ×EA +after penetration +Δx +anchor +tendon +x +
+ +그림 2.9.3 정착되어 있는 텐던에서의 프리스트레스 손실 + +그리고 정착구로부터 시작하여 식 (2.9.13)를 만족하는 텐던의 프리스트레스를찾게 된다. + +$$ +\int_ {L} \Delta P (x) d r \geq \Delta l E A \tag {2.9.13} +$$ + +식 (2.9.13)의 적분은 사다리꼴 공식(trapezoidal rule)을 사용해 수행하며, 식 + + + +(2.9.13)의 조건을 만족하는 적분점을 우선 찾아내게 된다. 만약 식 (2.9.13)의조건을 만족시키는 점이 텐던 내에 존재하지 않으면, 텐던의 손실을 구하는 과정은 정상적으로 수행되지 않는다. + +텐던의 양쪽 끝에서 동시에 긴장하게 되는 경우는 두 단계에 걸쳐서 텐던의 응력을 구하게 된다. 이 때의 프리스트레스 손실에 의한 영향은 양단에 대해서 각각 계산된다. 그림 2.9.4는 양단에서 정착구의 이동에 의한 프리스트레스 손실을보이고 있다. + +![](images/page-167_5ad9bae6243c3e8e7d31e17faaf2f4cc8980675a1258bcd2a0d69386d36f88b2.jpg) + +
+line + +| penetration level | tendon | tensioning | penetration at 1 | penetration at 2 | penetration at 3 | +| ----------------- | ------ | --------- | ---------------- | ---------------- | ---------------- | +| limited penetration | 2 | P | P | P | P | +| limited penetration | 1 | P | P | P | P | +| large penetration | 2 | P | P | P | P | +| large penetration | 1 | P | P | P | P | +| excessive penetration | 2 | P | P | P | P | +
+ +그림 2.9.4 정착장치의 이동에 따른 프리스트레스의 손실 + +그림 2.9.4 1번 위치의 정착구로부터 시작한 프리스트레스와 2번 위치의 정착구로부터 시작한 프리스트레스를 각각 구한다. 이 때 구해진 프리스트레스는 정착구의 이동에 의한 손실을 반영하지 않은 프리스트레스다. + +양단의 정착구의 이동에 의한 손실을 각각 계산한 다음, 양단의 프리스트레스분포를 조합하여 최종적인 프리스트레스 분포를 구하게 된다. + + + +Part 1 Element Library + + + +# Chapter 3. Interface Elements + +# 3-1 개요 + +계면요소(interface element)는 동질재료에서의 균열면 또는 이질재료간의 경계면 활동을 해석하기 위한 요소이다. 계면요소는 콘크리트 균열을 묘사하는 이산균열(discrete-cracking), 철근과 콘크리트 사이의 부착슬립(bond-slip), 철근과콘크리트의 접합면이나 조적조의 접합면 등과 같은 구조물의 접합부를 해석하는데 주로 사용된다. + +계면요소는 일반적인 유한요소 정식화과정을 사용하지만, 요소의 두께는 0으로 가정한다. 두께가 0인 계면요소를 수치 해석적으로 정의하기 위해서 벌칙강성(penaltystiffness)을 적용한다. 만약 벌칙강성이 너무 크게 되면 수치적인 문제가 야기되고,너무 적은 경우에도 계면요소 상대변위의 정확한 결과값을 얻을 수가 없다. 그러므로사용자는 적절한 벌칙강성값을 입력하여야 한다. midas FEA에서는 벌칙강성을k E d = × × 1000 와 같이 추천한다. 이때 E 의 값은 모델의 요소 중 가장 큰 영 계수이고, d 는 대표되는 요소 크기이다. + +수치해석을 수행하기 위하여 벌칙방법에 근거하여, 상대변위(relative displacement)∆u 와 계면력(traction) t 의 관계는 식 (3.1.1)과 같은 구성방정식으로 정의된다. + +$$ +\mathbf {t} = \mathbf {D} \cdot \Delta \mathbf {u} \tag {3.1.1} +$$ + +2차원의 경우 t , D , ∆u 값은 식 (3.1.2)와 같으며, 경계면상에 존재하는 적분점에서의 상대변위와 계면력을 그림 3.1.1과 같이 나타낼 수 있다. + +$$ +\mathbf {t} = \left\{ \begin{array}{l} t _ {n} \\ t _ {t} \end{array} \right\}, \quad \mathbf {D} = \left[ \begin{array}{c c} k _ {n} & 0 \\ 0 & k _ {t} \end{array} \right], \quad \Delta \mathbf {u} = \left\{ \begin{array}{l} \Delta u _ {n} \\ \Delta u _ {t} \end{array} \right\} \tag {3.1.2} +$$ + +![](images/page-169_49e4f9ab8d371435a438e048cb5beec7019bb1c0fbfd0e135798846c4a7628fc.jpg) +(a) 상대변위 + +![](images/page-169_69afd5716fc0d80db17b86065fd50b03b0090ac33e75178ead3af4842c576c89.jpg) + +
+text_image + +t_n +t_t +
+ +(b) 계면력 +그림 3.1.1 2차원 계면요소에서 상대변위와 계면력 + + + +그리고 3차원인 경우는 각 성분들을 식 (3.1.3)과 그림 3.1.2와 같이 나타낼 수있다. + +$$ +\mathbf {t} = \left\{ \begin{array}{l} t _ {n} \\ t _ {s} \\ t _ {t} \end{array} \right\}, \mathbf {D} = \left[ \begin{array}{c c c} k _ {n} & 0 & 0 \\ 0 & k _ {s} & 0 \\ 0 & 0 & k _ {t} \end{array} \right], \quad \Delta \mathbf {u} = \left\{ \begin{array}{l} \Delta u _ {n} \\ \Delta u _ {s} \\ \Delta u _ {t} \end{array} \right\} \tag {3.1.3} +$$ + +![](images/page-170_3801be51d649202dee8331e5e87a9d0b2fcb34316199091c48a388d2e930b5b6.jpg) + +
+text_image + +t_n +t_t +t_s +
+ +(a) 상대변위 + +![](images/page-170_0ce08a47a7b0876971b0bf294b691813bfd8f5005a6999d3e6b3498cac909837.jpg) + +
+text_image + +Δuₙ +Δuₜ +Δu +
+ +(b) 계면력 +그림 3.1.2 3차원 계면요소에서 상대변위와 계면력 + +여기서, + +점선 : 경계면 + +$t _ { n }$ : 법선 계면력 + +$t _ { s }$ , $t _ { t }$ : 접선 계면력 + +$\Delta u _ { { _ n } }$ : 법선 상대변위 + +$\Delta u _ { s }$ , $\Delta u _ { \iota }$ : 접선 상대변위\` + +(3.1.2)와 (3.1.3)의 선형 구성방정식에서 상대변위와 계면력은 각 방향에 대해서상관관계를 가지지 않는다. 즉 법선방향의 계면력은 접선방향의 강성에 아무런영향을 미치지 않는다. diff --git a/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_018.md b/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_018.md new file mode 100644 index 00000000..f5e867ff --- /dev/null +++ b/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_018.md @@ -0,0 +1,452 @@ + + +# 3-2 좌표계 및 상대변위 + +계면요소에서의 ∆u 는 상단(top)과 하단(bottom)에 위치하는 요소의 변위 차이를 이용하여 산정한다. 이를 위하여 계면요소의 상단 절점에서의 변위는 다음과 같이 나타낼 수 있다. + +$$ +\mathbf {u} _ {i} ^ {t o p} = \left\{u _ {i} ^ {t o p} \quad v _ {i} ^ {t o p} \quad w _ {i} ^ {t o p} \right\} ^ {T} \tag {3.2.1} +$$ + +요소 내 임의의 좌표 x, y, z 와 이동변위 u, v, w 는 다음과 같이 나타낼 수 있다 + +$$ +x ^ {t o p} = \sum_ {i = 1} ^ {N} N _ {i} ^ {t o p} x _ {i} ^ {t o p}, y ^ {t o p} = \sum_ {i = 1} ^ {N} N _ {i} ^ {t o p} y _ {i} ^ {t o p}, z ^ {t o p} = \sum_ {i = 1} ^ {N} N _ {i} ^ {t o p} z _ {i} ^ {t o p} +$$ + +$$ +u ^ {t o p} = \sum_ {i = 1} ^ {N} N _ {i} ^ {t o p} u _ {i} ^ {t o p}, v ^ {t o p} = \sum_ {i = 1} ^ {N} N _ {i} ^ {t o p} v _ {i} ^ {t o p}, w ^ {t o p} = \sum_ {i = 1} ^ {N} N _ {i} ^ {t o p} w _ {i} ^ {t o p} \tag {3.2.2} +$$ + +또한 계면요소의 하단 절점에서의 변위 및 좌표도 동일한 방법으로 나타낼 수있다. 즉, 식 (3.2.1)과 (3.2.2)에서 위첨자 “top”을 위첨자 “bot”로 치환하여식 (3.2.3)과 (3.2.4)와 같이 나타낼 수 있다. + +$$ +\mathbf {u} _ {i} ^ {b o t} = \left\{u _ {i} ^ {b o t} \quad v _ {i} ^ {b o t} \quad w _ {i} ^ {b o t} \right\} ^ {T} \tag {3.2.3} +$$ + +$$ +x ^ {b o t} = \sum_ {i = 1} ^ {N} N _ {i} ^ {b o t} x _ {i} ^ {b o t}, y ^ {b o t} = \sum_ {i = 1} ^ {N} N _ {i} ^ {b o t} y _ {i} ^ {b o t}, z ^ {b o t} = \sum_ {i = 1} ^ {N} N _ {i} ^ {b o t} z _ {i} ^ {b o t} +$$ + +$$ +u ^ {b o t} = \sum_ {i = 1} ^ {N} N _ {i} ^ {b o t} u _ {i} ^ {b o t}, v ^ {b o t} = \sum_ {i = 1} ^ {N} N _ {i} ^ {b o t} v _ {i} ^ {b o t}, w ^ {b o t} = \sum_ {i = 1} ^ {N} N _ {i} ^ {b o t} w _ {i} ^ {b o t} \tag {3.2.4} +$$ + +구조해석에서의 변위는 경계면의 요소좌표계를 사용하지만, 계면요소의 강성은 법선-접선방향으로 이루어진 좌표계를 사용한다. 그리고 계면요소는 회전강성을 전달하지 않기 때문에 식 (3.2.5)와 같이 요소축을 따라 발생하는 상대이동변위만으로 나타낸다. + +$$ +\Delta \mathbf {u} = \mathbf {u} ^ {t o p} - \mathbf {u} ^ {b o t} \tag {3.2.5} +$$ + + + +여기서, + +$$ +\mathbf {u} ^ {t o p} = \left\{u _ {n} ^ {t o p} \quad u _ {s} ^ {t o p} \quad u _ {t} ^ {t o p} \right\} ^ {T} +$$ + +$$ +\mathbf {u} ^ {b o t} = \left\{u _ {n} ^ {b o t} \quad u _ {s} ^ {b o t} \quad u _ {t} ^ {b o t} \right\} ^ {T} +$$ + +따라서 계면요소에서의 상대변위를 상대변위-요소변위 행렬을 이용하여 다음과 같이정의할 수 있다. + +$$ +\Delta \mathbf {u} = \mathbf {B} \mathbf {u} \tag {3.2.6} +$$ + +여기서, + +$$ +\mathbf {B} = \left[ \begin{array}{c c} \mathbf {N} ^ {b o t} & \mathbf {N} ^ {t o p} \end{array} \right] +$$ + +$$ +\mathbf {u} = \left\{\mathbf {u} ^ {b o t} \quad \mathbf {u} ^ {t o p} \right\} ^ {T} +$$ + + + +# 3-3 점계면요소 + +![](images/page-173_91bd4ff49aad435d57e4d0a8cb62e79b3c5e0de04fa9a87e893ad771501a815f.jpg) + +
+text_image + +u_n^top +↑ +u_t^top +→ +u_s^top +n +↑ +t +s +Z +Y +X +u_n^bot +↑ +u_t^bot +→ +u_s^bot +
+ +그림 3.3.1 점계면요소 + +점계면요소(point interface element)는 그림 3.3.1과 같이 경계면에서의 요소축과 상단과 하단의 절점으로 정의할 수 있다. 이때 상단 절점과 하단 절점은 같은 위치에정의될 수 있으므로 사용자가 직접 경계면의 요소축을 설정할 수 있도록 하였다. 그림 3.3.1에서는 설명상의 편의를 위해서 두 절점을 분리하여 나타내었다. 여기서, 주의할 점은 midas FEA에서는 사용자가 정의한 요소축에 따라서 계면요소축이 결정된다. 즉, 요소축의 x, y, z축 순으로 계면요소축의 법선(nomal), 전단(shear), 접선(tangential) 방향이 결정되므로, 점계면요소가 그림 3.3.1과 같이 떨어진 경우에는하단절점에서 상단절점으로 향하는 방향을 요소의 x축으로 설정하여야 사용자는 원하는 결과를 얻을 수 있다. + +요소의 임의의 점에서의 전체 좌표는 형상함수 ( Ni)를 이용하여 식 (3.3.1)과 같이 정의된다. + +$$ +\boldsymbol {x} ^ {b o t} = N _ {1} ^ {b o t} \cdot \boldsymbol {x} _ {1} ^ {b o t} +$$ + +$$ +x ^ {t o p} = N _ {2} ^ {t o p} \cdot x _ {2} ^ {t o p} \tag {3.3.1} +$$ + + + +그리고 어떤 점에서의 변위는 식 (3.3.2)와 같이 등매개변수 형상함수를 이용하여 정의된다. + +$$ +\boldsymbol {u} ^ {b o t} = N _ {1} ^ {b o t} \cdot \boldsymbol {u} _ {1} ^ {b o t} +$$ + +$$ +u ^ {t o p} = N _ {2} ^ {t o p} \cdot u _ {2} ^ {t o p} \tag {3.3.2} +$$ + +점계면요소에 대한 등매개변수 형상함수는 식 (3.3.3)과 같이 정의된다. + +$$ +N _ {1} ^ {b o t} = N _ {2} ^ {t o p} = 1 \tag {3.3.3} +$$ + +따라서 상대변위-요소변위 행렬을 다음과 같이 정의할 수 있다. + +$$ +\mathbf {B} = \left[ \begin{array}{c c c c c c} - N _ {1} ^ {b o t} & 0 & 0 & N _ {2} ^ {t o p} & 0 & 0 \end{array} \right] \tag {3.3.4} +$$ + + + +# 3-4 선계면요소 + +![](images/page-175_aee27feb0ed751f46f522fa2cc6f0edaa5d03eb7109b4792a170351a8a0d55c3.jpg) + +
+text_image + +u_n^top u_t^top u_s^top +n t +s +4 +3 +2 +u_n^bot u_t^bot u_s^bot +1 +Z Y +X +u_n^top u_t^top u_s^top +6 +n t +s +4 +3 +u_n^bot u_t^bot u_s^bot +5 +2 +Z Y +X +
+ +그림 3.4.1 선계면요소 + +midas FEA에서 선계면요소(line interface element)는 그림 3.4.1과 같이 저차와 고차요소 두가지를 제공한다. 선계면요소는 평면요소와 평면요소 사이와 평면요소와 선요소 사이에서 상대거동을 나타내기 위해서 많이 사용된다. + +강성행렬 D 는 식 (3.1.2)와 동일하다. + +선계면요소의 임의의 점에서의 전체 좌표는 형상함수를 이용하여 식 (3.4.1)과같이 정의된다. + +$$ +x ^ {b o t} = N _ {1} ^ {b o t} \cdot x _ {1} ^ {b o t} + N _ {2} ^ {b o t} \cdot x _ {2} ^ {b o t} \left(+ N _ {5} ^ {b o t} \cdot x _ {5} ^ {b o t}\right) +$$ + +$$ +x ^ {t o p} = N _ {3} ^ {t o p} \cdot x _ {3} ^ {t o p} + N _ {4} ^ {t o p} \cdot x _ {4} ^ {t o p} \left(+ N _ {6} ^ {t o p} \cdot x _ {6} ^ {t o p}\right) +$$ + +$$ +y ^ {b o t} = N _ {1} ^ {b o t} \cdot y _ {1} ^ {b o t} + N _ {2} ^ {b o t} \cdot y _ {2} ^ {b o t} \left(+ N _ {5} ^ {b o t} \cdot y _ {5} ^ {b o t}\right) +$$ + +$$ +y ^ {t o p} = N _ {3} ^ {t o p} \cdot y _ {3} ^ {t o p} + N _ {4} ^ {t o p} \cdot y _ {4} ^ {t o p} \left(+ N _ {6} ^ {t o p} \cdot y _ {6} ^ {t o p}\right) \tag {3.4.1} +$$ + + + +여기서, 괄호안의 좌표는 고차요소를 나타낸다. + +그리고 어떤 점에서의 전체 변위는 식 (3.4.2)와 같이 나타낸다. + +$$ +\boldsymbol {u} ^ {\text { bot }} = N _ {1} ^ {\text { bot }} \cdot \boldsymbol {u} _ {1} ^ {\text { bot }} + N _ {2} ^ {\text { bot }} \cdot \boldsymbol {u} _ {2} ^ {\text { bot }} \left(+ N _ {5} ^ {\text { bot }} \cdot \boldsymbol {u} _ {5} ^ {\text { bot }}\right) +$$ + +$$ +u ^ {t o p} = N _ {3} ^ {t o p} \cdot u _ {3} ^ {t o p} + N _ {4} ^ {t o p} \cdot u _ {4} ^ {t o p} \left(+ N _ {6} ^ {t o p} \cdot u _ {6} ^ {t o p}\right) +$$ + +$$ +v ^ {b o t} = N _ {1} ^ {b o t} \cdot v _ {1} ^ {b o t} + N _ {2} ^ {b o t} \cdot v _ {2} ^ {b o t} \left(+ N _ {5} ^ {b o t} \cdot v _ {5} ^ {b o t}\right) +$$ + +$$ +v ^ {t o p} = N _ {3} ^ {t o p} \cdot v _ {3} ^ {t o p} + N _ {4} ^ {t o p} \cdot v _ {4} ^ {t o p} \left(+ N _ {6} ^ {t o p} \cdot v _ {6} ^ {t o p}\right) \tag {3.4.2} +$$ + +선계면요소에 대한 등매개변수 형상함수는 식 (3.4.3)과 같이 정의된다. + +$$ +N _ {1} ^ {b o t} (\xi) = N _ {3} ^ {t o p} (\xi) = \frac {1}{2} (1 - \xi), +$$ + +$$ +N _ {2} ^ {b o t} (\xi) = N _ {4} ^ {t o p} (\xi) = \frac {1}{2} (1 + \xi) \tag {3.4.3} +$$ + +그리고 고차요소는 식 (3.4.4)와 같이 나타난다. + +$$ +N _ {1} ^ {b o t} (\xi) = N _ {3} ^ {t o p} (\xi) = - \frac {1}{2} (1 - \xi) \xi , +$$ + +$$ +N _ {2} ^ {b o t} (\xi) = N _ {4} ^ {t o p} (\xi) = \frac {1}{2} (1 + \xi) \xi , +$$ + +$$ +N _ {5} ^ {b o t} (\xi) = N _ {6} ^ {t o p} (\xi) = \left(1 - \xi^ {2}\right) \tag {3.4.4} +$$ + +(5, 6절점은 2차원 선계면요소의 고차요소이다.) + +상대변위-요소변위 행렬 B 는 계면의 법선-접선방향과 요소좌표계의 상이한 점을 고려하고 있다. + + + +# 3-5 면계면요소 + +![](images/page-177_70a588710ce31362f7f2021eeff58e2aafc3f087eba9a8a9382f42d214d1af25.jpg) +그림 3.5.1 면계면요소 + +midas FEA에서 면계면요소(surface interface element)는 위 그림 3.5.1과 같이 저차, 고차의 삼각형 또는 사각형 요소를 제공한다. 입체요소(solid element)사이 혹은 판요소(shell element)와 입체요소 사이 등에서 계면거동을 해석할 때적용한다. + +사각형 요소의 임의의 점에서의 전체 좌표는 형상함수를 이용하여 식 (3.5.1)과 같이정의된다. + +$$ +x ^ {b o t} = N _ {1} ^ {b o t} \cdot x _ {1} ^ {b o t} + N _ {2} ^ {b o t} \cdot x _ {2} ^ {b o t} + N _ {3} ^ {b o t} \cdot x _ {3} ^ {b o t} + N _ {4} ^ {b o t} \cdot x _ {4} ^ {b o t} +$$ + +$$ +\left(+ N _ {9} ^ {b o t} \cdot x _ {9} ^ {b o t} + N _ {1 0} ^ {b o t} \cdot x _ {1 0} ^ {b o t} + N _ {1 1} ^ {b o t} \cdot x _ {1 1} ^ {b o t} + N _ {1 2} ^ {b o t} \cdot x _ {1 2} ^ {b o t}\right) +$$ + + + +$$ +x ^ {t o p} = N _ {5} ^ {t o p} \cdot x _ {5} ^ {t o p} + N _ {6} ^ {t o p} \cdot x _ {6} ^ {t o p} + N _ {7} ^ {t o p} \cdot x _ {7} ^ {t o p} + N _ {8} ^ {t o p} \cdot x _ {8} ^ {t o p} +$$ + +$$ +\left(+ N _ {1 3} ^ {t o p} \cdot x _ {1 3} ^ {t o p} + N _ {1 4} ^ {t o p} \cdot x _ {1 4} ^ {t o p} + N _ {1 5} ^ {t o p} \cdot x _ {1 5} ^ {t o p} + N _ {1 6} ^ {t o p} \cdot x _ {1 6} ^ {t o p}\right) +$$ + +$$ +y ^ {b o t} = N _ {1} ^ {b o t} \cdot y _ {1} ^ {b o t} + N _ {2} ^ {b o t} \cdot y _ {2} ^ {b o t} + N _ {3} ^ {b o t} \cdot y _ {3} ^ {b o t} + N _ {4} ^ {b o t} \cdot y _ {4} ^ {b o t} +$$ + +$$ +\left(+ N _ {9} ^ {b o t} \cdot y _ {9} ^ {b o t} + N _ {1 0} ^ {b o t} \cdot y _ {1 0} ^ {b o t} + N _ {1 1} ^ {b o t} \cdot y _ {1 1} ^ {b o t} + N _ {1 2} ^ {b o t} \cdot y _ {1 2} ^ {b o t}\right) +$$ + +$$ +y ^ {t o p} = N _ {5} ^ {t o p} \cdot y _ {5} ^ {t o p} + N _ {6} ^ {t o p} \cdot y _ {6} ^ {t o p} + N _ {7} ^ {t o p} \cdot y _ {7} ^ {t o p} + N _ {8} ^ {t o p} \cdot y _ {8} ^ {t o p} +$$ + +$$ +\left(+ N _ {1 3} ^ {t o p} \cdot y _ {1 3} ^ {t o p} + N _ {1 4} ^ {t o p} \cdot y _ {1 4} ^ {t o p} + N _ {1 5} ^ {t o p} \cdot y _ {1 5} ^ {t o p} + N _ {1 6} ^ {t o p} \cdot y _ {1 6} ^ {t o p}\right) +$$ + +$$ +z ^ {b o t} = N _ {1} ^ {b o t} \cdot z _ {1} ^ {b o t} + N _ {2} ^ {b o t} \cdot z _ {2} ^ {b o t} + N _ {3} ^ {b o t} \cdot z _ {3} ^ {b o t} + N _ {4} ^ {b o t} \cdot z _ {4} ^ {b o t} +$$ + +$$ +\left(+ N _ {9} ^ {b o t} \cdot z _ {9} ^ {b o t} + N _ {1 0} ^ {b o t} \cdot z _ {1 0} ^ {b o t} + N _ {1 1} ^ {b o t} \cdot z _ {1 1} ^ {b o t} + N _ {1 2} ^ {b o t} \cdot z _ {1 2} ^ {b o t}\right) +$$ + +$$ +z ^ {t o p} = N _ {5} ^ {t o p} \cdot z _ {5} ^ {t o p} + N _ {6} ^ {t o p} \cdot z _ {6} ^ {t o p} + N _ {7} ^ {t o p} \cdot z _ {7} ^ {t o p} + N _ {8} ^ {t o p} \cdot z _ {8} ^ {t o p} \tag {3.5.1} +$$ + +$$ +\left(+ N _ {1 3} ^ {t o p} \cdot z _ {1 3} ^ {t o p} + N _ {1 4} ^ {t o p} \cdot z _ {1 4} ^ {t o p} + N _ {1 5} ^ {t o p} \cdot z _ {1 5} ^ {t o p} + N _ {1 6} ^ {t o p} \cdot z _ {1 6} ^ {t o p}\right) +$$ + +그리고 어떤 점에서의 전체 변위는 식 (3.5.2) 와 같이 등매개변수 형상함수를이용하여 정의된다. + +$$ +u ^ {b o t} = N _ {1} ^ {b o t} \cdot u _ {1} ^ {b o t} + N _ {2} ^ {b o t} \cdot u _ {2} ^ {b o t} + N _ {3} ^ {b o t} \cdot u _ {3} ^ {b o t} + N _ {4} ^ {b o t} \cdot u _ {4} ^ {b o t} +$$ + +$$ +\left(+ N _ {9} ^ {b o t} \cdot u _ {9} ^ {b o t} + N _ {1 0} ^ {b o t} \cdot u _ {1 0} ^ {b o t} + N _ {1 1} ^ {b o t} \cdot u _ {1 1} ^ {b o t} + N _ {1 2} ^ {b o t} \cdot u _ {1 2} ^ {b o t}\right) +$$ + +$$ +\boldsymbol {u} ^ {\text {top}} = N _ {5} ^ {\text {top}} \cdot \boldsymbol {u} _ {5} ^ {\text {top}} + N _ {6} ^ {\text {top}} \cdot \boldsymbol {u} _ {6} ^ {\text {top}} + N _ {7} ^ {\text {top}} \cdot \boldsymbol {u} _ {7} ^ {\text {top}} + N _ {8} ^ {\text {top}} \cdot \boldsymbol {u} _ {8} ^ {\text {top}} +$$ + +$$ +\left(+ N _ {1 3} ^ {t o p} \cdot u _ {1 3} ^ {t o p} + N _ {1 4} ^ {t o p} \cdot u _ {1 4} ^ {t o p} + N _ {1 5} ^ {t o p} \cdot u _ {1 5} ^ {t o p} + N _ {1 6} ^ {t o p} \cdot u _ {1 6} ^ {t o p}\right) +$$ + +$$ +\boldsymbol {v} ^ {b o t} = N _ {1} ^ {b o t} \cdot \boldsymbol {v} _ {1} ^ {b o t} + N _ {2} ^ {b o t} \cdot \boldsymbol {v} _ {2} ^ {b o t} + N _ {3} ^ {b o t} \cdot \boldsymbol {v} _ {3} ^ {b o t} + N _ {4} ^ {b o t} \cdot \boldsymbol {v} _ {4} ^ {b o t} +$$ + +$$ +\left(+ N _ {9} ^ {b o t} \cdot v _ {9} ^ {b o t} + N _ {1 0} ^ {b o t} \cdot v _ {1 0} ^ {b o t} + N _ {1 1} ^ {b o t} \cdot v _ {1 1} ^ {b o t} + N _ {1 2} ^ {b o t} \cdot v _ {1 2} ^ {b o t}\right) +$$ + +$$ +\nu^ {t o p} = N _ {5} ^ {t o p} \cdot \nu_ {5} ^ {t o p} + N _ {6} ^ {t o p} \cdot \nu_ {6} ^ {t o p} + N _ {7} ^ {t o p} \cdot \nu_ {7} ^ {t o p} + N _ {8} ^ {t o p} \cdot \nu_ {8} ^ {t o p} +$$ + +$$ +\left(+ N _ {1 3} ^ {t o p} \cdot v _ {1 3} ^ {t o p} + N _ {1 4} ^ {t o p} \cdot v _ {1 4} ^ {t o p} + N _ {1 5} ^ {t o p} \cdot v _ {1 5} ^ {t o p} + N _ {1 6} ^ {t o p} \cdot v _ {1 6} ^ {t o p}\right) +$$ + +$$ +w ^ {b o t} = N _ {1} ^ {b o t} \cdot w _ {1} ^ {b o t} + N _ {2} ^ {b o t} \cdot w _ {2} ^ {b o t} + N _ {3} ^ {b o t} \cdot w _ {3} ^ {b o t} + N _ {4} ^ {b o t} \cdot w _ {4} ^ {b o t} +$$ + +$$ +\left(+ N _ {9} ^ {b o t} \cdot w _ {9} ^ {b o t} + N _ {1 0} ^ {b o t} \cdot w _ {1 0} ^ {b o t} + N _ {1 1} ^ {b o t} \cdot w _ {1 1} ^ {b o t} + N _ {1 2} ^ {b o t} \cdot w _ {1 2} ^ {b o t}\right) +$$ + +$$ +w ^ {t o p} = N _ {5} ^ {t o p} \cdot w _ {5} ^ {t o p} + N _ {6} ^ {t o p} \cdot w _ {6} ^ {t o p} + N _ {7} ^ {t o p} \cdot w _ {7} ^ {t o p} + N _ {8} ^ {t o p} \cdot w _ {8} ^ {t o p} \tag {3.5.2} +$$ + +$$ +\left(+ N _ {1 3} ^ {t o p} \cdot w _ {1 3} ^ {t o p} + N _ {1 4} ^ {t o p} \cdot w _ {1 4} ^ {t o p} + N _ {1 5} ^ {t o p} \cdot w _ {1 5} ^ {t o p} + N _ {1 6} ^ {t o p} \cdot w _ {1 6} ^ {t o p}\right) +$$ + +3차원 면계면요소의 등매개변수 형상함수는 아래의 식과 같이 정의된다. + +8절점 형상의 계면요소의 형상함수는 + + + +$$ +N _ {1} ^ {b o t} (\xi , \eta) = N _ {5} ^ {t o p} (\xi , \eta) = \frac {1}{4} (1 - \xi) (1 - \eta), +$$ + +$$ +N _ {2} ^ {b o t} (\xi , \eta) = N _ {6} ^ {t o p} (\xi , \eta) = \frac {1}{4} (1 + \xi) (1 - \eta), +$$ + +$$ +N _ {3} ^ {b o t} (\xi , \eta) = N _ {7} ^ {t o p} (\xi , \eta) = \frac {1}{4} (1 + \xi) (1 + \eta), +$$ + +$$ +N _ {4} ^ {b o t} (\xi , \eta) = N _ {8} ^ {t o p} (\xi , \eta) = \frac {1}{4} (1 - \xi) (1 + \eta) \tag {3.5.3} +$$ + +16절점 형상의 계면요소의 형상함수는 + +$$ +N _ {1} ^ {b o t} (\xi , \eta) = N _ {5} ^ {t o p} (\xi , \eta) = \frac {1}{4} (1 - \xi) (1 - \eta) (- \xi - \eta - 1), +$$ + +$$ +N _ {2} ^ {b o t} (\xi , \eta) = N _ {6} ^ {t o p} (\xi , \eta) = \frac {1}{4} (1 + \xi) (1 - \eta) (\xi - \eta - 1) +$$ + +$$ +N _ {3} ^ {b o t} (\xi , \eta) = N _ {7} ^ {t o p} (\xi , \eta) = \frac {1}{4} (1 + \xi) (1 + \eta) (\xi + \eta - 1) +$$ + +$$ +N _ {4} ^ {b o t} (\xi , \eta) = N _ {8} ^ {t o p} (\xi , \eta) = \frac {1}{4} (1 - \xi) (1 + \eta) (- \xi + \eta - 1) +$$ + +$$ +N _ {9} ^ {b o t} (\xi , \eta) = N _ {1 3} ^ {t o p} (\xi , \eta) = \frac {1}{2} (1 - \xi^ {2}) (1 - \eta), +$$ + +$$ +N _ {1 0} ^ {b o t} (\xi , \eta) = N _ {1 4} ^ {t o p} (\xi , \eta) = \frac {1}{2} (1 + \xi) (1 - \eta^ {2}), +$$ + +$$ +N _ {1 1} ^ {b o t} (\xi , \eta) = N _ {1 5} ^ {t o p} (\xi , \eta) = \frac {1}{2} (1 - \xi^ {2}) (1 + \eta), +$$ + +$$ +N _ {1 2} ^ {b o t} (\xi , \eta) = N _ {1 6} ^ {t o p} (\xi , \eta) = \frac {1}{2} (1 - \xi) (1 - \eta^ {2}) \tag {3.5.4} +$$ + +6절점 삼각형 형상의 계면요소의 형상함수는 + + + +$$ +N _ {1} ^ {b o t} (\xi , \eta) = N _ {4} ^ {t o p} (\xi , \eta) = 1 - \xi - \eta , +$$ + +$$ +N _ {_ {2}} ^ {b o t} \left(\xi , \eta\right) = N _ {_ {5}} ^ {t o p} \left(\xi , \eta\right) = \xi , +$$ + +$$ +N _ {3} ^ {b o t} (\xi , \eta) = N _ {6} ^ {t o p} (\xi , \eta) = \eta \tag {3.5.5} +$$ + +12절점 삼각형 형상의 계면요소의 형상함수는 + +$$ +N _ {1} ^ {b o t} (\xi , \eta) = N _ {4} ^ {t o p} (\xi , \eta) = (1 - 2 \xi - 2 \eta) (1 - \xi - \eta) +$$ + +$$ +N _ {2} ^ {b o t} (\xi , \eta) = N _ {5} ^ {t o p} (\xi , \eta) = (2 \xi - 1) \xi +$$ + +$$ +N _ {3} ^ {b o t} (\xi , \eta) = N _ {6} ^ {t o p} (\xi , \eta) = (2 \eta - 1) \eta +$$ + +$$ +N _ {7} ^ {b o t} (\xi , \eta) = N _ {1 0} ^ {t o p} (\xi , \eta) = 4 (1 - \xi - \eta) \xi +$$ + +$$ +N _ {8} ^ {b o t} (\xi , \eta) = N _ {1 1} ^ {t o p} (\xi , \eta) = 4 \xi \eta +$$ + +$$ +N _ {9} ^ {b o t} (\xi , \eta) = N _ {1 2} ^ {t o p} (\xi , \eta) = 4 \eta (1 - \xi - \eta) \tag {3.5.6} +$$ + +midas FEA에서는 상단요소와 하단요소 사이에 존재하는 계면에 적분점 위치가존재하며, 이 때 적분법은 뉴튼-코츠법(Newton-Cotes method)을 사용하기 때문에 적분점 위치는 절점에 존재한다. + +상대변위-요소변위 행렬은 2차원에서 확장된 B 행렬과 동일하다. diff --git a/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_019.md b/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_019.md new file mode 100644 index 00000000..6ca40fda --- /dev/null +++ b/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_019.md @@ -0,0 +1,387 @@ + + +# 3-6 유한요소 정식화 + +국부좌표계에서 계면력은 국부좌표계에서의 계면요소 상대변위 ∆ux , ∆uy , ∆uz 와강성행렬 D 로부터 유도될수 있다. + +$$ +\left\{ \begin{array}{l} \mathbf {t} _ {x} \\ \mathbf {t} _ {y} \\ \mathbf {t} _ {z} \end{array} \right\} = \mathbf {D} \left\{ \begin{array}{l} \Delta \mathbf {u} _ {x} \\ \Delta \mathbf {u} _ {y} \\ \Delta \mathbf {u} _ {z} \end{array} \right\} \tag {3.6.1} +$$ + +계면요소의 변형율 에너지(strain energy) 식을 변분하여 얻어지는 강성행렬 식은 다음과 같다. + +$$ +\mathbf {K} _ {\text { inter }} = \int_ {\Gamma} \mathbf {B} _ {\text { inter }} ^ {T} \mathbf {D} \mathbf {B} _ {\text { inter }} d \Gamma \tag {3.6.2} +$$ + +그리고 내력(internal force)식은 다음과 같다. + +$$ +\mathbf {F} _ {\text { inter }} = \int_ {\Gamma} \mathbf {B} _ {\text { inter }} ^ {T} \mathbf {t} d \Gamma \tag {3.6.3} +$$ + +식 (3.6.2)를 수치적분 식으로 다시 전개하면 다음과 같은 계면요소의 강성행렬의 식으로 나타낼 수 있다. + +$$ +\mathbf {K} _ {\text { inter }} = \sum_ {j = 1} ^ {N i p} \mathbf {B} _ {\text { inter }} ^ {j} {} ^ {T} \mathbf {D} \mathbf {B} _ {\text { inter }} ^ {j} \det \mathbf {J} ^ {j} \tag {3.6.4} +$$ + +여기서, + +Nip : 계면요소에 대한 적분점의 수 + + + +# 3-7 계면요소 해석결과 + +계면요소의 해석 결과로는 절점에서의 계면력과 상대변위를 출력하며, 부호와 방향은 요소좌표계를 따른다. 계면력과 상대변위는 계면요소의 절점에서 정의되며, 한쌍을 이루는 두 절점사이의 결과값은 동일하다. + +계면요소에서 출력되는 계면요소의 상대변위의 종류는 다음과 같다. + +\- 계면력 성분 + +$$ +t _ {x}, t _ {y}, t _ {z} +$$ + +• 상대변위 성분 + +$$ +\Delta u _ {x}, \Delta u _ {y}, \Delta u _ {z} +$$ + +절점에서의 계면력과 상대변위는 뉴튼-코츠 적분법을 이용하였기 때문에 적분점에서 계산된 결과가 절점의 결과와 공유되어 산출된다. 계면요소의 적분점은 다음과 같다. + +• 2절점 점계면요소 : 1절점 뉴튼-코츠 적분법 +• 4절점 선계면요소 : 2 절점 뉴튼-코츠 적분법 +- 6절점 선계면요소 : 3절점 뉴튼-코츠 적분법 +- 6절점 삼각형 면계면요소 : 3절점 뉴튼-코츠 적분법 +• 8절점 사각형 면계면요소 : 4절점 뉴튼-코츠 적분법 +• 12절점 삼각형 면계면요소 : 6절점 뉴튼-코츠 적분법 +• 16절점 사각형 면계면요소 : 8절점 뉴튼-코츠 적분법 + + + +# Chapter 4. Geometric Nonlinearity + +# 4-1 개요 + +구조물에 변형 또는 회전이 크게 발생하면 기하학적 형상이 변하게 되며, 이로인해 변형률과 변위의 관계가 비선형이 될 수 있다. 그림 4.1.1에서 상태 (A)에있던 구조물이 하중(추)에 의해 상태 (B)와 같이 크게 변형된다면 하중에 대한변위의 상관 관계는 비선형이 된다. + +![](images/page-183_1758dff7cea3d32f7c22ebe6480caf843fd5d8ad575d9c8fc1b62a681879518c.jpg) + +
+text_image + +A +B +F_TIP +A +B +U_TIP +
+ +그림 4.1.1 비선형 해석의 예시 + +기하비선형 해석에서는 변형이 생기는 동안 구조물의 기하학적 형상이 변하므로변형 전의 형상(초기 형상)과 변형 후의 형상은 서로 다르다는 가정을 이용하며,변형률과 회전이 미소하다는 가정은 이용하지 않는다. 따라서 기하비선형 해석에서 강성 행렬 (K) 는 변위 (u) 의 함수가 된다. + +일반적으로 기하비선형 해석에서 사용하는 정식화 방법은 TLF(total Lagrangianformulation)와 ULF(updated Lagrangian formulation)이 있다. TLF에서 사용되는 응력과 변형률은 각각 PK2 응력(2nd Piola-Kirchhoff stress), Green 변형률(Green-Lagrange strain)이며, ULF에서 사용되는 응력과 변형률은 각각Cauchy 응력(Cauchy stress), 선형변형률증분(linear Euler strain increment)이다. TLF에서 변형률과 응력은 변형전의 형상에 대해 표현하며, ULF에서는 변형 후의 형상에 대해 표현한다. 예를 들어 1차원 해석의 경우에 PK2 응력은 힘과 변형전의 면적 (F / A0) 의 관계를 나타내나, Cauchy 응력은 힘과 변형 후 면적 + + + +( ) F / A 의 관계를 나타낸다. + +어떤 물체가 시간에 따라 변형을 일으키는 상태를 그림 4.1.2와 같이 나타낼 수있다. 그림 4.1.2에서 초기 상태(변형전의 상태)를 $\Omega _ { 0 }$ 로 표시하며, 현재 상태(변형후의 상태)를 Ω 로 표시하였다. 그리고 X 는 초기상태의 공간 좌표, x 는 t 만큼 시간이 경과된 후의 공간 좌표를 나타낸다. + +![](images/page-184_cf3f679151060908eb8cdd626b9371334737d71c00e0b3b16cb6e099b0a901f2.jpg) + +
+text_image + +y, Y +Ø (X̄, t) +u +X̄ +Ω₀ +x̄ +Γ₀ +Ω +Γ +x, X +
+ +그림 4.1.2 물체의 초기 및 현재 형상 + +Green 변형률은 다음 식 (4.1)과 같이 변형전의 좌표를 사용하여 정의한다. + +$$ +\begin{array}{l} \mathbf {E} = \frac {1}{2} \left(\mathbf {F} ^ {T} \mathbf {F} - \mathbf {I}\right) \text {or} E _ {i j} = \frac {1}{2} \left(u _ {i, j} + u _ {j, i} + u _ {k, i} u _ {k, j}\right) \\ \mathcal {O} (\overline {{{\mathbf {X}}}} _ {i, j}) \end{array} \tag {4.1.1} +$$ + +$$ +\mathbf {F} = \frac {\partial \mathbf {x}}{\partial \overline {{\mathbf {X}}}} = \frac {\partial (\overline {{\mathbf {X}}} + \mathbf {u})}{\partial \overline {{\mathbf {X}}}} = \mathbf {I} + \frac {\partial \mathbf {u}}{\partial \overline {{\mathbf {X}}}} +$$ + +여기서, + +E or $E _ { i j }$ : Green 변형률(Green-Lagrange strain) + +F : 변형 구배(deformation gradient) + +1차원 구조물의 경우에 식 (4.1.1)을 식 (4.1.2)와 같이 정리할 수 있다. + +$$ +E _ {1 1} = \frac {1}{2} \frac {L ^ {2} - \overline {{L}} ^ {2}}{\overline {{L}} ^ {2}} \tag {4.1.2} +$$ + + + +식 (4.1.2)에서 L 는 변형전의 길이, L 은 변형후의 길이를 표시한다. 식 (4.1.2)에서 변형이 미소하다면, Green 변형률은 식 (4.1.3)과 같이 선형 변형률로 회귀하는 것을 확인할 수 있다. + +$$ +\begin{array}{l} E _ {1 1} = \frac {1}{2} \frac {(L - \bar {L}) (L + \bar {L})}{\bar {L} ^ {2}} = \frac {1}{2} \frac {L - \bar {L}}{\bar {L}} \left(\frac {L - \bar {L}}{\bar {L}} + \frac {2 \bar {L}}{\bar {L}}\right) = \frac {1}{2} \left(\frac {L - \bar {L}}{\bar {L}}\right) ^ {2} + \frac {L - \bar {L}}{\bar {L}} \tag {4.1.3} \\ \cong \frac {L - \overline {{L}}}{\overline {{L}}} \\ \end{array} +$$ + +일반적인 3차원 연속체에서 PK2 응력의 의미를 알아보기 위해 Cauchy 응력과의 관계를 유도한다. 에너지는 Cauchy 응력 혹은 PK2 응력 등 어떤 응력을 사용하더라도 에너지는 동일하므로 가상 일은 식 (4.1.4)와 같다. + +$$ +r = \int \mathbf {S}: \delta \mathbf {E} d V _ {0} = \int \boldsymbol {\sigma}: \delta \boldsymbol {\varepsilon} d V = \int J \boldsymbol {\sigma}: \delta \boldsymbol {\varepsilon} d V _ {0}, \quad J = \det (\mathbf {F}) \tag {4.1.4} +$$ + +여기서, + +S : PK2 응력 텐서 + +σ : Cauchy 응력 텐서 + +8 $\boldsymbol { \varepsilon } = \frac { 1 } { 2 } \big ( u _ { i } , _ { j } + u _ { j } , _ { i } \big )$ + +먼저 Green 변형률과 변형구배(deformation gradient)를 변분하면 각각 다음 식과 같다. + +$$ +\delta \mathbf {E} = \frac {1}{2} \left(\delta \mathbf {F} ^ {T} \mathbf {F} + \mathbf {F} ^ {T} \delta \mathbf {F}\right) \tag {4.1.5} +$$ + +$$ +\delta \mathbf {F} = \frac {\partial \delta \mathbf {u}}{\partial \overline {{{\mathbf {X}}}}} = \frac {\partial \delta \mathbf {u}}{\partial \mathbf {x}} \frac {\partial \mathbf {x}}{\partial \overline {{{\mathbf {X}}}}} = \frac {\partial \delta \mathbf {u}}{\partial \mathbf {x}} \mathbf {F} \tag {4.1.6} +$$ + +식 (4.1.6)을 식 (4.1.5)에 대입하면, 아래 식과 같이 정리할 수 있다. + +$$ +\delta \mathbf {E} = \frac {1}{2} \left(\left(\frac {\partial \delta \mathbf {u}}{\partial \mathbf {x}} \mathbf {F}\right) ^ {T} \mathbf {F} + \mathbf {F} ^ {T} \left(\frac {\partial \delta \mathbf {u}}{\partial \mathbf {x}} \mathbf {F}\right)\right) = \mathbf {F} ^ {T} \left(\frac {1}{2} \left(\frac {\partial \delta \mathbf {u} ^ {T}}{\partial \mathbf {x}} + \frac {\partial \delta \mathbf {u}}{\partial \mathbf {x}}\right)\right) \mathbf {F} = \mathbf {F} ^ {T} \delta \boldsymbol {\varepsilon} \mathbf {F} \tag {4.1.7} +$$ + +식 (4.1.7)을 식 (4.1.4)에 대입하여 정리하면 식 (4.1.8)과 같다. + +$$ +r = \int \mathbf {S}: \delta \mathbf {E} d V _ {0} = \int \mathbf {S}: \left(\mathbf {F} ^ {T} \delta \boldsymbol {\varepsilon} \mathbf {F}\right) d V _ {0} = \int \mathbf {F} \mathbf {S} \mathbf {F} ^ {T}: \delta \boldsymbol {\varepsilon} d V _ {0} = \int J \boldsymbol {\sigma}: \delta \boldsymbol {\varepsilon} d V _ {0} \tag {4.1.8} +$$ + + + +따라서 Cauchy 응력과 PK2 응력의 관계를 식 (4.1.9)와 같이 표현할 수 있다. + +$$ +\boldsymbol {\sigma} = \frac {1}{J} \mathbf {F} \mathbf {S} \mathbf {F} ^ {T}, \quad \mathbf {S} = J \mathbf {F} ^ {- 1} \boldsymbol {\sigma} \mathbf {F} ^ {- T} \tag {4.1.9} +$$ + +시간 t t + ∆ 일때 가상 일의 정리를 PK2 응력과 Green 변형률로 표현하면 식(4.1.10)과 같다. 여기서, r 는 외부 힘에 의한 가상일이다. + +$$ +{ } ^ { t + \Delta t } r = \int ^ { t + \Delta t } S _ { i j } \delta ^ { t + \Delta t } E _ { i j } d V _ { 0 } \tag {4.1.10} +$$ + +$$ +{ } ^ { t + \Delta t } S _ { i j } = { } ^ { t } S _ { i j } + \Delta S _ { i j } +$$ + +$$ +{ } ^ { t + \Delta t } E _ { i j } = { } ^ { t } E _ { i j } + \Delta E _ { i j } \tag {4.1.11} +$$ + +$$ +\delta^ {t + \Delta t} E _ {i j} = \frac {1}{2} \Big (\delta u _ {i, j} + \delta u _ {j, i} + ^ {t} u _ {k, i} \delta u _ {k, j} + \delta u _ {k, i} ^ {t} u _ {k, j} + \delta u _ {k, i} \Delta u _ {k, j} + \delta u _ {k, j} \Delta u _ {k, i} \Big) +$$ + +위 식 (4.1.11)을 (4.1.10)에 대입하여 정리하면 다음 식을 얻을 수 있다. + +$$ +\int \Delta S _ {i j} \delta e _ {i j} d V _ {0} + \int^ {t} S _ {i j} \delta \eta_ {i j} d V _ {0} = ^ {t + \Delta t} r - \int^ {t} S _ {i j} \delta e _ {i j} d V _ {0} \tag {4.1.12} +$$ + +여기서, + +$$ +\delta e _ {i j} \quad : \frac {1}{2} \left(\delta u _ {i, j} + \delta u _ {j, i} + ^ {t} u _ {k, i} \delta u _ {k, j} + \delta u _ {k, i} ^ {t} u _ {k, j}\right) \text { 선형항 } +$$ + +$$ +\delta \eta_ {i j} \quad : \frac {1}{2} \left(\delta u _ {k, i} \Delta u _ {k, j} + \delta u _ {k, j} \Delta u _ {k, i}\right) \text { 비선형항 } +$$ + +∆Sij 는 테일러 급수를 이용하여 다음과 같이 전개한다. + +$$ +\Delta S _ {i j} = ^ {t + \Delta t} S _ {i j} - ^ {t} S _ {i j} = \left(^ {t} S _ {i j} + \frac {\partial^ {t} S _ {i j}}{\partial^ {t} E _ {k l}} \Delta E _ {k l} + h i g h o r d e r\right) - ^ {t} S _ {i j} \tag {4.1.13} +$$ + +식 (4.1.13)을 식 (4.1.12)의 첫 번째 항에 대입하고, ∆Ekl 의 2차 이상의 항을 무시하면, 첫 번째 항은 식 (4.1.14)와 같이 선형화된 식이 된다. + + + +$$ +\int \Delta S _ {i j} \delta e _ {i j} d V _ {0} = \int \left(\frac {\partial^ {t} S _ {i j}}{\partial^ {t} E _ {k l}} \Delta E _ {k l}\right) \delta e _ {i j} d V _ {0} = \int D _ {i j k l} \Delta E _ {k l} \delta e _ {i j} d V _ {0} \tag {4.1.14} +$$ + +$$ +\Delta E _ {i j} = \Delta e _ {i j} = \frac {1}{2} \left(\Delta u _ {i, j} + \Delta u _ {j, i} + ^ {t} u _ {k, i} \Delta u _ {k, j} + \Delta u _ {k, i} ^ {t} u _ {k, j}\right) +$$ + +또한 $^{t}S_{ij} = ^{t}S_{ji}$ 인 성질을 이용하여 식 (4.1.12)의 두 번째 항을 정리하면 아래와 같다. + +$$ +\int^ {t} S _ {i j} \delta \eta_ {i j} d V _ {0} = \int^ {t} \mathbf {S}: (\delta \mathbf {L} ^ {T} \Delta \mathbf {L}) d V _ {0}, \quad L _ {i j} = \frac {\partial u _ {i}}{\partial X _ {j}} \tag {4.1.15} +$$ + +식 (4.1.12)의 우변에서 각 항은 식 (4.1.16)과 같이 표현할 수 있다. + +$$ +\begin{array}{l} ^ {t + \Delta t} r = \delta \mathbf {u} ^ {T} {} ^ {t + \Delta t} \mathbf {f} _ {\text {ext}} \\ \int_ {t} \mathbf {s} _ {t} \mathbf {s} _ {t} - \mathbf {W} _ {t} - \int_ {t} \mathbf {s} _ {t} \mathbf {s} _ {t} - \mathbf {W} _ {t} - \mathbf {s} _ {t} ^ {T} t \mathbf {s} _ {t} \end{array} \tag {4.1.16} +$$ + +$$ +\int^ {t} S _ {i j} \delta e _ {i j} d V _ {0} = \int^ {t} \mathbf {S}: \delta \mathbf {e} d V _ {0} = \delta \mathbf {u} ^ {T} {} ^ {t} \mathbf {f} _ {\text { int }} +$$ + +여기서, + +$$ +\delta \mathbf {u} \quad : \left[ \begin{array}{l l l l l l l l l l l} \delta u _ {1} & \delta v _ {1} & \delta w _ {1} & \delta u _ {2} & \delta v _ {2} & \delta w _ {2} & \dots & \delta u _ {N} & \delta v _ {N} & \delta w _ {N} \end{array} \right] ^ {T} +$$ + +따라서 식 (4.1.12)의 우변항을 식 (4.1.16)을 이용해서 정리하면 다음 식 (4.1.17)과 같다. + +$$ +r - \int^ {t} \mathbf {S}: \delta \mathbf {e} d V _ {0} = \delta \mathbf {u} ^ {T} \left(^ {t + \Delta t} \mathbf {f} _ {e x t} - ^ {t} \mathbf {f} _ {\text {int}}\right) \tag {4.1.17} +$$ + +식 (4.1.12)에 식 (4.1.14–15,17)을 대입하면 식 (4.1.18)과 같다. 식 (4.1.18)에서 좌변의 첫째 항은 선형변형률에 의해 발생되는 가상 일이며, 두 번째 항은 비선형변형률에 의한 가상 일이다. + +$$ +\int \delta \mathbf {e} ^ {T} \mathbf {D} \Delta \mathbf {e} d V _ {0} + \int^ {t} \mathbf {S}: \delta \mathbf {L} ^ {T} \Delta \mathbf {L} d V _ {0} = \delta \mathbf {u} ^ {T} \left(^ {t + \Delta t} \mathbf {f} _ {\text { ext }} - ^ {t} \mathbf {f} _ {\text { int }}\right) \tag {4.1.18} +$$ + +식 (4.1.18)는 식 (4.1.19)과 같이 표현되며, 강성 행렬 $K_{L}^{e}$ 과 $K_{NL}^{e}$ 은 요소의 형상함수에 따라 결정되며 자세한 것은 다음 절을 참조하라. + +$$ +\left(^ {t} \mathbf {K} _ {L} ^ {e} + ^ {t} \mathbf {K} _ {N L} ^ {e}\right) \Delta \mathbf {u} = ^ {t + \Delta t} \mathbf {f} _ {e x t} - ^ {t} \mathbf {f} _ {\text {int}} \tag {4.1.19} +$$ + + + +# 4-2 트러스요소 + +트러스(Truss) 요소는 요소좌표계에서 이동변위 u , v , w 를 가지며, 변위는 형상함수 Ni 를 이용하여 다음과 같이 나타낸다. + +$$ +u = \sum_ {i = 1} ^ {N} N _ {i} u _ {i}, v = \sum_ {i = 1} ^ {N} N _ {i} v _ {i}, w = \sum_ {i = 1} ^ {N} N _ {i} w _ {i} \tag {4.2.1} +$$ + +트러스요소에서 사용되는 응력과 변형률은 다음 식 (4.2.2)과 같다 + +$$ +\mathbf {S} = \left\{S _ {x x} \right\}, \quad \mathbf {E} = \left\{E _ {x x} \right\} \tag {4.2.2} +$$ + +여기서, + +x : 요소 좌표계 + +식 (4.1.18)에서 가상 변형률의 선형 항 δe 는 다음과 같이 정리할 수 있다. + +$$ +\delta \mathbf {e} = \left\{\delta u _ {, x} \right\} + \left\{\delta u _ {, x} ^ {t} u _ {, x} + \delta v _ {, x} ^ {t} v _ {, x} + \delta w _ {, x} ^ {t} w _ {, x} \right\} \tag {4.2.3} +$$ + +식 (4.2.3)은 다음과 같이 가상변위 항 δu 와 행렬 BL 의 곱으로 표현된다. + +$$ +\delta \mathbf {e} = \mathbf {B} _ {L 0} \delta \mathbf {u} + \mathbf {B} _ {L 1} \delta \mathbf {u} = \mathbf {B} _ {L} \delta \mathbf {u} \tag {4.2.4} +$$ + +증분 변형률의 선형 항 역시 유사한 형태로 표현할 수 있다. + +$$ +\Delta \mathbf {e} = \mathbf {B} _ {L 0} \Delta \mathbf {u} + \mathbf {B} _ {L 1} \Delta \mathbf {u} = \mathbf {B} _ {L} \Delta \mathbf {u} \tag {4.2.5} +$$ + +식 (4.2.4)와 (4.2.5)에서 변위-변형률 관계행렬은 다음과 같다. + +$$ +\mathbf {B} _ {L 0} = \left\{N _ {1, x} \quad 0 \quad 0 \quad N _ {2, x} \quad 0 \quad 0 \right\} \tag {4.2.6} +$$ + +$$ +\mathbf {B} _ {L 1} = \left\{^ {t} u _ {, x} N _ {1, x} \quad {} ^ {t} v _ {, x} N _ {1, x} \quad {} ^ {t} w _ {, x} N _ {1, x} \quad {} ^ {t} u _ {, x} N _ {2, x} \quad {} ^ {t} v _ {, x} N _ {2, x} \quad {} ^ {t} w _ {, x} N _ {2, x} \right\} \tag {4.2.7} +$$ + +식 (4.1.18)에서 가상 변형률의 비선형 항을 구성하는 δL 은 다음과 같다. + + + +$$ +\delta \mathbf {L} = \left\{\delta u _ {, x} \quad \delta v _ {, x} \quad \delta w _ {, x} \right\} ^ {T} \tag {4.2.8} +$$ + +식 (4.2.8)은 가상변위 항 δu 와 행렬 ${ \bf { B } } _ { N L }$ 의 곱으로 표현된다. + +$$ +\delta \mathbf {L} = \mathbf {B} _ {N L} \delta \mathbf {u} \tag {4.2.9} +$$ + +유사한 방법으로 ∆L 또한 다음과 같이 나타낼 수 있다. + +$$ +\Delta \mathbf {L} = \mathbf {B} _ {N L} \Delta \mathbf {u} \tag {4.2.10} +$$ + +${ \bf { B } } _ { N L }$ 은 다음과 같다. + +$$ +\mathbf {B} _ {N L} = \left[ \begin{array}{c c c c c c} N _ {1, x} & 0 & 0 & N _ {2, x} & 0 & 0 \\ 0 & N _ {1, x} & 0 & 0 & N _ {2, x} & 0 \\ 0 & 0 & N _ {1, x} & 0 & 0 & N _ {2, x} \end{array} \right] \tag {4.2.11} +$$ + +식 (4.2.4-5)와 (4.2.9-10)을 식 (4.1.19)에 대입하여 정리하면 선형화된 평형 방정식을 얻을 수 있다. + +$$ +\delta \mathbf {u} ^ {T} \left(^ {t} \mathbf {K} _ {L} ^ {e} + ^ {t} \mathbf {K} _ {N L} ^ {e}\right) \Delta \mathbf {u} = \delta \mathbf {u} ^ {T} \left(^ {t + \Delta t} \mathbf {f} _ {e x t} ^ {e} - ^ {t} \mathbf {f} _ {\text { int }} ^ {e}\right) \tag {4.2.12} +$$ + +식 (4.2.12)의 각 항은 다음과 같다. + +$$ +{ } ^ { t } \mathbf { K } _ { L } ^ { e } = \int _ { L _ { e } } A \mathbf { B } _ { L } ^ { T } \mathbf { D } \mathbf { B } _ { L } d L +$$ + +$$ +{ } ^ { t } \mathbf { K } _ { N L } ^ { e } = \int _ { L _ { e } } A ^ { t } \mathbf { B } _ { N L } ^ { T } { } ^ { t } \hat { \mathbf { S } } ^ { t } \mathbf { B } _ { N L } d L \tag {4.2.13} +$$ + +$$ +{ } ^ { t } \mathbf { f } _ { \text { i n t } } ^ { e } = \int _ { L _ { e } } A ^ { t } \mathbf { B } _ { L } ^ { T } { } ^ { t } \mathbf { S } d L +$$ + +여기서, + +A : 단면적(section area) + +응력 성분으로 구성된 행렬 t ˆS 는 다음과 같다. + + + +$$ +{ } ^ { t } \hat { \mathbf { S } } = \left[ \begin{array} { c c c } { } ^ { t } \mathbf { S } & \mathbf { 0 } & \mathbf { 0 } \\ \mathbf { 0 } & { } ^ { t } \mathbf { S } & \mathbf { 0 } \\ \mathbf { 0 } & \mathbf { 0 } & { } ^ { t } \mathbf { S } \end{array} \right] \quad { } ^ { t } \mathbf { S } = \left[ \begin{array} { c } { } ^ { t } S _ { x x } \end{array} \right] \tag {4.2.14} +$$ + +트러스요소의 해석 결과로는 선형 해석과 같이 절점 응력과 요소내력이 있으며,절점에서의 결과 이외에 적분점에서의 응력을 표를 통해 볼 수 있다. diff --git a/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_020.md b/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_020.md new file mode 100644 index 00000000..90ce180c --- /dev/null +++ b/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_020.md @@ -0,0 +1,309 @@ + + +# 4-3 평면응력요소 + +기하비선형성을 고려한 평면 응력(plane stress) 요소는 등매개변수(isoparametric) 요소로 구성되어 있으며, 3절점, 4절점, 6절점, 8절점 요소가 있다. 각 요소는 선형 요소와 동일한 형상함수를 사용하며, 비적합모드는 사용하지 않는다. + +평면 응력 요소는 요소좌표계에서 이동변위 u, v 를 가지며, 형상함수 $N_{i}$ 를 이용하여 다음과 같이 나타낸다. + +$$ +u = \sum_ {i = 1} ^ {N} N _ {i} u _ {i}, v = \sum_ {i = 1} ^ {N} N _ {i} v _ {i} \tag {4.3.1} +$$ + +평면 응력 요소에서 사용되는 응력과 변형률은 다음과 같다. + +$$ +\mathbf {S} = \left\{S _ {x x} \quad S _ {y y} \quad S _ {x y} \right\} ^ {T}, \mathbf {E} = \left\{E _ {x x} \quad E _ {y y} \quad E _ {x y} \right\} ^ {T} \tag {4.3.2} +$$ + +식 (4.1.18)에서 가상 변형률의 선형 항 δe 는 다음과 같이 정리할 수 있다. + +$$ +\delta \mathbf {e} = \left\{ \begin{array}{c} \delta u _ {, x} \\ \delta v _ {, y} \\ \delta u _ {, y} + \delta v _ {, x} \end{array} \right\} + \left\{ \begin{array}{c} \delta u _ {, x} ^ {t} u _ {, x} + \delta v _ {, x} ^ {t} v _ {, x} \\ \delta u _ {, y} ^ {t} u _ {, y} + \delta v _ {, y} ^ {t} v _ {, y} \\ \delta u _ {, x} ^ {t} u _ {, y} + \delta v _ {, x} ^ {t} v _ {, y} + \delta u _ {, y} ^ {t} u _ {, x} + \delta v _ {, y} ^ {t} v _ {, x} \end{array} \right\} \tag {4.3.3} +$$ + +식 (4.3.3)은 다음과 같이 가상변위 항 $\delta u$ 와 행렬 $B_{L}$ 의 곱으로 표현된다. + +$$ +\delta \mathbf {e} = \mathbf {B} _ {L 0} \delta \mathbf {u} + \mathbf {B} _ {L 1} \delta \mathbf {u} = \mathbf {B} _ {L} \delta \mathbf {u} \tag {4.3.4} +$$ + +증분 변형률의 선형 항 역시 유사한 형태로 표현할 수 있다. + +$$ +\Delta \mathbf {e} = \mathbf {B} _ {L 0} \Delta \mathbf {u} + \mathbf {B} _ {L 1} \Delta \mathbf {u} = \mathbf {B} _ {L} \Delta \mathbf {u} \tag {4.3.5} +$$ + +식 (4.3.4)와 (4.3.5)에서 변위-변형률 관계행렬은 다음과 같다. + +$$ +\mathbf {B} _ {L 0} = \left[ \begin{array}{c c c c c} N _ {1, x} & 0 & \dots & N _ {N, x} & 0 \\ 0 & N _ {1, y} & \dots & 0 & N _ {N, y} \\ N _ {1, y} & N _ {1, x} & \dots & N _ {N, y} & N _ {N, x} \end{array} \right] \tag {4.3.6} +$$ + + + +$$ +\mathbf {B} _ {L 1} = \left[ \begin{array}{c c c} ^ {t} u _ {, x} N _ {1, x} & ^ {t} v _ {, x} N _ {1, x} & \dots \\ ^ {t} u _ {, y} N _ {1, y} & ^ {t} v _ {, y} N _ {1, y} & \dots \\ \left(^ {t} u _ {, x} N _ {1, y} + ^ {t} u _ {, y} N _ {1, x}\right) & \left(^ {t} v _ {, x} N _ {1, y} + ^ {t} v _ {, y} N _ {1, x}\right) & \dots \\ ^ {t} u _ {, x} N _ {n, x} & ^ {t} v _ {, x} N _ {n, x} \\ ^ {t} u _ {, y} N _ {n, y y} & ^ {t} v _ {, y} N _ {n, y} \\ \left(^ {t} u _ {, x} N _ {n, y} + ^ {t} u _ {, y} N _ {n, x}\right) & \left(^ {t} v _ {, x} N _ {n, y} + ^ {t} v _ {, y} N _ {n, x}\right) \end{array} \right] \tag {4.3.7} +$$ + +식 (4.1.18)에서 가상 변형률의 비선형 항을 구성하는 δL 은 다음과 같다. + +$$ +\delta \mathbf {L} = \left\{\delta u _ {, x} \quad \delta u _ {, y} \quad \delta v _ {, x} \quad \delta v _ {, y} \right\} ^ {T} \tag {4.3.8} +$$ + +식 (4.3.8)는 가상변위 항 $\delta u$ 와 행렬 $B_{NL}$ 의 곱으로 표현된다. + +$$ +\delta \mathbf {L} = \mathbf {B} _ {N L} \delta \mathbf {u} \tag {4.3.9} +$$ + +유사한 방법으로 ΔL 또한 다음과 같이 나타낼 수 있다. + +$$ +\Delta \mathbf {L} = \mathbf {B} _ {N L} \Delta \mathbf {u} \tag {4.3.10} +$$ + +$B_{NL}$ 은 다음과 같다. + +$$ +\mathbf {B} _ {N L} = \left[ \begin{array}{c c c c c} N _ {1, x} & 0 & \dots & N _ {N, x} & 0 \\ N _ {1, y} & 0 & \dots & N _ {N, y} & 0 \\ 0 & N _ {1, x} & \dots & 0 & N _ {N, x} \\ 0 & N _ {1, y} & \dots & 0 & N _ {N, y} \end{array} \right] \tag {4.3.11} +$$ + +식 (4.3.4-5)과 (4.3.9-10)을 식 (4.1.18)에 대입하여 정리하면 선형화된 평형 방정식을 얻을 수 있다. + +$$ +\delta \mathbf {u} ^ {T} \left(^ {t} \mathbf {K} _ {L} ^ {e} + ^ {t} \mathbf {K} _ {N L} ^ {e}\right) \Delta \mathbf {u} = \delta \mathbf {u} ^ {T} \left(^ {t + \Delta t} \mathbf {f} _ {\text { ext }} ^ {e} - ^ {t} \mathbf {f} _ {\text { int }} ^ {e}\right) \tag {4.3.12} +$$ + +식 (4.3.12)의 각 항은 다음과 같다. + + + +$$ +{ } ^ { t } \mathbf { K } _ { L } ^ { e } = \int _ { A _ { e } } t \mathbf { B } _ { L } ^ { T } \mathbf { D } \mathbf { B } _ { L } d A +$$ + +$$ +{ } ^ { t } \mathbf { K } _ { N L } ^ { e } = \int _ { A _ { e } } t ^ { t } \mathbf { B } _ { N L } ^ { T } { } ^ { t } \hat { \mathbf { S } } ^ { t } \mathbf { B } _ { N L } d A \tag {4.3.13} +$$ + +$$ +{ } ^ { t } \mathbf { f } _ { \text { i n t } } ^ { e } = \int _ { A _ { e } } t { } ^ { t } \mathbf { B } _ { L } ^ { T } { } ^ { t } \mathbf { S } d A +$$ + +여기서, + +t : 두께(thickness) + +응력 성분으로 구성된 행렬 $^{t}$ S 는 다음과 같다 + +$$ +{ } ^ { t } \hat { \mathbf { S } } = \left[ \begin{array} { c c } { } ^ { t } \mathbf { S } & \mathbf { 0 } \\ \mathbf { 0 } & { } ^ { t } \mathbf { S } \end{array} \right] \quad { } ^ { t } \mathbf { S } = \left[ \begin{array} { c c } { } ^ { t } S _ { x x } & { } ^ { t } S _ { x y } \\ { } ^ { t } S _ { x y } & { } ^ { t } S _ { y y } \end{array} \right] \tag {4.3.14} +$$ + +평면 응력 요소의 해석 결과로는 선형 해석과 같이 절점 응력과 변형률이 있으며, 절점에서의 결과 이외에 적분점에서의 응력과 변형률을 표를 통해 볼 수 있다. 적분 차수는 다음과 같다. + +• 3절점 삼각형 : 1 점 가우스 적분 +• 4절점 사각형 : 4 점 가우스 적분 +- 6절점 삼각형 : 3 점 가우스 적분 +• 8절점 사각형 : 9 점 가우스 적분 + + + +# 4-4 판요소 + +기하 비선형성을 고려한 판요소는 3, 4절점 요소의 경우 “감절점 이론(degenerated shell approach)”을, 6, 8절점 고차요소의 경우 “연속체 셸이론(continuum shell approach)”이용하며, 평면 응력 상태의 면내변형과 휨/전단으로 이루어진 면외변형을 고려할 수 있다. 기하 비선형을 고려한 판요소는면에 수직한 방향의 회전(drilling) 자유도를 고려하지 않으며, Mindlin 판이론(Mindlin plate theory)를 기초로 한 전단 변형을 고려한다. 요소의 종류에는 선형 해석의 경우와 같이 3절점, 4절점, 6절점, 8절점 요소가 있다. + +기하 비선형성을 고려한 판요소는 정식화 과정이 매우 복잡하므로 연속체 셸이론에 대한 사항만 간단하게 설명하고자 한다. 판요소에서는 두께방향의 인장 응력을 무시하는 것이 일반적이지만, 연속체 셸이론에 근거한 정식화 과정에서는모든 응력과 변형률 성분을 고려한다. + +$$ +\mathbf {S} = \left\{S _ {x x} \quad S _ {y y} \quad S _ {z z} \quad S _ {x y} \quad S _ {y z} \quad S _ {z x} \right\} ^ {T}, \quad \mathbf {E} = \left\{E _ {x x} \quad E _ {y y} \quad E _ {z z} \quad E _ {x y} \quad E _ {y z} \quad E _ {z x} \right\} ^ {T} \tag {4.4.1} +$$ + +요소 내부의 임의 위치에 대한 이동 변위는 다음과 같이 중립면의 이동변위와회전에 의한 효과로 구분하여 표현할 수 있다. + +$$ +\mathbf {U} = \mathbf {u} _ {0} + \mathbf {t} \tag {4.4.2} +$$ + +여기서, 벡터 t 는 회전에 의한 효과를 고려한 항이며, 다음과 같이 표현할 수있다. + +$$ +\mathbf {t} = \frac {t}{2} \zeta (\mathbf {T} - \overline {{\mathbf {T}}}) \tag {4.4.3} +$$ + +여기서, + +T : 변형후 판의 수직벡터(deformed unit shell normalvector) + +T : 변형전 판의 수직벡터(undeformed unit shell normalvector) + + + +식 (4.1.18)에서 가상 변형률의 선형 항 δe 는 아래와 같이 표현할 수 있다. + +$$ +\delta \mathbf {e} = \left\{ \begin{array}{c} \delta \left(u _ {0, x} + t _ {x, x}\right) \\ \delta \left(v _ {0, y} + t _ {y, y}\right) \\ \delta \left(w _ {0, z} + t _ {z, z}\right) \\ \delta \left(u _ {0, y} + t _ {x, y}\right) + \delta \left(v _ {0, x} + t _ {y, x}\right) \\ \delta \left(v _ {0, z} + t _ {y, z}\right) + \delta \left(w _ {0, y} + t _ {z, y}\right) \\ \delta \left(w _ {0, x} + t _ {z, x}\right) + \delta \left(u _ {0, z} + t _ {x, z}\right) \end{array} \right\} \tag {4.4.4} +$$ + +$$ ++ \left\{ \begin{array}{c} \delta (\mathbf {u} _ {0, x} + \mathbf {t} _ {, x}) ^ {T} \left(^ {t} \mathbf {u} _ {0, x} + ^ {t} \mathbf {t} _ {, x}\right) \\ \delta (\mathbf {u} _ {0, y} + \mathbf {t} _ {, y}) ^ {T} \left(^ {t} \mathbf {u} _ {0, y} + ^ {t} \mathbf {t} _ {, y}\right) \\ \delta (\mathbf {u} _ {0, z} + \mathbf {t} _ {, z}) ^ {T} \left(^ {t} \mathbf {u} _ {0, z} + ^ {t} \mathbf {t} _ {, z}\right) \\ \delta (\mathbf {u} _ {0, x} + \mathbf {t} _ {, x}) ^ {T} \left(^ {t} \mathbf {u} _ {0, y} + ^ {t} \mathbf {t} _ {, y}\right) + \delta (\mathbf {u} _ {0, y} + \mathbf {t} _ {, y}) ^ {T} \left(^ {t} \mathbf {u} _ {0, x} + ^ {t} \mathbf {t} _ {, x}\right) \\ \delta (\mathbf {u} _ {0, y} + \mathbf {t} _ {, y}) ^ {T} \left(^ {t} \mathbf {u} _ {0, z} + ^ {t} \mathbf {t} _ {, z}\right) + \delta (\mathbf {u} _ {0, z} + \mathbf {t} _ {, z}) ^ {T} \left(^ {t} \mathbf {u} _ {0, y} + ^ {t} \mathbf {t} _ {, y}\right) \\ \delta (\mathbf {u} _ {0, z} + \mathbf {t} _ {, z}) ^ {T} \left(^ {t} \mathbf {u} _ {0, x} + ^ {t} \mathbf {t} _ {, x}\right) + \delta (\mathbf {u} _ {0, x} + \mathbf {t} _ {, x}) ^ {T} \left(^ {t} \mathbf {u} _ {0, z} + ^ {t} \mathbf {t} _ {, z}\right) \end{array} \right\} +$$ + +식 (4.1.18)에서 가상 변형률의 비선형 항을 구성하는 δL 은 다음과 같다. + +$$ +\delta \mathbf {L} = \left\{ \begin{array}{l} \delta \left(u _ {0, x} + t _ {x, x}\right) \\ \delta \left(u _ {0, y} + t _ {x, y}\right) \\ \delta \left(u _ {0, z} + t _ {x, z}\right) \\ \delta \left(v _ {0, x} + t _ {y, x}\right) \\ \delta \left(v _ {0, y} + t _ {y, y}\right) \\ \delta \left(v _ {0, z} + t _ {y, z}\right) \\ \delta \left(w _ {0, x} + t _ {z, x}\right) \\ \delta \left(w _ {0, y} + t _ {z, y}\right) \\ \delta \left(w _ {0, z} + t _ {z, z}\right) \end{array} \right\} \tag {4.4.5} +$$ + +실제 계산에 필요한 행렬 $\mathbf { B } _ { L }$ 과 ${ \bf { B } } _ { N L }$ 은 본 절에서 설명하지 않는다. 요소 강성또는 내력의 계산에서는 (4.4.1)의 응력과 변형률을 요소 중립면에 접하는 좌표계로 변환하여 사용한다. 3, 4절점 요소에 사용하는 감절점 이론에서는 변형률성분을 요소 중립면에 접하는 좌표계에 대하여 정의함으로써 $E _ { z z }$ 를 제외한 5개의 성분을 직접 구할 수 있다. 회전에 의한 변위의 구현 방법과 절점 개수 별요소의 특징은 다음과 같다. + + + +![](images/page-196_b3868be5f720c3eb5e27e54ae23558a33db8af49f2bab3da13147ebadcc312f9.jpg) + +
+text_image + +T +U +u₀ +x̄ +O (Ref. Frame) +
+ +그림 4.4.1 판의 초기 및 현재 형상 + +3절점 또는 4절점을 가진 평면판은 전체좌표계에서 3개의 이동변위와 3개의 회전변위를 모두 고려한다. 단, 회전 증분변위에서 두께방향 벡터와 수직한 성분은직교화 과정을 통하여 제외된다. + +$$ +\mathbf {u} _ {i} = \left\{u _ {i} \quad v _ {i} \quad w _ {i} \quad \theta_ {X i} \quad \theta_ {Y i} \quad \theta_ {Z i} \right\} ^ {T} \tag {4.4.6} +$$ + +4절점 판요소는 라그랑지(Lagrangian) 형상함수를 이용하며, 유한회전에 의해변형된 두께방향 벡터는 T R RT= δ 와 같이 회전의 누적에 의해 계산한다. δR은 다음과 같다. + +$$ +\delta \mathbf {R} = I + \frac {\sin \delta \theta}{\delta \theta} \mathbf {S} (\delta \theta) + \frac {1 - \cos \delta \theta}{\delta \theta^ {2}} \mathbf {S} (\delta \theta) \mathbf {S} (\delta \theta) \tag {4.4.7} +$$ + +여기서, + +S( ) δθ : 교대행렬(skew symmetric matrix) + +midas FEA에서는 회전행렬을 사원수(quaternion)로부터 계산하는 방법을 이용하여 계산에 효율을 기하였다. 판요소의 대회전(large rotation)은 회전에 의한변위 t 가 절점자유도에 대해 선형이 아님을 의미한다. 그러므로 δt 와 ∆t 가 명시적으로 구분되지 않기 때문에 강성의 구성에 있어서 식(4.4.4)와 (4.4.5) 이외에 δ ( ) ∆t 항이 추가됨에 주의해야 한다. + + + +전단변형에 의한 변형률은 그림 4.4.2와 같이 4 변에서의 ANS(assumednatural shear strain)로부터 계산하여 잠김(locking) 현상을 방지한다. + +$$ +\gamma_ {\zeta \xi} = \frac {1}{2} (1 - \eta) \gamma_ {\zeta \xi} (0, - 1) + \frac {1}{2} (1 + \eta) \gamma_ {\zeta \xi} (0, 1) \tag {4.4.8} +$$ + +$$ +\gamma_ {\zeta \eta} = \frac {1}{2} (1 - \xi) \gamma_ {\zeta \eta} (- 1, 0) + \frac {1}{2} (1 + \xi) \gamma_ {\zeta \eta} (1, 0) \tag {4.4.9} +$$ + +$$ +\left\{ \begin{array}{l} \gamma_ {z \alpha} \\ \gamma_ {z \beta} \end{array} \right\} = \mathbf {P} \left\{ \begin{array}{l} \gamma_ {\zeta \xi} \\ \gamma_ {\zeta \eta} \end{array} \right\} \tag {4.4.10} +$$ + +여기서, + +P : 좌표변화행렬(coordinate transform matrix) + +![](images/page-197_eb22ba2b3776eecdfca1fd0a23630d86bd77d6a957182c5b9a3fb1616204fd0b.jpg) + +
+text_image + +4 +γης +γξς +3 +γης +1 +γξς +2 +
+ +그림 4.4.2 ANS의 내삽 점 + +3절점 요소의 계산에는 4절점 요소의 4번째 절점을 3번째 절점과 같은 좌표를가지도록 하는 기법을 이용한다. + +6절점 또는 8절점을 가진 곡면판은 전체좌표계에서 3개의 이동변위와 절점에서정의된 면내 두 방향 $\mathbf { V } _ { 1 i } , \ \mathbf { V } _ { 2 i }$ 에 대한 회전변위를 갖는다. $\mathbf { V } _ { 1 i } , \ \mathbf { V } _ { 2 i }$ 의 정의는 선형 요소의 정식화를 참조하고, 본 절에서는 생략한다. + + + +$$ +\mathbf {u} _ {i} = \left\{u _ {i} \quad v _ {i} \quad w _ {i} \quad \theta_ {1 i} \quad \theta_ {2 i} \right\} ^ {T} \tag {4.4.11} +$$ + +곡면판은 라그랑지(Lagrangian) 형상함수를 이용하며, 유한회전에 의해 변형된 두께방향 벡터는 평면판 요소의 경우와 동일하게 회전행렬의 누적에 의해 계산한다. 단, 회전 증분변위의 중에서 $V_{1i}$ , $V_{2i}$ 방향의 성분만을 고려함으로써 암시적으로 직교화 효과를 얻을 수 있다. + +기하비선형 판요소의 해석 결과로는 선형 해석과 같이 절점 응력/변형률, 요소 내력 등이 있으며 절점에서의 결과 이외에 적분점에서의 응력과 변형률을 표를 통해 볼 수 있다. 응력과 변형률은 $z$ 축을 따라 두 지점에서 계산하며, 면내거동에 대한 두께를 기준으로 상단( $z=t/2$ )과 하단( $z=-t/2$ )이 기본 위치이다. 기하비선형을 고려한 판 요소는 두께 방향으로 3점 심슨 적분(Simpson integration)을 수행하며, 면적에 대한 적분 차수는 다음과 같다. + +• 3절점 삼각형 : 1 점 가우스 적분 +• 4절점 사각형 : 4 점 가우스 적분 +- 6절점 삼각형 : 3 점 가우스 적분 +• 8절점 사각형 : 4 점 가우스 적분 + + + +# 4-5 평면변형률요소 + +기하비선형성을 고려한 평면변형률(plane strain) 요소는 등매개변수 (isoparametric) 요소로 구성되어 있으며, 3절점, 4절점, 6절점, 8절점 요소가 있다. 각 요소는 선형 요소와 동일한 형상함수를 사용하며, 비적합모드를 사용하지 않는다. + +평면변형률 요소는 요소좌표계에서 이동변위, u, v 를 가지며 평면응력 요소와 전개과정이 같다. 최종적인 평면변형률 요소의 선형화된 평형 방정식은 다음과 같다. + +$$ +\delta \mathbf {u} ^ {T} \left(^ {t} \mathbf {K} _ {L} ^ {e} + ^ {t} \mathbf {K} _ {N L} ^ {e}\right) \Delta \mathbf {u} = \delta \mathbf {u} ^ {T} \left(^ {t + \Delta t} \mathbf {f} _ {\text { ext }} ^ {e} - ^ {t} \mathbf {f} _ {\text { int }} ^ {e}\right) \tag {4.5.1} +$$ + +식 (4.5.1)의 각 항은 응력-변형률의 관계를 정의하는 행렬 D 를 제외하고, 평면 응력 요소와 동일하다. + +평면변형률 요소의 해석 결과로는 선형 해석과 같이 절점 응력과 변형률이 있으며, 절점에서의 결과 이외에 적분점에서의 응력과 변형률을 표를 통해 볼 수 있다. 적분 차수는 다음과 같다. + +• 3절점 삼각형 : 1 점 가우스 적분 +• 4절점 사각형 : 4 점 가우스 적분 +- 6절점 삼각형 : 3 점 가우스 적분 +• 8절점 사각형 : 9 점 가우스 적분 + + + +# 4-6 축대칭요소 + +기하비선형성을 고려한 축대칭(axisymmetric) 요소는 등매개변수(isoparametric)요소로 구성되어 있으며, 3절점, 4절점, 6절점, 8절점 요소가 있다. 각 요소는선형 요소와 동일한 형상함수를 사용한다. + +평면응력 요소는 요소좌표계에서 이동변위 u , v 를 가지며, 변위는 형상함수 Ni를 이용하여 다음과 같이 나타낸다. + +$$ +u = \sum_ {i = 1} ^ {N} N _ {i} u _ {i}, v = \sum_ {i = 1} ^ {N} N _ {i} v _ {i} \tag {4.6.1} +$$ + +축대칭 요소에서 사용되는 응력과 변형률은 다음과 같다. + +$$ +\mathbf {S} = \left\{S _ {x x} \quad S _ {y y} \quad S _ {x y} \quad S _ {z z} \right\} ^ {T}, \quad \mathbf {E} = \left\{E _ {x x} \quad E _ {y y} \quad E _ {x y} \quad E _ {z z} \right\} ^ {T} \tag {4.6.2} +$$ + +식 (4.1.18)에서 가상 변형률의 선형 항 δe 는 다음과 같이 정리할 수 있다. + +$$ +\delta \mathbf {e} = \left\{ \begin{array}{c} \delta u _ {, x} \\ \delta v _ {, y} \\ \delta u _ {, y} + \delta v _ {, x} \\ \frac {\delta u}{r} \end{array} \right\} + \left\{ \begin{array}{c} \delta u _ {, x} ^ {t} u _ {, x} + \delta v _ {, x} ^ {t} v _ {, x} \\ \delta u _ {, y} ^ {t} u _ {, y} + \delta v _ {, y} ^ {t} v _ {, y} \\ \delta u _ {, x} ^ {t} u _ {, y} + \delta v _ {, x} ^ {t} v _ {, y} + \delta u _ {, y} ^ {t} u _ {, x} + \delta v _ {, y} ^ {t} v _ {, x} \\ \frac {\delta u ^ {t} u}{r ^ {2}} \end{array} \right\} \tag {4.6.3} +$$ + +식 (4.6.3)은 다음과 같이 가상변위 항 δe 와 행렬 B 의 곱으로 표현된다. + +$$ +\delta \mathbf {e} = \left\{ \begin{array}{c} \delta u _ {, x} \\ \delta v _ {, y} \\ \delta u _ {, y} + \delta v _ {, x} \\ \frac {\delta u}{r} \end{array} \right\} + \left\{ \begin{array}{c} \delta u _ {, x} ^ {t} u _ {, x} + \delta v _ {, x} ^ {t} v _ {, x} \\ \delta u _ {, y} ^ {t} u _ {, y} + \delta v _ {, y} ^ {t} v _ {, y} \\ \delta u _ {, x} ^ {t} u _ {, y} + \delta v _ {, x} ^ {t} v _ {, y} + \delta u _ {, y} ^ {t} u _ {, x} + \delta v _ {, y} ^ {t} v _ {, x} \\ \frac {\delta u ^ {t} u}{r ^ {2}} \end{array} \right\} \tag {4.6.4} +$$ + +증분 변형률의 선형 항 역시 유사한 형태로 표현할 수 있다. + +$$ +\delta \mathbf {e} = \mathbf {B} _ {L 0} \delta \mathbf {u} + \mathbf {B} _ {L 1} \delta \mathbf {u} = \mathbf {B} _ {L} \delta \mathbf {u} \tag {4.6.5} +$$ + +식 (4.6.4)와 (4.6.5)에서 변위-변형률 관계행렬은 다음과 같다. diff --git a/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_021.md b/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_021.md new file mode 100644 index 00000000..45a9f036 --- /dev/null +++ b/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_021.md @@ -0,0 +1,276 @@ + + +$$ +\mathbf {B} _ {L 0} = \left[ \begin{array}{c c c c c} N _ {1, x} & 0 & \dots & N _ {N, x} & 0 \\ 0 & N _ {1, y} & \dots & 0 & N _ {N, y} \\ N _ {1, y} & N _ {1, x} & \dots & N _ {N, y} & N _ {N, x} \\ \frac {N _ {1}}{r} & 0 & \dots & \frac {N _ {N}}{r} & 0 \end{array} \right] \tag {4.6.6} +$$ + +$$ +\mathbf {B} _ {L 1} = \left[ \begin{array}{c c c} ^ {t} u _ {, x} N _ {1, x} & ^ {t} v _ {, x} N _ {1, x} & \dots \\ ^ {t} u _ {, y} N _ {1, y} & ^ {t} v _ {, y} N _ {1, y} & \dots \\ \left(^ {t} u _ {, x} N _ {1, y} + ^ {t} u _ {, y} N _ {1, x}\right) & \left(^ {t} v _ {, x} N _ {1, y} + ^ {t} v _ {, y} N _ {1, x}\right) & \dots \\ ^ {t} u \frac {N _ {1}}{r ^ {2}} & 0 & \dots \end{array} \right. \tag {4.6.7} +$$ + +$$ +\left. \begin{array}{c c} ^ {t} u _ {, x} N _ {N, x} & ^ {t} v _ {, x} N _ {N, x} \\ ^ {t} u _ {, y} N _ {N, y} & ^ {t} v _ {, y} N _ {N, y} \\ \left(^ {t} u _ {, x} N _ {N, y} + ^ {t} u _ {, y} N _ {N, x}\right) & \left(^ {t} v _ {, x} N _ {N, y} + ^ {t} v _ {, y} N _ {N, x}\right) \\ ^ {t} u \frac {N _ {N}}{r ^ {2}} & 0 \end{array} \right] +$$ + +식 (4.1.18)에서 가상 변형률의 비선형 항을 구성하는 δL 은 다음과 같다. + +$$ +\delta \mathbf {L} = \left\{\delta u _ {, x} \quad \delta u _ {, y} \quad \delta v _ {, x} \quad \delta v _ {, y} \quad \frac {\delta u}{r} \right\} ^ {T} \tag {4.6.8} +$$ + +식 (4.6.8)은 가상변위 항 $\delta u$ 와 행렬 $B_{NL}$ 의 곱으로 표현된다. + +$$ +\delta \mathbf {L} = \mathbf {B} _ {N L} \delta \mathbf {u} \tag {4.6.9} +$$ + +유사한 방법으로 ΔL 또한 다음과 같이 나타낼 수 있다. + +$$ +\Delta \mathbf {L} = \mathbf {B} _ {N L} \Delta \mathbf {u} \tag {4.6.10} +$$ + +$B_{NL}$ 은 다음과 같다. + +$$ +\mathbf {B} _ {N L} = \left[ \begin{array}{c c c c c} N _ {1, x} & 0 & \dots & N _ {N, x} & 0 \\ N _ {1, y} & 0 & \dots & N _ {N, y} & 0 \\ 0 & N _ {1, x} & \dots & 0 & N _ {N, x} \\ 0 & N _ {1, y} & \dots & 0 & N _ {N, y} \\ \frac {N _ {1}}{r} & 0 & \dots & \frac {N _ {N}}{r} & 0 \end{array} \right] \tag {4.6.11} +$$ + + + +식 (4.6.4-5)와 (4.6.9-10)을 식 (4.1.18)에 대입하여 정리하면 선형화된 평형 방정식을 얻을 수 있다. + +$$ +\delta \mathbf {u} ^ {T} \left(^ {t} \mathbf {K} _ {L} ^ {e} + ^ {t} \mathbf {K} _ {N L} ^ {e}\right) \Delta \mathbf {u} = \delta \mathbf {u} ^ {T} \left(^ {t + \Delta t} \mathbf {f} _ {\text { ext }} ^ {e} - ^ {t} \mathbf {f} _ {\text { int }} ^ {e}\right) \tag {4.6.12} +$$ + +식 (4.6.12)의 각 항은 다음과 같다. + +$$ +{ } ^ { t } \mathbf { K } _ { L } ^ { e } = \int _ { A _ { e } } r \mathbf { B } _ { L } ^ { T } \mathbf { D } \mathbf { B } _ { L } d A +$$ + +$$ +{ } ^ { t } \mathbf { K } _ { N L } ^ { e } = \int _ { A _ { e } } r ^ { t } \mathbf { B } _ { N L } ^ { T } { } ^ { t } \hat { \mathbf { S } } ^ { t } \mathbf { B } _ { N L } d A \tag {4.6.13} +$$ + +$$ +{ } ^ { t } \mathbf { f } _ { \text { i n t } } ^ { e } = \int _ { A _ { e } } r ^ { t } \mathbf { B } _ { L } ^ { T } { } ^ { t } \mathbf { S } d A +$$ + +그리고, + +$$ +{ } ^ { t } \hat { \mathbf { S } } = \left[ \begin{array} { c c c } { } ^ { t } S _ { x x } & { } ^ { t } S _ { x y } & \mathbf { 0 } \\ { } ^ { t } S _ { x y } & { } ^ { t } S _ { y y } & \mathbf { 0 } \\ \mathbf { 0 } & \mathbf { 0 } & { } ^ { t } \mathbf { S } \end{array} \right] \quad { } ^ { t } \mathbf { S } = \left[ \begin{array} { c c c } { } ^ { t } S _ { x x } & { } ^ { t } S _ { x y } & 0 \\ { } ^ { t } S _ { x y } & { } ^ { t } S _ { y y } & 0 \\ 0 & 0 & { } ^ { t } S _ { z z } \end{array} \right] \tag {4.6.14} +$$ + +축대칭 요소의 해석 결과로는 선형 해석과 같이 절점 응력과 변형률이 있으며, 절점에서의 결과 이외에 적분점에서의 응력과 변형률을 표를 통해 볼 수 있다. 적분 차수는 다음과 같다. + +• 3절점 삼각형 : 1 점 가우스 적분 +• 4절점 사각형 : 4 점 가우스 적분 +- 6절점 삼각형 : 3 점 가우스 적분 +• 8절점 사각형 : 9 점 가우스 적분. + + + +# 4-7 입체요소 + +기하비선형성을 고려한 입체(solid) 요소는 등매개변수(isoparametric) 요소로 구성되어 있으며, 4절점, 6절점, 8절점, 10절점, 15절점, 20절점 요소가 있다. 각요소는 선형 요소와 동일한 형상함수를 사용하며, 비적합모드를 사용하지 않는다. + +입체요소는 요소좌표계에서 이동변위, u , v , w 를 가지며, 변위는 형상함수 Ni를 이용하여 다음과 같이 나타낸다. + +$$ +u = \sum_ {i = 1} ^ {N} N _ {i} u _ {i}, v = \sum_ {i = 1} ^ {N} N _ {i} v _ {i}, w = \sum_ {i = 1} ^ {N} N _ {i} w _ {i} \tag {4.7.1} +$$ + +입체 요소에서 사용되는 응력과 변형률은 다음과 같다. + +$$ +\mathbf {S} = \left\{S _ {x x} \quad S _ {y y} \quad S _ {z z} \quad S _ {x y} \quad S _ {y z} \quad S _ {z x} \right\} ^ {T}, \quad \mathbf {E} = \left\{E _ {x x} \quad E _ {y y} \quad E _ {z z} \quad E _ {x y} \quad E _ {y z} \quad E _ {z x} \right\} ^ {T} \tag {4.7.2} +$$ + +식 (4.1.18)에서 가상 변형률의 선형 항 는 다음과 같이 정리할 수 있다. + +$$ +\delta \mathbf {e} = \left\{ \begin{array}{c} \delta u _ {, x} \\ \delta v _ {, y} \\ \delta w _ {, z} \\ \delta u _ {, y} + \delta v _ {, x} \\ \delta v _ {, z} + \delta w _ {, y} \\ \delta w _ {, z} + \delta u _ {, z} \end{array} \right\} + \left\{ \begin{array}{c} \delta u _ {, x} ^ {t} u _ {, x} + \delta v _ {, x} ^ {t} v _ {, x} + \delta w _ {, x} ^ {t} w _ {, x} \\ \delta u _ {, y} ^ {t} u _ {, y} + \delta v _ {, y} ^ {t} v _ {, y} + \delta w _ {, y} ^ {t} w _ {, y} \\ \delta u _ {, z} ^ {t} u _ {, z} + \delta v _ {, z} ^ {t} v _ {, z} + \delta w _ {, z} ^ {t} w _ {, z} \\ \delta u _ {, x} ^ {t} u _ {, y} + \delta v _ {, x} ^ {t} v _ {, y} + \delta w _ {, x} ^ {t} w _ {, y} + \delta u _ {, y} ^ {t} u _ {, x} + \delta v _ {, y} ^ {t} v _ {, x} + \delta w _ {, y} ^ {t} w _ {, x} \\ \delta u _ {, y} ^ {t} u _ {, z} + \delta v _ {, y} ^ {t} v _ {, z} + \delta w _ {, y} ^ {t} w _ {, z} + \delta u _ {, z} ^ {t} u _ {, y} + \delta v _ {, z} ^ {t} v _ {, y} + \delta w _ {, z} ^ {t} w _ {, y} \\ \delta u _ {, z} ^ {t} u _ {, x} + \delta v _ {, z} ^ {t} v _ {, x} + \delta w _ {, z} ^ {t} w _ {, x} + \delta u _ {, x} ^ {t} u _ {, z} + \delta v _ {, x} ^ {t} v _ {, z} + \delta w _ {, x} ^ {t} w _ {, z} \end{array} \right\} \tag {4.7.3} +$$ + +식 (4.7.3)은 다음과 같이 가상변위 항 δu 와 행렬 B 의 곱으로 표현된다. + +$$ +\delta \mathbf {e} = \mathbf {B} _ {L 0} \delta \mathbf {u} + \mathbf {B} _ {L 1} \delta \mathbf {u} = \mathbf {B} _ {L} \delta \mathbf {u} \tag {4.7.4} +$$ + +증분 변형률의 선형 항 역시 유사한 형태로 표현할 수 있다. + +$$ +\Delta \mathbf {e} = \mathbf {B} _ {L 0} \Delta \mathbf {u} + \mathbf {B} _ {L 1} \Delta \mathbf {u} = \mathbf {B} _ {L} \Delta \mathbf {u} \tag {4.7.5} +$$ + + + +식 (4.7.4)과 (4.7.5)에서 변위-변형률 관계행렬은 다음과 같다. + +$$ +\mathbf {B} _ {L 0} = \left[ \begin{array}{c c c c c c c c} N _ {1, x} & 0 & 0 & N _ {2, x} & \dots & N _ {N, x} & 0 & 0 \\ 0 & N _ {1, y} & 0 & 0 & \dots & 0 & N _ {N, y} & 0 \\ 0 & 0 & N _ {1, z} & 0 & \dots & 0 & 0 & N _ {N, z} \\ N _ {1, y} & N _ {1, x} & 0 & N _ {2, y} & \dots & N _ {N, y} & N _ {N, x} & 0 \\ 0 & N _ {1, z} & N _ {1, y} & 0 & \dots & 0 & N _ {N, z} & N _ {N, y} \\ N _ {1, z} & 0 & N _ {1, x} & N _ {2, z} & \dots & N _ {N, z} & 0 & N _ {N, x} \end{array} \right] \tag {4.7.6} +$$ + +$$ +\mathbf {B} _ {L 1} = \left[ \begin{array}{c c c c c c} ^ {\prime} u _ {, x} N _ {1, x} & ^ {\prime} v _ {, x} N _ {1, x} & ^ {\prime} w _ {, x} N _ {1, x} & ^ {\prime} u _ {, x} N _ {2, x} & \dots & ^ {\prime} w _ {, x} N _ {N, 1} \\ ^ {\prime} u _ {, y} N _ {1, y} & ^ {\prime} v _ {, y} N _ {1, y} & ^ {\prime} w _ {, y} N _ {1, y} & ^ {\prime} u _ {, y} N _ {2, y} & \dots & ^ {\prime} w _ {, y} N _ {N, 2} \\ ^ {\prime} u _ {, z} N _ {1, z} & ^ {\prime} v _ {, z} N _ {1, z} & ^ {\prime} w _ {, z} N _ {1, z} & ^ {\prime} u _ {, z} N _ {2, z} & \dots & ^ {\prime} w _ {, z} N _ {N, 3} \\ ^ {\prime} u _ {, x} N _ {1, y} + ^ {\prime} u _ {, x} N _ {1, x} & ^ {\prime} v _ {, x} N _ {1, y} + ^ {\prime} v _ {, y} N _ {1, x} & ^ {\prime} w _ {, x} N _ {1, y} + ^ {\prime} w _ {, y} N _ {1, x} & ^ {\prime} u _ {, x} N _ {2, y} + ^ {\prime} u _ {, y} N _ {2, x} & \dots & ^ {\prime} w _ {, x} N _ {N, y} + ^ {\prime} w _ {, y} N _ {N, x} \\ ^ {\prime} u _ {, y} N _ {1, z} + ^ {\prime} u _ {, z} N _ {1, y} & ^ {\prime} v _ {, y} N _ {1, z} + ^ {\prime} v _ {, z} N _ {1, y} & ^ {\prime} w _ {, y} N _ {1, z} + ^ {\prime} w _ {, z} N _ {1, y} & ^ {\prime} u _ {, y} N _ {2, z} + ^ {\prime} u _ {, z} N _ {2, y} & \dots & ^ {\prime} w _ {, y} N _ {N, z} + ^ {\prime} w _ {, z} N _ {N, y} \\ ^ {\prime} u _ {, x} N _ {1, z} + ^ {\prime} u _ {, z} N _ {1, x} & ^ {\prime} v _ {, x} N _ {1, z} + ^ {\prime} v _ {, z} N _ {1, x} & ^ {\prime} w _ {, x} N _ {1, z} + ^ {\prime} w _ {, z} N _ {1, x} & ^ {\prime} u _ {, x} N _ {3, z} + ^ {\prime} u _ {, z} N _ {3, x} & \dots & ^ {\prime} w _ {, x} N _ {N, z} + ^ {\prime} w _ {, z} N _ {N, x} \end{array} \right] \tag {4.7.7} +$$ + +식 (4.1.18)에서 가상 변형률의 비선형 항을 구성하는 δL 은 다음과 같다. + +$$ +\delta \mathbf {L} = \left\{\delta u _ {, x} \quad \delta u _ {, y} \quad \delta u _ {, z} \quad \delta v _ {, x} \quad \delta v _ {, y} \quad \delta v _ {, z} \quad \delta w _ {, x} \quad \delta w _ {, y} \quad \delta w _ {, z} \right\} ^ {T} \tag {4.7.8} +$$ + +식 (4.7.8)은 가상변위 항 $\delta u$ 와 행렬 $B_{NL}$ 의 곱으로 표현된다. + +$$ +\delta \mathbf {L} = \mathbf {B} _ {N L} \delta \mathbf {u} \tag {4.7.9} +$$ + +유사한 방법으로 ΔL 또한 다음과 같이 나타낼 수 있다. + +$$ +\Delta \mathbf {L} = \mathbf {B} _ {N L} \Delta \mathbf {u} \tag {4.7.10} +$$ + +여기서, + +$$ +\begin{array}{l} \mathbf {B} _ {N L} = \left[ \begin{array}{c c c c c c c c} \tilde {\mathbf {B}} _ {1} & \mathbf {0} & \mathbf {0} & \tilde {\mathbf {B}} _ {2} & \dots & \tilde {\mathbf {B}} _ {N} & \mathbf {0} & \mathbf {0} \\ \mathbf {0} & \tilde {\mathbf {B}} _ {1} & \mathbf {0} & \mathbf {0} & \dots & \mathbf {0} & \tilde {\mathbf {B}} _ {N} & \mathbf {0} \\ \mathbf {0} & \mathbf {0} & \tilde {\mathbf {B}} _ {1} & \mathbf {0} & \dots & \mathbf {0} & \mathbf {0} & \tilde {\mathbf {B}} _ {N} \end{array} \right] \\ \tilde {\mathbf {B}} _ {i} = \left\{N _ {i, x} \quad N _ {i, y} \quad N _ {i, z} \right\} ^ {T} \end{array} \tag {4.7.11} +$$ + +식 (4.7.4-5)와 (4.7.9-10)을 식 (4.1.18)에 대입하여 정리하면 선형화된 평형 방 + + + +정식을 얻을 수 있다. + +$$ +\delta \mathbf {u} ^ {T} \left(^ {t} \mathbf {K} _ {L} ^ {e} + ^ {t} \mathbf {K} _ {N L} ^ {e}\right) \Delta \mathbf {u} = \delta \mathbf {u} ^ {T} \left(^ {t + \Delta t} \mathbf {f} _ {e x t} ^ {e} - ^ {t} \mathbf {f} _ {\text { int }} ^ {e}\right) \tag {4.7.12} +$$ + +식 (4.7.12)의 각 항은 다음과 같다. + +$$ +{ } ^ { t } \mathbf { K } _ { L } ^ { e } = \int _ { V _ { e } } \mathbf { B } _ { L } ^ { T } \mathbf { D } \mathbf { B } _ { L } d V +$$ + +$$ +{ } ^ { t } \mathbf { K } _ { N L } ^ { e } = \int _ { V _ { e } } { } ^ { t } \mathbf { B } _ { N L } ^ { T } { } ^ { t } \hat { \mathbf { S } } ^ { t } \mathbf { B } _ { N L } d V \tag {4.7.13} +$$ + +$$ +{ } ^ { t } \mathbf { f } _ { \text { i n t } } ^ { e } = \int _ { V _ { e } } { } ^ { t } \mathbf { B } _ { L } ^ { T } { } ^ { t } \mathbf { S } d V +$$ + +여기서, + +$$ +{ } ^ { t } \hat { \mathbf { S } } = \left[ \begin{array} { c c c } { } ^ { t } \mathbf { S } & \mathbf { 0 } & \mathbf { 0 } \\ \mathbf { 0 } & { } ^ { t } \mathbf { S } & \mathbf { 0 } \\ \mathbf { 0 } & \mathbf { 0 } & { } ^ { t } \mathbf { S } \end{array} \right] \quad { } ^ { t } \mathbf { S } = \left[ \begin{array} { c c c } { } ^ { t } S _ { x x } & { } ^ { t } S _ { x y } & { } ^ { t } S _ { z x } \\ { } ^ { t } S _ { x y } & { } ^ { t } S _ { y y } & { } ^ { t } S _ { y z } \\ { } ^ { t } S _ { z x } & { } ^ { t } S _ { y z } & { } ^ { t } S _ { z z } \end{array} \right] \tag {4.7.14} +$$ + +입체요소의 해석 결과로는 선형 해석과 같이 절점 응력과 변형률이 있으며, 절점에서의 결과 이외에 적분점에서의 응력과 변형률을 표를 통해 볼 수 있다. 적분 차수는 다음과 같다. + +• 4절점 4면체 : 1 점 가우스 적분 +• 6절점 5면체 : 6 점 가우스 적분 +• 8절점 6면체 : 8 점 가우스 적분 +• 10절점 4면체 : 4 점 가우스 적분 +• 15절점 5면체 : 9 점 가우스 적분 +• 20절점 6면체 : 27 점 가우스 적분 + + + +Part 1 Element Library +![](images/page-206_0e36928321c838c323b9bfdaeb9fd1da27f9b93a497afd0dc2d7e3f3659dda75.jpg) + +
+natural_image + +Blank white image with no visible content, text, or symbols +
+ + + +# Analysis and Algorithm Manual + +# Part 2 Material Library + +Chapter 1. Material Models + +Chapter 2. Total Strain Crack + +Chapter 3. Interface Nonlinearities + + + + + +# Chapter 1. Material Models + +# 1-1 개요 + +# 1-1-1 서론 + +재료의 소성거동은 탄성거동과 달리 재하된 하중을 제거하여도 구조물에 영구변형이발생한다. 이러한 거동적 특성을 나타내기 위해 변형률은 다음과 같이 탄성과소성성분으로 나누는 변형분리가정에 따라 정식화한다. + +$$ +\boldsymbol {\varepsilon} = \boldsymbol {\varepsilon} ^ {\mathrm{e}} + \boldsymbol {\varepsilon} ^ {\mathrm{p}} \tag {1.1.1} +$$ + +여기서, + +ε : 총 변형률 + +$\pmb { \varepsilon } ^ { \mathrm { e } }$ : 탄성 변형률 (elastic strain) + +$\mathbf { \varepsilon } _ { \mathbf { \varepsilon } _ { \mathbf { \varepsilon } } } ^ { \mathrm { p } }$ : 소성 변형률 (plastic strain) + +후크의 법칙(Hook’s law)은 탄성범위에서의 변형과 응력의 관계를 정의하며, 이를식 (1.1.1)에 적용하면 응력은 다음과 같이 정의할 수 있다. + +$$ +\boldsymbol {\sigma} = \mathbf {D} \boldsymbol {\varepsilon} ^ {\mathrm{e}} = \mathbf {D} \left(\boldsymbol {\varepsilon} - \boldsymbol {\varepsilon} ^ {\mathrm{p}}\right) \tag {1.1.2} +$$ + +여기서, + +σ : 응력벡터 + +D : 재료강성 행렬 (material stiffness matrix) + +하중을 받는 구조물 내 임의의 한 점에 발생하는 응력은 그 점의 탄-소성상태를정의하는 척도가 된다. 탄성 및 소성을 정의하는 이러한 기준은 강재나 콘크리트등 재료의 특성에 따라 다르게 정의되며, 이를 항복기준(yield criterion)이라 한다.재료의 항복기준은 다양한 응력상태에 대한 실험을 통해 정의되며, 소성흐름(plastic flow)을 유발하는 시점의 응력값들은 응력공간 상의 함수형태로 다음과 같이 나타낼 수 있다. + + + +$$ +f (\boldsymbol {\sigma}, \kappa) = 0 \tag {1.1.3} +$$ + +여기서, + +f : 항복함수 (yield function) + +κ : 경화변수 (hardening parameter) + +항복함수 f 가 0보다 작을 경우 소성흐름은 발생하지 않으며, f 가 0보다 클 경우 소성흐름이 발생한다. + +# 1-1-2 소성흐름법칙(Plastic flow rule) + +재료의 향복은 소성흐름을 유발하며 이러한 소성흐름은 재료의 평형상태를 유지하기위해 응력의 재분배를 유발한다. 이러한 소성흐름의 계산은 비선형 형태로 이루어지며 이를 위해 일반적으로 증분형태를 사용하여 정식화된다. 재료의 탄소성해석에서 소성흐름의 계산에 사용되는 일반적인 값들 중 대표적인 것은 증분응력방향, 소성변형률방향이며 이중 증분응력방향은 다음과 같다. + +$$ +\mathbf {n} _ {\mathrm{i}} = \frac {\partial f _ {\mathrm{i}}}{\partial \boldsymbol {\sigma}} \tag {1.1.4} +$$ + +여기서, n 은 항복면에 수직한 증분응력의 방향을 나타내는 기울기벡터를 나타내며 i 는 항복함수가 여러 개로 이루어진 경우의 항복함수개수를 의미한다. + +증분소성변형률은 코이터의 법칙(Koiter's law)에 따라 다음과 같이 크기와 방향성분으로 분리하여 나타낸다. + +$$ +\dot {\pmb {\varepsilon}} _ {\mathrm{p}} = \sum_ {\mathrm{i} = 1} ^ {\mathrm{n}} \dot {\lambda} _ {\mathrm{i}} \frac {\partial \mathbf {g} _ {\mathrm{i}}}{\partial \pmb {\sigma}} = \sum_ {\mathrm{i} = 1} ^ {\mathrm{n}} \dot {\lambda} _ {\mathrm{i}} \mathbf {m} _ {\mathrm{i}} \tag {1.1.5} +$$ + +여기서, $g_{i}$ 는 소성포텐셜함수(plastic potential function)로써 응력과 경화변수 $\kappa$ 에 의한 함수 $g_{i}(\sigma,\kappa)$ 이며 일반적으로 재료실험을 통해 얻어진다. $\dot{\lambda}_{i}$ 는 소성승수(plastic multiplier)이며, 다음과 같이 쿠-터커 조건(Kuhn-Tucker condition)을 만족해야 한다. diff --git a/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_022.md b/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_022.md new file mode 100644 index 00000000..405c488d --- /dev/null +++ b/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_022.md @@ -0,0 +1,373 @@ + + +$$ +f \leq 0 \quad ; \quad \dot {\lambda} _ {1} \geq 0 \quad ; \quad \dot {\lambda} _ {1} f = 0 \tag {1.1.6} +$$ + +위의 조건으로부터 항복함수 f 가 0보다 작은 경우 $\dot{\lambda}_{i}$ 은 항상 0이 되며, 소성흐름이 발생하지 않음을 알 수 있다. + +식 (1.1.6)에서 m 은 증분소성변형률의 방향을 정의하는 벡터이다. 이때 소성포텐 셔함수 g 를 사용하지 않고 항복함수 f 를 사용한 $\partial f/\partial \sigma$ 으로 증분소성변형률 방향을 정의하는 방법을 상관소성흐름법칙(associated flow rule)이라 하며, 소성포텐 셔함수를 사용하여 $\partial g/\partial \sigma$ 으로 증분소성변형률방향을 정의하는 방법을 비상관소성흐름법칙(non-associated flow rule)이라 한다. 일반적으로 von Mises나 Tresca 모델과 같이 응력공간 상에서 등압축(hydrostatic axis)을 따라 일정한 편차응력(deviatoric stress)의 분포를 나타내는 형태의 재료모델인 경우는 상관소성흐름법칙을 사용하는 것이 일반적이다. + +그러나 Mohr-Coulomb과 Drucker-Prager 모델과 같이 응력공간 상에서 편차응력이 등압축을 따라 변하는 형태의 재료모델인 경우 비상관소성흐름법칙을 사용한다. 등압축에 따라 편차응력이 변화하는 재료모델에 비상관소성흐름법칙을 적용하는 경우 응력방향과 변형률방향의 불일치로 인하여 발생되는 과도한 체적팽창현상을 억제하는 효과가 있다. 그러나 강성행렬이 비대칭이 되어 비대칭 솔버를 이용해서 계산해야 하기 때문에, 수렴속도가 느려지는 단점이 있다. 따라서 일반적인 경우 상관소성흐름법칙을 사용한다. + +예외적으로 콘크리트에 띠철근이나 강관 또는 강박스에 의한 구속현상이 크게 발생되는 경우 구속효과가 소성해석에 민감하게 작용할 수 있으므로, 이 때는 비상관소성흐름해석을 사용할 것을 권장한다. + +# 1-1-3 경화거동(Hardening behavior) + +Elasto-plastic모델에서 경화거동(hardening behavior)을 정의하는 방법으로는 변형경화가정(strain hardening hypothesis)과 소성일경화가정(plastic work hardening hypothesis)의 두 가지 방법이 있다. 변형경화가정은 등가소성변형률이 증가함에 따라 경화가 진행된다고 가정하는 방법이며 소성일경화가정은 소성일이 + + + +증가함에 따라 경화가 진행된다고 가정하는 방법이다. + +변형경화에 사용되는 경화변수 $\kappa$ 는 무차원의 등가소성변형률(equivalent plastic strain) $\overline{\varepsilon^{p}}$ 를 사용함으로써 다음과 같이 정의할 수 있다. + +$$ +\overline {{{\varepsilon}}} ^ {\mathrm{p}} = \kappa = \sqrt {\frac {2}{3} \left(\boldsymbol {\varepsilon} ^ {\mathrm{p}}\right) ^ {\mathrm{T}} \mathbf {Q} \boldsymbol {\varepsilon} ^ {\mathrm{p}}} \tag {1.1.7} +$$ + +여기서, + +$$ +\boldsymbol {\varepsilon} ^ {\mathrm{p}} = \left\{ \begin{array}{l} \varepsilon_ {x x} ^ {\mathrm{p}} \\ \varepsilon_ {y y} ^ {\mathrm{p}} \\ \varepsilon_ {z z} ^ {\mathrm{p}} \\ \tau_ {x y} ^ {\mathrm{p}} \\ \tau_ {y z} ^ {\mathrm{p}} \\ \tau_ {z x} ^ {\mathrm{p}} \end{array} \right\}, \quad \mathbf {Q} = \left[ \begin{array}{c c c c c c} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac {1}{2} & 0 & 0 \\ 0 & 0 & 0 & 0 & \frac {1}{2} & 0 \\ 0 & 0 & 0 & 0 & 0 & \frac {1}{2} \end{array} \right] +$$ + +이때 일반적으로 경화변수는 주소성변형률로부터 계산하는 경우가 편리하기 때문에 많이 쓰이며 이때의 수식은 다음과 같다. + +$$ +\overline {{{\varepsilon}}} ^ {\mathrm{p}} = \kappa = \sqrt {\frac {2}{3} \varepsilon^ {\mathrm{p}} \varepsilon^ {\mathrm{p}}} \tag {1.1.8} +$$ + +여기서, + +$$ +\boldsymbol {\varepsilon} ^ {p} = \left\{ \begin{array}{l} \varepsilon_ {1} ^ {\mathrm{p}} \\ \varepsilon_ {2} ^ {\mathrm{p}} \\ \varepsilon_ {3} ^ {\mathrm{p}} \end{array} \right\} +$$ + +소성일경화는 응력단위의 소성일 $W^{p}$ (plastic work)를 경화변수 $\kappa$ 로 사용하여 경화거동을 다음과 같이 정의한다. + +$$ +W ^ {p} = \kappa = \boldsymbol {\sigma} ^ {\mathrm{T}} \boldsymbol {\varepsilon} ^ {\mathrm{p}} \tag {1.1.9} +$$ + +위 두 가지 형태의 경화 모두 시간 t 에 대해 적분함으로써 현재 하중단계까지의 경화변수를 계산한다. + +$$ +\kappa = \int \dot {\kappa} d t \tag {1.1.10} +$$ + + + +경화변수 κ 를 사용하여 표현되는 경화거동은 항복면의 거동적 특성에 따라 등방경화(isotropic hardening), 이동경화(kinematic hardening), 그리고 두 가지 경화의 조합인 혼합경화(mixed hardening)로 분류된다. 등방경화는 그림 1.1.1(a)와 같이 경화변수에 따라 항복면이 등방팽창 또는 등방수축하는 거동을 보이며, 이동경화는 그림 1.1.1(b)와 같이 항복면의 팽창이나 수축없이 항복면의 원점이 이동하는거동을 보인다. 혼합경화는 그림 1.1.1(c)와 같이 위의 두 거동을 조합한 거동적 특징을 갖는다. 등방경화는 Rankine 모델, Mohr-Coulomb 모델, Drucker-Prager모델 등과 같은 취성재료에 사용되며 이동경화모델이나 혼합경화모델은 vonMises와 같은 연성거동모델에 사용된다. + +midas FEA에서는 현재 변형경화가정을 사용한 등방경화기능만을 지원하고 있다. + +![](images/page-213_f8051616e8b8cd9f914a3f1d4f51eec6485d7c08b795980bc70654e1e0a569f0.jpg) + +
+text_image + +σ₁ +P₁ +P₂ +P₃ +O +σ₂ +σ₃ +
+ +(a) 등방경화 + +![](images/page-213_cc4a89f7912cbf20fef4decb7618c22fbdc22e3a865fb018dd9f5698a2587f9d.jpg) + +
+text_image + +σ₁ +P +O₂ +P +O₁ +σ₂ +σ₃ +P +O₃ +
+ +(b) 이동경화 +![](images/page-213_eeed7bceb1f98c2a2077db90f501d4a68efbbfd9889d5213195c1b7f86426e86.jpg) + +
+text_image + +σ₁ +P₂ +P₃ +O₃ +P₁ +O +Q₁ +σ₂ +σ₃ +
+ +(c) 혼합경화 +그림 1.1.1 함복면의 경화거동 + + + +# 1-1-4 선형화된 적합 조건(Linearized consistency condition) + +탄-소성 물질의 내적변화 상태는 미소증가 형태로 표현되는 점증적 구성관계에 의하여 정의된다. 소성흐름은 항복조건에 도달할 때 시작되며, κ와 같은 소성상태변수에 의해 제어된다. + +비상관소성흐름에서 미소변형상태에 대한 구성관계식은 식 (1.1.2)에 기초하여 다음과 같이 나타낼 수 있다. + +$$ +\dot {\boldsymbol {\sigma}} = \mathbf {E} \left(\dot {\boldsymbol {\varepsilon}} - \dot {\boldsymbol {\varepsilon}} ^ {\mathrm{p}}\right) = \mathbf {E} \left(\dot {\boldsymbol {\varepsilon}} - \dot {\lambda} \mathbf {m}\right) \tag {1.1.11} +$$ + +응력상태가 항복면상에 놓일 때 선형화된 일관성조건(linearized consistency condition)은 함수를 테일러 급수(Taylor series)의 1차 확장을 통해 다음과 같이 정의한다. + +$$ +\dot {f} (\boldsymbol {\sigma}, \kappa) = \left(\frac {\partial f}{\partial \boldsymbol {\sigma}}\right) ^ {\mathrm{T}} \dot {\boldsymbol {\sigma}} + \frac {\partial f}{\partial \kappa} \frac {\partial \kappa}{\partial \lambda} \dot {\lambda} = \mathbf {n} ^ {\mathrm{T}} \dot {\boldsymbol {\sigma}} - h \dot {\lambda} = 0 \tag {1.1.12} +$$ + +여기서, $n=\frac{\partial f}{\partial\sigma}$ , h 는 경화계수(hardening modulus)라고 하며, $h=-\frac{\partial f}{\partial\kappa}\frac{\partial\kappa}{\partial\lambda}$ 이다. +이때 $\partial f/\partial\kappa$ 는 실험으로부터 $f-\kappa$ 관계를 도출하여 계산한다. $\partial\kappa/\partial\lambda$ 는 재료의 +향복모델에 따라 다르며 1.2절에서 항복모델별로 정의된 $\kappa-\lambda$ 관계식을 사용하여 +계산한다. + +위의 식 (1.1.12)에 식 (1.1.11)을 대입하여, 소성승수 $\dot{\lambda}$ 에 대하여 정리하면 다음과 같다. + +$$ +\dot {\lambda} = \frac {\mathbf {n} ^ {\mathrm{T}} \mathbf {E}}{h + \mathbf {n} ^ {\mathrm{T}} \mathbf {E} \mathbf {m}} \dot {\varepsilon} \tag {1.1.13} +$$ + +식 (1.1.13)을 식 (1.1.11)에 대입하면 응력과 변형률의 증분형태관계식으로 나타낼 수 있다. + +$$ +\dot {\boldsymbol {\sigma}} = \left(\mathbf {E} - \frac {\mathbf {E} \mathbf {m} \mathbf {n} ^ {\mathrm{T}} \mathbf {E}}{h + \mathbf {n} ^ {\mathrm{T}} \mathbf {E} \mathbf {m}}\right) \dot {\boldsymbol {\varepsilon}} = \mathbf {E} ^ {\mathrm{ep}} \dot {\boldsymbol {\varepsilon}} \tag {1.1.14} +$$ + +여기서, $E^{ep}$ 는 탄-소성 접선 연산자로써 재료의 접선강성행렬이다. 이때 비상관소성 + + + +흐름법칙을 도입하면 m n ≠ 이므로, ep E 는 비대칭행렬이 된다. + +# 1-1-5 증분방정식의 적분(Integration of rate form) + +증분형태 방정식에 대한 적분방법에는 크게 명시적 전방 오일러 방법(explicitforward euler algorithm with sub-incrementation)과 암시적 후방 오일러 방법(implicit backward euler algorithm)으로 나뉜다. + +명시적 전방 오일러 방법에서 소성흐름의 방향은 증분응력과 항복면이 만나는 A점(그림 1.1.2와 1.1.3)에서 계산되지만, 암시적 후방 오일러 방법에서는 최종 응력 지점인 B점(그림 1.1.4)에서 계산된다. + +명시적 전방 오일러 방법은 상대적으로 단순하고 적분점 (integration point)에서반복계산과정이 필요하지 않지만, 다음과 같은 단점이 있다. + +-. 조건에 따라서만 안정적이다. +-. 허용 가능한 정확도를 위해 부증분(sub-increment)이 필요하다. (그림 1.1.3). +-. 항복면으로 되돌리기 위한 추가적 보정이 필요하다. (그림 1.1.3). + +또한, 명시적 전방 오일러 방법은 적합 강성행렬(consistent tangent stiffnessmatrix)을 구성할 수 없다. 암시적 후방 오일러 방법은 부증분이나 인위적 회귀 없이 충분히 정확한 결과를 도출하고 조건에 관계없이 안정적이지만, 적분점에서 반복계산이 필요하다. 그러나 이 방법을 사용하면 적합 강성행렬(consistent tangentstiffness matrix)을 구성할 수 있기 때문에 명시적 전방 오일러 방법보다 효율적이다. + + + +![](images/page-216_78ba4ad7561fbc8e1872908f666e35075fc3017c93866147d2571df0bb40b32f.jpg) + +
+text_image + +Δσₑ +A +B +X +Yield surface +
+ +(a) 교차 점 A의 위치 + +![](images/page-216_de77fcf03935c85230036e29654bc03b063f33dea75678c4fb4f4747b63778ad.jpg) + +
+text_image + +(1-r)Δσe +rΔσe +-ΔλmA +A +B +C +D +X +Yield surface +
+ +(b) A에서 접선방향으로 C로 이동한 후 D위치로 보정 +그림 1.1.2 명시적 전방 오일러 방법 + + + +![](images/page-217_9213b3cf0a537d0565086e3e0d6122dadce50250bdfb9a1b4f3432599eac8100.jpg) + +
+text_image + +-ΔλA mA +-ΔλB mB +-ΔλB mB' +A +B +B' +C +D +X +Yield surface +
+ +그림 1.1.3 명시적 전방 오일러 방법에서의 부증분 + +# 1-1-6 명시적 전방 오일러 방법(Explicit forward Euler method) + +명시적 전방 오일러 방법에서는 먼저 탄성 변형을 가정한 탄성 증분응력을 산정한다. (그림 1.1.2(a)의 B점). + +$$ +\Delta \boldsymbol {\sigma} ^ {\mathrm{e}} = \mathbf {E} \Delta \varepsilon \tag {1.1.15} +$$ + +$$ +\pmb {\sigma} _ {\mathrm{B}} = \pmb {\sigma} _ {\mathrm{X}} + \Delta \pmb {\sigma} ^ {\mathrm{e}} +$$ + +다음으로 탄성한계를 정의하는 증분응력량을 계산한다. 초기 탄성 증분응력은 탄성범위 내의 증분응력인 허용 증분 응력 ( ) e 1− r ∆σ 과 허용 불가능한 항복함수 바깥쪽의 증분응력 e r∆σ 으로 나뉜다. 탄성한계를 정의하는 증분응력은 다음 식을이용하여 계산한다(그림 1.1.2(a)의 A점). + +$$ +\begin{array}{c} f \left(\boldsymbol {\sigma} _ {X} + (1 - r) \Delta \boldsymbol {\sigma} ^ {e}, \kappa\right) = 0 \\ f _ {r} \end{array} \tag {1.1.16} +$$ + +$$ +r = \frac {f _ {\mathrm{B}}}{f _ {\mathrm{B}} - f _ {\mathrm{X}}} +$$ + +식 (1.1.14)와 식 (1.1.15)의 첨자는 그림 1.1.2를 참조. + +부증분방법을 사용하기 위해서는 항복면을 벗어난 증분응력 e r∆σ 를 k개의 작은부증분응력으로 나누어 근사화한다(그림 1.1.3). 부증분의 개수는 오차의 크기에 직접적으로 관계되며, 식 (1.1.17)과 같이 계산한다. + +$$ +k = \operatorname{INT} \left[ 8 \left(\sigma_ {\text { effB }} - \sigma_ {\text { effA }}\right) / \sigma_ {\text { effA }} \right] + 1 \tag {1.1.17} +$$ + + + +여기서, $\sigma _ { \mathrm { e f f A } }$ 와 $\sigma _ { \mathrm { e f f B } }$ 는 그림 1.1.2(a)의 A점과 B점에서의 유효응력을 나타낸다.최종 응력상태가 항복면상에 있지 않을 경우, 다음의 인위적 회귀 방법을 사용하여 항복면상으로 응력상태가 옮겨지도록 한다 (그림 1.1.3의 D점). + +$$ +\Delta \lambda_ {\mathrm{C}} = \frac {f _ {\mathrm{C}}}{h + \mathbf {n} _ {\mathrm{C}} ^ {\mathrm{T}} \mathbf {E} \mathbf {m} _ {\mathrm{C}}} \tag {1.1.18} +$$ + +$$ +\boldsymbol {\sigma} _ {\mathrm{D}} = \boldsymbol {\sigma} _ {\mathrm{C}} - \Delta \lambda_ {\mathrm{C}} \mathbf {E} \mathbf {m} _ {\mathrm{C}} +$$ + +# 1-1-7 암시적 후방 오일러 방법(Implicit backward Euler method) + +![](images/page-218_42c3765179bd545b5c3f398e8aed76e30763b7edfda2d587dd93d8669114cb01.jpg) + +
+text_image + +Δσ^e +B +-Δλm_B +C +D +X +Yield surface +
+ +그림 1.1.4 암시적 후방 오일러 방법 + +명시적 전방 오일러 방법은 교차점 A에서의 항복면에 수직한 방향성분을 이용하여다음 응력값을 추정하기 때문에 교차점의 계산이 반드시 요구된다. 만약 항복면에수직한 방향성분을 그림 1.1.4와 같이 B점에서 추정한다면 교차점의 계산이 불필요해진다. + +$$ +\boldsymbol {\sigma} _ {\mathrm{C}} = \boldsymbol {\sigma} _ {\mathrm{X}} + \Delta \boldsymbol {\sigma} ^ {\mathrm{e}} - \Delta \lambda \mathbf {E} \mathbf {m} _ {\mathrm{B}} = \boldsymbol {\sigma} _ {\mathrm{B}} - \Delta \lambda \mathbf {E} \mathbf {m} _ {\mathrm{B}} \tag {1.1.19} +$$ + +이에 따라 항복함수 f 를 테일러급수를 사용하여 B점에 대해 1차항까지 확장하면 + + + +다음과 같다. + +$$ +f = f _ {\mathrm{B}} + \left(\frac {\partial f}{\partial \boldsymbol {\sigma}}\right) ^ {\mathrm{T}} \Delta \boldsymbol {\sigma} + \frac {\partial f}{\partial \kappa} \Delta \kappa = f _ {\mathrm{B}} - \Delta \lambda \mathbf {n} _ {\mathrm{B}} ^ {\mathrm{T}} \mathbf {E} \mathbf {m} _ {\mathrm{B}} - h _ {\mathrm{B}} \Delta \lambda \tag {1.1.20} +$$ + +새로이 계산된 점에서의 항복함수값은 0이 되며 위의 수식을 Δλ 에 대해 다음과 같이 정리할 수 있다. + +$$ +\Delta \lambda = \frac {f _ {\mathrm{B}}}{h _ {\mathrm{B}} + \mathbf {n} _ {\mathrm{B}} ^ {\mathrm{T}} \mathbf {E} \mathbf {m} _ {\mathrm{B}}} \tag {1.1.21} +$$ + +이를 사용하여 암시적 후방 오일러 방법에서는 $\sigma_{c}$ 를 다음과 같이 계산한다. + +$$ +\boldsymbol {\sigma} _ {\mathrm{C}} = \boldsymbol {\sigma} _ {\mathrm{B}} - \Delta \lambda \mathbf {E} \mathbf {m} _ {\mathrm{B}} \tag {1.1.22} +$$ + +이 때 C점에서의 응력 $\sigma_{c}$ 는 항상 항복면상에 존재하지 않으며, 이를 처리하기 위해 C점을 새로운 응력값의 추정을 위한 기준점으로 하여 위의 계산을 반복수행한다. 이러한 계산은 응력값이 항복면 위에 존재할 때까지 반복된다. + +# 1-1-8 구성행렬(constitutive matrix) + +소성 구성방정식(plastic constitutive equation)을 구성하는 방법은 다음과 같다. +미소 증분응력은 미소 증분변형률 벡터의 탄성 부분에 의하여 결정된다. 즉, + +$$ +\dot {\boldsymbol {\sigma}} = \mathbf {E} \left(\dot {\boldsymbol {\varepsilon}} - \dot {\boldsymbol {\varepsilon}} ^ {\mathrm{p}}\right) = \mathbf {E} \dot {\boldsymbol {\varepsilon}} - \dot {\lambda} \mathbf {E} \mathbf {m} \tag {1.1.23} +$$ + +현재 응력은 항상 항복면 상에 위치해야 하기 때문에 적합조건(consistency condition)을 만족해야 한다. 식 (1.1.23)에 식 (1.1.13)을 대입하고 이를 미소 증분 변형률에 대하여 정리하면, 미소 증분응력은 식 (1.1.24)와 같이 계산할 수 있다. + +$$ +\dot {\boldsymbol {\sigma}} = \left(\mathbf {D} - \frac {\mathbf {D m n} ^ {\mathrm{T}} \mathbf {D}}{h + \mathbf {n} ^ {\mathrm{T}} \mathbf {D m}}\right) \dot {\boldsymbol {\varepsilon}} = \mathbf {D} ^ {\mathrm{ep}} \dot {\boldsymbol {\varepsilon}} \tag {1.1.24} +$$ + +명시적 전방 오일러 방법에서 뉴튼 략슨(Newton-Raphson) 반복과정이 사용될 때 적합접선강성행렬(consistent tangent stiffness matrix)을 사용하면, 뉴튼 략슨 반복과정의 2차 수렴 특성으로 인하여 더욱 빠른 수렴해를 얻을 수 있다. 이러한 2 + + + +차 특성을 고려하기 위하여 식 (1.1.20)을 미분하면 다음과 같다. + +$$ +\dot {\boldsymbol {\sigma}} = \mathbf {D} \dot {\boldsymbol {\varepsilon}} - \dot {\lambda} \mathbf {D} \mathbf {m} - \Delta \lambda \mathbf {D} \frac {\partial \mathbf {m}}{\partial \boldsymbol {\sigma}} \dot {\boldsymbol {\sigma}} - \Delta \lambda \mathbf {D} \frac {\partial \mathbf {m}}{\partial \kappa} \frac {\partial \kappa}{\partial \lambda} \dot {\lambda} \tag {1.1.25} +$$ + +여기서, $\dot{\lambda}$ 은 $\Delta\lambda$ 의 변화량이다. + +식 (1.1.25)는 다음과 같이 정리할 수 있다. + +$$ +\mathbf {A} \dot {\boldsymbol {\sigma}} = \mathbf {E} \dot {\boldsymbol {\varepsilon}} - \dot {\lambda} \mathbf {E} \overline {{\mathbf {m}}} \tag {1.1.26} +$$ + +여기서, $A = I + \Delta \lambda E \frac{\partial m}{\partial \sigma}$ , $\bar{m} = m + \Delta \lambda \frac{\partial m}{\partial \kappa} \frac{\partial \kappa}{\partial \lambda}$ + +이때 $H = A^{-1}E$ 라하면 식 (1.1.26)은 다음과 같이 정리할 수 있다. + +$$ +\dot {\boldsymbol {\sigma}} = \mathbf {H} \left(\dot {\boldsymbol {\varepsilon}} - \dot {\lambda} \overline {{\mathbf {m}}}\right) \tag {1.1.27} +$$ + +식 (1.1.27)을 선형화된 적합조건(linearized consistency condition)을 사용하여 전체 변형률항으로 정리하면 다음의 식을 얻을 수 있다. + +$$ +\dot {\boldsymbol {\sigma}} = \left(\mathbf {H} - \frac {\mathbf {H} \overline {{\mathbf {m}}} \mathbf {n} ^ {\mathrm{T}} \mathbf {H}}{h + \mathbf {n} ^ {\mathrm{T}} \mathbf {H} \overline {{\mathbf {m}}}}\right) \dot {\boldsymbol {\varepsilon}} = \mathbf {C} ^ {\mathrm{ep}} \dot {\boldsymbol {\varepsilon}} \tag {1.1.28} +$$ + +식 (1.1.24)에서 $D^{ep}$ 는 접선강성행렬(tangent stiffness matrix)이라 하며, 식 (1.1.28)의 $C^{ep}$ 은 적합접선강성행렬이라 한다. + +midas FEA의 제한사항 + +Isotropic Plasticity Only + +Isotropic Hardening Only + +Strain Hardening Only diff --git a/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_023.md b/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_023.md new file mode 100644 index 00000000..e0140a7a --- /dev/null +++ b/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_023.md @@ -0,0 +1,383 @@ + + +# 1-2 항복기준(Yield Criteria) + +일반적으로 사용되는 다축상태의 응력성분으로 항복함수를 나타내는 데에는 기하학적이나 물리학적으로 많은 어려움이 따른다. 따라서 응력좌표축에 독립적인 성분을 사용하여 항복함수를 나타내는 것이 일반적이며, 다음과 같이 주응력을 사용하여 항복조건을 정의한다. + +$$ +f \left(\sigma_ {1}, \sigma_ {2}, \sigma_ {3}\right) = 0 \tag {1.2.1} +$$ + +항복함수를 표현하는 하나의 편리한 방법은 응력불변량을 사용하는 것이다. + +# 1-2-1 응력 불변량(stress invariants) + +# ■ 응력 불변량 + +재료 내 임의의 점에 발생하는 응력은 응력텐서 σ ij 를 사용하여 나타내며, 주응력방향을 정의하는 방향벡터 jn 를 사용하면 다음의 식이 성립된다. + +$$ +\left(\sigma_ {i j} - \sigma \delta_ {i j}\right) n _ {j} = 0 \tag {1.2.2} +$$ + +여기서, ij δ 은 크로네커 델타(Kronecker delta)이다. + +위의 식 (1.2.2)에서 0 jn ≠ 이며, 식 (1.2.2)를 만족하기 위한 필요충분조건은 다음과 같다. + +$$ +\left| \sigma_ {i j} - \sigma \delta_ {i j} \right| = 0 \tag {1.2.3} +$$ + +행렬식 (1.2.3)는 주응력에 대한 3차방정식으로 나타낼 수 있으며 다음과 같다. + +$$ +\sigma^ {3} - I _ {1} \sigma^ {2} + I _ {2} \sigma - I _ {3} = 0 \tag {1.2.4} +$$ + +여기서, + + + +$$ +I _ {1} = \sigma_ {x} + \sigma_ {y} + \sigma_ {z} = \sigma_ {i i} +$$ + +$$ +I _ {2} = \left(\sigma_ {x} \sigma_ {y} + \sigma_ {y} \sigma_ {z} + \sigma_ {z} \sigma_ {x}\right) - \left(\tau_ {x y} ^ {2} + \tau_ {y z} ^ {2} + \tau_ {z x} ^ {2}\right) = \frac {1}{2} \left(I _ {1} ^ {2} - \sigma_ {i j} \sigma_ {j i}\right) \tag {1.2.5} +$$ + +$$ +I _ {3} = \left| \begin{array}{c c c} \sigma_ {x} & \tau_ {x y} & \tau_ {x z} \\ \tau_ {y x} & \sigma_ {y} & \tau_ {y z} \\ \tau_ {z x} & \tau_ {z y} & \sigma_ {z} \end{array} \right| = \frac {1}{3} \sigma_ {i j} \sigma_ {j k} \sigma_ {k i} - \frac {1}{2} I _ {1} \sigma_ {i j} \sigma_ {j i} + \frac {1}{6} I _ {1} ^ {3} +$$ + +1I , 2I , 3I 를 주응력 σ1 , σ 2 , σ 3 을 사용하여 나타내면 다음과 같다. + +$$ +I _ {1} = \sigma_ {1} + \sigma_ {2} + \sigma_ {3} +$$ + +$$ +I _ {2} = \sigma_ {1} \sigma_ {2} + \sigma_ {2} \sigma_ {3} + \sigma_ {3} \sigma_ {1} \tag {1.2.6} +$$ + +$$ +I _ {3} = \sigma_ {1} \sigma_ {2} \sigma_ {3} +$$ + +# ■ 편차응력 불변량 + +응력텐서 $\sigma _ { i j }$ 는 정수압과 편차응력성분으로 나누어 다음과 같이 나타낼 수 있다. + +$$ +\sigma_ {i j} = s _ {i j} + \sigma_ {m} \delta_ {i j} \tag {1.2.7} +$$ + +여기서, $\sigma _ { { } _ { m } } = \left( \sigma _ { { } _ { x } } + \sigma _ { { } _ { y } } + \sigma _ { { } _ { z } } \right) / 3 = I _ { 1 } / 3$ 이며, 평균응력을 의미한다. 그리고$s _ { i j } = \sigma _ { i j } - \sigma _ { m } \delta _ { i j }$ 는 편차응력이라 하며, 순수전단상태를 나타낸다. + +식 (1.2.3)에서 주응력에 대한 불변량을 계산한 것처럼, 편차응력에 대한 불변량을계산하기 위해서는 다음과 같은 수식을 풀어야 한다. + +$$ +\left| s _ {i j} - s \delta_ {i j} \right| = 0 \tag {1.2.8} +$$ + +식 (1.2.8)를 방정식형태로 나타내면 다음과 같다. + +$$ +s ^ {3} - J _ {1} s ^ {2} + J _ {2} s - J _ {3} = 0 \tag {1.2.9} +$$ + +여기서, + + + +$$ +\begin{array}{l} J _ {1} = s _ {i i} = s _ {x} + s _ {y} + s _ {z} = 0 \\ J _ {2} = \frac {1}{2} s _ {i j} s _ {j i} \\ = \frac {1}{6} \left[ \left(\sigma_ {x} - \sigma_ {y}\right) ^ {2} + \left(\sigma_ {y} - \sigma_ {z}\right) ^ {2} + \left(\sigma_ {z} - \sigma_ {x}\right) ^ {2} \right] + \tau_ {x y} ^ {2} + \tau_ {y z} ^ {2} + \tau_ {z x} ^ {2} \tag {1.2.10} \\ \end{array} +$$ + +$$ +J _ {3} = \frac {1}{3} S _ {i j} S _ {j k} S _ {k i} = \left| \begin{array}{c c c} s _ {x} & \tau_ {x y} & \tau_ {x z} \\ \tau_ {y x} & s _ {y} & \tau_ {y z} \\ \tau_ {z x} & \tau_ {z y} & s _ {z} \end{array} \right| +$$ + +1J , 2J , 3J 를 편차주응력 $s _ { 1 }$ , $s _ { 2 }$ , $s _ { 3 }$ 로 나타내면 다음과 같다. + +$$ +\begin{array}{l} J _ {1} = s _ {1} + s _ {2} + s _ {3} = 0 \\ J _ {2} = \frac {1}{2} \left(s _ {1} ^ {2} + s _ {2} ^ {2} + s _ {3} ^ {2}\right) = \frac {1}{6} \left[ \left(\sigma_ {1} - \sigma_ {2}\right) ^ {2} + \left(\sigma_ {2} - \sigma_ {3}\right) ^ {2} + \left(\sigma_ {3} - \sigma_ {1}\right) ^ {2} \right] \tag {1.2.11} \\ \end{array} +$$ + +$$ +J _ {3} = \frac {1}{3} \left(s _ {1} ^ {3} + s _ {2} ^ {3} + s _ {3} ^ {3}\right) = s _ {1} s _ {2} s _ {3} +$$ + +1I , 2I , 3I , 1J , 2J , 3J 는 모두 스칼라로 표현되는 불변량으로써 좌표축에 독립적인 특성을 갖는다. 이중 항복함수를 기하학적으로 편리하게 나타내기 위하여 $I _ { 1 }$ ,$J _ { 2 \mathrm { ~ , ~ } } J _ { 3 }$ 의 세 불변량을 주로 사용하며 $I _ { 1 } \equiv 1 \bar { \pi } \mathsf { k }$ 불변량, $J _ { 2 }$ 는 2차 불변량, $J _ { 3 }$ 는3차 불변량이라 한다. + +# ■ 세 응력 불변량이 가지는 기하학적 의미 + +대부분의 재료모델들에서 항복은 편차응력에 의해 주로 지배된다. 따라서 항복함수를 정수압성분과 편차응력성분으로 나누어 나타내는 것은 항복함수의 기하학적형상을 정의하는데 매우 편리하게 사용된다. + +다음은 임의의 응력상태를 나타내는 한 점 $\mathbf { P } \left( \sigma _ { 1 } , \sigma _ { 2 } , \sigma _ { 3 } \right) \mathbf { \equiv }$ 등압축과 편차축 성분으로 나누어 표현하는 방법에 대해 설명한다. + + + +![](images/page-224_a6fb215ab96b2f2b0b0b57213c890cd461c75c37145c91123b2c06a35ec0c7d0.jpg) + +
+text_image + +σ₁ +P(σ₁, σ₂, σ₃) +θ₀ +r +ξ +N +σ₁ = σ₂ = σ₃ +e +O +σ₃ +σ₂ +
+ +그림 1.2.1 주응력공간에서의 응력상태 정의 + +그림 1.2.1에서와 같이 주응력 공간상에 임의의 응력상태로 표현되는 점$P ( \sigma _ { 1 } , \sigma _ { 2 } , \sigma _ { 3 } )$ 가 정의되는 경우 벡터 OP 를 정의할 수 있다. 벡터 OP 는 정수압축을 따르는 벡터 ON 과 정수압축에 수직인 편차평면상에 존재하는 벡터 NP 로나누어 질 수 있으며, 그 크기는 다음과 같다. + +$$ +\left| \mathbf {O N} \right| = \xi = \frac {1}{\sqrt {3}} I _ {1} \tag {1.2.12} +$$ + +$$ +\left| \mathbf {N P} \right| = r = \sqrt {2 J _ {2}} +$$ + +![](images/page-224_71ae198a544f90b27097a345d55085fab5268022ef4aded26e74c2df41c6e262.jpg) + +
+text_image + +σ₁ +P(σ₁, σ₂, σ₃) +θ₀ +r +N +σ₂ +σ₃ +
+ +그림 1.2.2 편차평면상에서의 응력상태 정의 + + + +그림 1.2.2는 등압축에 수직인 평면인 편차평면을 나타낸다. 앞에서 정의된 벡터NP 는 편차편차평면 상에서 점 P 를 정의하기 위해 $\sigma _ { 1 } \bar { \Xi } \underline { { { Q } } }$ 로부터 $\theta _ { 0 }$ 만큼 회전되어야 한다. 이때 $\theta _ { 0 } \equiv$ 상이각(similarity angle)이라 하며 다음과 같다. + +$$ +\theta_ {0} = \frac {1}{3} \cos^ {- 1} \left(\frac {3 \sqrt {3}}{2} \frac {J _ {3}}{J _ {2} ^ {3 / 2}}\right) \tag {1.2.13} +$$ + +이때 $\theta _ { 0 } \cong \mathtt { L } | \frac { \circ } { \boxtimes } \frac { \circ } { }$ 범위를 갖는다. + +$$ +0 \leq \theta_ {0} \leq \frac {\pi}{3} \tag {1.2.14} +$$ + +수치해석을 위해서는 $\theta _ { 0 }$ 보다는 로데의 각(Lode’s angle) θ 를 사용하는 것이 편리하며 다음과 같이 정의한다. + +$$ +\theta = \frac {1}{3} \sin^ {- 1} \left(- \frac {3 \sqrt {3}}{2} \frac {J _ {3}}{J _ {2} ^ {3 / 2}}\right) \tag {1.2.15} +$$ + +이때 $\theta = \theta _ { 0 } - \frac { \pi } { 6 }$ + +$$ +- \frac {\pi}{6} \leq \theta \leq \frac {\pi}{6} \tag {1.2.16} +$$ + +재료의 항복함수를 정의하는데에는 종종 주응력을 응력불변수로 나타내는 것이 편리할 때가 있으며 로데의 각을 사용하여 이를 정리하면 다음과 같다. + +$$ +\left\{ \begin{array}{l} \sigma_ {1} \\ \sigma_ {2} \\ \sigma_ {3} \end{array} \right\} = \frac {2 \sqrt {J _ {2}}}{\sqrt {3}} \left\{ \begin{array}{l} \sin \left(\theta + \frac {2}{3} \pi\right) \\ \sin (\theta) \\ \sin \left(\theta + \frac {4}{3} \pi\right) \end{array} \right\} + \frac {I _ {1}}{3} \left\{ \begin{array}{l} 1 \\ 1 \\ 1 \end{array} \right\} \tag {1.2.17} +$$ + + + +# 1-2-2 Rankine 모델 + +Rankine 모델은 취성재료의 인장거동을 정의하는데 사용되는 모델로서 토목분야에서는 주로 암석의 인장에 의한 파괴거동이나 콘크리트의 인장균열 또는 고체요소의 압축전담거동을 정의할 때 사용된다. Rankine 모델은 외력에 의한 최대 주응력(maximum principal stress) σ1 과 실험을 통해 정의된 인장강도(tensilestrength) σ t 를 사용하여 재료의 항복을 정의하는 모델로서 최대 주응력 항복기준(maximum principal stress criterion)이라고도 하며 최대 주응력이 인장강도를 초과하는 경우 항복이 진행된다고 가정하여 항복함수를 다음과 같이 나타낸다. + +$$ +f (\boldsymbol {\sigma}, \kappa) = \sigma_ {1} - \sigma_ {t} (\kappa) = 0 \tag {1.2.18} +$$ + +Rankine 모델은 주로 인장절단(tension cutoff)거동을 묘사하는데 사용되며, 뒤에서 거론될 전단거동을 정의하는 Mohr-Coulomb이나 Drucker-Prager 모델과 함께 복합항복모델로 사용되는 것이 일반적이다. + +식 (1. 2.18)을 불변량 1 2I J , , θ 및 경화변수 κ 를 사용하여 나타내면 다음과 같다. + +$$ +f \left(I _ {1}, J _ {2}, \theta , \kappa\right) = \frac {2}{\sqrt {3}} \sqrt {J _ {2}} \left(\sin \left(\theta + \frac {2}{3} \pi\right)\right) + \frac {I _ {1}}{3} - \sigma_ {t} (\kappa) = 0 \tag {1.2.19} +$$ + +또한 식 (1.2.18)을 ξ, , , r θ κ 의 항으로 나타내면 다음과 같다. + +$$ +f (\xi , r, \theta , \kappa) = \frac {2}{\sqrt {6}} r \left(\sin \left(\theta + \frac {2}{3} \pi\right)\right) + \frac {\xi}{\sqrt {3}} - \sigma_ {t} (\kappa) = 0 \tag {1.2.20} +$$ + +그림 1.2.3은 응력 공간상에서 Rankine 모델의 3차원 형상을 보여준다. 그림 1.2.4에서와 같이 π 평면 상에서의 형상은 정삼각형 형상이며, 메리디안 평면(meridianplane)상에서는 등압축(hydrostatic axis)에 대한 선형함수로 정의할 수 있다. + +# ■ 경화거동 + +Rankine 모델의 경화거동을 정의하기 위한 등가소성변형률의 계산에 사용되는 주소성변형률은 다음과 같다. + + + +$$ +\boldsymbol {\varepsilon} ^ {\mathrm{p}} = \lambda \mathbf {m} = \lambda \left\{ \begin{array}{l} 1 \\ 0 \\ 0 \end{array} \right\} \tag {1.2.21} +$$ + +식 (1.2.21)을 식 (1.1.8)에 대입하여 정리하면 다음과 같이 경화변수와 소성승수의관계를 구할 수 있다. + +$$ +\kappa = \sqrt {\frac {2}{3}} \lambda \tag {1.2.22} +$$ + +midas FEA에서는 Rankine 모델에 대해 인장응력 σ t 에 대한 경화거동을 지원하며, 경화거동은 다중선형함수를 사용하여 정의할 수 있다. + +![](images/page-227_7857b905c1dd384bf955cd4f8a01c481409bfd2bba1bc4ccb04954ea4bc3a23f.jpg) + +
+text_image + +-σ₁ +-σ₃ +-σ₂ +
+ +그림 1.2.3 주응력공간에서의 Rankine 항복면 형상 + + + +![](images/page-228_7868e17301a8388ed0e5976a1ccf5c53a46bcaac481bdf80d89cdafa410a3c39.jpg) + +
+text_image + +σ₁ +θ +r +rₜ +r꜀ +σ₂ +σ₃ +
+ +(a) π 평면에서의 항복면 형상 + +![](images/page-228_54358e2291b015869307d2e4bbd216a4d5a5b82e2148a13d63afac78e589c0ae.jpg) + +
+text_image + +θ = -π/6 +1 +√2 +r_t = √(3/2) f_t' +√3 f_t' +hydrostatic axis +θ = π/6 +1 +√2 +r_c = √6 f_t' +
+ +(b) $\theta = - { \frac { \pi } { 6 } }$ +그림 1.2.4 π 평면과 메리디안 평면에서의 Rankine 항복면 형상 + +# 1-2-3 Tresca 모델 + +Tresca 모델은 연성재료인 금속재료의 탄-소성 비선형 거동을 묘사하기 위해서개발되었으며, 지반에서는 연약지반의 전단강도를 정의하기 위한 φ = 0 해석에도사용된다. Tresca는 외력에 의해 물체의 한 점에서 발생하는 최대 전단응력이 재료의 전단강도 τ (κ ) 에 도달하였을 때 항복이 발생되는 것으로 간주하여 항복함수를 다음과 같이 정의한다. + +$$ +f (\sigma , \kappa) = \frac {1}{2} \left| \sigma_ {1} - \sigma_ {3} \right| - \tau_ {y} (\kappa) = 0 \tag {1.2.23} +$$ + +여기서, $\sigma _ { 1 } \geq \sigma _ { 2 } \geq \sigma _ { 3 }$ 이다. 식 (1.2.23)의 양변에 2를 곱하여 응력불변량 $J _ { 2 } , \theta$ 의항으로 나타내면 다음과 같다. + +$$ +f \left(J _ {2}, \theta\right) = \frac {2}{\sqrt {3}} \sqrt {J _ {2}} \left[ \sin \left(\theta + \frac {2}{3} \pi\right) - \sin \left(\theta + \frac {4}{3} \pi\right) \right] - 2 \tau_ {y} (\kappa) \tag {1.2.24} +$$ + +$$ += 2 \sqrt {J _ {2}} \cos \theta - \sigma_ {y} (\kappa) = 0 +$$ + +여기서, $\sigma _ { y } ( \kappa ) = 2 \tau _ { y } \left( \kappa \right)$ 이며, 일축항복응력을 나타낸다. + + + +또한 식 (1.2.23)을 불변량 r, θ 로 나타내면 다음과 같다. + +$$ +f (r, \theta) = \sqrt {2} r \cos \theta - \sigma_ {y} (\kappa) = 0 \tag {1.2.25} +$$ + +Tresca 모델에서는 항복면에 작용하는 정수압의 영향을 고려하지 않으므로, 식(1.2.24)에서와 같이 정수압을 정의하는 응력 불변수 I 에 무관하며, 그림 1.2.6(a)와 같이 편차평면 상에서 로데의 각 θ에 따라 편차응력 r 의 크기가 달라지는 형태를 가지게 된다. 정수압에 무관하다는 의미는 등방적으로 동일한 하중이 작용되는 경우 즉, 삼축 등인장이나 삼축 등압축이 작용하는 경우에는 하중의 크기가 무한대로 커진다고 할지라도 재료가 항복하지 않는다는 것을 뜻한다. 이는 정수압축과 항복면이 만나지 않는다는 것을 뜻하며, 이에 따라 Tresca 항복기준은 그림1.2.5와 같이 주응력 공간에서 정수압축에 평행한 정육면체 기둥이 되고, 편차평면에서는 정육각형으로 묘사된다. 그림 1.2.6은 π 평면과 메리디안 평면에서의 항복면 형상을 보여준다. + +# ■ 경화거동 + +Tresca 모델에서 소성변형률은 다음과 같다. + +$$ +\boldsymbol {\varepsilon} ^ {\mathrm{p}} = \lambda \mathbf {m} = \lambda \left\{ \begin{array}{l} 1 \\ 0 \\ - 1 \end{array} \right\} \tag {1.2.26} +$$ + +위의 식 (1.2.26)을 식 (1.1.8)에 대입하면, 경화를 정의하기 위한 소성승수 λ와 경화변수 κ 의 관계는 다음과 같이 정의할 수 있다. + +$$ +\kappa = \frac {2}{\sqrt {3}} \lambda \tag {1.2.27} +$$ + +midas FEA에서는 Tresca 모델에 대해 항복응력 σ y 에 대한 경화거동을 지원하며,경화거동은 다중선형함수를 사용하여 정의할 수 있다. + + + +![](images/page-230_d90f1d45b6a95e01281ef6e284326d81198af3292ad6fc2dba0b1c575518d772.jpg) + +
+text_image + +-σ₁ +hydrostatic axis +-σ₂ +-σ₃ +
+ +그림 1.2.5 주응력공간에서의 Tresca 항복면 형상 + +![](images/page-230_84e61f7b1961d0de0d5d9d16995175b2981cc29bf777aaf3c3084f8d53c7c60b.jpg) + +
+text_image + +σ₁ +θ +r_c +r_t +σ₂ +σ₃ +
+ +(a) π 평면에서의 항복면 형상 + +![](images/page-230_927050cd0d64ec21f3c99e4dbd48e0de7dfa94a6f13251a2ad918c9d1283caf1.jpg) + +
+text_image + +deviatoric axis +r_c +θ = -π/6 +hydrostatic axis +r_t +θ = π/6 +
+ +(b) $\theta = - { \frac { \pi } { 6 } }$ +그림 1.2.6 π 평면과 메리디안 평면에서의 Tresca 항복면 형상 diff --git a/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_024.md b/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_024.md new file mode 100644 index 00000000..e95bba52 --- /dev/null +++ b/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_024.md @@ -0,0 +1,347 @@ + + +# 1-2-4 Von Mises 모델 + +Von Mises 모델은 연성재료인 금속재료의 해석에서 가장 많이 사용되는 모델로서,정팔면체 전단응력 $\tau _ { o c t }$ 가 순수전단에서의 항복응력 $\tau _ { y }$ 에 도달했을 때 항복이 일어난다는 가정을 적용하여 다음과 같이 수식화한다. + +$$ +f \left(\tau_ {o c t}\right) = \tau_ {o c t} - \sqrt {\frac {2}{3}} \tau_ {y} = 0 \tag {1.2.28} +$$ + +일축압축의 경우를 고려하여 응력불변수를 사용하여 나타내면 다음과 같다. + +$$ +f \left(J _ {2}, \kappa\right) = \sqrt {3 J _ {2}} - \sigma_ {y} (\kappa) = 0 \tag {1.2.29} +$$ + +여기서, $\sigma _ { y } \left( \kappa \right) \in ~ \sigma _ { y } \left( \kappa \right) = \sqrt { 3 \tau _ { y } }$ 이며, 일축거동에 대한 항복응력을 나타낸다. + +가장 많이 사용되는 형태는 식 (1.2.29)이며, 응력불변량 $J _ { 2 }$ 에만 종속적인 모델로서 $J _ { 2 }$ theory라고도 불린다. Von Mises 항복면은 그림 1.2.7과 같이 주응력공간에서 정수압축에 평행한 원주형상을 나타낸다. 만약, von Mises와 Tresca 기준을압축과 신장자오선 즉, ( / 6) cr θ π = − 및 ( / 6) tr θ π = 에서 서로 일치시키면, 편차평면에서의 von Mises 곡면은 Tresca의 육각형을 외접하는 원이 된다. (그림1.2.8(a)). + +이 경우 예상되는 항복응력의 최대차는 순수전단자오선 ( 0) θ = 을 따라서 일어나며,von Mises와 Tresca 기준의 항복전단응력의 비는 $2 / \sqrt { 3 } = 1 . 1 5$ 이다. 다른 한편으로 만약 두 기준을 순수전단자오선에 일치시키면 von Mises원은 Tresca 육각형을내접하게 되며, 두 기준간의 최대 예상오차는 압축자오선 ( / 6) θ = −π 과 신장자오선$\left( \theta = \pi / 6 \right)$ 을 따라 생긴다(그림 1.1.8(b)). + + + +![](images/page-232_06ed7ede4ec3b2c072aadd6f165c7afd893b9be50d4713672b5cce85112d9a21.jpg) + +
+text_image + +-σ₁ +hydrostatic axis +-σ₃ +-σ₂ +
+ +그림 1.2.7 주응력공간에서의 von Mises 항복면 형상 + +![](images/page-232_f97d1c68a2e65fdb00c4349c4e9e7139a914937095e90ed05ef4d3f794275f6c.jpg) + +
+text_image + +Von mises +Tresca +θ +rc +ri +σ₁ +σ₂ +σ₃ +
+ +(a) π 평면에서 외접한 경우 Tresca와의 관계 + +![](images/page-232_11558c63f70214e0ca2e153a75a255d30ac2452437ad7b768a46edf6a9cc1bf2.jpg) + +
+text_image + +Tresca +Von mises +θ +r +σ₁ +σ₂ +σ₃ +
+ +(b) π 평면에서 내접한 경우 Tresca와의 관계 + + + +![](images/page-233_61062f757048f70d1a2738f8aa35a94e64467278287a000c2efaa88f54b1aaa0.jpg) + +
+text_image + +deviatoric axis +r_c +θ = - π/6 +hydrostatic axis +r_t +θ = π/6 +
+ +(c) $\theta = - { \frac { \pi } { 6 } }$ +그림 1.2.8 π 평면과 메리디안 평면에서의 von Mises 항복면 형상 + +# ■ 경화거동 + +Von Mises 모델의 변형경화거동을 정의하기 위한 소성 주 변형률은 다음과 같다. + +$$ +\boldsymbol {\varepsilon} ^ {\mathrm{p}} = \lambda \mathbf {m} = \lambda \frac {1}{2 \sigma_ {e}} \left\{ \begin{array}{c c c} 2 \sigma_ {1} & - \sigma_ {2} & - \sigma_ {3} \\ - \sigma_ {1} & + 2 \sigma_ {2} & - \sigma_ {3} \\ - \sigma_ {1} & - \sigma_ {2} & + 2 \sigma_ {3} \end{array} \right\} \tag {1.2.30} +$$ + +여기서, + +$$ +\sigma_ {e} = \sqrt {\frac {1}{2} \pmb {\sigma} ^ {T} \mathbf {P} \pmb {\sigma}} +$$ + +$$ +\boldsymbol {\sigma} = \left\{ \begin{array}{l} \sigma_ {x x} \\ \sigma_ {y y} \\ \sigma_ {z z} \\ \tau_ {x y} \\ \tau_ {y z} \\ \tau_ {z x} \end{array} \right\}, \quad \mathbf {P} = \left[ \begin{array}{c c c c c c} 2 & - 1 & - 1 & 0 & 0 & 0 \\ - 1 & 2 & - 1 & 0 & 0 & 0 \\ - 1 & - 1 & 2 & 0 & 0 & 0 \\ 0 & 0 & 0 & 6 & 0 & 0 \\ 0 & 0 & 0 & 0 & 6 & 0 \\ 0 & 0 & 0 & 0 & 0 & 6 \end{array} \right] +$$ + +윗 식을 적절히 정리하면 다음과 같다. + + + +$$ +\kappa = \lambda \tag {1.2.31} +$$ + +midas FEA에서는 von Mises 모델에 대해 항복응력 σ y 에 대한 경화거동을 지원하며, 경화거동은 다중선형함수를 사용하여 정의할 수 있다. + +# 1-2-5 Mohr-Coulomb 모델 + +Mohr(1900)의 기준에 의하면 재료의 항복은 다음과 같이 정의된다. + +$$ +\left| \tau \right| = F (\sigma) \tag {1.2.32} +$$ + +여기서, 임의 평면에서의 한계전단응력 τ 는 동일평면상의 수직응력 σ 에만 관계된다고 가정한다. 식 (1.2.32)는 대응하는 Mohr원의 항복면을 나타내며 항복함수F( ) σ 는 실험으로 결정되는 함수이다. Mohr의 기준에 의하면 재료의 항복은 가장큰 Mohr원이 Coulomb의 항복면에 접하는 순간 발생된다고 가정한다. 이것은 중간주응력 $\sigma _ { { } _ { 2 } } ( \sigma _ { { } _ { 1 } } \geq \sigma _ { { } _ { 2 } } \geq \sigma _ { { } _ { 3 } } )$ 가 항복조건에 영향을 미치지 않는다는 것을 의미한다.Coulomb 항복면의 가장 간단한 형상은 직선이며 이 직선 포락선의 방정식은 다음과 같다. + +$$ +\left| \tau \right| = c + \sigma \tan \phi \tag {1.2.33} +$$ + +여기서, c, φ = 재료의 강도변수 + +$$ +c = \text { 점착력 } +$$ + +$$ +\phi = \text { 내부마찰각 } +$$ + +식 (1.2.33)의 항복기준을 Mohr-Coulomb 기준이라 하며, 간단하고 정확성이 높은장점 때문에 압축응력에 따라 전단강도가 달라지는 재료모델에 현재까지 널리 사용되고 있다. + +Mohr-Coulomb 식을 주응력의 항으로 나타내면, 다음과 같이 정리할 수 있다. + +$$ +\sigma_ {1} \frac {(1 + \sin \phi)}{2 c \cos \phi} - \sigma_ {3} \frac {(1 - \sin \phi)}{2 c \cos \phi} = 1 \tag {1.2.34} +$$ + + + +식 (1.2.34)를 1 2I J, 및 θ 의 항으로 나타내면 다음과 같다. + +$$ +f \left(I _ {1}, J _ {2}, \theta\right) = - \frac {1}{3} I _ {1} \sin \phi + \sqrt {J _ {2}} \left(\cos \theta + \frac {1}{\sqrt {3}} \sin \theta \sin \phi\right) - c \cos \phi = 0 \tag {1.2.35} +$$ + +Mohr-Coulomb 기준은 그림 1.2.9에서 보는 바와 같이 주응력공간에서 불규칙 육각형 피라미드 형상이고, 자오선은 직선이다. 그리고 π 평면 $( \sigma _ { 1 } + \sigma _ { 2 } + \sigma _ { 3 } = 0 )$ 상의편차형상은 그림 1.2.10(a)와 같이 불규칙 육각형이 된다. 불규칙 육각형을 그리기위해서는 θ = −π / 6 에서의 메리디안 평면에 대한 그림 1.2.10(b)에서와 같이 $r _ { t 0 }$ 와$r _ { c 0 }$ 의 길이가 필요하며, 이들 길이는 다음과 같다. + +$$ +r _ {t 0} = \frac {2 \sqrt {6} c \cos \phi}{3 + \sin \phi} \tag {1.2.36} +$$ + +$$ +r _ {c 0} = \frac {2 \sqrt {6} c \cos \phi}{3 - \sin \phi} \tag {1.2.37} +$$ + +식 (1.2.36)과 식 (1.2.37)로부터 $r _ { t 0 } / r _ { c 0 }$ 는 다음과 같다. + +$$ +\frac {r _ {t 0}}{r _ {c 0}} = \frac {3 - \sin \phi}{3 + \sin \phi} \tag {1.2.38} +$$ + +등압축을 따라 Mohr-Coulomb 항복면의 편차평면형상은 모두 기하학적으로 닮은꼴이기 때문에 임의의 편차평면에 대한(즉, I 또는 ξ 의 다른 값에 대한) $r _ { t } / r _ { c }$ 비는 항상 일정하게 유지된다. + +$$ +\frac {r _ {t}}{r _ {c}} = \frac {r _ {t 0}}{r _ {c 0}} = \frac {3 - \sin \phi}{3 + \sin \phi} \tag {1.2.39} +$$ + +또한 과도한 체적팽창현상을 제어할 목적으로 비상관소성흐름법칙을 적용할 수 있으며, 이를 위해 소성포텐셜함수는 내부마찰각 φ 대신 팽창각 ψ 를 사용하여 다음과 같이 정의할 수 있다. + +$$ +g \left(I _ {1}, J _ {2}, \theta\right) = - \frac {1}{3} I _ {1} \sin \psi + \sqrt {J _ {2}} \left(\cos \theta + \frac {1}{\sqrt {3}} \sin \theta \sin \psi\right) \tag {1.2.40} +$$ + + + +![](images/page-236_6f4f348090a62dd59e55c4e60d3bf2c324144de40421589dd8a65e171190395a.jpg) + +
+text_image + +-σ₁ +hydrostatic axis +-σ₃ +-σ₂ +
+ +그림 1.2.9 주응력공간에서의 Mohr-Coulomb 항복면 형상 + +![](images/page-236_a37aca018fec57ff0862098462f2f2d6d3d58fcb0c3c31867401be53b62d873a.jpg) + +
+text_image + +σ₁ +θ +r₀ +r₀ +x +σ₂ +
+ +(a) π 평면 항복면 형상 + +![](images/page-236_a0f04694831b4229a2285f02260e86e4f09975920a98eda6f40ac560a11f69f9.jpg) + +
+text_image + +θ = -π/6 +r₁₀ = 2√6c cosφ/3 + sinφ +√3c cotφ +hydrostatic axis +r_c0 = 2√6c cosφ/3 - sinφ +σ₃ θ = π/6 +
+ +(b) $\theta = - { \frac { \pi } { 6 } }$ +그림 1.2.10 π 평면과 메리디안 평면에서의 Mohr-Coulomb 항복면 형상 + + + +# ■ 경화거동 + +Morh-Coulomb 모델에서 소성변형률은 다음과 같다. + +$$ +\boldsymbol {\varepsilon} ^ {\mathrm{p}} = \lambda \mathbf {m} = \lambda \left\{ \begin{array}{l} \frac {1}{2} (1 + \sin \psi) \\ 0 \\ - \frac {1}{2} (1 - \sin \psi) \end{array} \right\} \tag {1.2.41} +$$ + +위의 식 (1.2.41)을 식 (1.1.8)에 대입하면, 경화를 정의하기 위한 소성승수 λ 와 경화변수 κ 의 관계는 다음과 같이 정의된다. + +$$ +\kappa = \lambda \sqrt {\frac {1}{3} \left(1 + \sin^ {2} \psi\right)} \tag {1.2.42} +$$ + +midas FEA에서는 Mohr-Coulomb 모델에 대해 점착력 c , 내부마찰각 φ , 팽창각수 있다. + + + +# 1-2-6 Drucker-Prager 모델 + +Drucker-Prager 모델은 Drucker와 Prager(1952)가 von Mises 모델을 수정하여확장시킨 파괴 기준으로써 실무 문제에 널리 적용되고 있으며, 확장 von Mises(extended von Mises)기준이라고도 불리운다. Drucker-Prager모델의 항복함수와소성포텐셜함수를 응력 불변량 $I _ { 1 }$ 및 $J _ { 2 }$ 의 항으로 나타내면 다음과 같다. + +$$ +f \left(I _ {1}, J _ {2}\right) = \sqrt {3 J _ {2}} - \alpha I _ {1} - \beta c = 0 \tag {1.2.43} +$$ + +$$ +g \left(I _ {1}, J _ {2}\right) = \sqrt {3 J _ {2}} - \gamma I _ {1} +$$ + +여기서, $\alpha = \frac { 2 \sin \phi } { 3 - \sin \phi } ~ , ~ \beta = \frac { 6 \cos \phi } { 3 - \sin \phi } ~ , ~ \gamma = \frac { 2 \sin \psi } { 3 - \sin \psi }$ + +이때, 재료상수 α 와 k 는 응력상태를 일치시킴으로써, Mohr-Coulomb 기준의 점착력과 내부마찰각인 c 와 φ 에 관련지어 나타낼 수 있다. α 가 ‘0(zero)’이면 식(1.2.43)은 von Mises 항복기준으로 환원된다. 소성포텐셜함수 g 의 β 값을 정의하기 위해서는 팽창각 ψ 값이 추가로 필요하다. + +Drucker-Prager 항복면을 주응력공간에 나타내면 그림 1.2.11과 같다. 이 항복면은 공간대각선(정수압면 응력축, $\sigma _ { 1 } = \sigma _ { 2 } = \sigma _ { 3 }$ )을 축으로 하는 정원추형 형상을 나타낸다. + +Drucker-Prager 항복면은 콘크리트나 지반재료의 거동에 맞추어 von Mises 항복면을 정수압에 따라 변화하도록 확장한 것으로 생각할 수 있다. + + + +![](images/page-239_48c4042e195ce5dc605b87a91fdee471f6b254dfcac885476e4da79f6e64f624.jpg) + +
+text_image + +-σ₁ +hydrostatic axis +-σ₃ +-σ₂ +
+ +그림 1.2.11 주응력공간에서의 Drucker-Prager 항복면 형상 + +![](images/page-239_d2669f4fd259d3df6758145eb22bb70b392ffa0c85bad8f73fefb9470ba64cf6.jpg) + +
+text_image + +σ₁ +θ +r +r₀ +σ₂ +σ₃ +
+ +(a) π 평면에서의 항복면 형상 + +![](images/page-239_4c1f1d32be3dbcf53d36b89ef6537cbf6f34121db29de75103119629ba1fa2ae.jpg) + +
+text_image + +θ = - π/6 +√6α +1 +r₀ = √2βc +hydrostatic axis +βc/√3α +r₀ = √2βc +θ = π/6 +√6α +1 +
+ +(b) 메리디안 평면에서의 항복면 형상 +그림 1.2.12 π 평면과 메리디안 평면에서의 Drucker-Prager 항복면 형상 + + + +# ■ 경화거동 + +Drucker-Prager 모델에서 소성변형률은 다음과 같다. + +$$ +\boldsymbol {\varepsilon} ^ {\mathrm{p}} = \lambda \mathbf {m} = \lambda \left(\frac {1}{2 \sigma_ {e}} \left\{ \begin{array}{l l l} 2 \sigma_ {1} & - \sigma_ {2} & - \sigma_ {3} \\ - \sigma_ {1} & + 2 \sigma_ {2} & - \sigma_ {3} \\ - \sigma_ {1} & - \sigma_ {2} & + 2 \sigma_ {3} \end{array} \right\} + \gamma \left\{ \begin{array}{l} 1 \\ 1 \\ 1 \end{array} \right\}\right) \tag {1.2.44} +$$ + +여기서, + +$$ +\sigma_ {e} = \sqrt {\frac {1}{2} \boldsymbol {\sigma} ^ {T} \mathbf {P} \boldsymbol {\sigma}} +$$ + +$$ +\boldsymbol {\sigma} = \left\{ \begin{array}{l} \sigma_ {x x} \\ \sigma_ {y y} \\ \sigma_ {z z} \\ \tau_ {x y} \\ \tau_ {y z} \\ \tau_ {z x} \end{array} \right\}, \quad \mathbf {P} = \left[ \begin{array}{c c c c c c} 2 & - 1 & - 1 & 0 & 0 & 0 \\ - 1 & 2 & - 1 & 0 & 0 & 0 \\ - 1 & - 1 & 2 & 0 & 0 & 0 \\ 0 & 0 & 0 & 6 & 0 & 0 \\ 0 & 0 & 0 & 0 & 6 & 0 \\ 0 & 0 & 0 & 0 & 0 & 6 \end{array} \right] +$$ + +위의 식 (1.2.44)를 식 (1.1.8)에 대입하면, 경화를 정의하기 위한 소성승수 λ와 경화변수 κ 의 관계는 다음과 같이 정의된다. + +$$ +\kappa = \lambda \sqrt {1 + 2 \gamma^ {2}} \tag {1.2.45} +$$ + +midas FEA에서는 Drucker-Prager 모델에 대해 점착력 c , 내부마찰각 φ , 팽창각 ψ 에 대한 경화거동을 지원하며, 각 경화거동은 다중선형함수를 사용하여 정의할 수 있다. diff --git a/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_025.md b/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_025.md new file mode 100644 index 00000000..bd69e1a9 --- /dev/null +++ b/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_025.md @@ -0,0 +1,253 @@ + + +# Chapter 2. Total Strain Crack + +# 2-1 개요 + +# 2.1.1 콘크리트 균열모델 + +콘크리트의 균열에 대한 해석 방법은 그림 2.1.1과 같이 이산균열모델(discretecrack model)과 분산균열모델(smeared crack model)로 구분할 수 있다. 이산균열모델은 균열을 경계로 분리된 유한요소를 사용하는 방법이며, 분산균열모델은균열이 분산 분포된 것으로 가정하여 균열 위치에 분리된 요소를 사용하지 않는방법이다. + +![](images/page-241_d3cdff7be71c6228b17f28e2ae437ce2b9dc4b949874e1aa2385883ec2f99950.jpg) + +
+text_image + +Discrete Crack +fct σnσ +Gf w +Smeared Crack +σnσ +fct σnσ +Gf/h εnσ +
+ +그림 2.1.1 콘크리트 균열 모델 + +이산균열모델은 콘크리트 균열에 의한 물리적인 불연속 및 철근의 파단과 미끄러짐 등의 거동을 구체적으로 모사할 수 있는 장점이 있다. 그러나 해석에 필요한 물성치에 따라 해석의 정밀도가 크게 좌우되거나, 유한요소 모델링이 상당히 복잡하다는 단점이 있다. 모델링 방법으로는 균열 발생 위치에서 요소를 자동으로 분할하게 하거나, 균열이 예상되는 부위에 계면 요소(interface element)를 미리 추가하는 방법 등이 있다. + +분산균열모델은 국부적으로 발생하는 균열이 넓은 면에 고르게 분산된 것으로 가정하는 방법을 사용한다. 일반적으로 철근이 많이 배근된 철근 콘크리트 구조물의해석에 적합하다고 알려져 있으며, 유한요소 모델링이 비교적 간단하다. 분산균열 + + + +모델은 발생 균열의 각도를 가정하는 방법에 따라 균열 방향이 서로 직교하는 직교 균열(orthogonal crack)모델과 균열 방향이 서로 직교하지 않는 비직교 균열(non-orthogonal crack)모델로 구분하기도 한다. 또한 균열에 대한 수치적 해석방법에 따라 변형률 분해 모델(decomposed-strain model), 전변형률 모델(totalstrain model) 및 기타 다양한 모델로 구분된다. + +분산균열모델에서의 변형률분해 모델은 전변형률을 재료변형률(material strain)과균열변형률(crack strain)로 분리하여 계산을 수행한다. 재료변형률은 선형변형률(elastic strain), 소성변형률(plastic strain), 크리프변형률(creep strain), 열변형률(thermal strain) 등을 포함할 수 있기 때문에 다양한 확장성을 가지고 있다. 그리고 균열 변형률은 다른 각도를 가진 여러 개의 균열변형률을 포함할 수 있어 비직교 다방향균열모델(non-othogonal multi-directional crack model)로의 확장도가능하다. 그러나 알고리즘이 복잡하고, 물성치 선정이 어려우며, 수렴 성능이 낮아질 수 있는 단점이 있다. + +분산균열모델에서의 전변형률 모델은 변형률 성분을 분리하지 않고 전변형률을 사용하므로 비교적 간단한 방법으로 수식화할 수 있다. 또한 균열을 포함한 인장 및압축 거동 모두에 대하여 각각 한가지만의 응력-변형률 관계를 사용하기 때문에알고리즘의 이해가 쉽다. 그리고 비선형 거동을 정의하기 위한 물성치의 입력이비교적 단순하여 실무에서 사용하기 쉬운 장점이 있다. + + + +# 2.1.2 전변형률 균열모델 + +midas FEA에서는 분산균열모델의 전변형률 균열모델(total strain crack model)을 사용하고 있다. 그리고 그림 2.1.2와 같이 균열 축을 취급하는 방법에 따라 고정균열모델(fixed crack model) 및 회전균열모델(rotating crack model)로 구분되는 두 가지 방법을 제공한다. 전자는 균열 축이 한번 결정되면 변화하지 않는 것으로 가정하는 방법이며, 후자는 주변형률의 변화에 따라 균열 방향이 계속해서 회전한다고 가정하는 방법이다. + +![](images/page-243_5f4e26dde7cccfd9cc82dea7956e96b44ae07bbe954a8613b3defee68834f5c7.jpg) + +
+text_image + +σc2 +τ +ε1 +ε2 +τ +σc1 +y +x +
+ +(a）fixed crack model + +![](images/page-243_c9fa4a91aa1e50e465ff8be16b57dfa2b109afd85ede4870ebad3029f9d101c1.jpg) + +
+text_image + +σc2 ε2 +σc1 ε1 +y +x +
+ +(b）rotating crack model +그림 2.1.2 직교균열모델 + +고정균열모델과 회전균열모델 모두 적분점에서의 첫번째 균열은 항상 주변형률 방향에서 발생한다. 콘크리트는 균열 발생 전에는 등방성(isotropic)의 특성을 가지지만 균열 발생 이후에는 이방성(anisotropic) 특징을 가지게 된다. midas FEA에서는 균열 이후 콘크리트의 특성을 직교이방성(orthotropic)의 재료로 취급하며 균열면에서 수직응력(normal stress)과 전단응력(shear stress)을 산정한다. 고정균열모델에서는 초기 균열의 방향이 변하지 않는 것으로 가정하기 때문에 그림2.1.2(a)와 같이 균열면에 수직응력과 전단응력이 존재한다. 그러나 회전균열모델에서는 전단계에서 발생한 균열을 무시하고, 현재의 주변형률 방향에서 새로운 균열이 발생하는 것으로 가정한다. 따라서 그림 2.1.2(b)와 같이 균열 면에서 수직응 + + + +력만이 발생한다. midas FEA의 고정균열모델과 회전균열모델에서는 균열각이 수직인 경우만을 고려하기 때문에 직교균열모델로 분류할 수 있다. + +고정균열모델은 균열 현상에 대한 물리적 특성을 구체적으로 반영할 수 있는 있지만 직교균열모델인 경우 비직교균열모델에 비하여 강성과 강도를 약간 과대 평가하는 경향이 있다. 반면에 회전균열모델은 이전의 균열 상태를 기억할 필요가 없으므로 알고리즘이 비교적 단순하며 수렴성도 좋다. 이러한 장점으로 인하여 회전균열모델은 철근콘크리트 구조물의 비선형 해석 방법으로 오랫동안 사용되어 왔다.전변형률을 바탕으로 한 구성 모델(constitutive model)은 Vecchio와 Collins1 가제안한 수정압축장 이론(modified compression field theory)에 이론적인 바탕을두고 있다. 이 이론은 기본적으로 2차원 기반이다. 따라서 3차원으로의 확장 모델은 Selby와 Vecchio2 가 제안한 이론에 준하여 프로그램을 구현하였다. + +1 Vecchio, F. J., and Collins, M. P. “The modified compression field theory for reinforced concrete elements subjected to shear”. ACI Journal 83, 22 (1986), 219–231. + +2 Selby, R. G., and Vecchio, F. J., “Three-dimensional Constitutive Relations for Reinforced Concrete”. Tech. Rep. 93-02, Univ. Toronto, dept. Civil Eng., Toronta, Canada, 1993. + + + +# 2-2 기본특성 + +전변형률 균열모델 해석을 수행하기 위해서는 다음과 같은 값을 정의하여야 한다. + +- 균열모델타입(crack model type) +- 일반적인 콘크리트 특성(general concrete properties ) +- 인장거동(tensile behavior) +- 압축거동(compression behavior) +- 전단거동(shear behavior) +- 횡방향영향(lateral influences) + +# 2.2.1 균열모델타입 + +midas FEA에서 구현하고 있는 전변형률 균열모델은 분산 고정균열모델 과 분산회전균열모델 두 종류이다. 회전균열모델과 고정균열모델 은 일반적인 철근 콘크리트 구조물의 균열 거동 모사에 탁월한 것으로 알려져 있으며, 특히 고정균열모델은 콘크리트 균열의 물리적인 거동을 적절히 모사하고 있는 것으로 평가 받고있다. 두 모델의 차이는 균열 방향을 결정하는 과정에 있다. + +# 2.2.2 일반 콘크리트 특성 + +균열해석에 필요한 물성치는 직접 수치로 입력할 수도 있고, 시방서에서 제안하는값을 사용할 수도 있다.(현재 midas FEA 에서는 CEB-FIP 1990기준을 적용하고있음). + +# 1) 사용자입력 + +사용자가 직접 수치로 입력해야 하는 경우, 다음과 같은 물성치를 입력하여야 한다. + +(1) 영 계수(Young’s modulus) + + + +(2) 포아송비(Poisson’s ratio) [default = 0. 0] +(3) 인장강도(tensile strength) +(4) 압축강도(compressive strength) +(5) 파괴에너지(fracture energy) [if necessary] + +# 2) 설계 기준 입력 + +설계기준에서 제시하는 값을 적용하고자 할 때 사용하는 방식이며, 현재는 CEB-FIP 1990 기준으로 물성치를 계산하여 해석에 적용한다. 전변형률 균열모델에서필요한 물성치를 적절히 입력하기 위해서 사용자는 콘크리트 강도등급과 콘크리트최대 골재크기를 입력해야 하며 입력된 두 값을 이용하여 CEB-FIP 1990에서 제시하는 압축강도와 인장강도 사이의 관계식 뿐만 아니라, 압축강도와 파괴에너지사이의 관계식을 통해 물성치를 계산한다. + +$$ +\begin{array}{l l} \text {Grade} & : \text {콘크리트 강도등급} \\ D _ {\max} & : \text {콘크리트 최대골재 크기} \end{array} +$$ + +콘크리트 강도등급은 특성압축강도, ckf (characteristic compressive strength)로 구분하며 다음과 같다. + +$$ +\text { 콘크리트 강도등급 }: \mathrm{C12,C20,C30,C40,C50,C60,C70,C80} +$$ + +$\Theta \vert 7 \vert k \vert ~ \subset 6 0 \{ \underline { { \circ } } _ { k } ~ f _ { c k } = 6 0 [ \mathsf { N / \bar { \ m } } ] ~ \equiv ~ \underline { { \underline { { \ o } } } } | \mathsf { D } | \bar { \underline { { \varphi } } } \vdash \mathsf { L } .$ + +CEB-FIP 1990에서는 입력된 콘크리트 등급과 최대골재 크기를 사용하여 영 계수,평균압축강도(mean compressive strength), 평균인장강도(mean tensile strength),그리고 파괴에너지를 산정하게 된다. + +평균압축강도는 다음과 같이 계산된다. + +$$ +f _ {c m} = f _ {c k} + \Delta f \tag {2.2.1} +$$ + + + +여기서 $\Delta f = 8$ [MPa] 이다. + +영 계수는 평균압축강도를 이용하여 아래의 식으로 구하게 된다. + +$$ +E _ {c} = E _ {c 0} \left(\frac {f _ {c m}}{f _ {c m 0}}\right) ^ {\frac {1}{3}} \tag {2.2.2} +$$ + +여기서 $E_{c0}=2.15\times10^{4}$ [N/mm²]이며, 기준 평균압축강도, fcm0 는 10 [N/mm²]이다. + +평균인장강도는 아래와 같이 계산된다. + +$$ +f _ {c t, m} = f _ {c t k 0, m} \left(\frac {f _ {c k}}{f _ {c k 0}}\right) ^ {\frac {2}{3}} \tag {2.2.3} +$$ + +여기서, $f_{ctk0}$ 는 1.40 $[N/mm^{2}]$ , $f_{ck0}$ 는 10 $[N/mm^{2}]$ 이다. + +파괴에너지는 압축강도와 최대골재 크기와 관계가 있으며, 아래와 같은 관계식을 사용하고 있다. + +$$ +G _ {f} = G _ {f 0} \left(\frac {f _ {c m}}{f _ {c m 0}}\right) ^ {0. 7} \tag {2.2.4} +$$ + +여기서, $f_{cm0}$ 는 10 $[N/mm^{2}]$ , $G_{f0}$ 는 아래 표와 같이 최대 골재 크기와 관련 있다. +
Dmax[mm]Gf0[J/m2]
825
1630
3258
+ + + +# 2-3 재하 및 제하 + +midas FEA에서 구현하고 있는 전변형률 균열모델은 균열(cracking)과 압괴(crushing)와 같은 재료적 극한 상태를 모두 모사하고 있으며, 특히 전단 거동은전단 응력과 전단 변형률간의 관계를 통해 명시적으로 모사할 수 있다. 전변형률균열모델은 하중이 제하(unloading)되는 경우 응력-변형률 관계에서 원점을 지향하도록 하였다. 해석적인 구현 과정을 자세히 살펴보면 다음과 같다. + +하중을 받는 콘크리트는 인장력이나 압축력에 저항하게 되며, 균열과 압괴 같은재료적 극한 상태에 이르기도 한다. 고정균열모델을 사용할 경우 전단 거동을 명시적으로 모사할 수 있다. 콘크리트의 균열과 압괴에 의한 재료의 열화는 여섯 개의 손상 변수(damage variables) k (  1,...,2  strk n ( nstr = 주응력 개수))에 의해평가될 수 있다. 내부 손상 변수들은 벡터 α 를 구성한다. 최대 인장 변형률에 관한 변수들 $( \mathbf { \nabla } k = 1 , . . , n _ { s t r } ) \supseteq \mathbf { \nabla } ^ { \circ } \cdot \mathbf { 0 } ^ { \circ }$ 이상의 값을 갖게 된다. 최소 압축 변형률을 파악해주는 변수들 $( \textit { \textbf { k } } = 1 + n _ { s t r } , . . . , 2 \times n _ { s t r } \textit { \textbf { ) } } \stackrel { \mathrm { { O } } } { = } \textit { \mathrm { { \Omega } } } ^ { , }$ 이하의 값을 갖는다. 손상의 복원(damage recovery)은 일어날 수 없다고 가정하기 때문에 손상 변수들의 절대값은증가하기만 한다. + +하중의 재하-제하-재재하(loading-unloading-reloading) 조건은 추가적인 제하제한(unloading constraints)변수 kr 에 의해 파악한다. 제하 제한 변수는 인장과 압축 영역 각각에서 결정되고, 각 영역의 강성 저감(stiffness degradation)을 모사하는데 사용된다. + +인장영역에서 제하 제한 변수는 다음과 같다. + +$$ +r _ {k} = \left\{ \begin{array}{l l} 0 & \text { if } \quad \begin{array}{c} t + \Delta t \\ i + 1 \end{array} \varepsilon_ {k} > \alpha_ {k} \\ 1 & \text { if } \quad \begin{array}{c} t + \Delta t \\ i + 1 \end{array} \varepsilon_ {k} \leq \alpha_ {k} \end{array} \right. (k = 1,..., n _ {s t r}) \tag {2.3.1} +$$ + +압축영역에서 제하 제한 변수는 다음과 같다. + + + +$$ +r _ {k} = \left\{ \begin{array}{l l} 0 & \text { if } \quad \begin{array}{c} t + \Delta t \\ i + 1 \end{array} \varepsilon_ {k - 3} < \alpha_ {k} \\ 1 & \text { if } \quad \begin{array}{c} t + \Delta t \\ i + 1 \end{array} \varepsilon_ {k - 3} \geq \alpha_ {k} \end{array} \right. (k = 1 + n _ {s t r}, \dots , 2 \times n _ {s t r}) \tag {2.3.2} +$$ + +그리고 내부 변수들의 갱신은 다음과 같이 수행된다. + +$$ +{ } _ { i + 1 } ^ { t + \Delta t } \boldsymbol { \alpha } = { } ^ { t } \boldsymbol { \alpha } + \mathbf { W } \Delta \boldsymbol { \varepsilon } \tag {2.3.3} +$$ + +여기서, + +$$ +\mathbf {W} = \left\{ \begin{array}{c c} W _ {k, k} = 1 - r _ {k} & k = 1, \dots , n _ {s t r} \\ W _ {k + n _ {s t r}, k} = 1 - r _ {k} & k = 1 + n _ {s t r}, \dots , 2 \times n _ {s t r} \end{array} \right. \tag {2.3.4} +$$ + +손상 복원이 일어나지 않는다는 가정하에, j 방향 응력은 다음과 같이 나타낼 수 있다. + +$$ +\sigma_ {j} = f _ {j} (\boldsymbol {\alpha}, \boldsymbol {\varepsilon} _ {n s t}) \cdot g _ {j} (\boldsymbol {\alpha}, \boldsymbol {\varepsilon} _ {n s t}) \tag {2.3.5} +$$ + +일축 응력-변형률 관계를 나타내는 j 방향의 내부 변수 j 뿐만 아니라 나머지 방향들의 내부 변수들과 변형률에 의한 영향을 받기 때문에 위와 같이 α와 ε의 함수로 표현된다. 제하와 재재하가 각 방향의 최대, 최소 변형률을 고려한 할선 (secant) 방식으로 모사가 된다면, $g_{j}$ 로 명명된 재하-제하 함수는 다음과 같이 나타낼 수 있다. + +$$ +g _ {j} = \left\{ \begin{array}{c c c} 1 - \frac {\alpha_ {j} - \varepsilon_ {j}}{\alpha_ {j}} = \frac {\varepsilon_ {j}}{\alpha_ {j}} & \text { if } & \varepsilon_ {j} > 0 \\ 1 - \frac {\alpha_ {j + n s t r} - \varepsilon_ {j}}{\alpha_ {j + n s t r}} = \frac {\varepsilon_ {j}}{\alpha_ {j + n s t r}} & \text { if } & \varepsilon_ {j} < 0 \end{array} \right. \quad \left[ 0 \leq g \leq 1 \right] \tag {2.3.6} +$$ + +일축 응력-변형률 관계식은 균열 방향의 기본 강도를 의미하는 재하-제하 함수를 곱하여 얻어진다. midas FEA에서 구현된 재료 모델은 구속 효과(confinement effect)와 횡균열(lateral cracking)이 강도에 미치는 영향을 고려하고 있다. 왜냐하면 이들 요인이 최대 강도 및 응력-변형률 곡선의 개형에 영향을 미치기 때문에 적절히 고려될 필요가 있기 때문이다. + + + +# 2-4 균열변형률 변화 + +전변형률 균열모델의 구성 모델은 응력을 전변형률의 함수로서 정의하고 있다.midas FEA에서는 하중재하와 제하를 각기 다른 경로상에서 일어나는 것으로 모사되고 있으며 특히 하중제하는 할선 기울기를 가지고 일어난다고 본다. 전변형률 균열모델의 응력-변형률 관계을 위해서는 midas FEA가 제공하는 이력 모델을사용할 수 있다. + +본 절에서 설명할 직교균열모델개념은 균열 해석에서 널리 사용되고 있는 대표적인 방식으로서 변형률 벡터의 주방향(principal direction)에 따라 응력-변형률 관계를 정의한다. + +회전균열모델은 철근콘크리트 구조물의 거동 예측에 탁월한 것으로 알려져 있다.한편 고정균열모델은 초기 균열 방향에 따른 고정된 좌표 축을 기초로 응력-변형률 관계를 정의한다. 이상 두 가지 개념은 균열 방향들이 고정되어 있는지, 회전하는지의 차이가 있을 뿐 공통적으로 비슷한 틀 안에서 비교적 용이하게 설명될 수있다. + +전변형률 균열모델의 기본적인 개념은 균열 방향들을 고려하여 응력들이 산정된다는 점이다. 요소 좌표계에서의 변형률은 변형률 증분량 xyz 을 고려하여 다음과같이 갱신된다. + +$$ +{ } _ { i + 1 } ^ { t + \Delta t } \varepsilon _ { x y z } = { } ^ { t } \varepsilon _ { x y z } + { } _ { i + 1 } ^ { t + \Delta t } \Delta \varepsilon _ { x y z } \tag {2.4.1} +$$ + +균열 방향의 변형률은 요소좌표계의 변형률에 변환행렬을 곱하여 다음과 같이 구해진다. + +$$ +\mathbf {\Sigma} _ {i + 1} ^ {t + \Delta t} \varepsilon_ {n s t} = \mathbf {T} _ {i + 1} ^ {t + \Delta t} \varepsilon_ {x y z} \tag {2.4.2} +$$ + +회전균열모델에서 변환 행렬 T 는 다음과 같이 현재 변형률에 의해 결정된다. + +$$ +\mathbf {T} = \mathbf {T} \left( \begin{array}{c} t + \Delta t \\ i + 1 \end{array} \varepsilon_ {x y z}\right) \tag {2.4.3} +$$ + +반면 고정균열모델에서 변환 행렬은 초기 균열 방향에 의해 고정되며 변형률 텐서(strain tensor)가 다음과 같을 때, diff --git a/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_026.md b/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_026.md new file mode 100644 index 00000000..20a46441 --- /dev/null +++ b/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_026.md @@ -0,0 +1,319 @@ + + +$$ +\mathbf {E} = \left[ \begin{array}{l l l} \varepsilon_ {x x} & \varepsilon_ {x y} & \varepsilon_ {x z} \\ \varepsilon_ {y x} & \varepsilon_ {y y} & \varepsilon_ {y z} \\ \varepsilon_ {z x} & \varepsilon_ {z y} & \varepsilon_ {z z} \end{array} \right] \tag {2.4.4} +$$ + +고유벡터(eigenvector)는 아래와 같은 회전 행렬(rotation matrix) R 에 저장된다. + +$$ +\mathbf {R} = \left[ \begin{array}{l l l} \mathbf {n} & \mathbf {s} & \mathbf {t} \end{array} \right] = \left[ \begin{array}{l l l} c _ {x n} & c _ {x s} & c _ {x t} \\ c _ {y n} & c _ {y s} & c _ {y t} \\ c _ {z n} & c _ {z s} & c _ {z t} \end{array} \right] \tag {2.4.5} +$$ + +여기서 $c_{xn} = \cos\phi_{ij}$ 로서 i 축과 j 축 사이의 코사인을 의미한다. 그리고 적절한 연산과정을 통해 다음과 같은 일반 3-D기준의 변환 행렬 T 를 얻게 된다. + +$$ +\mathbf {T} = \left[ \begin{array}{c c c c c c} c _ {x n} ^ {2} & c _ {y n} ^ {2} & c _ {z n} ^ {2} & c _ {x n} c _ {y n} & c _ {y n} c _ {z n} & c _ {z n} c _ {x n} \\ c _ {x s} ^ {2} & c _ {y s} ^ {2} & c _ {z s} ^ {2} & c _ {x s} c _ {y s} & c _ {y s} c _ {z s} & c _ {z s} c _ {x s} \\ c _ {x t} ^ {2} & c _ {y t} ^ {2} & c _ {z t} ^ {2} & c _ {x t} c _ {y t} & c _ {y t} c _ {z t} & c _ {z t} c _ {x t} \\ 2 c _ {x n} c _ {x s} & 2 c _ {y n} c _ {y s} & 2 c _ {z n} c _ {z s} & c _ {x n} c _ {y s} + c _ {y n} c _ {x s} & c _ {y n} c _ {z s} + c _ {z n} c _ {y s} & c _ {z n} c _ {x s} + c _ {x n} c _ {z s} \\ 2 c _ {x s} c _ {x t} & 2 c _ {y s} c _ {y t} & 2 c _ {z s} c _ {z t} & c _ {x s} c _ {y t} + c _ {y s} c _ {x t} & c _ {y s} c _ {z t} + c _ {z s} c _ {y t} & c _ {z s} c _ {x t} + c _ {x s} c _ {z t} \\ 2 c _ {x t} c _ {x n} & 2 c _ {y t} c _ {y n} & 2 c _ {z t} c _ {z n} & c _ {x t} c _ {y n} + c _ {y t} c _ {x n} & c _ {y t} c _ {z n} + c _ {z t} c _ {y n} & c _ {z t} c _ {x n} + c _ {x t} c _ {z n} \end{array} \right] \tag {2.4.6} +$$ + +구성 모델은 균열 좌표계를 기준으로 구성되며 다음과 같이 표현된다. + +$$ +\sigma_ {n s t} ^ {t + \Delta t} = \sigma \left(_ {i + 1} ^ {t + \Delta t} \varepsilon_ {n s t}\right) \tag {2.4.7} +$$ + +최종적으로 균열 좌표계 기준의 응력벡터는 요소 좌표계 기준으로 변환, 갱신된다. + +$$ +\mathbf {\Sigma} _ {i + 1} ^ {t + \Delta t} \sigma_ {x y z} = \mathbf {T} _ {i + 1} ^ {T} \mathbf {\Sigma} _ {n s t} ^ {t + \Delta t} \sigma_ {n s t} \tag {2.4.8} +$$ + +회전균열모델에서는 변환 행렬 T 가 현재 변형률에 의해 $\mathbf{T}^{T}\left(\begin{array}{c}t+\Delta t\\ i+1\end{array}\varepsilon_{xyz}\right)$ 와 같이 결정된다. 한편 고정균열모델에서는 초기 균열(incipient cracking)에 의해 정의된 변환 행렬을 그대로 준용한다. + + + +# 2-5 강성 행렬 + +콘크리트 균열해석은 비선형 해석이므로 반복 해법을 사용한다. 증분 반복 해법을통한 해석에서 외력 벡터와 내력 벡터의 평형은 뉴튼 랩슨(Newton-Raphson)법과같은 방식을 통해 찾게 된다. 이때 평형 조건을 만족하기 위해 구성 모델을 적절한강성 행렬로 정의해야 한다. + +midas FEA에서는 강성 행렬을 구하기 위하여 할선강성(secant stiffness) 기법과접선강성(tangent stiffness) 기법을 사용한다. 할선강성 기법은 균열이 광범위하게발생하는 철근 콘크리트 구조물의 해석에 특히 탁월하고 안정적인 해를 구하는 것으로 알려져 있다. 접선강성 기법은 국부적인 균열이나 균열 전파(crackpropagation)와 같은 해석에 탁월한 것으로 알려져 있다. + +# 2.5.1 접선강성 행렬 + +요소 좌표계에서 접선강성행렬은 다음과 같다. + +$$ +\mathbf {D} = \mathbf {T} ^ {T} \mathbf {D} _ {\text { tangent }} \mathbf {T} \tag {2.5.1} +$$ + +여기서, T 는 변형률 변환행렬을 의미하고, $\mathbf { D } _ { \mathrm { t a n g e n t } }$ 는 균열 좌표계 기준 접선강성행렬을 의미한다. + +접선강성행렬은 다음과 같이 네 개의 부분 행렬로 분할할 수 있다. + +$$ +\mathbf {D} _ {\text {tangent}} = \left[ \begin{array}{l l} \mathbf {D} _ {\mathrm{nn}} & \mathbf {D} _ {\mathrm{n} \theta} \\ \mathbf {D} _ {\theta \mathrm{n}} & \mathbf {D} _ {\theta \theta} \end{array} \right] \tag {2.5.2} +$$ + +여기서 $\mathsf { D } _ { \mathsf { n } \mathsf { n } } \mathsf { \stackrel { \circ } { = } }$ 접선강성성분 중 국부균열 변형률의 수직 성분을 나타내는 부분행렬이다. $\sf D _ { \sf \theta \Theta } \equiv$ 접선강성성분 중 국부균열 변형률의 전단 성분을 의미한다. $\mathsf { D } _ { \mathsf { n } \theta } .$ 와 $\mathsf { D } _ { \mathsf { { \theta } n } }$ 는 접선 강성성분 중 수직과 전단 변형률 간의 커플링 관계를 나타내는 부분 행렬이다. + + + +여러 연구자들 $^{3}$ 에 의하면 $D_{\theta\theta}$ 가 주응력 성분에만 영향을 받으며 커플링 부분 행렬은 없다고 본다. 따라서 다음과 같이 나타낼 수 있다. + +$$ +\mathbf {D} _ {\theta \theta} = \left[ \begin{array}{c c c} \frac {\sigma_ {1} - \sigma_ {2}}{2 \left(\varepsilon_ {1} - \varepsilon_ {2}\right)} & 0 & 0 \\ 0 & \frac {\sigma_ {2} - \sigma_ {3}}{2 \left(\varepsilon_ {2} - \varepsilon_ {3}\right)} & 0 \\ 0 & 0 & \frac {\sigma_ {3} - \sigma_ {1}}{2 \left(\varepsilon_ {3} - \varepsilon_ {1}\right)} \end{array} \right] \tag {2.5.3} +$$ + +위의 관계식에서 알 수 있듯이 전단강성은 주응력들의 영향을 받는 것을 알 수 있다. 이것은 주방향 좌표계의 회전에 의한 직접적 결과이다. + +한편 고정균열모델에서는 커플링 부분행렬이 0 이 아닐 수가 있으며, 전단 지연 (shear-retention)과 수직 변형률 성분간의 관계에 의해 결정된다. 일반적으로 전단 성분이 수직 응력 성분에 미치는 영향은 없다고 보기 때문에 $D_{n\theta}$ 은 0 이 된다. 반면에 $D_{n\theta}$ 은 다음과 같이 주어진다. + +$$ +\mathbf {D} _ {\theta \mathrm{n}} = \left[ \begin{array}{c c c} \frac {\partial \sigma_ {n s}}{\partial \varepsilon_ {n n}} & \frac {\partial \sigma_ {n s}}{\partial \varepsilon_ {s s}} & \frac {\partial \sigma_ {n s}}{\partial \varepsilon_ {t t}} \\ \frac {\partial \sigma_ {s t}}{\partial \varepsilon_ {n n}} & \frac {\partial \sigma_ {s t}}{\partial \varepsilon_ {s s}} & \frac {\partial \sigma_ {s t}}{\partial \varepsilon_ {t t}} \\ \frac {\partial \sigma_ {t n}}{\partial \varepsilon_ {n n}} & \frac {\partial \sigma_ {t n}}{\partial \varepsilon_ {s s}} & \frac {\partial \sigma_ {t n}}{\partial \varepsilon_ {t t}} \end{array} \right] \tag {2.5.4} +$$ + +3 Crisfield, M. A., and Wills, J., “Analysis of R/C panels using different concrete models”, J. Eng. Mech. Div., ASCE 115, 3 (1989), 578–597. + +Feenstra, P. H., “Computational Aspects of Biaxial Stress in Plain and Reinforced Concrete”, PhD thesis, Delft University of Technology, 1993. + +Rots, J. G., “Computational Modeling of Concrete Fracture”, PhD thesis, Delft University of Technology, 1988. + +Willam, K. J., Pramono, E., and Sture, S., “Fundamental issues of smeared crack models”, In Proc. SEM/RILEM Int. Conf. on Fracture of Concrete and Rock, Houston 1987 (New York, 1989), S. P. Shah and S. E. Schwartz, Eds., Springer-Verlag, pp. 142–157. + + + +위 행렬은 전단 지연이 수직 균열 변형률에 독립적인 경우에는 0 이 된다. +한편 접선강성행렬의 전단 항들은 다음과 같이 나타낼 수 있다. + +$$ +\mathbf {D} _ {0 0} = \left[ \begin{array}{c c c} \frac {\partial \sigma_ {n s}}{\partial \gamma_ {n s}} & 0 & 0 \\ 0 & \frac {\partial \sigma_ {s t}}{\partial \gamma_ {s t}} & 0 \\ 0 & 0 & \frac {\partial \sigma_ {t n}}{\partial \gamma_ {t n}} \end{array} \right] \tag {2.5.5} +$$ + +수직 강성 항들을 의미하는 $D_{nn}$ 성분은 다음과 같이 편미분 항들로 구성 된다. $D_{nn}$ 은 횡방향 변형률에 의한 상호 영향이 주 응력의 연산에 반영되기 때문에, 다음과 같이 비대각항들이 0이 아니며 비대칭 행렬이 된다. + +$$ +\mathbf {D} _ {\mathrm{nn}} = \left[ \begin{array}{c c c} \frac {\partial \sigma_ {n n}}{\partial \varepsilon_ {n n}} & \frac {\partial \sigma_ {n n}}{\partial \varepsilon_ {s s}} & \frac {\partial \sigma_ {n n}}{\partial \varepsilon_ {t t}} \\ \frac {\partial \sigma_ {s s}}{\partial \varepsilon_ {n n}} & \frac {\partial \sigma_ {s s}}{\partial \varepsilon_ {s s}} & \frac {\partial \sigma_ {s s}}{\partial \varepsilon_ {t t}} \\ \frac {\partial \sigma_ {t t}}{\partial \varepsilon_ {n n}} & \frac {\partial \sigma_ {t t}}{\partial \varepsilon_ {s s}} & \frac {\partial \sigma_ {t t}}{\partial \varepsilon_ {t t}} \end{array} \right] \tag {2.5.6} +$$ + +편미분을 통해 강성 행렬을 유도하기 위하여 먼저 응력-변형률 관계를 다시 적어 보면 다음과 같다. + +$$ +\sigma_ {i} = f _ {i} (\boldsymbol {\alpha}, \boldsymbol {\varepsilon} _ {n s t}) \cdot \boldsymbol {g} _ {i} (\boldsymbol {\alpha}, \boldsymbol {\varepsilon} _ {n s t}) \tag {2.5.7} +$$ + +주변형률 벡터 $\varepsilon_{nst}$ 에 대해 편미분을 취하면, + +$$ +\begin{array}{l} \frac {\partial \sigma_ {i}}{\partial \boldsymbol {\varepsilon} _ {n s t}} = \boldsymbol {g} _ {i} (\boldsymbol {\alpha}, \boldsymbol {\varepsilon} _ {n s t}) \left\{\frac {\partial \boldsymbol {\alpha} ^ {\mathrm{T}}}{\partial \boldsymbol {\varepsilon} _ {n s t}} \frac {\partial f _ {i}}{\partial \boldsymbol {\alpha}} + \frac {\partial f _ {i}}{\partial \boldsymbol {\varepsilon} _ {n s t}} \right\} + f _ {i} (\boldsymbol {\alpha}, \boldsymbol {\varepsilon} _ {n s t}) \left\{\frac {\partial \boldsymbol {\alpha} ^ {\mathrm{T}}}{\partial \boldsymbol {\varepsilon} _ {n s t}} \frac {\partial \boldsymbol {g} _ {i}}{\partial \boldsymbol {\alpha}} + \frac {\partial \boldsymbol {g} _ {i}}{\partial \boldsymbol {\varepsilon} _ {n s t}} \right\} \\ = \mathbf {g} _ {i} (\boldsymbol {\alpha}, \boldsymbol {\varepsilon} _ {n s t}) \left\{\mathbf {W} ^ {\mathrm{T}} \frac {\partial f _ {i}}{\partial \boldsymbol {\alpha}} + \frac {\partial f _ {i}}{\partial \boldsymbol {\varepsilon} _ {n s t}} \right\} + f _ {i} (\boldsymbol {\alpha}, \boldsymbol {\varepsilon} _ {n s t}) \left\{\mathbf {W} ^ {\mathrm{T}} \frac {\partial g _ {i}}{\partial \boldsymbol {\alpha}} + \frac {\partial g _ {i}}{\partial \boldsymbol {\varepsilon} _ {n s t}} \right\} \tag {2.5.8} \\ \end{array} +$$ + +여기서, $W = \frac{\partial \alpha^{T}}{\partial \varepsilon_{nst}}$ + +위의 식을 정리하면 다음과 같다. + + + +$$ +\mathbf {D} _ {n s t} = \left[ \begin{array}{c c c} \left\{m _ {1} r _ {1} + \left(1 - m _ {1}\right) r _ {4} \right\} \overline {{E _ {1}}} & 0 & 0 \\ 0 & \left\{m _ {2} r _ {2} + \left(1 - m _ {2}\right) r _ {5} \right\} \overline {{E _ {2}}} & 0 \\ 0 & 0 & \left\{m _ {3} r _ {3} + \left(1 - m _ {3}\right) r _ {6} \right\} \overline {{E _ {3}}} \end{array} \right] +$$ + +$$ ++ \left[ \begin{array}{c c c} g _ {1} & 0 & 0 \\ 0 & g _ {2} & 0 \\ 0 & 0 & g _ {3} \end{array} \right] \left[ \begin{array}{c c c} \frac {\partial f _ {1}}{\partial \varepsilon_ {1}} & \frac {\partial f _ {1}}{\partial \varepsilon_ {2}} & \frac {\partial f _ {1}}{\partial \varepsilon_ {3}} \\ \frac {\partial f _ {2}}{\partial \varepsilon_ {1}} & \frac {\partial f _ {2}}{\partial \varepsilon_ {2}} & \frac {\partial f _ {2}}{\partial \varepsilon_ {3}} \\ \frac {\partial f _ {3}}{\partial \varepsilon_ {1}} & \frac {\partial f _ {3}}{\partial \varepsilon_ {2}} & \frac {\partial f _ {3}}{\partial \varepsilon_ {3}} \end{array} \right] +$$ + +$$ ++ \left[ \begin{array}{c c c} g _ {1} & 0 & 0 \\ 0 & g _ {2} & 0 \\ 0 & 0 & g _ {3} \end{array} \right] \left[ \begin{array}{c c c c} \frac {\partial f _ {1}}{\partial \alpha_ {1}} & \frac {\partial f _ {1}}{\partial \alpha_ {2}} & \dots & \frac {\partial f _ {1}}{\partial \alpha_ {6}} \\ \frac {\partial f _ {2}}{\partial \alpha_ {1}} & \frac {\partial f _ {2}}{\partial \alpha_ {2}} & \dots & \frac {\partial f _ {2}}{\partial \alpha_ {6}} \\ \frac {\partial f _ {3}}{\partial \alpha_ {1}} & \frac {\partial f _ {3}}{\partial \alpha_ {2}} & \dots & \frac {\partial f _ {3}}{\partial \alpha_ {6}} \end{array} \right] \left[ \begin{array}{c c c} 1 - r _ {1} & 0 & 0 \\ 0 & 1 - r _ {2} & 0 \\ 0 & 0 & 1 - r _ {3} \\ 1 - r _ {4} & 0 & 0 \\ 0 & 1 - r _ {5} & 0 \\ 0 & 0 & 1 - r _ {6} \end{array} \right] +$$ + +(2.5.9) + +여기서 상태 표시를 의미하는 $m_{i}$ 가 추가되었는데, 이는 변형률의 상태에 따라 다음과 같은 값을 갖는다. + +$$ +m _ {i} = \left\{ \begin{array}{l l} 1 & \text { if } \quad \varepsilon_ {i} > 0 \\ 0 & \text { if } \quad \varepsilon_ {i} < 0 \end{array} \right. \tag {2.5.10} +$$ + +또한 인장과 압축 영역 각각에 대해 할선강선항은 다음과 같이 정의 된다. + +$$ +\overline {{{E}}} _ {j} = \frac {f _ {j} (\boldsymbol {\alpha} , \boldsymbol {\varepsilon} _ {n s t})}{\alpha_ {j}} \quad , \quad \overline {{{E}}} _ {j} = \frac {f _ {j} (\boldsymbol {\alpha} , \boldsymbol {\varepsilon} _ {n s t})}{\alpha_ {j + n s t r}} \tag {2.5.11} +$$ + +접선강성 항들은 전향차분(forward-difference) 기법을 통해 계산된다. 즉 j 번째 성분은 미소한 값 h을 사용하여 다음과 같이 나타낼 수 있다. + + + +$$ +\begin{array}{l} \frac {\partial f _ {i}}{\partial \varepsilon_ {j}} = \frac {f _ {i} \left(\boldsymbol {\alpha} , \boldsymbol {\varepsilon} _ {n s t} + h \boldsymbol {\varepsilon} _ {j}\right) - f _ {i} \left(\boldsymbol {\alpha} , \boldsymbol {\varepsilon} _ {n s t}\right)}{h} \tag {2.5.12} \\ \frac {\partial f _ {i}}{\partial \alpha_ {j}} = \frac {f _ {i} (\mathbf {a} + h \mathbf {a} _ {j} , \boldsymbol {\varepsilon} _ {n s t}) - f _ {i} (\mathbf {a} , \boldsymbol {\varepsilon} _ {n s t})}{h} \\ \end{array} +$$ + +여기서 벡터 $a_{j}$ , $e_{j}$ 는 j번째 항을 제외하고는 모두 0이다. + +강성 행렬 $D_{nst}$ 의 대각 항들은 다음과 같이 명료하게 표현할 수 있다. + +$$ +\begin{array}{l} \frac {\partial \sigma_ {i}}{\partial \varepsilon_ {i}} \\ = \left\{m _ {i} r _ {i} + (1 - m _ {i}) r _ {i + n s t r} \right\} \overline {{E _ {i}}} + g _ {i} \left\{(1 - r _ {i}) \frac {\partial f _ {i}}{\partial \alpha_ {i}} + (1 - r _ {i + n s t r}) \frac {\partial f _ {i}}{\partial \alpha_ {i + n s t r}} \right\} \end{array} \tag {2.5.13} +$$ + +인장 상태, 즉 $m_{i}=1$ 그리고 $r_{i+nsrt}=1$ 인 경우에 강성 항은 다음과 같이 축약된다. + +$$ +\frac {\partial \sigma_ {i}}{\partial \varepsilon_ {i}} = r _ {i} \overline {{{E _ {i}}}} + g _ {i} (1 - r _ {i}) \frac {\partial f _ {i}}{\partial \alpha_ {i}} \tag {2.5.14} +$$ + +$$ +\frac {\partial \sigma_ {i}}{\partial \varepsilon_ {i}} = \left\{ \begin{array}{l l} \overline {{E _ {i}}} & \text { if unloading, } \left(r _ {i} = 1, g _ {i} \leq 1\right) \\ \frac {\partial f _ {i}}{\partial \alpha_ {i}} & \text { if loading, } \left(r _ {i} = 0, g _ {i} = 1\right) \end{array} \right. \tag {2.5.15} +$$ + +압축 상태, 즉 $m_{i}=1$ 그리고 $r_{i+nsrt}=1$ 인 경우에 강성 항은 다음과 같이 축약된다. + +$$ +\frac {\partial \sigma_ {i}}{\partial \varepsilon_ {i}} = r _ {i + n s t r} \overline {{E _ {i}}} + g _ {i} (1 - r _ {i + n s t r}) \frac {\partial f _ {i}}{\partial \alpha_ {i + n s t r}} \tag {2.5.16} +$$ + +$$ +\frac {\partial \sigma_ {i}}{\partial \varepsilon_ {i}} = \left\{ \begin{array}{l l} \overline {{E _ {i}}} & \text { if unloading, } \left(r _ {i + n s t r} = 1, g _ {i} \leq 1\right) \\ \frac {\partial f _ {i}}{\partial \alpha_ {i + n s t r}} & \text { if loading, } \left(r _ {i + n s t r} = 0, g _ {i} = 1\right) \end{array} \right. \tag {2.5.17} +$$ + + + +# 2.5.2 할선강성 행렬 + +할선강성 방식은 모든 방향의 포아송비를 0으로 갖는 직교 이방성 재료 (orthotropic material)의 강성행렬을 사용한다. 따라서 강성 행렬은 주 방향 좌표계(principal coordinate system)에서 다음과 같이 표현할 수 있다. + +$$ +\mathbf {D} _ {\text {secant}} = \left[ \begin{array}{c c c c c c} \overline {{E}} _ {1} & 0 & 0 & 0 & 0 & 0 \\ & \overline {{E}} _ {2} & 0 & 0 & 0 & 0 \\ & & \overline {{E}} _ {3} & 0 & 0 & 0 \\ & & & \overline {{G}} _ {1 2} & 0 & 0 \\ & \text {sym.} & & & \overline {{G}} _ {2 3} & 0 \\ & & & & & \overline {{G}} _ {3 1} \end{array} \right] \tag {2.5.18} +$$ + + + +# 2-6 압축모델 + +# 2.6.1 전변형률 균열모델의 압축 모델 + +압축 응력하의 콘크리트가 횡방향 구속이 되면, 등방성 응력(isotropic stress)들이증가하고, 이로 인하여 강도(strength)와 연성(ductility)이 증가하게 된다. 이러한등방성 응력의 영향을 반영하기 위해 압축 응력-변형률 관계가 적절히 수정된다.즉, 압축 응력-변형률 함수의 인자들인 피크 응력(peak stress) fcf와 피크 변형률(peak strain) p 들은 파괴를 일으키는 압축 응력들을 나타내는 파괴 함수(failurefunction)로부터 결정된다. 이 파괴 함수는 횡방향 구속응력들의 함수이다. 압축응력의 직교방향으로 균열이 발생하면 피크 변형률과 피크 응력이 감소하게 된다.피크 변형률은 계수 $\beta _ { \mathrm { { \varepsilon c r } } }$ 를 사용하여 감소되며, 피크 응력은 계수 $\beta _ { \mathrm { o c r } }$ 을 사용하여감소되게 된다. 이들 감소계수에 대한 자세한 식은 이후 절 횡방향 영향에서 설명하고 있다. + +$$ +f _ {p} = \beta_ {\sigma_ {c r}} \cdot f _ {c f} \quad \alpha_ {p} = \beta_ {\varepsilon_ {c r}} \cdot \varepsilon_ {p} \tag {2.6.1} +$$ + +압축부의 기본 함수는 $\mathrm { f _ { p } } \underline { { \boldsymbol { \mathcal { R } } } } \mathrm { | }$ $\alpha _ { \mathrm { { p } } }$ 들로 표현되며, 기 정의된 곡선들을 활용할 수 있다.기 정의된 곡선의 종류로는 불변(constant), 선형(linear), 선형경화(linearhardening), 포화경화(saturation hardening), 다중선형(multi-linear) 곡선들이 있으며 압축부에 사용 가능한 경화-연화(hardening-softening) 곡선은 포물선(parabolic), Thorenfeldt et. al.4 의 경화 곡선들이다. + +4 Thorenfeldt, E., Tomaszewicz, A., and Jensen, J. J., “Mechanical properties of high-strength concrete and applications in design”, In Proc. Symp. Utilization of High-Strength Concrete (Stavanger, Norway) (Trondheim, 1987), Tapir. + + + +![](images/page-259_cbaeba8345436fc03c4abb247a5f0f0e68a4cba73e4f378b648f18344cbe3b47.jpg) + +
+text_image + +σ +ε +
+ +(a) elastic + +![](images/page-259_b03a8ac99d961b684c3a83a21e511d28d84d041fb33b1db42b9c164d81d76d10.jpg) +(b) constant + +![](images/page-259_1a48be4030fdd27feaedbfe24af421ffb2352453e167c259d8e26beccb11d21d.jpg) +(c) Thorenfeldt + +![](images/page-259_7f72ffc5bb2b7b401d91b10347c862617b5c9f17ebcb7e22c596164312e4f03e.jpg) + +
+text_image + +σ +ε +E_har +f_t +
+ +(d) linear + +![](images/page-259_8f87b20c483cc00e26da09b3e82e2d12b8f7c830587ff028e9e6829054417e61.jpg) + +
+line + +| ε | σ | +|-------|-------| +| (ε₀, σ₀) | 0 | +| (ε₁, σ₁) | 0 | +| (ε₂, σ₂) | 0 | +| (εₙ, σₙ) | 0 | +
+ +(e) mult-linear + +![](images/page-259_18d4e4e657039c9aab9a903a93585471079fb3d2135f02ba2c2cd9de1c37e107.jpg) + +
+line + +| ε | σ | +| ------- | ------- | +| -1/γ | -1/γ | +| ε | σ | +| f_c0 | f_c0 | +| f_c∞ | f_c∞ | +
+ +(f) saturation type + +![](images/page-259_d0d92bb4c3881194afbde6b0118c910325f2eeda29ecd0686ccbe1733c252c93.jpg) +(g) parabolic +그림 2.6.1 압축 모델 + +# 탄성 모델 + +일반적인 탄성 모델은 영 계수를 사용한다. 그림 2.6.1(a). + +# 불변 모델 + +압축강도를 초과 하면 더 이상 압축응력 증가가 없는 모델이다. 그림 2.6.1(b). + +압축강도 : fc  0.0 의 값을 입력한다. + +# Thorenfeldt 모델 + +압축강도 : fc  0.0 의 값을 입력한다. (그림 2.6.1(c)). + + + +![](images/page-260_67d80a133bc3f18e1f5ab3f8043c9b5c371792f52605b1e6e607f1c0e9d73e3d.jpg) + +
+text_image + +αp +f +α +fp +
+ +그림2.6.2 Thorenfeldt 압축곡선 + +Thorenfeldt 곡선 식은 다음과 같다. + +$$ +f = - f _ {p} \frac {\alpha_ {i}}{\alpha_ {p}} \left(\frac {n}{n - 1 + \left(\frac {\alpha_ {i}}{\alpha_ {p}}\right) ^ {n k}}\right) \tag {2.6.2} +$$ + +$$ +\text { 에기서 } n = 0. 8 0 + \frac {f _ {c c}}{1 7}, k = \left\{ \begin{array}{l l} 1 & \text { if } 0 > \alpha > \alpha_ {p} \\ 0. 6 7 + \frac {f _ {c c}}{6 2} & \text { if } \alpha \leq \alpha_ {p} \end{array} \right. +$$ + +# - 선형경화 모델 + +1차 압축 항복 후 저감된 강성, $E_{har}$ 으로 압축거동을 계속하는 모델이다. (그림 2.6.1(d)). + +$$ +f _ {c} > 0. 0, E _ {h a r} > 0. 0 \text { 의 값을 입력한다. } +$$ + +# - 다중 선형경화 모델 + +사용자가 임의의 응력-변형률 값들을 입력하여 응력 곡선을 정의할 수 있다. 최대 diff --git a/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_027.md b/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_027.md new file mode 100644 index 00000000..17b77284 --- /dev/null +++ b/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_027.md @@ -0,0 +1,388 @@ + + +30개의 좌표를 입력할 수 있으며, 최초 좌표는 0.d0 이어야 한다. (그림 2.6.1(e)).[조건 < 0.d0 , 변형률은 단조 감소의 값을 입력하여야 한다.] + +# 포화 모델 + +본 모델을 사용하기 위해 입력해야 하는 물성치들은 다음과 같다. (그림 2.6.1(f)). + +초기압축강도(initial compressive strength) : fco  0.0 + +극한압축강도(ultimate compressive strength) : fc  0.0 + +불변경화계수(constant hardening modulus) : Ehar  0.0 + +붕괴 계수(decaying factor) :   0.0 + +# 포물선 모델 + +포물선(parabolic) 모델은 Feenstra5 가 제안한 모델로서 파괴 에너지에 근거하여유도되었다. 본 곡선은 아래 그림과 같이 세 개의 특성 값으로 표현될 수 있다.(그림 2.6.1(g)). + +압축강도(compressive strength) : fc  0.0 + +압축파괴에너지(compressive fracture energy) : Gc  0.0 + +특성요소길이(characteristic element length) : h  0.0 + +5 FEENSTRA, P. H., “Computational Aspects of Biaxial Stress in Plain and Reinforced Concrete”, PhD thesis, Delft University of Technology, 1993. + + + +![](images/page-262_8314a3e01f47a96e740a5b06a2182c182d4d4d88a76a3268a9deaae17ae5336a.jpg) + +
+line + +| Point | f | Gc/h | +|-------|-------|-------| +| αu | - | - | +| αc | - | - | +| αc/3 | - | - | +| fc/3 | - | - | +
+ +그림 2.6.3 포물선 압축곡선 + +최대 압축강도 $f_{c}$ 의 1/3 에 도달하는 지점의 변형률은 다음과 같다. + +$$ +\alpha_ {c / 3} = \frac {1}{3} \frac {f _ {c}}{E} \tag {2.6.3} +$$ + +최대 압축 강도에 해당하는 변형률은 다음과 같다. + +$$ +\alpha_ {c} = - \frac {4}{3} \frac {f _ {c}}{E} = 4 \alpha_ {c / 3} \tag {2.6.4} +$$ + +위 값들은 요소 크기나 압축 파괴 에너지와는 무관하게 결정됨을 알 수 있다. 끝으로 압축부에서 연화가 완료된 상태를 의미하는 극한 변형률은 다음과 같다. + +$$ +\alpha_ {u} = \alpha_ {c} - \frac {3}{2} \frac {G _ {c}}{h f _ {c}} \tag {2.6.5} +$$ + +이상의 변수들을 바탕으로 아래와 같은 곡선 관계식을 얻을 수 있다. + +$$ +f = \left\{ \begin{array}{l l} - f _ {c} \frac {1}{3} \frac {\alpha_ {j}}{\alpha_ {c / 3}} & \text { if } \quad 0 \leq \alpha_ {j} < \alpha_ {c / 3} \\ - f _ {c} \frac {1}{3} \left(1 + 4 \left(\frac {\alpha_ {j} - \alpha_ {c / 3}}{\alpha_ {c} - \alpha_ {c / 3}}\right) - 2 \left(\frac {\alpha_ {j} - \alpha_ {c / 3}}{\alpha_ {c} - \alpha_ {c / 3}}\right) ^ {2}\right) & \text { if } \quad \alpha_ {c / 3} \leq \alpha_ {j} < \alpha_ {c} \\ - f _ {c} \left(1 - \left(\frac {\alpha_ {j} - \alpha_ {c}}{\alpha_ {u} - \alpha_ {c}}\right) ^ {2}\right) & \text { if } \quad \alpha_ {c} \leq \alpha_ {j} < \alpha_ {u} \\ 0 & \text { if } \quad \alpha_ {u} \leq \alpha_ {j} \end{array} \right. \tag {2.6.6} +$$ + + + +한편, 파괴에너지 Gc 와 특성요소길이 h 가 연화부분에서 지배적임을 다음과 같이유도할 수 있다. + +$$ +\int_ {\alpha_ {c}} ^ {\alpha_ {u}} f d \alpha_ {j} = f _ {c} \left(\alpha_ {j} - \frac {1}{3} \left(\frac {\alpha_ {j} - \alpha_ {c}}{\alpha_ {u} - \alpha_ {c}}\right) ^ {3}\right) \Bigg | _ {\alpha_ {c}} ^ {\alpha_ {u}} = \frac {G _ {c}}{h} \tag {2.6.7} +$$ + + + +# 2-7인장 모델 + +# 2.7.1 인장연화에 대한 배경이론 + +균열의 수직방향 응력과 변형률 사이의 관계는 다음과 같이 나타낼 수 있다. + +$$ +\sigma_ {n n} ^ {c r} (\mathcal {E} _ {n n} ^ {c r}) = f _ {t} \cdot y \left(\frac {\mathcal {E} _ {n n} ^ {c r}}{\mathcal {E} _ {n n . u l t} ^ {c r}}\right) \tag {2.7.1} +$$ + +여기서 $f_{t}$ 는 인장강도를 의미하고, $\varepsilon_{nn.ult}^{cr}$ 는 극한균열변형률을 의미한다. + +함수 $y(\cdot)$ 는 다양한 연화 도식(softening diagram)을 모사하게 된다. + +응력과 변형률 관계에 있어서 연화 거동이 균열폭(crack bandwidth) h 를 통해 Mode-I 파괴에너지, $G_{f}^{I}$ 와 연관이 있다고 보면, 아래와 같은 관계식을 얻을 수 있다. + +$$ +G _ {f} ^ {I} = h \int_ {\varepsilon_ {n n} ^ {c r} = 0} ^ {\varepsilon_ {n n} ^ {c r} = \infty} \sigma_ {n n} ^ {c r} (\varepsilon_ {n n} ^ {c r}) d \varepsilon_ {n n} ^ {c r} \tag {2.7.2} +$$ + +식 (2.45)을 식 (2.46)에 대입하면, + +$$ +G _ {f} ^ {I} = h f _ {t} \int_ {\varepsilon_ {n n} ^ {c r} = 0} ^ {\varepsilon_ {n n} ^ {c r} = \infty} y \left(\frac {\varepsilon_ {n n} ^ {c r}}{\varepsilon_ {n n . u l t} ^ {c r}}\right) d \varepsilon_ {n n} ^ {c r} \tag {2.7.3} +$$ + +여기서 $f_{t}$ 는 상수라고 보았다. + +식 (2.7.4)와 같이 치환을 하고 $\varepsilon_{nn.ult}^{cr}$ 을 유한하다고 가정하면, 식 (2.7.5) 를 얻을 수 있다. + +$$ +x = \frac {\varepsilon_ {n n} ^ {c r}}{\varepsilon_ {n n . u l t} ^ {c r}} \tag {2.7.4} +$$ + +$$ +G _ {f} ^ {I} = h f _ {t} \left(\int_ {x = 0} ^ {x = \infty} y (x) d x\right) \varepsilon_ {n n. u l t} ^ {c r} \tag {2.7.5} +$$ + +이상의 과정을 통해 최종적인 극한균열변형률, $\varepsilon_{nn.ult}^{cr}$ 을 식 (2.7.6)과 같이 얻을 수 있다. + + + +이 값은 해석 수행 도중에는 일정하다고 보며, 식 (2.7.6) 에 나타난 바와 같이 요소의 물성치와 인장강도, 파괴에너지, 균열폭을 고려하여 계산된다. + +$$ +\varepsilon_ {n n. u l t} ^ {c r} = \frac {1}{\alpha} \times \frac {G _ {f} ^ {I}}{h f _ {t}} \tag {2.7.6} +$$ + +$$ +\alpha = \int_ {x = 0} ^ {x = \infty} y (x) d x \tag {2.7.7} +$$ + +인장강도가 초과되거나 변형이 특정요소에 집중되는 경우 Mode-I 파괴에너지는소산되게 된다. 이러한 해석 기법에서는 유한 요소 망의 정밀도(mesh refinement)와 해석결과는 밀접한 관계를 가지게 된다. 경우에 따라 요소 분할이 너무 크게 되어 구성모델상에서 스냅백(snap-back) 문제가 야기될 수 있다. 결과적으로 본해석기법에서 가정된 객관적인 파괴에너지 개념이 만족되지 않을 수 있다. + +구성 모델에서의 스냅백 문제는 연화 도식(softening diagram)의 초기 기울기가영 계수보다 절대값으로 클 경우 발생할 수 있다. 여기서 전제로 된 가정은 인장연화 도식(tension softening diagram)의 초기 접선 기울기가 가장 큰 절대값을 가진다고 본 것이다. 이러한 조건을 수식으로 표현해 보면 아래와 같다. + +$$ +\left. \frac {d \sigma_ {n n} ^ {c r}}{d \varepsilon_ {n n} ^ {c r}} \right| _ {\varepsilon_ {n n} ^ {c r} = 0} \geq - E \tag {2.7.8} +$$ + +위 식은 식 (2.7.9) 와 같이 다시 표현될 수 있다. + +식 (2.7.9)에 식 (2.7.6)의 극한균열변형률을 대입하고, 다시 정리하면 식 (2.7.10)을 얻을 수 있다. + +$$ +\left. \frac {f _ {t}}{\varepsilon_ {n n . u l t} ^ {c r}} \frac {d y}{d x} \right| _ {x = 0} \geq - E \tag {2.7.9} +$$ + +$$ +\varepsilon_ {n n. u l t} ^ {c r} \geq - \frac {f _ {t}}{E} \left. \frac {d y}{d x} \right| _ {x = 0} = \varepsilon_ {n n. u l t. \min} ^ {c r} \tag {2.7.10} +$$ + +만약 식 (2.7.10)의 조건이 만족되지 못할 경우, 이를 해결하기 위한 몇 가지 대안이 있다. + +먼저 균열폭을 줄이는 방법이다. 그러나 이 값은 요소의 고유한 값이며 일정하게유지되어야 한다. 다음으로는 파괴에너지, I Gf 를 증가시키는 방법이 있다. 이것은 + + + +결국 재료의 연성을 증가시키는 역할을 하게 된다. 마지막으로는 인장 강도 tf 를감소시키는 방법이 있다. 이는 파괴에너지는 일정하게 하면서 재료의 연성을 증가시키는 역할을 하게 된다. + +이상 제시된 방안들 중 가장 효율적인 방법은 물리적인 의미를 고려할 때, 인장강도를 감소시키는 방법이다. 고려되는 영역이 클수록 인장강도가 감소될 확률은 높아진다. 이는 곳 요소가 큰 요소의 경우 적절히 인장강도가 감소되어야 함을 의미한다. 왜냐하면 이렇게 큰 요소의 경우에는 응력집중이 나타나기 힘들기 때문이다.따라서 만약 식 (2.7.10) 이 만족되지 않는 경우 다음과 같이 인장강도는 감소되어야 한다. + +$$ +f _ {t, r e d} ^ {2} = - \frac {G _ {f} ^ {\prime} E}{\alpha h \frac {d y}{d x} \Big | _ {x = 0}} \tag {2.7.11} +$$ + +다른 대안으로는 균열폭이 아래와 같은 값을 갖도록 요소 크기를 줄여주는 방법도있다. + +$$ +h _ {\max} = - \frac {G _ {f} ^ {I} E}{\left. \alpha f _ {t} ^ {2} \frac {d y}{d x} \right| _ {x = 0}} \tag {2.7.12} +$$ + + + +# 2.7.2 전변형률 균열모델의 인장 모델 + +전변형률 균열모델에서 제공하는 인장 거동 모델은 탄성(elastic), 불변(constant),취성(brittle), 선형(linear), 지수(exponential), Hordijk, 다중선형(multi-linear)이다.이들은 다음과 같이 이론적 차이를 바탕으로 구분할 수 있다. + +전변형률 균열모델에서는 파괴에너지에 근거한 연화 함수들을 구현하고 있다. 이연화함수들을 바탕으로 모델을 구분해보면, 선형 연화 곡선(linear softeningcurve), 지수 연화 곡선(exponential softening curve), 비선형 연화 곡선(nonlinear softening curve - Hordijk6 )이 있다. 이들 모델들은 분산균열모델에서와 마찬가지로 균열폭과 연관을 가진다. + +다음으로, 파괴에너지에 직접적으로 상관관계가 없는 인장 거동이 있다. 이들은 전변형률 개념 안에서 적절히 모사될 수 있다. 이들은 불변, 다중선형, 취성 거동들이다. + +![](images/page-267_7a4beb794317e29ef59ce0bf1f8dc549a5a39e3093419ee06852a56c75020c2b.jpg) + +
+text_image + +σ +ε +
+ +(a) elastic + +![](images/page-267_76d32a2bb7a5c4504dfa1566c5c5dabfd4b5c6602bd1ec237f4a7c7e78939729.jpg) + +
+text_image + +σ +fₜ +
+ +(b) constant + +![](images/page-267_bf495229118c44465802e8b91875065b192875e2c691843eafb9b29881023be8.jpg) + +
+line + +| ε | σ | +| ---- | ----- | +| 0 | 0 | +| 0.5 | 1 | +| 1 | 0 | +
+ +(c) brittle + +![](images/page-267_dad178ee85648c9be341462b497f970c98dc2b4ec2c6dea96c0ca3268a8f24e6.jpg) + +
+text_image + +σ +fₜ +Gₙ/h +ε +
+ +(d) linear + +![](images/page-267_1533ee71c9dd2e949dbe004e8abd28bf5b52c74ac1b9fac1c0b571d1cd35bc0c.jpg) + +
+line + +| f_t | σ | +|-----|-------| +| 0 | 0 | +| Peak| 1 | +| 1 | 0 | +| 2 | 0 | +
+ +(e) exponential + +![](images/page-267_77470b6bfa42532ed48404bef83d11f2dc785ba381e57d969d6a4481d928c79a.jpg) + +
+line + +| ε | σ | +| ------- | ----- | +| 0 | 0 | +| G_f^I / h | Peak | +| (σ₀, ε₀) | (σ₀, ε₀) | +
+ +(f) Hordijk + +![](images/page-267_51e8cd94122d2a362474dbf21a60cbd3e8387e129cb66fa6581f133324350dc8.jpg) + +
+line + +| ε | σ | +|---|---| +| (ε₁, σ₁) | | +| (ε₂, σ₂) | | +| (εₙ, σₙ) | | +
+ +(g) multi-linear +그림 2.7.1 인장 모델 + + + +# 탄성 모델 + +일반적인 탄성 모델로서 영 계수를 사용한다. (그림 2.7.1(a)). + +# 불변 모델 + +인장강도를 초과 하면 더 이상 인장응력 증가가 없는 모델이다. (그림 2.7.1(b)). + +인장강도 : ft  0.0 의 값을 입력한다. + +# 취성 모델 + +인장강도를 초과 하면 더 이상 인장응력 증가가 없고, 저항 응력이 0 이 되는 모델이다. (그림 2.7.1(c)). + +인장강도 : ft  0.0 의 값을 입력한다. + +그림 2.7.2 에서 알 수 있듯이 파괴 에너지 소산과 피크 변형률, peaknn  의 관계는다음 식과 같다. + +$$ +G _ {f} = \frac {1}{2} f _ {t} \varepsilon_ {n n} ^ {\text { peak }} h \tag {2.7.13} +$$ + +![](images/page-268_eee22d3212b54f5cc5a66c5dd396f6a9b1f27f217985af28bfdb44d17ec17ef3.jpg) + +
+line +| ε_nn | σ_nn | +|------|------| +| 0 | 0 | +| peak | 1 | +
+ +그림 2.7.2 취성 균열거동 + + + +# - 선형연화 모델 + +인장강도를 초과 하면 선형 연화(linear softening)가 일어난다고 본 모델이다. (그림 2.7.1(d)). + +입력한 파괴에너지와 균열폭을 바탕으로 연화의 기울기가 결정된다. + +인장강도: $f_{t} > 0.0$ 의 값을 입력한다. + +인장 파괴에너지: $G_{f}^{I} > 0.0$ + +균열폭: h > 0.0 + +![](images/page-269_af834288b88d4d0a91674c1b507ea9fece8d4676e37f92b45bff687d025e4fe1.jpg) + +
+line + +| ε_cr_nn | σ_cr_nn | +| ------- | ------- | +| f_t | σ_cr_nn | +| ε_cr_nn, ult | σ_cr_nn, G_f^I / h | +
+ +그림 2.7.3 선형 인장연화 + +균열응력(crack stress)의 관계식과 α 은 아래와 같이 주어진다. + +$$ +\frac {\sigma_ {n n} ^ {c r} \left(\varepsilon_ {n n} ^ {c r}\right)}{f _ {t}} = \left\{ \begin{array}{l l} 1 - \frac {\varepsilon_ {n n} ^ {c r}}{\varepsilon_ {n n . u l t} ^ {c r}} & \left(i f \quad 0 < \varepsilon_ {n n} ^ {c r} < \varepsilon_ {n n. u l t} ^ {c r}\right) \\ 0 & \left(i f \quad \varepsilon_ {n n. u l t} ^ {c r} < \varepsilon_ {n n} ^ {c r} < \infty\right) \end{array} \right. \tag {2.7.14} +$$ + +$$ +\alpha = \int_ {0} ^ {\infty} y (x) d x = \int_ {0} ^ {1} y (x) d x + \int_ {0} ^ {\infty} 0 d x = \int_ {0} ^ {1} (1 - x) d x = \frac {1}{2} \tag {2.7.15} +$$ + +이상의 유도 과정을 통해 극한균열변형률을 다음과 같이 얻을 수 있다. + + + +$$ +\varepsilon_ {n n. u l t} ^ {c r} = 2 \frac {G _ {f} ^ {I}}{h f _ {t}} \tag {2.7.16} +$$ + +여기서 간단히 식 (2.7.17) 을 유도할 수 있으며, 이를 통해 극한균열변형률의 최소값을 식 (2.7.18) 과 같이 얻을 수 있다. 또한 감소된 인장 강도도 식 (2.7.19) 와같이 얻을 수 있다. + +$$ +\left. \frac {d y}{d x} \right| _ {x = 0} = - 1 \tag {2.7.17} +$$ + +$$ +\varepsilon_ {\mathrm{nn.ult.min}} ^ {c r} = \frac {f _ {t}}{E} \tag {2.7.18} +$$ + +$$ +f _ {t} = \sqrt {2 \frac {G _ {f} ^ {I} E}{h}} \tag {2.7.19} +$$ + +# 지수연화 모델 + +인장강도를 초과하면 지수연화(exponential softening)가 일어난다고 본 모델이다.(그림 2.7.1(e)). + +입력한 파괴에너지와 균열폭을 바탕으로 연화의 기울기가 결정된다. + +인장강도 : ft  0.0 의 값을 입력한다. + +인장 파괴에너지 : 0.0 I Gf  + +균열폭 : h  0.0 + +# Hordijk 모델 + +인장강도를 초과하면 Hordijk이 제안한 연화가 일어난다고 본 모델이다. (그림2.7.1(f)). + +입력한 파괴에너지와 균열폭을 바탕으로 연화의 기울기가 결정된다. + +인장강도 : ft  0.0 의 값을 입력하면 된다. diff --git a/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_028.md b/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_028.md new file mode 100644 index 00000000..e9a304ca --- /dev/null +++ b/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_028.md @@ -0,0 +1,348 @@ + + +인장 파괴에너지 : 0.0 I Gf  + +균열폭 : h  0.0 + +본 모델의 연화거동은 Hordijk7 , Cornelissen 그리고 Reinhardt8 가 제안한 모델이다. + +![](images/page-271_889fb4d7afddb1ab2d64ac1195b105ce110f5b473adeaf573fe4a206be024e11.jpg) + +
+line + +| ε_cr_nn | σ_cr_nn | +| ------- | ------- | +| 0 | 0 | +| ε_cr_nn.ult | 0 | +
+ +그림 2.7.4 비선형 인장연화 (Hordijk et al.) + +본 모델의 기본 함수는 다음과 같다. + +7 HORDIJK, D. A. Local Approach to Fatigue of Concrete. PhD thesis, Delft University of Technology, 1991. +8 CORNELISSEN, H. A. W., HORDIJK, D. A., AND REINHARDT, H. W. + +Experimental determination of crack softening characteristics of normalweight and lightweight concrete. + +Heron 31, 2 (1986). + + + +$$ +\frac {\sigma_ {n n} ^ {c r} \left(\varepsilon_ {n n} ^ {c r}\right)}{f _ {t}} = \left\{ \begin{array}{c c} \left(1 + \left(c _ {1} \frac {\varepsilon_ {n n} ^ {c r}}{\varepsilon_ {n n . u l t} ^ {c r}}\right) ^ {3}\right) \exp \left(- c _ {2} \frac {\varepsilon_ {n n} ^ {c r}}{\varepsilon_ {n n . u l t} ^ {c r}}\right) \dots & \\ - \frac {\varepsilon_ {n n} ^ {c r}}{\varepsilon_ {n n . u l t} ^ {c r}} \left(1 + c _ {1} ^ {3}\right) \exp \left(- c _ {2}\right) & \left(i f 0 < \varepsilon_ {n n} ^ {c r} < \varepsilon_ {n n. u l t} ^ {c r}\right) \\ 0 & \left(i f \varepsilon_ {n n. u l t} ^ {c r} < \varepsilon_ {n n} ^ {c r} < 0\right) \end{array} \right\} \tag {2.7.20} +$$ + +여기서 $c_{1}=3,\quad c_{2}=6.93$ 이다. + +본 모델에서 α 는 다음과 같이 나타낼 수 있다. + +$$ +\begin{array}{l} \alpha = \int_ {0} ^ {\infty} y (x) d x = \int_ {0} ^ {1} y (x) d x + \int_ {1} ^ {\infty} 0 d x \\ = \int_ {0} ^ {1} \left(1 + \left(c _ {1} x\right) ^ {3}\right) \exp \left(- c _ {2} x\right) - x \left(1 + c _ {1} ^ {3}\right) \exp \left(- c _ {2}\right) d x \\ = \frac {- 1 2 c _ {1} ^ {3} - 1 2 c _ {1} ^ {3} c _ {2} - 6 c _ {1} ^ {3} c _ {2} ^ {2} - 2 c _ {2} ^ {3} - 2 c _ {1} ^ {3} c _ {2} ^ {3}}{\dots} \dots \tag {2.7.21} \\ \dots \frac {- c _ {2} ^ {4} - c _ {1} ^ {3} c _ {2} ^ {4} + 1 2 c _ {1} ^ {3} \exp (c _ {2}) + 2 c _ {2} ^ {3} \exp (c _ {2})}{\cdots 2 c _ {2} ^ {4} \exp (c _ {2})} \\ \end{array} +$$ + +위 관계식을 통해 $c_{1}=3,\quad c_{2}=6.93$ 인 경우에 $\alpha=0.195$ 를 얻을 수 있다. + +극한균열변형률은 다음과 같이 얻을 수 있다. + +$$ +\varepsilon_ {n n. u l t} ^ {c r} = 5. 1 3 6 \frac {G _ {f} ^ {I}}{h f _ {t}} \tag {2.7.22} +$$ + +Hordijk 등이 제안한 연화 도식(softening diagram)에 의해 다음과 같은 관계식을 얻을 수 있다. + +$$ +\left. \frac {d y}{d x} \right| _ {x = 0} = \left(3 c _ {1} \left(c _ {1} x\right) ^ {2} - c _ {2} \left(1 + \left(c _ {1} x\right) ^ {3}\right)\right) \exp \left(- c _ {2} x\right) - \left(1 + c _ {1} ^ {3}\right) \exp \left(- c _ {2}\right) \big | _ {x = 0} \tag {2.7.23} +$$ + +$$ += - c _ {2} - \left(1 + c _ {1} ^ {3}\right) \exp \left(- c _ {2}\right) +$$ + +이상의 관계식들을 통해, 최소 극한균열변형률과 감소된 인장강도는 다음과 같이 얻을 수 있다. + + + +$$ +\varepsilon_ {n n. u l t. \min} ^ {c r} = 6. 9 5 7 \frac {f _ {t}}{E} \tag {2.7.24} +$$ + +$$ +f _ {t} = \left(0. 7 3 9 \frac {G _ {f} ^ {I} E}{h}\right) ^ {\frac {1}{2}} \tag {2.7.25} +$$ + +# 다중선형모델 + +인장강도를 초과 하면 사용자가 입력한 연화가 일어난다고 본 모델이다, (그림2.7.1(g)). + +최대 30개의 좌표를 입력할 수 있으며, 최초 좌표는 무조건 0.d0 이어야 한다.[조건 > 0.d0 , 변형률은 단조 증가의 값을 입력하여야 한다. + +![](images/page-273_ed558afd9c590ef8fd35c46c8a78ce4df51a1309b70bf451a7487a5a3b77109f.jpg) + +
+line +| εₙₙᶜʳ | σₙₙᶜʳ | +| ------ | ------ | +| 0 | 0 | +| fₜ,₁ | εₙₙ,₁ᶜʳ | +| fₜ,ₙ | εₙₙ,ₙᶜʳ | +
+ +그림 2.7.5 다중 인장연화 + +초기 접선 기울기는 아래와 같은 조건을 만족해야 한다. + +$$ +\frac {f _ {t , 1} - f _ {t , 0}}{\varepsilon_ {n n , 1} ^ {c r}} \geq - E \tag {2.7.26} +$$ + + + +# 2-8 전단 모델 + +전단 거동의 모사는 균열발생 이후에 전단 강성이 일반적으로 감소하는 고정균열에서만 필요하며 회전균열에서는 효과가 없다. midas FEA에서 균열 발생 이후에전단 강성이 전단지연계수(shear retention factor) 만큼 저감되게 구현되어 있다.현재 midas FEA에서는 불변모델과 다중선형 모델이 구현되어 있다. + +# 2.8.1 전변형률 균열모델의 전단 모델 + +![](images/page-274_7d952ec0fe056e367593511e7ded047e083f87025df0009115de63d2a455e9b9.jpg) + +
+text_image + +Elastic Modulus G +τ +γ +
+ +![](images/page-274_4e029bc65e43fcec33c3889078418d3998c8cf1f7bbd65827ee9425498d4f8a4.jpg) + +
+text_image + +Elastic Modulus +τ +βG +γ +
+ +![](images/page-274_ef9a1542b41fe15ebf8895816abebe2d5fdcc735d00015503aa41d933e37f5ff.jpg) + +
+line + +| γ | τ | +| ------- | ------- | +| γ₀ | τ₀ | +| γ₁ | τ₁ | +| γ₂ | τ₂ | +
+ +![](images/page-274_ef78f02386f061b40ad39128c94767d464211c649a3dba0ba905110374732acf.jpg) + +
+line + +| Point | γ | β | +|---|---|---| +| (γ₀, β₀) | 0 | | +| (γ₁, β₁) | 0 | | +| (γ₂, β₂) | 0 | | +
+ +그림 2.8.1 전단모델 + + + +# 탄성모델 + +일반적인 전단 모델로 전단 강성의 감소가 없다. + +![](images/page-275_be6895cc8745e145aa683b3da3dffbbb0b748153a98607e51a76d41a10cd8746.jpg) + +
+text_image + +Elastic Modulus G +τ +γ +
+ +그림 2.8.2 탄성모델 + +# 불변모델 + +전단지연계수만큼 전단 강성이 감소한다. + +$$ +G ^ {c r} = \beta G [ 0 \leq \beta \leq 1 ] \tag {2.8.1} +$$ + +![](images/page-275_ccc49f72c274a087ccc2bf9285e151c09b24552c67bf02edc60423e4be7453e7.jpg) + +
+text_image + +Elastic Modulus +βG +τ +γ +
+ +그림 2.8.3 불변모델 + +여기서 는 전단지연계수이다. + + + +# 다중선형모델 + +midas FEA는 두 가지 다중선형 모델을 제공한다. 하나는 전단응력과 전단변형률이 다중 선형으로 표현한 것이고 다른 하나는 전단지연계수와 전단변형률의 관계를 표현한 것이다. + +![](images/page-276_9b3d835d699b54cc003b1821a34abaaf3ddfb7a69730b57c0cfbceba0bf33a59.jpg) + +
+line + +| γ | β | +| ------- | ------- | +| (γ₀, β₀) | (γ₀, β₀) | +| (γ₁, β₁) | (γ₁, β₁) | +| (γ₂, β₂) | (γ₂, β₂) | +
+ +그림 2.8.4 다중선형(전단변형률, 전단지연계수) + +![](images/page-276_2756fe9bce73d1341ef8bfc5be49400f95e0e734112cd5272b14e54b657fab3c.jpg) + +
+line + +| γ | τ | +| ------- | ------- | +| (γ₀, τ₀) | (γ₀, τ₀) | +| (γ₁, τ₁) | (γ₁, τ₁) | +| (γ₂, τ₂) | (γ₂, τ₂) | +
+ +그림 2.8.5 다중선형(전단변형률, 전단응력) + + + +# 2-9 횡방향 영향 + +# 2.9.1 포아송비에 의한 횡방향 확장 + +포아송 효과(Poisson effect)는 일축 인장이나 압축력을 받는 시편의 횡방향 변형을 결정한다. 횡 변형이 구속된 시편에는 횡방향 구속효과가 발생하게 되며 이러한 효과는 철근콘크리트 부재의 3차원적 모사에 있어서 상당히 중요하다. Selby와 Vecchio $^{9}$ 는 횡방향 팽창효과를 구조물에 추가적으로 가해지는 외력으로 대체해 설명했으며 이 개념은 유한요소 해석 절차의 적절한 수정을 요하게 된다. + +포아송 효과는 등가의 일축 변형률 개념을 통해 해석에 반영되었다. 선형-탄성 (linear-elastic) 거동의 경우 3차원 응력-변형률 관계는 다음과 같다. + +$$ +\boldsymbol {\sigma} _ {n s t} = \frac {E}{(1 + \nu) (1 - 2 \nu)} \left[ \begin{array}{c c c} 1 - \nu & \nu & \nu \\ \nu & 1 - \nu & \nu \\ \nu & \nu & 1 - \nu \end{array} \right] \boldsymbol {\varepsilon} _ {n s t} \tag {2.9.1} +$$ + +이 식은 아래와 같이 표현될 수 있다. + +$$ +\boldsymbol {\sigma} _ {n s t} = \left[ \begin{array}{l l l} E & 0 & 0 \\ 0 & E & 0 \\ 0 & 0 & E \end{array} \right] \times \left[ \begin{array}{c c c} \frac {1 - \nu}{(1 + \nu) (1 - 2 \nu)} & \frac {\nu}{(1 + \nu) (1 - 2 \nu)} & \frac {\nu}{(1 + \nu) (1 - 2 \nu)} \\ \frac {\nu}{(1 + \nu) (1 - 2 \nu)} & \frac {1 - \nu}{(1 + \nu) (1 - 2 \nu)} & \frac {\nu}{(1 + \nu) (1 - 2 \nu)} \\ \frac {\nu}{(1 + \nu) (1 - 2 \nu)} & \frac {\nu}{(1 + \nu) (1 - 2 \nu)} & \frac {1 - \nu}{(1 + \nu) (1 - 2 \nu)} \end{array} \right] \boldsymbol {\varepsilon} _ {n s t} \tag {2.9.2} +$$ + +위 관계식은 등가 일축 변형률을 사용하여 다음과 같이 표현할 수 있다. + + + +$$ +\boldsymbol {\sigma} _ {n s t} = \left[ \begin{array}{l l l} E & 0 & 0 \\ 0 & E & 0 \\ 0 & 0 & E \end{array} \right] \tilde {\boldsymbol {\varepsilon}} _ {n s t} \tag {2.9.3} +$$ + +여기서 등가 일축 변형률 벡터는 다음과 같이 정의된다. + +$$ +\begin{array}{l} \tilde {\boldsymbol {\varepsilon}} _ {n s t} = \left\{ \begin{array}{c} \tilde {\mathcal {E}} _ {1} \\ \tilde {\mathcal {E}} _ {2} \\ \tilde {\mathcal {E}} _ {3} \end{array} \right\} \\ = \left[ \begin{array}{c c c} \frac {1 - \nu}{(1 + \nu) (1 - 2 \nu)} & \frac {\nu}{(1 + \nu) (1 - 2 \nu)} & \frac {\nu}{(1 + \nu) (1 - 2 \nu)} \\ \frac {\nu}{(1 + \nu) (1 - 2 \nu)} & \frac {1 - \nu}{(1 + \nu) (1 - 2 \nu)} & \frac {\nu}{(1 + \nu) (1 - 2 \nu)} \\ \frac {\nu}{(1 + \nu) (1 - 2 \nu)} & \frac {\nu}{(1 + \nu) (1 - 2 \nu)} & \frac {1 - \nu}{(1 + \nu) (1 - 2 \nu)} \end{array} \right] \left\{ \begin{array}{l} \varepsilon_ {1} \\ \varepsilon_ {2} \\ \varepsilon_ {3} \end{array} \right\} \tag {2.9.4} \\ \end{array} +$$ + +$$ +\tilde {\boldsymbol {\varepsilon}} _ {n s t} = \mathbf {P} \quad \boldsymbol {\varepsilon} _ {n s t} \tag {2.9.5} +$$ + +이 개념은 비선형 재료 모델에도 적용되었다. 주 방향 좌표계에서 응력 벡터는 주변형률에 의해 계산되는 것이 아니라, 위에서 설명한 등가의 일축 변형률 벡터에 의해 계산된다. 등가의 일축 변형률 벡터는 포아송비와 주변형률을 통해 결정될 수 있다. + +접선 강성 행렬의 하위 행렬인 $D_{nst}$ 는 등가 일축 변형률 개념에서 다음과 같이 약간의 수정이 필요하게 된다. + +$$ +\mathbf {D} _ {n s t} = \frac {\partial \boldsymbol {\sigma} _ {n s t}}{\partial \boldsymbol {\varepsilon} _ {n s t}} = \frac {\partial \boldsymbol {\sigma} _ {n s t}}{\partial \tilde {\boldsymbol {\varepsilon}} _ {n s t}} \mathbf {P} \tag {2.9.6} +$$ + + + +# 2.9.2 횡방향 구속하의 압축거동 + +등방성 응력(isotropic stress)의 증가에 따른 강도의 증가는 4개의 인자를 가지는 Hsieh-Ting-Chen 파괴면(failure surface)에 의해 모사된다. 이 파괴면의 정의는 다음과 같다. + +$$ +f = 2. 0 1 0 8 \frac {J _ {2}}{f _ {c c} ^ {2}} + 0. 9 7 1 4 \frac {\sqrt {J _ {2}}}{f _ {c c}} + 9. 1 4 1 2 \frac {f _ {c 1}}{f _ {c c}} + 0. 2 3 1 2 \frac {I _ {1}}{f _ {c c}} - 1 = 0 \tag {2.9.7} +$$ + +여기서 응력 불변수 J₂와 I₁은 다음과 같이 콘크리트 주 응력들로 표현된다. + +$$ +J _ {2} = \frac {1}{6} \left\{\left(\sigma_ {c 1} - \sigma_ {c 2}\right) ^ {2} + \left(\sigma_ {c 2} - \sigma_ {c 3}\right) ^ {2} + \left(\sigma_ {c 3} - \sigma_ {c 1}\right) ^ {2} \right\} \tag {2.9.8} +$$ + +$$ +I _ {1} = \sigma_ {c 1} + \sigma_ {c 2} + \sigma_ {c 3} \tag {2.9.9} +$$ + +한편 $f_{c1}$ 은 다음과 같이 콘크리트의 최대 인장응력이 아닌 최대 주응력을 의미한다. $^{10}$ + +$$ +f _ {c 1} = \max (\sigma_ {c 1}, \sigma_ {c 2}, \sigma_ {c 3}) \tag {2.9.10} +$$ + +위 파괴면의 인자들은 콘크리트 시편의 일축 인장, 압축 강도, 이축 압축 강도, 3 축 실험 자료들의 보정을 통해 결정된다. 응력 $f_{c3}$ 는 파괴를 일으킨다고 가정되고, 아래와 같이 선형 탄성 응력 벡터에 적절한 비례계수를 곱하여 얻게 된다. 이때 앞서 Hsieh-Ting-Chen의 파괴면 조건식을 만족해야 한다. + +$$ +\boldsymbol {\sigma} _ {c} = s E \varepsilon_ {n s t} \tag {2.9.11} +$$ + +그렇다면 다축 응력 하에서 압축 파괴 응력은 다음과 같다. + +$$ +f _ {c 3} = s \cdot \min (\sigma_ {c 1}, \sigma_ {c 2}, \sigma_ {c 3}) \tag {2.9.12} +$$ + +만약 비례 계수 s 가 음수이고 결과적으로 양의 파괴 응력 $f_{c3}$ 이 얻어지게 되면, + + + +응력 벡터는 파괴면의 인장부를 기준으로 비례 조정되고, 파괴 강도는 상당히 큰음수 값인 (-30fcc) 로 맞추어 지게 된다. 파괴강도 fcf는 아래와 같이 얻어진다. + +$$ +f _ {c f} = - f _ {c 3} \tag {2.9.13} +$$ + +피크 응력 계수 $K _ { \sigma } \cong \mathsf { S e l b y } ^ { 1 1 } \supset \mathsf { I }$ 제안한 식에 의하면 아래와 같다. + +$$ +K _ {\sigma} = \frac {f _ {c f}}{f _ {c c}} \geq 1 \tag {2.9.14} +$$ + +한편 피크 변형률 계수는 아래와 같다. + +$$ +K _ {\varepsilon} = K _ {\sigma} \tag {2.9.15} +$$ + +비 구속된 압축 상태에서 피크위치의 값들은 일축 압축 강도와 같다고 보고, 피크응력 계수는 ‘1’로 본다. 이런 조건 하에서 압축 응력-변형률 함수의 인자들은 다음과 같다. + +$$ +f _ {c f} = K _ {\sigma} f _ {c c}, \quad \varepsilon_ {p} = K _ {\sigma} \varepsilon_ {0} \tag {2.9.16} +$$ + +여기서 초기 변형률은 다음과 같은 관계식을 가진다. + +$$ +\varepsilon_ {0} = - \frac {n}{n - 1} \times \frac {f _ {c c}}{E _ {c}} \tag {2.9.17} +$$ + +위의 관계식은 구속된 압축 하중 하에서 최대 강도의 점진적인 증가를 반영하며,응력-변형률 그래프의 초기 기울기를 영 계수와 같게 만든다. 완전한 삼축 응력상태에서 파괴면까지의 도달은 가능하지 않으며, 아래 그림과 같이 선형 응력-변형률 관계가 얻어지게 된다. diff --git a/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_029.md b/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_029.md new file mode 100644 index 00000000..968927ed --- /dev/null +++ b/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_029.md @@ -0,0 +1,291 @@ + + +![](images/page-281_791782409efb1be341195390c0721f308af885f27de90661a4169f337ec509b1.jpg) + +
+line + +| Category | f | +| ------------------------- | ----- | +| unconfined | 0.0 | +| low lateral confinement | 0.0 | +| medium lateral confinement| 0.0 | +| triaxial loading | 0.0 | +
+ +그림 2.9.1 응력-변형률 곡선의 횡방향 구속영향 + +구속 콘크리트 연성의 증가는 아래 식과 같이 Thorenfeldt 곡선의 하강 지점을 선형 보간하여 표현하게 된다. + +$$ +f _ {j} = - f _ {p} \left(1 - (1 - r) \frac {\alpha_ {j} - \alpha_ {p}}{\alpha_ {u} - \alpha_ {p}}\right) \leq - r f _ {p} \tag {2.9.18} +$$ + +![](images/page-281_b8dbb94f1df1c50f81768cada44e374cd595b455021ef85b844553c8e3a0be38.jpg) + +
+text_image + +α_u +α_p +f +α +rf_p +f_p +
+ +그림 2.9.2 횡 향 구속하의 압축거동 + +여기서 계수 r은 재료의 잔존 강도(residual strength)를 표현하기 위한 값이다. 압축부에서 극한 변형률은 피크 변형률에 다음과 같은 피크 강도와 압축 강도의 비를 곱하여 결정한다. + + + +$$ +\alpha_ {u} = \left(\frac {f _ {p}}{f _ {c c}}\right) ^ {\gamma} \alpha_ {p} \tag {2.9.19} +$$ + +숫자 γ 는 따로 결정되어야 하지만 여기서는 ‘3’으로 가정되었다. 잔존 강도 또한 피크 강도와 압축 강도의 비에 다음과 같이 영향을 받는다. + +$$ +r = \left(\frac {f _ {p}}{f _ {c c}}\right) ^ {\gamma} r _ {0} \tag {2.9.20} +$$ + +잔존강도의 초기 값인 r₀는 여기서 '0.1'로 가정되었다. + +만약 피크강도가 압축 강도에 비해 현저히 큰 경우(fp / fcc > 1.05), 선형 압축-연화(compression-softening) 관계식은 Thorenfeldt 곡선에만 적용될 수 있다. 횡 압축과 횡 균열로 인하여 fp / fcc < 1.05 가 되게 되면, 재료의 연성은 증가하지 않는다. + + + +# 2.9.3 횡방향 균열과 압축거동 + +균열이 발생한 콘크리트에서 주 압축 방향에 수직으로 발생한 큰 인장 변형률은 콘크리트의 압축 강도를 감소시킨다. 결과적으로 압축 강도 $f_{p}$ 는 내부 변수 $\alpha_{j}$ 의 함수일 뿐만 아니라 횡방향으로 인장 손상(tensile damage)를 야기하게 되는 내부 변수들인 $\alpha_{1,1}$ $\alpha_{1,2}$ 의 함수이기도 하다. 횡 균열에 의한 감소 계수들은 다음과 같다. + +$$ +\beta_ {\varepsilon c r} = \beta_ {\varepsilon c r} (\alpha_ {l a t}), \quad \beta_ {\sigma c r} = \beta_ {\sigma c r} (\alpha_ {l a t}) \tag {2.9.21} +$$ + +여기서 평균 횡방향 손상 변수 $\alpha_{lat}$ 는 다음과 같다. + +$$ +\alpha_ {l a t} = \sqrt {\alpha_ {l , 1} ^ {2} + \alpha_ {l , 2} ^ {2}} \tag {2.9.22} +$$ + +횡 균열에 의한 감소 관계식은 Vecchio와 Collins $^{12}$ 가 제안한 다음의 모델을 채용한다. + +$$ +\beta_ {\sigma c r} = \frac {1}{1 + K _ {c}} \leq 1 \tag {2.9.23} +$$ + +여기서 $K_{c}=0.27\left(-\frac{\alpha_{lat}}{\varepsilon_{0}}-0.37\right)$ 이며, $\beta_{\varepsilon cr}=1$ 로 본다. + +![](images/page-283_bd4bff7d272e578c17de355d07e4328a78c21ad0ec979001f54ae9d177e08a81.jpg) + +
+line +| x | y | +|---|------| +| 0 | 1.0 | +| 1 | 0.85 | +| 2 | 0.75 | +| 3 | 0.65 | +| 4 | 0.55 | +| 5 | 0.45 | +| 6 | 0.35 | +| 7 | 0.3 | +| 8 | 0.25 | +
+ +그림 2.9.3 횡방향 균열에 의한 감소계수 + + + +Part 2 Material Library + + + +# Chapter 3. Interface Nonlinearities + +# 3-1 개요 + +midas FEA에서는 그림 3.1.1과 같이 점요소, 선요소, 면요소 3가지 계면요소(interface element)를 제공한다. 이 때 점요소를 제외한 선요소와 면요소는 고차와 저차요소로 사용이 가능하다. + +![](images/page-285_9e95abc279419eed1236ecc47f12496a61d456ad467ef572979b7148b81edfe9.jpg) +그림 3.1.1 midas FEA 계면요소 + +본 매뉴얼에서는 이해의 편의성을 위해 2차원 해석에 대한 수식 전개를 기본으로한다. 또한 첨자 n 은 법선방향을 정의하며, t , s 는 접선방향을 정의한다.계면요소에 발생하는 계면력 벡터(traction vector)는 법선과 접선방향으로 나누어 + + + +식 (3.1.1)과 같이 정의된다. 이때 계면력의 단위는 “힘/면적”이며 응력 단위와 같다. + +$$ +\boldsymbol {t} = \left\{ \begin{array}{l} t _ {n} \\ t _ {t} \end{array} \right\} \tag {3.1.1} +$$ + +계면요소에 발생하는 상대변위(relative displacement)는 다음과 같으며 단위는 길이이다. + +$$ +\Delta u = \left\{ \begin{array}{c} \Delta u _ {n} \\ d t \end{array} \right\} \tag {3.1.2} +$$ + +여기서, 계면력 벡터와 상대변위와의 관계를 선형구성관계로 정의할 때 다음과 같이 나타낼 수 있다. + +$$ +\left\{ \begin{array}{l} t _ {n} \\ t _ {t} \end{array} \right\} = \left[ \begin{array}{c c} k _ {n} & 0 \\ 0 & k _ {t} \end{array} \right] \left\{ \begin{array}{l} \Delta u _ {n} \\ d t \end{array} \right\} \tag {3.1.3} +$$ + +여기서, + +$$ +k _ {n} \quad : \text { 법선방향 벌칙강성(penalty stiffness) } (\text { 단위 }: N / m ^ {3}) +$$ + +$$ +k _ {t} \quad : \text { 접선방향 벌칙강성(단위 : } N / m ^ {3}) +$$ + +계면요소에서 비선형 조건은 절점에 적용되므로 가우스 적분법을 사용하면 정확한거동을 표현할 수 없으며 결과값이 계면을 따라 진동하게 된다. midas FEA에서는이를 극복하기 위해 뉴튼-코츠(Newton-Cotes) 적분법을 사용하였다. + +위의 초기 연속체 선형구성관계에 의한 계면력 벡터와 상대변위 관계식을 증분형태로 나타내면 다음과 같다. + +$$ +\boldsymbol {t} = \boldsymbol {D} \Delta \boldsymbol {u} \tag {3.1.4} +$$ + +여기서, + +t : 계면력 벡터 + +∆u : 상대변위의 증분량 + +이 때 구성행렬 D 는 식 (3.1.5)와 같다. + +$$ +\boldsymbol {D} = \left[ \begin{array}{l l} D _ {1 1} & D _ {1 2} \\ D _ {2 1} & D _ {2 2} \end{array} \right] \tag {3.1.5} +$$ + + + +# 3-2 이산균열 + +![](images/page-287_495a30ad6a1d4c04238b8bdadd8121f4d6a0c775febd982f593b710f90944327.jpg) + +
+text_image + +f_n +f_t +k_n +Δu_n +
+ +그림 3.2.1 이산균열에서 상대변위와 계면력의 관계 + +이산균열(discrete cracking)의 구성방정식은 전체 변위를 사용하여 계면력을 정의하는 총 변형이론(total deformation theory)을 기초로 한다. 총 변형이론은 법선 방향과 접선 방향의 비선형 거동을 모두 정의할 수 있으며 이는 식 (3.2.1)과같이 비선형 함수로 정의한다. + +$$ +t _ {n} = f _ {n} \left(\Delta u _ {n}\right) \tag {3.2.1} +$$ + +$$ +t _ {t} = f _ {t} (d t) +$$ + +위의 식은 결국 다음과 같이 접선강성계수들을 사용하여 식 (3.2.2)와 같이 정의할수 있다. + +$$ +\left[ \begin{array}{c c} D _ {1 1} = \frac {\partial f _ {n}}{\partial \Delta u _ {n}} & D _ {1 2} = 0 \\ D _ {2 1} = 0 & D _ {2 2} = \frac {\partial f _ {t}}{\partial d t} \end{array} \right] \tag {3.2.2} +$$ + + + +일반적으로 법선 방향 계면력 nt 은 모드-I형식의 균열을 유발하며, 주로 인장 연화관계식에 의해 지배된다. midas FEA에서는 다음과 같이 세 가지 형태의 인장연화 모델을 지원하고 있습니다. + +1. 취성 균열모델 (brittle cracking model) +2. 선형 인장연화모델 (linear tension softening model) +3. 비선형 인장연화모델 (nonlinear tension softening model) + +# 3-2-1 취성 균열모델 + +취성 균열모델은 일반적인 완전취성모델을 사용하였으며 최대응력점 이후 거동을그림 3.2.2와 같이 정의한다. + +![](images/page-288_6b156c2e12a45b810c19b46dce58b6a6d02438ee781dd930158d05f0469f4e14.jpg) + +
+text_image + +tₙ +fₜ +Δuₙ +
+ +그림 3.2.2 취성 균열 거동 + +$$ +\frac {f _ {n} \left(\Delta u _ {n}\right)}{f _ {t}} = \left\{ \begin{array}{l l} 1 & \text { if } \quad \Delta u _ {n} \leq 0 \\ 0 & \text { if } \quad 0 < \Delta u _ {n} < \infty \end{array} \right. \tag {3.2.3} +$$ + + + +# 3-2-2 선형 인장연화모델 + +선형 인장연화모델은 법선 방향의 균열 응력과 변위와의 관계는 그림 3.2.3과 같이 표현된다. + +![](images/page-289_876b5528c4fb9d9db6e25ccf8077a0acaa0453ec0d9d859ab2959082d48ff47e.jpg) + +
+text_image + +t_n +f_t +G_f^I +Δu_n,ult +Δu_n +
+ +그림 3.2.3 선형 인장 연화 거동 + +$$ +\frac {f _ {n} \left(\Delta u _ {n}\right)}{f _ {t}} = \left\{ \begin{array}{c c} 1 - \frac {\Delta u _ {n}}{\Delta u _ {n , u l t}} & \text { if } 0 < \Delta u _ {n} < \Delta u _ {n, u l t} \\ 0 & \text { if } \Delta u _ {n, u l t} < \Delta u _ {n} < \infty \end{array} \right. \tag {3.2.4} +$$ + +여기서, + +$$ +\Delta u _ {n, u l t} \quad := 2 \frac {G _ {f} ^ {I}}{f _ {t}} +$$ + +$$ +G _ {f} ^ {I} \quad : \text { 모드 - 1 파괴 에너지 } +$$ + +$$ +f _ {t} \quad : \text { 인장강도 } +$$ + +선형 연화모델에서 제하와 재하는 할선접근방법(secant approach)과 탄성접근방법(elastic approach)의 두가지 모델을 제공한다. 할선접근방법은 제하시 원점을지나는 직선의 강성으로 제하되며 원점을 지나서는 초기강성이 복원되는 것으로 + + + +가정하는 방법이다. 탄성접근방법은 제하시 바로 초기강성값으로 제하되는 것을가정하는 방법이다. + +# 3-2-3 비선형 인장연화모델 + +Hordijk, Cornelissen과 Reinhardt는 콘크리트 연화거동을 그림 3.2.4와 같이 지수함수 형태로 나타내었다. + +![](images/page-290_755f8fb708c280826e8a6888c79af02d586fdd772a0d10770aed2a8c77227338.jpg) + +
+line + +| Δuₙ | tₙ | +| --- | --- | +| 0 | fₜ | +| Δuₙ,ult | 0 | +
+ +그림 3.2.4 비선형 인장 연화 거동 + +$$ +\frac {f _ {n} \left(\Delta u _ {n}\right)}{f _ {t}} = \left\{ \begin{array}{l l} \left(1 + \left(c _ {1} \frac {\Delta u _ {n}}{\Delta u _ {n , u l t}}\right) ^ {3}\right) \exp \left(- c _ {2} \frac {\Delta u _ {n}}{\Delta u _ {n , u l t}}\right) \\ - \frac {\Delta u _ {n}}{\Delta u _ {n , u l t}} \left(1 + c _ {1} ^ {3}\right) \exp \left(- c _ {2}\right) & \text { if } 0 < \Delta u _ {n} < \Delta u _ {n, u l t} \\ 0 & \text { if } \Delta u _ {n, u l t} < \Delta u _ {n} < \infty \end{array} \right. \tag {3.2.5} +$$ diff --git a/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_030.md b/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_030.md new file mode 100644 index 00000000..11388a19 --- /dev/null +++ b/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_030.md @@ -0,0 +1,267 @@ + + +여기서, + +$$ +\Delta u _ {n, u l t} \quad := 5. 1 3 6 \frac {G _ {f} ^ {I}}{f _ {t}} +$$ + +$$ +c _ {1} \quad := 3 +$$ + +$$ +c _ {2} \quad := 6. 9 3 +$$ + +재하와 제하에서 할선접근방법과 탄성접근방법외 아래그림에 표현된 이력모델(hysteresis model)을 사용할 수 있다. + +![](images/page-291_320294bd313d9c8a0d497a1b68a582dfb2162f5d6bb74acf6d89adb59b74bdba.jpg) + +
+line +| Δu_n | t_n | +|------|-----| +| 0 | 1 | +| Δu_n,ult | 0 | +
+ +그림 3.2.5 이력모델 + + + +# 3-2-4 전단지연법(Shear Retention) + +모드-I형태의 균열 후 전단 계면력 tt 값은 줄어들게 되며 이는 다음과 같이 나타낸다. + +$$ +t _ {t} = \left\{ \begin{array}{l l} k _ {t} d t & \text { if } \Delta u _ {n} < \frac {t _ {t}}{k _ {n}} \\ \beta k _ {t} d t & \text { if } \Delta u _ {n} \geq \frac {t _ {t}}{k _ {n}} \end{array} \right. \tag {3.2.6} +$$ + +여기서, + +$\beta k _ { t }$ : 감소 전단강성(reduced shear stiffness) + +$\beta \supseteq ~ 0 \leq \beta \leq 1$ 는 β = 0 이며 콘크리트와 같이 매끈하지 않을 경우는 $0 < \beta \leq 1 0 |$ 된다. 일반적으로 β는 0.1에서 0.3사이의 값을 사용한다. + + + +# 3-3 균열팽창 + +계면에서의 균열은 법선방향 상대변위 $\Delta u _ { n }$ 값이 $\Delta u _ { _ { n , u l t } }$ 보다 클 경우 발생한다. 또한 콘크리트와 같이 균열면이 매끈하지 않은 재질에서는 거친 균열면의 골재 맞물림 현상으로 접선방향의 상대변위로 인해서 법선방향의 팽창(dilatancy)이 발생한다. 따라서 법선방향의 변위와 접선방향의 변위는 독립적이지 않으며, 강성행렬은대각방향 이외에도 값을 가지게 된다. + +![](images/page-293_0fca90ba71db0b76f818278cd92b2d5481a774a4016d233675e0863955813f9b.jpg) + +
+text_image + +dt +t_n +t_t +Δu_n +t_t +t_n +n +t +
+ +그림 3.3.1 거친 균열(Rough crack) + +$$ +\left\{ \begin{array}{l} t _ {n} = f _ {n} \left(\Delta u _ {n}, d t\right) \\ t _ {t} = f _ {t} \left(\Delta u _ {n}, d t\right) \end{array} \right. \tag {3.3.1} +$$ + +식 (3.3.1)를 상대변위에 대해 편미분할 경우 기울기 함수는 식 (3.3.2).와 같다. + + + +$$ +\left[ \begin{array}{l l} D _ {1 1} = \frac {\partial f _ {n}}{\partial \Delta u _ {n}} & D _ {1 2} = \frac {\partial f _ {n}}{\partial d t} \\ D _ {2 1} = \frac {\partial f _ {t}}{\partial \Delta u _ {n}} & D _ {2 2} = \frac {\partial f _ {t}}{\partial d t} \end{array} \right] \tag {3.3.2} +$$ + +균열 팽창 모델은 수식이 복잡하여, 여러 학자들에 의해 제안되었으며, 크게 두가지 형태로 분류된다. 첫 번째는 주로 실험결과에 의존한 경험적 균열모델(empirical crack models)이며, 다른 하나는 균열면의 형상을 가정하는데 기초한이론적 수치모델인 물리적 균열 모델(physical crack models)이다. midas FEA에서 지원하는 균열 팽창모델은 다음과 같다. + +# 경험적 균열모델 : + +1. 거친 균열모델 I(rough crack model (Bazant와 Gambarova)) +2. 거친 균열모델 II(rough crack model (Gambarova와 Karakoc)) +3. 골재 맞물림모델(aggregate interlock model (Walraven과 Reinhardt)) + +# 물리적 균열모델 : + +1. 2상모델(two-phase model (Walraven)) +2. 접촉 밀도모델(contact density model (Li et al.)) + + + +# 3-3-1 거친 균열모델 I (Bazant와 Gambarova) + +Bazant와 Gambarova는 균열면을 균일한 사다리꼴 형상의 요철로 가정한 거친균열모델을 제안하였다. 그림 3.3.2.는 거친 균열모델의 응답 그래프를 나타내며,아래에 열거한 가정사항을 바탕으로 한다. + +1. 계면의 쐐기 효과로 인하여, 전단 응력은 변위 비율( / nr dt u = ∆ )과 가장 큰연관관계를 지닌다. +2. 대변위 비율에 대한 전단력은 골재 주위 모르타르의 미소 균열 때문에 근사값을 사용한다. +3. 법선방향의 균열 대변위에 대해서 계면의 접촉효과는 고려하지 않는다. + +$$ +\left(\Delta u _ {n} > \frac {1}{2} D _ {\max}, \text { 여기서, } D _ {\max} \text { 는 최대 골재크기이다. }\right) +$$ + +![](images/page-295_2ae8e8aac5e33f58335a5e2c4ccb8f00fe3f37e18afab1356b3a92c3d9c6beeb.jpg) +그림 3.3.2 거친 균열모델 + + + +Paulay와 Loeber의 테스트 결과를 기초로 한 수식은 식 (3.3.3).과 같다. + +$$ +f _ {t} = \tau_ {u} r \frac {a _ {3} + a _ {4} | r | ^ {3}}{1 + a _ {4} r ^ {4}} \tag {3.3.3} +$$ + +$$ +f _ {n} = - \frac {a _ {1}}{\Delta u _ {n}} \left(a _ {2} \mid f _ {t} \mid\right) ^ {p} +$$ + +여기서, + +$$ +p = 1. 3 0 \times \left(1 - \frac {0 . 2 3 1}{1 + 0 . 1 8 5 \Delta u _ {n} + 5 . 6 3 \left(\Delta u _ {n}\right) ^ {2}}\right) +$$ + +$$ +r = \frac {\delta t}{\Delta u _ {n}} +$$ + +$$ +\tau_ {u} = \frac {\tau_ {0} a _ {0}}{a _ {0} + \left(\Delta u _ {n}\right) ^ {2}} +$$ + +$$ +a _ {0} = 0. 0 1 D _ {\mathrm{max}} ^ {2} +$$ + +$$ +a _ {1} = 0. 0 0 0 5 3 4 +$$ + +$$ +a _ {2} = 1 4 5. 0 +$$ + +$$ +a _ {3} = \frac {2 . 4 5}{\tau_ {0}} +$$ + +$$ +a _ {4} = 2. 4 4 \times \left(1 - \frac {4}{\tau_ {0}}\right) +$$ + +$$ +\tau_ {0} = 0. 2 4 5 f _ {c} = 0. 1 9 5 f _ {c c} +$$ + +그리고, + +cf : 원통 공시체의 일축압축강도 + +cc f : 입방 공시체의 일축압축강도 + + + +3-3-2 거친 균열 모델 II (Gambarova와 Karakoc) +![](images/page-297_a08185e6b30f7bfc620cc56c9e8346f143df34c443dbe8034108f86ca838b86a.jpg) +그림 3.3.3 거친 균열모델 II + +Bazant와 Gambarova 모델을 좀 더 향상시킨 것으로 Daschner와 Kupfer의 일정구속압상태에서의 실험결과에 맞추어 법선 계면력과 균열변위사이의 관계를 개선했기 때문에 Bazant와 Gambarova 모델 보다 더 좋은 결과를 얻을 수 있다고 알려져 있다. 추가적으로 이 수식은 골재 크기효과가 고려된다. + +$$ +f _ {t} = \tau_ {0} \left(1 - \sqrt {\frac {2 \Delta u _ {n}}{D _ {\max}}}\right) r \frac {a _ {3} + a _ {4} | r | ^ {3}}{1 + a _ {4} r ^ {4}} \tag {3.3.4} +$$ + +$$ +f _ {n} = - a _ {1} a _ {2} \sqrt {\Delta u _ {n}} \frac {r}{\left(1 + r ^ {2}\right) ^ {0 . 2 5}} f _ {t} +$$ + + + +여기서, + +$$ +a _ {1} a _ {2} = 0. 6 2 +$$ + +$$ +a _ {3} = \frac {2 . 4 5}{\tau_ {0}} +$$ + +$$ +a _ {4} = 2. 4 4 \times \left(1 - \frac {4}{\tau_ {0}}\right) +$$ + +$$ +\tau_ {0} = 0. 2 5 f _ {c} = 0. 2 f _ {c c} +$$ + +# 3-3-3 골재 맞물림 (Walraven과 Reinhardt) + +Walraven과 Reinhardt는 경량, 자갈 콘크리트 실험결과를 통해 계면력과 상대 변위의 선형관계를 도출하였다. 이 모델은 자갈 콘크리트에 사용할 경우 매우 정확한 결과를 도출하는 것으로 알려져 있지만, 자갈 콘크리트 모델에만 적용되는 한계가 있다. + +![](images/page-298_e26bd2d055f0e96130a2fdd6a2b8efb25395a69b9672a93ba4f1001538e4d7c8.jpg) +그림 3.3.4 골재 맞물림 거동 + + + +$$ +f _ {t} = - \frac {f _ {c c}}{3 0} + \left(1. 8 \Delta u _ {n} ^ {- 0. 8 0} + \left(0. 2 3 4 \Delta u _ {n} ^ {- 0. 7 0 7} - 0. 2 0\right) f _ {c c}\right) d t \tag {3.3.5} +$$ + +$$ +f _ {n} = \frac {f _ {c c}}{2 0} - \left(1. 3 5 \Delta u _ {n} ^ {- 0. 6 3} + \left(0. 1 9 1 \Delta u _ {n} ^ {- 0. 5 5 2} - 0. 1 5\right) f _ {c c}\right) d t +$$ + +여기서, + +$$ +d t \geq 0 +$$ + +$$ +f _ {t} \geq 0 +$$ + +$$ +f _ {n} \leq 0 +$$ + +# 3-3-4 2상 모델(Walraven) + +Walraven이 제기한 2상(two-phase) 모델은 다음의 가정에 기초한다. + +1. 콘크리트는 강체의 원형 함유물과 완전소성체의 2상재료로 가정한다. +2. 골재의 입도는 Fuller 곡선과 일치한다. +3. 함유물(inclusion)과 모체(matrix) 사이의 활동 접촉 면적은 계면 변위와 기하적인 형태와 관련이 있고, 골재분포의 통계를 고려한다. +4. 모체에 대한 압축 접촉강도는 콘크리트 강도와 관련이 있다. 반면에 전단 접촉강도는 마찰계수를 고려한 압축 접촉 강도와 선형 관계가 있다. + +Walraven은 철근이 포함되어있지 않은 균열 내 골재의 맞물림 현상을 이론적으로전개하여 나타내었으며 식 (3.3.6)과 같다. + + + +![](images/page-300_2d59ec1b579c4df7f31ab317f959a8e065f80591c3fc6941ab6afa0bcadc248a.jpg) +그림 3.3.5 2상 모델 + +$$ +f _ {t} = \sigma_ {p u} \left(A _ {n} + \mu A _ {t}\right) \tag {3.3.6} +$$ + +$$ +f _ {n} = - \sigma_ {p u} \left(A _ {t} - \mu A _ {n}\right) +$$ + +여기서, + +An, At : n 과 t 방향에 대한 함유물과 모체사이의 평균 접촉면적 + +$\sigma _ { p u }$ Opt ： + +$\mu$ ： + +이 모델은 접선강성항이 전단방향 균열변위 dt 와 법선방향 균열변위 $\Delta u _ { n }$ 및 골재 분포에 대한 함수로 이루어진 특징을 지닌다. diff --git a/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_031.md b/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_031.md new file mode 100644 index 00000000..faaa6e0a --- /dev/null +++ b/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_031.md @@ -0,0 +1,245 @@ + + +# 3-3-5 접촉 밀도모델(Li et al.) + +접촉 밀도모델은 아래와 같은 가정을 바탕으로 하였다. + +1. 균열면은 여러 경사면의 조합으로 이루어져 있다. 경사각(θ )은 접촉면 밀도 확률 함수 ( Ω( ) θ )로 표현 가능하다. +2. 접촉 응력의 방향은 초기 접촉 방향에 법선 방향으로 고정되어있다. +3. 밀도 함수 ( Ω( ) θ )는 삼각법에 의한 함수로 가정되었고, 골재의 크기와 입도 그리고 굵은 골재의 강도와 종류와는 독립적이다. +4. 접촉력은 완전 탄소성 모델에 의해서 산정된다. +5. 유효 접촉면적 비율 $K ( \Delta u _ { n } )$ 은 균열면의 거칠기에 비해 법선 방향의 균열 변위$( \Delta u _ { n }$ )가 충분히 클 때 접촉면에 손실을 야기한다. + +![](images/page-301_41d5952e83a2e3eddb0fa3b6d5ed2c85f9779015b0b54174afecdeaf3e3c5fe2.jpg) +그림 3.3.6 접촉밀도모델 + + + +접촉밀도모델에 대한 수식은 식 (3.3.7)과 같다. + +$$ +f _ {t} = \int_ {- \frac {1}{2} \pi} ^ {\frac {1}{2} \pi} \sigma_ {c o n} K (\Delta u _ {n}) A _ {t} \Omega (\theta) \sin \theta d \theta \tag {3.3.7} +$$ + +$$ +f _ {n} = \int_ {- \frac {1}{2} \pi} ^ {\frac {1}{2} \pi} \sigma_ {c o n} K (\Delta u _ {n}) A _ {t} \Omega (\theta) \cos \theta d \theta +$$ + +여기서, + +At : 균열 단면적 (균열면의 1.27배) + + + +# 3-4 부착슬립 + +철근콘크리트 구조물에서 철근과 콘크리트 사이의 계면거동은 이차 종, 횡방향 균열에 지배된다. 이러한 거동은 부착슬립(bond-slip)으로써 규정할 수 있으며 이는두께가 0인 계면요소를 사용하여 모델링하는 것이 일반적이다. + +midas FEA에서는 총 변형이론에 근거한 구성모델을 사용하여 부착슬립모델을 표현한다. 이때 법선방향 거동은 선형탄성거동, 접선방향거동은 비선형거동을 하는것으로 가정하며 이를 수식화 하면 다음과 같다. + +$$ +\begin{array}{l} t _ {n} = k _ {n} \Delta u _ {n} \\ \text { and } f (t _ {n}) \end{array} \tag {3.4.1} +$$ + +$$ +t _ {t} = f _ {t} (d t) +$$ + +식 (3.19)의 우변을 상대변위에 대해 일차 미분하여 계산한 접선 강성은 다음과 같다. + +$$ +\left[ \begin{array}{l l} D _ {1 1} = k _ {n} & D _ {1 2} = 0 \\ D _ {2 1} = 0 & D _ {2 2} = \frac {\partial f _ {t}}{\partial d t} \end{array} \right] \tag {3.4.2} +$$ + + + +# 3-4-1 입방함수(Cubic Function (Dorr)) + +Dorr은 전단응력과 슬립의 관계를 그림 3.4.1과 같이 표현하였으며, 수식은 식(3.4.3)와 같다. + +![](images/page-304_7abf5edba6d944f7cb173762ac71264479cd3e3af258abdde6f60bca31628d3e.jpg) +그림 3.4.1 부착슬립의 입방함수 + +$$ +f _ {t} = \left\{ \begin{array}{c c} c \left(5 \left(\frac {d t}{d t ^ {0}}\right) - 4. 5 \left(\frac {d t}{d t ^ {0}}\right) ^ {2} + 1. 4 \left(\frac {d t}{d t ^ {0}}\right) ^ {3}\right) & i f \quad 0 \leq d t \leq d t ^ {0} \\ 1. 9 c & i f \quad d t \geq d t ^ {0} \end{array} \right. \tag {3.4.3} +$$ + +여기서, + +c : 실험데이타에 의한 상수 + +0 dt : 전단슬립(shear slip) + +제하와 재하는 할선 접근방법을 사용한다. + + + +# 3-4-2 거듭제곱 법칙(Power Law (Noakowski)) + +Noakowski 모델은 식 (3.4.4)와 같이 지수형태로 부착슬립거동을 정의하였다. 그림 3.4.2와 같이 슬립이 초기 변위 ( 0 dt ) 보다 작은 경우에는 비정상적으로 초기전단강성 값이 커지는 것을 방지하기 위해 선형 거동하는 것으로 가정한다. + +![](images/page-305_97defe82786e52fe4a7616d66a4fe1e569c2690e61547e0d3e6fd4605f2d42f6.jpg) +그림 3.4.2 부착슬립의 거듭제곱 법칙 + +$$ +f _ {t} = \left\{ \begin{array}{l l} a (d t) ^ {b} & \text { if } \quad d t \geq d t ^ {0} \\ a (d t) ^ {b - 1} \Delta u _ {t} & \text { if } \quad 0 \leq d t \leq d t ^ {0} \end{array} \right. \tag {3.4.4} +$$ + +여기서, + +$$ +a, b \quad : \text { 실험데이터에 의한 상수 } (b < 1) +$$ + +$$ +d t ^ {0} \quad : \text { 전단슬립(shear slip) } +$$ + +제하와 재하는 할선 접근방법을 사용한다. + + + +# 3-5 쿨롱마찰 + +midas FEA에서는 마찰 거동에 지배되는 이질적 또는 동질적인 재료의 경계면을표현하기 위해, 계면요소의 거동을 쿨롱마찰이론(coulomb friction theory)을 사용하여 정의한다. 미소 상대변위 ∆u 를 변위분리 가정에 따라 탄성과 소성에 대한미소증분형태로 분리하여 나타내면 다음과 같다. + +$$ +\Delta \dot {\mathbf {u}} = \Delta \dot {\mathbf {u}} ^ {e} + \Delta \dot {\mathbf {u}} ^ {p} \tag {3.5.1} +$$ + +계면에 발생하는 응력의 증분량은 다음과 같이 정의된다. + +$$ +\dot {\mathbf {t}} = \mathbf {D} ^ {e} \Delta \dot {\mathbf {u}} ^ {e} = \mathbf {D} ^ {e} \left(\Delta \dot {\mathbf {u}} - \Delta \dot {\mathbf {u}} ^ {p}\right) \tag {3.5.2} +$$ + +![](images/page-306_f513d290a6621dd4d54341a0ce544ce83e8b829b5f08c4de2089d5484d6f1aaf.jpg) + +
+text_image + +t_t +c / tanø +c +Ø +t_n +
+ +그림 3.5.1 쿨롱마찰 기준 + +쿨롱마찰모델의 파괴함수 f 와 포텐셜함수 g 는 마찰각을 나타내는 φ 와 점착력 + + + +을 나타내는 c 값으로 구성된 2계수를 사용하여 식 (3.5.3)과 같이 정의하며 응력면상에서의 형상은 그림 3.5.1과 같다. + +$$ +\left\{ \begin{array}{l} f = \sqrt {t _ {t} ^ {2}} + t _ {n} \tan \phi (\kappa) - c (\kappa) = 0 \\ g = \sqrt {t _ {t} ^ {2}} + t _ {n} \tan \varphi \end{array} \right. \tag {3.5.3} +$$ + +여기서, + +k）： + +c k( ) : 내부변수 κ 의 함수로 정의되는 점착계수 + +소성상대변위 p ∆u 는 다음과 같이 크기를 나타내는 소성승수와 소성방향을 나타내는 성분으로 구분하여 정의할 수 있다. + +$$ +\Delta \dot {\mathbf {u}} ^ {p} = \dot {\lambda} \frac {\partial g}{\partial \mathbf {t}} \tag {3.5.4} +$$ + +외력에 저항하는 구조체의 소성응답거동을 정의하기 위해 Taylor series 1차 확장을 사용하여 파괴함수 f 를 확장하면 다음과 같이 증가률형태로 수식화할 수 있다. + +$$ +\dot {f} = \frac {\partial f ^ {T}}{\partial \mathbf {t}} \dot {\mathbf {t}} + \frac {\partial f}{\partial \kappa} \dot {\kappa} = 0 \tag {3.5.5} +$$ + +이때 소성승수 λ 는 식(3.5.5)에 식 (3.5.2), (3.5.4)를 대입하여 λ 에 대해 정리하면 다음과 같다. + +$$ +\dot {\lambda} = \frac {\frac {\partial f ^ {T}}{\partial \mathbf {t}} \mathbf {D} ^ {e}}{- \frac {\partial f}{\partial \kappa} + \frac {\partial f ^ {T}}{\partial \mathbf {t}} \mathbf {D} ^ {e} \frac {\partial g}{\partial \mathbf {t}}} \Delta \dot {\mathbf {u}} = \frac {\frac {\partial f ^ {T}}{\partial \mathbf {t}} \mathbf {D} ^ {e}}{- h + \frac {\partial f ^ {T}}{\partial \mathbf {t}} \mathbf {D} ^ {e} \frac {\partial g}{\partial \mathbf {t}}} \Delta \dot {\mathbf {u}} \tag {3.5.6} +$$ + +내부변수 증가량 κ 과 소성승수 증가량 λ 의 관계는 다음과 같다. + + + +$$ +\dot {\kappa} = \left| \Delta \dot {\mathbf {u}} _ {t} ^ {p} \right| = \dot {\lambda} \left| \frac {\partial g}{\partial \mathbf {t}} \right| = \dot {\lambda} \sqrt {\left(\frac {\partial g}{\partial \mathbf {t}}\right) ^ {T} \cdot \left(\frac {\partial g}{\partial \mathbf {t}}\right)} = \dot {\lambda} \sqrt {1 + \tan^ {2} \varphi} = \dot {\lambda} \quad \because \tan \varphi < < 1 \tag {3.5.7} +$$ + +여기서, + +$$ +\partial \mathbf {g} / \partial \mathbf {t} = \left\{\tan \varphi \quad t _ {t} / \mid t _ {t} \mid \right\} +$$ + +최종적으로 응력증가량은 식(3.5.2)에 식(3.5.4), (3.5.6)을 대입하여 정리하면 다음과 같이 식 (3.5.8)로 나타낼 수 있다. + +$$ +\dot {\mathbf {t}} = \left\{\mathbf {D} ^ {e} - \frac {\mathbf {D} ^ {e} \frac {\partial \mathbf {g}}{\partial \mathbf {t}} \frac {\partial f ^ {T}}{\partial \mathbf {t}} \mathbf {D} ^ {e}}{- h + \frac {\partial f ^ {T}}{\partial \mathbf {t}} \mathbf {D} ^ {e} \frac {\partial \mathbf {g}}{\partial \mathbf {t}}} \right\} \Delta \dot {\mathbf {u}} \tag {3.5.8} +$$ + +여기서, + +$$ +\mathbf {D} ^ {e} = \left[ \begin{array}{c c} k _ {n} & 0 \\ 0 & k _ {t} \end{array} \right] +$$ + +$$ +h = \frac {\partial f}{\partial \kappa} = \frac {\partial f}{\partial \mathbf {t}} \frac {\partial \mathbf {t}}{\partial \Delta \mathbf {u} _ {t} ^ {p}} \frac {\partial \Delta \mathbf {u} _ {t} ^ {p}}{\partial \kappa} +$$ + +$$ +\frac {\partial g}{\partial \mathbf {t}} = \left\{\tan \varphi \frac {t _ {t}}{\left| t _ {t} \right|} \right\} +$$ + +$$ +\frac {\partial f ^ {T}}{\partial \mathbf {t}} = \left\{\tan \phi (k) \frac {t _ {t}}{\left| t _ {t} \right|} \right\} +$$ + +위의 식을 정리하여 다시 쓰면 다음과 같다. + +$$ +\dot {\mathbf {t}} = \frac {1}{- h + k _ {n} \tan \phi \tan \varphi + k _ {t}} \left[ \begin{array}{c c} k _ {n} (h + k _ {t}) & - k _ {n} k _ {t} \tan \varphi \frac {t _ {t}}{\left| t _ {t} \right|} \\ - k _ {n} k _ {t} \tan \phi (k) \frac {t _ {t}}{\left| t _ {t} \right|} & k _ {t} (h + k _ {n} \tan \phi \tan \varphi) \end{array} \right] \Delta \dot {\mathbf {u}} \tag {3.5.9} +$$ + + + +이때 φ ≠ ϕ 인 경우 식 (3.5.9)에서 대괄호⎡ ⎤ ⎣ ⎦ 안의 행렬은 비대칭이 되어 비상관소성흐름이 발생하며 φ = ϕ 인 경우 식 (3.5.9)에서 대괄호⎡ ⎤ ⎣ ⎦ 안의 행렬은 대칭이 되어 상관소성흐름이 발생한다. 상관소성흐름해석을 수행할 경우 계면에 수직한 방향으로 과도한 열림현상이 발생하게 되며 이는 실제거동과 맞지 않게 된다. 그러나 비상관소성흐름해석을 수행할 경우 강성행렬의 저장량이 커짐에 따라 메모리와수행속도가 증가하여 해석이 느리게 진행될 수 있다. 특히 φ 값과 ϕ 값의 차이가클 경우, 즉 비상관성이 큰 해석의 경우에는 수렴이 잘 되지 않는 현상이 관찰된다. 이를 위해 midas FEA에서는 φ −ϕ ≤ ° 20 가 되도록 권장하고 있다. + + + +# 3-6 복합파괴 모델 (Combined Cracking-Shearing-Crushing) + +계면 재료 모델 중 ‘복합파괴모델(Combined Cracking-Shearing-Crushing)’은조적식 구조물의 결합부와 같은 모델의 파괴, 마찰슬립(frictional slip)과압괴파괴(crushing) 거동을 해석하는데 적합하다. 그림 3.6.1 과 같은 일반적인경우에 단위 벽돌은 선형 탄성 또는 점탄성으로 모델링하고, 모르타르와 같은결합부는 Lourenco 와 Rots 와 van Zijl 이 제시한 ‘복합파괴모델’ 계면요소로모델링한다. + +![](images/page-310_5e46c1e4128f813a1da17720b70e09ed120e0eb6ad8c612a68545f252863a9e8.jpg) + +
+text_image + +mortar +interface +interface +brick +potential brick crack +
+ +![](images/page-310_19d4344dc5337c855529bad20aba56a6b5b8104b8f80e437a95a607e377b4b32.jpg) + +
+text_image + +joint / interface +joint / interface +brick +potential brick crack +
+ +그림 3.6.1 조적식 구조물의 모델링 방법 + +# 3-6-1 2차원 계면모델 + +midas FEA에서 2차원 계면모델은 Lourenco와 Rots 모델과 이를 개선한 van Zijl의 모델을 이용하여 정의하였다. 이 모델은 인장한계(tension cutoff) 및 타원형의압축 캡모델(Elliptical compression cap model)과 결합된 쿨롬 마찰모델에 기초하여 다중면 소성모델을 사용한다. 그림 3.6.12와 같이 연화과정은 쿨롱 마찰모드,인장모드와 캡모드 3방향으로 일어나며, 경화과정은 캡모드에서만 고려된다. 계면모델에서 응력, 변형률 벡터는 다음과 같다. diff --git a/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_032.md b/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_032.md new file mode 100644 index 00000000..4051dbf1 --- /dev/null +++ b/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_032.md @@ -0,0 +1,346 @@ + + +$$ +\mathbf {t} = \binom {t _ {n}} {t _ {t}} = \binom {\sigma} {\tau}, \quad \Delta \mathbf {u} = \binom {\Delta u _ {n}} {d t} \tag {3.6.1} +$$ + +여기서, + +nt , un : 계면 법선방향의 응력과 상대변위 + +tt , dt : 접선방향의 전단력과 상대변위 + +![](images/page-311_f8cd791a35d67dff7606542b127adcef1bf386316c2f0df4c793fee21538d2d2.jpg) + +
+text_image + +compression cap mode +coulomb friction mode +|τ| +θ +tension mode +intermediate yield surface +residual yield surface +initial yield surface +σ +
+ +그림 3.6.2 2 차원 계면 모델 + +탄성영역에서 구성방정식은 다음과 같다. + +$$ +\mathbf {t} = \mathbf {D} \Delta \mathbf {u} \tag {3.6.2} +$$ + +여기서, 강성 행렬의 대각행렬은 다음과 같이 구성된다. + +$$ +\mathbf {D} = \operatorname{diag} \left[ \begin{array}{l l} k _ {n} & k _ {s} \end{array} \right] \tag {3.6.3} +$$ + + + +# 전단 슬립 + +초기 쿨롱 마찰항복한계(coulomb friction yield criterion)는 다음과 같다. + +$$ +f = \left| t _ {t} \right| + t _ {n} \Phi - c \tag {3.6.4} +$$ + +여기서, + +tt : 접선방향의 전단슬립응력 + +nt : 경계면에 수직방향의 법선 응력 + +Φ = tanφ : 마찰각 + +c : 점착력 + +쿨롱 마찰항복한계에서 연화거동은 내부변수 κ 를 사용하여 점착력 c 와 마찰계수Φ 값을 감소시킴으로써 나타내며 다음과 같이 수식화한다. + +$$ +c \left(t _ {n}, \kappa\right) = c _ {0} e ^ {- \frac {c _ {0}}{G _ {f} ^ {l l}} \kappa}, \quad \Phi \left(t _ {n}, \kappa\right) = \Phi_ {0} + \left(\Phi_ {r} - \Phi_ {0}\right) \frac {c _ {0} - c}{c _ {0}} \tag {3.6.5} +$$ + +여기서, + +$c _ { 0 }$ : 초기 점착력 + +$G _ { f } ^ { I I }$ : 모드-II 형식의 전단슬립 파괴에너지 + +$\Phi _ { 0 }$ Φ : 초기 마찰계수 + +$\Phi _ { r }$ : 잔류 마찰계수 + +여기서, 파괴에너지는 법선방향의 구속 응력과 선형관계가 있으며, 둘 사이의 관계는 다음과 같다. + +$$ +G _ {f} ^ {I I} = \left\{ \begin{array}{c c} a t _ {n} + b & \text { if } t _ {n} < 0 \\ b & \text { if } t _ {n} \geq 0 \end{array} \right. \tag {3.6.6} +$$ + +여기서, a , b 는 실험 데이터를 바탕으로 선형 회기 분석에 의해 결정된다. + + + +# 팽창 + +소성상대변위 p ∆u 는 식 (3.5.4)와 같이 포텐셜 함수로 나타낼 수 있다. + +$$ +\Delta \dot {\mathbf {u}} ^ {p} = \dot {\lambda} \frac {\partial g}{\partial \mathbf {t}} \tag {3.6.7} +$$ + +그리고 포텐셜 함수는 식 (3.6.8)과 같다. + +$$ +\frac {\partial g}{\partial t _ {n}} = \left\{ \begin{array}{c} \Psi \\ \text { sign } (t _ {t}) \end{array} \right\} \tag {3.6.8} +$$ + +위 식 (3.6.7)과 (3.6.8)을 이용하여 팽창 계수(ψ = tanϕ )를 나타내면 식 (3.6.9)와같다. + +$$ +\psi = \frac {\Delta \dot {u} _ {n} ^ {p}}{\Delta \dot {u} _ {t} ^ {p}} \text { sign } (t _ {t}) \tag {3.6.9} +$$ + +실험에 의해 팽창 계수가 구속응력과 전단 슬립에 의한 함수임이 입증되었으며,식 (3.6.10)과 같이 나타낼 수 있다. + +$$ +\psi = \psi_ {1} (\Delta \dot {u} _ {n} ^ {p}) \psi_ {2} (t _ {t} ^ {p}) \tag {3.6.10} +$$ + +따라서 포텐셜 함수는 다음과 같이 나타난다. + +$$ +g = \int \left(\frac {\partial g}{\partial t _ {n}}\right) ^ {T} d \mathbf {t} = \left| t _ {t} \right| + \psi_ {2} \left(\Delta \dot {u} _ {t} ^ {p}\right) \int \psi_ {1} \left(t _ {n}\right) d t _ {n} \tag {3.6.11} +$$ + +접선 슬립에 의한 법선 방향의 상대 변위는 다음과 같다. + +$$ +\Delta \dot {\mathbf {u}} ^ {p} = \left\{ \begin{array}{c c} 0 & \text { if } t _ {n} < \sigma_ {u} \\ \frac {\psi_ {0}}{\delta} \left(1 - \frac {t _ {n}}{\sigma_ {u}}\right) \left(1 - e ^ {- \delta \Delta u _ {t} ^ {p}}\right) & \text { if } \sigma_ {u} \leq t _ {n} < 0 \\ \frac {\psi_ {0}}{\delta} \left(1 - e ^ {- \delta \Delta u _ {t} ^ {p}}\right) & \text { if } t _ {n} \geq 0 \end{array} \right. \tag {3.6.12} +$$ + +그리고 미분 이후 팽창 계수는 다음과 같이 유도된다. + + + +$$ +\psi = \left\{ \begin{array}{c c} 0 & \text { if } t _ {n} < \sigma_ {u} \\ \psi_ {0} \left(1 - \frac {t _ {n}}{\sigma_ {u}}\right) \left(1 - e ^ {- \delta \Delta \dot {u} _ {t} ^ {p}}\right) & \text { if } \sigma_ {u} \leq t _ {n} < 0 \\ \psi_ {0} \left(1 - e ^ {- \delta \Delta \dot {u} _ {t} ^ {p}}\right) & \text { if } t _ {n} \geq 0 \end{array} \right. \tag {3.6.13} +$$ + +여기서, + +$\psi _ { 0 }$ 초기 팽창 계수 + +$\sigma _ { u }$ O ： + +$\delta$ + +# 연화거동 + +변형연화가정을 따르며 전단슬립에 의해 정의되는 연화는 다음과 같다. + +$$ +\Delta \kappa = \left| \Delta \dot {u} _ {t} ^ {p} \right| = \Delta \lambda \tag {3.6.14} +$$ + +소성 변형률 증가( ∆κ, ∆λ)는 뉴튼 랩슨법에 의해 산정된다. + +# 인장 한계 거동 + +인장 한계에 대한 파괴규준은 다음과 같이 랭킨 기준(Rankine criterion)을 사용한다. + +$$ +f _ {2} = t _ {n} - \sigma_ {t} \tag {3.6.15} +$$ + +여기서, $\sigma _ { t }$ 는 벽돌 모르타르 부착강도(brick-mortar bond strength)이며 다음과같다. + + + +$$ +\sigma_ {t} = f _ {t} e ^ {- \frac {f _ {t}}{G _ {f} ^ {I}} \kappa_ {2}} \tag {3.6.16} +$$ + +여기서, + +$$ +f _ {t} \quad : \text { 부착 강도 } +$$ + +$$ +G _ {f} ^ {I} \quad : \text { 모드-1 파괴 에너지 } +$$ + +이때 내부변수 κ 2 는 다음과 같다. + +$$ +\Delta \kappa_ {2} = \left| \Delta u _ {p} \right| \tag {3.6.17} +$$ + +상관소성흐름법칙을 적용하면 다음과 같다. + +$$ +\Delta u _ {p} = \Delta \lambda_ {2} \frac {\partial f _ {2}}{\partial t} \tag {3.6.18} +$$ + +그러므로 결과적으로 다음의 결론을 도출한다. + +$$ +\Delta \kappa = \Delta \lambda_ {2} \tag {3.6.19} +$$ + +# 압축 캡거동 + +압축 캡에 대한 파괴규준은 다음과 같다. + +$$ +f _ {3} = t _ {n} ^ {2} + C _ {s} t _ {t} ^ {2} - \sigma_ {c} ^ {2} \tag {3.6.20} +$$ + +여기서, + +$$ +C _ {s} \quad : \text { 파괴시 전단응력 분포를 나타내는 계수 } +$$ + +$$ +\sigma_ {c} \quad : \text { 압축강도 } +$$ + +압축 캡거동에 사용되는 내부변수 κ 3는 다음과 같다. + + + +$$ +\Delta \kappa_ {3} = \sqrt {\Delta u _ {p} ^ {T} \Delta u _ {p}} \tag {3.6.21} +$$ + +여기서, + +$$ +\Delta \dot {u} _ {n} ^ {p} = \Delta \lambda_ {3} \frac {\partial f _ {3}}{\partial t _ {n}} +$$ + +그러므로 + +$$ +\Delta \kappa_ {3} = 2 \Delta \lambda_ {3} \sqrt {t _ {n} ^ {2} + \left(C _ {s} t _ {t}\right) ^ {2}} \tag {3.6.22} +$$ + +![](images/page-316_aabf0971053fb445ab9428d88cf655ab41425a26f52ba1881abf74f050a100d6.jpg) + +
+line +| Point | K3 | σc | +|-------|----|----| +| σ₁ | Kp | f_c | +| σ₂ | Km | σ_m | +| σ₃ | Km | σ_r | +| σ_i | Kp | σ_i | +
+ +그림 3.6.3 계면의 압축 캡에서 경화-연화 법칙 + +캡모델은 그림 3.6.3에서와 같이 최대 강도( f )까지는 경화거동을 하며, 그 이후부터는 연화거동을 한다. + + + +위 그림과 같이 3가지 영역에 따라 내부변수에 따른 응력은 식 (3.6.23)과 같다. + +$$ +\overline {{{{\sigma_ {1}}}}} \left(\kappa_ {3}\right) = \overline {{{{\sigma_ {i}}}}} + \left(f _ {c} - \overline {{{{\sigma_ {i}}}}}\right) \sqrt {\frac {2 \kappa_ {3}}{\kappa_ {p}} - \frac {\kappa_ {3} ^ {2}}{\kappa_ {p} ^ {2}}} +$$ + +$$ +\overline {{{\sigma_ {2}}}} \left(\kappa_ {3}\right) = f _ {c} + \left(\overline {{{\sigma_ {m}}}} - f _ {c}\right) \left(\frac {\kappa_ {3} - \kappa_ {p}}{\kappa_ {m} - \kappa_ {p}}\right) ^ {2} \tag {3.6.23} +$$ + +$$ +\overline {{\sigma_ {3}}} \left(\kappa_ {3}\right) = \overline {{\sigma_ {r}}} + \left(\overline {{\sigma_ {m}}} - \overline {{\sigma_ {r}}}\right) \exp \left(2 \left(\frac {\overline {{\sigma_ {m}}} - f _ {c}}{\kappa_ {m} - \kappa_ {p}}\right) \left(\frac {\kappa_ {3} - \kappa_ {m}}{\overline {{\sigma_ {m}}} - \overline {{\sigma_ {r}}}}\right)\right) +$$ + +식 (3.54)에서 $\overline { { \sigma _ { i } } } = \frac { 1 } { 3 } f _ { c } , \overline { { \sigma _ { m } } } = \frac { 1 } { 2 } f _ { c } , \overline { { \sigma _ { r } } } = \frac { 1 } { 1 0 } f _ { c }$ 이다. + +# 교차점(Corner) 거동 + +쿨롱 마찰한계과 인장한계 또는 압축 캡과의 교차지점에서 소성 변위 증가율은 식(3.6.24)와 같다. + +$$ +\Delta \dot {u} _ {n} ^ {p} = \Delta \lambda_ {1} \frac {\partial g _ {1}}{\partial t _ {n}} + \Delta \lambda_ {i} \frac {\partial g _ {i}}{\partial t _ {n}} \tag {3.6.24} +$$ + +여기서,, + +아래 첨자 1 : 전단 한계 + +i가 2인 경우 : 인장한계 + +i가 3인 경우 : 압축 캡 + +# 3-6-2 3차원 계면모델 + +3차원 문제에 있어서 midas FEA에는 인장한계모델만이 고려되며 압축 캡모델은고려하지 않는다. 삼차원문제는 단순히 2차원 문제의 확장으로써 다음과 같이 나타낸다. + + + +$$ +\boldsymbol {t} = \left\{ \begin{array}{l} t _ {n} \\ t _ {s} \\ t _ {t} \end{array} \right\}, \quad \boldsymbol {u} = \left\{ \begin{array}{l} \Delta u _ {n} \\ \Delta u _ {s} \\ \Delta u _ {t} \end{array} \right\}, \quad \boldsymbol {D} = \left[ \begin{array}{c c c} k _ {n} & 0 & 0 \\ 0 & k _ {s} & 0 \\ 0 & 0 & k _ {t} \end{array} \right] \tag {3.6.25} +$$ + +![](images/page-318_5eb01301ae07753fdc8abd7450f0464bb4e3fc3e78e69e9211c9d9da65d24b30.jpg) + +
+text_image + +τs +C0 +τt ← C0 +
+ +![](images/page-318_d8f14bb63cc5c2a87b2889c5d73078dccedf1fbc4f8eaf388c6e7706b366bbef.jpg) + +
+text_image + +τs +ft +σ +τt +
+ +그림 3.6.4 3차원 계면 항복 함수 + +파괴함수는 다음과 같다. + +$$ +f = \sqrt {t _ {s} ^ {2} + t _ {t} ^ {2}} + t _ {n} \Phi - c \tag {3.6.26} +$$ + +$$ +\Delta \dot {\boldsymbol {u}} ^ {p} = \Delta \lambda \frac {\partial g}{\partial \boldsymbol {t}} = \Delta \lambda \left[ \begin{array}{c} \varphi \\ \frac {t _ {s}}{\sqrt {t _ {s} ^ {2} + t _ {t} ^ {2}}} \\ \frac {t _ {t}}{\sqrt {t _ {s} ^ {2} + t _ {t} ^ {2}}} \end{array} \right] \tag {3.6.27} +$$ + +위 식 (3.6.28)에서 ϕ 는 비상관 팽창각이며 ∆κ은 내부변수 증가량으로써 다음과같다. + +$$ +\Delta \kappa = \sqrt {\left(\Delta \dot {u} _ {s} ^ {p}\right) ^ {2} + \left(\Delta \dot {u} _ {t} ^ {p}\right) ^ {2}} = \Delta \lambda \tag {3.6.28} +$$ + + + +# Analysis and Algorithm Manual + +# Part 3 General Algorithms + +Chapter 1. Load and Boundary + +Chapter 2. Equation Solver + +Chapter 3. Iteration Methods + + diff --git a/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_033.md b/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_033.md new file mode 100644 index 00000000..888069bc --- /dev/null +++ b/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_033.md @@ -0,0 +1,270 @@ + + +# Chapter 1. Load and Boundary + +# 1-1 자유도 구속조건 + +자유도 구속(constraint) 기능은 임의 절점의 변위를 구속시키거나 자유도가 부족한 요소(truss, plane stress 요소 등)끼리 접합될 때 해당 자유도성분을 구속하는데 사용된다. 자유도 구속조건은 전체좌표계(global coordinate system) 또는 절점좌표계(nodal coordinate system)를 기준으로 절점당 6개 자유도에 대해 입력된다. + +예를 들어 그림 1.1.1과 같은 평면골조모델에 자유도 구속조건을 부여하는 방법은다음과 같다. 이 모델은 전체좌표계 X Z− 평면내에서만 거동이 허용되는 2차원모델이기 때문에 모든 절점에 대해 전체좌표계 Y 방향 변위자유도와 X 방향 및Z 방향에 대한 회전자유도를 구속해야 한다. + +![](images/page-321_10e88087016c348fe05a487d7a875259e7d23d1fc62c4487b58c8e5ead7b70c3.jpg) + +
+text_image + +N2 +N4 +N6 +N1 +N3 +z +y +x +angle of +inclination +Y +X +• : fixed support condition +: pinned support condition +: roller support condition +
+ +그림 1.1.1 자유도 구속조건이 고려된 평면골조모델 + +고정지지조건인 절점 N1은 전체좌표계 X , Z 방향 변위자유도와 Y 방향에 대한회전자유도를 추가로 구속한다. 핀접합이면서 롤러 지지조건인 N3에 대해서는 Z + + + +방향 변위자유도를 추가로 구속한다. 절점좌표계에 대해 로울러 지지조건이 부여된 절점 N5에 대해서는 전체좌표계 X 축에 대해 경사각만큼 회전한 절점좌표계를설정한 다음, 전체좌표계 Z 축 방향 변위자유도를 구속한다. 절점좌표계가 선언되어 있는 절점에 입력되는 구속조건은 절점좌표계를 따라 구속을 수행하게 된다. + +절점변위를 구속하는 기능은 변위를 무시할 수 있는 지지조건(supports) 등에 주로 이용되며 임의 절점에 대해 구속조건이 주어지면 해당절점에 대한 반력이 발생하게 된다. 절점에서의 반력은 전체좌표계를 기준으로 출력된다. + +그림 1.1.2는 구속조건이 부족한 자유도에 구속조건을 사용한 예 이다. 그림 1.1.2(a)의 경우는 트러스요소가 축방향의 변위자유도만 가지기 때문에 연결절점에서의X 방향 변위와 모든 회전방향 변위성분은 구속되어있다. 그림 1.1.2 (b)는 상·하부플랜지를 보요소로 대신한 예로써 보요소가 절점당 6개의 자유도를 가지기 때문에보요소와 연결되는 절점에서는 별도의 구속조건이 필요 없다. 그러나 평면응력요소만으로 구성된 절점에 대해서는 평면응력요소가 면내의 거동에 대한 자유도만가지고 있기 때문에 면외변위성분인 Y 방향 변위자유도와 모든 회전자유도를 구속하여야 한다. 구조물이 충분한 강성을 가지지 못하는 경우에는 원활한 해석을 위해서 전체구조물의 해석에 영향을 거의 주지 않은 작은 강성을 부여할 수도 있다. + + + +![](images/page-323_24bce34b260fa921ea3c8beb173e812ba02f21c77fab94a310dc667a2382b9a5.jpg) + +
+text_image + +connecting node +(DX, RX, RY and RZ are constrained) +supports (all degrees of +freedom are constrained) +
+ +(a) 트러스요소끼리 접합된 경우 + +![](images/page-323_e1d4479804093cafc46522a94d102ac21f508425d911f92d3a84f4b16e29c9f9.jpg) + +
+text_image + +supports (all degrees of freedom are constrained) +top flange (beam element) +web (plane stress element) +Y +Z +X +bottom flange (beam element) +in-plane vertical load +● : nodes without constrains +○ : DY, RX, RY and RZ are constrained +DX : displacement in the GCS X direction +DY : displacement in the GCS Y direction +DZ : displacement in the GCS Z direction +RX : rotation about the GCS X-axis +RY : rotation about the GCS Y-axis +RZ : rotation about the GCS Z-axis +
+ +(b) H형 캔틸레버를 상/하부 플랜지를 보요소로 모델링하고 웨브를 평면응력요소로 모델링한 경우 +그림 1.1.2 자유도 구속조건의 사용 예 + + + +# 1-2 경사 지지조건 + +그림 1.2.1과 같이 경사지지(skewed support) 조건을 갖는 구조물의 경계조건은자유도 구속조건과 절점좌표계를 사용하여 모델링 한다. 구속 자유도는 자유도 구속조건에서 입력된 방향을 사용하고, 해당자유도의 좌표축은 절점좌표계를 따른다.경사지지조건이 입력된 절점의 반력은 전체좌표계를 기준으로 출력된다. + +![](images/page-324_3ad19cbeb0cc1f80a00375c30dadabcde6a5750a6e65a6e4e0c06aa8ad12305f.jpg) + +
+text_image + +Z +N +X +N +X +X +θ +Y +X +X +
+ +그림 1.2.1 경사지지조건의 예 + + + +# 1-3 구속 방정식 + +구속방정식(constraint equation)은 구조물의 특정 절점이 다른 절점들의 거동에종속적으로 움직이도록 구속하는 기능이다. 여기서, 구속하고자 하는 특정 절점을구속절점(constraint node)이라 하고, 자유도가 독립적인 절점을 독립절점(independent node)이라 한다. 구속방정식에 의한 구속절점과 독립절점간의 관계는 다음과 같다. + +$$ +U _ {M, m} = a _ {1} U _ {I, i} + a _ {2} U _ {J, j} + \dots + b _ {1} R _ {I, i} + b _ {2} R _ {J, j} + \dots \quad - \text { 이동변위 구속 } \tag {1.3.1} +$$ + +$$ +R _ {M, m} = c _ {1} U _ {I, i} + c _ {2} U _ {J, j} + \dots + d _ {1} R _ {I, i} + d _ {2} R _ {J, j} + \dots \quad - \text { 회전변위 구속 } \tag {1.3.2} +$$ + +여기서, + +$$ +U _ {M, m} \quad : \text { 구속절점 } m \text { 의 } M \text { 방향 이동변위 } +$$ + +$$ +U _ {I, i} \quad : \text { 독립절점 } i \text { 의 } I \text { 방향 이동변위 } +$$ + +$$ +R _ {M, m} \quad : \text { 구속절점 } m \text { 의 } M \text { 방향 회전변위 } +$$ + +$$ +R _ {I, i} \quad : \text { 독립절점 } i \text { 의 } I \text { 방향 회전변위 } +$$ + +$$ +a _ {i}, b _ {i}, c _ {i}, d _ {i} \quad : \text { 자유도 간의 상호관계를 정의하는 계수 } +$$ + +구속방정식은 식 (1.3.1) 또는 식 (1.3.2)와 같이 임의 절점, 임의의 자유도를 상호구속할 수 있기 때문에 적용 범위가 매우 넓다. 구속방정식은 전체좌표계에서 정의된 자유도에 대해 적용된다. + +그림 1.3.1은 입체요소로 이루어진 3차원 구조물과 판요소로 이루어진 얇은 판의접합 부분에 적용한 예이다. 입체요소는 회전자유도에 대한 강성이 없기 때문에접합되어 있는 판의 회전거동을 구속하지 못한다. 구속방정식을 이용하여 접합 부분의 회전자유도를 식(1.3.3)과 같이 구속하게 되면, 판요소가 접합면에 대체로 수 + + + +직을 이룬 상태에서 거동하게 된다. + +$$ +R _ {Y, 3} = \frac {1}{h} U _ {X, 1} - \frac {1}{h} U _ {X, 2} \tag {1.3.3} +$$ + +![](images/page-326_e0970ea76f76092a9481f85bdb1c952e4f211fe4500b9a53961392beb23c600b.jpg) + +
+natural_image + +3D wireframe diagram of a curved surface with a dotted circular feature and arrow indicating direction (no text or symbols) +
+ +![](images/page-326_0ca2a6b7659a5255620cb8c843ae9af5eff3d161bb5f666168fb7a34acde67fd.jpg) + +
+text_image + +U_{X,1} +N1 +Ry_3 +N3 +N2 +U_{X,2} +
+ +그림 1.3.1 구속방정식의 적용 예 + +구속방정식은 강체연결(rigid link)과 혼동을 일으키기 쉽기 때문에 상호간의 차이점을 정확하게 인지해야 한다. 강체연결에 의한 연결은 하나의 절점 거동에 의해다른 여러 절점이 종속적으로 움직이도록 하는 반면, 구속방정식에 의한 구속조건 + + + +은 하나의 절점이 다른 여러 절점의 거동에 종속되도록 한다. + +구속방정식을 부가하는 방법에는 두 가지가 있다. 식 (1.3.1) 또는 식 (1.3.2)의 모든 자유도와 계수를 직접 입력하여 구속방정식을 생성하는 방법을 명시적 방법(explicit method)이라 하고, 구속절점의 움직임을 독립절점의 평균 변위와 같게하는 방법을 가중변위 방법(weighted displacement)라 한다. 각각의 방법으로 구속방정식을 부가하는 과정은 다음과 같다. + +# 명시적 방법 + +- 구속절점 및 자유도(복수 선택 불가) 선택 +- 독립절점 및 자유도(복수 선택 불가) 선택, 계수( ,i ia b 또는 ,i ic d ) 입력 +독립절점에 대한 정보는 반복적으로 입력할 수 있으며, 이러한 과정으로 식(1.3.1)또는 식(1.3.2)를 생성한다. + +# 가중변위 방법 + +- 구속절점 선택 +- 구속절점과 독립절점에 공통으로 적용할 자유도(복수 선택 가능) 선택 +- 독립절점 선택, 가중치 w 입력 + +독립절점에 대한 정보는 반복적으로 입력할 수 있으며, 다음과 같은 구속방정식이생성된다. + +$$ +U _ {I, m} = \sum_ {i} \frac {w _ {i}}{S} U _ {I, i} \text { 또는 } R _ {I, m} = \sum_ {i} \frac {w _ {i}}{S} R _ {I, i} +$$ + +여기서, + +S : 가중치( w )를 모두 더한 값 + +이 방법을 이용하면 이동변위와 회전변위간의 연관(coupling)이 발생하지 않는다. + + + +# 1-4 절점하중 + +절점하중(nodal load)은 가장 기본적인 하중으로 절점별로 6개 방향의 하중값을입력할 수 있다. 절점하중의 방향은 임의의 좌표계에 대하여 정의할 수 있다. + + + +# 1-5 요소압력하중 + +요소압력 하중(element pressure load)은 요소의 면(surface)이나 변(edge)에 분포하중의 형태로 입력된다. 2차원 요소(판요소, 평면응력요소, 평면변형요소, 축대칭요소)와 3차원 요소(입체요소)에 입력이 가능하고 요소강성 구성 시에 절점자유도의 하중으로 변환되어 해석에 사용된다. 하중의 입력방향은 요소좌표계나 전체좌표계 방향으로 입력할 수 있고, 임의 방향인 벡터 형태의 입력도 가능하다. 그림1.5.1은 요소압력하중의 예를 나타내고 있다. + +![](images/page-329_866770365b45ca21d755fc3d05c45494ef039431856e41304a8e0e54fc95f5da.jpg) + +
+text_image + +P₁ +P₁ +N₄ +Edge 3 +N₃ +P₂ +Edge 4 +Edge 2 +N₁ +Edge 1 +N₂ +P₁ +P₂ +P₁ +P₂ +
+ +(a) 판요소, 평면응력요소, 평면변형요소, 축대칭요소의 각 변에수직방향으로 작용하는 압력하중 + + + +![](images/page-330_355a625c023af1db0932e82f21a441147474d0bd00484e804148eb8a5f90bdfb.jpg) + +
+natural_image + +Isometric diagram of a rectangular frame with vertical arrows indicating force or direction (no text or symbols) +
+ +(b) 판요소와 평면응력요소의 면에 작용하는 압력하중 +![](images/page-330_f78a2765e1fd7e3f2e49acce44f32a795b83ea758f7c2c93addd8c29097fac23.jpg) + +
+natural_image + +Isometric line drawing of a 3D cube with internal vertical supports and hatched walls (no text or symbols) +
+ +(c) 입체요소의 각 면에 수직방향으로 작용하는 압력하중 +그림 1.5.1 각 요소에 작용하는 압력하중 diff --git a/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_034.md b/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_034.md new file mode 100644 index 00000000..dfd9d7b6 --- /dev/null +++ b/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_034.md @@ -0,0 +1,212 @@ + + +# 1-6 체적력 + +체적력(body force)은 주로 구조물의 자중이나 관성력을 모델링하는데 사용된다.모든 요소에 재하가 가능하고, 전체좌표계 X , Y , Z 에 대하여 입력이 가능하다.체적력의 기본 적분식은 식 (1.6.1)와 같고, , , ωx y zω ω 의 크기는 밀도와 방향별 중력 계수(gravitational force factor)로부터 계산된다. 형상함수( Ni )는 요소별 강성계산에 사용한 것과 동일하다. + +$$ +\mathbf {F} _ {i} = \int_ {V _ {e}} N _ {i} \left\{ \begin{array}{l} \omega_ {x} \\ \omega_ {y} \\ \omega_ {z} \end{array} \right\} d V \tag {1.6.1} +$$ + +여기서, + +$$ +\begin{array}{l} \mathbf {F} _ {i} \quad : \text { 체적력 } \\ \omega_ {x} \quad : x \text { 방향 단위중량 } \\ \omega_ {y} \quad : y \text { 방향 단위중량 } \\ \omega_ {z} \quad : z \text { 방향 단위중량 } \\ N _ {i} \quad : \text { 형상함수 } \\ \end{array} +$$ + + + +# 1-7 강제변위 + +강제변위(prescribed displacement)는 특정 절점에 대한 변형 후 위치를 알고 있을 때 사용된다. 강제변위는 기본적으로 전체좌표계에 대하여 정의되지만, 절점좌표계와 함께 사용하면 임의의 방향으로의 변위를 모사할 수 있다. 강제변위는 구조물의 변형을 발생시키기 때문에 하중으로 분류되어 있으나, 반력이 발생하는 등경계조건과 유사한 측면이 있다. 특히 강제변위가 적용된 절점과 해당 요소의 강성이 변화되기 때문에, 여러 하중조건을 동시에 푸는 경우 각별한 주의가 필요하다. 예를 들어, 동일한 구조물과 경계조건을 이용하여 여러 하중조건을 동시에 해석하는 경우, 특정 하중조건에서 사용된 임의 절점의 강제변위는 다른 하중조건에의한 결과에서도 그 절점의 변위가 구속된 것과 같은 효과를 야기한다. + + + +# Chapter 2. Equation Solver + +# 2-1 개요 + +선형연립방정식의 해법은 식 (2.1.1)과 같은 선형 행렬식의 해 u 를 구하는 방법이다. + +$$ +\mathbf {K} \mathbf {u} = \mathbf {p} \tag {2.1.1} +$$ + +선형연립방정식 해법은 선형 해석뿐만 아니라, 고유치/좌굴 해석, 동적 해석, 비선형 해석 등 거의 모든 해석군에 이용되며, 해법은 크게 직접해법(direct method)과 반복해법(iterative method)으로 분류할 수 있다. midas FEA에서 사용할 수 있는 방법과 선택사항은 다음과 같다. +
직접해법스카이라인(skyline)
다중프런트법(multi frontal)
반복해법컬레 구배법(conjugate gradient)ILUT 예조건화
자코비 예조건화
GMRES(generalized minimal residual)ILUT 예조건화
자코비 예조건화
+ + + +# 2-2 직접해법 + +직접해법에서는 방정식의 해를 두 단계에 걸쳐 구하게 된다. 첫 번째 단계는 행렬분해(decomposition)이고, 두 번째 단계는 전-후진 대입(forward-backwardsubstitution : FBS) 과정이다. + +행렬 분해 과정에서는 대칭인 강성행렬 K 를 T LL 또는 T LDL 형태로 분해한다. + +$$ +\mathbf {L} \mathbf {L} ^ {T} \mathbf {u} = \mathbf {p} \quad \text { or } \quad \mathbf {L} \mathbf {D} \mathbf {L} ^ {T} \mathbf {u} = \mathbf {p} \tag {2.2.1} +$$ + +여기서, L 은 하삼각행렬(lower-triangular matrix)이고 D 는 대각행렬(diagonalmatrix)이다. 일반적으로 D 가 포함된 분해법은 행렬이 양의 정부호(positivedefinite)가 아닌 경우에 필요하다. 분해된 행렬 K ( T = LL 또는 T = LDL )는 FBS과정을 통하여 해를 구하는데 이용된다. 먼저 T v L u = 또는 T v DL u = 로 치환하면식 (2.2.1)을 다음과 같이 나타낼 수 있고 + +$$ +\mathbf {L} \mathbf {v} = \mathbf {p} \tag {2.2.2} +$$ + +Lv p = 의 해 v 는 L 의 하삼각 특성을 이용하여 구할 수 있다. 그리고 v 가 계산되면 T v L u = 또는 T v DL u = 를 이용하여 u 를 계산한다. + +직접해법 적용시 중요한 점은 행렬의 희소성(sparsity)을 적절히 이용해야 하는 것이다. 일반적으로 유한요소해석 시에 발생하는 행렬 K 는 희소행렬(sparsematrix)이며, 희소성을 활용하는 방법에 따라 계산 시간과 요구되는 메모리 양이현저하게 달라진다. midas FEA에서는 단일 프런트를 이용하는 스카이라인 방법과다중 프런트 행렬을 이용하는 다중프런트법을 제공하고 있다. + +# 2-2-1 스카이라인 방법 + +스카이라인 방법은 단일 프런트 행렬을 이용하여 행렬 분해를 수행하며, 행렬의저장 방식으로 스카이라인 형태를 이용한다. + + + +![](images/page-335_42b24dca799115a6ecc69f0c5271c0d7a90fcaad7ab5887086dd976e50d7330f.jpg) + +
+text_image + +P₁ P₂ P₃ +
+ +그림 2.2.1 3 영역으로 구분되어 있는 요소망 + +그림 2.2.1과 같은 요소망의 강성행렬을 3개의 부분으로 나누어 표현하면 식(2.2.3)과 같고 + +$$ +\mathbf {K} = \left[ \begin{array}{c c c} \mathbf {K} _ {1 1} & \mathbf {K} _ {2 1} ^ {T} & \mathbf {0} \\ \mathbf {K} _ {2 1} & \mathbf {K} _ {2 2} & \mathbf {K} _ {3 2} ^ {T} \\ \mathbf {0} & \mathbf {K} _ {3 2} & \mathbf {K} _ {3 3} \end{array} \right] \tag {2.2.3} +$$ + +이를 P P P 1 2 3→ → 순서로 분해한다고 가정하면, P1의 분해와 이에 따른 P2 의 갱신은 다음과 같다. + +$$ +\mathbf {K} _ {1 1} = \mathbf {L} _ {1 1} \mathbf {L} _ {1 1} ^ {T} +$$ + +$$ +\mathbf {L} _ {2 1} \mathbf {L} _ {1 1} ^ {T} = \mathbf {K} _ {2 1} \tag {2.2.4} +$$ + +$$ +\mathbf {K} _ {2 2} - \mathbf {L} _ {2 1} \mathbf {L} _ {2 1} ^ {T} = \mathbf {K} _ {2 2} ^ {*} (= \mathbf {L} _ {2 2} \mathbf {L} _ {2 2} ^ {T}) +$$ + +같은 방법으로 P 의 분해와 P 의 갱신 및 분해를 할 수 있다. 이와 같이 단일 프런트 행렬에 의한 순차적인 행렬의 분해는 그 자유도의 재배치(renumbering)에 따라 계산의 효율성이 크게 좌우된다. 그림 2.2.2는 직사각형 요소망에서 효과적인계산 순서를 도식적으로 표현한 것이다. 자유도 재배치를 구현하기 위한 알고리즘은 Sloan1 이 제안한 프런트 감소법(wavefront reduction method)을 이용한다. 전 + + + +진 대입(forward substitution)은 행렬의 분해와 같은 순서로 이루어지고, 후진 대입(backward substitution)은 그 역순으로 계산하게 된다. + +![](images/page-336_45f064d22f5ebedd3e7707867a077afe015776a79f5144ad43c3d3d9aeb9addc.jpg) + +
+text_image + +1 +5 +9 +
+ +그림 2.2.2 스카이라인 방법에서 사용하는 자유도 배치 순서 + +# 2-2-2 다중프런트법 + +다중프런트법은 여러 개의 프런트 행렬을 동시에 고려하여 행렬의 분해를 수행하며, 행렬을 요소 단위로 저장하여 계산하는 방법2 을 이용한다. 그림 2.2.1과 같은요소망에 대하여 식 (2.2.5)과 같이 배치하고 + +$$ +\mathbf {K} = \left[ \begin{array}{c c c} \mathbf {K} _ {1 1} & \mathbf {0} & \mathbf {K} _ {2 1} ^ {T} \\ \mathbf {0} & \mathbf {K} _ {3 3} & \mathbf {K} _ {2 3} ^ {T} \\ \mathbf {K} _ {2 1} & \mathbf {K} _ {2 3} & \mathbf {K} _ {2 2} \end{array} \right] \tag {2.2.5} +$$ + +이를 1 3 2( , ) P P P → 순서로 분해하게 되면, 아래의 식 (2.2.6), (2.2.7)과 같이 표현할 수 있다. + + + +$$ +\mathbf {K} _ {1 1} = \mathbf {L} _ {1 1} \mathbf {L} _ {1 1} ^ {T} +$$ + +$$ +\mathbf {L} _ {2 1} \mathbf {L} _ {1 1} ^ {T} = \mathbf {K} _ {2 1} \tag {2.2.6} +$$ + +$$ +\mathbf {K} _ {2 2} - \mathbf {L} _ {2 1} \mathbf {L} _ {2 1} ^ {T} = \mathbf {K} _ {2 2} ^ {*} +$$ + +$$ +\mathbf {K} _ {3 3} = \mathbf {L} _ {3 3} \mathbf {L} _ {3 3} ^ {T} +$$ + +$$ +\mathbf {L} _ {2 3} \mathbf {L} _ {3 3} ^ {T} = \mathbf {K} _ {2 3} \tag {2.2.7} +$$ + +$$ +\mathbf {K} _ {2 2} ^ {*} - \mathbf {L} _ {2 3} \mathbf {L} _ {2 3} ^ {T} = \mathbf {K} _ {2 2} ^ {* *} +$$ + +다중 프런트 행렬에 의한 행렬의 분해 역시 자유도 재배치가 필요하다. 그림 2.2.3은 직사각형 요소망에서 효과적인 계산 순서를 도식적으로 표현한 것이다. 자유도재배치를 구현하기 위한 알고리즘은 재귀 이분할법(recursive bisection)을 이용한다. 전진 대입은 행렬의 분해와 같은 순서로 이루어지고, 후진 대입은 그 역순으로계산하게 된다. + +![](images/page-337_5019a032ee9225c49a340450fe71e92287e75acb143a1bed244ffee349fc6fb0.jpg) + +
+text_image + +1 +2 1 4 1 2 +3 +3 +1 2 1 1 2 1 +
+ +그림 2.2.3 다중프런트법에서 사용하는 자유도 분할법 + + + +# 2-3 반복해법 + +반복해법(iterative method)은 현 단계의 해 ui 와 잉여값( r p - Ku i i= )을 이용하여식 (2.1.1)의 정해에 가까운 값을 찾아가는 방법이다. ui+1을 계산하는 방법을 간략하게 표현하면 식 (2.3.1)와 같다. + +$$ +\mathbf {u} _ {i + 1} = \mathbf {u} _ {i} + \boldsymbol {\Gamma} _ {i} \mathbf {Q} (\mathbf {p} - \mathbf {K} \mathbf {u} _ {i}) \tag {2.3.1} +$$ + +여기서, Γi 는 반복법의 종류에 따라 결정되는 행렬 연산자이고, Q 는 수렴 속도를 좌우하는 예조건화 기법(pre-conditioner)이다. 반복해법에서 반복 계산의 지속여부는 최대 반복 회수(maximum iteration)와 잉여값 r p - Ku i i= 의 수렴 허용범위(convergence tolerance)를 기준으로 결정한다. midas FEA 에서는 이 값을 사용자가 결정할 수 있으며, 그 기본값은 다음과 같다. +
최대 반복 회수 $N/4 >1000$ $N/4$
$N/4 < 1000$ 1000
수렴 허용범위 $\|\mathbf{r}_{i}\| < 10^{-6} \|\mathbf{f}\|$
+ +# 2-3-1 켤레구배법과 GMRES + +midas FEA에서는 행렬이 대칭이며 양의 정부호인 경우에 사용하는 켤레구배법3 (CG:conjugate gradient) 방법과 일반적인 행렬에 적합한 GMRES4 (generalizedminimal residual) 방법을 사용할 수 있다. 켤레구배법과 GMRES 방법 모두 잉여값 ir 가 Krylov 부분공간(subspace) $s p a n \{ \mathbf { r } _ { 0 } , \mathbf { K } \mathbf { r } _ { 0 } , \mathbf { K } ^ { 2 } \mathbf { r } _ { 0 } , \ldots \}$ 내에서 직교하도록 해 + + + +의 탐색 방향을 설정한다. 따라서 수치적으로 오차가 없는 계산을 하게 되면 이론적으로 N(구조물의 자유도 개수)회의 반복 계산을 통해 정해를 구할 수 있다. + +켤레구배법은 켤레 방향(conjugated direction : $\mathbf { d } _ { i } = \mathbf { r } _ { i } + \beta _ { i } \mathbf { d } _ { i - }$ −1)을 탐색 방향으로설정하여 해를 찾아가는 방법이다. 켤레구배법은 K 의 대칭성을 이용하여 잉여값의 직교성을 쉽게 얻을 수 있는 특징이 있다. GMRES 방법에서는 잉여값 ir 가Krylov 부분공간 에서 직교하는 성질을 가지도록 반복계산의 매 단계마다 Gram-Schmidt 직교화(orthogonalization)을 수행한다. Gram-Schmidt 직교화는 반복 회수가 늘어남에 따라 계산량이 증가하기 때문에 적정 회수의 반복계산 이후에Krylov 부분공간을 새롭게 구축하는 방법을 이용한다. + +# 2-3-2 예조건화 기법 + +식 (2.3.1)에서 $\mathbf { K } ^ { - 1 } \equiv \mathbf { 0 }$ 대신 사용하게 되면 1회 반복 계산으로 정해를 얻을 수있는 것과 같이 예조건화 기법은 적은 양의 계산으로 −1 K 와 유사한 행렬$( { \bf \nabla Q } \approx { \bf K } ^ { - 1 }$ )을 얻음으로써 수렴 속도를 증가시키는 역할을 한다. midas FEA에서는ILUT 5 (incomplete LU decomposition with threshold) 예조건화 기법과 자코비(Jacobi) 예조건화 기법을 이용할 수 있다. ILUT 기법은 행렬의 분해 과정에서 발행하는 Fill-In에 허용치를 적용한 불완전 LU 분해를 이용하는 것이다. 자코비 예조건화 기법은 행렬 K 의 대각항만을 고려하여 Q K = diag( ) 를 이용한다. + + + +# 2-4 특징 + +선형연립방정식의 해법은 다음과 같이 해석하고자 하는 문제에 따라 midas FEA프로그램이 자동으로 결정하는 경우가 있다. + +
좌굴해석 또는 Lanczos 방법 이용다중프런트법
반복법을 이용한 동해석다중프런트법
구속방정식(constraint equation)이 포함된 문제다중프런트법
비대칭 강성 발생(반복법을 사용할 경우)GMRES
+ +선형 연립방정식의 해법은 해석하고자 하는 문제에 따라 크게 성능이 달라지므로각각의 경우에 적절한 해법을 선택하는 것이 필요하다. 각 해법의 특징 및 성능은다음과 같다. + +직접해법 : 행렬의 상태(condition)에 관계 없이 안정적으로 해를 계산하지만, 메모리 요구량이 많다. 사용자의 시스템 메모리가 부족한 경우에는 자동적으로 디스크저장소(disk storage)를 이용한다. + +- 스카이라인 방법 : 1차원 형상의 구조물 해석에 적합 +- 다중프런트법 : 2,3차원 형상의 구조물 해석에 적합 + +반복해법 : 메모리 요구량이 적은 반면, 행렬의 상태에 따라 계산 시간이 매우 상이하다. 입체요소가 많은 경우에 적합하다. + +\- ILUT 예조건화 : 반복 회수가 적음 + +-자코비 예조건화 : 예조건화 작업이 빠르고 메모리 요구량이 적음 diff --git a/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_035.md b/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_035.md new file mode 100644 index 00000000..3524c7a1 --- /dev/null +++ b/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_035.md @@ -0,0 +1,306 @@ + + +# Chapter 3. Iteration Methods + +# 3-1 개요 + +선형해석에서는 변위가 미소변위이고 재료의 특성이 선형이라는 가정이 필요하다.변위가 대변위이고 재료가 비선형성을 가지게 되면 선형해석에서와 같이 한번의해석으로 해를 구할 수 없게 된다. 따라서 비선형해석에서 올바른 해를 얻기 위해반복법(iteration method)을 사용한다. 반복법에서 비선형해석의 해가 되는 변위는매 반복해석 증분변위의 누적으로 나타나게 된다. + +![](images/page-341_723abcd3a0a0da9b9613c2c3ffd76aa78f433b4340156e144f9b26d8a20a9535.jpg) + +
+line + +| u | f | +| ------- | ----- | +| t | f_ext | +| t+Δt | f_ext | +| g_i | f_int,i | +| K_{i+1} | K_{i+1} | +| Δu_i | Δu_{i+1} | +| δu_{i+1} | δu_{i+1} | +| t+Δt | t+Δt | +
+ +그림 3.1.1 반복과정 + +반복계산은 그림 3.1.1에서와 같이 진행된다. ${ } ^ { t } \mathbf { f } _ { e x t }$ 와 t t ext+∆ f 는 각각 시간 t 와 시간t t+∆ 에서의 외력을 의미한다. 시간 t 와 시간 t t+∆ 사이에서의 변위 및 변위 증분의 관계는 다음과 같이 나타낼 수 있다. + +$$ +{ } ^ { t + \Delta t } \mathbf { u } = { } ^ { t } \mathbf { u } + \Delta \mathbf { u } \tag {3.1.1} +$$ + + + +여기서, + +$$ +\Delta \mathbf {u} \quad : \text { 시간증분 } \Delta t \text { 사이에서 발생하는 전체변위증분 } (\text { total displacement increment }) +$$ + +시간증분 ∆t 구간에서 비선형 해석을 위한 반복계산이 이루어지게 되며, 이는 다음과같이 나타낼 수 있다. + +$$ +\Delta \mathbf {u} = \sum_ {i = 1} ^ {n} \delta \mathbf {u} _ {i} \quad \text { or } \quad \Delta \mathbf {u} _ {i + 1} = \Delta \mathbf {u} _ {i} + \delta \mathbf {u} _ {i + 1} \tag {3.1.2} +$$ + +여기서, + +$$ +\begin{array}{l l} \Delta \mathbf {u} _ {i} & : i \text { 번째 까지의 누적증분변위 } \\ & \text {(cumulated incremental displacement)} \\ \delta \mathbf {u} _ {i + 1} & : i + 1 \text { 번째의 반복계산에서 발생하는 증분변위 } \\ & \text {(incremental displacement)} \end{array} +$$ + +증분변위 $\delta \mathbf { u } _ { i + 1 }$ 는 접선강성행렬(tangent stiffness matrix) $\mathbf { K } _ { i + 1 }$ 을 사용하여 다음과 같은 선형해석을 통하여 구해진다. + +$$ +\delta \mathbf {u} _ {i + 1} = \mathbf {K} _ {i + 1} ^ {- 1} \mathbf {g} _ {i} \tag {3.1.3} +$$ + +여기서, + +$$ +\mathbf {g} _ {i} \quad : i \text { 번째 단계에서의 불평형력(out - of - balance force) } +$$ + +불평형력 gi는 다음과 같이 계산된다. + +$$ +\mathbf {g} _ {i} = ^ {t + \Delta t} \mathbf {f} _ {\text { ext }} - \mathbf {f} _ {\text { int }, i} \tag {3.1.4} +$$ + +최종적인 누적증분변위(total incremental displacement)는 시간 t 에서 시간t t + ∆ 까지 증분변위 ∆u 이며, 시간 t t + ∆ 까지 모든 증분변위를 더한 값 u 가 외력 ${ { t + \Delta t } _ { } } { { \bf \Delta f } _ { e x t } }$ 이 가해졌을 때의 전체변위이다. 이 때 $\mathbf { f } _ { \mathrm { i n t } , i }$ 는 내력을 의미하며, 이는 재료의 성질에 따른 경로의 존재(path-dependent)특성을 가지고 있다. 따라서 요소의 변형률을 시간 t 로부터 누적된 변위를 사용하여 구하여야 하며, 최종적인 수렴 + + + +상태인 t t+∆ 에 도달하였을 때 결정된 상태(status)를 저장하게 된다. 이상 일련의반복계산에서 사용자가 지정한 수렴 조건(convergence tolerance)를 통과하게 되면 반복계산을 멈춘다. + +midas FEA에서 사용 가능한 반복법으로는 초기강성법(initial stiffness method),뉴튼 랩슨법(Newton-Raphson method), 수정 뉴튼 랩슨법, 호장법(arc-lengthmethod)이 있다. + + + +# 3-2 초기강성법 + +초기강성법은 비선형 수치해석에 있어서 다른 반복법을 사용할 경우 불안정한 경향을 보이는 해석에 대해서 사용된다. 해가 안정적으로 구해지지만, 증분구간의 크기가 뉴튼 랩슨법이나 수정 뉴튼 랩슨법보다는 상대적으로 작아서 수렴속도가 느리다. 반복법이 진행되어 가면서 접선강성이 업데이트 될 때 상태가 변하는 요소가 포함되어 있는 모델의 경우 접선강성 업데이트가 무의미해 질 수 있다. + +![](images/page-344_0cd1f2eced78a23c6c46509ee561146c6bcb363bd0335be93a67ba24b1c3472f.jpg) + +
+line + +| u | f_ext | f_int,i | Δu_i | δu_{i+1} | Δu_{i+1} | +|---------|-------|---------|------|----------|----------| +| t+Δt | f_ext | f_int,i | Δu_i | δu_{i+1} | Δu_{i+1} | +| t+Δt_t | f_ext | f_int,i | Δu_i | δu_{i+1} | Δu_{i+1} | +
+ +그림 3.2.1 초기강성법의 반복해석 + +초기강성법에서는 그림 3.2.1과 같이 해석의 전 과정에서 초기 강성을 사용한다.따라서 최초의 해석단계에서 만들어진 강성을 수정하지 않고 사용하게 된다. 따라서 접선강성의 업데이트를 위해 수행되는 계산이 필요 없다는 장점이 있다. + + + +# 3-3 뉴튼 랩슨법 + +뉴튼 랩슨법은 그림 3.3.1과 같이 매 반복단계마다 접선강성으로 강성이 갱신된다.따라서 수렴속도가 빠르고 적은 수의 반복을 통해서도 수렴이 가능하다. + +![](images/page-345_46a45693ff48d5e040b10a41bea7537428787770ebf9356d4c9575674a2fb9fe.jpg) + +
+line + +| u | f_ext | f_int,i | g_i | +|---------|-------|---------|---------| +| t_u | 0 | 0 | 0 | +| t+Δt | 1 | 1 | 1 | +
+ +그림 3.3.1 뉴튼 랩슨법의 반복해석 + +뉴튼 랩슨법은 해석모델이 클 경우 접선강성을 구하기 위해 많은 계산을 필요로한다. 최초의 반복단계에서 구해진 해와 최종적으로 구해진 해의 경향이 많이 다를 경우 수렴에 실패할 수도 있다. 따라서 뉴튼 랩슨법은 수렴하는데 드는 반복 회수는 적은 반면에, 각 반복 단계에서 드는 계산비용이 초기강성법이나 수정 뉴튼랩슨법보다 상대적으로 높다. + + + +# 3-4 수정 뉴튼 랩슨법 + +수정 뉴튼 랩슨법은 뉴튼 랩슨법의 계산비용을 줄이기 위한 동기에서 도입되었다.뉴튼 랩슨법에서는 매 증분변위를 구할 때마다 새로운 접선강성과 내력을 구하는것에 반해, 수정 뉴튼 랩슨법은 접선강성을 다시 구하는 과정을 생략하고 내력만을 갱신 한다. 수정 뉴튼 랩슨법은 뉴튼 랩슨법보다 더 많은 반복단계를 거쳐야 하므로 수렴속도가 더 느리지만, 각각의 반복단계에서 걸리는 시간이 더 짧은 장점이 있다. + +![](images/page-346_9bebe6780ec4a8d4e8fbbff3281eccb2106135d3bfca6432680af1e7d9f604c6.jpg) + +
+line + +| t | f | +|-------|-------| +| t+Δt | f_ext | +| f_int,i | f_int,i | +| g_i | f_int,i | +| t+Δt | f_ext | +| f_ext | f_int,i | +| t+Δt | f_ext | +| Δu_i | f_ext | +| δu_{i+1} | f_ext | +| Δu_{i+1} | f_ext | +
+ +그림 3.4.1 수정 뉴튼 랩슨법의 반복해석 + +뉴튼 랩슨법을 사용하여 정상적으로 수렴하지 못하는 문제에 대해서 수정 뉴튼 랩슨법이 유용하게 사용될 수 있다. 초기강성법은 매 하중단계에서 같은 접선강성을사용하지만 수정 뉴튼 랩슨법은 매 하중단계의 최초 반복에서 업데이트한 강성을사용하게 된다. + + + +# 3-5 호장법 + +![](images/page-347_1450762538d55c5f722981c48c7f8f8485f2de3a54a3e83d205f64cadb509783.jpg) + +
+text_image + +A +B +C +
+ +(a) 스냅스루 + +![](images/page-347_75921e25308a0b650c7a275a018cf7df4809bbe0adaafb796915c37cc5a94fed.jpg) + +
+text_image + +A +B +C +
+ +(b): 스냅백 +그림 3.5.1 변위-하중 곡선 + +비선형 해석에서 변위-하중 곡선의 최대점 근처에서 그림 3.5.1과 같은 스냅스루(snap-through)나 스냅백(snap-back)이 발생할 경우 접선강성을 사용한 반복법으로는 국부최대 점에 도달한 다음 올바른 방향을 찾아가지 못하게 된다. + +![](images/page-347_dd583585a3bbd2171c964f94b03d704c51928e0bc6b3de40a41f5731891638ce.jpg) + +
+line + +| Load Step | f | +| --------- | ----- | +| Load Step1 | 1/2 | +| Load Step2 | 1/2 | +| Load Step3 | 1/2 | +
+ +그림 3.5.2 호장법의 반복해석 진행과정 + + + +그림 3.5.2는 호장법의 진행과정을 간략하게 보여주고 있다. 3개의 하중단계(number of load step)를 진행하며 반복해석을 수행하며, 초기하중계수(initialload step)는 0.5을 사용하고 있다. 호장법에서는 하중계수와 외력을 곱한 벡터의크기의 반지름을 가지는 호와 변위-하중 곡선이 교차하는 점을 반복을 통해 찾으며 수렴조건을 만족하게 되면 수렴한 위치에서 새로운 하중계수를 산정한 다음 하중단계로 넘어가게 된다. 하중계수의 크기는 고정되어 있지 않고, 다음 하중단계에서 사용자가 입력한 적정 반복횟수(desired number of iterations)에 근접한 반복을 하도록 적절한 크기로 산정된다. + +하중 단계의 증분 단계(incremental step)내에서 평형을 이루기 위해 계산되는 반복해석 시 i 번째 반복계산에 대한 구조물의 선형대수방정식은 다음과 같이 계산된다. + +$$ +\delta \mathbf {u} _ {i + 1} = \mathbf {K} _ {i + 1} ^ {- 1} \mathbf {g} _ {i} \tag {3.5.1} +$$ + +여기서, + +$$ +\delta \mathbf {u} _ {i + 1} \quad : i + 1 \text { 번째 반복계산에서의 증분변위 } +$$ + +$$ +\mathbf {g} _ {i} \quad : i \text { 번째 반복계산에서의 불평형력 } +$$ + +불평형력 ig 은 다음과 같이 나타낼 수 있다. + +$$ +\mathbf {g} _ {i} = ^ {t + \Delta t} \mathbf {f} _ {e x t} - \mathbf {f} _ {\text { int }, i} + \delta \lambda_ {i} \mathbf {f} _ {e q} \tag {3.5.2} +$$ + +여기서, + +$$ +\mathbf {f} _ {e q} \quad : \text { 사용자 선택에 의한 단위하중(reference force) } +$$ + +$$ +\delta \lambda_ {i} \quad : \text { 하중계수증분 } +$$ + +위 식 (3.5.2)를 식 (3.5.1)에 대입하면 현재 반복계산에서의 증분변위는 다음과 같다. + + + +$$ +\begin{array}{l} \delta \mathbf {u} _ {i + 1} = \mathbf {K} _ {i + 1} ^ {- 1} \left(^ {t + \Delta t} \mathbf {f} _ {e x t} - \mathbf {f} _ {\text { int }, i} + \delta \lambda_ {i} \mathbf {f} _ {e q}\right) \\ = \mathbf {K} _ {i + 1} ^ {- 1} \left(^ {t + \Delta t} \mathbf {f} _ {e x t} - \mathbf {f} _ {\text { int }, i}\right) + \delta \lambda_ {i} \mathbf {K} _ {i + 1} ^ {- 1} \mathbf {f} _ {e q} \tag {3.5.3} \\ = \mathbf {u} _ {I} + \delta \lambda_ {i} \mathbf {u} _ {I I} \\ \end{array} +$$ + +하중증분과 반복계산을 수행할 때 하중과 변위의 증분량과의 관계는 다음과 같다. + +$$ +\Delta \mathbf {u} _ {i + 1} = \Delta \mathbf {u} _ {i} + \delta \mathbf {u} _ {i + 1} \tag {3.5.4} +$$ + +따라서 + +$$ +\Delta \mathbf {u} _ {i + 1} = \Delta \mathbf {u} _ {i} + \mathbf {u} _ {I} + \delta \lambda_ {i} \mathbf {u} _ {I I} \tag {3.5.5} +$$ + +반복해석 시 부여되는 변위에 대한 구속조건은 다음과 같이 정의할 수 있다. + +$$ +\Delta \mathbf {u} _ {i + 1} ^ {T} \Delta \mathbf {u} _ {i + 1} = \Delta l ^ {2} \tag {3.5.6} +$$ + +위 식 (3.5.6)에 식 (3.5.5)를 대입하여 정리하면 i 번째 반복계산에서 δλi 에 대한2차방정식을 얻을 수 있다. + +$$ +a _ {1} \delta \lambda_ {i} ^ {2} + a _ {2} \delta \lambda_ {i} + a _ {3} = 0 \tag {3.5.7} +$$ + +여기서, + +$$ +\begin{array}{l} a _ {1} \quad : \quad \mathbf {u} _ {I I} ^ {T} \mathbf {u} _ {I I} \\ a _ {2} \quad : \quad 2 \mathbf {u} _ {I I} ^ {T} \left(\Delta \mathbf {u} _ {i} + \mathbf {u} _ {I}\right) \\ a _ {3} \quad : \left(\Delta \mathbf {u} _ {i} + \mathbf {u} _ {I}\right) ^ {T} \left(\Delta \mathbf {u} _ {i} + \mathbf {u} _ {I}\right) - \Delta l ^ {2} \\ \end{array} +$$ + +식 (3.5.7)은 $a _ { \scriptscriptstyle 1 } = 0$ 인 경우 $\delta \lambda _ { _ i } = - a _ { 3 } / a _ { 2 }$ 의 선형근을 가지며 $a _ { \mathrm { 1 } } \neq 0$ 인 경우 두 개의 실근, 중근 혹은 허근의 경우를 가지게 된다. 허근을 가질 경우는 초기 호길이를 반으로 줄여서 다시 해석을 수행하도록 하고 있으며, 두 개의 실근을 가질 경우는 다음의 식을 사용하여 $\Delta \pmb { u } _ { i }$ 와 $\Delta \pmb { u } _ { i + }$ +1사이의 각이 최소가 되는 해를 찾도록 하고있다. + + + +$$ +\cos \theta = \frac {\Delta \mathbf {u} _ {i} ^ {T} \Delta \mathbf {u} _ {i + 1}}{\Delta l ^ {2}} = \frac {\Delta \mathbf {u} _ {i} ^ {T} \left(\Delta \mathbf {u} _ {i} + \mathbf {u} _ {I}\right)}{\Delta l ^ {2}} + \delta \lambda_ {i} \frac {\Delta \mathbf {u} _ {i} ^ {T} \mathbf {u} _ {I I}}{\Delta l ^ {2}} \tag {3.5.8} +$$ + +$$ += \frac {a _ {4} + a _ {5} \delta \lambda_ {i}}{\Delta l ^ {2}} +$$ + +여기서, + +$$ +a _ {4} \quad : \Delta \mathbf {u} _ {i} ^ {T} \mathbf {u} _ {I} + \Delta \mathbf {u} _ {i} ^ {T} \Delta \mathbf {u} _ {i} +$$ + +$$ +a _ {5} \quad : \Delta \mathbf {u} _ {i} ^ {T} \mathbf {u} _ {I I} +$$ + +식 (3.5.7)에서 두 개의 실근이 나올 경우 각각의 해를 δλi,1 , δλi,2 라고 하면 + +$$ +\Delta l ^ {2} \cos \theta_ {1} = a _ {4} + a _ {5} \delta \lambda_ {i, 1} \tag {3.5.9} +$$ + +$$ +\Delta l ^ {2} \cos \theta_ {2} = a _ {4} + a _ {5} \delta \lambda_ {i, 2} +$$ + +이므로 + +$$ +\left\{ \begin{array}{l} \delta \lambda_ {i} = \delta \lambda_ {i, 1} \quad \left(\Delta l ^ {2} \cos \theta_ {1} > \Delta l ^ {2} \cos \theta_ {2}\right) \\ \delta \lambda_ {i} = \delta \lambda_ {i, 1} = \delta \lambda_ {i, 2} \left(\Delta l ^ {2} \cos \theta_ {1} = \Delta l ^ {2} \cos \theta_ {2}\right) \\ \delta \lambda_ {i} = \delta \lambda_ {i, 2} \quad \left(\Delta l ^ {2} \cos \theta_ {1} < \Delta l ^ {2} \cos \theta_ {2}\right) \end{array} \right. \tag {3.5.10} +$$ + +식 (3.5.10)을 사용하여 δλi 을 구할 수 있고, 식 (3.5.5)를 이용하여 현재의 변위를구할 수 있다. diff --git a/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_036.md b/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_036.md new file mode 100644 index 00000000..78245f61 --- /dev/null +++ b/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_036.md @@ -0,0 +1,258 @@ + + +# 3-6 수렴조건 + +비선형해석에서 해는 반복법을 사용하여 얻어진다. 이 때 일정한 기준을 통과하게되면 반복법을 사용하여 얻어진 해가 수렴하였다고 간주하게 된다. 이러한 수렴판단기준을 제공하기 위해서 수렴조건을 적용하게 된다. 수렴조건으로는 부재력기준(force norm), 변위 기준(displacement norm), 에너지 기준(energy norm) 등이 있으며, 이들 값이 허용값보다 작아지면 해가 수렴되었다고 판단하게 된다. + +$$ +\text { Force norm ratio } = \frac {\sqrt {\delta \mathbf {g} _ {i} ^ {T} \delta \mathbf {g} _ {i}}}{\sqrt {\Delta \mathbf {g} _ {i} ^ {T} \Delta \mathbf {g} _ {i}}} \tag {3.6.1} +$$ + +$$ +\text { Displacement norm ratio } = \frac {\sqrt {\delta \mathbf {u} _ {i} ^ {T} \delta \mathbf {u} _ {i}}}{\sqrt {\Delta \mathbf {u} _ {i} ^ {T} \Delta \mathbf {u} _ {i}}} \tag {3.6.2} +$$ + +$$ +\text { Energy norm ratio } = \left| \frac {\left(\delta \mathbf {u} _ {i} \delta \mathbf {g} _ {i}\right) \left(\delta \mathbf {u} _ {i} \delta \mathbf {g} _ {i}\right) ^ {T}}{\left(\Delta \mathbf {u} _ {i} \Delta \mathbf {g} _ {i}\right) \left(\Delta \mathbf {u} _ {i} \Delta \mathbf {g} _ {i}\right) ^ {T}} \right| \tag {3.6.3} +$$ + +수렴여부를 판단하기 위하여 선택하는 기준은 해석종류에 따라 다르다. 구조물이자유롭게 변형될 수 있는 구조로 되어 있는 경우 부재력 기준은 부적절할 수 있다.재료의 연화(softening)거동은 경화(hardening)거동보다 수렴조건이 더 까다롭다.어떤 모델을 해석하는 경우에도 수렴기준이 되는 수렴조건의 선택은 타당한 이유를 가지고 있어야 한다. 그리고 비선형모델에서 수렴조건을 선택하는데 어려움이있다면, 두 개의 수렴조건을 사용하여 각각의 결과를 비교해 보는 것이 좋다. + + + +# 3-7 자동전환기능 + +midas FEA에서는 반복해석을 진행하는 도중 호장법으로 전환하여 계속적으로 진행시키는 자동전환(auto-switching)기능을 제공한다. 반복해석을 진행하는 과정에서 수렴성의 문제가 발생하는 경우에 자동으로 호장법으로 반복법을 바꿔주는 것은 매우 유용하다. + +자동전환기능을 사용하기 위해서는 기울기 예측을 수행하여야 하며, 다음과 같은식이 사용된다. + +$$ +k _ {i} = \frac {\mathbf {g} _ {i} \Delta \mathbf {u} _ {i}}{\Delta \mathbf {u} _ {i} ^ {T} \Delta \mathbf {u} _ {i}} \tag {3.7.1} +$$ + +식 (3.7.1)에서 얻어진 $k _ { i }$ 와 초기 1k 값을 사용하여 강성매개변수 $C _ { s }$ 를 산정한다. + +$$ +C _ {s} = \frac {k}{k _ {1}} \tag {3.7.2} +$$ + +한 하중 구간 동안의 $C _ { s }$ 의 변화량을 $\Delta C _ { s }$ 라고 할 때, 현재의 하중 구간에서의 변화량을 $\Delta C _ { s d }$ 라고 하고, 초기 $C _ { s }$ 변화량을 $\Delta C _ { s 1 }$ 라고 하면 현재의 변화와 초기의변화의 비를 $\Delta C _ { s d } / \Delta C _ { s 1 }$ 로 표현할 수 있다. + +$\Delta C _ { s d } / \Delta C _ { s 1 }$ 값을 하중스텝에 따라 갱신하게 되면 최대값에 다가가게 되면서 $\Delta C _ { s d }$ 는 0에 근접하게 된다. $\Delta C _ { s d } / \Delta C _ { s 1 }$ 값이 사용자가 정의한 허용 오차허용범위 이내로 들어가게 되면 호장법으로 자동 전환된다. + + + +# Analysis and Algorithm Manual + +# Part 4 Linear Analysis + +Chapter 1. Linear Static Analysis + +Chapter 2. Modal Analysis + +Chapter 3. Time History Analysis + +Chapter 4. Response Spectrum Analysis + +Chapter 5. Linear Buckling Analysis + + + + + +# Chapter 1. Linear Static Analysis + +# 1-1 개요 + +midas FEA의 선형정적해석(linear static analysis)에 사용된 기본방정식은 다음과같다. 구조물의 강성, 하중, 변위는 전체좌표계를 기준으로 계산되고, 절점좌표계의변위는 전체좌표계의 변위를 변환하여 구한다. 구조해석시 강성이나 경계조건의부족으로 특이성오류(singular error)가 발생할 수 있으므로 입력에 주의해야 한다. + +$$ +\mathbf {K} \mathbf {u} = \mathbf {p} \tag {1.1.1} +$$ + +여기서, + +K : 구조물의 강성행렬 +u : 전체좌표계의 변위벡터 +p : 하중벡터 + + + +# 1-2 비선형요소의 선형정적해석 + +midas FEA에서는 인장 또는 압축전담 요소를 사용한 구조물의 선형정적해석은아래의 순서도와 같이 반복적인 해석으로 해를 구한다. 탄성연결요소(elastic link)의 인장 또는 압축 전담요소를 사용한 경우에는 프로그램 내부적으로 자동 반복선형해석을 수행하여 해를 얻는다. 비선형 부재의 강성은 해석 결과의 영향을 받기 때문에 반복 수렴과정을 통하여 구조물의 강성과 하중에 맞는 비선형부재의 강성을 결정해야 한다. 비선형 부재를 사용한 해석의 결과는 다른 하중조건의 결과와 선형조합을 하지 않아야 한다. 조합된 하중에 대한 결과를 구하고자 할 경우에는 하중을 조합하여 독립적으로 해석을 수행하여 결과를 얻어야 한다. + +![](images/page-356_5dee2ad7d127e18ee67fa96a8080619fc4826d2f666da1b23e6f1223a2ea252b.jpg) + +
+flowchart + +```mermaid +graph TD + A["해석모델 입력"] --> B["선형 및 비선형 부재
강성, 불평형력 계산"] + B --> C["선형 방정식의 해
(K + Kₙ,ₛ) ΔU = ΔP"] + C --> D{증분 변위를 +사용한 수렴여부 판단} + D -->|Yes| E["누적된 변위를 사용한결과 계산
U = ΣΔU"] + D -->|No| B +``` +
+ +여기서, + +K : 선형부재의 강성 + +${ \bf K } _ { n , s }$ K n s, : 비선형부재의 강성 + +∆U U, : 불평형력에 의한 증분변위와 누적변위 + +∆P : 불평형력(외력-내력) + + + +# Chapter 2. Modal Analysis + +# 2-1 개요 + +고유치해석은 구조물 고유의 동적 특성을 분석하는데 사용되며, 자유진동해석(free vibration analysis) 이라고도 한다. 비감쇠 자유진동(undamped free vibration) 조건하의 운동 방정식은 식 (2.1.1)과 같은 선형 2차 미분방정식이다. + +$$ +\mathbf {M} \ddot {\mathbf {u}} (t) + \mathbf {K} \mathbf {u} (t) = \mathbf {0} \tag {2.1.1} +$$ + +여기서, + +K : 구조물의 강성행렬 + +M : 구조물의 질량행렬 + +변위 u 를 형상 행렬 Φ와 시간의 함수로 이루어진 벡터 $\mathbf{Y}(t)$ 의 곱( $\mathbf{u}=\mathbf{\Phi}\mathbf{Y}(t)$ )으로 표현하면 다음과 같은 식을 얻을 수 있다. + +$$ +\mathbf {M} \boldsymbol {\Phi} \ddot {\mathbf {Y}} + \mathbf {K} \boldsymbol {\Phi} \mathbf {Y} = \mathbf {0} \tag {2.1.2} +$$ + +시간의 함수 $\mathbf{Y}(t)$ 를 다음과 같이 가정한다. + +$$ +\mathbf {Y} (t) = \left\{y _ {1} (t) \quad \dots \quad y _ {m} (t) \quad \dots \quad y _ {n} (t) \right\} ^ {T} \tag {2.1.3} +$$ + +여기서, + +n : 전체 자유도 개수 + +$$ +y _ {m} (t) \quad : \cos (\omega_ {m} t + \beta_ {m}) +$$ + +식 (2.1.3)을 (2.1.2)에 대입하면 다음과 같다. + +$$ +(- \mathbf {M} \boldsymbol {\Phi} \boldsymbol {\Lambda} + \mathbf {K} \boldsymbol {\Phi}) \mathbf {Y} = \mathbf {0} \tag {2.1.4} +$$ + +행렬 Λ와 Φ는 다음과 같이 구성된다. + + + +$$ +\boldsymbol {\Lambda} = \left[ \begin{array}{c c c c c} \lambda_ {1} & & & & \\ & \ddots & & & \\ & & \lambda_ {m} & & \\ & & & \ddots & \\ & & & & \lambda_ {n} \end{array} \right], \quad \lambda_ {m} = \omega_ {m} ^ {2} \tag {2.1.5} +$$ + +$$ +\boldsymbol {\Phi} = \left[ \begin{array}{l l l l l} \phi_ {1} & \dots & \phi_ {m} & \dots & \phi_ {n} \end{array} \right] \tag {2.1.6} +$$ + +식 (2.1.4)를 만족하는 해는 각각의 $y_{m}(t)$ 에 대하여 다음과 같이 나타낼 수 있다. + +$$ +\mathbf {K} \phi_ {m} - \lambda_ {m} \mathbf {M} \phi_ {m} = 0 \tag {2.1.7} +$$ + +식 (2.1.7)은 고유치 문제이며, 자유진동 운동방정식에서 변위가 의미 있는 해를 갖기 위해서는 식 (2.1.8)의 조건을 만족해야 한다. + +$$ +\left| \mathbf {K} - \lambda_ {m} \mathbf {M} \right| = 0 \tag {2.1.8} +$$ + +식 (2.1.8)은 행렬 크기인 n 개의 해를 가지며, 작은 값부터 $\lambda_{1}, \lambda_{2}, \ldots, \lambda_{m}$ 이라한다. 또한 m 번째 고유치 $\lambda_{m}$ 에는 이에 대응하는 고유모드를 $\phi_{m}$ 이 존재한다. 식 (2.1.7)에서 질량행렬 M 과 강성행렬 K 는 대칭(symmetric) 행렬이므로 고유치 $\lambda_{m}$ 와 고유모드 $\phi_{m}$ 은 실수이다. 또한 질량행렬은 양의 정부호(positive definite)이고, 강성행렬은 양의 준정부호(positive semi-definite)이므로 $\lambda_{m} \geq 0$ 이다. 그러므로 비감쇠 자유진동 조건에서의 회전진동수 $\omega_{m}$ 은 실수이다. + +외력이 존재하지 않는 상태에서 구조물은 고유모드 $\phi_{m}$ 의 형상으로 진동하며, 그 속도는 회전진동수(circular frequency) $\omega_{m}$ (radian/time) 과 같다. 구조물의 진동 속도는 고유진동수 $f_{m}$ (cycle/time) 또는 고유주기 $t_{m}$ (time/cycle) 으로 나타내기도 한다. $\omega_{m}$ , $f_{m}$ 과 $T_{m}$ 은 다음과 같은 관계를 가진다. + +$$ +T _ {m} = \frac {1}{f _ {m}}, \quad f _ {m} = \frac {\omega_ {m}}{2 \pi} \tag {2.1.9} +$$ + +일반적으로 고유치 $\lambda_{m}$ 은 m 차 모드 형상에서 운동 에너지에 대한 변형에너지의 비율이며, 그 값이 작은 것부터 1차 모드, 2차 모드, ..., n 차 모드라 한다. 그림 2.1.1은 캔틸레버(cantilever)의 진동모드를 1차모드부터 순차적으로 나타낸 것이다. + + + +![](images/page-359_6a338d0c26da11d7b403b0540596cea92b739955a16fc184db58245854293243.jpg) + +
+text_image + +u +t +
+ +![](images/page-359_b779d9fd422b3207ae78574c9aefd65c65d78779cbc19848dd40e00fa6ff6701.jpg) + +
+text_image + +u +t +
+ +![](images/page-359_82417a0a80ed3120c283a2a620d604cc18cf6a9cdb5d1daa0cb69c7ea4a5bb24.jpg) + +
+text_image + +u +t +
+ +그림 2.1.1 캔틸레버의 진동모드 + +midas FEA에서 구조물의 동특성을 나타내는 지표로 모드기여계수(modalparticipation factor), 모드참여질량(effective modal mass), 모드별 방향계수(modal direction factor)등을 출력하고 있다. + +모드기여계수의 방향별 값은 식 (2.1.10)과 같이 계산되고, 응답스펙트럼해석이나지진하중을 받는 구조물의 시간이력해석에 사용된다. + +$$ +\Gamma_ {m X} = \frac {\phi_ {m} ^ {T} \mathbf {M} \mathbf {1} _ {X}}{\phi_ {m} ^ {T} \mathbf {M} \phi_ {m}}, \Gamma_ {m Y} = \frac {\phi_ {m} ^ {T} \mathbf {M} \mathbf {1} _ {Y}}{\phi_ {m} ^ {T} \mathbf {M} \phi_ {m}}, \Gamma_ {m Z} = \frac {\phi_ {m} ^ {T} \mathbf {M} \mathbf {1} _ {Z}}{\phi_ {m} ^ {T} \mathbf {M} \phi_ {m}} \tag {2.1.10} +$$ + +$$ +\Gamma_ {m R X} = \frac {\phi_ {m} ^ {T} \mathbf {M 1} _ {R X}}{\phi_ {m} ^ {T} \mathbf {M} \phi_ {m}}, \Gamma_ {m R Y} = \frac {\phi_ {m} ^ {T} \mathbf {M 1} _ {R Y}}{\phi_ {m} ^ {T} \mathbf {M} \phi_ {m}}, \Gamma_ {m R Z} = \frac {\phi_ {m} ^ {T} \mathbf {M 1} _ {R Z}}{\phi_ {m} ^ {T} \mathbf {M} \phi_ {m}} +$$ + +$$ +\begin{array}{l} \Gamma_ {m X}, \Gamma_ {m Y}, \Gamma_ {m Z} \quad \begin{array}{l} \text { : } m \text { 번째 모드의 전체좌표계 } X, Y, Z \text { 이동 방향 } \\ \text { 모드기여계수 } \end{array} \\ \Gamma_ {m R X}, \Gamma_ {m R Y}, \Gamma_ {m R Z} \quad \begin{array}{l} \text { : } m \text { 번째 모드의 전체좌표계 } X, Y, Z \text { 회전 방향 } \\ \text { 모드기여계수 } \end{array} \\ \end{array} +$$ + + + +$$ +\begin{array}{l} \mathbf {1} _ {X}, \mathbf {1} _ {Y}, \mathbf {1} _ {Z} \quad : \text { 전체좌표계 } X, Y, Z \text { 이동 방향 자유도만 단위 } \text { 값을 갖는 방향 벡터 } \\ \mathbf {1} _ {R X}, \mathbf {1} _ {R Y}, \mathbf {1} _ {R Z} \quad : \text { 전체좌표계 } X, Y, Z \text { 회전 방향 자유도만 단위 } \text { 값을 갖는 방향 벡터 } \\ \phi_ {m} \quad : m \text { 번째 모드의 형상 } \\ \end{array} +$$ + +모드참여질량의 방향별 값은 모드형상에 따른 방향별 질량의 참여를 나타내는 것으로 식 (2.1.11)과 같이 계산된다. 모드의 부호가 계산에 포함되기 때문에 모드형상에 따라서 영의 값으로 계산될 수 있다. 모든 모드의 방향별 참여질량의 합은 구조물의 방향별 전체질량과 같다. 일반내진설계기준에서는 해석에 포함되는 모드참여질량의 방향별 합이 전체 질량의 90% 이상을 확보하도록 요구하고 있다. 이는해석결과에 영향을 주는 대부분의 주요 모드를 포함하도록 하기 위한 것이다. + +$$ +M _ {m X} ^ {*} = \frac {\left[ \phi_ {m} ^ {T} \mathbf {M} \mathbf {1} _ {X} \right] ^ {2}}{\phi_ {m} ^ {T} \mathbf {M} \phi_ {m}}, M _ {m Y} ^ {*} = \frac {\left[ \phi_ {m} ^ {T} \mathbf {M} \mathbf {1} _ {Y} \right] ^ {2}}{\phi_ {m} ^ {T} \mathbf {M} \phi_ {m}}, M _ {m Z} ^ {*} = \frac {\left[ \phi_ {m} ^ {T} \mathbf {M} \mathbf {1} _ {Z} \right] ^ {2}}{\phi_ {m} ^ {T} \mathbf {M} \phi_ {m}} \tag {2.1.11} +$$ + +$$ +M _ {m R X} ^ {*} = \frac {\left[ \phi_ {m} ^ {T} \mathbf {M} \mathbf {1} _ {R X} \right] ^ {2}}{\phi_ {m} ^ {T} \mathbf {M} \phi_ {m}}, M _ {m R Y} ^ {*} = \frac {\left[ \phi_ {m} ^ {T} \mathbf {M} \mathbf {1} _ {R Y} \right] ^ {2}}{\phi_ {m} ^ {T} \mathbf {M} \phi_ {m}}, M _ {m R Z} ^ {*} = \frac {\left[ \phi_ {m} ^ {T} \mathbf {M} \mathbf {1} _ {R Z} \right] ^ {2}}{\phi_ {m} ^ {T} \mathbf {M} \phi_ {m}} +$$ + +여기서, + +$$ +\begin{array}{l} M _ {m X} ^ {*}, M _ {m Y} ^ {*}, M _ {m Z} ^ {*} \quad : m \text { 번째 모드의 전체좌표계 } X, Y, Z \text { 이동방향 참여질량 } \\ M _ {m R X} ^ {*}, M _ {m R Y} ^ {*}, M _ {m R Z} ^ {*}: m \text { 번째 모드의 전체좌표계 } X, Y, Z \text { 회전방향 참여질량 } \\ \end{array} +$$ + +모드별 방향계수는 해당모드의 전체참여질량에 대한 방향별 참여질량의 비율을 나타낸 것이다. midas FEA에서는 식 (2.1.7)과 같은 고유치문제를 해석하는데 있어서 대형구조물의 해석에 적합한 Lanczos반복법1 과 Subspace반복법2 을 이용한다. diff --git a/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_037.md b/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_037.md new file mode 100644 index 00000000..bf8f4961 --- /dev/null +++ b/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_037.md @@ -0,0 +1,270 @@ + + +# 2-2 Lanczos 반복법 + +Lanczos반복법은 Krylov 부분공간(subspace) span( $V_1, V_2, ..., V_k$ ) 을 생성하는 과정을 통하여 발생하는 삼중 대각행렬 $T_k$ 를 이용하여 고유치의 근사값을 구하는 방법이다. 구조물의 진동해석에서 발생하는 고유치 문제 식 (2.1.7)에 Lanczos반복법을 효과적으로 적용하기 위해서는 고유치 $\lambda_m$ 을 $\lambda_m = \sigma + 1/\theta_m$ 로 치환하여야 한다. 이러한 방법을 Shift-invert 기법이라 하며, $\sigma$ 는 첫번째 고유치로 예상되는 값이다. Shift-invert 기법을 적용한 Lanczos 반복계산 과정을 간단하게 요약하면 다음과 같다. + +- 첫 번째 반복계산의 경우 블록(block) 벡터의 초기값 $V_{1}$ 을 가정 +- 질량 행렬 곱셈 $U_{k} = MV_{k}$ +- 선형 연립 방정식 풀이 $(\mathbf{K}-\sigma\mathbf{M})\mathbf{W}_{k}=\mathbf{U}_{k}$ +- 직교화 $\mathbf{W}_{k}^{*} = \mathbf{W}_{k} - \mathbf{V}_{k-1} \mathbf{B}_{k-1}^{T}$ +- 행렬 $C_{k}$ 계산 $C_{k}=V_{k}MW_{k}^{*}$ +- 직교화 $\mathbf{W}_{k}^{**} = \mathbf{W}_{k}^{*} - \mathbf{V}_{k} \mathbf{C}_{k}$ +- 블록 벡터 내 정규화 $\mathbf{W}_{k}^{**} = \mathbf{V}_{k+1} \mathbf{B}_{k}$ + +midas FEA에서는 효과적인 고유치 계산을 위하여 블록 벡터 $V_{m}$ 를 이용하며, 이를 블록 Lanczos 방법이라 한다. 위의 반복계산 과정을 통하여 발생하는 블록 삼중 대각 행렬 $T_{k}$ 는 다음과 같다. + +$$ +\mathbf {T} _ {k} = \left[ \begin{array}{c c c c c} \mathbf {C} _ {1} & \mathbf {B} _ {1} ^ {T} & & & \\ \mathbf {B} _ {1} & \mathbf {C} _ {2} & \ddots & & \\ & \ddots & \ddots & \ddots & \\ & & \ddots & \mathbf {C} _ {k - 1} & \mathbf {B} _ {k - 1} ^ {T} \\ & & & \mathbf {B} _ {k - 1} ^ {T} & \mathbf {C} _ {k} \end{array} \right] \tag {2.2.1} +$$ + +$T_{k}$ 를 이용하여 고유치 문제 $T_{k}\psi_{m}=\theta_{m}^{*}I$ 를 풀게 되면 $\lambda_{m}^{*}=\sigma+1/\theta_{m}^{*}$ 을 이용하여 $\lambda_{m}^{*}$ 을 구할 수 있으며, $\lambda_{m}^{*}$ 은 본래의 고유치 문제 식 (2.1.7)의 근사값이다. $V_{k}$ 의 블록 크기를 $N_{b}$ 이라 할 때, 반복계산 과정의 반복 회수가 늘어날 수록 $T_{k}$ 의 크 + + + +기는 Nb 만큼 커지게 되며, \* λm 는 λm 으로 수렴한다. 고유모드 φ m 의 근사값 \* φm 는다음 식을 통하여 구할 수 있으며, \* λ 과 함께 수렴한다. + +$$ +\phi_ {m} ^ {*} = \left[ \begin{array}{l l l l l} \mathbf {V} _ {1} & \mathbf {V} _ {2} & \mathbf {V} _ {3} & \dots & \mathbf {V} _ {k} \end{array} \right] \psi_ {m} ^ {*} \tag {2.2.2} +$$ + +Lanczos 반복법으로 계산된 고유치와 고유모드의 수렴 여부는 다음과 같이 확인할 수 있다. + +$$ +\frac {\left\| \mathbf {K} \phi_ {m} ^ {*} - \lambda_ {m} ^ {*} \mathbf {M} \phi_ {m} ^ {*} \right\|}{\left\| \mathbf {K} \right\|} \leq \varepsilon \tag {2.2.3} +$$ + +여기서, . 은 2-norm을 의미한다. midas FEA에서는 16 ε 2.22 10− = × 를 이용한다. + + + +# 2-3 Subspace 반복법 + +Subspace 반복법은 부분공간 $E_{k}$ 를 이루는 $N_{s}$ 개의 벡터 $X_{k}$ 가 반복계산과정을 통하여 고유모드 $\left[\phi_{1} \quad \phi_{2} \quad \ldots \quad \phi_{N_{s}}\right]$ 에 수렴하도록 하는 방법이다. 부분공간 $E_{k}$ 를 구성하는 벡터 $X_{k}$ 를 반복적으로 계산해 나가는 과정은 다음과 같다. + +\- 첫 번째 반복계산의 경우 $E_{1}$ 을 구성하는 $N_{s}$ 개의 초기 벡터 $X_{1}$ + +-선형 연립 방정식 풀이 $\mathbf{K}\mathbf{Y}_{k}=\mathbf{M}\mathbf{X}_{k}$ + +\- 강성 행렬의 투영(projection) $\mathbf{K}_{k+1}=\mathbf{Y}_{k}^{T}\mathbf{K}\mathbf{Y}_{k}$ + +\- 질량 행렬의 투영(projection) $\mathbf{M}_{k+1}=\mathbf{Y}_{k}^{T}\mathbf{M}\mathbf{Y}_{k}$ + +\- 투영된 고유치 문제 풀이 $K_{k+1}Q_{k+1}=M_{k+1}Q_{k+1}\Lambda_{k+1}$ + +$$ +\mathbf {Q} _ {k + 1} = \left[ \begin{array}{c c c c} \psi_ {1} ^ {*} & \psi_ {2} ^ {*} & \dots & \psi_ {N _ {s}} ^ {*} \end{array} \right] +$$ + +$$ +\boldsymbol {\Lambda} _ {k + 1} = \left[ \begin{array}{c c c c} \lambda_ {1} ^ {*} & & & \\ & \lambda_ {2} ^ {*} & & \\ & & \ddots & \\ & & & \lambda_ {N _ {s}} ^ {*} \end{array} \right] +$$ + +$$ +- \mathbf {X} _ {k + 1} \text { 의 계산 } \quad \mathbf {X} _ {k + 1} = \mathbf {Y} _ {k} \mathbf {Q} _ {k + 1} +$$ + +위의 반복계산과정을 통하여 계산되는 $\lambda_{n}^{*}$ 와 $X_{k}$ 는 다음과 같이 고유치와 고유모드로 각각 수렴하게 된다. + +$$ +\lambda_ {m} ^ {*} \rightarrow \lambda_ {m}, \quad \mathbf {X} _ {k} \rightarrow \left[\begin{array}{c c c c}\phi_ {1}&\phi_ {2}&\dots&\phi_ {N _ {s}}\end{array}\right] \tag {2.3.1} +$$ + +$X_{k}$ 가 $N_{s}$ 개의 벡터로 이루어져 있을 때, 반복계산과정의 반복 회수와 관계 없이 $K_{k}$ 와 $M_{k}$ 의 크기는 $N_{s} \times N_{s}$ 로 고정된다. Subspace 반복법으로 계산된 고유치의 수렴 여부는 근사값 $\lambda_{m}^{*}$ 이 변화하는 정도에 의하여 다음과 같이 판단한다. + + + +Part 4 Linear Analysis + +$$ +\left\| \frac {\lambda_ {m} ^ {(k + 1)} - \lambda_ {m} ^ {(k)}}{\lambda_ {m} ^ {(k + 1)}} \right\| \leq \varepsilon \tag {2.3.2} +$$ + + + +# 2-4 관련기능 + +# 2-4-1 질량 행렬 + +고유치 해석에서 사용할 수 있는 질량행렬의 종류에는 분포(consistent) 질량과 집중(lumped) 질량이 있다. 분포 질량을 사용하게 되면 이론 해에 비하여 고유치가크게 계산되며, 집중 질량의 경우는 일반적으로 고유치가 작게 계산된다. 분포질량은 집중질량에 비하여 요소수에 따른 고유치의 수렴성이 더 좋은 것으로 알려져있으나, 계산량과 메모리 요구량이 많은 단점이 있다. 이와 같은 특징들은 그림2.4.1을 통하여 알 수 있다. + +![](images/page-365_2151cb534eec83186770c6f7025ac8f727cf11cb295d4d8f0888221e23fa0354.jpg) + +
+line + +| Number of Element | Lumped Mass | Consistent Mass | +| ----------------- | ----------- | --------------- | +| 0 | Low | High | +| 1 | Medium | Medium | +| 2 | High | Low | +| 3 | High | Low | +| 4 | High | Low | +| 5 | High | Low | +| 6 | High | Low | +| 7 | High | Low | +| 8 | High | Low | +| 9 | High | Low | +| 10 | High | Low | +
+ +그림 2.4.1 집중질량과 분포질량의 진동수 비교 + + + +# 2-4-2 Lanczos 반복법 관련기능 + +midas FEA에서 사용하는 Lanczos 반복법은 두 개의 진동수 1f , 2f 에 의해 계산하고자 하는 진동수의 관심영역(range of interest)을 결정할 수 있다. 그리고 진동수의 관심 영역은 Shift-invert 기법에서 고유치 예상값을 $\sigma = ( 2 \pi f _ { 1 } ) ^ { 2 }$ 로 설정함으로써 계산에 반영할 수 있다. 그림 2.4.2는 1f 과 2f 의 관계에 따라 계산하게 되는고유치의 영역과 순서를 그린 것이다. 계산하고자 하는 고유치 개수를 $N _ { f }$ 이라 할때, 1f , 2f 의 크기에 따라 다음과 같이 고유치를 계산한다. + +- 1 2f = f 인 경우 : 1f 에 가까운 진동수 $N _ { f }$ 개를 계산 +- 1 2f < f 인 경우 : 영역 1 2[ , ] f f 에서 1f 에 가까운 진동수 $N _ { f }$ 개를 계산 +- 1 2f > f 인 경우 : 영역 2 1[ , ] f f 에서 1f 에 가까운 진동수 $N _ { f }$ 개를 계산 + +![](images/page-366_f7cc228ecf2ac917ddf6037ba874c30bb1dc477738c0bdcd512cb6faba515316.jpg) + +
+text_image + +f₁=f₂ +④ ①②③ +←→ +f +
+ +![](images/page-366_b00ef754b887d00ff6247caade26a7e2c9f4ee27399bf6aa4b55eb69353c7449.jpg) + +
+text_image + +f₁ +①② +③④ +f₂ +f +
+ +![](images/page-366_d2c28fe27574e31158106430a02f4a9d10b1bdcba1e694f166a4de4e8f5448bd.jpg) + +
+text_image + +f₂ +④ +③②① +f₁ +f +
+ +그림 2.4.2 고유치 검색 방향 + +진동수의 관심영역 1f , 2f 의 기본값은 f f 1 2= = 0 이다. + + + +# 2-4-2-1 Sturm Sequence + +반복법을 이용하여 고유치를 계산하게 되면 낮은 차수의 고유치보다 높은 차수의고유치가 먼저 수렴하는 경우가 발생한다. midas FEA에서는 이와 같은 현상을 고려하여 진동수의 누락을 방지하는 기능을 제공하며, 그 방법은 다음과 같다.Lanczos 반복계산 과정 중에는 다음과 같은 연립 방정식의 해법이 필요하다. + +$$ +(\mathbf {K} - \sigma \mathbf {M}) \mathbf {W} _ {k} = \mathbf {U} _ {k} \tag {2.4.1} +$$ + +이 방정식의 풀이를 위하여 T LDL 행렬 분해를 수행하고, 행렬 D의 대각항 중 음수의 개수를 계산한다. 이 개수가 σ 보다 작은 고유치 λn 의 개수이다. 예를 들어$\boldsymbol { \sigma } = \lambda _ { N _ { f } } + \delta \underline { { \boldsymbol { \circ } } } \underline { { \boldsymbol { \mathbf { \Pi } } } }$ 두고 행렬을 분해하면 $\lambda _ { N _ { f } } + \delta$ 보다 작은 고유치 개수를 알 수 있다. $\lambda _ { N _ { f } } + \delta$ 가 보다 작은 고유치 개수가 $N _ { f }$ 보다 크면, 누락된 고유치가 모두 수렴할 때까지 Lanczos 반복계산 과정을 지속한다. 이 때 δ는 다음의 관계를 만족하도록 결정한다. + +$$ +\lambda_ {N _ {f}} < \lambda_ {N _ {f}} + \delta < \lambda_ {N _ {f} + 1} \tag {2.4.2} +$$ + +midas FEA에서는 “Sturm Sequence Check”를 수행할 경우 Lanczos 블록 벡터$\mathbf { V } _ { k }$ 의 크기를 $N _ { b } = 7$ 로 하여 계산 속도를 높이도록 하였다. 그러나 T LDL 분해를추가적으로 수행해야 하는 계산 부담을 고려하여 “Sturm Sequence Check”를 생략할 수 있도록 선택사항을 제공하고 있으며, 이 경우에는 $N _ { b } = 1$ 로 계산하기 때문에 고유치의 누락을 최소화할 수 있다. + +# 2-4-2-2 강체 모드 + +진동 특성을 분석하고자 하는 구조물의 경계조건이 부족한 경우에는 K 가 특이(singular) 행렬이 되기 때문에 σ = 0 으로 계산을 진행할 수 없다. 이러한 경우에는 0보다 작은 σ 를 가정하여 Shift-invert 기법을 적용하며, 계산된 고유치는 부족한 경계조건 개수 만큼의 0을 포함한다. midas FEA에서는 행렬 K 와 M 의 대각항을 이용하여 1차 모드로 예상되는 진동수 $\overline { { \lambda } } _ { 1 } \equivq$ 예측하고, $\sigma = - \overline { { \lambda } } _ { 1 }$ 로 설정한다. + + + +# 2-4-3 Subspace 반복법 관련기능 + +midas FEA에서 사용하는 Subspace 반복법에서는 Subspace $\mathbf { X } _ { k } \ \underline { { \circ } } | \ \exists 7 | \ N _ { s }$ ,최대 반복 회수 $N _ { I }$ , 그리고 식 (2.3.2)의 수렴 허용오차 $\varepsilon \ { \stackrel { \circ } { \equiv } }$ 설정할 수 있다.$N _ { s }$ 의 기본값은 0이며, 실제로 계산에 사용되는 Subspace의 크기 $ { N _ { s } } \equiv \ { \mathbb { L } } | \mathbf { { \frac { \mathrm { ~ o ~ } } { \mathrm { ~ \scriptstyle { \Xi ~ } } } } } { \mathbf { \vec { \imath } } } |$ 같다. + +$$ +N _ {s} = \max \left\{N _ {s}, \min (2 N _ {f}, N _ {f} + 8) \right\} \tag {2.4.3} +$$ + +Subspace 반복계산 과정의 최대 반복 회수 기본값은 $N _ { \mathit { I } } = 3 0 \ 0 | \mathbb { H }$ , 수렴 허용오차 기본값은 $\varepsilon = 1 . 0 \times 1 0 ^ { - 6 } 0$ 다. + + + +# Chapter 3. Time History Analysis + +# 3-1 개요 + +시간이력해석은 구조물에 동적하중이 작용할 때 식(3.1.1)과 같은 동적 평형방정식의 해를 구하는 것이다. + +$$ +\mathbf {M} \ddot {\mathbf {u}} (t) + \mathbf {C} \dot {\mathbf {u}} (t) + \mathbf {K} \mathbf {u} (t) = \mathbf {F} (t) \tag {3.1.1} +$$ + +여기서, + +M : 질량행렬 + +C : 감쇠행렬 + +K : 강성행렬 + +$\mathbf{F}(t)$ : 동적하중 + +u(t) : 상대변위 + +$\dot{\mathbf{u}}(t)$ : 속도 + +$\ddot{\mathbf{u}}(t)$ : 가속도 + +식 (3.1.1)에서 $\mathbf{F}(t)=0$ 이면 자유진동에 대한 방정식이 되고, C=0 인 조건을 추가하면 비감쇠 자유진동방정식이 된다. 그리고 $\mathbf{F}(t)$ 가 임의 시간에 대한 기진력(또는 기진변위, 속도, 가속도 등)이면 강제진동 해석문제가 된다. midas FEA에서는 식(3.1.1)로 표현되는 시간이력해석을 위해 모드중첩법(modal superposition method)과 직접적분법(direct integration method)을 사용한다 + + + +# 3-2 모드중첩법 + +모드종첩법에서는 고유모드의 직교성을 이용하여, 식(3.1.1)을 모드별로 서로 독립된 방정식으로 분해하여 계산을 수행한다. 이를 위하여 모드종첩법에서는 감쇠행렬은 질량행렬과 강성행렬의 선형조합으로 이루어진다는 가정을 사용한다. 그리고 모드별 응답이 얻어지면 이들을 식(3.2.1)와 같이 선형조합하여 해를 구한다. + +$$ +\mathbf {u} (t) = \boldsymbol {\Phi} \mathbf {Y} (t) = \sum_ {i = j} ^ {n} \boldsymbol {\phi} _ {i} Y _ {i} (t) \tag {3.2.1} +$$ + +식(3.2.1)을 식(3.1.1)에 대입하면 식(3.2.2)와 같이 나타낼 수 있다. + +$$ +\mathbf {M} \boldsymbol {\Phi} \ddot {\mathbf {Y}} (t) + \mathbf {C} \boldsymbol {\Phi} \dot {\mathbf {Y}} (t) + \mathbf {K} \boldsymbol {\Phi} \mathbf {Y} (t) = \mathbf {F} (t) \tag {3.2.2} +$$ + +식(3.2.2)에 $\phi_{m}^{T}$ (m번째 모드형상)을 곱하면 식(3.2.3)가 얻어진다. + +$$ +\boldsymbol {\phi} _ {m} ^ {T} \mathbf {M} \boldsymbol {\Phi} \ddot {\mathbf {Y}} (t) + \boldsymbol {\phi} _ {m} ^ {T} \mathbf {C} \boldsymbol {\Phi} \dot {\mathbf {Y}} (t) + \boldsymbol {\phi} _ {m} ^ {T} \mathbf {K} \boldsymbol {\Phi} \mathbf {Y} (t) = \boldsymbol {\phi} _ {m} ^ {T} \mathbf {F} (t) \tag {3.2.3} +$$ + +고유모드의 질량행렬과 강성행렬에 대한 직교성은 다음과 같이 표현할 수 있다. + +$$ +\phi_ {i} ^ {T} \mathbf {M} \phi_ {j} = m _ {i j} \delta_ {i j} \tag {3.2.4} +$$ + +$$ +\boldsymbol {\phi} _ {i} ^ {T} \mathbf {K} \boldsymbol {\phi} _ {j} = k _ {i j} \boldsymbol {\delta} _ {i j} +$$ + +여기서, + +$$ +\delta_ {i j} \quad : \text { 크로네커 델타(Kronecker delta) } \quad (i = j; \delta_ {i j} = 1, i \neq j; \delta_ {i j} = 0) +$$ + +고유모드의 질량 및 강성행렬에 대한 직교성을 이용하면, 동적평형방정식(3.2.2)은식(3.2.5)과 같이 서로 독립된 n개의 독립된 방정식으로 분해할 수 있다. + +$$ +\boldsymbol {\phi} _ {m} ^ {T} \mathbf {M} \boldsymbol {\phi} _ {m} \ddot {Y} (t) + \boldsymbol {\phi} _ {m} ^ {T} \mathbf {C} \boldsymbol {\phi} _ {m} \dot {Y} (t) + \boldsymbol {\phi} _ {m} ^ {T} \mathbf {K} \boldsymbol {\phi} _ {m} Y (t) = \boldsymbol {\phi} _ {m} ^ {T} \mathbf {F} (t) \quad (m = 1, 2, \dots \dots , n) \tag {3.2.5} +$$ + +따라서 모드별로 분해된 m개의 식은 일반화 좌표계(general coordinate)에서의 단자유도계에 대한 동적평형방정식이 된다. 그리고 식(3.2.5)를 변형하면 식(3.2.6) diff --git a/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_038.md b/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_038.md new file mode 100644 index 00000000..961af575 --- /dev/null +++ b/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_038.md @@ -0,0 +1,340 @@ + + +과 같이 나타낼 수 있다. + +$$ +\ddot {Y} _ {m} (t) + 2 \xi_ {m} \omega_ {m} \dot {Y} _ {m} (t) + \omega_ {m} ^ {2} Y _ {m} (t) = \frac {\boldsymbol {\phi} _ {m} ^ {T} \mathbf {F} (t)}{\boldsymbol {\phi} _ {m} ^ {T} \mathbf {M} \boldsymbol {\phi} _ {m}} +$$ + +$$ +2 \xi_ {m} \omega_ {m} \dot {Y} _ {m} (t) = \frac {\boldsymbol {\phi} _ {m} ^ {T} \mathbf {C} \boldsymbol {\phi} _ {m}}{\boldsymbol {\phi} _ {m} ^ {T} \mathbf {M} \boldsymbol {\phi} _ {m}} \tag {3.2.6} +$$ + +$$ +\omega_ {m} ^ {2} Y _ {m} (t) = \frac {\boldsymbol {\phi} _ {m} ^ {T} \mathbf {K} \boldsymbol {\phi} _ {m}}{\boldsymbol {\phi} _ {m} ^ {T} \mathbf {M} \boldsymbol {\phi} _ {m}} +$$ + +여기서, + +$$ +\xi_ {m} \quad : m \text { 차 모드의 감쇠비 } +$$ + +$$ +\omega_ {m} \quad : m \text { 차 모드의 고유진동수 } +$$ + +$$ +q _ {m} (t) \quad : m \text { 차 모드의 일반화 변위 } +$$ + +$$ +\dot {q} _ {m} (t) \quad : m \text { 차 모드의 속도 } +$$ + +$$ +\ddot {q} _ {m} (t) \quad : m \text { 차 모드의 가속도 } +$$ + +일반화 좌표계에서의 변위응답은 식(3.2.7)에 의해 구할 수 있다. + +$$ +\begin{array}{l} q _ {m} (t) = e ^ {- \xi_ {m} \omega_ {m} t} \left[ q _ {m} (0) \cos \omega_ {D m} t + \frac {\xi_ {m} \omega_ {m} q _ {m} (0) + q _ {m} (0)}{\omega_ {D m}} \sin \omega_ {D m} t \right] \\ + \frac {1}{m _ {m} \omega_ {D m}} \int_ {0} ^ {t} P _ {m} (\tau) e ^ {- \xi_ {m} \omega_ {m} (t - \tau)} \sin \omega_ {D m} (t - \tau) d \tau \tag {3.2.7} \\ \end{array} +$$ + +$$ +\omega_ {D m} = \omega_ {m} \sqrt {1 - \xi_ {m} ^ {2}} +$$ + +구조물의 변위응답은 단일자유도계 방정식(3.2.7)에 의해 구한 각 모드별 일반화 변위를 식(3.2.1)에 대입하여 구할 수 있다. 모드종첩법에서의 변위응답의 정확성은 사용하는 모드 수에 영향을 받는다. 모드종첩법은 구조해석 프로그램에서 가장 많이 사용되는 것으로 대형구조물의 선형 동적해석에 매우 효과적인 방법이다. 그러나 비선형 동적해석이나 특별한 감쇠장치가 포함되어 감쇠를 강성과 질량의 선형조합으로 가정할 수 없을 경우에는 사용할 수 없다. + + + +# 3-3 직접적분법 + +직접적분법은 전체해석의 시간 구간을 여러 개의 미소 시간 구간으로 분할한 후, 각 시간구간에서 동적평형방정식에 대한 수치적분(numerical integration)을 수행하는 방법이다. 직접적분법은 강성이나 감쇠의 비선형성을 고려한 비선형 해석에도 적용 가능하며, 모든 시간 단계에 대하여 해석을 수행하기 때문에 시간 단계의 수에 비례하여 해석시간이 소요된다. + +수치적분법은 사용방법에 따라 다양한 적분방법이 사용될 수 있다. midas FEA에서는 Newmark-β법의 평균가속도법(average acceleration method)을 사용한다. 평균가속도법에서는 시간 구간 $t_{i} < t < t_{i+1}$ 에서의 가속도 $\ddot{\mathbf{u}}(t)$ 는 식(3.3.1)와 같이 $\ddot{\mathbf{u}}_{i}$ 과 $\ddot{\mathbf{u}}_{i+1}$ 의 평균치로서 일정하다고 가정한다. + +$$ +\ddot {\mathbf {u}} (t) = \frac {\ddot {\mathbf {u}} _ {i} + \ddot {\mathbf {u}} _ {i + 1}}{2} = c o n s t. \tag {3.3.1} +$$ + +따라서 시간 $t_{i+1}$ 에서의 속도와 변위는 다음과 같이 나타낼 수 있다. + +$$ +\dot {\mathbf {u}} _ {i + 1} = \dot {\mathbf {u}} _ {i} + \frac {\ddot {\mathbf {u}} _ {i} + \ddot {\mathbf {u}} _ {i + 1}}{2} \Delta t \tag {3.3.2} +$$ + +$$ +\mathbf {u} _ {i + 1} = \mathbf {u} _ {i} + \dot {\mathbf {u}} _ {i} \Delta t + \frac {\ddot {\mathbf {u}} _ {i} + \ddot {\mathbf {u}} _ {i + 1}}{4} \Delta t ^ {2} \tag {3.3.3} +$$ + +식(3.3.2)와 (3.3.3)을 Newmark-β법의 적분변수에 의해 표현하면 다음과 같다. + +$$ +\dot {\mathbf {u}} _ {i + 1} = \dot {\mathbf {u}} _ {i} + (1 - \gamma) \Delta t \ddot {\mathbf {u}} _ {i} + \gamma \Delta t \ddot {\mathbf {u}} _ {i + 1} \tag {3.3.4} +$$ + +$$ +\mathbf {u} _ {i + 1} = \mathbf {u} _ {i} + \Delta t \dot {\mathbf {u}} _ {i} + \left(\frac {1}{2} - \beta\right) \Delta t ^ {2} \ddot {\mathbf {u}} _ {i} + \beta \Delta t ^ {2} \ddot {\mathbf {u}} _ {i + 1} \tag {3.3.5} +$$ + +여기서, + +$$ +\beta \quad : 0. 2 5 +$$ + +$$ +\gamma \quad : 0. 5 +$$ + + + +식(3.3.5)을 가속도에 관해서 정리하면 다음과 같다. + +$$ +\ddot {\mathbf {u}} _ {i + 1} = \frac {1}{\beta \Delta t ^ {2}} \left\{\mathbf {u} _ {i + 1} - \mathbf {u} _ {i} - \Delta t \dot {\mathbf {u}} _ {i} - \left(\frac {1}{2} - \beta\right) \Delta t ^ {2} \ddot {\mathbf {u}} _ {i} \right\} \tag {3.3.6} +$$ + +식(3.3.6)을 식(3.3.4)에 대입하여 속도에 관해 정리하면 다음과 같다. + +$$ +\dot {\mathbf {u}} _ {i + 1} = \frac {\gamma}{\beta \Delta t} \mathbf {u} _ {i + 1} - \frac {\gamma}{\beta \Delta t} \mathbf {u} _ {i} + \left(1 - \frac {\gamma}{\beta}\right) \dot {\mathbf {u}} _ {i} + \left(1 - \frac {\gamma}{2 \beta}\right) \Delta t \ddot {\mathbf {u}} _ {i} \tag {3.3.7} +$$ + +식(3.3.6)과 (3.3.7)을 동적평형방정식에 대입하여, 시간 $t_{i+1}$ 에서의 변위응답 $u_{i+1}$ 예관해 정리하면 다음과 같다. + +$$ +\begin{array}{l} \left(\frac {1}{\beta \Delta t ^ {2}} \mathbf {M} + \frac {\gamma}{\beta \Delta t} \mathbf {C} + \mathbf {K}\right) \mathbf {u} _ {i + 1} \\ = \mathbf {F} + \mathbf {M} \left\{\frac {1}{\beta \Delta t ^ {2}} \mathbf {u} _ {i} + \frac {1}{\beta \Delta t} \dot {\mathbf {u}} _ {i} + \left(\frac {1}{2 \beta} - 1\right) \ddot {\mathbf {u}} _ {i} \right\} + \mathbf {C} \left\{\frac {\gamma}{\beta \Delta t} \mathbf {u} _ {i} + \left(\frac {\gamma}{\beta} - 1\right) \dot {\mathbf {u}} _ {i} + \left(\frac {\gamma}{2 \beta} - 1\right) \Delta t \ddot {\mathbf {u}} _ {i} \right\} \\ \end{array} +$$ + +(3.3.8) + +식 (3.3.8)에 의해 구한 시간 $t_{i+1}$ 에서의 변위 $u_{i+1}$ 를 식 (3.3.6)과 (3.3.7)에 대입하여, 속도와 가속도를 구할 수 있다. + +직접적분법에서 감쇠는 식 (3.3.9)과 같이 Rayleigh 감쇠를 사용한다. + +$$ +\text { Rayleigh 감쇠 }: \mathbf {C} = a _ {0} \mathbf {M} + a _ {1} \mathbf {K} \tag {3.3.9} +$$ + +여기서, + +$a_{0}$ : 감쇠계산을 위한 질량 + +$a_{1}$ : 감쇠계산을 위한 강성의 비례상수 + +식 (3.3.8)에 식(3.3.9)를 대입하여 정리하면 동적평형방적식은 다음과 같이 표현된다. + + + +$$ +\begin{array}{l} \left\{\left(\frac {1}{\beta \Delta t ^ {2}} + \frac {\mathbf {a} _ {1} \gamma}{\beta \Delta t}\right) \mathbf {M} + \left(\frac {\mathbf {a} _ {2} \gamma}{\beta \Delta t} + 1\right) \mathbf {K} \right\} \mathbf {u} _ {i + 1} \\ = \mathbf {F} + \mathbf {M} \left\{\frac {1}{\beta \Delta t ^ {2}} \mathbf {u} _ {i} + \frac {1}{\beta \Delta t} \dot {\mathbf {u}} _ {i} + \left(\frac {1}{2 \beta} - 1\right) \ddot {\mathbf {u}} _ {i} + \mathbf {a} _ {1} \overline {{\mathbf {D}}} \right\} + \mathbf {a} _ {2} \mathbf {K} \overline {{\mathbf {D}}} \tag {3.3.10} \\ \end{array} +$$ + +$$ +\overline {{{\mathbf {D}}}} = \frac {\gamma}{\beta \Delta t} \mathbf {u} _ {i} + \left(\frac {\gamma}{\beta} - 1\right) \dot {\mathbf {u}} _ {i} + \left(\frac {\gamma}{2 \beta} - 1\right) \Delta t \ddot {\mathbf {u}} _ {i} +$$ + +시간이력해석을 통하여 구하는 동적 평형방정식의 해는 상대변위 $\mathbf{u}(t)$ ，상대속도 $\dot{\mathbf{u}}(t)$ ，상대가속도 $\ddot{\mathbf{u}}(t)$ 이다. + +지반가속도와 같은 동적하중을 받을 때 구조물의 절대응답은 식 (3.3.11)에 나타낸 것과 같이 상대응답과 지반의 응답을 더하여 구할 수 있다. + +$$ +\ddot {\mathbf {u}} _ {g, i + 1} + \ddot {\mathbf {u}} _ {i + 1}: \text { 절대가속도 } +$$ + +$$ +\dot {\mathbf {u}} _ {g, i + 1} + \dot {\mathbf {u}} _ {i + 1}: \text {절대속도} \tag {3.3.11} +$$ + +$$ +\mathbf {u} _ {g, i + 1} + \mathbf {u} _ {i + 1}: \text { 절대변위 } +$$ + +여기서, + +$$ +\ddot {\mathbf {u}} _ {g, i + 1} \quad : \text { 지반가속도 } +$$ + +$$ +\dot {\mathbf {u}} _ {g, i + 1} \quad : \text { 속도 } +$$ + +$$ +\mathbf {u} _ {g, i + 1} \quad : \text { 변위 } +$$ + +midas FEA에서 지반가속도에 의한 지반속도, 변위는 선형가속도법에 의해 식(3.3.12)로 계산되며, 모드종첩법과 직접적분법의 절대응답계산시에 적용된다. + +$$ +\dot {\mathbf {u}} _ {g, i + 1} = \dot {\mathbf {u}} _ {g, i} + \Delta t \ddot {\mathbf {u}} _ {g, i} + \frac {1}{2} \frac {\ddot {\mathbf {u}} _ {g , i + 1} - \ddot {\mathbf {u}} _ {g , i}}{\Delta t} \Delta t ^ {2} \tag {3.3.12} +$$ + +$$ +\mathbf {u} _ {g, i + 1} = \mathbf {u} _ {g, i} + \Delta t \dot {\mathbf {u}} _ {g, i} + \frac {1}{2} \Delta t ^ {2} \ddot {\mathbf {u}} _ {g, i} + \frac {1}{6} \frac {\ddot {\mathbf {u}} _ {g , i + 1} - \ddot {\mathbf {u}} _ {g , i}}{\Delta t} \Delta t ^ {3} +$$ + + + +# 3-4 감쇠 + +midas FEA는 동적해석의 해석방법에 따라서, 다음의 감쇠방법을 사용한다. + +응답 스펙트럼 해석 및 모드중첩법에 의한 시간이력해석의 감쇠설정 + +\- Modal + +\- 질량 & 강성 비례 (Rayleigh 감쇠) + +직접 적분법에 의한 시간이력해석의 감쇠설정 + +\- 질량 & 강성 비례 (Rayleigh 감쇠) + +# 3-4-1 Rayleigh 감쇠 + +Rayleigh 감쇠는 그림 3.4.1(a)에 나타낸 것과 같이 감쇠행렬을 구조물의 질량행렬과 강성행렬의 선형합으로써 구성한다. r차모드의 감쇠정수 ξr 와 고유진동수 ωr및 s차 모드의 감쇠정수 ξ s 와 고유진동수 ωs 가 주어졌을 때 Rayleigh 감쇠의 감쇠행렬은 다음과 같이 표현된다. 단, r, s차 모드는 구조물의 주요한 2개의 모드를의미한다. + +$$ +\mathbf {C} = a _ {0} \mathbf {M} + a _ {1} \mathbf {K} \tag {3.4.1} +$$ + +$$ +\xi_ {i} = \frac {1}{2} \left(\frac {a _ {0}}{\omega_ {i}} + a _ {1} \cdot \omega_ {i}\right) \tag {3.4.2} +$$ + +여기서 + +$$ +a _ {0} \quad : \frac {2 \cdot \omega_ {r} \cdot \omega_ {s} \left(\xi_ {r} \cdot \omega_ {s} - \xi_ {s} \cdot \omega_ {r}\right)}{\left(\omega_ {s} ^ {2} - \omega_ {r} ^ {2}\right)} +$$ + +$$ +a _ {1} \quad : \frac {2 \left(\xi_ {s} \cdot \omega_ {s} - \xi_ {r} \cdot \omega_ {r}\right)}{\left(\omega_ {s} ^ {2} - \omega_ {r} ^ {2}\right)} +$$ + + + +![](images/page-376_e005572ff6e98e04ca1bbc1b442ba618db783d69a96c3a39265ef53a4226b822.jpg) + +
+line + +| Natural frequencies ωi | Mass Proportional ξi | Stiffness Proportional ξi | +| --------------------- | --------------------- | -------------------------- | +| ω₁ | ξi = a₀M | ξi = a₀/2ωi | +| ω₂ | ξi = a₁K | ξi = a₁·ωi/2 | +| ω₃ | ξi = a₁K | ξi = a₁·ωi/2 | +| ω₄ | ξi = a₁K | ξi = a₁·ωi/2 | +
+ +(a) 질량 비례 감쇠와 + +![](images/page-376_862e610a71647901230a897b54ee998ec3bc9008f0a7ea22d12e07b67f4696d3.jpg) + +
+line + +| Natural frequencies ωi | Rayleigh Damping ξi | Rayleigh Damping ξi | +| --------------------- | ------------------- | ------------------- | +| ωr | High | Low | +| ωs | Low | High | +
+ +(b) Rayleigh 감쇠 +강성 비례 감쇠 + +그림 3.4.1 모드별 감쇠율과 고유진동수와의 관계 + +# 3-4-2 Modal 감쇠 + +모드별 감쇠는 각 모드별로 사용자가 직접 감쇠비를 정의하고 정의된 모드별 감쇠비에 따라서 모드별 응답을 계산한다. 모드별 감쇠는 응답 스펙트럼해석 및 모드중첩법에 의한 시간이력해석에서 사용이 가능하다. + +응답스펙트럼해석과 모드중첩법에 의한 해석에서는 구조물의 운동 방정식을 모드별로 분해하여, 각 모드의 운동 방정식에 사용자가 직접 입력한 모드별 감쇠비를적용하여 해를 구한다. + + + +# 3-5 주의사항 + +해석에 사용되는 시간 간격은 결과의 정확도에 상당한 영향을 미칠 수 있으며, 시간 간격이 부적절한 경우 부정확한 결과를 나타낼 수 있다. 또한 시간 간격의 크기는 구조물의 고차모드의 주기나 하중의 주기와 밀접한 관계를 갖는다. 일반적으로해석 시간 간격은 고려하고자 하는 최고차 모드 주기의 1/10 정도의 시간 간격이타당하며, 입력된 하중의 시간 간격보다는 작아야 한다. 그리고 동적하중은 전체하중 변화를 충분히 나타낼 수 있어야 한다. midas FEA에서 입력되지 않은 시간에서의 하중 값은 선형보간하여 사용한다. + + + +Part 4 Linear Analysis + + + +# Chapter 4. Response Spectrum Analysis + +# 4-1 개요 + +응답스펙트럼해석(response spectrum analysis)은 지진하중에 의한 구조물의 응답을 평가하기 위한 방법중의 하나로 내진설계시 사용하는 가장 보편화된 방법이다. 이는 다자유도 구조물의 응답을 단자유도계의 응답스펙트럼함수(response spectrum function)를 이용하여 근사적으로 구하는 방법으로 크게 다음의 두 단계로 설명할 수 있다. + +- 1단계 : 전체 구조물에 대한 하나의 다자유도 동적 평형방정식을 여러 개의 단자유도 평형방정식으로 분리한다. +- 2단계 : 응답스펙트럼함수를 이용해 단자유도 평형방정식의 최대응답을 구하고 이를 조합하여 다자유도에 대한 최대응답을 구한다. + +응답스펙트럼해석을 위한 지반운동이 가해지는 구조물에 대한 동적 평형방정식은 식 (4.1.1)과 같다. + +$$ +\mathbf {M} [ \ddot {\mathbf {u}} (t) + \mathbf {r} \ddot {u} _ {g} (t) ] + \mathbf {C} \dot {\mathbf {u}} (t) + \mathbf {K} \mathbf {u} (t) = \mathbf {0} \tag {4.1.1} +$$ + +$$ +\mathbf {M} \ddot {\mathbf {u}} (t) + \mathbf {C} \dot {\mathbf {u}} (t) + \mathbf {K} \mathbf {u} (t) = - \mathbf {M} \mathbf {r} \ddot {u} _ {g} (t) +$$ + +여기서, + +M : 질량행렬 (mass matrix) +C : 감쇠행렬 (damping matrix) +K : 강성행렬 (stiffness matrix) +r : 지반가속도의 방향벡터 +$\ddot{u}_{g}(t)$ : 지반가속도의 시간이력 +u(t) : 상대변위 +$\dot{\mathbf{u}}(t)$ : 속도 +$\ddot{\mathbf{u}}(t)$ : 가속도 + + + +비감쇠 자유진동해석에서 얻은 고유모드 형상(Φ)을 사용하여 변위 u(t) 를 모드 변위 y(t) 의 조합으로 나타내면 식 (4.1.2)와 같다. + +$$ +\mathbf {u} (t) = \boldsymbol {\Phi} \mathbf {y} (t) \tag {4.1.2} +$$ + +식 (4.1.2)를 식 (4.1.1)에 대입하고, 양변에 $\Phi^{T}$ 를 곱하여 정리하면 식 (4.1.3)과 같이 된다. + +$$ +\boldsymbol {\Phi} ^ {T} \mathbf {M} \boldsymbol {\Phi} \ddot {\mathbf {y}} (t) + \boldsymbol {\Phi} ^ {T} \mathbf {C} \boldsymbol {\Phi} \dot {\mathbf {y}} (t) + \boldsymbol {\Phi} ^ {T} \mathbf {K} \boldsymbol {\Phi} \mathbf {y} (t) = - \boldsymbol {\Phi} ^ {T} \mathbf {M} \mathbf {r} \ddot {u} _ {g} (t) \tag {4.1.3} +$$ + +고유벡터 $\phi$ 는 직교성으로 인하여 다음과 같은 관계를 가진다. + +$$ +\phi_ {i} (\mathbf {M} \text { or } \mathbf {C} \text { or } \mathbf {K}) \phi_ {j} = 0 (i \neq j) \tag {4.1.4} +$$ + +따라서 질량에 대하여 무차원화( $\Phi^{T}M\Phi=1$ )된 고유모드 형상과 $\zeta_{m}=\phi_{m}^{T}C\phi_{m}$ , $\omega_{m}=\phi_{m}^{T}K\phi_{m}$ 을 식 (4.1.3)에 적용하면, 식 (4.1.5)와 같이 모드 별로 독립적인 연립 미분방정식을 얻을 수 있다. + +$$ +\left[ \begin{array}{c c c c c} 1 & & & & \\ & \ddots & & & \\ & & 1 & & \\ & & & \ddots & \\ & & & & 1 \end{array} \right] \dot {\mathbf {y}} (t) + \left[ \begin{array}{c c c c c} 2 \xi_ {1} \omega_ {1} & & & & \\ & \ddots & & & \\ & & 2 \xi_ {m} \omega_ {m} & & \\ & & & \ddots & \\ & & & & 2 \xi_ {n} \omega_ {n} \end{array} \right] \dot {\mathbf {y}} (t) + \left[ \begin{array}{c c c c c} \omega_ {1} ^ {2} & & & & \\ & \ddots & & & \\ & & \omega_ {m} ^ {2} & & \\ & & & \ddots & \\ & & & & \omega_ {n} ^ {2} \end{array} \right] +$$ + +$$ +\mathbf {y} (t) = - \boldsymbol {\Phi} ^ {T} \mathbf {M} \mathbf {r} \ddot {u} _ {g} (t) \tag {4.1.5} +$$ + +그리고 식 (4.1.5)에서 m 번째 모드에 대한 수식을 정리하면 식 (4.1.6)과 같다. + +$$ +\ddot {y} _ {m} (t) + 2 \xi_ {m} \omega_ {m} \dot {y} _ {m} (t) + \omega_ {m} ^ {2} y _ {m} (t) = - \Gamma_ {m} \ddot {u} _ {g} (t) \tag {4.1.6} +$$ + +$$ +\Gamma_ {m} = \boldsymbol {\phi} _ {m} ^ {T} \mathbf {M} \mathbf {r} +$$ + +식 (4.1.6)에서 모드기여계수 $\Gamma_{m}$ 는 질량에 대하여 무차원화된 모드형상과 질량 그리고 지반가속도의 방향벡터의 곱으로 정의된다. 지반가속도의 방향벡터는 지반의 diff --git a/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_039.md b/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_039.md new file mode 100644 index 00000000..137b0063 --- /dev/null +++ b/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_039.md @@ -0,0 +1,293 @@ + + +가속도 방향의 자유도만 1을 갖는 벡터이다. 지반가속도가 작용하는 구조물의 동적 평형방정식의 해는 식 (4.1.6)과 같은 n 개의 방정식의 해를 계산하여 식 (4.1.7)과 같은 방법으로 조합하여 구한다. + +$$ +\mathbf {u} (t) = \boldsymbol {\Phi} \mathbf {y} (t), \dot {\mathbf {u}} (t) = \boldsymbol {\Phi} \dot {\mathbf {y}} (t), \ddot {\mathbf {u}} (t) = \boldsymbol {\Phi} \ddot {\mathbf {y}} (t) \tag {4.1.7} +$$ + +응답스펙트럼해석에서 지반가속도를 받는 동적 평형방정식의 개념은 식 (4.1.1)\~식 (4.1.7)을 따른다. 응답스펙트럼 해석에서는 지진가속도를 특정한 함수로 정의하지 않고, 스펙트럼 함수를 사용하여 모드별 결과를 구한다. 일반적으로 스펙트럼 함수는 식 (4.1.5)의 주기별 최대값을 의미하는데, 시방서나 기준에서는 지진가속도의 여러 가지 가능성과 지역의 특성 또는 구조물의 중요도 등을 고려하여 정한다. 식 (4.1.6)과 같은 임의 모드 m 번째에 해당하는 단자유도시스템의 해는 스펙트럼 함수로부터 구한 변위, 속도, 가속도에 해당 차수의 모드기여계수를 곱하여 식 (4.1.8)과 같이 구한다. + +$$ +S _ {d m} = \frac {S _ {a m}}{\omega_ {m} ^ {2}}, \quad S _ {v m} = \frac {S _ {a m}}{\omega_ {m}} \tag {4.1.8} +$$ + +$$ +y _ {m} = \Gamma_ {m} S _ {d m}, \dot {y} _ {m} = \Gamma_ {m} S _ {v m}, \ddot {y} _ {m} = \Gamma_ {m} S _ {a m} +$$ + +각 모드별 결과는 식 (4.1.7)의 결과에 모드 형상을 곱하여 식 (4.1.9)와 같이 계산한다. + +$$ +\mathbf {u} _ {m} = \phi_ {m} \Gamma_ {m} S _ {d m}, \quad \dot {\mathbf {u}} _ {m} = \phi_ {m} \Gamma_ {m} S _ {v m}, \quad \ddot {\mathbf {u}} _ {m} = \phi_ {m} \Gamma_ {m} S _ {a m} \tag {4.1.9} +$$ + +여기서, + +$S_{dm}$ : m번째 모드의 변위 스펙트럼 값 + +$S_{vm}$ : m번째 모드의 속도 스펙트럼 값 + +$S_{am}$ : m번째 모드의 가속도 스펙트럼 값 + +모드별 해석결과는 최대값만을 가지고 있기 때문에 시간이력해석과 같이 모드별선형조합을 할 수가 없다. 그러므로 응답스펙트럼해석의 최종적인 결과는 식 (4.1.9)의 각 모드별 해석 결과를 적절한 방법으로 조합하여 구한다. + + + +midas FEA에서는 전체좌표계 X-Y 평면의 임의의 방향과 Z방향에 대한 응답스펙트럼해석이 가능하다. 그리고 모드별 해석결과의 조합(modal combination)은 사용자의 선택에 따라 ABS(absolute sum)방법, SRSS(square root of the sum ofthe squares)방법과 CQC(complete quadratic combination)방법 등을 사용할 수있다. + +ABS (absolute sum) + +$$ +R _ {\max} = \left| R _ {1} \right| + \left| R _ {2} \right| + \dots + \left| R _ {n} \right| \tag {4.1.10} +$$ + +SRSS (square root of the sum of the squares) + +$$ +R _ {\max} = \left[ R _ {1} ^ {2} + R _ {2} ^ {2} + \dots + R _ {n} ^ {2} \right] ^ {1 / 2} \tag {4.1.11} +$$ + +CQC (complete quadratic combination) + +$$ +R _ {\max} = \left[ \sum_ {i = 1} ^ {N} \sum_ {j = 1} ^ {N} R _ {i} \rho_ {i j} R _ {j} \right] ^ {1 / 2} \tag {4.1.12} +$$ + +$$ +\rho_ {i j} = \frac {8 \sqrt {\xi_ {i} \xi_ {j}} \left(\xi_ {i} + r _ {i j} \xi_ {m}\right) r _ {i j} ^ {3 / 2}}{\left(1 - r _ {i j} ^ {2}\right) ^ {2} + 4 \xi_ {i} \xi_ {j} r _ {i j} \left(1 + r _ {i j} ^ {2}\right) + 4 \left(\xi_ {i} ^ {2} + \xi_ {j} ^ {2}\right) r _ {i j} ^ {2}} +$$ + +$$ +\rho_ {i j} = \frac {8 \xi^ {2} \left(1 + r _ {i j}\right) r _ {i j} ^ {3 / 2}}{\left(1 - r _ {i j} ^ {2}\right) ^ {2} + 4 \xi^ {2} r _ {i j} \left(1 + r _ {i j}\right) ^ {2}} \quad \left(\xi_ {i} = \xi_ {i} = \xi\right) +$$ + +$$ +r _ {i j} = \frac {\omega_ {i}}{\omega_ {j}} \quad \omega_ {j} > \omega_ {i} +$$ + +$$ +0 \leq \rho_ {i j} \leq 1 \quad \rho_ {i j} = 1 (i = j) +$$ + +여기서, + +$R _ { \mathrm { m a x } }$ Rmax : 최종 결과 값 + +$R _ { i }$ : 임의 i 번째 모드의 스펙트럼 값 + +$r _ { i j }$ : i 번째 모드에 대한 j 번째 모드의 고유치 비율 + + + +$$ +\omega_ {i}, \omega_ {j} \quad : i, j \text { 번째 모드의 고유치 값 } +$$ + +$$ +\xi_ {i}, \xi_ {j} \quad : i, j \text { 번째 모드의 감쇠비 } +$$ + +식 (4.1.12)에서 i j = 이면, 감쇠비( ξi , ξ j )에 관계없이 ρij = 1이 되고, 감쇠비가 0인 경우 CQC와 SRSS의 결과가 동일한 값을 갖는다. 모드별 조합 방법 중에서ABS가 가장 큰 조합치를 산출한다. SRSS는 고유진동수들이 근접한 값을 가질 경우, 조합결과가 과대 또는 과소평가 되는 경향이 있다. 예를 들어, 감쇠비가 0.05이고 3개의 자유도를 가진 구조물의 고유진동수와 각 모드별 변위가 다음과 같이계산되었을 경우, SRSS와 CQC의 적용결과를 비교하면 다음과 같다. + +고유진동수 + +$$ +\omega_ {1} = 0. 4 6, \omega_ {2} = 0. 5 2, \omega_ {3} = 1. 4 2 +$$ + +모드별 응답스펙트럼 값 : $D _ { i j }$ ( j 번째 모드에 대한 i 자유도의 변위성분) + +$$ +\left[ D _ {i j} \right] = \left[ \begin{array}{c c c} 0. 0 3 6 & 0. 0 1 2 & 0. 0 1 9 \\ - 0. 0 1 2 & 0. 0 4 4 & - 0. 0 0 5 \\ 0. 0 4 9 & 0. 0 0 2 & - 0. 0 1 7 \end{array} \right] +$$ + +SRSS 방법의 결과 + +$$ +R _ {\max} = \left[ R _ {1} ^ {2} + R _ {2} ^ {2} + R _ {3} ^ {2} \right] ^ {1 / 2} = \left\{0. 0 4 2 \quad 0. 0 4 6 \quad 0. 0 5 2 \right\} +$$ + +CQC 방법의 결과 + +$$ +\rho_ {1 2} = \rho_ {2 1} = 0. 3 9 8 5 +$$ + +$$ +\rho_ {1 3} = \rho_ {3 1} = 0. 0 0 6 1 +$$ + +$$ +\rho_ {2 3} = \rho_ {3 2} = 0. 0 0 8 0 +$$ + + + +$$ +R _ {\max} = \left[ R _ {1} ^ {2} + R _ {2} ^ {2} + R _ {3} ^ {2} + 2 \rho_ {1 2} R _ {1} R _ {2} + 2 \rho_ {1 3} R _ {1} R _ {3} + 2 \rho_ {2 3} R _ {2} R _ {3} \right] ^ {1 / 2} +$$ + +$$ += \left\{ \begin{array}{l l l} 0. 0 4 6 & 0. 0 4 1 & 0. 0 5 3 \end{array} \right\} +$$ + +위의 두 가지 결과를 비교해보면 SRSS 방법을 사용할 경우가 CQC에 비해 첫 번째 자유도 성분에 대해서는 과소평가되고, 두 번째 자유도 성분에 대해서는 과대평가 되었다고 할 수 있다. 따라서 고유진동수들이 상대적으로 근접한 값을 가질때 SRSS 방법은 과소 또는 과대평가 된 결과를 산출함을 알 수 있다. + + + +# 4-2 스펙트럼 함수 + +![](images/page-385_5d072bb2a20043d4d437c39727f23d58cd837dc7613d6757a9e57873c8a6d94d.jpg) + +
+line + +| Period(Sec) | Spectral Data | +|---|---| +| T1 | S1 | +| T2 | S2 | +| T3 | S3 | +| T4 | S4 | +| T5 | S5 | +| T6 | S6 | +| T7 | S7 | +| Sn-1 | Sn-1 | +| Tn | Sn | +S6 = (S7 - S6)/(T7 - T6) × (Tx - T6) + S6 +S7 = (S7 - S6)/(T7 - T6) × (Tx - T6) + S6 +
+ +그림 4.2.1 스펙트럼함수와 임의 주기에 대한 스펙트럼 값의 보간 방법 + +응답스펙트럼해석에서는 각 모드별 해석결과를 스펙트럼 함수(spectrum function)를 사용하여 구한다. 일반적인 의미에서의 스펙트럼함수는 식 (4.1.6)의 시간이력해석결과 중에서의 최대값으로 구성된다. 구조물의 감쇠(ξ)와 시간에 따른 지진하중 ( $\ddot{u}_{g}(t)$ )이 정해지면 구조물의 고유주기(ω)에 따라 식 (4.1.6)의 해를 구할 수 있다. 구조물의 주기를 가로축으로 하고 변위, 속도, 가속도 등의 결과를 세로축에 표시하면 그림 4.2.1과 같은 스펙트럼 함수를 얻을 수 있다. 응답스펙트럼해석에 사용되는 스펙트럼함수는 주로 각종 기준에서 제공하는 값을 사용하게 된다. midas FEA에서는 설계응답스펙트럼함수(design response spectrum function)생 + + + +성기능을 이용하여 지진해석시 사용되는 스펙트럼함수를 설계기준에 따라 동적계수, 지반계수, 지역계수, 중요도계수, 반응수정계수 등의 입력으로 쉽게 생성할 수있다. 구조물의 고유주기에 해당하는 스펙트럼 값을 구하기 위해 보간법(linear orlog scale interpolation)을 사용하기 때문에 스펙트럼 값의 변화가 심한 부분은가능한 세분화된 데이터를 사용하는 것이 바람직하다. 그리고 스펙트럼 함수의 범위는 고유치 해석에서 산출된 최대, 최소 주기범위를 포함할 수 있도록 입력되어야 한다. midas FEA에서는 고유치 해석의 주기가 입력된 스펙트럼 함수의 범위를초과하는 경우에는 스펙트럼 함수의 최대 또는 최소값을 사용한다. + + + +# Chapter 5. Linear Buckling Analysis + +# 5-1 개요 + +선형좌굴해석(linear buckling analysis) 기능은 구조물의 임계하중계수(criticalload factor)와 그에 해당하는 좌굴모드형상(buckling mode shape)을 구하는데사용된다. 선형좌굴해석을 위하여 응력에 의한 기하강성(geometric stiffness)을고려한 구조물의 평형방정식을 기술하면 다음과 같다. + +$$ +\mathbf {K} \mathbf {u} + \overline {{\mathbf {K}}} _ {G} \mathbf {u} = \overline {{\mathbf {p}}} \tag {5.1.1} +$$ + +여기서, + +$\mathbf { K }$ : 탄성강성행렬 + +$\bar { \mathbf { K } } _ { G }$ : 응력에 의한 기하강성행렬 + +$\mathbf { u }$ : 구조물의 전체변위 + +$\overline { { \mathbf { p } } }$ : 구조물에 작용하는 하중 + +선형해석에서 구조물의 응력은 하중에 비례하고, 기하강성행렬은 응력에 비례한다.그러므로 하중 p 가 기준하중 p 에 비례한다고 가정하면, 기하강성행렬 $\overline { { \mathbf { K } } } _ { G } \triangleqq$ 다음과 같이 나타낼 수 있다. + +$$ +\overline {{{\mathbf {K}}}} _ {G} = \alpha \mathbf {K} _ {G} \tag {5.1.2} +$$ + +$$ +\overline {{{\mathbf {p}}}} = \alpha \mathbf {p} \tag {5.1.3} +$$ + +여기서, + +$\mathbf { p }$ : 기준하중 + +KG $\mathbf { K } _ { G }$ : 기준하중에 대응하는 기하강성 행렬 + +$\alpha$ ： + +식 (5.1.2)와 (5.1.3)을 식 (5.1.1)에 대입하면 다음과 같다. + + + +$$ +\mathbf {K} \mathbf {u} + \alpha \mathbf {K} _ {G} \mathbf {u} = \alpha \mathbf {p} \tag {5.1.4} +$$ + +식 (5.1.4)와 같은 평형 상태는 하중계수 α 의 크기에 따라 안정하거나 불안정할수 있다. 안정성(stability)을 판단하기 위하여 평형상태 u 에 섭동(perturbation)u. + +$$ +\mathbf {K} (\mathbf {u} + \delta \mathbf {u}) + \alpha \mathbf {K} _ {G} (\mathbf {u} + \delta \mathbf {u}) = \alpha \mathbf {p} \tag {5.1.5} +$$ + +평형방정식 (5.1.4)를 이용하여 식 (5.1.5)에서 섭동 이외의 항을 소거하면 다음과같은 고유치 문제를 얻을 수 있다. + +$$ +(\mathbf {K} + \alpha \mathbf {K} _ {G}) \delta \mathbf {u} = \mathbf {0} \tag {5.1.6} +$$ + +여기서, 평형상태의 안정성은 다음과 같이 행렬식을 통해 판단할 수 있다. + +$$ +\left| \mathbf {K} + \alpha \mathbf {K} _ {G} \right| > 0 \quad : \text { 안정한 상태 } +$$ + +$$ +\left| \mathbf {K} + \alpha \mathbf {K} _ {G} \right| \leq 0 \quad : \text { 불안정한 상태 } +$$ + +그러므로 식 (5.1.6)을 만족하는 고유치 α 는 평형상태의 불안정성이 시작되는 임계하중계수(critical load factor)라 할 수 있으며, 이에 대응하는 고유모드중(critical load)은 임계하중계수와 기준하중을 고려하여 αp 로 표현할 수 있다.그림 5.1.1은 압축 하중을 받는 기둥의 평형상태와 임계하중에 의한 좌굴 형상을나타내고 있다. + + + +![](images/page-389_9862038f871deddbb9815debedf92e98929b7d0f0b5df4c94c3d700842a9b8f0.jpg) + +
+text_image + +αP +u +αP +δu +δu +
+ +그림. 5.1.1 압축하중을 받는 기둥의 좌굴 + +선형좌굴해석에는 기하강성행렬 $\mathbf { K } _ { G } \ \underline { { \circ } } \mathbf { | }$ 계산 과정과 고유치 문제 식 (5.1.6)의 해법이 필요하다. midas FEA에서는 요소 별로 응력 또는 요소내력에 기반한 기하강성을 이용하며, 고유치 문제는 Lanczos 반복(iteration)법으로 계산한다. + + + +# 5-2 기하강성 + +midas FEA에서 응력에 의한 기하강성을 고려하는 요소는 다음과 같다. + +트러스요소, 보요소, 평면응력요소, 판요소, 입체요소 + +구조물의 유한요소 모델에 기하강성을 고려하는 요소가 존재하지 않으면 좌굴해석을 위한 고유치 문제가 성립되지 않음에 주의해야 한다. + +각각의 요소에 대한 기하강성의 일반적인 형태는 다음과 같다. + +$$ +\mathbf {K} _ {G} ^ {e} = \int \mathbf {G} ^ {T} \mathbf {S} \mathbf {G} d V \tag {5.2.1} +$$ + +여기서, S 는 응력 또는 요소내력이고, G 는 절점 변위와 변위 도함수(displacement derivative)의 관계를 정의하는 행렬이다. + +# 5-2-1 트러스요소 + +트러스요소의 기하강성 계산할 때는 요소좌표계에서 y , z 방향 이동변위만을 고려한다. + +$$ +\mathbf {u} _ {i} = \left\{v _ {i} \quad w _ {i} \right\} ^ {T} \tag {5.2.2} +$$ + +$$ +v = \sum_ {i = 1} ^ {2} N _ {i} v _ {i}, w = \sum_ {i = 1} ^ {2} N _ {i} w _ {i} \tag {5.2.3} +$$ + +여기서, + +Ni : 2 절점 형상함수 + +S 의 구성에 있어서는 축방향 응력 σ xx 만을 고려한다. diff --git a/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_040.md b/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_040.md new file mode 100644 index 00000000..be52907c --- /dev/null +++ b/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_040.md @@ -0,0 +1,257 @@ + + +$$ +\mathbf {S} = \left[ \begin{array}{c c} \sigma_ {x x} & 0 \\ 0 & \sigma_ {x x} \end{array} \right] \tag {5.2.4} +$$ + +절점변위-변위도함수의 관계 Gi 는 식 (5.2.5)와 같다. + +$$ +\mathbf {G} _ {i} = \left[ \begin{array}{c c} \frac {\partial N _ {i}}{\partial x} & 0 \\ 0 & \frac {\partial N _ {i}}{\partial x} \end{array} \right] \tag {5.2.5} +$$ + +행렬 Gi 를 이용하여 트러스요소의 기하강성을 표현하면 다음과 같다. + +$$ +\mathbf {K} _ {G i j} = A \int_ {L _ {e}} \mathbf {G} _ {i} ^ {T} \mathbf {S} \mathbf {G} _ {j} d L \tag {5.2.6} +$$ + +여기서, + +$$ +\begin{array}{l} A \quad : \text { 단면적 } \\ L _ {e} \quad : \text { 요소 길이 } \\ \end{array} +$$ + +식 (5.2.6)을 정리하여 트러스요소의 요소강성행렬을 다음과 같이 계산할 수 있다. + +$$ +\mathbf {K} _ {G} = \frac {N _ {x}}{L _ {e}} \left[ \begin{array}{c c c c} 1 & 0 & - 1 & 0 \\ 0 & 1 & 0 & - 1 \\ - 1 & 0 & 1 & 0 \\ 0 & - 1 & 0 & 1 \end{array} \right] \tag {5.2.7} +$$ + +여기서, + +Nx : 축방향 내력 + + + +# 5-2-2 보요소 + +선형좌굴해석에서 보요소의 기하강성 중 축방향 힘 Nx 를 고려한 부분은 트러스요소와 유사한 과정으로 계산할 수 있다. 보요소에서는 y , z 방향 이동변위를 다음과 같이 회전을 고려한 절점 자유도로 표현한다. + +$$ +\mathbf {u} _ {i} = \left\{v _ {i} \quad w _ {i} \quad \theta_ {y i} \quad \theta_ {z i} \right\} ^ {T} \tag {5.2.8} +$$ + +$$ +v = \sum_ {i = 1} ^ {2} (H _ {0 i} v _ {i} + H _ {1 i} \theta_ {z i}) , \quad w = \sum_ {i = 1} ^ {2} (H _ {0 i} w _ {i} - H _ {1 i} \theta_ {y i}) \tag {5.2.9} +$$ + +$H _ { 0 i }$ 과 $H _ { 1 i }$ 는 다음과 같은 Hermite 3차 형상함수이다. + +$$ +H _ {0 1} = 1 - 3 \xi^ {2} + 2 \xi^ {3}, \quad H _ {0 2} = 3 \xi^ {2} - 2 \xi^ {3}, \quad H _ {1 1} = L _ {e} (\xi - 2 \xi^ {2} + \xi^ {3}), +$$ + +$$ +H _ {0 2} = L _ {e} (- x ^ {2} + x ^ {3}) +$$ + +여기서, + +$$ +\xi \quad : 0 \leq \xi \leq 1 +$$ + +$$ +L _ {e} \quad : \text { 요소 길이 } +$$ + +따라서 절점변위-변위도함수의 관계 $\mathbf { G } _ { i }$ 는 다음과 같다. + +$$ +\mathbf {G} _ {i} = \left[ \begin{array}{c c c c} \frac {\partial H _ {0 i}}{\partial x} & 0 & 0 & \frac {\partial H _ {1 i}}{\partial x} \\ 0 & \frac {\partial H _ {0 i}}{\partial x} & - \frac {\partial H _ {1 i}}{\partial x} & 0 \end{array} \right] \tag {5.2.10} +$$ + +행렬 S 는 트러스요소와 같으므로 보요소의 기하강성 행렬을 다음과 같이 계산할수 있다. + + + +$$ +\mathbf {K} _ {G} = N _ {x} \left[ \begin{array}{c c c c c c c c} \frac {6}{5 L _ {e}} & & & & & & \\ 0 & \frac {6}{5 L _ {e}} & & & & \text {symm.} \\ 0 & - \frac {1}{1 0} & \frac {2 L _ {e}}{1 5} & & & \\ \frac {1}{1 0} & 0 & 0 & \frac {2 L _ {e}}{1 5} & & \\ - \frac {6}{1 5 L _ {e}} & 0 & 0 & - \frac {1}{1 0} & \frac {6}{5 L _ {e}} & & \\ 0 & - \frac {6}{1 5 L _ {e}} & \frac {1}{1 0} & 0 & 0 & \frac {6}{5 L _ {e}} & & \\ 0 & - \frac {1}{1 0} & - \frac {L _ {e}}{3 0} & 0 & 0 & \frac {1}{1 0} & \frac {2 L _ {e}}{1 5} & \\ \frac {1}{1 0} & 0 & 0 & - \frac {L _ {e}}{3 0} & - \frac {1}{1 0} & 0 & 0 & \frac {2 L _ {e}}{1 5} \end{array} \right] +$$ + +(5.2.11) + +보요소의 기하강성에는 축방향 내력 이외에 휨모멘트, 전단력, 비틀림모멘트 등에의해 발생하는 부분이 있다. midas FEA에서는 이와 같은 형태의 하중에 의한 다양한 좌굴(lateral-torsional, axial-torsional) 형태를 고려한 해석을 수행할 수 있다. + +# 5-2-3 평면응력요소 + +평면응력요소의 기하강성 계산에서는 요소좌표계에서 x, , y z 방향 이동변위를 모두고려한다. + +$$ +\mathbf {u} _ {i} = \left\{u _ {i} \quad v _ {i} \quad w _ {i} \right\} ^ {T} \tag {5.2.12} +$$ + +$$ +u = \sum_ {i = 1} ^ {n} N _ {i} u _ {i}, v = \sum_ {i = 1} ^ {n} N _ {i} v _ {i}, w = \sum_ {i = 1} ^ {n} N _ {i} w _ {i} \tag {5.2.13} +$$ + + + +여기서, + +n : 절점 개수 + +$N _ { i }$ : 절점 개수에 따른 형상함수 + +S 의 구성에 있어서는 면내방향 응력을 고려한다. + +$$ +\mathbf {S} = \left[ \begin{array}{l l l} \overline {{{\mathbf {S}}}} & \mathbf {0} & \mathbf {0} \\ \mathbf {0} & \overline {{{\mathbf {S}}}} & \mathbf {0} \\ \mathbf {0} & \mathbf {0} & \overline {{{\mathbf {S}}}} \end{array} \right], \quad \overline {{{\mathbf {S}}}} = \left[ \begin{array}{l l} \sigma_ {x x} & \tau_ {x y} \\ \tau_ {x y} & \sigma_ {y y} \end{array} \right] \tag {5.2.14} +$$ + +절점변위-변위도함수의 관계 Gi 는 식 (5.2.15)과 같다. + +$$ +\mathbf {G} _ {i} = \left[ \begin{array}{c c c c c c} \frac {\partial N _ {i}}{\partial x} & \frac {\partial N _ {i}}{\partial y} & 0 & 0 & 0 & 0 \\ 0 & 0 & \frac {\partial N _ {i}}{\partial x} & \frac {\partial N _ {i}}{\partial y} & 0 & 0 \\ 0 & 0 & 0 & 0 & \frac {\partial N _ {i}}{\partial x} & \frac {\partial N _ {i}}{\partial y} \end{array} \right] ^ {T} \tag {5.2.15} +$$ + +행렬 $\mathbf { G } _ { i }$ 를 이용하여 평면응력요소의 기하강성을 표현하면 다음과 같다. + +$$ +\mathbf {K} _ {G i j} = t \int_ {A _ {o}} \mathbf {G} _ {i} ^ {T} \mathbf {S} \mathbf {G} _ {j} d A \tag {5.2.16} +$$ + +여기서, + +t : 두께 + +Ae : 요소 면적 + +# 5-2-4 판요소 + +판요소에서는 요소좌표계의 x, , y z 방향 이동변위를 다음과 같이 회전을 고려한 절점 자유도로 표현한다. + +$$ +\mathbf {u} _ {i} = \left\{u _ {i} \quad v _ {i} \quad w _ {i} \quad \theta_ {x i} \quad \theta_ {y i} \right\} ^ {T} \tag {5.2.17} +$$ + +$$ +u = \sum_ {i = 1} ^ {n} (N _ {i} u _ {i} + z N _ {i} \theta_ {y i}), v = \sum_ {i = 1} ^ {n} (N _ {i} v _ {i} - z N _ {i} \theta_ {x i}), w = \sum_ {i = 1} ^ {n} N _ {i} w _ {i} \tag {5.2.18} +$$ + + + +여기서, + +n : 절점 개수 + +$N _ { i }$ : 절점 개수에 따른 형상함수 + +식 (5.2.18)는 요소의 곡률을 고려하고 있지 않으므로 저차 판요소에만 적용할 수있다. + +S 의 구성에 있어서는 $\sigma _ { z z } \frac { \equiv } { \equiv }$ 제외한 모든 성분의 응력을 고려한다. + +$$ +\mathbf {S} = \left[ \begin{array}{l l l} \overline {{{\mathbf {S}}}} & \mathbf {0} & \mathbf {0} \\ \mathbf {0} & \overline {{{\mathbf {S}}}} & \mathbf {0} \\ \mathbf {0} & \mathbf {0} & \overline {{{\mathbf {S}}}} \end{array} \right], \quad \overline {{{\mathbf {S}}}} = \left[ \begin{array}{l l l} \sigma_ {x x} & \tau_ {x y} & \tau_ {z x} \\ \tau_ {x y} & \sigma_ {y y} & \tau_ {y z} \\ \tau_ {z x} & \tau_ {y z} & 0 \end{array} \right] \tag {5.2.19} +$$ + +절점변위-변위도함수의 관계 $\mathbf { G } _ { i } \equiv$ 식 (5.2.20)과 같다. + +$$ +\mathbf {G} _ {i} = \left[ \begin{array}{c c c c c c c c c} \frac {\partial N _ {i}}{\partial x} & \frac {\partial N _ {i}}{\partial y} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac {\partial N _ {i}}{\partial x} & \frac {\partial N _ {i}}{\partial y} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & \frac {\partial N _ {i}}{\partial x} & \frac {\partial N _ {i}}{\partial y} & 0 \\ 0 & 0 & 0 & - z \frac {\partial N _ {i}}{\partial x} & - z \frac {\partial N _ {i}}{\partial y} & - N _ {i} & 0 & 0 & 0 \\ z \frac {\partial N _ {i}}{\partial x} & z \frac {\partial N _ {i}}{\partial y} & N _ {i} & 0 & 0 & 0 & 0 & 0 & 0 \end{array} \right] ^ {T} \tag {5.2.20} +$$ + +행렬 Gi 를 이용하여 판요소의 기하강성을 표현하면 다음과 같다. + +$$ +\mathbf {K} _ {G i j} = \int_ {V _ {e}} \mathbf {G} _ {i} ^ {T} \mathbf {S} \mathbf {G} _ {j} d V \tag {5.2.21} +$$ + +고차 판요소의 경우에는 요소의 곡률을 고려하며, 절점마다 회전 자유도에 대한좌표계가 정의된다는 점 이외에는 유사한 과정으로 기하강성을 계산할 수 있다. + + + +# 5-2-5 입체요소 + +입체요소의 기하강성 계산시에는 요소좌표계에서 x, , y z 방향 이동변위를 모두 고려한다. + +$$ +\mathbf {u} _ {i} = \left\{u _ {i} \quad v _ {i} \quad w _ {i} \right\} ^ {T} \tag {5.2.22} +$$ + +$$ +u = \sum_ {i = 1} ^ {n} N _ {i} u _ {i}, v = \sum_ {i = 1} ^ {n} N _ {i} v _ {i}, w = \sum_ {i = 1} ^ {n} N _ {i} w _ {i} \tag {5.2.23} +$$ + +여기서, + +n : 절점 개수 + +N i : 절점 개수에 따른 형상함수 + +S 의 구성에 있어서는 모든 성분의 응력을 고려한다. + +$$ +\mathbf {S} = \left[ \begin{array}{l l l} \overline {{{\mathbf {S}}}} & \mathbf {0} & \mathbf {0} \\ \mathbf {0} & \overline {{{\mathbf {S}}}} & \mathbf {0} \\ \mathbf {0} & \mathbf {0} & \overline {{{\mathbf {S}}}} \end{array} \right], \quad \overline {{{\mathbf {S}}}} = \left[ \begin{array}{l l l} \sigma_ {x x} & \tau_ {x y} & \tau_ {z x} \\ \tau_ {x y} & \sigma_ {y y} & \tau_ {y z} \\ \tau_ {z x} & \tau_ {y z} & \sigma_ {z z} \end{array} \right] \tag {5.2.24} +$$ + +절점변위-변위도 함수의 관계 Gi 는 식 (5.2.25)와 같다. + +$$ +\mathbf {G} _ {i} = \left[ \begin{array}{c c c c c c c c c} \frac {\partial N _ {i}}{\partial x} & \frac {\partial N _ {i}}{\partial y} & \frac {\partial N _ {i}}{\partial z} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac {\partial N _ {i}}{\partial x} & \frac {\partial N _ {i}}{\partial y} & \frac {\partial N _ {i}}{\partial z} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & \frac {\partial N _ {i}}{\partial x} & \frac {\partial N _ {i}}{\partial y} & \frac {\partial N _ {i}}{\partial z} \end{array} \right] ^ {T} \tag {5.2.25} +$$ + +행렬 Gi 를 이용하여 입체요소의 기하강성을 표현하면 다음과 같다. + +$$ +\mathbf {K} _ {G i j} = \int_ {V _ {e}} \mathbf {G} _ {i} ^ {T} \mathbf {S} \mathbf {G} _ {j} d V \tag {5.2.26} +$$ + + + +# 5-3 임계하중계수 추출방법 + +선형좌굴해석에서의 고유치 문제인 식 (5.1.6)은 다음과 같은 형태로 간략하게 표현할 수 있다. + +$$ +(\mathbf {K} + \lambda_ {m} \mathbf {K} _ {G}) \phi_ {m} = \mathbf {0} \tag {5.3.1} +$$ + +여기서, + +$\lambda _ { m }$ ： + +$\phi _ { m }$ + +midas FEA에서는 식 (5.3.1)과 같은 고유치 문제를 해석할 때 Lanczos 반복(iteration)법을 이용한다. 그리고 Lanczos 반복법은 고유치 해석에서 설명하고 있다. + +식 (5.3.1)은 자유진동해석의 고유치 문제와 유사하지만, 질량 행렬과는 다르게 기하강성 ${ \bf K } _ { G } \equiv \mathbf { \Delta } _ { \mathbf { O } } ^ { \mathrm { o p s o l } }$ 정부호(positive definite)가 아니다. 따라서 선형좌굴해석에서는 $\lambda _ { m } = \sigma \theta _ { m } / ( 1 - \theta _ { m } ) \underline { { \circ } } \underline { { \Xi } }$ 치환하여 Shift-invert 기법을 적용한 Lanczos 반복법을이용한다. 계산된 임계하중계수들은 절대값이 작은 것부터 순차적으로 출력된다. + + + +# 5-4 관련 기능 + +선형좌굴해석에서 임계하중계수 α 는 기준하중 p 의 방향에 따라 “+” 또는 “-“ 부호를 갖는다. 유한요소 모델과 하중이 복잡한 경우에는 임계하중계수의 부호가좌굴모드에 따라 바뀌는 경우도 발생한다. 따라서 midas FEA에서는 필요에 따라“+” 부호를 갖는 임계하중만을 계산할 수 있도록 하는 기능을 제공하고 있다. + +자중(self-weight)과 같은 특정 하중은 일반적으로 기준하중 p 와는 다르게 고정된 크기를 유지하지만, 응력을 발생시키기 때문에 기하강성을 유발한다. midasFEA에서는 하중의 종류를 선택하여 기준하중 또는 고정하중(constant load)으로사용할 수 있는 기능을 제공한다. 고정하중으로 선택된 하중은 하중계수 α 와 무관하게 일정한 값을 가지며, 이에 따른 기하강성만을 발생시킨다. 고정하중이 포함된 선형좌굴해석은 다음과 같은 형태의 고유치 문제가 된다. + +$$ +(\mathbf {K} + \mathbf {K} _ {G} ^ {*} + \lambda_ {m} \mathbf {K} _ {G}) \phi_ {m} = \mathbf {0} \tag {5.4.1} +$$ + +여기서, + +\* KG : 고정하중으로부터 발생한 응력에 대한 기하강성 + +z 방향 회전자유도를 고려하지 않은 판요소는 요소좌표계 z 방향 회전에 대한 탄성강성과 기하강성이 모두 존재하지 않는다. 이러한 경우 Lanczos 반복계산 과정중 수치오류가 발생하여 의미 없는(trivial) 임계하중계수 α 가 발생할 수 있다.midas FEA에서는 인접한 판요소 간의 각도를 고려하여 z 방향 회전에 대한 강성이 발생하지 않으면 이를 구속하는 기능을 제공한다. + + + +# Analysis and Algorithm Manual + +# Part 5 Construction Stage Analysis + +Chapter 1. Construction Stage Analysis + + diff --git a/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_041.md b/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_041.md new file mode 100644 index 00000000..3da77607 --- /dev/null +++ b/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_041.md @@ -0,0 +1,208 @@ + + +# Chapter 1. Construction Stage Analysis + +# 1-1 개요 + +구조물은 여러 단계의 시공과정을 거치면서 완성된다. 시공단계에서는 구조물의형상, 하중, 경계조건의 변화뿐 만 아니라, 시간에 따른 구조부재의 물리적 성질변화까지도 발생한다. 구조물의 시공과정에서 필요로 하는 임시 구조물의 설치와제거, 구조부재의 단계적인 설치 등이 구조계의 변화를 유발한다. 그리고 콘크리트의 경우에는 시간에 따른 물성의 변화를 나타낸다. 이와 같이 시공의 진행에 따라지속적으로 구조계가 변화할 경우에 실제 구조물의 거동은 시공단계를 고려하지않은 경우와 다를 수 있다. 또한 부재설계에 사용되는 최대 부재력이 시공이 완료된 시점이 아니라 시공 중간에도 발생할 수 있다. + +구조물이 완성되기까지 다양한 시공과정을 거치는 구조물의 경우에는 시공단계를고려한 시간의존해석(time-dependent analysis)을 수행하여 완성된 후의 부재력뿐 아니라 시공과정에서 발생하는 부재력을 설계에 포함하여야 한다. 콘크리트의시간의존 특성에는 탄성계수의 변화, 크리이프(creep), 건조수축(shrinkage) 등이있다. 이러한 시공단계를 고려해야 하는 경우는 신규 구조물뿐 만 아니라 기존 구조물의 보수나 보강과정에서도 발생한다. + +midas FEA의 시공단계 해석에서 고려하는 구조계의 변화는 다음과 같다. + +\- 구조 부재의 생성 및 제거 + +\- 하중의 재하 및 제거 + +\- 경계조건의 변화 + +midas FEA의 시공단계 해석에서 고려하는 콘크리트의 시간의존 특성은 다음과같다. + +\- 시간에 따른 콘크리트 부재의 강도발현 + +\- 콘크리트 부재의 크리이프 변형 + + + +\- 콘크리트 부재의 건조수축 변형 + +midas FEA에서 콘크리트의 시간의존 특성을 고려한 시공단계 해석을 위한 절차는 다음과 같다. + +1. 구조물을 모델링한다. 이때 임의의 시공단계에서 함께 생성 또는 제거할 요소,하중 및 경계조건들을 그룹으로 지정한다. +2. 크리이프나 건조수축과 같은 시간의존적 재질 특성을 정의한다. 시간의존적 재질은 ACI209나 CEB-FIP 등과 같은 기준을 선택하여 생성하거나 사용자가 직접 정의할 수 있다. +3. 실제 시공순서에 따른 시공단계를 구성한다.요소그룹, 경계조건그룹, 하중그룹을 이용하여 시공단계를 정의한다. +4. 선택사항들을 지정하여 해석의 조건을 구성하고 구조해석을 수행한다. + + + +# 1-2 시공단계의 구성 + +시공단계 모델의 구성에 필요한 것은 기본모델(base model)의 생성과 세부 시공단계(construction stage)의 구성이다. + +# 1-2-1 기본모델의 생성 + +구재부재가 단계별로 생성과 제거 가능하도록 전체 구조물을 요소그룹, 경계조건그룹, 하중그룹으로 분리하여 모델링한다. + +# 1-2-2 시공단계의 구성 + +각각의 시공단계를 구성하는 부분으로 요소그룹, 경계조건그룹, 하중그룹의 추가와삭제를 통하여 사용자가 원하는 시공단계를 구성한다. 시공단계는 하위의 시간구간을 추가할 수 있고, 각 구간에는 하중 그룹의 추가와 삭제가 가능하다. 구조계의변화를 유발하는 요소그룹과 경계조건그룹의 추가나 삭제는 시공단계의 첫 번째시간구간에만 입력이 가능하다. 시간의존 특성을 반영하지 않아도 되는 구조물인경우에는 시공단계에 시간을 입력할 필요는 없다. 그러나 한 개의 시공단계에 여러 개의 하중그룹을 순차적으로 재하 할 경우에는 시간구간을 입력하여야 한다. + +각 시공단계는 요소그룹, 경계조건그룹, 하중그룹의 활성화(activation)와 비활성화(deactivation) 정의에 의하여 구성된다. 각 시공단계는 이전 시공단계의 구조물에추가나 제거되는 요소, 경계조건, 하중 그룹을 누적한 구조물로 구성된다. + +각각의 시공단계별로 반영할 수 있는 내용은 다음과 같다. + +- 임의의 재령을 가지는 부재의 생성 및 제거 +- 임의의 재하시점을 가지는 하중의 재하 및 제거 + + + +\- 경계조건의 변화 + +midas FEA에서 사용하는 시공단계 구성의 개념도는 그림 1.2.1과 같다. 시공단계는 각 단계별 기간(duration)만 가지고 쉽게 정의된다. 기간이 ‘0’인 시공단계도 가능하며, 시공단계가 정의되면 기본적으로 시간구간인 first step과 last step이 생성된다. 실질적인 요소, 경계조건 및 하중의 생성과 제거는 각각의 시간구간에서이루어진다. + +![](images/page-404_b97e9c670dcc1f56ea2d7312ac210cd7c38b11a2f4159c00c63074e97df18afb.jpg) + +
+flowchart + +```mermaid +graph TD + A["0days"] --> B["10days"] + B --> C["20days"] + C --> D["30days"] + D --> E["40days"] + F["Element, Load, Boundary Activate & Deactivate"] --> G["Element, Load, Boundary Activate & Deactivate"] + H["Additional Load Activate & Deactivate"] --> I["Additional Load Activate & Deactivate"] + J["Element, Load, Boundary Activate & Deactivate"] --> K["Element, Load, Boundary Activate & Deactivate"] + L["First Step"] --> M["Additional Step"] + N["Last Step"] --> O["First Step"] + P["Last Step"] --> Q["..."] + style A fill:#f9f,stroke:#333 + style B fill:#f9f,stroke:#333 + style C fill:#f9f,stroke:#333 + style D fill:#f9f,stroke:#333 + style E fill:#f9f,stroke:#333 + style F fill:#f9f,stroke:#333 + style G fill:#f9f,stroke:#333 + style H fill:#f9f,stroke:#333 + style I fill:#f9f,stroke:#333 + style J fill:#f9f,stroke:#333 + style K fill:#f9f,stroke:#333 +``` +
+ +그림 1.2.1 시공단계 구성의 개념 + +기본적으로 요소의 생성 및 제거, 경계조건의 변화, 하중의 재하 및 제거 등의 모든 변경사항은 매 시공단계의 first step에서 이루어진다. 따라서 실제 시공중에 여러가지 원인에 의하여 구조계의 변화가 발생하면, 구조계의 변화가 발생하는 시기를 반영하는 시공단계를 생성하여야 한다. 즉 구조계의 변화가 잦을수록 시공단계의 수는 많아지게 된다. + +요소 및 경계조건 등 구조계의 변화는 매 시공단계의 first step에서만 이루어진다.그러나 하중의 변화는 해석의 편의를 위하여 시공단계 내에 추가적인 단계를 만들 + + + +어서 하중을 재하 및 제거할 수 있다. 즉, 임의의 시공단계에서 지연시간을 가지는하중을 가할 수 있다. 이 기능을 사용하면 구조계의 변화 없이 가설재의 설치나 소거로 인한 하중의 변화를 새로운 시공단계를 만들지 않고 쉽게 고려할 수 있다.또한 크리이프와 건조수축을 고려한 시간의존해석에서 시공단계 내에 추가적인 시간구간을 많이 정의하면 보다 정확한 해석결과를 얻을 수 있다. 그러나 추가 시간구간을 너무 많이 정의하면 해석시간이 증가하여 비효율적일 수 있으므로 주의해야 한다. 특히 시공단계해석 조건에서 시간의존적 특성을 고려하지 않도록 설정하고 해석을 수행하면 추가 시간구간이 많더라도 해석결과에는 영향을 주지 않는다. + +임의의 시공단계에서 지정한 재령을 가진 요소는 생성된 후에 매 시공단계마다 지속기간 만큼의 재령을 추가로 얻게 된다. 콘크리트 부재의 재령에 따른 물리적 특성의 변화는 각 시공단계에서 누적된 재령을 기준으로 계산하여 고려한다. + +임의의 시공단계에서 요소를 생성하는 경우에는 생성되는 요소의 재령을 지정하여야 한다. 재령이 ‘0’인 요소를 생성한다는 것은 콘크리트의 타설 순간부터 모사를하는 것이다. 그러나 일반적으로 구조물을 해석할 때 거푸집 등의 가설구조물은모델에 포함시키지 않기 때문에 경화되지 않은 상태의 콘크리트를 해석한다는 것은 의도하지 않은 결과를 가져올 수 있다. 특히 재령이 ‘0’인 요소를 생성시키고시간에 따른 강도발현을 고려하여 해석을 하면, 콘크리트 타설 후 24시간까지는강도를 발현하지 못하므로 의미 없는 큰 변위가 계산될 수 있다. 일반적으로 경화되기 전의 콘크리트는 가설 구조물과 함께 하중으로 고려하고, 거푸집을 제거한후에 실질적인 요소가 생성된다고 가정하는 것이 적절하다. + +임의의 시공단계에서 생성되는 요소는 이전 시공단계에서 발생한 변위나 내부응력에 영향을 받지 않는다. 즉, 새롭게 생성되는 요소는 그 시공단계에서 구조물이 이전에 어떠한 하중을 받고 있는지에 관계없이 요소의 내부 응력이 ‘0’인 상태에서생성된다. + +임의의 시공단계에서 하중을 재하하면 이후의 시공단계에서는 재하된 하중을 소거하지 않는 한 계속해서 하중이 가해진 상태가 된다. 요소의 생성도 임의의 시공단 + + + +계에서 필요한 모든 요소를 생성하는 것이 아니라, 그 시공단계에 필요한 요소만생성한다. 요소는 한번 생성이 되면 다시 생성할 수 없으며, 이미 생성된 요소만을제거할 수 있다. + + + +# 1-3 요소의 생성 및 제거 + +요소의 추가 및 제거는 시공단계 구조물의 강성과 자중 및 경계조건에 영향을 준다. 여기서 자중 및 경계조건의 영향은 이어지는 장에서 설명하며, 이 장에서는 요소의 생성 및 제거에 대해서만 설명하도록 한다. + +현재 시공단계의 구조물 강성은 식 (1.3.1)과 같이 현재 시공단계에서 사용되는 요소들의 강성을 조합하여 계산한다. 전체 시공단계에 걸쳐 사용되는 모든 요소는데이타베이스로 저장되며, 현재 시공단계에 사용되는 요소정보는 전체 시공단계의요소 데이타베이스를 검색하여 초기단계부터 현재단계까지 활성화된 모든 요소를추가하고 비활성화된 모든 요소를 제거함으로써 정의한다. + +$$ +\mathbf {K} = \sum_ {i = 1} ^ {n} \mathbf {L} _ {i} ^ {T} \mathbf {k} _ {i} \mathbf {L} _ {i} \tag {1.3.1} +$$ + +여기서, + +K : 현재 시공단계의 구조물 강성 + +$\mathbf { L } _ { \boldsymbol { i } }$ : 구조물 강성 내 요소 강성의 위치를 나타내는 행렬 + +$\mathbf { k } _ { i }$ : 현재 시공단계에 활성화된 요소 강성 + +n : 현재 시공단계에 활성화된 요소 개수 + +요소 강성은 요소가 가지고 있는 모든 절점자유도 사이의 강성관계로 이산화되어계산된다. 계산된 요소강성은 식 (1.3.1)과 같이 동일한 절점자유도를 가지는 인접요소의 해당 강성에 누적함으로써 전체 구조물 강성으로 조합된다. 이때 인접 요소와의 절점자유도 관계를 나타내는 정보는 식 (1.3.1)의 행렬 $\mathbf { L } _ { i }$ 로 정의한다. 행렬 L 는 현재 시공단계에서 사용되는 절점의 절점자유도정보를 이용하여 요소 별로 정의되며, 매 시공단계마다 요소가 추가 및 제거됨으로써 새로이 갱신된다. + +현재 시공단계에서 새롭게 활성화된 요소는 내부적으로 초기응력이 0으로 초기화되며, 전 단계에서 이미 활성화된 요소의 초기응력은 전 단계에서 해석된 최종 응력값을 초기응력(initial stress)으로 사용한다. + + + +# 1-4 하중의 재하 및 제거와 적용 + +시공단계해석은 단계별 증분형태의 해석으로 구성되며, 이러한 증분해석을 위해현재 시공단계의 증분하중이 요구된다. 현재 시공단계에서 증분하중 current ∆F 은 현재 단계의 총 하중 current F 와 이전단계의 총 하중 previous F 의 차를 통해 계산되며,다음과 같다. + +$$ +\Delta \mathbf {F} ^ {\text { current }} = \mathbf {F} ^ {\text { current }} - \mathbf {F} ^ {\text { previous }} \tag {1.4.1} +$$ + +전체 시공단계에 걸쳐 사용되는 모든 하중은 데이타베이스로 저장되며, 현재 시공단계의 총 하중 current F 은 전체 시공단계의 하중 데이타베이스를 검색하여 초기단계부터 현재단계까지 활성화된 모든 하중를 추가하고 비활성화된 모든 하중를 제거함으로써 정의한다. + +현재 시공단계의 총 하중은 다음과 같이 표면력(surface force)와 체적력(bodyforce)으로 구분할 수 있다. + +$$ +\mathbf {F} ^ {\text { current }} = \mathbf {F} _ {\text { surface }} ^ {\text { current }} + \mathbf {F} _ {\text { body }} ^ {\text { current }} \tag {1.4.2} +$$ + +이때 표면력에는 절점하중이나 분포하중등과 같이 절점과 관련된 하중이 포함되며,체적력에는 자중과 같이 요소와 관련된 하중이 포함된다. 표면력의 경우는 하중이작용되는 절점의 절점자유도 정보가 추가로 입력되기 때문에 이를 사용하여 조합하지만 체적력의 경우는 요소의 추가 및 제거와 관련되기 때문에 현재 시공단계에서 계산된 식 (1.3.1)의 행렬 Li 를 사용하여 조합한다. + +하나의 시공단계 내에서 정의되는 부증분 하중단계(incremental load step)는 식(1.4.1)에서 계산된 시공단계 증분하중 current ∆F 에 미리 정의된 현재 시공단계 내에서의 하중계수(load factor) α 를 곱하여 계산하며, 이를 임의의 하중단계 i 에 대해 수식으로 표현하면 다음과 같다. + + + +$$ +\mathbf {F} _ {i} ^ {\text { current }} = \mathbf {F} ^ {\text { previous }} + \alpha_ {i} ^ {\text { current }} \cdot \Delta \mathbf {F} ^ {\text { current }} \tag {1.4.3} +$$ + +임의의 하중단계 i 에서 구조물의 응답을 계산하기 위한 정해석(static analysis)의선형대수방정식(linear algebraic equation)은 다음과 같다. + +$$ +\mathbf {F} _ {i} ^ {\mathrm{ubf}} = \mathbf {K} _ {i} \mathbf {d} _ {i} \tag {1.4.4} +$$ + +이때 $\mathbf { K } _ { i } \equiv \mathbf { \rho } _ { i }$ 번째 하중단계에서의 구조물 강성이며, di 는 i 번째 하중단계에서의절점변위를 나타낸다. $\mathbf { F } _ { i } ^ { \mathrm { u b f } }$ 은 i 번째 단계에서의 불평형력(unbalance force)이라하며, 현재 하중단계의 $\hat { \textmd s }$ 하중에서 이전단계 총 내력을 뻬줌으로써 다음과 같이정의한다. + +$$ +\mathbf {F} _ {i} ^ {\mathrm{ubf}} = \mathbf {F} _ {i} ^ {\text { current }} - \mathbf {F} _ {i - 1} ^ {\text { internal }} \tag {1.4.5} +$$ + +식 (1.4.5)에서 $\mathbf { F } _ { i } ^ { \mathrm { c u r r e n t } }$ 은 식 (1.4.3)에서 이미 정의되었으며, $\mathbf { F } _ { i - 1 } ^ { \mathrm { i n t e r n a l } }$ 은 이전 하중단계에 대한 구조물의 총 내력으로써 이전 하중단계의 요소 응력을 사용하여 다음과같이 계산된다. + +$$ +\mathbf {F} _ {i - 1} ^ {\text { internal }} = \sum_ {j = 1} ^ {n} \mathbf {L} _ {j} ^ {\mathrm{T}} \mathbf {f} _ {j} = \sum_ {j = 1} ^ {n} \mathbf {L} _ {j} ^ {T} \left(\int_ {V} \mathbf {B} _ {j} ^ {\mathrm{T}} \boldsymbol {\sigma} _ {j} \mathrm{d} V\right) \tag {1.4.6} +$$ + +여기서, + +$\mathbf { f } _ { j }$ : 요소의 부재력 + +$\mathbf { L } _ { j }$ : 총하중 내 요소 하중의 위치를 나타내는 행렬 + +$\mathbf { B } _ { j }$ : 변위-변형률 기울기행렬 + +${ \pmb { \sigma } } _ { j }$ : 요소의 응력 + +$V$ : 체적 + +n : 현재 시공단계에 활성화된 요소 개수 + +이때 i = 1인 경우 $\mathbf { F } _ { 0 } ^ { \mathrm { i n t e m a l } }$ 은 이전 시공단계의 최종 하중단계에 대한 총 내력을 나타낸다. 또한 요소의 응력은 시공단계 내에서 각 하중단계의 계산이 끝나는 시점에업데이트된다. + + + +# 1-5 경계조건의 추가 및 제거 + +현재 시공단계의 경계조건 정보는 요소 및 하중과 마찬가지로 모든 시공단계의 경계조건을 데이타베이스로 만들고 초기단계부터 현재단계까지 검색하여 활성화된경계조건을 추가하고 비활성화된 경계조건을 제거함으로써 정의한다. 이때 현재단계에 추가된 경계조건과 제거된 요소에 의해 어떤 요소에도 포함되지 않는 절점인 자유절점(free node)이 발생할 수 있으며, 이러한 자유절점은 활성화된 요소에포함된 절점정보를 사용하여 제거함으로써 해석에 불필요한 절점자유도를 제거하여 해석 속도를 향상시킨다. diff --git a/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_042.md b/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_042.md new file mode 100644 index 00000000..1f4993d3 --- /dev/null +++ b/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_042.md @@ -0,0 +1,328 @@ + + +# Analysis and Algorithm Manual + +# Part 6 Potential Flow Analysis + +Chapter 1. General Heat Transfer Analysis + +Chapter 2. Heat of Hydration Analysis + + + + + +# Chapter 1. General Heat Transfer Analysis + +# 1-1 개요 + +토목 및 건축구조물 중에서 고온에 노출 가능성이 있는 구조물에는 시간에 따른온도 변화 및 구배(gradient)가 심하게 발생한다. 열전달(heat transfer) 해석은 이러한 구조물의 시간에 따른 온도 분포를 계산함으로써 구조물의 열전달 특성을 파악하고 나아가 온도 변화에 기인한 구조적 반응(변형, 응력)을 알아내기 위한 것이다. 짧은 기간 안에 막대한 양의 콘크리트를 타설할 때 발생하는 수화열 또는 화재등으로 인한 구조물의 국소적 가열은 열전달 해석을 필요로 하는 대표적인 예이다. + +열전달의 메커니즘(mechanism)에는 전도(conduction), 대류(convection) 그리고복사(radiation) 현상이 있다. 분자나 전자의 진동이 연쇄반응에 의해 고온에서 저온구역으로 에너지를 전달하는 현상을 전도라 하며, 액체나 기체와 같이 매질의이동에 의해 에너지가 전달되는 것을 대류라 한다. 서로 떨어져 있는 물체는 그 사이에 매질이 존재하지 않아도 전자기파 형태의 에너지를 교환하며 이를 복사 현상이라 한다. + +midas FEA 에서는 고체 물질에서 가장 활발한 메커니즘인 전도에 기초하여 열전도 방정식을 유한요소법으로 해석하고, 대류 현상은 경계조건 또는 하중의 형태로고려한다. + + + +# 1-2 열전달 방정식 + +전도에 의해 지배되는 열전달 방정식은 에너지 보존 법칙에 근거한다. 예를 들어일정한 부피 V 와 이를 둘러 싸고 있는 면적 S 가 있을 때, 에너지 보존은 다음과같이 만족한다. + +$$ +\text { 열증가율 } (V) = \text { 열유량 } (S) + \text { 발열률 } (V) +$$ + +부피 V 에서의 열증가율은 비열(specific heat)과 밀도(density)에 의해 다음과 같이 계산할 수 있다. + +$$ +\text { rate of increase of heat in } V = \int_ {V} \rho c \frac {\partial T}{\partial t} d V \tag {1.2.1} +$$ + +여기서, + +$$ +c \quad : \text { 비열 } (J / k g \cdot {} ^ {\circ} C) +$$ + +$$ +\rho \quad : \text { 밀도 } (k g / m ^ {3}) +$$ + +전도에 의해 전달되는 열유량(heat flow rate)은 Fourier의 법칙에 의해 온도구배(temperature gradient)와 열전도율(thermal conductivity)를 이용하여 계산된다. + +$$ +\text { rate of heat conduction across } S = \int_ {S} k (\nabla T) \cdot \mathbf {n} d S \tag {1.2.2} +$$ + +여기서, + +$$ +k \quad : \text { 열전도율 } (J / m \cdot h r \cdot {} ^ {\circ} C) +$$ + +$$ +\nabla T \quad : \text { 온도구배 } (^ {\circ} C / m) +$$ + +단위 부피당 발열률을 Q 라 하고 열전도율이 공간상에서 일정하고 방향 별로 같다고 가정하면 식 (1.2.1)과 (1.2.2)에 의해 열전달 지배방정식을 다음과 같이 얻을수 있다. + + + +$$ +\rho c \frac {\partial T}{\partial t} = k \nabla^ {2} T + Q \tag {1.2.3} +$$ + +위 식을 유한요소법에 의한 공간이산화 하면 다음과 같이 시간에 대한 행렬 미분방정식이 되며 해석결과는 각 시간 별 절점 온도이다. + +$$ +\mathrm{CT} + \mathrm{KT} = \mathrm{R} \tag {1.2.4} +$$ + +여기서, + +C : 열용량(capacitance) 행렬 + +K : 열전도(conduction) 행렬 + +R : 열하중 벡터 + +열용량 행렬은 밀도와 비열에 의해 계산되는 요소별 행렬이며 midas FEA에서는분포행렬(consistent matrix) 형태를 사용하고 있다. 열전도 행렬은 일반적으로 열전도율에 의해 계산되는 요소별 행렬이며, 대류 경계조건에 의한 영향이 반영된다.열하중 벡터는 발열(heat source), 열속(heat flux), 열유량 하중에 의해 계산되며대류 경계조건에 의한 영향이 반영된다. + +midas FEA에서는 정상상태(steady state) 온도응답과 과도(transient)응답의 해를모두 계산할 수 있다. 정상상태의 해는 식 (1.2.4)에서 열용량 행렬의 효과를 무시하면 간단하게 계산할 수 있다. + +$$ +\mathrm{KT} = \mathrm{R} \tag {1.2.5} +$$ + +과도응답의 해는 다음과 같은 시간에 대한 수치적분을 통하여 계산한다. 시간 it 에서의 온도상태( Ti )와 시간 i+1t 에서의 온도상태( Ti+1 )가 식 (1.2.4)를 각각 만족한다고 가정하면 다음과 같다. + +$$ +\mathrm{CT} _ {i} + \mathrm{KT} _ {i} = \mathrm{R} _ {i} \tag {1.2.6} +$$ + +$$ +\mathrm{CT} _ {i + 1} + \mathrm{KT} _ {i + 1} = \mathrm{R} _ {i + 1} +$$ + +식 (1.2.6)을 α 와 1− α 로 가중평균하면 다음과 같다. + + + +$$ +\mathrm{C} \left(\alpha \dot {\mathrm{T}} _ {i + 1} + (1 - \alpha) \dot {\mathrm{T}} _ {i}\right) + \mathrm{K} \left(\alpha \mathrm{T} _ {i + 1} + (1 - \alpha) \mathrm{T} _ {i}\right) \tag {1.2.7} +$$ + +$$ += \alpha \mathrm{R} _ {i + 1} + (1 - \alpha) \mathrm{R} _ {i} +$$ + +식 (1.2.7)에 다음의 관계를 적용하면 Ti 와 Ti+1 로 이루어진 방정식을 얻을 수 있다. + +$$ +\alpha \dot {\mathrm{T}} _ {i + 1} + (1 - \alpha) \dot {\mathrm{T}} _ {i} = \frac {\mathrm{T} _ {i + 1} - \mathrm{T} _ {i}}{\Delta t _ {i + 1}} \tag {1.2.8} +$$ + +$$ +\left[ \mathrm{C} + \alpha \Delta t _ {i + 1} \mathrm{K} \right] \mathrm{T} _ {i + 1} = \tag {1.2.9} +$$ + +$$ +\left[ \mathrm{C} - (1 - \alpha) \Delta t _ {i + 1} \mathrm{K} \right] \mathrm{T} _ {i} + \Delta t _ {i + 1} \left[ \alpha R _ {i + 1} + (1 - \alpha) R _ {i} \right] +$$ + +또는 다음과 같이 등가의 행렬로 치환하여 간단하게 표현할 수 있다. + +$$ +\overline {{{\mathrm{K}}}} \mathrm{T} _ {i + 1} = \overline {{{R}}} +$$ + +$$ +\overline {{{\mathbf {K}}}} = \left[ \mathbf {C} + \alpha \Delta t _ {i + 1} \mathbf {K} \right] \mathrm{T} _ {i + 1} \tag {1.2.10} +$$ + +$$ +\overline {{{\mathrm{R}}}} = \left[ \mathrm{C} - (1 - \alpha) \Delta t _ {i + 1} \mathrm{K} \right] \mathrm{T} _ {i} + \Delta t _ {i + 1} \left[ \alpha \mathrm{R} _ {i + 1} + (1 - \alpha) \mathrm{R} _ {i} \right] +$$ + +식 (1.2.10)은 도입한 적분변수(α )에 따라서 아래와 같이 구분이 되고 적분변수에따라서 수렴조건이 달라진다. + +$$ +\alpha = 0. 0: \text { 전향 차분(forward difference) } (\Delta t \text { 의 크기에 따라 수렴 }) +$$ + +$$ +\alpha = 0. 5: \text {Crank - Nicolson (무조건 수렴)} +$$ + +$$ +\alpha = 2 / 3: \text { Galerkin 법 (무조건 수렴) } +$$ + +$$ +\alpha = 1. 0: \text { 후방 차분(backward difference) } (\text { 무조건 수렴 }) +$$ + +midas FEA 에서는 α = 0.0인 경우를 제외하고 모든α 값을 사용할 수 있다. α에 따른 1차 미분방정식의 비정상해석 결과의 특징은 다음과 같은 간단한 1차원확산방정식(1-dimensional diffusion equation)의 해를 비교함으로써 확인할 수있다. + +$$ +1 \text { 차원 확산방정식 }: \frac {\partial^ {2} \phi}{\partial x ^ {2}} = \frac {\partial \phi}{\partial t} +$$ + +$$ +\text { 경계조건 }: \frac {\partial \phi (0 , t)}{\partial x} = 0 (t < 0), - \frac {\partial \phi (0 , t)}{\partial x} = 1 (t > 0), \frac {\partial \phi (L , t)}{\partial x} = 0 +$$ + + + +초기조건 : φ( , 0) 0 x = + +다음의 그래프는 무조건 수렴이라는 특성을 갖는 세 가지의 알고리즘에 대해 두개의 시간증분에 대한 결과를 비교한 것이며, 그 특징은 다음과 같이 정리할 수 있다. + +\- Crank-Niconsol법 ( α = 0.5) + +시간증분이 클 경우, 시간이 지남에 따라 사라지기는 하지만 계산초기에 상대적으로 큰 오실레이션(oscillation) 현상을 보인다. 시간증분을 줄일 경우에는 결과의 정확도가 크게 개선된다. + +\- Galerkin법 ( α = 2/3) + +시간증분이 클 경우, 계산초기에 역시 오실레이션 현상을 보이기는 하지만Crank-Nicolson법에 비해 그 정도는 약하다. 시간증분과 무관하게 계산초기 결과의 정확도는 가장 좋다. + +\- 후방차분법 ( α = 1.0) + +오실레이션 현상을 보이지는 않지만, 상대적으로 결과를 과소평가한다. 시간증분을 줄일 경우에는 결과의 정확도가 개선되지만 상대적으로 미소하다. + +4가지 알고리즘 중에서 α =0.5인 Crank-Nicolson법이 무조건 수렴하고 시간증분에 대해 2차의 수렴도( 2 O t ( ) ∆ )를 제공한다는 특징으로 인해 일반적으로 가장 많이 사용된다. 하지만, 2차 요소를 사용하거나 파이프쿨링(pipe cooling)을 적용할경우에는 오실레이션 현상을 발생할 수 있으며, 이러한 오실레이션 현상을 방지하기 위하여 후방차분법을 사용하는 것이 좋다. + + + +![](images/page-418_b656882326d99b85b260452d2b24532b40179c2a28cbe40acd3297668ccce544.jpg) + +
+line + +| t | Analytical Solution | Crank-Nicolson (α=0.5) | Galerkin (α=2/3) | Backward Difference (α=1.0) | +| ---- | ------------------- | ---------------------- | ---------------- | --------------------------- | +| 0.1 | 0.38 | 0.43 | 0.38 | 0.32 | +| 0.2 | 0.48 | 0.48 | 0.48 | 0.45 | +| 0.3 | 0.60 | 0.65 | 0.62 | 0.58 | +| 0.4 | 0.70 | 0.70 | 0.70 | 0.68 | +| 0.5 | 0.80 | 0.82 | 0.80 | 0.78 | +| 0.6 | 0.88 | 0.88 | 0.88 | 0.85 | +| 0.7 | 0.95 | 0.95 | 0.95 | 0.92 | +| t | 1.00 | 1.00 | 1.00 | 1.00 | +
+ +(a) Δt=0.10의 경우 + +![](images/page-418_d0f61c50bcb990bdc05af3b7e606c55ba7e73fdd32a0151e5f91e9262c60a905.jpg) + +
+line + +| t | Analytical Solution | Crank-Nicolson (α=0.5) | Galerkin (α=2/3) | Backward Difference (α=1.0) | +| ---- | ------------------- | ---------------------- | ---------------- | --------------------------- | +| 0.1 | 0.35 | - | 0.35 | 0.35 | +| 0.2 | 0.50 | - | 0.50 | 0.50 | +| 0.3 | 0.60 | - | 0.60 | 0.60 | +| 0.4 | 0.70 | - | 0.70 | 0.70 | +| 0.5 | 0.80 | - | 0.80 | 0.80 | +| 0.6 | 0.90 | - | 0.90 | 0.90 | +| 0.7 | 0.95 | - | 0.95 | 0.95 | +| 1.0 | 1.00 | - | 1.00 | 1.00 | +
+ +(b) Δt=0.05의 경우 +그림 1.2.1. α값에 따른 이론치와의 해석결과 비교 + + + +# 1-3 요소 + +열전달 해석에 사용되는 요소는 1차원, 2차원 그리고 3차원 요소로 구분되며 특수한 효과를 얻기 위해 탄성연결(elastic link) 요소와 계면(interface) 요소를 사용할수 있다. + +# 1차원 요소 + +열전달 해석에 사용되는 1차원 요소는 트러스요소, 보요소이다. 열전달 해석에서두 요소는 동일한 성질을 가지게 되며 열용량행렬과 열전도행렬은 다음과 같다. + +$$ +\mathrm{C} _ {i j} = \int_ {L _ {e}} \rho c A N _ {i} N _ {j} d L \tag {1.3.1} +$$ + +$$ +\mathrm{K} _ {i j} = \int_ {L _ {e}} k A \frac {\partial N _ {i}}{\partial x} \frac {\partial N _ {j}}{\partial x} d L \tag {1.3.2} +$$ + +여기서, + +A : 요소의 단면적 + +# 2차원 요소 + +열전달 해석에 사용되는 2차원 요소는 평면응력요소, 평면변형요소, 축대칭요소 그리고 판요소이다. 열전달 해석에서 3가지 요소는 모두 동일한 성질을 가지게 되며열용량행렬과 열전도행렬은 다음과 같다. + +$$ +\mathrm{C} _ {i j} = \int_ {A _ {e}} \rho c t N _ {i} N _ {j} d A \tag {1.3.3} +$$ + +$$ +\mathrm{K} _ {i j} = \int_ {A _ {e}} k t \left(\frac {\partial N _ {i}}{\partial x} \frac {\partial N _ {j}}{\partial x} + \frac {\partial N _ {i}}{\partial y} \frac {\partial N _ {j}}{\partial y}\right) d L \tag {1.3.4} +$$ + +여기서, + +t : 요소의 두께 + +축대칭 요소의 경우 두께 대신 반지름을 이용한다. + + + +# 3차원 요소 + +열전달 해석에 사용되는 3차원 요소는 입체요소이다. 열전달 해석에서 가장 많이사용되는 요소이며 열용량행렬과 열전도행렬은 다음과 같다. + +$$ +C _ {i j} = \int_ {V _ {e}} \rho c N _ {i} N _ {j} d V \tag {1.3.5} +$$ + +$$ +\mathrm{K} _ {i j} = \int_ {V _ {e}} k \left(\frac {\partial N _ {i}}{\partial x} \frac {\partial N _ {j}}{\partial x} + \frac {\partial N _ {i}}{\partial y} \frac {\partial N _ {j}}{\partial y} + \frac {\partial N _ {i}}{\partial z} \frac {\partial N _ {j}}{\partial z}\right) d V \tag {1.3.6} +$$ + +여기서, + +$$ +V _ {e} \quad : \text { 요소의 체적 } +$$ + +# 기타 요소 + +열전달 해석에서 추가적으로 사용할 수 있는 요소는 탄성연결요소와 계면요소이다.탄성연결요소는 1차원 요소와 유사하지만, 열전도도(thermal conductance)를 입력해야 하고 열용량이 없다는 점이 다르다. 탄성연결요소의 열전도행렬은 다음과같다. + +$$ +\mathrm{K} _ {i i} = k _ {A} +$$ + +$$ +\mathrm{K} _ {i j} = - k _ {A} (i \neq j) \tag {1.3.7} +$$ + +여기서, + +$$ +k _ {e} \quad : \text { 열전도도 } (J / h r \cdot {} ^ {\circ} C) +$$ + +계면요소 역시 열용량이 없으며 대류계수(convection coefficient)를 입력 받아 열전도행렬에 반영한다. 계면요소의 열전도행렬은 다음과 같다. + +$$ +\mathrm{K} _ {i j} = \int_ {A _ {e}} h N _ {i} N _ {j} d A \quad (i, j \text { 가 같은 면의 절점일 때 }) +$$ + +$$ +\mathrm{K} _ {i j} = - \int_ {A _ {e}} h N _ {i} N _ {j} d A \quad (i, j \text { 가 다른 면의 절점일 때 }) \tag {1.3.8} +$$ + +여기서, + +$$ +h \quad : \text { 대류계수 } (J / m ^ {2} \cdot h r \cdot {} ^ {\circ} C) +$$ diff --git a/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_043.md b/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_043.md new file mode 100644 index 00000000..eb660b72 --- /dev/null +++ b/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_043.md @@ -0,0 +1,214 @@ + + +탄성연결요소와 계면요소는 고정온도 조건과 함께 대류경계조건을 모사할 수 있으며, 구조물의 특정 부분에 대한 열흐름을 제어하는데 사용한다. + + + +# 1-4 하중과 경계조건 + +열전달 해석에서 하중과 경계조건은 열하중벡터에 반영되는 조건, 열하중벡터와열전도행렬에 반영되는 조건 그리고 절점온도에 직접 반영되는 조건으로 구분할수 있다. 하중과 경계조건이 전혀 반영되지 않은 부분은 단열조건이 된다. midasFEA 에서는 하중과 경계조건이 시간에 따라 변하는 효과를 고려할 수 있다. 절점에 입력되는 하중/경계조건의 경우 시간축의 정의는 해석의 시작 시점부터의 시간을 의미하며 요소에 부가되는 하중/경계조건의 경우에는 요소의 생성 시점부터의시간을 시간축으로 정의한다. 그림 1.4.1는 midas FEA에서 사용되는 열하중과 경계조건의 종류를 나타내고 있다. + +![](images/page-422_f14bf3ba692844d67c2225c8610b47141c19297fabfd9fdd39d376dcdf482fb9.jpg) + +
+text_image + +Convection +Cooling +Flux +Heat generated internally +Insulated +Temperature prescribed +
+ +그림 1.4.1. 고체 내/외부에서 발생할 수 있는 열하중과 경계조건 + + + +# 초기온도 + +구조물이 생성될 당시의 온도로 해석의 초기조건이 된다. 단계별 초기온도와 절점별 초기온도를 사용할 수 있다. 절점별 초기온도가 설정되지 않은 경우 단계별 초기온도가 반영된다. + +# 고정온도 + +고정온도(prescribed temperature)는 열전달해석의 경계조건을 구성하게 되며, 해당 절점은 항상 주어진 온도를 유지하게 된다. 시간에 따라 변하는 고정온도의 경우 시간축의 정의는 해석의 시작 시점부터의 시간이다. + +# 열유량과 열속 + +열유량하중은 절점에 직접 입력되는 하중으로 식 (1.2.4)의 R 에 해당하는 값이다.시간에 따라 변하는 열유량하중인 경우 시간축의 정의는 해석의 시작 시점부터의시간이다. + +열속하중은 요소의 특정 면을 통해 열의 유입이 이루어짐을 의미한다. 일반적으로 열속하중은 다음과 같이 R 를 통해 반영된다. + +$$ +\mathrm{R} _ {i} = \int_ {S} F _ {s} N _ {i} d S \tag {1.4.1} +$$ + +여기서, + +$$ +F _ {s} \quad : \text { 열속 } (J / m ^ {2} \cdot h r) +$$ + +시간에 따라 변하는 열속하중인 경우 시간축의 정의는 해당 요소의 생성 시점부터의 시간이다. + +# 대류 + +액체 또는 기체에서 더운 부분과 차가운 부분이 순차적으로 자리바꿈하여 이루어지는 열전달 현상을 대류라 한다. 유체를 표면위로 강제로 흐르게 하는 경우처럼유체유동을 인위적으로 일으키는 경우를 강제대류(forced convection)라 하고, 유체의 유동이 유체내의 온도차에 의해 생기는 밀도차에 의한 부력효과 때문에 일어 + + + +나는 열전달을 자유대류(free convection) 라고 한다. midas FEA에서는 자유대류를 경계조건으로 고려하고 있으며 파이프쿨링(pipe cooling) 형태의 강제대류를하중으로 사용할 수 있다. + +자유대류는 외기온도(ambient temperature) T 와 대류계수 h 를 통하여 정의하며, 고체 표면에서 외기로 빠져나가는 열유량은 다음과 같다. + +$$ +\mathrm{R} _ {i} = \int_ {s} h N _ {i} (T - T _ {\infty}) d S \tag {1.4.2} +$$ + +여기서, + +$$ +T \quad : \text { 고체 표면온도 } \left(\sum_ {j} T _ {j} N _ {j}\right) +$$ + +위 식에서 첫 번째 항은 열전도행렬에 포함되며 외기온도에 관련된 나머지 항이열하중벡터에 포함된다. 시간에 따라 변하는 대류계수 및 외기온도인 경우 시간축의 정의는 요소의 생성시점부터의 시간이다. + +강제대류의 일종인 파이프쿨링은 파이프 속으로 흐르는 유체의 대류와 파이프 표면과 유체 사이의 열전달을 모사하는 것이다. 유체와 파이프 사이의 열전달량은다음과 같이 두 가지 방법으로 계산할 수 있다. + +$$ +q _ {c o n v} = h _ {p} A _ {s} (T _ {s} - T _ {m}) = h _ {P} A _ {s} \left(\frac {T _ {s , i} + T _ {s , o}}{2} - \frac {T _ {m , i} + T _ {m , o}}{2}\right) \tag {1.4.3} +$$ + +$$ +q _ {c o n v} = \dot {m} c _ {m} (T _ {m, o} - T _ {m, i}) +$$ + +여기서, + +$$ +h _ {p} \quad : \text { 파이프의 유수대류계수 } +$$ + +$$ +A _ {s} \quad : \text { 파이프의 표면적 } +$$ + +$$ +T _ {s}, T _ {m} \quad : \text { 파이프 표면과 냉각수의 온도 } +$$ + +$$ +T _ {s, i}, T _ {s, o} \quad : \text {파이프의 유입구와 출구에서의 온도} +$$ + +$$ +T _ {m i} T _ {m o} \quad : \text { 냉각수의 유입구와 출구에서의 온도 } +$$ + +$$ +\dot {m} \quad : \text { 단위 시간당 냉각수 유입량 } (k g / h r) +$$ + +$$ +c _ {m} \quad : \text { 냉각수 비열 } +$$ + + + +식 (1.4.3)에 의해 냉각수의 온도를 계산할 수 있고, 냉각수의 온도를 이용하여 파이프 표면과 유체 사이의 열전달량을 알 수 있다. 파이프쿨링 역시 열전도행렬과열하중벡터에 반영된다. + +# 발열 + +발열(heat source)은 고체 내부에서 발생하는 열량을 모사하기 위한 것이다. 발열은 다음과 같이 하중벡터에 포함된다. + +$$ +\mathrm{R} _ {i} = \int_ {V} Q N _ {i} d V \tag {1.4.4} +$$ + +여기서, + +$Q : = \frac { \Xi \mid Q \mid } { \Xi \mid } \Xi \mid _ { \Xi } ^ { \Xi \mid } \Xi \stackrel { \sqcup \mid } { = } \Xi \stackrel { \cong } { = } \big ( J / m ^ { 3 } \cdot h r \big )$ + +발열에 의한 하중은 단위부피당 발열율을 직접 입력하지 않고 단열온도 상승식을통하여 간접적으로 입력할 수 있다. 단열온도 상승식은 다음과 같이 발열율로 환산된다. + +$$ +Q = \rho c \frac {\partial T}{\partial t} \tag {1.4.5} +$$ + +시간에 따라 변하는 발열인 경우 시간축의 정의는 해석의 요소의 생성시점부터의시간이다. + + + +# 1-5 단계별 열전달 해석과 결과 + +midas FEA에서 열전달 해석은 기본적으로 시공단계(construction stage)에 따라수행하게 된다. 단계에 따라 열전달 요소와 하중 경계조건이 변경될 수 있고, 단계별로 정상상태 해석과 과도응답 해석을 선택할 수 있다. 각 단계는 여러 개의 시간단계로 구성되고, 요소와 경계조건 및 하중은 추가와 제거가 모두 가능하다. 단계의 정의는 시공단계 해석과 동일한 정의를 사용한다. 또한, 열전달 해석을 통해 계산된 온도 이력(temperature history)은 시공단계 해석에서 온도하중으로 반영할수 있다. + +열전달 해석 결과는 절점별 온도와 요소별 온도구배/열속/열유량 등이며 모든 결과들은 시간단계별로 확인할 수 있다. 다음은 각 요소별 출력 결과이다. + +# 1차원 요소 + +온도구배 $\mathrm { ~ : ~ } \frac { \partial T } { \partial x }$ + +열속 $\mathbf { \Psi } : - k \frac { \partial T } { \partial x }$ + +# 2차원 요소 + +온도구배 $\mathrm { ~ : ~ } \frac { \partial T } { \partial x } , \frac { \partial T } { \partial y }$ + +열속 $\mathbf { \epsilon } : \mathbf { \epsilon } - k \frac { \hat { \sigma } T } { \hat { \sigma } x } , - k \frac { \hat { \sigma } T } { \hat { \sigma } y }$ + +평면변형요소와 축대칭 요소는 전체좌표계(global coordinate system)에 대하여 + + + +결과가 표현되며, 판요소와 평면응력요소는 요소좌표계에 대한 결과가 계산된다. + +3차원 요소 + +온도구배 $: ~ \frac { \partial T } { \partial x } , \frac { \partial T } { \partial y } , \frac { \partial T } { \partial z }$ + +열속 $\mathbf { \partial } : - k \frac { \hat { \sigma } T } { \hat { \sigma } x } , - k \frac { \hat { \sigma } T } { \hat { \sigma } y } , - k \frac { \hat { \sigma } T } { \hat { \sigma } z }$ + +입체요소의 결과는 전체좌표계에 대해 계산된다. + +기타 요소 + +$\begin{array} { r l } { \underline { { \mathsf { E } } } \sharp \underline { { \mathsf { M } } } \supseteq \underline { { \mathsf { Z } } } \underline { { \mathsf { d } } } \underline { { \mathsf { Q } } } \qquad \therefore \ { i } \ \underline { { \mathsf { E } } } \mathsf { I } \mathsf { M } \mathsf { A } \quad j \ \underline { { \mathsf { E } } } \mathsf { I } \underline { { \mathsf { O } } } \Xi \equiv \underline { { \mathsf { E } } } \equiv \underline { { \mathsf { S } } } \underline { { \mathsf { G } } } \underline { { \mathsf { P } } } \bar { \mathsf { E } } \quad k _ { A } ( T _ { i } - T _ { j } ) } \end{array}$ + +계면요소 : i 면에서 j 면으로 흐르는 열속 ( ) i j h T T− + + + +Part 6 Potential Flow Analysis + + + +# Chapter 2. Heat of Hydration Analysis + +# 2-1 개요 + +매스콘크리트의 수화열(heat of hydration)에 의한 온도 응력은 구조물에 균열을발생시켜 구조물의 내구성뿐만 아니라 구조적인 안전을 저해할 수 있다. 이러한문제를 해결하기 위해 매스콘크리트의 타설시 온도와 응력의 분포를 계산하여 균열을 적절히 제어하고자 하는 목적으로 수화열해석을 수행한다. 수화열해석을 수행하여야 할 매스콘크리트 구조물의 치수는 구조형식, 사용재료, 시공조건에 따라다르지만 대략 슬래브는 800∼1000mm 이상, 하단이 구속되어 있는 벽체는 두께500mm 이상을 기준으로 한다. + +수화열에 의한 균열은 초기에 표면부와 중심부의 온도차이에 의해 발생하는 표면균열과 시멘트 수화열에 의해 상승된 온도가 하강할 때 구조물의 수축이 외적으로구속되어 발생하는 관통균열로 구분할 수 있다. 이러한 수화열 해석은 크게 시멘트의 수화과정에서 발생하는 발열, 대류, 전도 등에 의한 열전달(heat transfer) 해석과 발생한 온도, 재령에 의한 탄성계수의 변화, 크리프(creep) 및 건조수축(shrinkage) 등에 의한 열응력(thermal stress) 해석으로 구분할 수 있다. + + + +# 2-2 열전달 해석 + +열전달 해석에서는 시멘트의 수화과정에서 발생하는 발열, 전도, 대류 등에 의한시간에 따른 절점온도 변화를 계산한다. 일반적으로 포화된 콘크리트의 열전도율는 1.21\~3.11 정도이며, 열전도율의 단위는 /( ) o kcal m h C ⋅ ⋅ 이다. 콘크리트의 열전도율은 온도가 증가하면서 감소하는 경향을 보이지만, 대기온도의 범위에서는 큰영향을 보이지 않는다. 수화열 해석에서 열전달 작용을 하는 요소와 사용 가능한하중/경계조건은 다음과 같다. + +# 요소 + +수화열 해석에서는 입체요소(solid element)와 트러스요소(truss element)만이 열전달 작용을 하게 된다. 열응력 해석에서는 철근요소(reinforcement element)도포함된다. + +# 초기온도 + +콘크리트 타설시의 온도로 물, 시멘트, 골재의 평균온도이며 해석의 초기조건이 된다. 시공단계별로 다른 초기온도를 입력할 수 있으나 절점별로 다른 초기온도를사용할 수 없다. + +# 고정온도 + +고정온도(prescribed temperature)는 열전달해석의 경계조건을 구성하게 되며, 수화열 해석에서는 대체로 일정한 온도를 유지하게 된다. + +# 대류 + +수화열 해석에서는 외기와의 열교환으로 이루어지는 자유대류(free convection)와파이프쿨링(pipe cooling) 형태의 강제대류(forced convection)을 고려할 수 있다.자유대류에서 외기온도(ambient temperature)는 콘크리트의 타설 후 양생과정에서의 외기온도를 의미한다. 일정한 온도나 Sine 함수 또는 시간에 대한 온도 형태 diff --git a/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_044.md b/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_044.md new file mode 100644 index 00000000..24fe1893 --- /dev/null +++ b/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_044.md @@ -0,0 +1,322 @@ + + +의 입력이 가능하다. 일반적으로 매스콘크리트의 온도해석에서 사용되는 대류 문제는 콘크리트 표면과 대기의 열교환 형태로 이루어지므로, 대류계수를 식 (2.2.1)과 같은 경험식으로 계산하기도 한다. + +$$ +h _ {c} (k c a l / (m ^ {2} \cdot h \cdot {} ^ {o} C)) = h _ {n} + h _ {f} = 5. 2 (k c a l / (m ^ {2} \cdot h \cdot {} ^ {o} C)) + 3. 2 V (m / \sec) \tag {2.2.1} +$$ + +파이프쿨링은 콘크리트 구조물 속에 파이프를 매설하고, 파이프 속으로 온도가 낮은 유체를 흐르게 하여 수화열로 인한 온도상승을 감소시키는 방법이다. + +# 발열 + +발열(heat source)은 수화과정에서 발생하는 열량을 모델하기 위한 것이다. 매스 콘크리트에서 수화발열에 의한 단위시간당 단위부피의 내부 발열량은 단열온도 상승식을 미분하고, 비열과 밀도를 곱하여 얻을 수 있다. 일반적으로 콘크리스의 수화발열에 의한 단열온도 상승식과 내부발열량은 다음과 같은 형태로 표현된다. + +단위시간당 단위부피의 내부발열량 (kcal/(m³·h)) + +$$ +g = \frac {1}{2 4} \rho c K \alpha e ^ {- \alpha t / 2 4} \tag {2.2.2} +$$ + +단열온도 상승식(°C) + +$$ +T = K (1 - e ^ {- \alpha t}) \tag {2.2.3} +$$ + +여기서 + +$$ +\begin{array}{l} T \quad : \text { 단열온도( } ^ {\circ} \text { C) } \\ K \quad : \text { 단열최고상승온도( } ^ {\circ} \text { C) } \\ \alpha \quad : \text { 반응속도 } \\ t \quad : \text { 시간(days) } \\ \end{array} +$$ + + + +# 2-3 열응력 해석 + +열전달 해석에서 얻어진 절점온도 분포, 시간과 온도에 따른 재질의 변화, 시간에따른 건조수축, 시간과 응력에 따른 크리프 변형 등을 고려하여 매스콘크리트의각 단계별 응력을 계산한다. 수화열 해석의 열응력 계산 단계에서 사용할 수 있는요소와 하중의 종류 그리고 콘크리트의 특성은 다음과 같다. + +# 요소 + +열응력 해석 단계에서는 입체요소와 트러스요소 그리고 철근요소를 사용할 수 있다. 철근요소는 계산에 반영되지만 요소 결과를 출력하지 않는다. 철근요소의 온도변화는 모재요소(mother element)와 같다고 가정하며 일반적으로 열팽창률 또한모재요소와 같다. + +# 하중 + +수화열 해석에서 사용할 수 있는 하중은 자중과 프리스트레스(prestress)이다. 자중은 해석에 포함된 모든 요소에 대해 적용되며 요소의 생성과 동시에 발생한다. + +# 온도와 시간에 의한 등가재령 + +콘크리트의 경화 과정에서 발생하는 재질특성의 변화는 온도와 시간의 함수 형태로 나타나게 된다. 동일한 단계에 타설된 절점이라도 온도분포가 다르게 되면 발현하는 재료의 성질이 달라지게 된다. 이러한 현상을 반영하기 위해 시간과 온도를 사용하여 등가의 재령(equivalent age)을 만들고, 이를 사용하여 콘크리트의강성발현 등 재료변화를 반영한다. 등가재령은 CEB-FIP Model Code 90를 사용하여 산정한다. + +CEB-FIP Model Code 90 에서의 등가재령 계산식 + +$$ +t _ {e q} = \sum_ {i = 1} ^ {n} \Delta t _ {i} \exp \left[ 1 3. 6 5 - \frac {4 0 0 0}{2 7 3 + T \left(\Delta t _ {i}\right) / T _ {0}} \right] +$$ + + + +여기서 + +$$ +t _ {e q} \quad : \text { 등가재령 (days) } +$$ + +$$ +\Delta t _ {i} \quad : \text { 각 해석단계에서의 시간간격 (days) } +$$ + +$$ +T \left(\Delta t _ {i}\right) \quad : \text { 각 해석단계에서의 온도 } \left(^ {\circ} \mathrm{C}\right) +$$ + +$$ +T _ {0} \quad : 1 \text { day } +$$ + +일본 도로교 시방서에 따른 콘크리트 모델의 경우에는 다음과 같이 등가재령을 산정한다. + +$$ +t _ {e q} = \sum_ {i = 1} ^ {n} \Delta t _ {i} \frac {1 0 + T (\Delta t _ {i})}{3 0} +$$ + +midas FEA에서는 콘크리트의 강도/강성 발현에 등가재령을 반영하며 크리프와 건조수축 계산에는 반영하지 않는다. 단, 일본 콘크리트 표준시방서와 도로교시방서에 따른 콘크리트 모델의 경우에는 크리프와 건조수축의 계산에만 등가재령을 사용한다. + +# 콘크리트 압축강도 계산방법 + +콘크리트의 압축강도를 계산하는 방법은 아래와 같이 국가별 시방서에 따른다. 압축강도를 기준으로 하여 탄성계수, 인장강도 등을 계산하여 사용한다. + +# 〈한국 콘크리트 구조설계기준〉 + +$$ +\sigma_ {c} (t) = \frac {t}{a + b t _ {e q}} \sigma_ {c (9 1)} +$$ + +여기서 + +$$ +a, b \quad : \text { 시멘트종류에 따른 계수 } +$$ + +$$ +\sigma_ {c (9 1)} \quad : 9 1 \text {일 압축강도 } +$$ + + + +< ACI 209(1995) > + +$$ +\sigma_ {c} (t) = \frac {t}{a + b t _ {e q}} \sigma_ {c (2 8)} +$$ + +여기서 + +a, b : 시멘트종류에 따른 계수 + +$\sigma _ { c ( 2 8 ) }$ c(28) ：28 + +ACI committee 209, 1992년, Materials and general properties of concrete prediction of creep, shrinkage and temperature effects in concrete structures. + +ACI manual of concrete practice 1995, Part I, 209R-4 + +< CEB-FIP Model Code 90 > + +$$ +\sigma_ {c} (t) = \exp \left\{s \left[ 1 - \left(\frac {2 8}{t _ {e q} / t _ {1}}\right) ^ {1 / 2} \right] \right\} \sigma_ {c (2 8)} +$$ + +여기서 + +s : 시멘트 종류에 따른 계수 + +$\sigma _ { c ( 2 8 ) }$ σ c(28) : 28일 압축강도 + +1t : 1 day + +COMITE EURO-INTERNATIONAL DU ETON, 1991년 + +CEB-FIP Model Code, p51 + +< 일본 콘크리트 표준시방서 > + +$$ +\sigma_ {c} (t) = \frac {d t}{a + b t} \sigma_ {c (2 8)} +$$ + +$$ +\sigma_ {t e n s i l e} (t) = c \sqrt {\sigma_ {c} (t)} +$$ + + + +여기서 + +c(28) ：28 + +c : 인장강도 발현 계수( 0.44 ) + +# 온도의 변화에 의한 변형률 + +열전달해석을 통해서 구해진 각 단계별 절점온도의 변화를 사용하여 온도에 의한변형률을 산정하고, 강성에 따른 등가하중을 계산하여 해석을 수행한다. + +# 건조수축에 의한 변형 + +콘크리트의 초기 양생이 끝나게 되면 거푸집을 떼어내게 되는데 이때부터 건조수축이 시작되고 이로 인해 변형으로 응력이 추가로 발생하게 된다. midas FEA에서는 한국 콘크리트 구조설계기준, ACI 209, CEB-FIP Model Code 90, JAPANCode 등을 사용하여 시멘트의 종류, 구조물의 형상, 시간에 따른 건조수축률을 계산하여 열응력 해석에 포함한다. + +# 크리프에 의한 변형 + +콘크리트에 응력이 발생하게 되면 시간이 흐름에 따라 크리프 변형이 발생하게 되고, 구조물에 추가적인 변형률과 응력을 발생하게 된다. midas FEA에서는 한국콘크리트 구조설계기준, ACI 209, CEB-FIP Model Code 90, JAPAN Code 등을사용하여 크리프에 의한 효과를 고려한다. + + + +# 2-4 시공단계를 고려한 수화열 해석 + +수화열 해석은 시간에 따른 해석이고, 시공단계를 고려한 해석이 필요하다. 시공단계에 따라 구조물이나 경계조건이 변경될 수 있고, 프리스트레스나 추가적인 하중이 도입될 수도 있다. 그림 2.4.1은 시공단계에 따른 구조모델의 변화를 나타내고있다. 시공단계에 따라 구조물이 추가되고 열 경계조건이 변화하지만, 동일한 시공단계에서는 같은 구조모델과 경계조건을 가져야 한다. 각 시공단계는 여러 개의시간 단계로 구성되고, 각 단계별로 하중을 추가하거나 제거할 수 있다. 자중은 새롭게 생성되는 구조모델에 대하여 각 시공단계 첫 번째 시간 단계에 반영된다.각 시공단계에서 입력하는 사항에는 추가되는 구조모델, 구조와 열 하중/경계조건이 있다. 구조모델과 구조 경계조건은 추가만 가능하고, 하중은 시공단계의 시간단계별로 추가와 삭제가 가능하다. 열 하중/경계조건은 추가와 제거가 모두 가능하다. + +![](images/page-436_4c89a593bff73ceeae1ac1ab5fbc9a33fb7a067c8b66a52d2cf611cd47ec5ef8.jpg) + +
+flowchart + +```mermaid +graph TD + A["structure 1"] -->|convection boundary| B["structure 2"] + B -->|prescribed temperature| C["soil"] + style A fill:#f9f,stroke:#333 + style B fill:#f9f,stroke:#333 + style C fill:#f9f,stroke:#333 + note1["(a) 첫번째 시공단계 모델"] --> A + note2["(b) 두번째 시공단계 모델"] --> B +``` +
+ +그림 2.4.1. 수화열 시공단계 해석 모델 + + + +# 2-5 시간의존적 재질특성 + +midas FEA에서는 콘크리트의 시간의존적 특징으로 크리프(creep), 건조수축(shrinkage), 강도증가(aging) 등을 고려할 수 있다. + +# 2-5-1 크리프 + +그림 2.5.1와 같이 실제 구조물에서 크리프는 건조수축과 함께 발생한다. 따라서건조수축, 탄성변형, 크리프를 각각 분리해서 고려할 수는 없다. 그러나 실제 해석및 설계에서는 편의상 이들을 분리하여 고려한다. + +그림 2.5.1에서 실제 탄성변형(true elastic strain)이란 시간과 더불어 증가되는 탄성계수의 증대로 인해 감소되는 탄성변형을 나타낸 것이다. 일반적인 경우는 겉보기 탄성변형(apparent elastic strain)을 탄성변형으로 보지만 해석에서는 콘크리트의 강도발현을 고려할 수 있으므로 실제 탄성변형으로 해석할 수도 있다. + +크리프 변형률은 하중재하시의 탄성변형률에 비례하며, 동일한 응력에서는 고강도콘크리트가 저강도 콘크리트보다 작은 크리프 변형률을 나타낸다. 크리프 변형률은 탄성변형률의 1.5\~3배 정도에 이르며, 재하 후 첫 몇 개월 동안에 크리프 변형이 크게 진행되고, 그 이후에 완만하게 진행되다가 약 5년 내에 대부분 변형이 발생한다. + +크리프 현상은 대부분의 재료가 가지고 있는 성질이지만, 특히 콘크리트는 다른재료에 비하여 그 값이 커서 변형의 시간적 증가 원인의 하나가 되기 때문에 설계에서 무시할 수 없다. 보통의 콘크리트 구조물에서는 주로 자중과 외력에 의하여크리프 현상이 일어나지만, 프리스트레스를 도입하는 경우에는 프리스트레스 응력에 의한 크리프 현상이 추가로 발생한다. + + + +![](images/page-438_7e040b78a21c21aa0ec5a29ad9e7d0e790647f9a8fcd9caec869aaf580ea9d38.jpg) + +
+line + +| Time Point | Creep strain | Shrinkage strain | True elastic strain | Apparent elastic strain | +| ---------- | ------------ | ---------------- | ------------------- | ----------------------- | +| to | Low | Low | Low | Low | +| Time to | High | Low | Low | Low | +
+ +그림 2.5.1 시간경과에 따른 콘크리트의 변형률 + +일축방향 응력상태에서 콘크리트 탄성변형률과 크리프 변형률의 합은 다음과 같이나타낼 수 있다. + +$$ +\varepsilon (t) = \varepsilon_ {i} (\tau) + \varepsilon_ {c} (t, \tau) = \sigma \cdot J (t, \tau) \tag {2.5.1} +$$ + +여기서 + +J ( , ) t τ : 단위 응력이 작용할 때의 총 변형률, 크리프 함수(creep function) + +T + +t : 변형률을 계산하고자 하는 임의의 시간 + +![](images/page-438_ef82774f3ed8b22e716c1466e5f8a92e6b78752b15973480289b2a5f24af632f.jpg) + +
+line +| Time | σ | +|------|-------| +| τ | 1 | +
+ +![](images/page-438_1c17a57533716ba859587d496f4a86c48c96b8f89b9fd45125dc44ec694c749f.jpg) + +
+line + +| Time | J(t, τ) | +|------|---------| +| τ | 1/(E(τ)) | +| t | J(t, τ) curve | +
+ +그림 2.5.2 크리프 함수 및 특성 크리프의 정의 + + + +그림 2.5.2에서 보듯이 크리프 함수 J ( , ) t τ 를 초기탄성변형과 크리프 변형의 합으로 나타내면 다음과 같다. + +$$ +J (t, \tau) = \frac {1}{E (\tau)} + C (t, \tau) \tag {2.5.2} +$$ + +여기서 + +E( ) τ : 하중 재하시의 탄성계수 + +C t( , ) τ : 재령 t에서의 크리프 변형, 특성크리프(specific creep) + +또한 크리프 함수J ( , ) t τ 를 탄성변형과의 비율로 나타내면 다음과 같다. + +$$ +J (t, \tau) = \frac {1 + \phi (t , \tau)}{E (\tau)} \tag {2.5.3} +$$ + +여기서 + +(t,τ）： (creep coefficient)， + +위의 두 식으로부터 특성크리프와 크리프 계수 사이에는 다음과 같은 관계가 성립한다. + +$$ +\phi (t, \tau) = E (\tau) \cdot C (t, \tau) \tag {2.5.4} +$$ + +$$ +C (t, \tau) = \frac {\phi (t , \tau)}{E (\tau)} \tag {2.5.5} +$$ + +midas FEA에서는 크리프 계수나 건조수축 변형률의 계산식으로 CEB-FIP나 ACI등에서 정하고 있는 식들을 사용할 수 있고, 사용자가 실험에 의한 값을 직접 입력하여 사용할 수도 있다. 사용자 정의는 크리프 계수(creep coefficient), 크리프 함수(creep function), 특성 크리프(specific creep)의 세가지 값 중 사용자가 원하는 형식으로 입력이 가능하다. + + + +![](images/page-440_8316e8a49473232905a0f9b6079e8d5ec1fbf5185e0195a4c506c865689e54df.jpg) + +
+line + +| Time (day) | Value | +| ---------- | ------ | +| 1 | 13.00 | +| 2 | 17.00 | +| 3 | 23.00 | +| 4 | 31.00 | +| 5 | 42.00 | +| 6 | 56.00 | +| 7 | 74.00 | +| 8 | 100.00 | +
+ +그림 2.5.3 사용자 정의 크리프 계수 지정 대화상자 + +콘크리트의 크리프 함수는 그림 2.5.4와 같이 하중이 가해지는 시간에 따라서 각기 다른 형상을 나타내게 된다. 즉, 요소의 재령이 커지면 콘크리트의 강도증가(aging) 효과에 의하여 탄성계수가 증가하기 때문에, 콘크리트의 즉시 변형은 하중의 재하시기가 늦을수록 작아진다. 그리고 하중의 재하시간으로부터 임의의 시간후의 변형은 하중의 재하시기가 늦은 경우에 더 작아지게 된다. + +재하시간이 늦어질수록 즉시 변형과 크리프 변형이 감소하는 것은 콘크리트의 수화정도와 강도발현 때문이다. 따라서 사용자 정의를 사용하여 크리프 함수를 입력할 경우에는 콘크리트의 강도발현 특성이 잘 반영될 수 있도록 하여야 한다. 즉,크리프 함수에서 재하시간의 범위는 시간의존해석에서 존재하는 요소의 재령(재하시간)을 포함하여야 하고, 서로 다른 재하시간의 크리프 함수를 많이 입력할수록정확한 해석결과를 얻을 수 있다. diff --git a/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_045.md b/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_045.md new file mode 100644 index 00000000..987f91b0 --- /dev/null +++ b/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_045.md @@ -0,0 +1,464 @@ + + +![](images/page-441_9b3d77cf26d782d999b326735768d0ec257bda28aca8255629b468be5d773b89.jpg) + +
+line + +| Time t | Creep Function J (t, τ) | +| ------ | ------------------------ | +| τ₁ | Low | +| τ₂ | Medium | +| τ₃ | Medium | +| τ₄ | High | +
+ +그림 2.5.4 하중 재하 시간의 차이에 따른 크리프 함수 + +# 2-5-2 크리프의 계산 방법 + +크리프 계산에 사용되는 크리프 함수는 특정 응력이 작용하는 시간(τ )과 현재시간( t )의 함수이다. 즉 특정응력이 작용하는 시점이 다른 경우에 그림 2.5.4와 같이다른 형태의 크리프 함수를 사용하여야 한다. 따라서 시간에 따라 응력이 변화하는 경우에 각 시간에서의 증감하는 응력은 독자적인 크리프 함수를 필요로 한다.임의의 시점에서의 크리프 변형은 응력이 변화하는 시점에서 증감하는 응력들에의한 변형률을 독자적으로 계산하고 이 값들을 중첩(superposition)하여 산정한다.이와 같은 중첩법을 사용하기 위해서는 모든 부재에 대한 응력이력을 저장하고,매 단계마다 모든 응력에 대하여 초기단계부터 현재의 시점까지의 변형률을 계산할 수 있어야 한다. 따라서 중첩법은 많은 양의 데이터의 저장과 많은 계산을 필요로 한다. midas FEA에서는 응력의 전체 이력을 저장하지 않고, 계산의 효율을 높이기 위하여 다음과 같은 적분방법을 사용한다. + +크리프 계수를 사용하여 시간 τ j 에서 발생한 응력에 의한 시간 t 의 크리프 변형계산 방법은 다음과 같다. + +$$ +\varepsilon_ {c} \left(t, \tau_ {j}\right) = \phi \left(t, \tau_ {j}\right) \varepsilon \left(\tau_ {j}\right): \text { 크리프 변형률 } \tag {2.5.6} +$$ + + + +$$ +P _ {c r} = \int_ {A} E (t) \varepsilon_ {c} \left(t, \tau_ {j}\right) d A: \text {크리프 변형에 의한 하중} \tag {2.5.7} +$$ + +여기서 + +$$ +\varepsilon (\tau_ {j}) \quad : \text { 시간 } \tau_ {j} \text { 에서의 응력에 의한 변형률 } +$$ + +$$ +\phi (t, \tau_ {j}) \quad : \text { 시간 } \tau_ {j} \text { 에서 } t \text { 까지의 크리프 계수 } +$$ + +$$ +\varepsilon_ {c} (t, \tau_ {j}) \quad : \text { 시간 } \tau_ {j} \text { 에서의 탄성변형에 의한 시간 } t \text { 에서 } +$$ + +의 크리프 변형률 + +다음은 크리프의 특성함수를 수식화하여 응력과 시간에 대한 적분을 사용하는 방법이다. 특정 시간 τ 에서 임의의 시간 t 까지의 전체 크리프량은 각 단계마다 발생하는 응력에 의한 크리프의 중첩적분으로 나타내면 다음식과 같다. + +$$ +\varepsilon_ {c} (t) = \int_ {0} ^ {t} C (t, \tau) \frac {\partial \sigma (\tau)}{\partial \tau} d \tau \tag {2.5.8} +$$ + +여기서 + +$$ +\varepsilon_ {c} (t) \quad : \text { 시간 } t \text { 에서의 크리프 변형률 } +$$ + +$$ +C (t, \tau) \quad : \text { 특성크리프 } (\text { specific creep }) +$$ + +$$ +\tau \quad : \text { 하중재하시점 } +$$ + +위의 식에서 응력이 각 단계에서 일정하다고 가정하면 다음과 같이 전체 변형률을단계별로 구분된 변형률의 합으로 표현할 수 있다. + +$$ +\varepsilon_ {c, n} = \sum_ {j = 1} ^ {n - 1} \Delta \sigma_ {j} C (t _ {n}, \tau_ {j}) \tag {2.5.9} +$$ + +위 식을 사용하여 시간 − −t t 사이에서 발생하는 크리프 변형률의 증분( cn,∆ε )을정리하여 나타내면 식 (2.5.10)과 같다. + +$$ +\Delta \varepsilon_ {c, n} = \varepsilon_ {c, n} - \varepsilon_ {c, n - 1} = \sum_ {j = 1} ^ {n - 1} \Delta \sigma_ {j} C (t _ {n}, \tau_ {j}) - \sum_ {j = 1} ^ {n - 2} \Delta \sigma_ {j} C (t _ {n - 1}, \tau_ {j}) \tag {2.5.10} +$$ + +특성크리프를 다음과 같이 Dirichlet 급수의 “Degenerate Kernel” 로 표현하면 응력의 전체 이력을 저장할 필요없이 크리프에 의한 증분변형률을 계산할 수 있다. + + + +$$ +C (t, \tau) = \sum_ {i = 1} ^ {m} a _ {i} (\tau) \left[ 1 - e ^ {- (t - \tau) / \Gamma_ {i}} \right] \tag {2.5.11} +$$ + +여기서 + +$a_{i}(\tau)$ : 하중 재하시간 $\tau$ 에 관련된 특성크리프의 형상 계수 + +$\Gamma_{i}$ : 시간의 경과에 따른 특성트리프 곡선의 형상에 관한 값 + +특성크리프 곡선의 초기형상에 관련한 계수( $a_{i}(\tau)$ )값은 아래와 같은 방법으로 계산할 수 있다. + +1. 계산에 사용할 m 과 $\Gamma_{i}$ 를 정한다. + +재하재령( $\tau_{j}$ )을 정한다. 재하재령은 전체해석 시간내에서 적절히 분포되어야 한다. + +2. 크리프 발생량을 계산하고자 하는 시점( $t_{i}$ )을 선택한다. 크리프 계산은 재하재령( $\tau_{j}$ )을 기준으로 하기 때문에 계산시점( $t_{i}$ )은 재하재령보다 항상 커야 한다. + +3. 계측한 값이나 국가별 기준을 사용하여 $C(t_{i}, \tau_{j})$ 값을 계산한다. 이때 사용하는 데이터 개수는 전체해석시간에 고르게 분포해야 하고 충분한 많아야 한다. + +4. 3\~4 절차에서 구한 값을 사용하여 다음과 같은 식을 구성한다. + +$$ +\left[ \begin{array}{c c c c} 1 - e ^ {- \left(t _ {1} - \tau_ {j}\right) / \Gamma_ {1}} & 1 - e ^ {- \left(t _ {1} - \tau_ {j}\right) / \Gamma_ {2}} & \dots & 1 - e ^ {- \left(t _ {1} - \tau_ {j}\right) / \Gamma_ {m}} \\ 1 - e ^ {- \left(t _ {2} - \tau_ {j}\right) / \Gamma_ {1}} & 1 - e ^ {- \left(t _ {2} - \tau_ {j}\right) / \Gamma_ {2}} & \dots & 1 - e ^ {- \left(t _ {2} - \tau_ {j}\right) / \Gamma_ {m}} \\ \vdots & \vdots & \vdots & \vdots \\ 1 - e ^ {- \left(t _ {n} - \tau_ {j}\right) / \Gamma_ {1}} & 1 - e ^ {- \left(t _ {n} - \tau_ {j}\right) / \Gamma_ {2}} & \dots & 1 - e ^ {- \left(t _ {n} - \tau_ {j}\right) / \Gamma_ {m}} \end{array} \right] \left\{ \begin{array}{c} a _ {1} \left(\tau_ {j}\right) \\ a _ {2} \left(\tau_ {j}\right) \\ \vdots \\ a _ {m} \left(\tau_ {j}\right) \end{array} \right\} = \left\{ \begin{array}{c} C \left(t _ {1}, \tau_ {j}\right) \\ C \left(t _ {2}, \tau_ {j}\right) \\ \vdots \\ C \left(t _ {n}, \tau_ {j}\right) \end{array} \right\} +$$ + +$$ +A _ {i, 1} = \Delta \sigma_ {0} a _ {i} \left(\tau_ {j}\right) \tag {2.5.12} +$$ + +$$ +\mathbf {A} _ {n \times m} a _ {m \times 1} = C _ {n \times 1} (n > m) +$$ + +5. 최소자승법(least square method)을 사용하여 위 식의 해를 구한다. + +$$ +\mathbf {A} ^ {T} \mathbf {A} \mathbf {a} = \mathbf {A} ^ {T} \mathbf {C} +$$ + +$$ +\mathbf {a} = \left(\mathbf {A} ^ {T} \mathbf {A}\right) ^ {- 1} \left(\mathbf {A} ^ {T} \mathbf {C}\right) +$$ + + + +최적의 m 과 Γ 값을 구하기 위해서는 아래와 같은 조건을 만족해야 한다. + +- 최소자승의 에러를 최소화해야 한다. +- $- \sum a _ { i } ( \tau _ { j } ) \frac { \ d s } { \ d s }$ 를 사용하여 계산한 최종 크리프 변형(ultimate creep strain)이예측하는 최종 크리프 변형에 거의 일치해야 한다. +- 각각 $a _ { i } ( \tau _ { j } ) \{ 1 - e ^ { - ( t - \tau _ { j } ) / \Gamma _ { i } } \}$ 값의 기여도가 유사해야 한다. + +서로 다른 재하재령 $\tau _ { j } )$ 를 사용하여 3\~5의 과정을 진행하고, 해당하는 $a _ { i } ( \tau _ { j } ) \equiv _ { \equiv }$ 구한다. 여러 개의 재하재령( $\tau _ { j }$ )에 대한 $a _ { i } ( \tau _ { j } )$ 값을 구하여 저장하고, 재하재령이일치하지 않는 경우에는 계산된 재하재령( $\tau _ { j }$ )과 $a _ { i } ( \tau _ { j } )$ 를 보간하여 사용한다. + +위의 특성크리프 수식을 도입하여 증분 변형률을 다시 정리하면 다음과 같다. + +$$ +\Delta \varepsilon_ {c, n} = \sum_ {i = 1} ^ {m} \left[ \sum_ {j = 1} ^ {n - 2} \Delta \sigma_ {j} a _ {i} \left(\tau_ {j}\right) e ^ {- \left(t _ {n - 1} - \tau_ {j}\right) / \Gamma_ {i}} + \Delta \sigma_ {n - 1} a _ {i} \left(\tau_ {n - 1}\right) \right] \left[ 1 - e ^ {- \Delta t _ {n} / \Gamma_ {i}} \right] +$$ + +$$ +\Delta \varepsilon_ {c, n} = \sum_ {i = 1} ^ {m} A _ {i, n} \left[ 1 - e ^ {- \Delta t _ {n} / \Gamma_ {i}} \right] +$$ + +$$ +A _ {i, n} = \sum_ {j = 1} ^ {n - 2} \Delta \sigma_ {j} a _ {i} (\tau_ {j}) e ^ {- (t _ {n - 1} - \tau_ {j}) / \Gamma_ {i}} + \Delta \sigma_ {n - 1} a _ {i} (\tau_ {n - 1}) \tag {2.5.13} +$$ + +위의 식 $A _ { i , n } \underline { { \underline { { O } } } }$ 로 부터 $A _ { i , n - 1 }$ 을 다음과 같이 나타낼 수 있다. + +$$ +A _ {i, n - 1} = \sum_ {j = 1} ^ {n - 3} \Delta \sigma_ {j} a _ {i} (\tau_ {j}) e ^ {- (t _ {n - 2} - \tau_ {j}) / \Gamma_ {i}} + \Delta \sigma_ {n - 2} a _ {i} (\tau_ {n - 2}) \tag {2.5.14} +$$ + +따라서, $A _ { i , n }$ 과 $A _ { i , n - }$ 1의 관계는 다음과 같다. + +$$ +A _ {i, n} = A _ {i, n - 1} e ^ {- \Delta t _ {n - 1} / \Gamma_ {i}} + \Delta \sigma_ {n - 1} a _ {i} (\tau_ {n - 1}) +$$ + +$$ +A _ {i, 1} = \Delta \sigma_ {0} a _ {i} \left(\tau_ {0}\right) \tag {2.5.15} +$$ + +각 시간구간의 응력이 선형적으로 변한다고 가정한다면, 다음과 같이 (2.5.13)과유사한 관계식을 얻을 수 있다. + + + +$$ +\Delta \varepsilon_ {c, n} = \sum_ {i = 1} ^ {m} A _ {i, n} \left[ 1 - e ^ {- \left(\Delta t _ {n}\right) / \Gamma_ {i}} \right] + R \Delta \sigma_ {n} +$$ + +$$ +\phi_ {n} = \Gamma_ {i} (1 - e ^ {- \Delta t _ {n} / \Gamma_ {i}}) / \Delta t _ {n} +$$ + +$$ +R _ {n} = \sum_ {i = 1} ^ {m} a _ {i} \left(\tau_ {n}\right) \left(1 - \phi_ {n}\right) +$$ + +$$ +A _ {i, n} = A _ {i, n - 1} e ^ {- \left(\Delta t _ {n - 1}\right) / \Gamma_ {i}} + \phi_ {n - 1} \Delta \sigma_ {n - 1} a _ {i} \left(\tau_ {n - 1}\right) \tag {2.5.16} +$$ + +위와 같은 방법을 사용하면 각 단계에서의 요소의 증분변형률은 이전단계에서 발생하는 응력과 이전단계까지 수정된 응력의 누적값( ∆σ )만을 사용하여 계산할수 있다. 따라서 매 단계마다 모든 부재의 응력 이력을 저장하여 초기단계부터 적분을 수행하여야 하는 문제가 발생하지 않으며 응력의 변화를 고려한 비교적 정확한 해석을 할 수 있다. 식 (2.5.13)과 (2.5.16)은 시간 간격 내에서 응력 변화를 고려하는지 여부에 따라 달라진다. 수화열 해석에서는 온도변화에 의한 응력이 시간에 따라 선형적으로 변한다 가정하여 식 (2.5.16)을 사용하는 반면, 일반 시공단계해석에서는 식 (2.5.13)을 사용한다. + +한 단계에서 큰 시간 간격을 사용하고자 하는 경우에는 내부적인 시간간격을 만들어서 크리프의 효과를 적절하게 계산할 수 있도록 해야 하며 크리프 변형의 발생특성상 시간간격은 로그(log) 스케일로 분할하는 것이 바람직하다. midas FEA에서는 간격 수만 입력하면 시간간격을 자동으로 로그 스케일로 분할하여 시간구간을 생성하는 기능을 가지고 있다. 타당한 시간간격의 개수는 정해져 있지 않지만많이 세분할수록 정해에 수렴하게 된다. 따라서 큰 시간 간격이 도입되는 단계에서는 적당한 간격으로 분할해주는 것이 필요하다. + + + +# 2-5-3 건조수축 + +건조수축은 콘크리트 부재가 시간에 따라 수축하는 현상으로 각종 시방서의 건조수축 특성 곡선을 사용하여 해석에 반영하고 있다. 건조수축은 부재에 발생하는응력과는 무관한 시간의 함수이며, 일반적으로 시간 $t _ { 0 }$ 에서 t 까지 발생한 건조수축에 의한 변형률은 다음과 같이 나타낸다. + +$$ +\varepsilon_ {s h} (t, t _ {0}) = \varepsilon_ {\infty} \cdot f (t, t _ {0}) \tag {2.5.17} +$$ + +여기서 + +$\varepsilon _ { \infty }$ £ + +$f ( t , t _ { 0 } )$ : 시간에 따른 발생 함수 + +t : 관측 시점 + +$t _ { 0 }$ : 건조수축 발생시점 + +midas FEA에서 건조수축 해석은 CEB-FIP Model Code, ACI209, 도로교시방서,실험데이터를 사용한 사용자 정의 건조수축 특성 곡선 등을 사용할 수 있다. 건조수축 특성 곡선을 사용하여 시공단계의 해당 단계에서의 건조수축 변형률을 다음과 같이 계산한다. + +$$ +\varepsilon_ {s h} (t _ {2}, t _ {1}) = \varepsilon_ {s h} (t _ {2}, t _ {0}) - \varepsilon_ {s h} (t _ {1}, t _ {0}) \tag {2.5.18} +$$ + +$\mathcal { E } _ { s h } ( t _ { 2 } , t _ { 1 } )$ : 시공단계 1t 에서 $t _ { 2 }$ 까지의 건조수축 변형률 + +$\varepsilon _ { s h } ( t _ { 1 } , t _ { 0 } )$ : 건조수축 발생시점 $t _ { 0 }$ 에서 $t _ { 1 }$ 까지의 건조수축 변형률 + +$\mathcal { E } _ { s h } ( t _ { 2 } , t _ { 0 } )$ : 건조수축 발생시점 $t _ { 0 }$ 에서 $t _ { 2 }$ 까지의 건조수축 변형률 + +건조수축 변형은 온도, 크리프에 의한 변형과 같이 비역학적(non-mechanical) 변형이기 때문에 부재력( F ) 계산시의 변형률은 변위에 의한 변형률에서 건조수축에의한 변형률을 감하여 계산한다. + +$$ +F = E A \left(\varepsilon - \varepsilon_ {s h}\right) \tag {2.5.19} +$$ + + + +그러므로 축 방향 구속이 없는 구조물에서의 건조수축에 의한 효과는 부재력을 만들지 않고 변위만을 발생시키게 된다. 또한 외부하중이 없더라도 건조수축에 의해발생하는 부재력은 크리프 변형을 유발할 수 있다. 따라서 건조수축 변형은 구속조건과 시간에 영향을 받는다. + +# 2-5-4 시간에 따른 탄성계수의 변화 + +콘크리트의 압축강도와 탄성계수는 시간에 따라 변화한다. 실제 콘크리트 구조물이나 교량의 시공에서 콘크리트의 초기 재령을 정확하게 예측하여 계획된 구조물의 형상과 강도를 지니도록 하기 위해서는, 이러한 효과를 합리적으로 묘사하는것이 필수적이라 할 수 있다. + +midas FEA에서는 콘크리트 부재의 재령에 따른 탄성계수의 변화를 고려함으로써강도발현 효과를 포함하여 해석할 수 있다. 그림 2.5.5와 같이 ACI209, CEB-FIP,또는 콘크리트 표준시방서 등의 기준에 따른 콘크리트의 강도발현 함수를 정의하거나 사용자가 직접 입력할 수도 있다. 이와 같이 정의된 강도발현 함수를 참조하여, 각각의 시공단계에 정의된 시간의 경과에 따른 콘크리트의 강도변화를 자동으로 계산해서 해석을 수행한다. + +![](images/page-447_94842f2d0ad0defb6890ab6dc48f3bfc83cd973297cf164cd7257a623ec08cc2.jpg) + +
+text_image + +Add/Modify Time Dependent Material (Comp. Strength) +Name +TdMat1 +Scale Factor +1.0 +Type +Code User +Development of Strength +Code : ACI +S = teq * S28 / (a + b * teq) +Concrete Compressive Strength +at 28 Days(S28) +28 kN/m² +Concrete Compressive Strength +Factor(a, b) +a : 4.5 b : 0.85 +Redraw Graph +Graph Options +X-axis log scale Y-axis log scale +30 +25 +20 +15 +10 +5 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +OK Cancel +
+ +그림 2.5.5 규준에 따른 콘크리트의 강도발현 함수정의 + + + +# 2-5-5 크리프/건조수축 모델 + +midas FEA에서 사용할 수 있는 주요 기준은 다음과 같다. + +# 〈 한국 콘크리트 구조설계기준〉 + +\- 크리프 계산식 + +$$ +\varepsilon_ {c \sigma} = f _ {c} (t ^ {\prime}) / E _ {c i} \cdot \phi (t, t ^ {\prime}) +$$ + +$$ +\varepsilon_ {c \sigma} (t, t ^ {\prime}) = f _ {c} (t ^ {\prime}) \left[ \frac {1}{E _ {c i} (t ^ {\prime})} + \frac {\phi (t , t ^ {\prime})}{E _ {c i}} \right] +$$ + +$$ +\phi (t, t ^ {\prime}) = \phi_ {0} \beta_ {c} (t - t ^ {\prime}) +$$ + +$$ +\phi_ {0} = \phi_ {R H} \beta (f _ {c u}) \beta (t ^ {\prime}) +$$ + +$$ +\phi_ {R H} = 1 + \frac {1 - 0 . 0 1 R H}{0 . 1 0 \sqrt [ 3 ]{h}} +$$ + +$$ +\beta \left(f _ {c u}\right) = \frac {1 6 . 8}{\sqrt {f _ {c u}}} +$$ + +$$ +\beta (t ^ {\prime}) = \frac {1}{0 . 1 + (t ^ {\prime}) ^ {0 . 2}} +$$ + +$$ +\beta_ {c} (t - t ^ {\prime}) = \left[ \frac {(t - t ^ {\prime})}{\beta_ {H} + (t - t ^ {\prime})} \right] ^ {0. 3} +$$ + +$$ +\beta_ {H} = 1. 5 \left\{1 + (0. 0 1 2 R H) ^ {1 8} \right\} h + 2 5 0 \leq 1 5 0 0 +$$ + +$$ +h = \frac {2 A _ {c}}{u} +$$ + +$\varepsilon_{cc}$ : 콘크리트의 크리프 변형 + +$\sigma_{c}(t')$ : 재하 재령 $(t')$ 지속 응력 + +$E_{ci}$ : 28일 콘크리트의 초기 접선 탄성 계수 + +$\phi(t,t')$ : 콘크리트의 크리프 계수 + +$\phi_{o}$ : Notional 크리프 계수 + +$f_{cu}$ : 28일 평균 압축강도 (MPa) + +RH : 상대습도(%) + + + +h : 부재의 notional size (mm) + +$A_{c}$ : 부재의 단면적 ( $mm^{2}$ ) + +u : 대기와 접하는 단면둘레 (mm) + +$t'$ : 지속하중 재하시 재령 (day) + +t : 계측 시간 (day) + +\- 건조수축 계산식 + +$$ +\varepsilon_ {s h} (t, t _ {s}) = \varepsilon_ {s h o} \cdot \beta_ {s} (t - t _ {s}) +$$ + +$$ +\varepsilon_ {s h o} = \varepsilon_ {s} (f _ {c u}) \beta_ {R H} +$$ + +$$ +\varepsilon_ {s} \left(f _ {c u}\right) = \left[ 1 6 0 + 1 0 \beta_ {s c} \left(9 - \frac {f _ {c u}}{1 0}\right) \right] \times 1 0 ^ {- 6} +$$ + +$$ +\beta_ {R H} = \left\{ \begin{array}{l l} - 1. 5 5 \left[ 1 - \left(\frac {R H}{1 0 0}\right) ^ {3} \right] & (40 \% \leq R H \leq 99 \%) \\ 0. 2 5 & (R H \geq 99 \%) \end{array} \right. +$$ + +$$ +\beta_ {s} \left(t - t _ {s}\right) = \sqrt {\frac {\left(t - t _ {s}\right)}{0 . 0 3 5 h ^ {2} + \left(t - t _ {s}\right)}} +$$ + +$\varepsilon_{cs}(t,t_{s})$ : 건조수축 시작시간( $t_{s}$ )에서 + +임의 시간(t)까지의 건조수축 변형률 + +$\varepsilon_{cso}$ : Notional 건조수축 변형률 + +$\beta_{s}(t-t_{s})$ : 건조수축 발현 함수 + +$\beta_{sc}$ : 시멘트 종류에 따른 계수 + +$\beta_{sc}=4$ 2종 시멘트 + +$\beta_{sc}=5\ 1,5종 시멘트$ + +$\beta_{sc}=6$ 3종 시멘트 + +사단법인 한국콘크리트학회, 2003년개정, 콘크리트 구조설계기준, p38\~p44 + + + +< ACI209 (1995) > + +-크리프 계산식 + +$$ +\varepsilon_ {c} (t, t ^ {\prime}) = \sigma_ {c} (t ^ {\prime}) +$$ + +$$ +J \left(t, t ^ {\prime}\right) = \frac {1}{E _ {c i} \left(t ^ {\prime}\right)} \left[ 1 + \phi \left(t, t ^ {\prime}\right) \right] +$$ + +$$ +\phi (t, t ^ {\prime}) = \frac {(t - t ^ {\prime}) ^ {0 . 6}}{1 0 + (t - t ^ {\prime}) ^ {0 . 6}} \phi_ {u} +$$ + +$$ +\phi_ {u} = 2. 3 5 C _ {c u} C _ {h} C _ {t} C _ {s} C _ {f} C _ {a} +$$ + +1) 양생조건( $C_{cu}$ ) + +$$ +C _ {c u} = \left\{ \begin{array}{l l} 1. 2 5 \left(t ^ {\prime}\right) ^ {- 0. 1 1 8} & \text {(moist cured)} \\ 1. 1 3 \left(t ^ {\prime}\right) ^ {- 0. 0 9 4} & \text {(steam cured)} \end{array} \right. +$$ + +2) 상대습도( $C_{h}$ ) + +$$ +C _ {h} = \left\{ \begin{array}{l l} 1. 2 7 - 0. 0 0 6 7 H & (H > 4 0) \\ 1. 0 & (H \leq 4 0) \end{array} \right. +$$ + +3) 체적-표면적비( $C_{t}$ ) + +$$ +C _ {t} = \frac {2}{3} \left[ 1 + 1. 1 3 e ^ {- 0. 0 2 1 3 v / s} \right] \quad (\text { unit }: \mathrm{mm}) +$$ + +4) 슬럼프치( $C_{s}$ ) + +$$ +C _ {S} = 0. 8 2 + 0. 0 0 2 6 4 S \quad (\text { unit }: \mathrm{mm}) +$$ + +5) 잔골재의 비율( $C_{f}$ ) + +$$ +C _ {f} = 0. 8 8 + 0. 0 0 2 4 \psi +$$ + +6) 공기량( $C_{a}$ ) + +$$ +C _ {a} = 0. 4 6 + 0. 0 9 A \geq 1. 0 +$$ diff --git a/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_046.md b/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_046.md new file mode 100644 index 00000000..bfa9de92 --- /dev/null +++ b/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_046.md @@ -0,0 +1,504 @@ + + +t' : 재하시의 콘크리트재령 + +H : 상대습도(%) + +v/s : 체적-표면적비 + +S : 슬럼프값 + +ψ : 전체골재에 대한 잔골재의 중량비율 + +A : 공기량(%) + +\- 건조수축 계산식 + +$$ +\varepsilon_ {s h} (t, t _ {0}) = \left\{ \begin{array}{l l} \frac {(t - t _ {0})}{3 5 + (t - t _ {0})} \varepsilon_ {s h, u} & (\text { moist cured }) \\ \frac {(t - t _ {0})}{5 5 + (t - t _ {0})} \varepsilon_ {s h, u} & (\text { steam cured }) \end{array} \right. +$$ + +$$ +\varepsilon_ {s h, u} = - 7 8 0 \times 1 0 ^ {- 6} C _ {s h} +$$ + +$$ +C _ {s h} = C _ {c p} C _ {h} C _ {t} C _ {s} C _ {f} C _ {a} C _ {c} +$$ + +1) 양생기간에 따른 보정( $C_{cp}$ ) +
습윤양생기간(일)137142890
보정계수( $C_{cp}$ )1.21.11.00.930.860.75
+ +2) 상대습도( $C_{h}$ ) + +$$ +C _ {h} = \left\{ \begin{array}{l l} 1. 4 0 - 0. 0 1 0 H & (4 0 \leq H \leq 8 0) \\ 3. 0 0 - 0. 0 3 0 H & (8 0 \leq H \leq 1 0 0) \end{array} \right. +$$ + +3) 체적-표면적비( $C_{t}$ ) + +$$ +C _ {t} = 1. 2 e ^ {- 0. 0 0 4 7 2 v / s} \quad (\text { unit }: \mathrm{mm}) +$$ + +4) 슬럼프치( $C_{s}$ ) + +$$ +C _ {S} = 0. 8 9 + 0. 0 0 1 6 1 S \quad (\text { unit }: \mathrm{mm}) +$$ + + + +5) 잔골재의 $\mathsf { H } | \mathop { \cong } ( C _ { f } )$ + +$$ +C _ {f} = \left\{ \begin{array}{l l} 0. 3 0 + 0. 0 1 4 \psi & (\psi \leq 5 0) \\ 0. 9 0 + 0. 0 0 2 \psi & (\psi \geq 5 0) \end{array} \right. +$$ + +6) 공기량( $C _ { a } )$ + +$$ +C _ {a} = 0. 9 5 + 0. 0 0 8 A +$$ + +7) 단위시멘트량( $C _ { c } )$ + +$$ +C _ {c} = 0. 7 5 + 0. 0 0 0 6 1 C \quad \left(\text { unit }: \mathrm{kg} / \mathrm{m} ^ {3}\right) +$$ + +$\mathcal { E } _ { s h } ( t , t _ { 0 } )$ ： $t _ { o }$ t발생한 건조수축 변형 + +$\mathcal { E } _ { s h , u }$ εsh,u ： + +$t _ { o }$ : 건조수축 시작 시간(day) + +$H$ : 상대습도(%) + +$\nu / s$ : 체적-표면적비 + +$S$ : 슬럼프값 + +$\psi$ + +$A$ : 공기량(%) + +C : 단위 시멘트량( $\mathsf { k g } / \mathsf { m } ^ { 3 } \ )$ + +ACI committee 209, 1992년, Materials and general properties of concrete prediction of creep, shrinkage and temperature effects in concrete structures. + +ACI manual of concrete practice 1995, Part I, 209R-6\~209R-9 + + + +# 〈 CEB-FIP Model Code 90 〉 + +\- 크리프 계산식 + +$$ +\varepsilon_ {c c} (t, t _ {0}) = \sigma_ {c} (t _ {0}) / E _ {c i} \cdot \phi (t, t _ {0}) +$$ + +$$ +\phi (t, t _ {0}) = \phi_ {0} \beta_ {c} (t - t _ {0}) +$$ + +$$ +\phi_ {0} = \phi_ {R H} \beta (f _ {c m}) \beta (t _ {0}) +$$ + +$$ +\phi_ {0} = \phi_ {R H} \beta (f _ {c m}) \beta (t _ {0}) +$$ + +$$ +\phi_ {R H} = I + \frac {I - R H / R H _ {o}}{0 . 4 6 (h / h _ {0}) ^ {1 / 3}} +$$ + +$$ +\beta \left(f _ {c m}\right) = \frac {5 . 3}{\left(f _ {c m} / f _ {c m o}\right) ^ {0 . 5}} +$$ + +$$ +\beta \left(t _ {0}\right) = \frac {1}{0 . 1 + \left(t _ {o} / t _ {1}\right) ^ {0 . 2}} +$$ + +$$ +h = \frac {2 A _ {c}}{u} +$$ + +$$ +\beta_ {c} \left(t - t _ {o}\right) = \left[ \frac {\left(t - t _ {o}\right) / t _ {I}}{\beta_ {H} + \left(t - t _ {o}\right) / t _ {I}} \right] ^ {0. 3} +$$ + +$$ +\beta_ {H} = 1 5 0 \left\{1 + \left(1. 2 \frac {R H}{R H _ {o}}\right) ^ {1 8} \right\} \frac {h}{h _ {o}} + 2 5 0 \leq 1 5 0 0 +$$ + +$\varepsilon_{cc}$ : 콘크리트의 크리프 변형 + +$\sigma_{c}(t_{o})$ : 재하 재령 $(t_{o})$ 지속응력 + +$E_{ci}$ : 28일 콘크리트의 탄성계수 + +$\phi(t,t_{o})$ : 콘크리트의 크리프 계수 + +$\phi_{o}$ : Notional 크리프 계수 + +$f_{cm}$ : 28일 평균 압축강도(MPa), $f_{cmo} = 10 MPa$ + +RH : 상대습도, $RH_{o}=100\%$ + +h : 부재의 notional size, $h_{o} = 100$ mm + + + +$$ +A _ {c} \quad : \text { 부재의 단면적 } (m m ^ {2}) +$$ + +$$ +u \quad : \text { 대기와 접하는 주변장 (mm) } +$$ + +$$ +t _ {o} \quad : \text {재하시 재령(day)} +$$ + +$$ +t \quad : \text { 계측 시간(day) }, t _ {1} = 1 \text { day } +$$ + +\- 건조수축 계산식 + +$$ +\varepsilon_ {c s} (t, t _ {s}) = \varepsilon_ {c s o} \cdot \beta_ {s} (t - t _ {s}) +$$ + +$$ +\varepsilon_ {c s o} = \varepsilon_ {s} (f _ {c m}) \beta_ {R H} +$$ + +$$ +\varepsilon_ {s} \left(f _ {c m}\right) = \left[ 1 6 0 + 1 0 \beta_ {s c} \left(9 - f _ {c m} / f _ {c m o}\right) \right] \times 1 0 ^ {- 6} +$$ + +$$ +\beta_ {R H} = - 1. 5 5 \beta_ {s R H} (40 \% \leq R H \leq 99 \%) +$$ + +$$ +\beta_ {R H} = + 0. 2 5 \quad (R H \geq 99 \% +$$ + +$$ +\beta_ {s R H} = 1 - \left(\frac {R H}{R H _ {o}}\right) ^ {3} +$$ + +$$ +\beta_ {s} \left(t - t _ {s}\right) = \left[ \frac {\left(t - t _ {s}\right) / t _ {l}}{3 5 0 \left(h / h _ {o}\right) ^ {2} + \left(t - t _ {s}\right) / t _ {l}} \right] ^ {0. 5} +$$ + +$$ +\varepsilon_ {c s} \left(t, t _ {s}\right) \quad : \text { 건조수축 시작시간 } \left(t _ {s}\right) \text { 에서 } +$$ + +임의 시간(t)까지의 건조수축 변형률 + +$\varepsilon_{cso}$ : Notional 건조수축 변형률 + +$$ +\beta_ {s} (t - t _ {s}) \quad : \text { 건조수축 발현 함수 } +$$ + +$\beta_{sc}$ : 시멘트 종류에 따른 계수 + +$\beta_{sc}=4$ Slowly hardening cements + +$\beta_{sc} = 5$ Normal or Rapid hardening cements + +$\beta_{sc} = 8$ Rapid hardening high strength cements + +COMITE EURO-INTERNATIONAL DU ETON, 1991년 + +CEB-FIP Model Code, p53\~p58 + + + +# 〈일본 콘크리트 표준시방서〉 + +\- 크리프 계산식 + +$$ +\varepsilon_ {c c} ^ {\prime} = \phi \sigma_ {c p} ^ {\prime} / E _ {c t} +$$ + +$$ +\varepsilon_ {c c} ^ {\prime} (t, t ^ {\prime}, t _ {0}) = \left[ 1 - e ^ {- 0. 0 9 (t - t ^ {\prime}) ^ {0. 6}} \right] \cdot \varepsilon_ {c r} ^ {\prime} +$$ + +$$ +\mathcal {E} _ {c r} ^ {\prime} = \mathcal {E} _ {b c} ^ {\prime} + \mathcal {E} _ {d c} ^ {\prime} +$$ + +$$ +\varepsilon_ {b c} ^ {\prime} = 1 5 (C + W) ^ {2. 0} (W / C) ^ {2. 4} \left(\log_ {e} t ^ {\prime}\right) ^ {- 0. 6 7} +$$ + +$$ +\varepsilon_ {d c} ^ {\prime} = 4 5 0 0 (C + W) ^ {1. 4} (W / C) ^ {4. 2} \left[ \log_ {e} (V / S / 1 0) \right] ^ {- 2. 2} (1 - R H / 1 0 0) ^ {0. 3 6} t _ {0} ^ {- 0. 3 0} +$$ + +$$ +\varepsilon_ {c c} ^ {\prime} \quad : \text { 콘크리트의 압축 크리프 변형 } +$$ + +$$ +\phi : \text { 크리프 계수 } +$$ + +$$ +\sigma_ {c p} ^ {\prime} \quad : \text { 압축 응력 } +$$ + +$$ +E _ {c t} \quad : \text {재하시 재령의 탄성계수} +$$ + +$$ +\varepsilon_ {c r} ^ {\prime} \quad : \text { 단위 응력당 크리프 변형의 최종값 } \left(\times 1 0 ^ {- 1 0} / (N / m m ^ {2})\right) +$$ + +$$ +\varepsilon_ {b c} ^ {\prime} \quad : \text { 단위 응력당 기본 크리프 변형의 최종값 } \left(\times 1 0 ^ {- 1 0} / (N / m m ^ {2})\right) +$$ + +$$ +\varepsilon_ {d c} ^ {\prime} \quad : \text { 단위 응력당 건조 크리프 변형의 최종값 } \left(\times 1 0 ^ {- 1 0} / (N / m m ^ {2})\right) +$$ + +$$ +C \quad : \text { 단위 시멘트량 } (k g / m ^ {3}) (2 6 0 k g / m ^ {3} \leq C \leq 5 0 0 k g / m ^ {3}) +$$ + +$$ +W \quad : \text { 단위 수량 } \left(k g / m ^ {3}\right) \left(1 3 0 k g / m ^ {3} \leq C \leq 2 3 0 k g / m ^ {3}\right) +$$ + +$$ +W / C \quad : \text { 물 - 시멘트 비 } (40 \% \leq C \leq 65 \%) +$$ + +$$ +R H \quad : \text { 상대습도 } (45 \% \leq C \leq 80 \%) +$$ + +$$ +V \quad : \text { 체적 } (m m ^ {3}) +$$ + +$$ +S \quad : \text { 대기와 접하는 면적 } (m m ^ {2}) +$$ + +$$ +V / S \quad : \text { 체적 - 표면적 비 } (m m) (1 0 0 m m \leq V / S \leq 3 0 0 m m) +$$ + +$$ +t _ {0} \quad : \text { 건조수축 시작 시간(day) } +$$ + +$$ +t ^ {\prime} \quad : \text {재하시 재령(day)} +$$ + +$$ +t \quad : \text { 계측 시간 (day) } +$$ + + + +\- 건조수축 계산식 + +$$ +\varepsilon_ {c s} ^ {\prime} (t, t _ {0}) = \left[ I - e ^ {- 0. 1 0 8 (t - t _ {0}) ^ {0. 5 6}} \right] \cdot \varepsilon_ {s h} ^ {\prime} +$$ + +$$ +\varepsilon_ {s h} ^ {\prime} = - 5 0 + 7 8 \left[ 1 - e ^ {(R H / 1 0 0)} \right] + 3 8 \log_ {e} W - 5 \left[ \log_ {e} (V / S / 1 0) \right] ^ {2} +$$ + +$$ +\varepsilon_ {s h} ^ {\prime} (t, t _ {0}) \quad : \text { 콘크리트 재령 } t _ {0} \text { 에서 } t \text { 까지의 } +$$ + +$$ +\text { 건조수축 변형률 } \left(\times 1 0 ^ {- 5}\right) +$$ + +$$ +\varepsilon_ {s h} ^ {\prime} \quad : \text { 건조수축 최종 변형률 } (\times 1 0 ^ {- 5}) +$$ + +# 〈일본 도로교시방서(평성 8년 12월)〉 + +\- 크리프 계산식 + +$$ +\varepsilon_ {c c} = \phi \sigma_ {c} / E _ {c t} +$$ + +$$ +\phi (t, t _ {0}) = \phi_ {d 0} \cdot \beta_ {d} (t - t _ {0}) + \phi_ {f 0} \left[ \beta_ {f} (t) - \beta_ {f} (t _ {0}) \right] +$$ + +$$ +h _ {t h} = \lambda \cdot \frac {A _ {c}}{u} +$$ + +$$ +\varepsilon_ {c c} \quad : \text { 콘크리트의 크리프 변형 } +$$ + +$$ +\sigma_ {c} \quad : \text { 지속하중 응력 } (N / m m ^ {2}) +$$ + +$$ +E _ {c} \quad : \text { 콘크리트의 탄성 계수 } (N / m m ^ {2}) +$$ + +$$ +\phi : \text { 콘크리트의 크리프 계수 } +$$ + +$$ +\phi_ {d 0} \qquad : \text {지속하중에 의한 지연탄성변형률에 대한 크리프 계수} +$$ + +$$ +\beta_ {d} (t - t _ {0}) : \text { 시간에 따른 } \phi_ {d 0} \text { 의 발현 함수 (그림 2.5.6) 참조) } +$$ + +$$ +\phi_ {f 0} \quad : \text { 복원되지 않는 크리프 변형에 대한 크리프 계수 } +$$ + +$$ +\beta_ {f} (t) \quad : \text { 시간에 따른 } \phi_ {f 0} \text { 의 발현 함수 (그림 2.5.7) 참조) } +$$ + +$$ +R H \quad : \text { 상대습도 } (45 \% \leq C \leq 80 \%) +$$ + +$$ +\lambda : \text { 상대습도에 따른 환경 조건에 관한 계수 (표 2.5.1 참조 } +$$ + + + +Ac : 부재의 표면적 ( 2 mm ) +u : 대기와 접하는 주변장 ( mm ) +$h _ { t h }$ : 부재의 가상두께 ( mm ) +$t _ { 0 }$ : 재하시 재령 (day) +t : 계측 시간 (day) + +![](images/page-457_0a38c90070011532f1dd1fa46ad1ccd0075a36c8ea234e26f4fa67c836ea3810.jpg) + +
+line + +| x | βd | +| ----- | ---- | +| 1 | 0.30 | +| 2 | 0.32 | +| 4 | 0.35 | +| 6 | 0.38 | +| 10 | 0.42 | +| 100 | 0.70 | +| 1000 | 0.90 | +| 10000 | 1.00 | +
+ +그림 2.5.6 콘크리트의 $\phi _ { d 0 }$ 발현함수 ( 0β d ( ) t t − ) + +![](images/page-457_941de31f9075cee35630615b25b1a0dbed2713294d15973e543f56a3d9703b0a.jpg) + +
+line + +| x | βf (h_th ≥ 50mm) | βf (h_th ≥ 100mm) | βf (h_th ≥ 200mm) | βf (h_th ≥ 400mm) | βf (h_th ≥ 800mm) | βf (h_th ≥ 1600mm) | +| ---- | ---------------- | ----------------- | ----------------- | ----------------- | ----------------- | ------------------ | +| 4 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | +| 8 | 0.35 | 0.35 | 0.35 | 0.35 | 0.35 | 0.35 | +| 10 | 0.45 | 0.45 | 0.45 | 0.45 | 0.45 | 0.45 | +| 100 | 0.60 | 0.60 | 0.60 | 0.60 | 0.60 | 0.60 | +| 1,000| 0.80 | 0.80 | 0.80 | 0.80 | 0.80 | 0.80 | +| 10,000| 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | +
+ +그림 2.5.7 콘크리트의 $\phi _ { f 0 }$ 발현함수 $( \beta _ { f } ( t ) )$ + + + +
환경조건 $\phi_{f0}$ $\lambda$
수 중0.860
상대습도 90%1.310
상대습도 70%2.03
상대습도 40%3.02
+ +표 2.5.1 환경조건에 의한 $\phi_{f0}$ 와 $\lambda$ 값 + +\- 건조수축 계산식 + +$$ +\varepsilon_ {c s} (t, t _ {0}) = \varepsilon_ {s 0} \cdot \beta_ {s} (t - t _ {0}) +$$ + +$\varepsilon_{cs}(t,t_{0})$ : 콘크리트 재령 $t_{0}$ 에서 t 까지의 건조수축 변형 + +$\varepsilon_{so}$ : 콘크리트의 기본 건조수축 최종변형률 + +$\beta_{s}(t)$ : 건조수축 발현 함수 +
환경조건 $\varepsilon_{so}$
수 중 $-10 \times 10^{-5}$
상대습도 90% $+10 \times 10^{-5}$
상대습도 70% $+25 \times 10^{-5}$
상대습도 40% $+50 \times 10^{-5}$
+ +표 2.5.2 환경조건에 의한 $\varepsilon_{so}$ 값 + + + +![](images/page-459_b47f91ff3e609178c17632a897c9b507e5e865b7468c992fd1708337002af819.jpg) + +
+line + +| x | h_th ≤ 50mm | 100 | 200 | 400 | 800 | h_th ≥ 1600mm | +| ---- | ----------- | ----- | ----- | ----- | ----- | ------------- | +| 1 | 0.2 | 0.1 | 0.05 | 0.02 | 0.01 | 0.005 | +| 10 | 0.6 | 0.3 | 0.15 | 0.08 | 0.04 | 0.02 | +| 100 | 1.2 | 0.8 | 0.4 | 0.2 | 0.1 | 0.05 | +| 1,000| 1.8 | 1.4 | 0.9 | 0.5 | 0.3 | 0.15 | +| 10,000| 1.9 | 1.5 | 1.05 | 0.75 | 0.6 | 0.3 | +
+ +그림 2.5.8 콘크리트의 건조수축 발현함수 ( β s ) + +사단법인 일본도로협회, 평성8년12월 + +도로교시방서(I공통편　III콘크리트교)　동해설, p37\~p44 + + + +# 2-6 수화열 해석 결과 + +수화열 해석은 열전달 해석과 열응력 해석을 통해서 얻은 결과를 가지고 있다. 열전달 해석을 수행하여 구한 결과는 절점별 온도이며 열응력 해석의 결과로는 절점별 변위, 요소별 응력과 발생응력에 대한 인장강도의 비율로 표현되는 균열지수(crack ratio) 등이 있다. 모든 결과들은 시공단계의 시간단계별로 확인할 수 있다. + +열전달 해석 결과 + +\- 절점별 온도 + +열응력 해석 결과 + +- 절점별 변위 +- 요소별 응력/변형률 +- 절점별 등가재령에 따른 인장 강도 +- 균열지수(crack ratio) + +Crack Ratio = 허용인장응력 / 발생인장응력 diff --git a/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_047.md b/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_047.md new file mode 100644 index 00000000..61e98181 --- /dev/null +++ b/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_047.md @@ -0,0 +1,321 @@ + + +# Analysis and Algorithm Manual + +# Part 7 Contact Analysis + +# Chapter 1. Static Contac Analysis + + + + + +# Chapter 1. Static Contact Analysis + +# 1-1 개요 + +접촉해석(contact analysis)은 공간상의 두 물체가 서로 맞닿을 수는 있으나, 관통할 수 없다는 가정에 따라 해석을 수행하는 것이다. 접촉의 종류에는 일반접촉(general contact) 혹은 접합접촉(weld contact) 해석 등 다양한 종류가 있을 수있으며, 이를 해석적으로 고려하기 위해서는 비선형해석을 수행하여야 한다. + +공간상에서 물체 A와 B가 서로 접촉할 가능성이 있는 경우, 물체 A를 주물체(master body), 물체 B를 종속물체(slave body)로 정의한다. 알고리즘 상에서는주물체/종속물체의 개념은 서로 뒤바뀌어도 상관없다. 그러나 수치해석에서는 주물체와 종속물체를 선택할 때에 강체, 상대적으로 밀도나 강성이 큰 물체, 또는 상대적으로 요소가 조밀하지 않는 물체를 주물체로 설정하면 더 나은 결과를 얻을수 있다. 접촉해석에서는 주물체에 접촉하는 절점을 종속절점(slave node), 주물체의 접촉면을 주접촉면(master surface)으로 지정하는 알고리즘을 주로 사용한다midas FEA에서는 접촉에 의한 비선형해석을 위하여 벌칙방법(penalty method)을사용한다. 이 기법은 접촉면과 그 면을 관통하는 절점 사이에 관통을 방지하기 위한 벌칙 스프링(penalty spring)을 사용하는 방법이다. 이 방법은 구현하기가 쉬우며, 동해석의 경우에 시간 증분에 영향을 받지 않는 장점이 있다. + + + +# 1-2 접촉 검색(Contact search) + +접촉해석의 과정에서 접촉이 발생할 위치를 미리 알 수 없기 때문에 접촉검색을실행하여야 한다. midas FEA는 접촉검색(contact search)을 위하여, 다음과 같은세 단계의 검색을 수행한다. + +(1) 전역검색(global search): 종속절점에 가장 가까운 주절점을 검색하는 단계이며, 해석의 효율을 위하여 버켓분류(bucket sort) 알고리즘을 사용한다. +(2) 국부검색(local search): 주절점에 연결되어 있는 접촉면(contact surface) 중에서 종속절점과 가장 가까운 접촉면을 검색한다. +(3) 접촉점 검색(contact search): 주접촉면 위에서 종속절점에 가장 가까운 접촉점(contact point)을 계산한다. + +# 1-2-1. 전역검색 + +전역검색을 위해서는 접촉의 가능성이 있는 N 개의 절점들을 대상으로 가장 가까운 절점을 찾기 위하여, 다음 식과 같은 계산을 N −1번 실행하여야 한다. + +$$ +l ^ {2} = \left(x _ {i} - x _ {j}\right) ^ {2} + \left(y _ {i} - y _ {j}\right) ^ {2} + \left(z _ {i} - z _ {j}\right) ^ {2} \tag {1.2.1} +$$ + +따라서 N N( ) −1 번 계산을 수행해야 하기 때문에 검색이 실제 해석 시간의 대부분을 차지할 수도 있다. 따라서 midas FEA에서는 효율적인 검색을 위해 버켓분류알고리즘을 사용한다. + +버켓분류 알고리즘은 절점들을 몇 개의 그룹(bucket)으로 분류하고, 가장 가까운그룹들끼리만 거리를 계산하는 방법이다. 예를 들면 1차원 해석일 경우에는 검색하여야 하는 그룹의 개수가 자신과 좌우에 인접한 그룹을 포함해서 총 3개이다. 이와 유사한 방법으로 2차원 해석의 경우 9개의 그룹, 3차원 해석의 경우 27개의 그룹을 검색하여야 한다. 그러므로 요구되는 계산량은 각각 다음과 같다. + + + +$$ +N \left(\frac {3 N}{N B _ {x}} - 1\right), \quad N \left(\frac {9 N}{N B _ {x} \cdot N B _ {y}} - 1\right), \quad N \left(\frac {2 7 N}{N B _ {x} \cdot N B _ {y} \cdot N B _ {z}} - 1\right) \tag {1.2.2} +$$ + +여기서, + +$N B _ { x } , N B _ { y } , N B _ { z } \quad \mathrm { ~ : ~ x , ~ y ~ , ~ z ~ } \stackrel { \forall \dag } { \sim } \stackrel { \triangledown } { \cong } \ 7 \sharp \widehat { \mp }$ + +# 1-2-2. 국부검색 + +3차원해석에서 3차원 요소가 종속절점과 만나게 되는 접촉면은 2차원의 평면이 되며, 이 면이 주접촉면(master surface)으로 정의된다. 국부검색은 종속절점으로부터 가장 가까운 곳에 위치하는 주접촉면을 찾는 과정이다. + +![](images/page-465_396bd5e401366a05b40017fa68240f5239410a1d4aafc784332f235d66550356.jpg) + +
+text_image + +c_{i+1} +x_{sr} +n_s +S +c_i +x_m +Z +Y +X +
+ +그림 1.2.1 국부 검색 + +국부검색을 위하여 그림 1.2.1에서와 같이 종속절점 sn 의 절점벡터 sx 와 주접촉면의 절점벡터 $\mathbf { X } _ { m }$ 는 다음과 같이 나타낼 수 있다. + +$$ +\mathbf {x} _ {s} = \left\{x _ {s} \quad y _ {s} \quad z _ {s} \right\} ^ {T} \tag {1.2.3} +$$ + + + +$$ +\mathbf {x} _ {m i} = \left\{x _ {m i} \quad y _ {m i} \quad z _ {m i} \right\} ^ {T} \tag {1.2.4} +$$ + +여기서, i ( ) =1,2,3,4 는 절점번호를 의미한다. 또한 주접촉면에 포함되어 있는 주절점 $n _ { { \scriptscriptstyle m } }$ 의 절점벡터는 다음과 같이 나타낼 수 있다. + +$$ +\mathbf {x} _ {m} = \left\{x _ {m} \quad y _ {m} \quad z _ {m} \right\} ^ {T} \tag {1.2.5} +$$ + +그리고 주절점 $n _ { m }$ 에서부터 종속절점 ns 까지의 벡터 $\mathbf { X } _ { s m }$ 는 다음과 같이 계산된다. + +$$ +\mathbf {X} _ {s m} = \mathbf {X} _ {s} - \mathbf {X} _ {m} \tag {1.2.6} +$$ + +주절점과 종속절점을 연결하는 벡터 $\mathbf { x } _ { \scriptscriptstyle s m }$ 을 주접촉면 위에 투영시킨 벡터 s 는 다음 과 산정할 수 있다. + +$$ +\mathbf {s} = \mathbf {x} _ {s m} - \left(\mathbf {x} _ {s m} \cdot \mathbf {m}\right) \mathbf {m} \tag {1.2.7} +$$ + +여기서, + +$\begin{array}{c} \begin{array}{c} \mathbf { c } _ { i } , \mathbf { c } _ { i + 1 } \quad \quad : \ \overset { \triangledown } \end{ \cong } \ \begin{array} { l } { \underline { { \ b { \ b { \ b { \ b { \ b { \ b { \ b { \ b { \ b { \ b { \ b { \ b { \ b { \ b { \ b { \ b { \ b { \ b { \ b { \ b } } } } } } } } } } } } } } } } } } } } } } \end{array} \ \begin{array} { l } \underline { { { \ b { \ b { \ b { \ b { \ b { \ b { \ b { \ b { \ b { \ b { \ b { \ b { \ b { \ b { \ b { \ b { \ b } } } } } } } } } } } } } } } } } } } \end{array} \ \begin{array} { l } { \underline { { { \ b { \ b { \ b { \ b { \ b { \ b { \ b { \ b { \ b { \ b { \ b { \ b { \ b { \ b { \ b } } } } } } } } } } } } } } } } } \end{array} \begin{array} { l } { \underline { { { \ b { \ b { \ b { \ b { \ b { \ b { \ b { \ b { \ b { \ b { \ b { \ b { \ b } } } } } } } } } } } } } } } \end{array} \begin{array} { l } { \underline { { { \ b { \ b { \ b { \ b { \ b { \ b { \ b { \ b { \ b { \ b { \ b { \ b } } } } } } } } } } } } } } \end{array} \begin{array} { l } { \underline { { { \ b { \ b { \ b { \ b { \ b { \ b { \ b { \ b { \ b { \ b { \ b } } } } } } } } } } } } } \end{array} } } \end{array} } } \end{array}$ + +$\begin{array} { r } { \textbf { m } \qquad : \mathbf { c } _ { i } , \mathbf { c } _ { i + 1 } \ 0 \nparallel \nleftrightarrow \pm \frac { \mathbf { \ell } _ { \mathbf { \hat { \mathbf { \phi } } } } | \bar { \mathbf { \phi } } _ { \mathbf { \phi } } \ y | } { \mathbf { c } _ { i } \times \mathbf { c } _ { i + 1 } } \ \frac { \mathbf { c } _ { i } \times \mathbf { c } _ { i + 1 } } { \left| \mathbf { c } _ { i } \times \mathbf { c } _ { i + 1 } \right| } } \end{array}$ + +따라서 다음 식 (1.2.8)을 이용해서 주절점 $n _ { { } _ { m } }$ 주위의 주접촉면 중에서 가장 가까운주접촉면 is 를 찾을 수 있다. + +$$ +\begin{array}{l} \left(\mathbf {c} _ {i} \times \mathbf {s}\right) \cdot \left(\mathbf {c} _ {i} \times \mathbf {c} _ {i + 1}\right) > 0 \\ \left(\mathbf {c} _ {i} \times \mathbf {s}\right) \cdot \left(\mathbf {s} \times \mathbf {c} _ {i + 1}\right) > 0 \tag {1.2.8} \\ \end{array} +$$ + +# 1-2-3. 접촉점 검색 + +midas FEA 에서는 주접촉면이 4절점으로 이루어진 등매개변수로 표현되는 것으로 가정하고, 접촉점 검색을 수행한다. + + + +![](images/page-467_54e930cb1838e38db6b4bbd8d31a5ecec3ba683626d2095b968e2ecdd1772c45.jpg) + +
+text_image + +n_s +4 +η +∂r/∂η +1 +∂r/∂ξ +3 +x_m +Y +ξ +Z +2 +X +
+ +그림 1.2.2 접촉 점의 위치 + +종속절점 $n _ { s }$ 의 절점변위 및 주접촉면 요소의 절점변위는 다음과 같이 나타낼 수있다. + +$$ +\mathbf {u} _ {s} = \left\{u _ {s} \quad v _ {s} \quad w _ {s} \right\} ^ {T} \tag {1.2.9} +$$ + +$$ +\mathbf {u} _ {m i} = \left\{u _ {m i} \quad v _ {m i} \quad w _ {m i} \right\} ^ {T} \tag {1.2.10} +$$ + +또한 주접촉면 요소 상의 임의의 점 $n _ { m }$ 에서 절점좌표는 형상함수 ( ) 1, 2,3, 4 N i i =를 이용하여 다음과 같이 나타낼 수 있다. + +$$ +x _ {m} = \sum_ {i = 1} ^ {4} N _ {i} x _ {m i}, y _ {m} = \sum_ {i = 1} ^ {4} N _ {i} y _ {m i}, z _ {m} = \sum_ {i = 1} ^ {4} N _ {i} z _ {m i} \tag {1.2.11} +$$ + +또한 주접촉면 요소 상의 임의의 점 $n _ { { } _ { m } }$ 에서 절점좌표는 형상함수 ( ) 1, 2,3, 4 N i i =를 이용하여 다음과 같이 나타낼 수 있다. + +$$ +\mathbf {X} = \left( \begin{array}{c c c c c} \mathbf {X} _ {s} & \mathbf {X} _ {m 1} & \mathbf {X} _ {m 2} & \mathbf {X} _ {m 3} & \mathbf {X} _ {m 4} \end{array} \right) \tag {1.2.12} +$$ + + + +$$ +\mathbf {p} = \left( \begin{array}{c c c c c} \mathbf {u} _ {s} & \mathbf {u} _ {m 1} & \mathbf {u} _ {m 2} & \mathbf {u} _ {m 3} & \mathbf {u} _ {m 4} \end{array} \right) +$$ + +절점변위를 고려한 현재상태에서 벡터 $\mathbf { X } _ { m }$ 와 $\mathbf { X } _ { s m }$ 에 대한 변분은 다음과 같다. + +$$ +\delta \mathbf {x} _ {m} = \mathbf {x} _ {m, \xi} \delta \xi + \mathbf {x} _ {m, \eta} \delta \eta + \sum_ {i = 1} ^ {4} N _ {i} \mathbf {I} \delta \mathbf {u} _ {m i} \tag {1.2.13} +$$ + +$$ +\delta \mathbf {x} _ {s m} = \delta \mathbf {x} _ {s} - \delta \mathbf {x} _ {m} = \delta \mathbf {u} _ {s} - \left(\mathbf {x} _ {m, \xi} \delta \xi + \mathbf {x} _ {m, \eta} \delta \eta + \sum N _ {i} \mathbf {I} \delta \mathbf {u} _ {i}\right) \tag {1.2.14} +$$ + +여기서, + +$$ +\mathbf {x} _ {m, \xi} \quad : \quad \sum_ {i = 1} ^ {4} N _ {i, \xi} \mathbf {x} _ {m i} +$$ + +$$ +\mathbf {x} _ {m, \eta} \quad : \quad \sum_ {i = 1} ^ {4} N _ {i, \eta} \mathbf {x} _ {m i} +$$ + +따라서 통합벡터를 이용하면 다음과 같이 나타낼 수 있다. + +$$ +\delta \mathbf {x} _ {m} = \mathbf {x} _ {m, \xi} \delta \xi + \mathbf {x} _ {m, \eta} \delta \eta + \mathbf {C} \delta \mathbf {p} \tag {1.2.15} +$$ + +$$ +\delta \mathbf {x} _ {s m} = \mathbf {F} \delta \mathbf {p} - \left(\mathbf {x} _ {m, \xi} \delta \xi + \mathbf {x} _ {m, \eta} \delta \eta\right) \tag {1.2.16} +$$ + +여기서, + +$$ +\mathbf {C} \quad : \left[ \begin{array}{c c c c c} \mathbf {0} & N _ {1} \mathbf {I} & N _ {2} \mathbf {I} & N _ {3} \mathbf {I} & N _ {4} \mathbf {I} \end{array} \right] +$$ + +$$ +\mathbf {F} \quad : \left[ \begin{array}{c c c c c} \mathbf {I} & - N _ {1} \mathbf {I} & - N _ {2} \mathbf {I} & - N _ {3} \mathbf {I} & - N _ {4} \mathbf {I} \end{array} \right] +$$ + +또한 $\delta \mathbf { x } _ { m }$ 의 ξ , η 에 대한 미분은 다음과 같다. + +$$ +\delta \mathbf {x} _ {m, \xi} = \mathbf {x} _ {m, \xi \eta} \delta \eta + \mathbf {C} _ {, \xi} \delta \mathbf {p} \tag {1.2.17} +$$ + +$$ +\delta \mathbf {x} _ {m, \eta} = \mathbf {x} _ {m, \xi \eta} \delta \xi + \mathbf {C} _ {, \eta} \delta \mathbf {p} \tag {1.2.18} +$$ + +종속절점 $n _ { s }$ 로부터 최단거리에 위치하는 주접촉면의 점인 주절점 $n _ { m }$ 의 좌표 $\mathbf { X } _ { m }$ + + + +을 알고 있다면, smx 은 m,ξ x 및 m,η x 에 각각 수직이 된다. 따라서 이는 다음과 같이 나타낼 수 있다. + +$$ +a = \mathbf {x} _ {m, \xi} ^ {T} \mathbf {x} _ {s m} = 0 \tag {1.2.19} +$$ + +$$ +b = \mathbf {x} _ {m, \eta} ^ {T} \mathbf {x} _ {s m} = 0 \tag {1.2.20} +$$ + +접촉점을 찾기 위해서는 뉴턴 랩슨(Newton-Raphson)법에 따른 반복법을 사용할수 있다. 이를 위하여 변분을 구한다. + +$$ +\delta a = \mathbf {x} _ {m, \xi} ^ {T} \left[ \mathbf {F} \delta \mathbf {p} - \mathbf {x} _ {m, \xi} \delta \xi - \mathbf {x} _ {m, \eta} \delta \eta \right] + \mathbf {x} _ {s m} ^ {T} \left(\mathbf {x} _ {m, \xi \eta} \delta \eta + \mathbf {C} _ {, \xi} \delta \mathbf {p}\right) \tag {1.2.21} +$$ + +$$ +\delta b = \mathbf {x} _ {m, \eta} ^ {T} \left[ \mathbf {F} \delta \mathbf {p} - \mathbf {x} _ {m, \xi} \delta \xi - \mathbf {x} _ {m, \eta} \delta \eta \right] + \mathbf {x} _ {s m} ^ {T} \left(\mathbf {x} _ {m, \xi \eta} \delta \xi + \mathbf {C} _ {, \eta} \delta \mathbf {p}\right) \tag {1.2.22} +$$ + +반복계산에 의해서 새로운 좌표 (ξ,η) 를 찾기 위하여 δp 0 = 으로 가정하면, 변분의 결과는 다음과 같다. + +$$ +\delta a = \mathbf {x} _ {m, \xi} ^ {T} \left[ - \mathbf {x} _ {m, \xi} \delta \xi - \mathbf {x} _ {m, \eta} \delta \eta \right] + \mathbf {x} _ {s m} ^ {T} \left(\mathbf {x} _ {m, \xi \eta} \delta \eta\right) \tag {1.2.23} +$$ + +$$ +\delta b = \mathbf {x} _ {m, \eta} ^ {T} \left[ - \mathbf {x} _ {m, \xi} \delta \xi - \mathbf {x} _ {m, \eta} \delta \eta \right] + \mathbf {x} _ {s m} ^ {T} \left(\mathbf {x} _ {m, \xi \eta} \delta \xi\right) \tag {1.2.24} +$$ + +따라서 이를 벡터로 나타내면 다음과 같다. + +$$ +\delta \mathbf {a} = \mathbf {D} \delta \boldsymbol {\xi} \tag {1.2.25} +$$ + +여기서, + +$$ +\begin{array}{l} \delta \mathbf {a} \quad : \left\{\delta a \quad \delta b \right\} ^ {T} \\ \mathbf {D} \quad : \left\{- \mathbf {A} ^ {T} \mathbf {A} + \mathbf {x} _ {s m} ^ {T} \mathbf {x} _ {m, \xi \eta} \left[ \begin{array}{c c} 0 & 1 \\ 1 & 0 \end{array} \right] \right\} \\ d \xi \quad : \left\{d \xi d \eta \right\} ^ {T} \\ \mathbf {A} \quad : \left[ \begin{array}{c c} \mathbf {x} _ {m, \xi} & \mathbf {x} _ {m, \eta} \end{array} \right] \\ \end{array} +$$ + + + +Truncated Taylor series 를 이용하여 ∆ξ 를 계산하고 (ξ,η) 를 갱신할 수 있다. + +$$ +\mathbf {a} _ {i + 1} = \mathbf {a} _ {i} + \mathbf {D} \Delta \boldsymbol {\xi} _ {i} = \mathbf {0} \tag {1.2.26} +$$ + +여기서, + +$$ +\mathbf {a} _ {i} \qquad \qquad : \quad \left\{\mathbf {X} _ {m, \xi} ^ {T} \mathbf {X} _ {s m} \quad \mathbf {X} _ {m, \eta} ^ {T} \mathbf {X} _ {s m} \right\} _ {i} +$$ + +종속절점이 주접촉면을 관통하는가를 판별하고, 관통이 발생하는 경우에는 접촉점에 접촉력(contact force)를 작용하게 한다. 이 때 적용되는 힘은 관통한 정도에비례한다. 다음 식에서 $g _ { N }$ 은 종속절점이 주접촉면을 관통하는 거리를 나타낸다. + +$$ +\mathbf {g} _ {N} = \mathbf {n} ^ {T} \mathbf {x} _ {s m} \tag {1.2.27} +$$ + +여기서, + +$$ +\mathbf {n} = \frac {\mathbf {X} _ {s m}}{\left| \mathbf {X} _ {s m} \right|} \quad : g _ {N} \text { 의 방향벡터 } +$$ + +만약 종속절점 $n _ { s }$ 가 주접촉면 ${ \cal { S } } _ { i } \equiv \equivq$ 관통했다면, 다음 식과 같은 힘을 종속절점과주접촉면 $s _ { i }$ 에 더해준다. + +$$ +\mathbf {f} _ {s} = - g _ {N} k _ {i} \mathbf {n} _ {i} \tag {1.2.28} +$$ + +여기서, + +$$ +k _ {i} = f _ {s i} \frac {K _ {i} A _ {i} ^ {2}}{V _ {i}} \quad : \text { 주접촉면 } s _ {i} \text { 의 강성 벡터 } +$$ + +$$ +K _ {i} \quad : \text { 벌크 계수(bulk modulus) } +$$ + +$$ +A _ {i} \quad : \text { 주접촉면 } s _ {i} \text { 의 면적 } +$$ + +$$ +V _ {i} \quad : \text { 부피 } +$$ + +$$ +f _ {s i} \quad : \text { 비례 계수(기본값은 } 0. 1 \text { 이다.) } +$$ diff --git a/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_048.md b/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_048.md new file mode 100644 index 00000000..81793988 --- /dev/null +++ b/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_048.md @@ -0,0 +1,152 @@ + + +종속절점에는 위 식 (1.2.28)을 바로 더할 수 있으나, 주접촉면에서는 식 (1.2.29)와같이 접촉력을 산정한다. + +$$ +\mathbf {f} _ {m} ^ {i} = N _ {i} (\xi , \eta) \mathbf {f} _ {s} \tag {1.2.29} +$$ + +여기서, + +i =1,2,3,4 : 주접촉면의 절점번호 + + + +# 1-3 기능 및 결과 + +# 1-3-1. 기능 + +midas FEA는 면과 면의 접합접촉(symmetric weld contact) 및 일반접촉(symmetric general contact) 기능을 제공한다. 다만, midas FEA는 주접속면과종속 접촉면이 같은 경우, 즉 캔이 찌그러지 듯이 물체가 자기자신과 접촉하는 해석은 현재 지원을 하지 않는다. + +사용자는 접촉 기능을 사용하기 위해서는 주접촉면/종속접촉면을 정의해야 하며,이때 두 면이 서로 마주보게 정의해야 한다. 또한 각 주접촉면과 종속접촉면의 강성 스케일 계수(stiffness scale factor)와 만약 마찰을 고려한다면 선형 마찰 계수(static friction coefficient)를 정의해야 한다. 앞 절에서 설명한 바와 같이 midasFEA는 벌칙 스프링을 이용한 벌칙 기법을 사용하며, 이 스프링에 해당하는 강성을자동으로 계산한다. 접촉면의 강성 스케일 계수(stiffness scale factor)는 계산되는 강성의 곱해지는 계수이며, 이 값을 변화시키면 결과가 얼마간 변할 수 있다. + +만약 사용자가 정의한 주접촉면/종속접촉면이 해석 전부터 서로 관통하고 있는 경우에는 midas FEA는 자동으로 종속접촉면의 절점들을 관통한 깊이만큼 움직여서관통하지 않은 상태로 만든다. + +일반적으로 접촉 해석기능은 비선행 해석에서 사용해야 하나 접합 접촉 기능의 경우 선형 시공단계에서도 사용이 가능하다. + +접합 접촉(symmetric weld contact) + +이 기능은 주접촉면/종속접촉면이 초기부터 붙어 있는 경우에 사용하며, 해석이진행되는 동안에 두 면이 분리되는 것을 허용하지 않는다. 특히 두 면이 붙어있으나 주/종속의 절점들이 서로 일치하지 않는 경우에 이 기능을 사용하는 것이 유용하다. 이 방법은 요소가 조밀한 쪽 혹은 상대적으로 강성이 작은 쪽을 종속절점으로 설정하고, 그렇지 않은 쪽을 주절점으로 설정하여야 한다. 주절점/종속절점의 + + + +설정을 반대로 하면 주접촉면이 종속면을 관통할 수 있다. + +일반접촉(symmetric general contact) + +접촉 해석은 면과 면(surface to surface)이 일반적인 접촉을 할 경우에 사용하는기능이다. 두 면이 해석 전에 붙어있거나, 해석 중에 두 면이 접촉과 분리를 반복해도 해석이 가능하다. + +# 1-3-2. 결과 + +midas FEA는 종속접촉면에서 전체 좌표계에 대한 접촉력을 출력한다. + + + +Part 7 Contact Analysis + + + +# Analysis and Algorithm Manual + +# Part 8 Fatigue Analysis + +# Chapter 1. Fatigue Analysis + + + + + +# Chapter 1. Fatigue Analysis + +# 1-1 개요 + +피로파괴는 부재의 항복강도 보다 낮은 하중이 반복하여 작용할 때 부재가 파괴되는 현상을 의미한다. 피로해석을 수행하는 방법에는 응력 기반의 응력-수명(stress-life) 방법과 변형률 기반의 변형률-수명(strain-life) 방법이 있다. 본 매뉴얼에서는 S-N선도를 이용하는 응력-수명 방법에 대하여 정리하였다. + +S-N선도는 구조물에 일정 진폭(constant amplitude)의 반복하중(reverseloading)이 작용할 때, 발생하는 응력진폭(stress amplitude, S)과 해당 진폭응력이 반복될 때 파괴에 이르게 하는 반복 회수(cycle to failure, N)의 관계를 나타낸곡선이다. + +피로해석을 위해서는 먼저 구조물에 대한 정적해석을 수행 한 후, 절점응력을 구한다. 이때 응력으로는 von Mises 응력 혹은 주응력의 최대값(principal stress)이사용될 수 있다. 구해진 절점응력을 사용해 응력진폭(stress amplitude)을 구한다.그리고 이를 S-N선도에 적용하면 피로파괴가 일어나기 까지 소요되는 하중의 반복 횟수 또는 손상도(damage)를 알 수가 있게 된다. + +모든 구조물은 구조물 고유의 S-N선도를 가지고 있으나, 모든 구조물에 대해서피로시험을 할 수 없다. 이 경우에는 표준 단축 피로시험을 통하여 얻어진 S-N선도에 수정계수를 적용한 수정 S-N선도를 사용할 수 있다. 또한 일반적으로 구조물에는 가변진폭(variable amplitude)의 반복하중이 작용하게 된다. 이 경우에 S-N선도를 이용하기 위해서는 레인플로집계(Rainflow-counting)기법을 사용한다. 레인플로집계는 가변진폭의 반복응력으로부터 단위 진폭의 응력과 그 횟수를 추출하여 S-N선도 에 적용하는 방법이다. + + + +# 1-2 반복하중 + +그림 1.2.1과 같이 일정 진폭의 응력이 규칙적으로 작용하는 경우 응력진폭(stressamplitude, σ a )과 평균응력(mean stress, $\sigma _ { m }$ )은 다음과 같이 계산할 수 있다. + +$$ +\sigma_ {a} = \frac {\sigma_ {\max} - \sigma_ {\min}}{2} \tag {1.2.1} +$$ + +$$ +\sigma_ {m} = \frac {\sigma_ {\max} + \sigma_ {\min}}{2} \tag {1.2.2} +$$ + +![](images/page-478_23c58da6f7468b03677f8c95d4825c6a77266887653839461760ff5a2b8acc54.jpg) + +
+line +| t | stress | +|-------|--------| +| 0 | 0 | +| t_max | 0.5 | +| t_min | 0.1 | +| t_max | 0.5 | +| t_min | 0.1 | +| t_max | 0.5 | +
+ +그림 1.2.1 응력진폭과 평균응력 + +응력-수명 방법에서는 평균응력이 영인 상태로 일정 진폭의 응력이 규칙적으로 반복되는 경우에 대하여 그림 1.2.2와 같은 S-N선도를 사용한다. + + + +![](images/page-479_11fa8b2bcfccae68bc148d259cf9463eae39971b5c09e23fb83ad9fbab370d65.jpg) + +
+line + +| Cycle to failure (Log N) | σₐ | Label | +| ------------------------ | ------ | --------------- | +| Endurance limit | Sₑ | Se | +| Endurance limit | Sₙ | Miner | +| Endurance limit | Sₙ | Haibach | +| Endurance limit | Sₙ | modified Miner | +
+ +그림 1.2.2 S-N선도 + +그림 1.2.2에서 Su 와 Se 는 최대진폭응력과 피로한계 진폭응력을 각각 의미한다.그림에서 Miner의 S-N선도는 피로한계(endurance limit) 이하의 반복응력은 피로수명에 영향을 미치지 않음을 보이고 있다. 그리고 수정 Miner선도는 피로한계 이하의 응력도 누적되면 피로파괴가 일어난다는 피로손상의 누적을 가정한 관계이다.그러나 반복하중 1000사이클 이하에서 피로파괴가 일어나는 피로해석에는 응력-수명 기반의 피로해석 방법을 사용할 수 없다. midas FEA에서는 Miner의 S-N선도를 사용하고 있다. 피로해석을 위한 물성치를 입력하면 S-N선도를 프로그램에서 자동적으로 생성하며 최대진폭응력( Su )의 90%에 해당하는 크기의 응력이1000회 반복되는 점과 피로한계진폭응력( S S e u= 0.5 )가 한계반복(cycles atendurance)횟수만큼 반복되는 점과 연결되고 진폭응력의 비가 유지되는 꺽인 직선의 형태를 가지고 있음을 알 수 있다. + + + +![](images/page-480_d5d6aaa0db5a0061764837b26142560665fbc05904e49af3877143d2425e3389.jpg) + +
+line + +| Life to failure (Log N) | Se/Su | +| ----------------------- | ----- | +| 3 | 0.9 | +| 6 | 0.5 | +
+ +그림 1.2.3 기본적인 S-N선도의 형상 + +그림 1.2.3과 같은 S-N선도에서 직선의 기울기를 b라고 하면, 특정 응력진폭 S가반복하여 작용할 때 피로파괴에 이르게 하는 응력의 반복 횟수 N을 다음과 같이계산 할 수 있다. + +$$ +b = - \frac {\left(\log S - \log S _ {e}\right)}{\log N _ {e} - \log N} \tag {1.2.3} +$$ + +$$ +N = N _ {e} \left(\frac {S}{S _ {e}}\right) ^ {\frac {1}{b}} \tag {1.2.4} +$$ + +응력-수명 방법의 장점 및 단점은 다음과 같다. + +- 비교적 간단한 알고리즘을 통해 피로해석을 수행할 수 있다. +- 계산이 간단하고 해석시간이 짧다. +- 진응력-변형률을 무시하고 모든 변형률을 탄성적으로 다루기 때문에 소성변형률이 큰 경우에 유효하지 않다. +- 긴 수명의 작은 주기적 소성변형률 성분을 갖는 경우에만 유효하다. diff --git a/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_049.md b/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_049.md new file mode 100644 index 00000000..e71b2b43 --- /dev/null +++ b/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_049.md @@ -0,0 +1,302 @@ + + +# 1-3 평균응력의 영향 + +구조물에 가해지는 응력진폭 $\sigma _ { a }$ 가 동일하여도 그림 1.3.1과 같이 평균응력 $\sigma _ { m }$ 이다르면 피로수명도 달라진다. 평균응력 $\sigma _ { m }$ 이 증가할수록 최대응력 $S _ { u }$ 와 피로한계응력 $S _ { e }$ 가 작아지며, 이러한 관계는 Haigh에 의해서 최초로 얻어졌다. 그림1.3.1 왼쪽 그림은 동일한 응력진폭에서 평균응력과 최대응력의 관계를 그래프로나타내고 있으며, 오른쪽 그림은 최대응력계수와 피로한계계수에 대한 평균응력의영향을 나타낸다. + +![](images/page-481_8951a1af6abf1a439203f84a2d0b27c8766e2ab9429a803685223511d5b3de25.jpg) + +
+line + +| N | σₐ (σₘ = 0) | σₐ (σₘ +) | +| ------- | ----------- | --------- | +| 10² | ~1.5 | ~1.0 | +| 10⁴ | ~0.8 | ~0.5 | +| 10⁶ | ~0.4 | ~0.2 | +| 10⁸ | ~0.2 | ~0.1 | +
+ +![](images/page-481_c978fa87322a84b67537420080de862479a51e5e916d39ac1536928cbc85b31b.jpg) + +
+text_image + +σa +Su +N = 1 +N+ +Se +0 +Su +σm +
+ +그림 1.3.1 평균응력의 영향 + +평균응력의 영향을 고려하기 위해서 Goodman과 Gerber는 그림 1.3.2로 표현되는다음과 같은 수식을 각각 제안하였다. + +$$ +\text { Goodman(England,1899) } \quad \frac {\sigma_ {a}}{S _ {e}} + \frac {\sigma_ {m}}{S _ {u}} = 1 \tag {1.3.1} +$$ + +$$ +\text { Gerber (Germany, 1874) } \quad \frac {\sigma_ {a}}{S _ {e}} + \left(\frac {\sigma_ {m}}{S _ {u}}\right) ^ {2} = 1 \tag {1.3.2} +$$ + + + +![](images/page-482_c63206d3b9a057ee1a345ed8d0bf9498c139fe87eb82b4278407cb516b4182f3.jpg) + +
+line + +| Stress Component | Stress Value | +| ---------------- | ------------ | +| σₐ | Sₑ | +| σₐ | Sᵤ | +
+ +그림 1.3.2 응력진폭과 평균응력의 관계 + +예를 들어 $\sigma _ { \operatorname* { m a x } } = 7 5 8 . 4 2 \mathrm { M P a }$ , $\sigma _ { \mathrm { { m i n } } } = 6 8 . 9 5 \mathrm { { M P a } }$ , $S _ { u } = 1 0 3 4 . 2 1 \mathrm { M P a }$ 인 경우$\sigma _ { a } = 3 4 4 . 7 4 \mathrm { M P a }$ 이고, $\sigma _ { m } = 4 1 3 . 6 9 \mathrm { M P a } 0 | \boldsymbol { \Xi } \boldsymbol { \Xi }$ , Goodman의 식을 이용면$S _ { e } = 5 7 4 . 5 7 \mathrm { { M P a } }$ 임을 알 수 있다. 여기서 Se 는 평균응력을 고려하여 수정 된 값이라는 것에 주의하여야 한다. + + + +# 1-4 수정계수 + +일반적으로 S-N선도는 표준 시험체를 대상으로 이상적인 반복 굽힘 하중(fullyreversed bending)에서 시험을 통하여 얻어낸다. 이 때 피로한도(endurancelimit)를 S′ 라고 하면, 실제 상황의 S 값은 수정을 통해서 얻어져야 한다. 강재의경우 실험에 의해서 근사적인 관계가 규명되어 있으므로, 간단한 수정을 통해 실제 상황에 적합한 S-N선도를 얻을 수 있다. 이 때 고려할 수 있는 요소는 다음과같다. + +- 부재 크기 및 형상(component size and shape) +- 하중의 종류(loading type) +- 표면마무리(surface finish) +- 표면처리(surface treatment) +- 온도(temperature) +- 환경(environment) + +수정계수를 사용하여 실제 피로한도를 구하면 다음과 같다. + +$$ +S _ {e} = S _ {e} ^ {\prime} C _ {s i z e} C _ {s u r} \dots \tag {1.4.1} +$$ + +피로강도 감소계수는 다음과 같이 정의된다. + +$$ +K _ {f} = \frac {1}{\left(C _ {\text { size }} C _ {\text { load }} C _ {\text { sur }} \dots\right)} \tag {1.4.2} +$$ + +수정계수는 보통 피로한계를 정하는 데 영향을 주며, S-N선도의 나머지 부분에대해서는 명확하게 정의가 되어있지 않다. 수정계수는 하중에 대한 보다 정확한안전율을 반영하기 위하여 사용된다. 즉 하중에 의한 효과는 6 10 사이클 에서의 피로한계에 영향을 주고, 때로는 1000사이클에서의 피로강도에도 영향을 줄 수 있다.이 경우 S-N선도는 그림 1.4.1과 같이 수정 될 수 있다. + + + +![](images/page-484_b40ce4708d97a5e287e6dbdac75aefc1d37e893994eb2d6a62f8a59e0572db38.jpg) + +
+line + +| Life to failure (Log N) | Stress Amplitude | +| ---------------------- | ---------------- | +| N₁₀₀₀ | S₁₀₀₀ = 0.9 Sᵤ | +| Nₑ | Sₑ' | +| Nₑ | Sₑ (modified - endurance limit) | +
+ +그림 1.4.1 수정계수가 S-N선도에 미치는 영향 + +# 1-4-1 부재크기 및 형상 + +피로 시험에서 다양한 직경을 사용한 경우 재료의 피로한도는 다음과 같은 식에의해서 수정이 된다. + +$$ +C _ {s i z e} = 1. 0 (d \leq 8 m m) \tag {1.4.3} +$$ + +$$ +C _ {s i z e} = 1. 1 8 9 d ^ {- 0. 0 9 7} (8 m m \leq d \leq 2 5 0 m m) \tag {1.4.4} +$$ + +시험편의 단면이 원형이 아니고 각이 진 사각형과 같은 모양의 경우, 등가직경$d _ { e q }$ 는 다음과 같은 식이 계산된다. + +$$ +d _ {e q} ^ {2} = 0. 6 5 w t \tag {1.4.5} +$$ + +여기서, + +w : 사각형 시험편의 너비 + +t : 사각형 시험편의 두께 + + + +# 1-4-2 하중의 종류 + +피로해석의 데이터는 시험편에 대한 반복 굽힘 하중 및 축하중 시험으로부터 얻어진다. 피로시험 데이터를 다른 하중에 의한 피로시험과 연계시킬 때 하중형태에대한 수정계수를 사용할 수 있다. 하중에 의한 영향은 $S _ { 1 0 0 0 }$ 값과 Se 값을 변화시킬 수 있다. $S _ { 1 0 0 0 }$ 의 하중 종류에 대한 수정계수는 다음과 같다. + +표 1.4.1 하중종류에 대한 $S _ { 1 0 0 0 }$ 의 수정계수 + +
Measured LoadingTarget Loading $C_{load}$
Axial toBending1.25
Axial toTorsion0.725
Bending toTorsion0.58
Bending toAxial0.8
Torsion toAxial1.38
Torsion toBending1.72
+ +1000 사이클에서의 $S _ { e }$ 의 하중종류에 대한 수정계수는 다음과 같다. + +표 1.4.2 하중종류에 대한 S 의 수정계수 + +
Measured LoadingTarget Loading $C_{load}$
Axial toTorsion0.82
Bending toTorsion0.82
Torsion toAxial1.22
Torsion toBending1.22
+ + + +# 1-4-3 표면마무리 + +재료 표면 위의 자국, 홈, 기계 가공흔적들은 부재의 기하학적 특성으로 이미 존재하고 있는 응력에 응력집중을 부가시킨다. 고강도강과 같은 균일하며 미세한 결정립으로 구성된 재료는 주철과 같이 조대화된 결정립으로 된 재료보다 거친 표면마무리에 의하여 더 큰 영향을 받는다. 연마(polished), 단조(Forged)와 같은 영향을포함하고 있다. + +다음은 표면의 마무리에 관한 그래프이다. 평균제곱근( $R _ { A }$ , the root meansquare), 산술평균(AA, arithmetic average)와 같은 정량적인 표면 마무리를 나타낸다. 기계적 가공으로 인한 표면마무리는 기계가공 및 제조편람에서 찾을 수 있다. 일반적인 강재의 표면 마무리와 $C _ { s u r }$ (surface factor), Su (tensile Strength)의관계는 그림 1.4.2와 같이 나타난다. + +![](images/page-486_ea42121132e01298efd824c5b79a061a8b5a6d4f790f14ea7bd28f6c77f47317.jpg) + +
+line + +| [Tensile Strength, Su (ksi)] | Surface Factor (μin.) | +| ---------------------------- | --------------------- | +| 40 | 1.0 | +| 60 | 0.95 | +| 80 | 0.9 | +| 100 | 0.85 | +| 120 | 0.8 | +| 140 | 0.75 | +| 160 | 0.7 | +| 180 | 0.65 | +| 200 | 0.6 | +| 220 | 0.55 | +| 240 | 0.5 | +
+ +그림 1.4.2 표면마무리의 영향 + +표면마무리는 고강도 강일수록 더 중요하다. 국부적인 표면의 불규칙성은 응력집중 요소로 작용하기 때문에 피로해석에 나쁜 영향을 미친다. + +# 1-4-4 표면처리 + +여러가지 형상작업으로 인한 표면마무리의 영향 이외에 표면처리 또한 피로수명에영향을 미친다. 표면처리로는 크게 기계적(mechanical)처리, 열(thermal)처리, 도 + + + +금(plating)처리 등이 있으며 표면의 잔류응력을 발생시킬 수 있다. 잔류응력이 존재하는 경우 외력에 의한 변형이 일어날 때 잔류응력이 표면의 인장응력에 영향을미치게 되어 피로수명에 영향을 준다. + +# 기계가공 (mechanical treatment) + +잔류응력이 발생하게 하는 대표적인 표면처리가 냉간압연과 쇼트 피닝(shotpeening)이다. 표면처리에 의해 표면에 가해진 하중으로 표면에 압축잔류응력이발생하여 피로수명이 더 나아지는 결과를 가져온다. + +![](images/page-487_82af530fbafe6da38ad5ba2fe6329fa7930fef5590ad61b497765289d2637620.jpg) + +
+text_image + +σ_max +Tension +M_B +Compression +-σ_max +M_B +
+ +(a) 굽힘에 의한 응력 + +![](images/page-487_e4593ed470ab40e426a0a74bbba2954c37b3a6f32281c96c8b2dcc8f3a6d3ebb.jpg) + +
+text_image + +σ_R +σ_R +
+ +(b) 소성변형에 의한 잔류응력 + +![](images/page-487_15293f9b70aea67f375a95ad34e39a65d5e093be54d319886cc3c294ec893b7e.jpg) + +
+text_image + +M_B +σ_max -σ_R +-σ_max +σ_R +M_B +
+ +(c) 잔류응력에 의한 최종응력 +그림 1.4.3 기계가공에 의한 영향 + +그림 1.4.3은 상단 표면의 휨 응력 값이 감소하는 것을 보여준다. 압연도 같은 효 + + + +과를 낼 수 있으므로 냉간압연으로 만들어진 볼트는 피로의 저항능력이 더 강하다.냉간 압연이나 쇼트 피닝(short peening)이 피로수명에 영향을 주는 것은 피로수명이 장수명일 때이며, 단수명일 경우에는 큰 영향을 주지 못한다. 피닝에 의한 수정계수는 1.5-2.0 사이의 적절한 값을 사용한다. + +# 도금 (plating treatment) + +크롬이나 니켈과 같은 재료로 도금을 하는 것은 피로한계를 향상시킨다. 기계가공처리한 후 도금을 하게 되면, 표면의 압축잔류응력을 감소시켜서 피로에 의한 저항력을 약하게 할 수 있다. + +# 열 (thermal treatment) + +열간 압연과 단조는 표면에 탈탄작용(decarbonization)을 일으킨다. 재료 표면에서 탄소 원자의 손실은 낮은 강도를 갖게 하고, 인장잔류응력을 생성할 수 있기 때문에 피로강도에 불리하게 작용한다. 용접, 연삭 그리고 화염절단과 같은 제조과정은 인장잔류응력을 생성시킬 수 있어 피로한계에 나쁜 영향을 미친다. + +# 1-4-5 온도 + +강재의 피로한계는 낮은 온도에서 증가하는 경향이 있다. 고온에서는 강재의 피로한도가 전위의 이동으로 인하여 나타나지 않는다. 또한 재료융점의 약 절반을 넘는 온도에서 크리이프(creep)가 중요하게 된다. 따라서 이 범위에서는 응력기반의피로수명방법이 더 이상 적용되지 않는다. 고온에서는 풀림(annealing)에 의해 압축잔류응력에 의한 이점이 제거될 수도 있다는 점이 중요하다. + +# 1-4-6 환경 + +부식환경에서 피로하중이 작용할 때에는 피로와 부식이 각각 작용된 경우보다 더해로운 결과를 나타낸다. 부식-피로(corrosion-fatigue)라 불리는 피로와 부식의 + + + +상호작용은 매우 복잡한 파괴기구를 나타낸다. 이 연구는 아직 단순 연구단계에머무르고 있으며 정량적인 자료나 유용한 이론적 방법은 매우 적다. + + + +# 1-5 레인플로집계 + +S-N선도는 일정진폭(constant amplitude)의 반복응력이 작용할 때 피로파괴에 이르게 하는 반복응력의 횟수를 나타내는 선도다. 그러나 실제의 경우 응력은 가변진폭(variable amplitude)의 특성을 보인다. + +가변진폭응력 때의 피로손상을 정의하기 위해서는 가변진폭응력을 여러 개의 일정진폭응력의 조합으로 바꾸어 S-N선도를 적용할 수 있다. + +midas FEA에서는 사이클 집계를 위해서 레인플로집계 방법을 사용한다. 레인플로집계 방법에서는 먼저 다음과 같이 국부적인 최대최소 점을 읽어 들인다. + +$$ +A (i - 1) \leq A (i + 1) \leq A (i) \leq A (i + 2) \tag {1.5.1} +$$ + +$$ +A (i - 1) \geq A (i + 1) \geq A (i) \geq A (i + 2) \tag {1.5.2} +$$ + +그리고 한 주기의 사이클을 계산하여 진폭을 S 라하며, 한 사이클씩 한 주기를 가지는 진폭들을 모두 집계한다. 식 (1.5.1)은 그림 1.5.1의 왼쪽 그림과 같은 형태를가지며, 식 (1.5.2)는 그림 1.5.1의 오른쪽 그림과 같은 형태를 가진다. + +![](images/page-490_911c7e7865312e1c719c52240c1ee049f179718bbcf977fa463bee0ff3fe6188.jpg) + +
+line +| Point | t | σ | +|---------|------|------| +| A(i-1) | 0 | 0 | +| A(i) | 1 | 1 | +| A(i+1) | 1 | 0 | +| A(i+2) | 2 | 1 | +
+ +![](images/page-490_ef3d36f2a05425285bcfd0074b02f72ac886674773d0e65bec6d95f6e1db2fe5.jpg) + +
+line + +| Point | t | σ | +|-------|------|------| +| A(i) | 0 | 0 | +| A(i-1)| 0 | 1 | +| A(i+1)| 1 | 0 | +| A(i+2)| 2 | 0 | +
+ +그림 1.5.1 한 주기의 사이클과 응력진폭 diff --git a/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_050.md b/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_050.md new file mode 100644 index 00000000..0897133e --- /dev/null +++ b/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_050.md @@ -0,0 +1,284 @@ + + +그 다음에는 레인플로집계를 수행하여 최종적으로 점감하거나 점증하는 그림 1.5.2와 같은 로드 사이클을 그린다. 이 때 각각의 S , S , S 은 한 사이클의 주기로본다. 이렇게 얻어진 단위 응력 진폭들에 Miner 법을 적용하면, 피로의 축적에 의한 손상을 고려하는 손상도를 얻을 수 있다. + +![](images/page-491_8f89c7a43a7857628faea375e1c2ac1d10c8e340b13d01a89a1e0475be57f421.jpg) + +
+line + +| t | S1 | S2 | S3 | +| ---- | ---- | ---- | ---- | +| 0 | 0 | 0 | 0 | +| 1 | 1 | 1 | 0 | +| 2 | 0 | 0 | 1 | +| 3 | 1 | 1 | 0 | +| 4 | 0 | 0 | 1 | +| 5 | 1 | 1 | 0 | +| 6 | 0 | 0 | 1 | +| 7 | 1 | 1 | 0 | +| 8 | 0 | 0 | 1 | +| 9 | 1 | 1 | 0 | +| 10 | 0 | 0 | 1 | +| 11 | 1 | 1 | 0 | +| 12 | 0 | 0 | 1 | +| 13 | 1 | 1 | 0 | +| 14 | 0 | 0 | 1 | +| 15 | 1 | 1 | 0 | +| 16 | 0 | 0 | 1 | +| 17 | 1 | 1 | 0 | +| 18 | 0 | 0 | 1 | +| 19 | 1 | 1 | 0 | +| 20 | 0 | 0 | 1 | +| 21 | 1 | 1 | 0 | +| 22 | 0 | 0 | 1 | +| 23 | 1 | 1 | 0 | +| 24 | 0 | 0 | 1 | +| 25 | 1 | 1 | 0 | +| 26 | 0 | 0 | 1 | +| 27 | 1 | 1 | 0 | +| 28 | 0 | 0 | 1 | +| 29 | 1 | 1 | 0 | +| 30 | 0 | 0 | 1 | +| 31 | 1 | 1 | 0 | +| 32 | 0 | 0 | 1 | +| 33 | 1 | 1 | 0 | +| 34 | 0 | 0 | 1 | +| 35 | 1 | 1 | 0 | +| 36 | 0 | 0 | 1 | +| 37 | 1 | 1 | 0 | +| 38 | 0 | 0 | 1 | +| 39 | 1 | 1 | 0 | +| 40 | 0 | 0 | 1 | +| 41 | 1 | 1 | 0 | +| 42 | 0 | 0 | 1 | +| 43 | 1 | 1 | 0 | +| 44 | 0 | 0 | 1 | +| 45 | 1 | 1 | 0 | +| 46 | 0 | 0 | 1 | +| 47 | 1 | 1 | 0 | +| 48 | 0 | 0 | 1 | +| 49 | 1 | 1 | 0 | +| 50 | 0 | 0 | 1 | +| 51 | 1 | 1 | 0 | +| 52 | 0 | 0 | 1 | +| 53 | 1 | 1 | 0 | +| 54 | 0 | 0 | 1 | +| 55 | 1 | 1 | 0 | +| 56 | 0 | 0 | 1 | +| 57 | 1 | 1 | 0 | +| 58 | 0 | 0 | 1 | +| 59 | 1 | 1 | 0 | +| 60 | 0 | 0 | 1 | +| 61 | 1 | 1 | 0 | +| 62 | 0 | 0 | 1 | +| 63 | 1 | 1 | 0 | +| 64 | 0 | 0 | 1 | +| 65 | 1 | 1 | 0 | +| 66 | 0 | 0 | 1 | +| 67 | 1 | 1 | 0 | +| 68 | 0 | 0 | 1 | +| 69 | 1 | 1 | 0 | +| 70 | 0 | 0 | 1 | +| 71 | 1 | 1 | 0 | +| 72 | 0 | 0 | 1 | +| 73 | 1 | 1 | 0 | +| 74 | 0 | 0 | 1 | +| 75 | 1 | 1 | 0 | +| 76 | 0 | 0 | 1 | +| 77 | 1 | 1 | 0 | +| 78 | 0 | 0 | 1 | +| 79 | 1 | 1 | 0 | +| 80 | 0 | 0 | 1 | +| 81 | 1 | 1 | 0 | +| 82 | 0 | 0 | 1 | +| 83 | 1 | 1 | 0 | +| 84 | 0 | 0 | 1 | +| 85 | 1 | 1 | 0 | +| 86 | 0 | 0 | 1 | +| 87 | 1 | 1 | 0 | +| 88 | 0 | 0 | 1 | +| 89 | 1 | 1 | 0 | +| 90 | 0 | 0 | 1 | +| 91 | 1 | 1 | 0 | +| 92 | 0 | 0 | 1 | +| 93 | 1 | 1 | 0 | +| 94 | 0 | 0 | 1 | +| 95 | 1 | 1 | 0 | +| 96 | 0 | 0 | 1 | +| 97 | 1 | 1 | 0 | +| 98 | 0 | 0 | 1 | +| 99 | 1 | 1 | 0 | +| 100 | 0 | 0 | 1 | +
+ +그림 1.5.2 반복하중의 응력진폭집계 + +그림 1.5.3과 같이 응력스펙트럼(stress spectrum)으로 정리할 수 있다. + +![](images/page-491_0ac010d6174ce9a7a24fe02fa30f1dd43e46f5eae18c9f2ee01040e982d4b225.jpg) + +
+line + +| S1 | S2 | S3 | +|----|----|----| +| n1 | n2 | n3 | +| n2 | n3 | n3 | +| n3 | n3 | n3 | +
+ +그림 1.5.3 반복하중의 응력진폭집계 스펙트럼 + +그리고 i 번째 반복하중에서 손상도는 다음과 같이 나타낼 수 있다. + + + +$$ +D _ {i} = \frac {n _ {i}}{N _ {i}} \tag {1.5.3} +$$ + +여기서, + +in : 해당 응력진폭의 반복횟수 + +Ni : 피로파괴까지의 반복횟수 + +손상의 최종 합을 수식으로 정리하면 다음과 같다. + +$$ +D _ {t o t a l} = \sum_ {i = 1} ^ {n} D _ {i} = \sum_ {i = 1} ^ {n} \frac {n _ {i}}{N _ {i}} \tag {1.5.4} +$$ + + + +# 1-6 피로해석의 단계 + +midas FEA 에서는 다음과 같은 순서로 피로해석을 수행한다. + +1. 탄성 응력 데이터를 해석결과로부터 읽어 들인다. +2. 불리한 하중조건 하에서의 절점응력의 절대값을 얻어낸다. +3. 응력 집중계수를 사용하여 탄성 응력을 스케일링한다. +4. 하중의 시간이력그래프를 레인플로집계로 분석해 낸다. +5. 수정계수를 사용해서 S-N선도의 응력진폭 값을 수정한다. +6. 평균응력의 영향을 고려해서 손상정도를 계산한다. +7. Miner 법을 사용하여 손상정도를 선형 합산한다. +8. 모든 절점에서의 피로수명이나 안전계수를 구한다. + + + +Part 8 Fatigue Analysis + +![](images/page-494_b0782609a23a12e9961e773ea3ac8dcaaee7d08e65b1f4913344b18bcc4c8e19.jpg) + +
+natural_image + +Blank white image with no visible content, text, or symbols +
+ + + +# Analysis and Algorithm Manual + +# Part 9 CFD(Computational Fluid Dynamic) Analysis + +Chapter 1. CFD Analysis + + + + + +# Chapter 1. CFD Analysis + +# 1-1 개요 + +전산유체역학(Computational Fluid Dynamics : CFD) 해석은 전산해석 기법을 유체역학에 적용하여 유체의 흐름을 파악하고, 구조물이 바람의 영향을 크게 받는경우 유체에 의한 하중을 예측하는 것이다. 토목구조물의 내풍 안정성 해석에서는주로 설계기준에 근거한 풍하중을 산정하여 반영하고 있으나, 전산유체해석을 통하여 풍하중의 상세한 분포 또는 설계 변경에 의한 풍하중의 변화 등을 예측할 수있다. 특히 교량과 같이 대체로 일정한 단면을 갖는 경우에는 2차원 해석을 통하여 충분한 유동의 정보를 계산할 수 있다. + +midas FEA는 구조화된(structured) 격자에서 2차원 유동을 해석할 수 있다. 2차원 압축성 점성유동을 기본으로 한 Navier-Stokes 방정식으로부터 Favre 평균을취한 RANS(Reynolds averaged Navier-Stokes) 방정식을 계산하며, 이때 사용하는 Favre 평균은 밀도가중(density weighted) 평균이다. 또한 2-방정식 난류모델을 적용할 수 있으며, 정상상태(steady state)와 비정상(unsteady) 상태에 대한 해석을 모두 수행할 수 있다. 전산유체기법으로는 밀도기반(density based) 시간전진법과 유한체적법(finite volume method)을 이용한 공간이산화를 수행한다. + +midas FEA에서 사용할 수 있는 전산유체해석의 경계조건, 난류모델 및 선택사항은 다음과 같다. + +
경계조건원방(far-field) 경계조건
벽면(solid wall) 경계조건
대칭(symmetric) 경계조건
난류모델 $q - \omega$ 모델 : 벽면함수(wall function) 사용가능
$k - \omega$ SST(Shear Stress Transport) 모델
$k - \omega$ BSL(Base Line) 모델
+ + + +# 1-2 RANS 방정식과 난류모델 + +난류 압축성 유동의 지배 방정식인 RANS 방정식과 난류모델은 다음과 같이 하나의 식으로 표현할 수 있다. + +$$ +\frac {\partial \mathbf {W}}{\partial t} + \frac {\partial \mathbf {E}}{\partial x} + \frac {\partial \mathbf {F}}{\partial y} = \frac {\partial \mathbf {E} _ {v}}{\partial x} + \frac {\partial \mathbf {F} _ {v}}{\partial y} + \mathbf {S} \tag {1.2.1} +$$ + +여기서, + +W : 보존형 유동변수벡터 $\left\{ \rho , \rho u , \rho \nu , e , \rho s _ { 1 } , \rho s _ { 2 } \right\} ^ { T }$ + +E 와 F 는 각각 x 방향과 y 방향 비점성(inviscid) 유량(flux) 벡터, $\mathbf { E } _ { \nu }$ 와 $\mathbf { F } _ { \nu } \equiv$ 점성유량벡터이다. + +$$ +\mathbf {E} = \left\{\rho u, \rho u ^ {2} + p, \rho u v, (e + p) u, \rho u s _ {1}, \rho u s _ {2} \right\} ^ {T} \tag {1.2.2} +$$ + +$$ +\mathbf {F} = \left\{\rho v, \rho u v, \rho v ^ {2} + p, (e + p) v, \rho v s _ {1}, \rho v s _ {2} \right\} ^ {T} \tag {1.2.3} +$$ + +$$ +\mathbf {E} _ {v} = \left\{0, \tau_ {x x}, \tau_ {x y}, \Omega_ {x}, (\mu_ {m} + \sigma_ {S 1} \mu_ {t}) \frac {\partial s _ {1}}{\partial x}, (\mu_ {m} + \sigma_ {S 2} \mu_ {t}) \frac {\partial s _ {2}}{\partial x} \right\} ^ {T} \tag {1.2.4} +$$ + +$$ +\mathbf {F} _ {v} = \left\{0, \tau_ {y x}, \tau_ {y y}, \Omega_ {y}, \left(\mu_ {m} + \sigma_ {S 1} \mu_ {t}\right) \frac {\partial s _ {1}}{\partial y}, \left(\mu_ {m} + \sigma_ {S 2} \mu_ {t}\right) \frac {\partial s _ {2}}{\partial y} \right\} ^ {T} \tag {1.2.5} +$$ + +여기서, + +$\rho$ ： + +$p$ : 압력 + +$e$ e : 총 에너지 + +$\tau _ { i j } , \Omega _ { i }$ : 점성응력, 총 에너지유량 + +$\mu _ { m } , \mu _ { t }$ ：， + + + +1s 과 2s 는 2-방정식 난류모델 방정식의 변수이며 난류모델에 따라 그 정의가 달라진다. S 는 난류모델의 원천항(source term)이며 역시 난류모델에 따라 그 정의가 다르다. + +midas FEA에서 사용할 수 있는 난류모델로는 앞서 설명한 바와 같이 Coakley1 의q − ω 모델과 Menter 2 의 k − ω BSL/SST 모델이 있다. q − ω 모델은 난류속도(turbulent velocity) 스케일 q 와 특성 소산율(specific dissipation rate) ω 에 관한 이동방정식을 이용하여 난류점성계수를 예측한다. 여기에서 ω 는 난류운동에너지(turbulence kinetic energy) k , 난류소산율(turbulent dissipation rate) ε 과다음과 같은 관계를 가진다. + +$$ +s _ {1} = q = \sqrt {k}, s _ {2} = \omega = \frac {\varepsilon}{k} \tag {1.2.6} +$$ + +Menter에 의해 개발된 k − ω BSL/SST 모델은 k − ε 모델과 과 k − ω모델의 장점을 결합한 혼합모델이다. 이 모델은 벽면 근처에서 k −ω모델을 사용하고 k −ε 모델을 그 밖의 영역에서 사용하는 난류모델이다. k −ω BSL/SST 모델의 변수는 다음과 같다. + +$$ +s _ {1} = k, s _ {2} = \omega = \frac {\varepsilon}{k} \tag {1.2.7} +$$ + +k − ω BSL 모델은 k − ω 모델과 유사한 특성을 보이지만 자유류(free stream)에대한 영향이 작으며, k −ω SST 모델은 난류전단응력의 전달을 포함하고 있어 역압력구배(adverse pressure gradient)에서 탁월한 성능을 보인다. + +난류 압축성 방정식인 식 (1.2.1)은 낮은 속도의 유동에 대해 해석할 경우 수치적인문제점으로 인하여 수렴된 해를 얻기 힘든 것으로 알려져 있다. 이러한 문제점을해결할 수 있는 방법으로 국소 예조건화 기법(local preconditioning method)이있다. 국소 예조건화 기법은 식 (1.2.1)의 시간항을 원시형변수(primitive variable) + + + +와 예조건화 행렬의 곱으로 표현하여 수렴의 경직성을 해결하는 방법이며 midasFEA에서는 Weiss와 Smith3 의 기법을 적용하였다. 국소 예조건화된 방정식은 다음과 같다. + +$$ +\Gamma \frac {\partial \mathbf {Q}}{\partial t} + \frac {\partial \mathbf {E}}{\partial x} + \frac {\partial \mathbf {F}}{\partial y} = \frac {\partial \mathbf {E} _ {v}}{\partial x} + \frac {\partial \mathbf {F} _ {v}}{\partial y} + \mathbf {S} \tag {1.2.8} +$$ + +여기서, + +Q : 원시형변수벡터 { } 1 2, , , , , Tp u v T s s + +Γ : 예조건화행렬 + +식 (1.2.8)은 정상해의 경우 수렴하면 ∂Q 0 / ∂ =t 이므로 정상해에 대한 예조건화행렬의 영향이 없다. + +midas FEA에서는 RANS 방정식과 난류방정식을 무차원화(non-dimensionalize)하여 계산한다. 무차원화는 자유류 조건을 기준으로 하며 무차원 방정식은 식(1.2.1)과 동일한 형태를 가진다. diff --git a/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_051.md b/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_051.md new file mode 100644 index 00000000..28df9bb4 --- /dev/null +++ b/.raw/MidasFEAAnalysisManual/MidasFEAAnalysisManual_051.md @@ -0,0 +1,212 @@ + + +# 1-3 공간이산화 + +예조건화된 방정식 (1.2.8)은 다음과 같이 간단하게 표현할 수 있다. + +$$ +\boldsymbol {\Gamma} \frac {\partial \mathbf {Q}}{\partial t} + \nabla \cdot \overline {{{\mathbf {F}}}} = \nabla \cdot \overline {{{\mathbf {F}}}} _ {v} + \mathbf {S} \tag {1.3.1} +$$ + +위 식을 그림 1.3.1과 같은 전산셀(computational cell)에 대해 적분하고 확산정리(divergence theorem)을 적용하면 다음과 같이 변환할 수 있다. + +$$ +\Gamma \frac {d}{d t} \int_ {V} \mathbf {Q} d V + \int_ {\Omega} \overline {{\mathbf {F}}} \cdot \mathbf {n} d \Omega = \int_ {\Omega} \overline {{\mathbf {F}}} _ {v} \cdot \mathbf {n} d \Omega + \int_ {V} \mathbf {S} d V \tag {1.3.2} +$$ + +midas FEA 에서는 구조화된 격자를 사용하기 때문에 셀의 배치가 그림 1.3.1과같음에 주의해야 한다. 각 셀의 크기가 작다고 가정하여 셀의 부피와 인접면에 대한 적분을 수행하면 다음과 같이 유한체적법으로 공간이산화한 준 이산화 방정식이 된다. + +$$ +\boldsymbol {\Gamma} \frac {d \mathbf {Q} _ {(i , j)}}{d t} + \mathbf {R} = \mathbf {0} \tag {1.3.3} +$$ + +![](images/page-501_e855280d1856fdd167990f0b9f0f11ec03705144cf377be78beb7b61231f0d8e.jpg) + +
+text_image + +Q_{ij-11} +Q_{ij-11} +Q_{11} +Q_{ij-11} +Q_{10-10} +Q_{10-0} +
+ +그림 1.3.1 전산셀의 배치와 지표(index) + + + +여기서, 잔류량 R 은 인접면에서의 총 유량 $\tilde { \mathbf { F } } _ { c }$ 에 의해 다음과 같이 표현할 수 있다. + +$$ +\mathbf {R} = \frac {1}{V _ {(i , j)}} (\sum \tilde {\mathbf {F}} _ {c} \cdot \mathbf {n} \Delta \Omega) + \mathbf {S} _ {(i, j)} \tag {1.3.4} +$$ + +안정적인 수치해석을 위해 유량벡터 $\tilde { \mathbf { F } } _ { c }$ 에서 비점성항은 ${ \mathsf { R o e } } ^ { 4 } { \underline { { \circ } } } |$ 수치유량벡터로대치하여 계산하며 특성치 수정(entropy correction) 방법을 적용하여 흔들림(wiggle)을 방지한다. 수치유량 계산에 단순한 상류차분(upwind difference)을 사용하게 되면 1차의 정확도만을 얻을 수 있으므로 보다 정확한 해를 얻기 위해 vanLeer5 의 MUSCL 외삽(extrapolation) 기법과 (limiter)를 적용하여 고차의 공간이산화를 얻음과 동시에 단조성(monotone)을 유지할 수 있다. + + + +# 1-4 정상유동 + +정상유동해석에서는 비정상 유동방정식에서 시간항이 0이 되는 Qp 를 구하게 된다. 식(1.2.1)을 시간차분 계수 θ를 이용하여 표현하면 다음과 같다. + +$$ +\Gamma \frac {\Delta \mathbf {Q}}{\Delta \tau} + \theta \mathbf {R} ^ {n + 1} + (1 - \theta) \mathbf {R} ^ {n} = \mathbf {0} \tag {1.4.1} +$$ + +midas FEA에서는 θ =1 을 사용한다. 위 식에서 n+1 R 을 선형화하여 다시 정리하면 다음과 같은 연립방정식을 얻을 수 있다. + +$$ +[ \mathbf {D} + \frac {\Delta \tau}{V} (\mathbf {A} + \mathbf {B}) ] \Delta \mathbf {Q} = - \Delta \tau \mathbf {R} ^ {n} \tag {1.4.2} +$$ + +대각행렬 D 는 다음과 같다. + +$$ +\mathbf {D} = \boldsymbol {\Gamma} - \Delta \tau \mathbf {K} \tag {1.4.3} +$$ + +여기서, + +K : 난류원천항의 자코비안(Jacobian) 행렬 + +A : 셀 경계 i ±1/ 2 에서의 F 자코비안 행렬 + +B : 셀 경계 j ±1/ 2 에서의 F 자코비안 행렬 + +원천항의 자코비안 행렬 K 는 난류모델에 따라 다른 형태를 가지며, midas FEA에서는 안정적인 수치해석을 위하여 원천항 중 감쇄항만을 포함하였다. + +식 (1.4.2)의 해는 AF-ADI(Approximate Factorization-Alternative DirectionImplicit) 방법으로 계산한다. + +$$ +[ \mathbf {D} + \frac {\Delta \tau}{V} \mathbf {A} ] \mathbf {D} ^ {- 1} [ \mathbf {D} + \frac {\Delta \tau}{V} \mathbf {B} ] \Delta \mathbf {Q} = - \Delta \tau \mathbf {R} ^ {n} \tag {1.4.4} +$$ + +위 식은 블록 삼중대각(block tri-diagonal) 행렬이므로 효과적으로 계산할 수 있다. + + + +# 1-5 비정상유동 + +비정상유동해석에서는 AF-ADI 방법을 적용할 때 발생하는 오차를 감소하기 위해개발된 이중시간 적분법6을 사용한다. 시간항에 대한 2차의 정확도를 갖고 “A-Stable” 한 “2-parameter family” 적분법7 에 의한 시간 이산화 방정식은 다음과같다. + +$$ +(1 + \frac {\phi}{2}) \frac {\Delta \mathbf {W} ^ {n}}{\Delta t} - \frac {\phi}{2} \frac {\Delta \mathbf {W} ^ {n - 1}}{\Delta t} + \theta \mathbf {R} ^ {n + 1} + (1 - \theta) \mathbf {R} ^ {n} = \mathbf {0} \tag {1.5.1} +$$ + +midas FEA에서는 φ θ =1, 1 = 을 사용한다. 위 식에 예조건화 행렬을 곱한 가상의시간항을 더하여 이중 시간적분법을 적용하면 다음과 같다. + +$$ +\frac {3}{2} \frac {\Delta \overline {{{\mathbf {W}}}}}{\Delta t} - \frac {1}{2} \frac {\Delta \mathbf {W} ^ {n - 1}}{\Delta t} + \boldsymbol {\Gamma} \frac {\Delta \mathbf {Q} ^ {l}}{\Delta \tau} + \mathbf {R} ^ {l + 1} = \mathbf {0} \tag {1.5.2} +$$ + +$$ +\Delta \overline {{{\mathbf {W}}}} = \mathbf {W} ^ {l + 1} - \mathbf {W} ^ {l} \tag {1.5.3} +$$ + +여기서, + +l : 이중시간의 반복(iteration ) 지표(index) + +n : 시간증분 지표 + +위 식에서 l +1 R 을 선형화하여 다시 정리하면 다음과 같은 연립방정식을 얻을 수있다. + +$$ +[ \mathbf {D} + \frac {\Delta \tau}{V} (\mathbf {A} + \mathbf {B}) ] \Delta \mathbf {Q} = - \Delta \tau \tilde {\mathbf {R}} \tag {1.5.4} +$$ + +대각행렬 D 는 다음과 같다. + + + +$$ +\mathbf {D} = \frac {3}{2} \mathbf {M} \frac {\Delta \tau}{\Delta t} + \mathbf {\Gamma} - \Delta \tau \mathbf {K} \tag {1.5.5.} +$$ + +$$ +\tilde {\mathbf {R}} = \frac {3}{2} \frac {\mathbf {W} ^ {l} - \mathbf {W} ^ {n}}{\Delta t} - \frac {1}{2} \frac {\mathbf {W} ^ {n} - \mathbf {W} ^ {n - 1}}{\Delta t} + \mathbf {R} ^ {l} \tag {1.5.6} +$$ + +여기서, + +R : 수정된 잔류량 + +M : Q 에서 W 로의 변환행렬 + +이중시간 적분법이 수렴하게 되면 R 0 = 이 되고 다음과 같은 관계를 만족하게 된다. + +$$ +\mathbf {W} ^ {l + 1} = \mathbf {W} ^ {l} = \mathbf {W} ^ {n + 1} \tag {1.5.7} +$$ + +식 (1.6.4)는 정상유동해석과 같이 AF-ADI 기법을 이용하여 계산할 수 있다. + + + +# 1-6 수치적 안정성 + +Navier-Stokes 방정식은 대류(convection)와 확산(diffusion)의 성질을 모두 가지고 있기 때문에 가상의 시간증분 ∆τ 을 다음과 같이 계산한다. + +$$ +\frac {1}{\Delta \tau} = \frac {1}{\Delta \tau_ {h}} + \frac {1}{\Delta \tau_ {p}} \tag {1.6.1} +$$ + +대류에 관한 시간증분 h∆τ 는 CFL 수에 의해 조절하며 확산에 관한 시간증분p ∆τ 는 von Neumann 수를 이용하여 조절한다. CFL 수와 von Neumann 수의기본값은 각각 10.0과 5.0 이다. + +수치유량의 비점성항 계산에 사용하는 Roe의 근사 리만해는 수치적인 흔들림(wiggle)을 발생시킬 수 있기 때문에 midas FEA에서는 특성치 수정방법을 사용한다. Roe의 수치유량벡터에 사용되는 수치점성항(numerical viscosity)의 특성치를 + +$$ +\left| \lambda \right| = \left| \lambda \right|, \text { if } \left| \lambda \right| \geq \varepsilon_ {1} +$$ + +$$ +\left| \lambda \right| = \frac {1}{2} \left\{\frac {\left| \lambda \right| ^ {2}}{\varepsilon_ {1}} + \varepsilon_ {1} \right\}, \text { if } \left| \lambda \right| < \varepsilon_ {1} \tag {1.6.2} +$$ + +일반적으로 ε 1 = 0.0 0.25 ∼ 의 값을 사용하며 클수록 해가 소산(dissipative)한 특성을 가진다. 정상유동 해석과 비정상유동 해석에서의 기본값은 각각 0.05와 0.0이다. + +midas FEA 에서는 난류 정상유동 해석에 있어서 수치적 안정성을 고려하여 사용자가 지정한 반복회수만큼을 층류(larminar) 유동으로 가정하여 계산한다. 또한 비정상유동의 안정적인 해석을 위해 초기 유동장을 정상해석 결과로부터 가져오게된다. + + + +# 1-7 전산유체 해석결과 + +midas FEA의 전산유체 해석 결과는 속도, 압력 등의 전산 격자에 대한 결과와 공기력계수(aerodynamic coefficient)가 있다. + +# 격자에 대한 결과 + +속도, 와도 : u v, [ m / sec ] ωz $\left[ \sec ^ { - 1 } \right]$ + +정압, 동압 : $ { p } _ { \mathrm { ~ \normalfont ~ \left. ~ \right.} } \left[ { N } / m ^ { 2 } \right]$ + +난류점성, 점성비 : $\mu _ { { } _ { t u r b } } \quad \ [ \ N \sec / m ^ { 2 } ] \quad \mu _ { { } _ { t u r b } } / \mu$ + +난류에너지, 강도 : $K E _ { t u r b } [ m ^ { 2 } / { \mathrm { s e c } } ^ { 2 } ] u _ { t u r b } / U [ \% ]$ + +# 공기력계수 + +양력계수 : CL $C _ { L }$ + +항력계수 : CD $C _ { D }$ + +모멘트계수 : CM $C _ { M }$ + +속도, 압력과 난류 관련 결과는 양이 많으므로 사용자가 지정한 시간 스텝에 대하여 출력하며, 공기력계수는 매 스텝마다 출력한다. + + + + + +midas FEA 프로그램이 + +한국 토목 구조분야의 기술신장과 대외 기술경쟁력의 확보에 + +다소나마 기여할 수 있기를 바랍니다. + +
강상욱강성규강은식강의영계만수고영현권병천권정우
김경환김규상김근영김동현김문성김상길김영민김영진
김용성김용수김우종김은아김정인김제현김종민 A김종민 B
김지웅김치원남궁계흥남궁용락남기수문정호박시형박영호
박임구박찬영배민제서기흥서충원선종복성실애손성용
신대석신미영심상우심후성양재석양진오염영종염정현
오진상우성운윤장호이 화이민영이샘이이유리이은숙 A
이은숙 B이재훈이정우이종협이창근이창렬이형우이형훈
이혜연이호정이호택정동진정선화정승식정진상정창진
조대현조훈석주민선주영태지영범최병현최성기최원호
하성문함성훈허문석AngshumanMaziar
+ +(가나다 순) + +제 작 : (주)마이다스아이티 MIDAS Information Technology Co.,LTD diff --git a/.raw/MidasFEAAnalysisManual/images/page-001_a124ac6f2e757115af2060d5e8a38ad7a55fc95bb93f1e2251c73b8d9878b134.jpg b/.raw/MidasFEAAnalysisManual/images/page-001_a124ac6f2e757115af2060d5e8a38ad7a55fc95bb93f1e2251c73b8d9878b134.jpg new file mode 100644 index 00000000..09a97e39 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-001_a124ac6f2e757115af2060d5e8a38ad7a55fc95bb93f1e2251c73b8d9878b134.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-004_363cfbdfcba930dd67f13feb91dc8d74259cb82201974bf2df47555a7382b78a.jpg b/.raw/MidasFEAAnalysisManual/images/page-004_363cfbdfcba930dd67f13feb91dc8d74259cb82201974bf2df47555a7382b78a.jpg new file mode 100644 index 00000000..f6f2de69 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-004_363cfbdfcba930dd67f13feb91dc8d74259cb82201974bf2df47555a7382b78a.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-019_258ab7ca72d297f5ce253777162e857f3475aff76812b71fd099fb8c77f13647.jpg b/.raw/MidasFEAAnalysisManual/images/page-019_258ab7ca72d297f5ce253777162e857f3475aff76812b71fd099fb8c77f13647.jpg new file mode 100644 index 00000000..5e30f22d Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-019_258ab7ca72d297f5ce253777162e857f3475aff76812b71fd099fb8c77f13647.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-020_11de2302999e20f698e1310e3be4c4370b8dbaf3fa44f64610155b16c16455b0.jpg b/.raw/MidasFEAAnalysisManual/images/page-020_11de2302999e20f698e1310e3be4c4370b8dbaf3fa44f64610155b16c16455b0.jpg new file mode 100644 index 00000000..a040722f Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-020_11de2302999e20f698e1310e3be4c4370b8dbaf3fa44f64610155b16c16455b0.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-021_f1e9bd7e6df32224fa23432a52688fa7aedb94d6a15e383855d587acf297775a.jpg b/.raw/MidasFEAAnalysisManual/images/page-021_f1e9bd7e6df32224fa23432a52688fa7aedb94d6a15e383855d587acf297775a.jpg new file mode 100644 index 00000000..b46615f9 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-021_f1e9bd7e6df32224fa23432a52688fa7aedb94d6a15e383855d587acf297775a.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-022_21b688d4db91767fdfa42b7ee20a8d6b207b7bb19e6dc6619fb1666181262da8.jpg b/.raw/MidasFEAAnalysisManual/images/page-022_21b688d4db91767fdfa42b7ee20a8d6b207b7bb19e6dc6619fb1666181262da8.jpg new file mode 100644 index 00000000..a3950a9e Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-022_21b688d4db91767fdfa42b7ee20a8d6b207b7bb19e6dc6619fb1666181262da8.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-023_2d7839039ed6e71221dd840c33b4c4be9b89a0b015ee66a27e4c0a0af2723811.jpg b/.raw/MidasFEAAnalysisManual/images/page-023_2d7839039ed6e71221dd840c33b4c4be9b89a0b015ee66a27e4c0a0af2723811.jpg new file mode 100644 index 00000000..91043a38 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-023_2d7839039ed6e71221dd840c33b4c4be9b89a0b015ee66a27e4c0a0af2723811.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-024_2ce31dfd94deb476f1ad31b273be1539a1081bd6b32f40bb0bae1fc93b157695.jpg b/.raw/MidasFEAAnalysisManual/images/page-024_2ce31dfd94deb476f1ad31b273be1539a1081bd6b32f40bb0bae1fc93b157695.jpg new file mode 100644 index 00000000..df39a783 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-024_2ce31dfd94deb476f1ad31b273be1539a1081bd6b32f40bb0bae1fc93b157695.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-025_757f37734eb6d78f3fd5c72c7ddbd71496f7a6fc66e43496564922382ad71c8d.jpg b/.raw/MidasFEAAnalysisManual/images/page-025_757f37734eb6d78f3fd5c72c7ddbd71496f7a6fc66e43496564922382ad71c8d.jpg new file mode 100644 index 00000000..4c338490 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-025_757f37734eb6d78f3fd5c72c7ddbd71496f7a6fc66e43496564922382ad71c8d.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-026_eebe2f4ec855cf914bc23694778fcffe1b239a9fc15d3a71e12bc95642d246e5.jpg b/.raw/MidasFEAAnalysisManual/images/page-026_eebe2f4ec855cf914bc23694778fcffe1b239a9fc15d3a71e12bc95642d246e5.jpg new file mode 100644 index 00000000..9508629c Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-026_eebe2f4ec855cf914bc23694778fcffe1b239a9fc15d3a71e12bc95642d246e5.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-027_029be22b0c7c20522c232c60598e04fb31218049eb7b1694488af93037f4892d.jpg b/.raw/MidasFEAAnalysisManual/images/page-027_029be22b0c7c20522c232c60598e04fb31218049eb7b1694488af93037f4892d.jpg new file mode 100644 index 00000000..3b876cee Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-027_029be22b0c7c20522c232c60598e04fb31218049eb7b1694488af93037f4892d.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-027_d00fcee6db61deae3e3e3f55ba32035767fa0b3f03ba804280661190b41f585b.jpg b/.raw/MidasFEAAnalysisManual/images/page-027_d00fcee6db61deae3e3e3f55ba32035767fa0b3f03ba804280661190b41f585b.jpg new file mode 100644 index 00000000..65ed1930 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-027_d00fcee6db61deae3e3e3f55ba32035767fa0b3f03ba804280661190b41f585b.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-028_55f2558dae8c5cb84c658de7b38d4ebefce6b4696441116045553aa02ccc8e77.jpg b/.raw/MidasFEAAnalysisManual/images/page-028_55f2558dae8c5cb84c658de7b38d4ebefce6b4696441116045553aa02ccc8e77.jpg new file mode 100644 index 00000000..c0df4b84 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-028_55f2558dae8c5cb84c658de7b38d4ebefce6b4696441116045553aa02ccc8e77.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-028_58ba63f3e5fcb0f869987416e7142b9e375a57298ee6eb92f205b533bbd4b09f.jpg b/.raw/MidasFEAAnalysisManual/images/page-028_58ba63f3e5fcb0f869987416e7142b9e375a57298ee6eb92f205b533bbd4b09f.jpg new file mode 100644 index 00000000..7f8c6605 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-028_58ba63f3e5fcb0f869987416e7142b9e375a57298ee6eb92f205b533bbd4b09f.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-028_6605ded1a8d137c4d9e57cafc1c66b127d6540a7fd415a93a104d1da90f71c1b.jpg b/.raw/MidasFEAAnalysisManual/images/page-028_6605ded1a8d137c4d9e57cafc1c66b127d6540a7fd415a93a104d1da90f71c1b.jpg new file mode 100644 index 00000000..30ce6c72 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-028_6605ded1a8d137c4d9e57cafc1c66b127d6540a7fd415a93a104d1da90f71c1b.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-028_6a897b8bdf7e28c0946ca85fd2ddee898d844df15325687339f53e5f6f19bc25.jpg b/.raw/MidasFEAAnalysisManual/images/page-028_6a897b8bdf7e28c0946ca85fd2ddee898d844df15325687339f53e5f6f19bc25.jpg new file mode 100644 index 00000000..88a7314a Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-028_6a897b8bdf7e28c0946ca85fd2ddee898d844df15325687339f53e5f6f19bc25.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-028_9949cfe2814ad62dd2527448bb6a9591980d50b9c1ae5720016dc1865e62f042.jpg b/.raw/MidasFEAAnalysisManual/images/page-028_9949cfe2814ad62dd2527448bb6a9591980d50b9c1ae5720016dc1865e62f042.jpg new file mode 100644 index 00000000..a2fcd391 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-028_9949cfe2814ad62dd2527448bb6a9591980d50b9c1ae5720016dc1865e62f042.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-028_c10a109fdbc0e1b3ed0b55e7dba236009bddf7372babfc9293855e84dae7cee9.jpg b/.raw/MidasFEAAnalysisManual/images/page-028_c10a109fdbc0e1b3ed0b55e7dba236009bddf7372babfc9293855e84dae7cee9.jpg new file mode 100644 index 00000000..28d94148 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-028_c10a109fdbc0e1b3ed0b55e7dba236009bddf7372babfc9293855e84dae7cee9.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-028_dd5332fdc22f02aa7f951e9434259751afc3bf9eb98d617df411c3ac2e0c02af.jpg b/.raw/MidasFEAAnalysisManual/images/page-028_dd5332fdc22f02aa7f951e9434259751afc3bf9eb98d617df411c3ac2e0c02af.jpg new file mode 100644 index 00000000..2d101387 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-028_dd5332fdc22f02aa7f951e9434259751afc3bf9eb98d617df411c3ac2e0c02af.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-028_e38fa8f9975391630552017cc1e9cdd80b070d21ebfc72aff0b7dec671dc06a8.jpg b/.raw/MidasFEAAnalysisManual/images/page-028_e38fa8f9975391630552017cc1e9cdd80b070d21ebfc72aff0b7dec671dc06a8.jpg new file mode 100644 index 00000000..a6073cf5 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-028_e38fa8f9975391630552017cc1e9cdd80b070d21ebfc72aff0b7dec671dc06a8.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-029_00087ec75ca407f063cdfc37840fe6be3849e07cab42b4b4f2d02725343039a9.jpg b/.raw/MidasFEAAnalysisManual/images/page-029_00087ec75ca407f063cdfc37840fe6be3849e07cab42b4b4f2d02725343039a9.jpg new file mode 100644 index 00000000..4a293646 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-029_00087ec75ca407f063cdfc37840fe6be3849e07cab42b4b4f2d02725343039a9.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-029_2e4c57e968ff8b57d890d93d0b36f41b424abd1ca8a72f3bd817dd6b41e6c448.jpg b/.raw/MidasFEAAnalysisManual/images/page-029_2e4c57e968ff8b57d890d93d0b36f41b424abd1ca8a72f3bd817dd6b41e6c448.jpg new file mode 100644 index 00000000..35939bfd Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-029_2e4c57e968ff8b57d890d93d0b36f41b424abd1ca8a72f3bd817dd6b41e6c448.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-029_33f9204a0087d405c92c504c93010068983e635510a7fd5471186be15dd7da27.jpg b/.raw/MidasFEAAnalysisManual/images/page-029_33f9204a0087d405c92c504c93010068983e635510a7fd5471186be15dd7da27.jpg new file mode 100644 index 00000000..d0abc96a Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-029_33f9204a0087d405c92c504c93010068983e635510a7fd5471186be15dd7da27.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-029_3ab07a53c9159625cc675e8a33a0e974db75d1f40666d58c03363a9c2fe3c463.jpg b/.raw/MidasFEAAnalysisManual/images/page-029_3ab07a53c9159625cc675e8a33a0e974db75d1f40666d58c03363a9c2fe3c463.jpg new file mode 100644 index 00000000..4aeb366a Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-029_3ab07a53c9159625cc675e8a33a0e974db75d1f40666d58c03363a9c2fe3c463.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-029_3d4a732ae5e3e6f767c6e95c170bd901fe80811c16457b6e3c29eecfff7488d3.jpg b/.raw/MidasFEAAnalysisManual/images/page-029_3d4a732ae5e3e6f767c6e95c170bd901fe80811c16457b6e3c29eecfff7488d3.jpg new file mode 100644 index 00000000..776627fe Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-029_3d4a732ae5e3e6f767c6e95c170bd901fe80811c16457b6e3c29eecfff7488d3.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-029_7627d9f7c24ce0bcf1dad5bdd2ccb15f67dd3da87200a2d24bce5d5577e87b28.jpg b/.raw/MidasFEAAnalysisManual/images/page-029_7627d9f7c24ce0bcf1dad5bdd2ccb15f67dd3da87200a2d24bce5d5577e87b28.jpg new file mode 100644 index 00000000..7eb89151 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-029_7627d9f7c24ce0bcf1dad5bdd2ccb15f67dd3da87200a2d24bce5d5577e87b28.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-029_e7bf7da485376897060de5904ce8f17816d6c950db4223d772aabe6db44ad9f4.jpg b/.raw/MidasFEAAnalysisManual/images/page-029_e7bf7da485376897060de5904ce8f17816d6c950db4223d772aabe6db44ad9f4.jpg new file mode 100644 index 00000000..3f046d5c Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-029_e7bf7da485376897060de5904ce8f17816d6c950db4223d772aabe6db44ad9f4.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-030_46987c42f2370d2186d5188ac1068ce566eb1d71b4857a3d5cdc45780d7ad0b0.jpg b/.raw/MidasFEAAnalysisManual/images/page-030_46987c42f2370d2186d5188ac1068ce566eb1d71b4857a3d5cdc45780d7ad0b0.jpg new file mode 100644 index 00000000..d8f32587 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-030_46987c42f2370d2186d5188ac1068ce566eb1d71b4857a3d5cdc45780d7ad0b0.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-030_7ab909c4bdfd78381ff3ce6760cf9f1b655428cf2c5e581e9636400fcf0b1bc4.jpg b/.raw/MidasFEAAnalysisManual/images/page-030_7ab909c4bdfd78381ff3ce6760cf9f1b655428cf2c5e581e9636400fcf0b1bc4.jpg new file mode 100644 index 00000000..0f33a0d1 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-030_7ab909c4bdfd78381ff3ce6760cf9f1b655428cf2c5e581e9636400fcf0b1bc4.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-031_44d5a7c712ab2d54adf66903d276612ea70420d5297f4cf2a132a2f27796c3a7.jpg b/.raw/MidasFEAAnalysisManual/images/page-031_44d5a7c712ab2d54adf66903d276612ea70420d5297f4cf2a132a2f27796c3a7.jpg new file mode 100644 index 00000000..d208d51d Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-031_44d5a7c712ab2d54adf66903d276612ea70420d5297f4cf2a132a2f27796c3a7.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-032_8f1bf99880c0df214845e729e97735e3d444702949fd06ca753b62256eb335c7.jpg b/.raw/MidasFEAAnalysisManual/images/page-032_8f1bf99880c0df214845e729e97735e3d444702949fd06ca753b62256eb335c7.jpg new file mode 100644 index 00000000..6e716442 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-032_8f1bf99880c0df214845e729e97735e3d444702949fd06ca753b62256eb335c7.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-032_d8795f79cd92d68729a4e7b013f31aaa1f11b8d4af9a040bc43c312e0eb2ff50.jpg b/.raw/MidasFEAAnalysisManual/images/page-032_d8795f79cd92d68729a4e7b013f31aaa1f11b8d4af9a040bc43c312e0eb2ff50.jpg new file mode 100644 index 00000000..91e63dd3 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-032_d8795f79cd92d68729a4e7b013f31aaa1f11b8d4af9a040bc43c312e0eb2ff50.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-032_d90885fb57d8337ff3803daf3916a59ea1bdb0c5eb0154cad90a14bb7427dd30.jpg b/.raw/MidasFEAAnalysisManual/images/page-032_d90885fb57d8337ff3803daf3916a59ea1bdb0c5eb0154cad90a14bb7427dd30.jpg new file mode 100644 index 00000000..92c86886 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-032_d90885fb57d8337ff3803daf3916a59ea1bdb0c5eb0154cad90a14bb7427dd30.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-032_e171529b15b9c8908f04655e41811bdbe62ea921b2dfafebccebc15308126067.jpg b/.raw/MidasFEAAnalysisManual/images/page-032_e171529b15b9c8908f04655e41811bdbe62ea921b2dfafebccebc15308126067.jpg new file mode 100644 index 00000000..5f0c804e Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-032_e171529b15b9c8908f04655e41811bdbe62ea921b2dfafebccebc15308126067.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-033_c0f6316ee0c9c54b5cb7b5c41133803694a69a98abf266eaafea6d58a0357301.jpg b/.raw/MidasFEAAnalysisManual/images/page-033_c0f6316ee0c9c54b5cb7b5c41133803694a69a98abf266eaafea6d58a0357301.jpg new file mode 100644 index 00000000..ee51d65e Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-033_c0f6316ee0c9c54b5cb7b5c41133803694a69a98abf266eaafea6d58a0357301.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-034_0819fd808d1cf870d3db1f248e1606efd35846377c0a93173b46824aec1cc20a.jpg b/.raw/MidasFEAAnalysisManual/images/page-034_0819fd808d1cf870d3db1f248e1606efd35846377c0a93173b46824aec1cc20a.jpg new file mode 100644 index 00000000..54418664 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-034_0819fd808d1cf870d3db1f248e1606efd35846377c0a93173b46824aec1cc20a.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-034_4a284522372b8626a91421f7dbb7cd5116e82cfb6af163700acfb707f21f2839.jpg b/.raw/MidasFEAAnalysisManual/images/page-034_4a284522372b8626a91421f7dbb7cd5116e82cfb6af163700acfb707f21f2839.jpg new file mode 100644 index 00000000..dfc3bbbc Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-034_4a284522372b8626a91421f7dbb7cd5116e82cfb6af163700acfb707f21f2839.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-034_a67da1b53a0521590c75a2886841420254feb276b34b60df3c781b7a69cdfe27.jpg b/.raw/MidasFEAAnalysisManual/images/page-034_a67da1b53a0521590c75a2886841420254feb276b34b60df3c781b7a69cdfe27.jpg new file mode 100644 index 00000000..9ee0e0b7 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-034_a67da1b53a0521590c75a2886841420254feb276b34b60df3c781b7a69cdfe27.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-034_cba36d96decf719618eb408c5b8a3dcc1dd86fd07a6dcf5e0d0c91442077a281.jpg b/.raw/MidasFEAAnalysisManual/images/page-034_cba36d96decf719618eb408c5b8a3dcc1dd86fd07a6dcf5e0d0c91442077a281.jpg new file mode 100644 index 00000000..19d54307 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-034_cba36d96decf719618eb408c5b8a3dcc1dd86fd07a6dcf5e0d0c91442077a281.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-035_3c453cd700412e1d03a95d9af34dfd4a08947a5accac9df225c0a5f52e298ffa.jpg b/.raw/MidasFEAAnalysisManual/images/page-035_3c453cd700412e1d03a95d9af34dfd4a08947a5accac9df225c0a5f52e298ffa.jpg new file mode 100644 index 00000000..ec6f0908 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-035_3c453cd700412e1d03a95d9af34dfd4a08947a5accac9df225c0a5f52e298ffa.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-035_b1c0ae61f3073d01b141f726258867b07ff542ec4934d5151acd21845008ca3c.jpg b/.raw/MidasFEAAnalysisManual/images/page-035_b1c0ae61f3073d01b141f726258867b07ff542ec4934d5151acd21845008ca3c.jpg new file mode 100644 index 00000000..83a6cdbe Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-035_b1c0ae61f3073d01b141f726258867b07ff542ec4934d5151acd21845008ca3c.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-036_2240beec2469d893f6dd366067fa1660de1733034e6d97dd912a02d0292faf94.jpg b/.raw/MidasFEAAnalysisManual/images/page-036_2240beec2469d893f6dd366067fa1660de1733034e6d97dd912a02d0292faf94.jpg new file mode 100644 index 00000000..0c0a0be2 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-036_2240beec2469d893f6dd366067fa1660de1733034e6d97dd912a02d0292faf94.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-036_349510807d78380ba812bd17e09d7c78fb232ed41f2bdd22586bcdc0748ba5c5.jpg b/.raw/MidasFEAAnalysisManual/images/page-036_349510807d78380ba812bd17e09d7c78fb232ed41f2bdd22586bcdc0748ba5c5.jpg new file mode 100644 index 00000000..50132e69 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-036_349510807d78380ba812bd17e09d7c78fb232ed41f2bdd22586bcdc0748ba5c5.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-036_588657910da9dcd72d53aac5bdec32d0efcde4331f7181b0e31c16b9203f5864.jpg b/.raw/MidasFEAAnalysisManual/images/page-036_588657910da9dcd72d53aac5bdec32d0efcde4331f7181b0e31c16b9203f5864.jpg new file mode 100644 index 00000000..608741f1 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-036_588657910da9dcd72d53aac5bdec32d0efcde4331f7181b0e31c16b9203f5864.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-036_7fd67bd8eab089258dce34a433c2d2e9c7c45bcc5d071f9ea534df00c21fa542.jpg b/.raw/MidasFEAAnalysisManual/images/page-036_7fd67bd8eab089258dce34a433c2d2e9c7c45bcc5d071f9ea534df00c21fa542.jpg new file mode 100644 index 00000000..cd80ad7a Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-036_7fd67bd8eab089258dce34a433c2d2e9c7c45bcc5d071f9ea534df00c21fa542.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-036_bce50900a69e9ece1e274c8b69233a63434f14108d9eb26b00a057a839e5fa46.jpg b/.raw/MidasFEAAnalysisManual/images/page-036_bce50900a69e9ece1e274c8b69233a63434f14108d9eb26b00a057a839e5fa46.jpg new file mode 100644 index 00000000..a065d267 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-036_bce50900a69e9ece1e274c8b69233a63434f14108d9eb26b00a057a839e5fa46.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-036_f045b68b887647545a141c6fab5e1cfe48354ac50bb228e671ad69c56374f07b.jpg b/.raw/MidasFEAAnalysisManual/images/page-036_f045b68b887647545a141c6fab5e1cfe48354ac50bb228e671ad69c56374f07b.jpg new file mode 100644 index 00000000..51ab98fa Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-036_f045b68b887647545a141c6fab5e1cfe48354ac50bb228e671ad69c56374f07b.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-037_43a75fe8f60056b66a2ce97456eb86cf2dc4590564fe38198f1759671828129f.jpg b/.raw/MidasFEAAnalysisManual/images/page-037_43a75fe8f60056b66a2ce97456eb86cf2dc4590564fe38198f1759671828129f.jpg new file mode 100644 index 00000000..d31345d0 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-037_43a75fe8f60056b66a2ce97456eb86cf2dc4590564fe38198f1759671828129f.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-037_6638277f1d43f7cdfbeb05a0be6910662a8d4a81fcc07408d8ab429cfc2b499f.jpg b/.raw/MidasFEAAnalysisManual/images/page-037_6638277f1d43f7cdfbeb05a0be6910662a8d4a81fcc07408d8ab429cfc2b499f.jpg new file mode 100644 index 00000000..84528ea3 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-037_6638277f1d43f7cdfbeb05a0be6910662a8d4a81fcc07408d8ab429cfc2b499f.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-037_95d024acc66c370414de0021bf7ebbe2c0bf5d8e0f33718b84cec284440c2812.jpg b/.raw/MidasFEAAnalysisManual/images/page-037_95d024acc66c370414de0021bf7ebbe2c0bf5d8e0f33718b84cec284440c2812.jpg new file mode 100644 index 00000000..d8f4824f Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-037_95d024acc66c370414de0021bf7ebbe2c0bf5d8e0f33718b84cec284440c2812.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-037_e284648c9397b5cd5488cbc4cf82985a041d064bd1ce5299ea6f9f4d7e23b785.jpg b/.raw/MidasFEAAnalysisManual/images/page-037_e284648c9397b5cd5488cbc4cf82985a041d064bd1ce5299ea6f9f4d7e23b785.jpg new file mode 100644 index 00000000..666fc094 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-037_e284648c9397b5cd5488cbc4cf82985a041d064bd1ce5299ea6f9f4d7e23b785.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-037_f262cb989711d4e67df47ef567e26302d403602914a74d65e6cd1af726d1dd0a.jpg b/.raw/MidasFEAAnalysisManual/images/page-037_f262cb989711d4e67df47ef567e26302d403602914a74d65e6cd1af726d1dd0a.jpg new file mode 100644 index 00000000..fff67e41 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-037_f262cb989711d4e67df47ef567e26302d403602914a74d65e6cd1af726d1dd0a.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-037_fb8e09c0d709807462bc46bec3be3365df8ef99e3a53bc00052e22444a504a5c.jpg b/.raw/MidasFEAAnalysisManual/images/page-037_fb8e09c0d709807462bc46bec3be3365df8ef99e3a53bc00052e22444a504a5c.jpg new file mode 100644 index 00000000..ad07903c Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-037_fb8e09c0d709807462bc46bec3be3365df8ef99e3a53bc00052e22444a504a5c.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-037_fe36b1c48d47060daf52a064f1945bee52152281ffa0afc053ea74bf3918e3d5.jpg b/.raw/MidasFEAAnalysisManual/images/page-037_fe36b1c48d47060daf52a064f1945bee52152281ffa0afc053ea74bf3918e3d5.jpg new file mode 100644 index 00000000..ae18bd0e Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-037_fe36b1c48d47060daf52a064f1945bee52152281ffa0afc053ea74bf3918e3d5.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-038_4243a85582002e6df86b43f846e999cf1f4d52bc63c188dbcdd564a33119b837.jpg b/.raw/MidasFEAAnalysisManual/images/page-038_4243a85582002e6df86b43f846e999cf1f4d52bc63c188dbcdd564a33119b837.jpg new file mode 100644 index 00000000..05c8e93f Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-038_4243a85582002e6df86b43f846e999cf1f4d52bc63c188dbcdd564a33119b837.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-038_4572d585ce81df700857fa25e728be3bf01b1b0b1f86af5ebad394a1dd3cf2b6.jpg b/.raw/MidasFEAAnalysisManual/images/page-038_4572d585ce81df700857fa25e728be3bf01b1b0b1f86af5ebad394a1dd3cf2b6.jpg new file mode 100644 index 00000000..3db1f526 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-038_4572d585ce81df700857fa25e728be3bf01b1b0b1f86af5ebad394a1dd3cf2b6.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-038_515ede14fee57449fc17aedae75e1ffb94a76e605e4c2f087e6092f2aaae524e.jpg b/.raw/MidasFEAAnalysisManual/images/page-038_515ede14fee57449fc17aedae75e1ffb94a76e605e4c2f087e6092f2aaae524e.jpg new file mode 100644 index 00000000..76ebbf7e Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-038_515ede14fee57449fc17aedae75e1ffb94a76e605e4c2f087e6092f2aaae524e.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-038_5ba63d63b2096b5cfa789c3541b3a4150a88936c97e668c3cdbc74db52ed240e.jpg b/.raw/MidasFEAAnalysisManual/images/page-038_5ba63d63b2096b5cfa789c3541b3a4150a88936c97e668c3cdbc74db52ed240e.jpg new file mode 100644 index 00000000..c73cb6fb Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-038_5ba63d63b2096b5cfa789c3541b3a4150a88936c97e668c3cdbc74db52ed240e.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-038_e2a6261f6a2fbc257a13b4d3fb804356dc20282129f94dc7ebe6b0f1d911e2b7.jpg b/.raw/MidasFEAAnalysisManual/images/page-038_e2a6261f6a2fbc257a13b4d3fb804356dc20282129f94dc7ebe6b0f1d911e2b7.jpg new file mode 100644 index 00000000..5dbd5d6e Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-038_e2a6261f6a2fbc257a13b4d3fb804356dc20282129f94dc7ebe6b0f1d911e2b7.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-038_ea9978dcd55059c98bf8a3d4c16e9ae4431d3ce0848edcd16c657722282e1bc6.jpg b/.raw/MidasFEAAnalysisManual/images/page-038_ea9978dcd55059c98bf8a3d4c16e9ae4431d3ce0848edcd16c657722282e1bc6.jpg new file mode 100644 index 00000000..a3b8d7d3 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-038_ea9978dcd55059c98bf8a3d4c16e9ae4431d3ce0848edcd16c657722282e1bc6.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-038_fac7b9793010ad4abad1f1e92032ad2ef411cfa36e4a0442d531b95bad47a406.jpg b/.raw/MidasFEAAnalysisManual/images/page-038_fac7b9793010ad4abad1f1e92032ad2ef411cfa36e4a0442d531b95bad47a406.jpg new file mode 100644 index 00000000..94e971bb Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-038_fac7b9793010ad4abad1f1e92032ad2ef411cfa36e4a0442d531b95bad47a406.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-039_739cd84637104811dcabb8a7d79c1dd60885eb7065ab97969659c4e2320ebf0e.jpg b/.raw/MidasFEAAnalysisManual/images/page-039_739cd84637104811dcabb8a7d79c1dd60885eb7065ab97969659c4e2320ebf0e.jpg new file mode 100644 index 00000000..73b36a72 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-039_739cd84637104811dcabb8a7d79c1dd60885eb7065ab97969659c4e2320ebf0e.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-040_01569af719e74e1b00af59dd8da04aadfb71796ceacff756fe3ec2d187de1636.jpg b/.raw/MidasFEAAnalysisManual/images/page-040_01569af719e74e1b00af59dd8da04aadfb71796ceacff756fe3ec2d187de1636.jpg new file mode 100644 index 00000000..426c3a98 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-040_01569af719e74e1b00af59dd8da04aadfb71796ceacff756fe3ec2d187de1636.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-040_4a6b85ff7490b350acfeefc6c726157e13eac77bac415abadf4e51a2e43f4cb6.jpg b/.raw/MidasFEAAnalysisManual/images/page-040_4a6b85ff7490b350acfeefc6c726157e13eac77bac415abadf4e51a2e43f4cb6.jpg new file mode 100644 index 00000000..5d3474b8 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-040_4a6b85ff7490b350acfeefc6c726157e13eac77bac415abadf4e51a2e43f4cb6.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-041_078b73fa20722b109a470ef2d125b42d71b64b9e40b802baaab9f339a66c3ea9.jpg b/.raw/MidasFEAAnalysisManual/images/page-041_078b73fa20722b109a470ef2d125b42d71b64b9e40b802baaab9f339a66c3ea9.jpg new file mode 100644 index 00000000..5aa54c24 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-041_078b73fa20722b109a470ef2d125b42d71b64b9e40b802baaab9f339a66c3ea9.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-041_2efc496fcce6253a1801368d3246fdb15b48cdc553d311ac99f59dff1b1fed79.jpg b/.raw/MidasFEAAnalysisManual/images/page-041_2efc496fcce6253a1801368d3246fdb15b48cdc553d311ac99f59dff1b1fed79.jpg new file mode 100644 index 00000000..ab61e0ce Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-041_2efc496fcce6253a1801368d3246fdb15b48cdc553d311ac99f59dff1b1fed79.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-041_5911aae7247a4b41fc1d59b11e7724fc9e8c9e2c0316e08415dc54a73dd1e579.jpg b/.raw/MidasFEAAnalysisManual/images/page-041_5911aae7247a4b41fc1d59b11e7724fc9e8c9e2c0316e08415dc54a73dd1e579.jpg new file mode 100644 index 00000000..f7d54b3a Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-041_5911aae7247a4b41fc1d59b11e7724fc9e8c9e2c0316e08415dc54a73dd1e579.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-041_93ffefcbd94e5716532124c74041ce20bb6081d4a847691a474bf71dc9fd4c85.jpg b/.raw/MidasFEAAnalysisManual/images/page-041_93ffefcbd94e5716532124c74041ce20bb6081d4a847691a474bf71dc9fd4c85.jpg new file mode 100644 index 00000000..bc4d414f Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-041_93ffefcbd94e5716532124c74041ce20bb6081d4a847691a474bf71dc9fd4c85.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-041_cd9ed6d3fd427498b17388992db8f28c93bd0f3e8b1c45ae3d699bb26ee25a33.jpg b/.raw/MidasFEAAnalysisManual/images/page-041_cd9ed6d3fd427498b17388992db8f28c93bd0f3e8b1c45ae3d699bb26ee25a33.jpg new file mode 100644 index 00000000..7bbee9f4 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-041_cd9ed6d3fd427498b17388992db8f28c93bd0f3e8b1c45ae3d699bb26ee25a33.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-041_d60c22822dee219f429e5e4c0b9ed9632e83e1ca04f931f7f4b89c84e43e7305.jpg b/.raw/MidasFEAAnalysisManual/images/page-041_d60c22822dee219f429e5e4c0b9ed9632e83e1ca04f931f7f4b89c84e43e7305.jpg new file mode 100644 index 00000000..784ee087 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-041_d60c22822dee219f429e5e4c0b9ed9632e83e1ca04f931f7f4b89c84e43e7305.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-041_d92b075eb35762e44855f2bd2d69dae2ca05cbb1b1d653a72833e8ec9d15b992.jpg b/.raw/MidasFEAAnalysisManual/images/page-041_d92b075eb35762e44855f2bd2d69dae2ca05cbb1b1d653a72833e8ec9d15b992.jpg new file mode 100644 index 00000000..ea9cca23 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-041_d92b075eb35762e44855f2bd2d69dae2ca05cbb1b1d653a72833e8ec9d15b992.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-042_98ac02bfad877d9c28a35c89b45da6a555207f5e8185680affd6213af145840a.jpg b/.raw/MidasFEAAnalysisManual/images/page-042_98ac02bfad877d9c28a35c89b45da6a555207f5e8185680affd6213af145840a.jpg new file mode 100644 index 00000000..0d4b3dd5 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-042_98ac02bfad877d9c28a35c89b45da6a555207f5e8185680affd6213af145840a.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-043_4d9cf0badba6b44d133c2756dbf402863c17e98dfabc6cf78f991fe2f56013de.jpg b/.raw/MidasFEAAnalysisManual/images/page-043_4d9cf0badba6b44d133c2756dbf402863c17e98dfabc6cf78f991fe2f56013de.jpg new file mode 100644 index 00000000..7c3e6230 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-043_4d9cf0badba6b44d133c2756dbf402863c17e98dfabc6cf78f991fe2f56013de.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-043_79745a40c065cb80b824051bada35d5e8a94d3c763e8841906f0a82e93a4cfbe.jpg b/.raw/MidasFEAAnalysisManual/images/page-043_79745a40c065cb80b824051bada35d5e8a94d3c763e8841906f0a82e93a4cfbe.jpg new file mode 100644 index 00000000..f5eacd23 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-043_79745a40c065cb80b824051bada35d5e8a94d3c763e8841906f0a82e93a4cfbe.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-043_b268c15508015868b130dae396fa2ad32205bed28cde91a8c1be5e0fd31589e4.jpg b/.raw/MidasFEAAnalysisManual/images/page-043_b268c15508015868b130dae396fa2ad32205bed28cde91a8c1be5e0fd31589e4.jpg new file mode 100644 index 00000000..6c7ea11c Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-043_b268c15508015868b130dae396fa2ad32205bed28cde91a8c1be5e0fd31589e4.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-044_fa9720b0eb3b79785a236e28dd68746799aff02baefbfbd565cb7158b52a3b52.jpg b/.raw/MidasFEAAnalysisManual/images/page-044_fa9720b0eb3b79785a236e28dd68746799aff02baefbfbd565cb7158b52a3b52.jpg new file mode 100644 index 00000000..6f4e631d Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-044_fa9720b0eb3b79785a236e28dd68746799aff02baefbfbd565cb7158b52a3b52.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-045_267359d37862d7954b69b6eb052afcd76c9917be6e6483b312a4b460ca46fd42.jpg b/.raw/MidasFEAAnalysisManual/images/page-045_267359d37862d7954b69b6eb052afcd76c9917be6e6483b312a4b460ca46fd42.jpg new file mode 100644 index 00000000..c6067925 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-045_267359d37862d7954b69b6eb052afcd76c9917be6e6483b312a4b460ca46fd42.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-045_2bcac7a7658c40f3b9975952f5daec4b1e574ec3c3fb4a6fbb3c69e89068e1b0.jpg b/.raw/MidasFEAAnalysisManual/images/page-045_2bcac7a7658c40f3b9975952f5daec4b1e574ec3c3fb4a6fbb3c69e89068e1b0.jpg new file mode 100644 index 00000000..34237c0f Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-045_2bcac7a7658c40f3b9975952f5daec4b1e574ec3c3fb4a6fbb3c69e89068e1b0.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-045_44a21e3fc9f4676d0bc38d7135b8fa9f5dbccc803585e7a7c0a9d6f2dc01c09d.jpg b/.raw/MidasFEAAnalysisManual/images/page-045_44a21e3fc9f4676d0bc38d7135b8fa9f5dbccc803585e7a7c0a9d6f2dc01c09d.jpg new file mode 100644 index 00000000..69166743 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-045_44a21e3fc9f4676d0bc38d7135b8fa9f5dbccc803585e7a7c0a9d6f2dc01c09d.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-045_83c253f888f6e4b32b1f2ebe8485c8e00d3da9383c3129d95a7eacd90547a76d.jpg b/.raw/MidasFEAAnalysisManual/images/page-045_83c253f888f6e4b32b1f2ebe8485c8e00d3da9383c3129d95a7eacd90547a76d.jpg new file mode 100644 index 00000000..f32be5fe Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-045_83c253f888f6e4b32b1f2ebe8485c8e00d3da9383c3129d95a7eacd90547a76d.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-045_fcd013f8aeb0749c0ed2d5bbf65fd718f7c766f918ef1f95a3814bede4cbe6e3.jpg b/.raw/MidasFEAAnalysisManual/images/page-045_fcd013f8aeb0749c0ed2d5bbf65fd718f7c766f918ef1f95a3814bede4cbe6e3.jpg new file mode 100644 index 00000000..f06399fa Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-045_fcd013f8aeb0749c0ed2d5bbf65fd718f7c766f918ef1f95a3814bede4cbe6e3.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-046_b44a70a0f00874ad1967653242fa8ba840bed9d1b91ed11483cec21ff69cb95b.jpg b/.raw/MidasFEAAnalysisManual/images/page-046_b44a70a0f00874ad1967653242fa8ba840bed9d1b91ed11483cec21ff69cb95b.jpg new file mode 100644 index 00000000..acce365b Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-046_b44a70a0f00874ad1967653242fa8ba840bed9d1b91ed11483cec21ff69cb95b.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-047_157f9a49327b73ac9fda7556dbf8fb59c8229a039b0a0d263da7e49b6b530f0b.jpg b/.raw/MidasFEAAnalysisManual/images/page-047_157f9a49327b73ac9fda7556dbf8fb59c8229a039b0a0d263da7e49b6b530f0b.jpg new file mode 100644 index 00000000..3e5332df Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-047_157f9a49327b73ac9fda7556dbf8fb59c8229a039b0a0d263da7e49b6b530f0b.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-048_e971f5a81d5a43516d32e0cbd7895bfd15bf8a68ea0c0041ea8bb90e97fb7801.jpg b/.raw/MidasFEAAnalysisManual/images/page-048_e971f5a81d5a43516d32e0cbd7895bfd15bf8a68ea0c0041ea8bb90e97fb7801.jpg new file mode 100644 index 00000000..d59f6576 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-048_e971f5a81d5a43516d32e0cbd7895bfd15bf8a68ea0c0041ea8bb90e97fb7801.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-048_ee2e3278a4ab7fc153531d151558a57a423fc2550142baec46b9a225fd7261b0.jpg b/.raw/MidasFEAAnalysisManual/images/page-048_ee2e3278a4ab7fc153531d151558a57a423fc2550142baec46b9a225fd7261b0.jpg new file mode 100644 index 00000000..93d3e5eb Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-048_ee2e3278a4ab7fc153531d151558a57a423fc2550142baec46b9a225fd7261b0.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-049_0b5e32feb99400ec9229ead7b0947bab995c575ea0700360d7bcb589612abc85.jpg b/.raw/MidasFEAAnalysisManual/images/page-049_0b5e32feb99400ec9229ead7b0947bab995c575ea0700360d7bcb589612abc85.jpg new file mode 100644 index 00000000..7291635e Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-049_0b5e32feb99400ec9229ead7b0947bab995c575ea0700360d7bcb589612abc85.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-049_7345ce977e36acbd3d559fed2bb3a0be000d27442a4483a39e473bf679216dfb.jpg b/.raw/MidasFEAAnalysisManual/images/page-049_7345ce977e36acbd3d559fed2bb3a0be000d27442a4483a39e473bf679216dfb.jpg new file mode 100644 index 00000000..b0668bda Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-049_7345ce977e36acbd3d559fed2bb3a0be000d27442a4483a39e473bf679216dfb.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-050_0107250c2922f939936a0525167b4fcd82a83c328d5c8abc89353277d142f74f.jpg b/.raw/MidasFEAAnalysisManual/images/page-050_0107250c2922f939936a0525167b4fcd82a83c328d5c8abc89353277d142f74f.jpg new file mode 100644 index 00000000..bc6dd4f4 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-050_0107250c2922f939936a0525167b4fcd82a83c328d5c8abc89353277d142f74f.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-050_073c8bab400442948daed95fe5e4a8a522520353a139af888e6392cb029d192f.jpg b/.raw/MidasFEAAnalysisManual/images/page-050_073c8bab400442948daed95fe5e4a8a522520353a139af888e6392cb029d192f.jpg new file mode 100644 index 00000000..94e03b33 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-050_073c8bab400442948daed95fe5e4a8a522520353a139af888e6392cb029d192f.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-051_28daafa3b29da5f5edebc735ff6bb62d4c7c34e9861669e6c6c3bf4b595baa9b.jpg b/.raw/MidasFEAAnalysisManual/images/page-051_28daafa3b29da5f5edebc735ff6bb62d4c7c34e9861669e6c6c3bf4b595baa9b.jpg new file mode 100644 index 00000000..5afcd120 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-051_28daafa3b29da5f5edebc735ff6bb62d4c7c34e9861669e6c6c3bf4b595baa9b.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-051_3e17a04464c2e63c3a70b5fd20f6451fc5dd8d9f10cb15e2c4f0a74df5363186.jpg b/.raw/MidasFEAAnalysisManual/images/page-051_3e17a04464c2e63c3a70b5fd20f6451fc5dd8d9f10cb15e2c4f0a74df5363186.jpg new file mode 100644 index 00000000..eea99b6d Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-051_3e17a04464c2e63c3a70b5fd20f6451fc5dd8d9f10cb15e2c4f0a74df5363186.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-051_7a3dfe3a7e951864fefbcadfc425b8640525ae4edc5f84b695dc43c7c193365d.jpg b/.raw/MidasFEAAnalysisManual/images/page-051_7a3dfe3a7e951864fefbcadfc425b8640525ae4edc5f84b695dc43c7c193365d.jpg new file mode 100644 index 00000000..b99f22a1 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-051_7a3dfe3a7e951864fefbcadfc425b8640525ae4edc5f84b695dc43c7c193365d.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-051_be5b0a8f0ecf490f89c1d49b53ca6aacb2ae02f74e5ee6f9e203a8f3f290aff1.jpg b/.raw/MidasFEAAnalysisManual/images/page-051_be5b0a8f0ecf490f89c1d49b53ca6aacb2ae02f74e5ee6f9e203a8f3f290aff1.jpg new file mode 100644 index 00000000..43ddfb24 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-051_be5b0a8f0ecf490f89c1d49b53ca6aacb2ae02f74e5ee6f9e203a8f3f290aff1.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-051_c19df79b085fc5a0c2d608e741f0b585cfe76cb8d140d2d26a84e41e81144680.jpg b/.raw/MidasFEAAnalysisManual/images/page-051_c19df79b085fc5a0c2d608e741f0b585cfe76cb8d140d2d26a84e41e81144680.jpg new file mode 100644 index 00000000..c504b41c Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-051_c19df79b085fc5a0c2d608e741f0b585cfe76cb8d140d2d26a84e41e81144680.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-051_da96f313a008a44c98b35738c1309f61f2e01ded810978b0ab3de94e26b099d1.jpg b/.raw/MidasFEAAnalysisManual/images/page-051_da96f313a008a44c98b35738c1309f61f2e01ded810978b0ab3de94e26b099d1.jpg new file mode 100644 index 00000000..b90ee33d Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-051_da96f313a008a44c98b35738c1309f61f2e01ded810978b0ab3de94e26b099d1.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-051_dd3ad91a3618cf72fb863e5a3a2942b3cbac0152889a610f3b21b2a106b68a0c.jpg b/.raw/MidasFEAAnalysisManual/images/page-051_dd3ad91a3618cf72fb863e5a3a2942b3cbac0152889a610f3b21b2a106b68a0c.jpg new file mode 100644 index 00000000..5dce2ae3 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-051_dd3ad91a3618cf72fb863e5a3a2942b3cbac0152889a610f3b21b2a106b68a0c.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-052_331c21ae39c4e6be1fc9bda4167f7f0e8658db08d1e17a9c8fe6d37891451043.jpg b/.raw/MidasFEAAnalysisManual/images/page-052_331c21ae39c4e6be1fc9bda4167f7f0e8658db08d1e17a9c8fe6d37891451043.jpg new file mode 100644 index 00000000..04da2266 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-052_331c21ae39c4e6be1fc9bda4167f7f0e8658db08d1e17a9c8fe6d37891451043.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-052_5679fe6a9f1cb3e7465931fe96e3825785194b9036478da7068a2c867b9d732c.jpg b/.raw/MidasFEAAnalysisManual/images/page-052_5679fe6a9f1cb3e7465931fe96e3825785194b9036478da7068a2c867b9d732c.jpg new file mode 100644 index 00000000..55998415 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-052_5679fe6a9f1cb3e7465931fe96e3825785194b9036478da7068a2c867b9d732c.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-052_69ed4f78a63b4517144e66049caa503cecca7e14eb73e54ad0b140400a4a4e85.jpg b/.raw/MidasFEAAnalysisManual/images/page-052_69ed4f78a63b4517144e66049caa503cecca7e14eb73e54ad0b140400a4a4e85.jpg new file mode 100644 index 00000000..f2d0a622 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-052_69ed4f78a63b4517144e66049caa503cecca7e14eb73e54ad0b140400a4a4e85.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-052_d6b767f5040c512d3893607ad72439182699d7b43b17b69f542cf063cc4eb29b.jpg b/.raw/MidasFEAAnalysisManual/images/page-052_d6b767f5040c512d3893607ad72439182699d7b43b17b69f542cf063cc4eb29b.jpg new file mode 100644 index 00000000..bd3e0b8c Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-052_d6b767f5040c512d3893607ad72439182699d7b43b17b69f542cf063cc4eb29b.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-052_df1fc4d67d394acf490f0162d7a9547ab8c6077fcd8bfd3f0f51afa6b0916f06.jpg b/.raw/MidasFEAAnalysisManual/images/page-052_df1fc4d67d394acf490f0162d7a9547ab8c6077fcd8bfd3f0f51afa6b0916f06.jpg new file mode 100644 index 00000000..9a1f24ad Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-052_df1fc4d67d394acf490f0162d7a9547ab8c6077fcd8bfd3f0f51afa6b0916f06.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-052_e08bb9b82268644c61ebe4f0e4bcb8e05f852e20506785f187976f198ada4971.jpg b/.raw/MidasFEAAnalysisManual/images/page-052_e08bb9b82268644c61ebe4f0e4bcb8e05f852e20506785f187976f198ada4971.jpg new file mode 100644 index 00000000..f25dd8f3 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-052_e08bb9b82268644c61ebe4f0e4bcb8e05f852e20506785f187976f198ada4971.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-052_ebc4ec8061ae77db61988c3334f8f5d9f7efbf45d61dcd5c08409fdd749fa052.jpg b/.raw/MidasFEAAnalysisManual/images/page-052_ebc4ec8061ae77db61988c3334f8f5d9f7efbf45d61dcd5c08409fdd749fa052.jpg new file mode 100644 index 00000000..0ad9b389 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-052_ebc4ec8061ae77db61988c3334f8f5d9f7efbf45d61dcd5c08409fdd749fa052.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-052_f24ce0f70530d27314addc340886e20c0e5c084aa1e347133e3733375ea2ebe8.jpg b/.raw/MidasFEAAnalysisManual/images/page-052_f24ce0f70530d27314addc340886e20c0e5c084aa1e347133e3733375ea2ebe8.jpg new file mode 100644 index 00000000..77e7703b Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-052_f24ce0f70530d27314addc340886e20c0e5c084aa1e347133e3733375ea2ebe8.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-053_14d712841e044c74fb3f22b8d5a850357cb162dc963baead22ab1a1c91464d8a.jpg b/.raw/MidasFEAAnalysisManual/images/page-053_14d712841e044c74fb3f22b8d5a850357cb162dc963baead22ab1a1c91464d8a.jpg new file mode 100644 index 00000000..d76de5ef Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-053_14d712841e044c74fb3f22b8d5a850357cb162dc963baead22ab1a1c91464d8a.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-053_1f8131f82d969c3b3fa6dd9029739074c3abf8c4ba86a115d8ccfda37d5a9b55.jpg b/.raw/MidasFEAAnalysisManual/images/page-053_1f8131f82d969c3b3fa6dd9029739074c3abf8c4ba86a115d8ccfda37d5a9b55.jpg new file mode 100644 index 00000000..0ed55ab7 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-053_1f8131f82d969c3b3fa6dd9029739074c3abf8c4ba86a115d8ccfda37d5a9b55.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-053_2db482bed747c33d366361fd09b08b6444fea6613708857980598c9553d80d9b.jpg b/.raw/MidasFEAAnalysisManual/images/page-053_2db482bed747c33d366361fd09b08b6444fea6613708857980598c9553d80d9b.jpg new file mode 100644 index 00000000..fd33689b Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-053_2db482bed747c33d366361fd09b08b6444fea6613708857980598c9553d80d9b.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-053_31849e5e2f6686836b646b89f93cbc9d3af6cd843fd06d910c621434586c3f7b.jpg b/.raw/MidasFEAAnalysisManual/images/page-053_31849e5e2f6686836b646b89f93cbc9d3af6cd843fd06d910c621434586c3f7b.jpg new file mode 100644 index 00000000..f0739d8d Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-053_31849e5e2f6686836b646b89f93cbc9d3af6cd843fd06d910c621434586c3f7b.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-053_796d1b306dc99a3a348a069ff6574496dfc91655433b62a6faaebd1bbf5a362e.jpg b/.raw/MidasFEAAnalysisManual/images/page-053_796d1b306dc99a3a348a069ff6574496dfc91655433b62a6faaebd1bbf5a362e.jpg new file mode 100644 index 00000000..2963a7dc Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-053_796d1b306dc99a3a348a069ff6574496dfc91655433b62a6faaebd1bbf5a362e.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-053_bc044adb57f2c83b6f1ca80b11ce57686e590ef1375615b27dd5c46c1d9637fc.jpg b/.raw/MidasFEAAnalysisManual/images/page-053_bc044adb57f2c83b6f1ca80b11ce57686e590ef1375615b27dd5c46c1d9637fc.jpg new file mode 100644 index 00000000..f27c70c7 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-053_bc044adb57f2c83b6f1ca80b11ce57686e590ef1375615b27dd5c46c1d9637fc.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-054_113de6c0e9f9dcf6c836589b7d9fd44277e28fdcdb8142d2ca148a4f892d1829.jpg b/.raw/MidasFEAAnalysisManual/images/page-054_113de6c0e9f9dcf6c836589b7d9fd44277e28fdcdb8142d2ca148a4f892d1829.jpg new file mode 100644 index 00000000..57d0fe85 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-054_113de6c0e9f9dcf6c836589b7d9fd44277e28fdcdb8142d2ca148a4f892d1829.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-054_70fe16dbacbfa6afd5da1007819df814482f7519a680b86f7ef6fa778ff0e4c5.jpg b/.raw/MidasFEAAnalysisManual/images/page-054_70fe16dbacbfa6afd5da1007819df814482f7519a680b86f7ef6fa778ff0e4c5.jpg new file mode 100644 index 00000000..d882cab5 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-054_70fe16dbacbfa6afd5da1007819df814482f7519a680b86f7ef6fa778ff0e4c5.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-054_7d67d2f49680bd06543cbeee0aa579f39ffddb68441a2e52e370c3c30b54835d.jpg b/.raw/MidasFEAAnalysisManual/images/page-054_7d67d2f49680bd06543cbeee0aa579f39ffddb68441a2e52e370c3c30b54835d.jpg new file mode 100644 index 00000000..34ed9461 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-054_7d67d2f49680bd06543cbeee0aa579f39ffddb68441a2e52e370c3c30b54835d.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-054_89db5cfa3809db5521f4f76f4174c426adb36b15fa6846907311ef8a61c798e0.jpg b/.raw/MidasFEAAnalysisManual/images/page-054_89db5cfa3809db5521f4f76f4174c426adb36b15fa6846907311ef8a61c798e0.jpg new file mode 100644 index 00000000..668d1f07 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-054_89db5cfa3809db5521f4f76f4174c426adb36b15fa6846907311ef8a61c798e0.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-054_8d4976e26ad410b08df08e9e1cb0e75faba91b07d100984b131907bf145b4d08.jpg b/.raw/MidasFEAAnalysisManual/images/page-054_8d4976e26ad410b08df08e9e1cb0e75faba91b07d100984b131907bf145b4d08.jpg new file mode 100644 index 00000000..4b78327a Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-054_8d4976e26ad410b08df08e9e1cb0e75faba91b07d100984b131907bf145b4d08.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-054_c3b5dc0b46b6937d117206ebbf58334daf9a410a740a4013ddf9680e0bd6335c.jpg b/.raw/MidasFEAAnalysisManual/images/page-054_c3b5dc0b46b6937d117206ebbf58334daf9a410a740a4013ddf9680e0bd6335c.jpg new file mode 100644 index 00000000..1a5bff63 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-054_c3b5dc0b46b6937d117206ebbf58334daf9a410a740a4013ddf9680e0bd6335c.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-055_43beb18930e52eececa2c7b44b575129afff564a9c2756f0761b746501e89111.jpg b/.raw/MidasFEAAnalysisManual/images/page-055_43beb18930e52eececa2c7b44b575129afff564a9c2756f0761b746501e89111.jpg new file mode 100644 index 00000000..eb979433 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-055_43beb18930e52eececa2c7b44b575129afff564a9c2756f0761b746501e89111.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-055_4609e6d3fc2aa98e7398ecc721ae8130fdca959eb9e9ba5fc9f80e8152687fb8.jpg b/.raw/MidasFEAAnalysisManual/images/page-055_4609e6d3fc2aa98e7398ecc721ae8130fdca959eb9e9ba5fc9f80e8152687fb8.jpg new file mode 100644 index 00000000..c162f7c2 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-055_4609e6d3fc2aa98e7398ecc721ae8130fdca959eb9e9ba5fc9f80e8152687fb8.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-055_723133bab20a9784e5a640d77ed1536b42c5735c3f851c8b4f2e188597f55c1b.jpg b/.raw/MidasFEAAnalysisManual/images/page-055_723133bab20a9784e5a640d77ed1536b42c5735c3f851c8b4f2e188597f55c1b.jpg new file mode 100644 index 00000000..5bf6f25b Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-055_723133bab20a9784e5a640d77ed1536b42c5735c3f851c8b4f2e188597f55c1b.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-055_831c815a8cd2f033e2d25a4a7e3486e6b8a8879bf32d053eaed1347cab9e8e53.jpg b/.raw/MidasFEAAnalysisManual/images/page-055_831c815a8cd2f033e2d25a4a7e3486e6b8a8879bf32d053eaed1347cab9e8e53.jpg new file mode 100644 index 00000000..8e4ed8f1 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-055_831c815a8cd2f033e2d25a4a7e3486e6b8a8879bf32d053eaed1347cab9e8e53.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-055_883748933b4f6cffbe03965da3db1f0d3a8d1679bb65554881b198912947516a.jpg b/.raw/MidasFEAAnalysisManual/images/page-055_883748933b4f6cffbe03965da3db1f0d3a8d1679bb65554881b198912947516a.jpg new file mode 100644 index 00000000..a304e833 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-055_883748933b4f6cffbe03965da3db1f0d3a8d1679bb65554881b198912947516a.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-055_ee00b40d7eb0bb8cc687d80256505d80e5a7eabdb4d7bce6ed23a9ac66aa41c3.jpg b/.raw/MidasFEAAnalysisManual/images/page-055_ee00b40d7eb0bb8cc687d80256505d80e5a7eabdb4d7bce6ed23a9ac66aa41c3.jpg new file mode 100644 index 00000000..c64ded5c Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-055_ee00b40d7eb0bb8cc687d80256505d80e5a7eabdb4d7bce6ed23a9ac66aa41c3.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-056_0d81cba5a3211038b35e6650adc2a1c811bd80e7717f5e80dec9b1ae19e623d5.jpg b/.raw/MidasFEAAnalysisManual/images/page-056_0d81cba5a3211038b35e6650adc2a1c811bd80e7717f5e80dec9b1ae19e623d5.jpg new file mode 100644 index 00000000..963572fd Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-056_0d81cba5a3211038b35e6650adc2a1c811bd80e7717f5e80dec9b1ae19e623d5.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-056_35b3882a984b021d5f2fda72acb066bb15b5858d61de59aecae0a195c1efc636.jpg b/.raw/MidasFEAAnalysisManual/images/page-056_35b3882a984b021d5f2fda72acb066bb15b5858d61de59aecae0a195c1efc636.jpg new file mode 100644 index 00000000..65f0a874 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-056_35b3882a984b021d5f2fda72acb066bb15b5858d61de59aecae0a195c1efc636.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-056_4ad36b724e59e45e6b91b139bdf6f9120f1c5af5d0d940d8832a7bd2c1329e53.jpg b/.raw/MidasFEAAnalysisManual/images/page-056_4ad36b724e59e45e6b91b139bdf6f9120f1c5af5d0d940d8832a7bd2c1329e53.jpg new file mode 100644 index 00000000..4fb79475 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-056_4ad36b724e59e45e6b91b139bdf6f9120f1c5af5d0d940d8832a7bd2c1329e53.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-056_892ad3bb606da13e12bcd77dcf2c28a70e2424ea974cab50300239b9224b795a.jpg b/.raw/MidasFEAAnalysisManual/images/page-056_892ad3bb606da13e12bcd77dcf2c28a70e2424ea974cab50300239b9224b795a.jpg new file mode 100644 index 00000000..37670cb4 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-056_892ad3bb606da13e12bcd77dcf2c28a70e2424ea974cab50300239b9224b795a.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-057_4541772a3362f1db3f26cef99aecbf386b324e2e72ddbf3d941c8ad312fd9524.jpg b/.raw/MidasFEAAnalysisManual/images/page-057_4541772a3362f1db3f26cef99aecbf386b324e2e72ddbf3d941c8ad312fd9524.jpg new file mode 100644 index 00000000..154c8578 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-057_4541772a3362f1db3f26cef99aecbf386b324e2e72ddbf3d941c8ad312fd9524.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-058_1bd90b077380e679825cbf7819fdf32a1d172cebd03c806f6e467013ccc63530.jpg b/.raw/MidasFEAAnalysisManual/images/page-058_1bd90b077380e679825cbf7819fdf32a1d172cebd03c806f6e467013ccc63530.jpg new file mode 100644 index 00000000..be1d5221 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-058_1bd90b077380e679825cbf7819fdf32a1d172cebd03c806f6e467013ccc63530.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-059_8172a1cd1f0e3947c4ffdb0a2183b55aa21b3797221165d1009fd5d87ae35d05.jpg b/.raw/MidasFEAAnalysisManual/images/page-059_8172a1cd1f0e3947c4ffdb0a2183b55aa21b3797221165d1009fd5d87ae35d05.jpg new file mode 100644 index 00000000..e87a13f9 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-059_8172a1cd1f0e3947c4ffdb0a2183b55aa21b3797221165d1009fd5d87ae35d05.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-059_9b6a161c691baf6281d677355967c3783cf3d504b7d80bf979c6e505080f4088.jpg b/.raw/MidasFEAAnalysisManual/images/page-059_9b6a161c691baf6281d677355967c3783cf3d504b7d80bf979c6e505080f4088.jpg new file mode 100644 index 00000000..bc0d8980 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-059_9b6a161c691baf6281d677355967c3783cf3d504b7d80bf979c6e505080f4088.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-059_c64d96092f1a79299410d05fea7a2e055b55459b78ef1bcb6f0d0427c2534460.jpg b/.raw/MidasFEAAnalysisManual/images/page-059_c64d96092f1a79299410d05fea7a2e055b55459b78ef1bcb6f0d0427c2534460.jpg new file mode 100644 index 00000000..fb23be43 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-059_c64d96092f1a79299410d05fea7a2e055b55459b78ef1bcb6f0d0427c2534460.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-060_061df440ce12761890a986cf325e5373c3efb0790dfbbc9bed4959348bbfd8b6.jpg b/.raw/MidasFEAAnalysisManual/images/page-060_061df440ce12761890a986cf325e5373c3efb0790dfbbc9bed4959348bbfd8b6.jpg new file mode 100644 index 00000000..5852c9bd Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-060_061df440ce12761890a986cf325e5373c3efb0790dfbbc9bed4959348bbfd8b6.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-060_4d174980d7f471bde2e24e3dcde866ca81490491613f81c1da991a07bedffa30.jpg b/.raw/MidasFEAAnalysisManual/images/page-060_4d174980d7f471bde2e24e3dcde866ca81490491613f81c1da991a07bedffa30.jpg new file mode 100644 index 00000000..046618d2 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-060_4d174980d7f471bde2e24e3dcde866ca81490491613f81c1da991a07bedffa30.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-061_4748343443fd2051ecd22b037f1109509ca912bacb33afa9414e825f2692f52d.jpg b/.raw/MidasFEAAnalysisManual/images/page-061_4748343443fd2051ecd22b037f1109509ca912bacb33afa9414e825f2692f52d.jpg new file mode 100644 index 00000000..cad7e5c2 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-061_4748343443fd2051ecd22b037f1109509ca912bacb33afa9414e825f2692f52d.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-061_563d61f9f0d5c24b8c76c56c96a203fe69213a864a10c1752d7c4be8353ea1c2.jpg b/.raw/MidasFEAAnalysisManual/images/page-061_563d61f9f0d5c24b8c76c56c96a203fe69213a864a10c1752d7c4be8353ea1c2.jpg new file mode 100644 index 00000000..b8fe32a7 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-061_563d61f9f0d5c24b8c76c56c96a203fe69213a864a10c1752d7c4be8353ea1c2.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-061_d4bd8a3d6e29adadb37fc3a3eecc5e82aa61cc4380fa1f201505a47f30f7349b.jpg b/.raw/MidasFEAAnalysisManual/images/page-061_d4bd8a3d6e29adadb37fc3a3eecc5e82aa61cc4380fa1f201505a47f30f7349b.jpg new file mode 100644 index 00000000..a140c9f5 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-061_d4bd8a3d6e29adadb37fc3a3eecc5e82aa61cc4380fa1f201505a47f30f7349b.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-061_eb91ecd641b01b8006ea7b7a72a3493183ed57ce56e9e923bf7db27f5ebe3529.jpg b/.raw/MidasFEAAnalysisManual/images/page-061_eb91ecd641b01b8006ea7b7a72a3493183ed57ce56e9e923bf7db27f5ebe3529.jpg new file mode 100644 index 00000000..f13816e4 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-061_eb91ecd641b01b8006ea7b7a72a3493183ed57ce56e9e923bf7db27f5ebe3529.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-064_146dcf26e3e59e9c0e29230a9ecb3f1729122829cfa5e3f9aa741ef2cfda65a8.jpg b/.raw/MidasFEAAnalysisManual/images/page-064_146dcf26e3e59e9c0e29230a9ecb3f1729122829cfa5e3f9aa741ef2cfda65a8.jpg new file mode 100644 index 00000000..7fd1721b Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-064_146dcf26e3e59e9c0e29230a9ecb3f1729122829cfa5e3f9aa741ef2cfda65a8.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-064_1c6228e364f717cfb859b8802ac23c950c7ce7f84565f3d82d420a0b1d95057d.jpg b/.raw/MidasFEAAnalysisManual/images/page-064_1c6228e364f717cfb859b8802ac23c950c7ce7f84565f3d82d420a0b1d95057d.jpg new file mode 100644 index 00000000..bf9c3c5a Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-064_1c6228e364f717cfb859b8802ac23c950c7ce7f84565f3d82d420a0b1d95057d.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-064_78e29c52aec42e672292d819521a45a06324b8292e87253676477bc23d178bbd.jpg b/.raw/MidasFEAAnalysisManual/images/page-064_78e29c52aec42e672292d819521a45a06324b8292e87253676477bc23d178bbd.jpg new file mode 100644 index 00000000..1b6c2a4c Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-064_78e29c52aec42e672292d819521a45a06324b8292e87253676477bc23d178bbd.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-064_7b723dd4aefa5ef4a47bc1f2bb11155bfa45f1f563bc05303f2d5ed621b6dd7f.jpg b/.raw/MidasFEAAnalysisManual/images/page-064_7b723dd4aefa5ef4a47bc1f2bb11155bfa45f1f563bc05303f2d5ed621b6dd7f.jpg new file mode 100644 index 00000000..e17bbece Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-064_7b723dd4aefa5ef4a47bc1f2bb11155bfa45f1f563bc05303f2d5ed621b6dd7f.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-064_8ddd7edfc150b5443945eb26b2a7a6ec5155687cfd2ea6c5e0e29e9eaa198941.jpg b/.raw/MidasFEAAnalysisManual/images/page-064_8ddd7edfc150b5443945eb26b2a7a6ec5155687cfd2ea6c5e0e29e9eaa198941.jpg new file mode 100644 index 00000000..0d41fc3d Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-064_8ddd7edfc150b5443945eb26b2a7a6ec5155687cfd2ea6c5e0e29e9eaa198941.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-064_d8245c15d146e1fd23bc6a7551c908cf01cf1bbce11f562e5aa4baa60fefcee6.jpg b/.raw/MidasFEAAnalysisManual/images/page-064_d8245c15d146e1fd23bc6a7551c908cf01cf1bbce11f562e5aa4baa60fefcee6.jpg new file mode 100644 index 00000000..3980c1df Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-064_d8245c15d146e1fd23bc6a7551c908cf01cf1bbce11f562e5aa4baa60fefcee6.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-064_f1e3b1baed55381a0683c5aed9f99b97c8396f182a1a9fa95770229701652b59.jpg b/.raw/MidasFEAAnalysisManual/images/page-064_f1e3b1baed55381a0683c5aed9f99b97c8396f182a1a9fa95770229701652b59.jpg new file mode 100644 index 00000000..fdf23bce Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-064_f1e3b1baed55381a0683c5aed9f99b97c8396f182a1a9fa95770229701652b59.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-064_fff707e7de5f5f85d09e6821df317076161d241d75b42218dab20277306c9b0f.jpg b/.raw/MidasFEAAnalysisManual/images/page-064_fff707e7de5f5f85d09e6821df317076161d241d75b42218dab20277306c9b0f.jpg new file mode 100644 index 00000000..fe1050f8 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-064_fff707e7de5f5f85d09e6821df317076161d241d75b42218dab20277306c9b0f.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-065_2fd67197ad7348394751d379c6d3b7426f0a464794b99df2076490d428b322ea.jpg b/.raw/MidasFEAAnalysisManual/images/page-065_2fd67197ad7348394751d379c6d3b7426f0a464794b99df2076490d428b322ea.jpg new file mode 100644 index 00000000..c9d1cf25 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-065_2fd67197ad7348394751d379c6d3b7426f0a464794b99df2076490d428b322ea.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-065_505a3722ac07ed6b28cc607510517b3d35d99a04a9cbebcbcc82114764620e05.jpg b/.raw/MidasFEAAnalysisManual/images/page-065_505a3722ac07ed6b28cc607510517b3d35d99a04a9cbebcbcc82114764620e05.jpg new file mode 100644 index 00000000..60b93b7c Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-065_505a3722ac07ed6b28cc607510517b3d35d99a04a9cbebcbcc82114764620e05.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-065_52296f7b1f6e62400b076d58bda361a16259331c7583c8f6f8b5b2c6b31d8ae3.jpg b/.raw/MidasFEAAnalysisManual/images/page-065_52296f7b1f6e62400b076d58bda361a16259331c7583c8f6f8b5b2c6b31d8ae3.jpg new file mode 100644 index 00000000..72827040 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-065_52296f7b1f6e62400b076d58bda361a16259331c7583c8f6f8b5b2c6b31d8ae3.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-065_75518c2965eafffff42a04745c6136b6524e40c8838b0577ecf178553da857c0.jpg b/.raw/MidasFEAAnalysisManual/images/page-065_75518c2965eafffff42a04745c6136b6524e40c8838b0577ecf178553da857c0.jpg new file mode 100644 index 00000000..3a80ba4a Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-065_75518c2965eafffff42a04745c6136b6524e40c8838b0577ecf178553da857c0.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-065_88fdec80545288577b123e8198a75270dd4882c39518996d1109be7b88813e81.jpg b/.raw/MidasFEAAnalysisManual/images/page-065_88fdec80545288577b123e8198a75270dd4882c39518996d1109be7b88813e81.jpg new file mode 100644 index 00000000..37c3ee69 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-065_88fdec80545288577b123e8198a75270dd4882c39518996d1109be7b88813e81.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-065_c5d560ec5711b2b299433108f6b313caaab36cd87ea66fea89874f6afb704280.jpg b/.raw/MidasFEAAnalysisManual/images/page-065_c5d560ec5711b2b299433108f6b313caaab36cd87ea66fea89874f6afb704280.jpg new file mode 100644 index 00000000..6c19d9d2 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-065_c5d560ec5711b2b299433108f6b313caaab36cd87ea66fea89874f6afb704280.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-066_15dbe7897d4be3d1c59f931ebc7cc648599f1452eff128b9e7e6e52a03c9b379.jpg b/.raw/MidasFEAAnalysisManual/images/page-066_15dbe7897d4be3d1c59f931ebc7cc648599f1452eff128b9e7e6e52a03c9b379.jpg new file mode 100644 index 00000000..9f25c57e Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-066_15dbe7897d4be3d1c59f931ebc7cc648599f1452eff128b9e7e6e52a03c9b379.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-066_5c5dbfcc3b548152bcfbebe085e3ef57ca830a4553211c59c57eb30c9b8b4fd1.jpg b/.raw/MidasFEAAnalysisManual/images/page-066_5c5dbfcc3b548152bcfbebe085e3ef57ca830a4553211c59c57eb30c9b8b4fd1.jpg new file mode 100644 index 00000000..bc4e257a Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-066_5c5dbfcc3b548152bcfbebe085e3ef57ca830a4553211c59c57eb30c9b8b4fd1.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-066_bea30dd794985b44de271ac52aa537f61f0c562a52a16391f1401ce02702f506.jpg b/.raw/MidasFEAAnalysisManual/images/page-066_bea30dd794985b44de271ac52aa537f61f0c562a52a16391f1401ce02702f506.jpg new file mode 100644 index 00000000..fc5eea4f Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-066_bea30dd794985b44de271ac52aa537f61f0c562a52a16391f1401ce02702f506.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-066_c3c7f9a3d895bb47f253c05fc42ec299b456ab844c3936f24a4c79ab83197b18.jpg b/.raw/MidasFEAAnalysisManual/images/page-066_c3c7f9a3d895bb47f253c05fc42ec299b456ab844c3936f24a4c79ab83197b18.jpg new file mode 100644 index 00000000..02bee5a4 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-066_c3c7f9a3d895bb47f253c05fc42ec299b456ab844c3936f24a4c79ab83197b18.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-066_d39248ec0f8b6225c26a6437fe93a5c11191552faca11243b0a12fe5db3ae12b.jpg b/.raw/MidasFEAAnalysisManual/images/page-066_d39248ec0f8b6225c26a6437fe93a5c11191552faca11243b0a12fe5db3ae12b.jpg new file mode 100644 index 00000000..d0494527 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-066_d39248ec0f8b6225c26a6437fe93a5c11191552faca11243b0a12fe5db3ae12b.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-067_474cfbe4b5b738f0a2b65768865bbb0f796d75444fce0603375f3999d8480f5f.jpg b/.raw/MidasFEAAnalysisManual/images/page-067_474cfbe4b5b738f0a2b65768865bbb0f796d75444fce0603375f3999d8480f5f.jpg new file mode 100644 index 00000000..056987b4 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-067_474cfbe4b5b738f0a2b65768865bbb0f796d75444fce0603375f3999d8480f5f.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-067_691ac5c80b62db93a5b0adf2a48d2287a1b38b407f5f182a8d98d0476b685491.jpg b/.raw/MidasFEAAnalysisManual/images/page-067_691ac5c80b62db93a5b0adf2a48d2287a1b38b407f5f182a8d98d0476b685491.jpg new file mode 100644 index 00000000..87a56db8 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-067_691ac5c80b62db93a5b0adf2a48d2287a1b38b407f5f182a8d98d0476b685491.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-067_7d406115403d89309229c924efa447f8bb803b1887f39d0e00525fa3199a6795.jpg b/.raw/MidasFEAAnalysisManual/images/page-067_7d406115403d89309229c924efa447f8bb803b1887f39d0e00525fa3199a6795.jpg new file mode 100644 index 00000000..43e65a8e Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-067_7d406115403d89309229c924efa447f8bb803b1887f39d0e00525fa3199a6795.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-067_9b2850535813d155a62693ae9f623e450dbe5ff71341cc27dcda081067aa2d86.jpg b/.raw/MidasFEAAnalysisManual/images/page-067_9b2850535813d155a62693ae9f623e450dbe5ff71341cc27dcda081067aa2d86.jpg new file mode 100644 index 00000000..4e779998 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-067_9b2850535813d155a62693ae9f623e450dbe5ff71341cc27dcda081067aa2d86.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-067_af87b8d64bb566082e844cbe1301d2aaeb6840278df76d182cb8cd9d848ab27a.jpg b/.raw/MidasFEAAnalysisManual/images/page-067_af87b8d64bb566082e844cbe1301d2aaeb6840278df76d182cb8cd9d848ab27a.jpg new file mode 100644 index 00000000..6ca0162e Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-067_af87b8d64bb566082e844cbe1301d2aaeb6840278df76d182cb8cd9d848ab27a.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-067_cb311871fabd96632cedeb9489d2d1d322a6cd3ef3398417e5639c02f0ee0f74.jpg b/.raw/MidasFEAAnalysisManual/images/page-067_cb311871fabd96632cedeb9489d2d1d322a6cd3ef3398417e5639c02f0ee0f74.jpg new file mode 100644 index 00000000..f51463e3 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-067_cb311871fabd96632cedeb9489d2d1d322a6cd3ef3398417e5639c02f0ee0f74.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-068_1b7d5a9ab2a9c050be8f485c72b1d0fbc394a753856182a1be1b161225fb0af4.jpg b/.raw/MidasFEAAnalysisManual/images/page-068_1b7d5a9ab2a9c050be8f485c72b1d0fbc394a753856182a1be1b161225fb0af4.jpg new file mode 100644 index 00000000..e7adb140 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-068_1b7d5a9ab2a9c050be8f485c72b1d0fbc394a753856182a1be1b161225fb0af4.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-068_29fbdaf34ab96e019c6b37fa489286419a738e64ebbc892972e72dd662512479.jpg b/.raw/MidasFEAAnalysisManual/images/page-068_29fbdaf34ab96e019c6b37fa489286419a738e64ebbc892972e72dd662512479.jpg new file mode 100644 index 00000000..c5e7ecfb Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-068_29fbdaf34ab96e019c6b37fa489286419a738e64ebbc892972e72dd662512479.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-068_703fafaa0205e63df233d61bc39006845867d470e5768b839a6a918e09226b39.jpg b/.raw/MidasFEAAnalysisManual/images/page-068_703fafaa0205e63df233d61bc39006845867d470e5768b839a6a918e09226b39.jpg new file mode 100644 index 00000000..6f19b324 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-068_703fafaa0205e63df233d61bc39006845867d470e5768b839a6a918e09226b39.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-068_9afefc5cbcddd31055210d3497ac90f4d6b5168b2db347fe6b84ff7365369778.jpg b/.raw/MidasFEAAnalysisManual/images/page-068_9afefc5cbcddd31055210d3497ac90f4d6b5168b2db347fe6b84ff7365369778.jpg new file mode 100644 index 00000000..d23ff5c9 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-068_9afefc5cbcddd31055210d3497ac90f4d6b5168b2db347fe6b84ff7365369778.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-068_a2f3b27dbca80a550bf2c49f3286a72c78e7397b71bc6d5f7479dc231a5c0a29.jpg b/.raw/MidasFEAAnalysisManual/images/page-068_a2f3b27dbca80a550bf2c49f3286a72c78e7397b71bc6d5f7479dc231a5c0a29.jpg new file mode 100644 index 00000000..82debdeb Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-068_a2f3b27dbca80a550bf2c49f3286a72c78e7397b71bc6d5f7479dc231a5c0a29.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-068_bf509de6ea35ed46a9bf98bc48f1ff3fcb45d89d45fa855a1e933d72a23773d9.jpg b/.raw/MidasFEAAnalysisManual/images/page-068_bf509de6ea35ed46a9bf98bc48f1ff3fcb45d89d45fa855a1e933d72a23773d9.jpg new file mode 100644 index 00000000..3386bf96 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-068_bf509de6ea35ed46a9bf98bc48f1ff3fcb45d89d45fa855a1e933d72a23773d9.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-068_ea10ebadc791d00fc7d514de3c6d8da6f88152f7c9992f4a8025fd8570d38c70.jpg b/.raw/MidasFEAAnalysisManual/images/page-068_ea10ebadc791d00fc7d514de3c6d8da6f88152f7c9992f4a8025fd8570d38c70.jpg new file mode 100644 index 00000000..b0756e9c Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-068_ea10ebadc791d00fc7d514de3c6d8da6f88152f7c9992f4a8025fd8570d38c70.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-069_11b5ab31d010324b413fe371bf98f6b56d360da9f3a871ca461e30722f2cf284.jpg b/.raw/MidasFEAAnalysisManual/images/page-069_11b5ab31d010324b413fe371bf98f6b56d360da9f3a871ca461e30722f2cf284.jpg new file mode 100644 index 00000000..41a55c4d Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-069_11b5ab31d010324b413fe371bf98f6b56d360da9f3a871ca461e30722f2cf284.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-069_189ebc631c61fa2dbf723835a9ee97a0af7d24269875acba1221a772653ee5c5.jpg b/.raw/MidasFEAAnalysisManual/images/page-069_189ebc631c61fa2dbf723835a9ee97a0af7d24269875acba1221a772653ee5c5.jpg new file mode 100644 index 00000000..86b22f29 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-069_189ebc631c61fa2dbf723835a9ee97a0af7d24269875acba1221a772653ee5c5.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-069_afd8f7d71ba1783001b9ae63da883e54265362bc27e33d78ef59ab01e2f54b93.jpg b/.raw/MidasFEAAnalysisManual/images/page-069_afd8f7d71ba1783001b9ae63da883e54265362bc27e33d78ef59ab01e2f54b93.jpg new file mode 100644 index 00000000..87857558 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-069_afd8f7d71ba1783001b9ae63da883e54265362bc27e33d78ef59ab01e2f54b93.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-069_bc63edd2577e0c9cfac07f75b30266fda5487e004fa853e341cebbe06aa360bb.jpg b/.raw/MidasFEAAnalysisManual/images/page-069_bc63edd2577e0c9cfac07f75b30266fda5487e004fa853e341cebbe06aa360bb.jpg new file mode 100644 index 00000000..81838216 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-069_bc63edd2577e0c9cfac07f75b30266fda5487e004fa853e341cebbe06aa360bb.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-069_ccb76d6e2949a011b690b113f3301637ed7090faeb13cfe15c1151ed0808c500.jpg b/.raw/MidasFEAAnalysisManual/images/page-069_ccb76d6e2949a011b690b113f3301637ed7090faeb13cfe15c1151ed0808c500.jpg new file mode 100644 index 00000000..94be14a6 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-069_ccb76d6e2949a011b690b113f3301637ed7090faeb13cfe15c1151ed0808c500.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-069_dd24bde7ba26e99c93e9fca4a757b338b8a9dcdb7f5273f166800fef22b05235.jpg b/.raw/MidasFEAAnalysisManual/images/page-069_dd24bde7ba26e99c93e9fca4a757b338b8a9dcdb7f5273f166800fef22b05235.jpg new file mode 100644 index 00000000..2ae7fac3 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-069_dd24bde7ba26e99c93e9fca4a757b338b8a9dcdb7f5273f166800fef22b05235.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-069_f77989941a96a202fec15acbec7c32bc8ebca260412fda5ab163d3e56dc44c1d.jpg b/.raw/MidasFEAAnalysisManual/images/page-069_f77989941a96a202fec15acbec7c32bc8ebca260412fda5ab163d3e56dc44c1d.jpg new file mode 100644 index 00000000..c4e2d953 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-069_f77989941a96a202fec15acbec7c32bc8ebca260412fda5ab163d3e56dc44c1d.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-070_254a39365f1db665b3184fd565b0601cfdd571a70493b403792773f23d27d351.jpg b/.raw/MidasFEAAnalysisManual/images/page-070_254a39365f1db665b3184fd565b0601cfdd571a70493b403792773f23d27d351.jpg new file mode 100644 index 00000000..6069b751 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-070_254a39365f1db665b3184fd565b0601cfdd571a70493b403792773f23d27d351.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-070_2d539c40b21f06760be58dec86d06360947cb654380f5f45eb138d473799d449.jpg b/.raw/MidasFEAAnalysisManual/images/page-070_2d539c40b21f06760be58dec86d06360947cb654380f5f45eb138d473799d449.jpg new file mode 100644 index 00000000..0c3ffb98 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-070_2d539c40b21f06760be58dec86d06360947cb654380f5f45eb138d473799d449.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-070_4a7b653114c2d3695cdf6f6bb56f7e9caeeb7883e29d95a0ec9abb748aab0814.jpg b/.raw/MidasFEAAnalysisManual/images/page-070_4a7b653114c2d3695cdf6f6bb56f7e9caeeb7883e29d95a0ec9abb748aab0814.jpg new file mode 100644 index 00000000..0db4b4a2 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-070_4a7b653114c2d3695cdf6f6bb56f7e9caeeb7883e29d95a0ec9abb748aab0814.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-070_748b93eb5ef4eb5b772f9c4767b568ae6590f7a99ae30f649f1f7d1acbff7c4a.jpg b/.raw/MidasFEAAnalysisManual/images/page-070_748b93eb5ef4eb5b772f9c4767b568ae6590f7a99ae30f649f1f7d1acbff7c4a.jpg new file mode 100644 index 00000000..5564eb4f Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-070_748b93eb5ef4eb5b772f9c4767b568ae6590f7a99ae30f649f1f7d1acbff7c4a.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-070_95158cf77dbd257d64bbbb4c0f46d77dc5b050b1d3f1775afe884c1acc981c2e.jpg b/.raw/MidasFEAAnalysisManual/images/page-070_95158cf77dbd257d64bbbb4c0f46d77dc5b050b1d3f1775afe884c1acc981c2e.jpg new file mode 100644 index 00000000..d1bcfa14 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-070_95158cf77dbd257d64bbbb4c0f46d77dc5b050b1d3f1775afe884c1acc981c2e.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-070_9d78cd08e73a800574f55383a2d0bafdf4259c4857823fcd67b155bfaccca187.jpg b/.raw/MidasFEAAnalysisManual/images/page-070_9d78cd08e73a800574f55383a2d0bafdf4259c4857823fcd67b155bfaccca187.jpg new file mode 100644 index 00000000..33e15ac1 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-070_9d78cd08e73a800574f55383a2d0bafdf4259c4857823fcd67b155bfaccca187.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-070_b296a61250298c3fcc0dc289f68a962d0e7f36e50306cfbc5aedd3fae653142f.jpg b/.raw/MidasFEAAnalysisManual/images/page-070_b296a61250298c3fcc0dc289f68a962d0e7f36e50306cfbc5aedd3fae653142f.jpg new file mode 100644 index 00000000..b02aa68f Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-070_b296a61250298c3fcc0dc289f68a962d0e7f36e50306cfbc5aedd3fae653142f.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-070_bd867ffa3a8e9091a5a159c4315628185ba8a4a60116b97f57d5c223b73f61c0.jpg b/.raw/MidasFEAAnalysisManual/images/page-070_bd867ffa3a8e9091a5a159c4315628185ba8a4a60116b97f57d5c223b73f61c0.jpg new file mode 100644 index 00000000..ba418d61 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-070_bd867ffa3a8e9091a5a159c4315628185ba8a4a60116b97f57d5c223b73f61c0.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-071_14c0b162433edf13af3e6f70f8983bf491dd67ed5b49deed7189e8ac6669f80d.jpg b/.raw/MidasFEAAnalysisManual/images/page-071_14c0b162433edf13af3e6f70f8983bf491dd67ed5b49deed7189e8ac6669f80d.jpg new file mode 100644 index 00000000..7f215660 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-071_14c0b162433edf13af3e6f70f8983bf491dd67ed5b49deed7189e8ac6669f80d.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-071_21f4671d1fdb1ea515394c66712872b1679f289d2adee1cac9ae722047c5e297.jpg b/.raw/MidasFEAAnalysisManual/images/page-071_21f4671d1fdb1ea515394c66712872b1679f289d2adee1cac9ae722047c5e297.jpg new file mode 100644 index 00000000..f9d74716 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-071_21f4671d1fdb1ea515394c66712872b1679f289d2adee1cac9ae722047c5e297.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-071_38212097799cc26f32166beaa1e676a164b54a8d0ab7ec4ffae03b12ae465db4.jpg b/.raw/MidasFEAAnalysisManual/images/page-071_38212097799cc26f32166beaa1e676a164b54a8d0ab7ec4ffae03b12ae465db4.jpg new file mode 100644 index 00000000..80424826 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-071_38212097799cc26f32166beaa1e676a164b54a8d0ab7ec4ffae03b12ae465db4.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-071_38c96584cdd2b9f5a2afcc456d722e243986a1ac96793fc2039cb2a6348f6f11.jpg b/.raw/MidasFEAAnalysisManual/images/page-071_38c96584cdd2b9f5a2afcc456d722e243986a1ac96793fc2039cb2a6348f6f11.jpg new file mode 100644 index 00000000..6d236d21 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-071_38c96584cdd2b9f5a2afcc456d722e243986a1ac96793fc2039cb2a6348f6f11.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-071_ac68aeafe18b2cd481d75a70760f2cdfa88c9a2facabe3a66d1e711f441f33a1.jpg b/.raw/MidasFEAAnalysisManual/images/page-071_ac68aeafe18b2cd481d75a70760f2cdfa88c9a2facabe3a66d1e711f441f33a1.jpg new file mode 100644 index 00000000..3c8d1aea Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-071_ac68aeafe18b2cd481d75a70760f2cdfa88c9a2facabe3a66d1e711f441f33a1.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-071_de2ed2f69072a844c7c977b7f618a828250623318731c8065847197e731dd313.jpg b/.raw/MidasFEAAnalysisManual/images/page-071_de2ed2f69072a844c7c977b7f618a828250623318731c8065847197e731dd313.jpg new file mode 100644 index 00000000..e5792f80 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-071_de2ed2f69072a844c7c977b7f618a828250623318731c8065847197e731dd313.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-072_42bbab7e0d99a31a134cfc544b9ec26a45c78c1a510665a281e7ef046fb860ad.jpg b/.raw/MidasFEAAnalysisManual/images/page-072_42bbab7e0d99a31a134cfc544b9ec26a45c78c1a510665a281e7ef046fb860ad.jpg new file mode 100644 index 00000000..90af4e81 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-072_42bbab7e0d99a31a134cfc544b9ec26a45c78c1a510665a281e7ef046fb860ad.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-072_59153913d1bb449060970d4b6b73931aa1685ecdfdf27fc4b9eca009ee076c37.jpg b/.raw/MidasFEAAnalysisManual/images/page-072_59153913d1bb449060970d4b6b73931aa1685ecdfdf27fc4b9eca009ee076c37.jpg new file mode 100644 index 00000000..04deca50 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-072_59153913d1bb449060970d4b6b73931aa1685ecdfdf27fc4b9eca009ee076c37.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-072_9481b74cfed980b376d3b2076d7c1592673a93418f2077ed9a23ebe6d402b418.jpg b/.raw/MidasFEAAnalysisManual/images/page-072_9481b74cfed980b376d3b2076d7c1592673a93418f2077ed9a23ebe6d402b418.jpg new file mode 100644 index 00000000..ca695f37 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-072_9481b74cfed980b376d3b2076d7c1592673a93418f2077ed9a23ebe6d402b418.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-072_a6e2282592dc989807c90bfef8161e51c8cd041c0b78925b1c2a7563d5ee06fa.jpg b/.raw/MidasFEAAnalysisManual/images/page-072_a6e2282592dc989807c90bfef8161e51c8cd041c0b78925b1c2a7563d5ee06fa.jpg new file mode 100644 index 00000000..5bd8ace0 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-072_a6e2282592dc989807c90bfef8161e51c8cd041c0b78925b1c2a7563d5ee06fa.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-072_cb5555726525015b6592f14bfd62d0d655e2d9948930b5325d216d6d53b1d4fe.jpg b/.raw/MidasFEAAnalysisManual/images/page-072_cb5555726525015b6592f14bfd62d0d655e2d9948930b5325d216d6d53b1d4fe.jpg new file mode 100644 index 00000000..cb302045 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-072_cb5555726525015b6592f14bfd62d0d655e2d9948930b5325d216d6d53b1d4fe.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-073_37a9020e4b6877bb8d3bf740dd724f4267b3f53584d425568dac7fd206670d31.jpg b/.raw/MidasFEAAnalysisManual/images/page-073_37a9020e4b6877bb8d3bf740dd724f4267b3f53584d425568dac7fd206670d31.jpg new file mode 100644 index 00000000..4f3849e7 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-073_37a9020e4b6877bb8d3bf740dd724f4267b3f53584d425568dac7fd206670d31.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-073_450bd276a32690f6e8389f72c54b7be5837a3a2b431cbd417dda4532cb92fd65.jpg b/.raw/MidasFEAAnalysisManual/images/page-073_450bd276a32690f6e8389f72c54b7be5837a3a2b431cbd417dda4532cb92fd65.jpg new file mode 100644 index 00000000..194a845e Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-073_450bd276a32690f6e8389f72c54b7be5837a3a2b431cbd417dda4532cb92fd65.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-073_66afab4498042f2b8352f9927b369d06fd61d9791e472cf3c8b6adac537d2f61.jpg b/.raw/MidasFEAAnalysisManual/images/page-073_66afab4498042f2b8352f9927b369d06fd61d9791e472cf3c8b6adac537d2f61.jpg new file mode 100644 index 00000000..5d4bfee1 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-073_66afab4498042f2b8352f9927b369d06fd61d9791e472cf3c8b6adac537d2f61.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-073_7093d651952dfa70a3fe7fd2739cdccc71e258da40f774d33338ee464f8e11dd.jpg b/.raw/MidasFEAAnalysisManual/images/page-073_7093d651952dfa70a3fe7fd2739cdccc71e258da40f774d33338ee464f8e11dd.jpg new file mode 100644 index 00000000..cf8a47d1 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-073_7093d651952dfa70a3fe7fd2739cdccc71e258da40f774d33338ee464f8e11dd.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-074_597d762ed1b204a45cba6cf9583a80be95f5547d9d944cd16328e977262f6655.jpg b/.raw/MidasFEAAnalysisManual/images/page-074_597d762ed1b204a45cba6cf9583a80be95f5547d9d944cd16328e977262f6655.jpg new file mode 100644 index 00000000..f42a3c6f Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-074_597d762ed1b204a45cba6cf9583a80be95f5547d9d944cd16328e977262f6655.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-074_5a51a624c72eb23a8280c4b0b7f7b4a4ce66f4583f4e7fe8eb031d9667a1f238.jpg b/.raw/MidasFEAAnalysisManual/images/page-074_5a51a624c72eb23a8280c4b0b7f7b4a4ce66f4583f4e7fe8eb031d9667a1f238.jpg new file mode 100644 index 00000000..82e56329 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-074_5a51a624c72eb23a8280c4b0b7f7b4a4ce66f4583f4e7fe8eb031d9667a1f238.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-074_61d9730cced1816c7956e53a21d4d9cc2c4a280576e3c2862bbbc2f39de550eb.jpg b/.raw/MidasFEAAnalysisManual/images/page-074_61d9730cced1816c7956e53a21d4d9cc2c4a280576e3c2862bbbc2f39de550eb.jpg new file mode 100644 index 00000000..e473ac1a Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-074_61d9730cced1816c7956e53a21d4d9cc2c4a280576e3c2862bbbc2f39de550eb.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-074_d0de99c4ac02ac0647249e3e0de2e21f1a1873112b80438102f060aebfa7c583.jpg b/.raw/MidasFEAAnalysisManual/images/page-074_d0de99c4ac02ac0647249e3e0de2e21f1a1873112b80438102f060aebfa7c583.jpg new file mode 100644 index 00000000..126c47b0 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-074_d0de99c4ac02ac0647249e3e0de2e21f1a1873112b80438102f060aebfa7c583.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-074_f709954d94828cc45a677d89d0d8fa26676916fbf97ec454e741f569f35852ab.jpg b/.raw/MidasFEAAnalysisManual/images/page-074_f709954d94828cc45a677d89d0d8fa26676916fbf97ec454e741f569f35852ab.jpg new file mode 100644 index 00000000..df65f7da Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-074_f709954d94828cc45a677d89d0d8fa26676916fbf97ec454e741f569f35852ab.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-075_270f1d8c3e7bd61751fbde2052e915e09d145459be1982454b4bf8c1fc6777c5.jpg b/.raw/MidasFEAAnalysisManual/images/page-075_270f1d8c3e7bd61751fbde2052e915e09d145459be1982454b4bf8c1fc6777c5.jpg new file mode 100644 index 00000000..19a738dd Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-075_270f1d8c3e7bd61751fbde2052e915e09d145459be1982454b4bf8c1fc6777c5.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-075_2e181711b24955513accf8191f93e7f36fcaa584d37485844c7f31cf327583bd.jpg b/.raw/MidasFEAAnalysisManual/images/page-075_2e181711b24955513accf8191f93e7f36fcaa584d37485844c7f31cf327583bd.jpg new file mode 100644 index 00000000..4a1a8da2 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-075_2e181711b24955513accf8191f93e7f36fcaa584d37485844c7f31cf327583bd.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-075_4b74fcdc7370d4e3b9769c3bac73695d783e55853ca8793d6676c2624bdbe3f8.jpg b/.raw/MidasFEAAnalysisManual/images/page-075_4b74fcdc7370d4e3b9769c3bac73695d783e55853ca8793d6676c2624bdbe3f8.jpg new file mode 100644 index 00000000..956c34d0 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-075_4b74fcdc7370d4e3b9769c3bac73695d783e55853ca8793d6676c2624bdbe3f8.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-075_69fb94e1c9d710f7e32cf4ec0a9f0ef69c10d53a604eddd7f7038da15a83fbb4.jpg b/.raw/MidasFEAAnalysisManual/images/page-075_69fb94e1c9d710f7e32cf4ec0a9f0ef69c10d53a604eddd7f7038da15a83fbb4.jpg new file mode 100644 index 00000000..47d793cd Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-075_69fb94e1c9d710f7e32cf4ec0a9f0ef69c10d53a604eddd7f7038da15a83fbb4.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-075_7f3b3936e3c3f5c5f8668ed459985d335d046cc88ee86a82ac0879882d8e9c72.jpg b/.raw/MidasFEAAnalysisManual/images/page-075_7f3b3936e3c3f5c5f8668ed459985d335d046cc88ee86a82ac0879882d8e9c72.jpg new file mode 100644 index 00000000..027b9338 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-075_7f3b3936e3c3f5c5f8668ed459985d335d046cc88ee86a82ac0879882d8e9c72.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-075_822d0a77a81165322e15c728b79aee10db7c26b4300b6f9dd3ff74db6d1e7058.jpg b/.raw/MidasFEAAnalysisManual/images/page-075_822d0a77a81165322e15c728b79aee10db7c26b4300b6f9dd3ff74db6d1e7058.jpg new file mode 100644 index 00000000..6f741fa6 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-075_822d0a77a81165322e15c728b79aee10db7c26b4300b6f9dd3ff74db6d1e7058.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-075_94fae97b7594d5219022f5eb9d4f29fa8427fa1334d5e918036f809a461fb7a9.jpg b/.raw/MidasFEAAnalysisManual/images/page-075_94fae97b7594d5219022f5eb9d4f29fa8427fa1334d5e918036f809a461fb7a9.jpg new file mode 100644 index 00000000..06c9757d Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-075_94fae97b7594d5219022f5eb9d4f29fa8427fa1334d5e918036f809a461fb7a9.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-075_ddd0d050d63adc6a8acb5beac57278b2c5cc52bebb3bfa5227c807a5b3d199ff.jpg b/.raw/MidasFEAAnalysisManual/images/page-075_ddd0d050d63adc6a8acb5beac57278b2c5cc52bebb3bfa5227c807a5b3d199ff.jpg new file mode 100644 index 00000000..0aa7ca71 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-075_ddd0d050d63adc6a8acb5beac57278b2c5cc52bebb3bfa5227c807a5b3d199ff.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-076_6f7f596d6789b9e7e57bad27c092b5452f4b30e34cecc0ba52071245b02a85de.jpg b/.raw/MidasFEAAnalysisManual/images/page-076_6f7f596d6789b9e7e57bad27c092b5452f4b30e34cecc0ba52071245b02a85de.jpg new file mode 100644 index 00000000..c55b3e29 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-076_6f7f596d6789b9e7e57bad27c092b5452f4b30e34cecc0ba52071245b02a85de.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-076_beacbeb3054d7d5fec38aa1a26a3bcea2083b9d3ab17dae20bc3b6a5985bee0d.jpg b/.raw/MidasFEAAnalysisManual/images/page-076_beacbeb3054d7d5fec38aa1a26a3bcea2083b9d3ab17dae20bc3b6a5985bee0d.jpg new file mode 100644 index 00000000..bd494d8d Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-076_beacbeb3054d7d5fec38aa1a26a3bcea2083b9d3ab17dae20bc3b6a5985bee0d.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-076_da16c85cb33d423325ed2ae3ea3a560088562f870a0b7c83190f42c827558bcb.jpg b/.raw/MidasFEAAnalysisManual/images/page-076_da16c85cb33d423325ed2ae3ea3a560088562f870a0b7c83190f42c827558bcb.jpg new file mode 100644 index 00000000..c46e371c Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-076_da16c85cb33d423325ed2ae3ea3a560088562f870a0b7c83190f42c827558bcb.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-077_3e761d7e1a4c8d7f026a61d3e7b05c3dd67ebcdd90940e7fcda404eac933eff1.jpg b/.raw/MidasFEAAnalysisManual/images/page-077_3e761d7e1a4c8d7f026a61d3e7b05c3dd67ebcdd90940e7fcda404eac933eff1.jpg new file mode 100644 index 00000000..49487500 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-077_3e761d7e1a4c8d7f026a61d3e7b05c3dd67ebcdd90940e7fcda404eac933eff1.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-077_4f21e12c36f104b1349310f7afc4eea361d4323c737b1f268e6c315d265908fe.jpg b/.raw/MidasFEAAnalysisManual/images/page-077_4f21e12c36f104b1349310f7afc4eea361d4323c737b1f268e6c315d265908fe.jpg new file mode 100644 index 00000000..d063c117 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-077_4f21e12c36f104b1349310f7afc4eea361d4323c737b1f268e6c315d265908fe.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-077_c57d968b30e8408ec68dfd119294b28ee2dd31b16fc1f415fd601c7deb9b2eb8.jpg b/.raw/MidasFEAAnalysisManual/images/page-077_c57d968b30e8408ec68dfd119294b28ee2dd31b16fc1f415fd601c7deb9b2eb8.jpg new file mode 100644 index 00000000..c8def715 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-077_c57d968b30e8408ec68dfd119294b28ee2dd31b16fc1f415fd601c7deb9b2eb8.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-077_cc4c6bce8d2a8148c4086ae09cbdd6410a64d3de474759c34ceff09326f3b8d6.jpg b/.raw/MidasFEAAnalysisManual/images/page-077_cc4c6bce8d2a8148c4086ae09cbdd6410a64d3de474759c34ceff09326f3b8d6.jpg new file mode 100644 index 00000000..c25a4b52 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-077_cc4c6bce8d2a8148c4086ae09cbdd6410a64d3de474759c34ceff09326f3b8d6.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-078_1159bdb4d1fd8e81658a56581639cd1f892d7c8cf015af1c9b00e00e4744847f.jpg b/.raw/MidasFEAAnalysisManual/images/page-078_1159bdb4d1fd8e81658a56581639cd1f892d7c8cf015af1c9b00e00e4744847f.jpg new file mode 100644 index 00000000..e4c239ab Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-078_1159bdb4d1fd8e81658a56581639cd1f892d7c8cf015af1c9b00e00e4744847f.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-078_24dee2b5697701b924d9739fa29843f1d398d2a87f131d07163d84edbbc96fe7.jpg b/.raw/MidasFEAAnalysisManual/images/page-078_24dee2b5697701b924d9739fa29843f1d398d2a87f131d07163d84edbbc96fe7.jpg new file mode 100644 index 00000000..03e19d15 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-078_24dee2b5697701b924d9739fa29843f1d398d2a87f131d07163d84edbbc96fe7.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-078_63935256514bb4b1bf55fe2b34b7e1c3047cc26bd80c120e4ade0d0000b02ed9.jpg b/.raw/MidasFEAAnalysisManual/images/page-078_63935256514bb4b1bf55fe2b34b7e1c3047cc26bd80c120e4ade0d0000b02ed9.jpg new file mode 100644 index 00000000..7eb1d57c Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-078_63935256514bb4b1bf55fe2b34b7e1c3047cc26bd80c120e4ade0d0000b02ed9.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-078_be7173e301fdfaa6290ad8b8262cf781a0584da2864aa63d7a8c9e77495932d4.jpg b/.raw/MidasFEAAnalysisManual/images/page-078_be7173e301fdfaa6290ad8b8262cf781a0584da2864aa63d7a8c9e77495932d4.jpg new file mode 100644 index 00000000..73103b43 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-078_be7173e301fdfaa6290ad8b8262cf781a0584da2864aa63d7a8c9e77495932d4.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-078_d8183247a96859520fc33a887f9c462a134390db65479f431012ea6ac87636d7.jpg b/.raw/MidasFEAAnalysisManual/images/page-078_d8183247a96859520fc33a887f9c462a134390db65479f431012ea6ac87636d7.jpg new file mode 100644 index 00000000..5f2ede2a Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-078_d8183247a96859520fc33a887f9c462a134390db65479f431012ea6ac87636d7.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-078_f3e373e9202b3a1b7324baefc8befe9f9b698f4c3a7e756592d46838d7442585.jpg b/.raw/MidasFEAAnalysisManual/images/page-078_f3e373e9202b3a1b7324baefc8befe9f9b698f4c3a7e756592d46838d7442585.jpg new file mode 100644 index 00000000..28bf0b2f Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-078_f3e373e9202b3a1b7324baefc8befe9f9b698f4c3a7e756592d46838d7442585.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-078_f4548b605411a755669ff14d05758c9b86652a42377929c2a19b5029423be6e5.jpg b/.raw/MidasFEAAnalysisManual/images/page-078_f4548b605411a755669ff14d05758c9b86652a42377929c2a19b5029423be6e5.jpg new file mode 100644 index 00000000..62817dc5 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-078_f4548b605411a755669ff14d05758c9b86652a42377929c2a19b5029423be6e5.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-079_b2d457f7503a729bce28e436bb04968e4cba48eb3f10c3bc1b28d07b7451f6b6.jpg b/.raw/MidasFEAAnalysisManual/images/page-079_b2d457f7503a729bce28e436bb04968e4cba48eb3f10c3bc1b28d07b7451f6b6.jpg new file mode 100644 index 00000000..16a7a09b Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-079_b2d457f7503a729bce28e436bb04968e4cba48eb3f10c3bc1b28d07b7451f6b6.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-080_79c13c5a67a1812d2d61a6d80ca76b55772cc3181569fcf6d894b211a7d43788.jpg b/.raw/MidasFEAAnalysisManual/images/page-080_79c13c5a67a1812d2d61a6d80ca76b55772cc3181569fcf6d894b211a7d43788.jpg new file mode 100644 index 00000000..39825321 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-080_79c13c5a67a1812d2d61a6d80ca76b55772cc3181569fcf6d894b211a7d43788.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-080_9c3a494f677699357d79205efab1fe8ed1fd5985c0597c74abf9fbdebdfaa592.jpg b/.raw/MidasFEAAnalysisManual/images/page-080_9c3a494f677699357d79205efab1fe8ed1fd5985c0597c74abf9fbdebdfaa592.jpg new file mode 100644 index 00000000..c6c84cbf Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-080_9c3a494f677699357d79205efab1fe8ed1fd5985c0597c74abf9fbdebdfaa592.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-081_c5405785c6503e0aad93d4da08e8a10f5a11bf9556292b0a4528fd966620bfae.jpg b/.raw/MidasFEAAnalysisManual/images/page-081_c5405785c6503e0aad93d4da08e8a10f5a11bf9556292b0a4528fd966620bfae.jpg new file mode 100644 index 00000000..818f4177 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-081_c5405785c6503e0aad93d4da08e8a10f5a11bf9556292b0a4528fd966620bfae.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-082_640ed361918410e4d3ddc398e2daeca5c9d2476b3bb3f2b1ababa5291d29ff1a.jpg b/.raw/MidasFEAAnalysisManual/images/page-082_640ed361918410e4d3ddc398e2daeca5c9d2476b3bb3f2b1ababa5291d29ff1a.jpg new file mode 100644 index 00000000..00185f94 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-082_640ed361918410e4d3ddc398e2daeca5c9d2476b3bb3f2b1ababa5291d29ff1a.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-082_9e661c721585385feeb5e0fb2255d6f46cb61ab93ff312cb6d8cdbbfe4c2a25f.jpg b/.raw/MidasFEAAnalysisManual/images/page-082_9e661c721585385feeb5e0fb2255d6f46cb61ab93ff312cb6d8cdbbfe4c2a25f.jpg new file mode 100644 index 00000000..81fe1b12 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-082_9e661c721585385feeb5e0fb2255d6f46cb61ab93ff312cb6d8cdbbfe4c2a25f.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-083_5bba54207b99f6a432fb743629f47f2737a2f67b4ca03ad530f609ffd585eb4d.jpg b/.raw/MidasFEAAnalysisManual/images/page-083_5bba54207b99f6a432fb743629f47f2737a2f67b4ca03ad530f609ffd585eb4d.jpg new file mode 100644 index 00000000..62f30658 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-083_5bba54207b99f6a432fb743629f47f2737a2f67b4ca03ad530f609ffd585eb4d.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-084_8279be80f4b105ca19f05cafeebe957e5c036040c292609f9c656f260670eb06.jpg b/.raw/MidasFEAAnalysisManual/images/page-084_8279be80f4b105ca19f05cafeebe957e5c036040c292609f9c656f260670eb06.jpg new file mode 100644 index 00000000..6434e8b4 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-084_8279be80f4b105ca19f05cafeebe957e5c036040c292609f9c656f260670eb06.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-085_1c542c5ba66aaddb35586bd2c2e3ea566a208fa93f4d819e8edd4f0effa35436.jpg b/.raw/MidasFEAAnalysisManual/images/page-085_1c542c5ba66aaddb35586bd2c2e3ea566a208fa93f4d819e8edd4f0effa35436.jpg new file mode 100644 index 00000000..7d8effa1 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-085_1c542c5ba66aaddb35586bd2c2e3ea566a208fa93f4d819e8edd4f0effa35436.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-086_189b1de82d9e422efe7188cf01ba909d8b2bedc0d939890352e3eb95ca748957.jpg b/.raw/MidasFEAAnalysisManual/images/page-086_189b1de82d9e422efe7188cf01ba909d8b2bedc0d939890352e3eb95ca748957.jpg new file mode 100644 index 00000000..27f15436 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-086_189b1de82d9e422efe7188cf01ba909d8b2bedc0d939890352e3eb95ca748957.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-086_27cae718d928e6e61e160e9009ae8bf4c044d1d9ee8596c252b8dbc3e6a3fdb7.jpg b/.raw/MidasFEAAnalysisManual/images/page-086_27cae718d928e6e61e160e9009ae8bf4c044d1d9ee8596c252b8dbc3e6a3fdb7.jpg new file mode 100644 index 00000000..a89ea1a7 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-086_27cae718d928e6e61e160e9009ae8bf4c044d1d9ee8596c252b8dbc3e6a3fdb7.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-087_22fcd872f3cec1dd8a1227d4ef07726bc24f80a2cfc98806cf5b73ceff95535e.jpg b/.raw/MidasFEAAnalysisManual/images/page-087_22fcd872f3cec1dd8a1227d4ef07726bc24f80a2cfc98806cf5b73ceff95535e.jpg new file mode 100644 index 00000000..10ad07e1 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-087_22fcd872f3cec1dd8a1227d4ef07726bc24f80a2cfc98806cf5b73ceff95535e.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-087_b79146df34f049bebf78dea3822b2f479351b8e92f4d078cb2514d0f24f5f555.jpg b/.raw/MidasFEAAnalysisManual/images/page-087_b79146df34f049bebf78dea3822b2f479351b8e92f4d078cb2514d0f24f5f555.jpg new file mode 100644 index 00000000..e7d6209f Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-087_b79146df34f049bebf78dea3822b2f479351b8e92f4d078cb2514d0f24f5f555.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-088_3ca372b976f00c6674a9f429560a6fc79a286c2e122585d54824e4d65027b8d0.jpg b/.raw/MidasFEAAnalysisManual/images/page-088_3ca372b976f00c6674a9f429560a6fc79a286c2e122585d54824e4d65027b8d0.jpg new file mode 100644 index 00000000..dd2d373f Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-088_3ca372b976f00c6674a9f429560a6fc79a286c2e122585d54824e4d65027b8d0.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-088_c479e4c229e79da8dd020d71c236f874da04b39f61b014c34697ccce9092d472.jpg b/.raw/MidasFEAAnalysisManual/images/page-088_c479e4c229e79da8dd020d71c236f874da04b39f61b014c34697ccce9092d472.jpg new file mode 100644 index 00000000..19c66efa Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-088_c479e4c229e79da8dd020d71c236f874da04b39f61b014c34697ccce9092d472.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-089_b2fbc89990c9ef9132294a1705f4cf499fa8eb79962fe9ae48bc8a7bec050c71.jpg b/.raw/MidasFEAAnalysisManual/images/page-089_b2fbc89990c9ef9132294a1705f4cf499fa8eb79962fe9ae48bc8a7bec050c71.jpg new file mode 100644 index 00000000..7ff09ab9 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-089_b2fbc89990c9ef9132294a1705f4cf499fa8eb79962fe9ae48bc8a7bec050c71.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-089_fafbbbee5cab31448fe838c89c41360668b5b99631f7ac04f82b58c826f11df1.jpg b/.raw/MidasFEAAnalysisManual/images/page-089_fafbbbee5cab31448fe838c89c41360668b5b99631f7ac04f82b58c826f11df1.jpg new file mode 100644 index 00000000..a602d29f Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-089_fafbbbee5cab31448fe838c89c41360668b5b99631f7ac04f82b58c826f11df1.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-090_d707853291b905e1c71c16f3f92c16ba59464910f4edb2cebef4b3fa3e6c533e.jpg b/.raw/MidasFEAAnalysisManual/images/page-090_d707853291b905e1c71c16f3f92c16ba59464910f4edb2cebef4b3fa3e6c533e.jpg new file mode 100644 index 00000000..12d7cf25 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-090_d707853291b905e1c71c16f3f92c16ba59464910f4edb2cebef4b3fa3e6c533e.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-091_e16925fdd6c0e26d1706c5d25ca7e8e0b409596bd73a9dd9bbaec42e10e3969b.jpg b/.raw/MidasFEAAnalysisManual/images/page-091_e16925fdd6c0e26d1706c5d25ca7e8e0b409596bd73a9dd9bbaec42e10e3969b.jpg new file mode 100644 index 00000000..d2b8cef0 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-091_e16925fdd6c0e26d1706c5d25ca7e8e0b409596bd73a9dd9bbaec42e10e3969b.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-092_679fb66e1b84ddf81a4d7194982db4ce0d777a078a179449b5f54dd4cf571374.jpg b/.raw/MidasFEAAnalysisManual/images/page-092_679fb66e1b84ddf81a4d7194982db4ce0d777a078a179449b5f54dd4cf571374.jpg new file mode 100644 index 00000000..04c6e0af Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-092_679fb66e1b84ddf81a4d7194982db4ce0d777a078a179449b5f54dd4cf571374.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-092_ec690adbdeb1cb2ec7073de3cdfe135031dd01db2b23628f792f85e6526f7763.jpg b/.raw/MidasFEAAnalysisManual/images/page-092_ec690adbdeb1cb2ec7073de3cdfe135031dd01db2b23628f792f85e6526f7763.jpg new file mode 100644 index 00000000..2f199806 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-092_ec690adbdeb1cb2ec7073de3cdfe135031dd01db2b23628f792f85e6526f7763.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-092_ff37aa22e79e21a47bfeef90b064b5517ad413640da98465ad5daeee49835806.jpg b/.raw/MidasFEAAnalysisManual/images/page-092_ff37aa22e79e21a47bfeef90b064b5517ad413640da98465ad5daeee49835806.jpg new file mode 100644 index 00000000..ef13a3ae Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-092_ff37aa22e79e21a47bfeef90b064b5517ad413640da98465ad5daeee49835806.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-093_2a2b6ab4960a9d45532e03876f7474c6c8ab3b5a99e4ab2974ce6897d0c4447f.jpg b/.raw/MidasFEAAnalysisManual/images/page-093_2a2b6ab4960a9d45532e03876f7474c6c8ab3b5a99e4ab2974ce6897d0c4447f.jpg new file mode 100644 index 00000000..ac59f903 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-093_2a2b6ab4960a9d45532e03876f7474c6c8ab3b5a99e4ab2974ce6897d0c4447f.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-093_d68827407c4f99fa7b9c95376e290c604fbb61f36be0231c718be04bc1b56c6e.jpg b/.raw/MidasFEAAnalysisManual/images/page-093_d68827407c4f99fa7b9c95376e290c604fbb61f36be0231c718be04bc1b56c6e.jpg new file mode 100644 index 00000000..6db8371b Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-093_d68827407c4f99fa7b9c95376e290c604fbb61f36be0231c718be04bc1b56c6e.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-094_81d3133e679e480f36a468028fd089e314d80d0323145911740002f77f08c7e0.jpg b/.raw/MidasFEAAnalysisManual/images/page-094_81d3133e679e480f36a468028fd089e314d80d0323145911740002f77f08c7e0.jpg new file mode 100644 index 00000000..c895a59f Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-094_81d3133e679e480f36a468028fd089e314d80d0323145911740002f77f08c7e0.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-094_96d4c33e4e5c9126414bc7f041d1d24ef7d75583a72f1110acfdf15a867fccb0.jpg b/.raw/MidasFEAAnalysisManual/images/page-094_96d4c33e4e5c9126414bc7f041d1d24ef7d75583a72f1110acfdf15a867fccb0.jpg new file mode 100644 index 00000000..c41c561a Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-094_96d4c33e4e5c9126414bc7f041d1d24ef7d75583a72f1110acfdf15a867fccb0.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-095_60ed38affcef1c40bfff01db1e3ceb0d55fb67014191cc6241973b800fafa9f9.jpg b/.raw/MidasFEAAnalysisManual/images/page-095_60ed38affcef1c40bfff01db1e3ceb0d55fb67014191cc6241973b800fafa9f9.jpg new file mode 100644 index 00000000..30d4a4f3 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-095_60ed38affcef1c40bfff01db1e3ceb0d55fb67014191cc6241973b800fafa9f9.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-095_ebfbdcf0393b0e913a3a651010279973f7d48d96e59c125cef8e41d9267285f4.jpg b/.raw/MidasFEAAnalysisManual/images/page-095_ebfbdcf0393b0e913a3a651010279973f7d48d96e59c125cef8e41d9267285f4.jpg new file mode 100644 index 00000000..002627c2 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-095_ebfbdcf0393b0e913a3a651010279973f7d48d96e59c125cef8e41d9267285f4.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-096_c04a003fd833954ba3a5814a68340bac6f2bb7a494e01e72e9c4cc0d9f3cac6c.jpg b/.raw/MidasFEAAnalysisManual/images/page-096_c04a003fd833954ba3a5814a68340bac6f2bb7a494e01e72e9c4cc0d9f3cac6c.jpg new file mode 100644 index 00000000..f1894d6d Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-096_c04a003fd833954ba3a5814a68340bac6f2bb7a494e01e72e9c4cc0d9f3cac6c.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-097_2d2d187b46e36066854b55c1006493b80f10ecc5e955548139fe24a8305adce8.jpg b/.raw/MidasFEAAnalysisManual/images/page-097_2d2d187b46e36066854b55c1006493b80f10ecc5e955548139fe24a8305adce8.jpg new file mode 100644 index 00000000..4ef8d5de Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-097_2d2d187b46e36066854b55c1006493b80f10ecc5e955548139fe24a8305adce8.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-097_30ec87144f196b619b37b0c933c145b3ccb9a8fd56588238c3bfae4f15693a22.jpg b/.raw/MidasFEAAnalysisManual/images/page-097_30ec87144f196b619b37b0c933c145b3ccb9a8fd56588238c3bfae4f15693a22.jpg new file mode 100644 index 00000000..de36c4bc Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-097_30ec87144f196b619b37b0c933c145b3ccb9a8fd56588238c3bfae4f15693a22.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-097_512cb05c8ddaca092242486b1e008748fe60eb6877bf2b948efc6723e3b4882d.jpg b/.raw/MidasFEAAnalysisManual/images/page-097_512cb05c8ddaca092242486b1e008748fe60eb6877bf2b948efc6723e3b4882d.jpg new file mode 100644 index 00000000..a93c3199 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-097_512cb05c8ddaca092242486b1e008748fe60eb6877bf2b948efc6723e3b4882d.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-097_62d07a4acbea336f92af353ce37b4e3dd6d75f48f68136c962d5a5fae6f9e3b1.jpg b/.raw/MidasFEAAnalysisManual/images/page-097_62d07a4acbea336f92af353ce37b4e3dd6d75f48f68136c962d5a5fae6f9e3b1.jpg new file mode 100644 index 00000000..a80be344 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-097_62d07a4acbea336f92af353ce37b4e3dd6d75f48f68136c962d5a5fae6f9e3b1.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-097_892f359ee09fd84e8a6f81203c10772499d7ada1113720e1beed5f637f548511.jpg b/.raw/MidasFEAAnalysisManual/images/page-097_892f359ee09fd84e8a6f81203c10772499d7ada1113720e1beed5f637f548511.jpg new file mode 100644 index 00000000..19a7aeed Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-097_892f359ee09fd84e8a6f81203c10772499d7ada1113720e1beed5f637f548511.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-097_961a39e18622f1f5ecd5e53f5d33fd2682dfa098124465e67d8790c75d1c6b2a.jpg b/.raw/MidasFEAAnalysisManual/images/page-097_961a39e18622f1f5ecd5e53f5d33fd2682dfa098124465e67d8790c75d1c6b2a.jpg new file mode 100644 index 00000000..fa2a1a24 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-097_961a39e18622f1f5ecd5e53f5d33fd2682dfa098124465e67d8790c75d1c6b2a.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-097_bc79d1cd286b9ecf315d438ffbb81acdec8d2363287c2e55022e251abbe59d2a.jpg b/.raw/MidasFEAAnalysisManual/images/page-097_bc79d1cd286b9ecf315d438ffbb81acdec8d2363287c2e55022e251abbe59d2a.jpg new file mode 100644 index 00000000..7f84d3ce Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-097_bc79d1cd286b9ecf315d438ffbb81acdec8d2363287c2e55022e251abbe59d2a.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-098_3a0757d184ffb67f66c66d270097c46e3a94e90e2d3d5629fdef21d1a06098d6.jpg b/.raw/MidasFEAAnalysisManual/images/page-098_3a0757d184ffb67f66c66d270097c46e3a94e90e2d3d5629fdef21d1a06098d6.jpg new file mode 100644 index 00000000..1ce05394 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-098_3a0757d184ffb67f66c66d270097c46e3a94e90e2d3d5629fdef21d1a06098d6.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-098_5a100fcae23e00270ec4daf22cf5440a72811f4a3eea1293dc840d41f3772a7a.jpg b/.raw/MidasFEAAnalysisManual/images/page-098_5a100fcae23e00270ec4daf22cf5440a72811f4a3eea1293dc840d41f3772a7a.jpg new file mode 100644 index 00000000..c79d2043 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-098_5a100fcae23e00270ec4daf22cf5440a72811f4a3eea1293dc840d41f3772a7a.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-098_5e2de30cc999d8d8266e24c9863458e76bdcb2c8c1eeef5b533e809b42d4f59e.jpg b/.raw/MidasFEAAnalysisManual/images/page-098_5e2de30cc999d8d8266e24c9863458e76bdcb2c8c1eeef5b533e809b42d4f59e.jpg new file mode 100644 index 00000000..ebb0e1b5 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-098_5e2de30cc999d8d8266e24c9863458e76bdcb2c8c1eeef5b533e809b42d4f59e.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-098_67d53920fb2ee53a17efe94d9211879e0c6a5a88ab7d1eb1f1646c59916baaad.jpg b/.raw/MidasFEAAnalysisManual/images/page-098_67d53920fb2ee53a17efe94d9211879e0c6a5a88ab7d1eb1f1646c59916baaad.jpg new file mode 100644 index 00000000..0c1ddb1b Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-098_67d53920fb2ee53a17efe94d9211879e0c6a5a88ab7d1eb1f1646c59916baaad.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-098_7043c1ffe6da3c3c06d0bcc9ef25bb24bbf35e220891fe8f8e87aaf45045cf36.jpg b/.raw/MidasFEAAnalysisManual/images/page-098_7043c1ffe6da3c3c06d0bcc9ef25bb24bbf35e220891fe8f8e87aaf45045cf36.jpg new file mode 100644 index 00000000..39d41e3e Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-098_7043c1ffe6da3c3c06d0bcc9ef25bb24bbf35e220891fe8f8e87aaf45045cf36.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-098_ae009da0da559c121fa907212fed3b578f79a61471a8ec2bcec4e1e861088a38.jpg b/.raw/MidasFEAAnalysisManual/images/page-098_ae009da0da559c121fa907212fed3b578f79a61471a8ec2bcec4e1e861088a38.jpg new file mode 100644 index 00000000..a8b90209 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-098_ae009da0da559c121fa907212fed3b578f79a61471a8ec2bcec4e1e861088a38.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-098_c0ae0535f4de1775ef13af1ee8c937180e94570e861a672e434341c499f25e39.jpg b/.raw/MidasFEAAnalysisManual/images/page-098_c0ae0535f4de1775ef13af1ee8c937180e94570e861a672e434341c499f25e39.jpg new file mode 100644 index 00000000..a62cbf59 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-098_c0ae0535f4de1775ef13af1ee8c937180e94570e861a672e434341c499f25e39.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-098_db13926d298d9c1e60972388486c942b682c67b780ab6529f451de2007548fef.jpg b/.raw/MidasFEAAnalysisManual/images/page-098_db13926d298d9c1e60972388486c942b682c67b780ab6529f451de2007548fef.jpg new file mode 100644 index 00000000..0ef7117a Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-098_db13926d298d9c1e60972388486c942b682c67b780ab6529f451de2007548fef.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-098_e3114e1ad744315d374b99603c95597b4ace0cbc7088a0cf059504534b724228.jpg b/.raw/MidasFEAAnalysisManual/images/page-098_e3114e1ad744315d374b99603c95597b4ace0cbc7088a0cf059504534b724228.jpg new file mode 100644 index 00000000..84ee14c0 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-098_e3114e1ad744315d374b99603c95597b4ace0cbc7088a0cf059504534b724228.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-099_e6d7a4b181c224ede49226fed7552b421277e886416b8bf9b6098fafee5643d2.jpg b/.raw/MidasFEAAnalysisManual/images/page-099_e6d7a4b181c224ede49226fed7552b421277e886416b8bf9b6098fafee5643d2.jpg new file mode 100644 index 00000000..4d54949f Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-099_e6d7a4b181c224ede49226fed7552b421277e886416b8bf9b6098fafee5643d2.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-100_0da1ef5c68f4f5c806b0da3fecbc6bd8a13d1d7ee1ff8fd1eea497975176dda1.jpg b/.raw/MidasFEAAnalysisManual/images/page-100_0da1ef5c68f4f5c806b0da3fecbc6bd8a13d1d7ee1ff8fd1eea497975176dda1.jpg new file mode 100644 index 00000000..a8e28930 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-100_0da1ef5c68f4f5c806b0da3fecbc6bd8a13d1d7ee1ff8fd1eea497975176dda1.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-100_1d33c4886a54d9e5e0ffaf3b6ad4fe65bf86c0eb883b166867cfaa3536d22b0b.jpg b/.raw/MidasFEAAnalysisManual/images/page-100_1d33c4886a54d9e5e0ffaf3b6ad4fe65bf86c0eb883b166867cfaa3536d22b0b.jpg new file mode 100644 index 00000000..e77de2ce Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-100_1d33c4886a54d9e5e0ffaf3b6ad4fe65bf86c0eb883b166867cfaa3536d22b0b.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-100_5649f6861cb3268cc2bdd4834327752d0767357d3a267914af7a8a36bf6750b9.jpg b/.raw/MidasFEAAnalysisManual/images/page-100_5649f6861cb3268cc2bdd4834327752d0767357d3a267914af7a8a36bf6750b9.jpg new file mode 100644 index 00000000..bda8cb10 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-100_5649f6861cb3268cc2bdd4834327752d0767357d3a267914af7a8a36bf6750b9.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-100_905968f8bc96e1e3bced9e4f6ffbfdfbb9401495005b82e355e062ded815a2bc.jpg b/.raw/MidasFEAAnalysisManual/images/page-100_905968f8bc96e1e3bced9e4f6ffbfdfbb9401495005b82e355e062ded815a2bc.jpg new file mode 100644 index 00000000..de29eeb0 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-100_905968f8bc96e1e3bced9e4f6ffbfdfbb9401495005b82e355e062ded815a2bc.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-100_c0f53f7e590431a6fbb3eb5ca550a4f664e6bf0864ce11afb34e8c22574bed29.jpg b/.raw/MidasFEAAnalysisManual/images/page-100_c0f53f7e590431a6fbb3eb5ca550a4f664e6bf0864ce11afb34e8c22574bed29.jpg new file mode 100644 index 00000000..fa274514 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-100_c0f53f7e590431a6fbb3eb5ca550a4f664e6bf0864ce11afb34e8c22574bed29.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-101_0a0ee70c26b0bd0d367c6c507b216fee82093b07270a8f5cf6a691d27609e6d3.jpg b/.raw/MidasFEAAnalysisManual/images/page-101_0a0ee70c26b0bd0d367c6c507b216fee82093b07270a8f5cf6a691d27609e6d3.jpg new file mode 100644 index 00000000..96ceed8d Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-101_0a0ee70c26b0bd0d367c6c507b216fee82093b07270a8f5cf6a691d27609e6d3.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-101_b81a5a5276f81e9ab5187243ea10ee774e102f6b8c081486e9dabeb5e015a2a7.jpg b/.raw/MidasFEAAnalysisManual/images/page-101_b81a5a5276f81e9ab5187243ea10ee774e102f6b8c081486e9dabeb5e015a2a7.jpg new file mode 100644 index 00000000..9f23e219 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-101_b81a5a5276f81e9ab5187243ea10ee774e102f6b8c081486e9dabeb5e015a2a7.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-102_5f880a4f4eee29c77e03eddebe90ccb228592e3b5b92a6c0feb24629c6c1eaf6.jpg b/.raw/MidasFEAAnalysisManual/images/page-102_5f880a4f4eee29c77e03eddebe90ccb228592e3b5b92a6c0feb24629c6c1eaf6.jpg new file mode 100644 index 00000000..df65c04c Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-102_5f880a4f4eee29c77e03eddebe90ccb228592e3b5b92a6c0feb24629c6c1eaf6.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-102_c2f28efd264b93a0e7d8dc10ce4dc1ac75f57480f6cd91186738ef3517ba9d33.jpg b/.raw/MidasFEAAnalysisManual/images/page-102_c2f28efd264b93a0e7d8dc10ce4dc1ac75f57480f6cd91186738ef3517ba9d33.jpg new file mode 100644 index 00000000..da77568d Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-102_c2f28efd264b93a0e7d8dc10ce4dc1ac75f57480f6cd91186738ef3517ba9d33.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-102_c622a48f238209186e98998b23e9ed90707d6b77ab5cc667facb63bd9733d696.jpg b/.raw/MidasFEAAnalysisManual/images/page-102_c622a48f238209186e98998b23e9ed90707d6b77ab5cc667facb63bd9733d696.jpg new file mode 100644 index 00000000..9465caa8 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-102_c622a48f238209186e98998b23e9ed90707d6b77ab5cc667facb63bd9733d696.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-103_461d3d1cd1dceee6c9b78326f8fd26835213a468126ee716b66d0f70f19284f5.jpg b/.raw/MidasFEAAnalysisManual/images/page-103_461d3d1cd1dceee6c9b78326f8fd26835213a468126ee716b66d0f70f19284f5.jpg new file mode 100644 index 00000000..67f4b684 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-103_461d3d1cd1dceee6c9b78326f8fd26835213a468126ee716b66d0f70f19284f5.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-103_b927e7fc78481ecd1cc18b07444bfa773c6cec727e65b2b799a1fa43964fc1d3.jpg b/.raw/MidasFEAAnalysisManual/images/page-103_b927e7fc78481ecd1cc18b07444bfa773c6cec727e65b2b799a1fa43964fc1d3.jpg new file mode 100644 index 00000000..626b7c3b Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-103_b927e7fc78481ecd1cc18b07444bfa773c6cec727e65b2b799a1fa43964fc1d3.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-105_660efba90efd5c38ce42615fcaee02f77540c489be80ef2ccbd67bda51d9ffb5.jpg b/.raw/MidasFEAAnalysisManual/images/page-105_660efba90efd5c38ce42615fcaee02f77540c489be80ef2ccbd67bda51d9ffb5.jpg new file mode 100644 index 00000000..821dd474 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-105_660efba90efd5c38ce42615fcaee02f77540c489be80ef2ccbd67bda51d9ffb5.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-105_679c0b39353a121b77e483a960a4a9f90e63d2071b4bacd786706674dfd347c6.jpg b/.raw/MidasFEAAnalysisManual/images/page-105_679c0b39353a121b77e483a960a4a9f90e63d2071b4bacd786706674dfd347c6.jpg new file mode 100644 index 00000000..0df3c8ad Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-105_679c0b39353a121b77e483a960a4a9f90e63d2071b4bacd786706674dfd347c6.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-105_77405d30d36b083170e2747acb3fa8cea4f589094e31efaba874dbb79c32ac88.jpg b/.raw/MidasFEAAnalysisManual/images/page-105_77405d30d36b083170e2747acb3fa8cea4f589094e31efaba874dbb79c32ac88.jpg new file mode 100644 index 00000000..61e290e0 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-105_77405d30d36b083170e2747acb3fa8cea4f589094e31efaba874dbb79c32ac88.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-105_9e8cfc635cae42b0fc2d808da373cb636d6c69163086b68c7cf5dbe12f82383c.jpg b/.raw/MidasFEAAnalysisManual/images/page-105_9e8cfc635cae42b0fc2d808da373cb636d6c69163086b68c7cf5dbe12f82383c.jpg new file mode 100644 index 00000000..1c58828d Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-105_9e8cfc635cae42b0fc2d808da373cb636d6c69163086b68c7cf5dbe12f82383c.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-105_bb07370b2e56cb266a80a2f5b878950399845a39ec0a81cafc124c9fe82a44c7.jpg b/.raw/MidasFEAAnalysisManual/images/page-105_bb07370b2e56cb266a80a2f5b878950399845a39ec0a81cafc124c9fe82a44c7.jpg new file mode 100644 index 00000000..8bfcfed5 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-105_bb07370b2e56cb266a80a2f5b878950399845a39ec0a81cafc124c9fe82a44c7.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-105_d134c86a8b2e963994a4d73738c88c8e885f82e18e17247a0cf8f736a07ead5e.jpg b/.raw/MidasFEAAnalysisManual/images/page-105_d134c86a8b2e963994a4d73738c88c8e885f82e18e17247a0cf8f736a07ead5e.jpg new file mode 100644 index 00000000..ecbb825c Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-105_d134c86a8b2e963994a4d73738c88c8e885f82e18e17247a0cf8f736a07ead5e.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-105_edf72b3997307122f1f3eef315c563b52d0ca37d4bb91811831a3d92aa2815d9.jpg b/.raw/MidasFEAAnalysisManual/images/page-105_edf72b3997307122f1f3eef315c563b52d0ca37d4bb91811831a3d92aa2815d9.jpg new file mode 100644 index 00000000..721a7a97 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-105_edf72b3997307122f1f3eef315c563b52d0ca37d4bb91811831a3d92aa2815d9.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-106_25874ebd1c4d794fa2fba1dd652cfff328c6d09adc7e8cbd56b1732c4db905f2.jpg b/.raw/MidasFEAAnalysisManual/images/page-106_25874ebd1c4d794fa2fba1dd652cfff328c6d09adc7e8cbd56b1732c4db905f2.jpg new file mode 100644 index 00000000..7d433c30 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-106_25874ebd1c4d794fa2fba1dd652cfff328c6d09adc7e8cbd56b1732c4db905f2.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-106_2bfd6d3d2ca9bf9e72af1fc9a56a7edbafa58f04d1e285a815eebdb7021f61a5.jpg b/.raw/MidasFEAAnalysisManual/images/page-106_2bfd6d3d2ca9bf9e72af1fc9a56a7edbafa58f04d1e285a815eebdb7021f61a5.jpg new file mode 100644 index 00000000..c529c5b9 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-106_2bfd6d3d2ca9bf9e72af1fc9a56a7edbafa58f04d1e285a815eebdb7021f61a5.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-106_309c9586399ebee9c5fbd9cc1c28b5dcf70692c94db6174788bb04165af8e6f8.jpg b/.raw/MidasFEAAnalysisManual/images/page-106_309c9586399ebee9c5fbd9cc1c28b5dcf70692c94db6174788bb04165af8e6f8.jpg new file mode 100644 index 00000000..87c9fe4a Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-106_309c9586399ebee9c5fbd9cc1c28b5dcf70692c94db6174788bb04165af8e6f8.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-106_43ac27b976dc7a06927b4d67c65f1962f77b3879657e4e1aa91d41964bc3f5da.jpg b/.raw/MidasFEAAnalysisManual/images/page-106_43ac27b976dc7a06927b4d67c65f1962f77b3879657e4e1aa91d41964bc3f5da.jpg new file mode 100644 index 00000000..672b59d0 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-106_43ac27b976dc7a06927b4d67c65f1962f77b3879657e4e1aa91d41964bc3f5da.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-106_8ca532285e84d84b264f54da8cf7f5150812bf5903b6400739e445da03f36cb2.jpg b/.raw/MidasFEAAnalysisManual/images/page-106_8ca532285e84d84b264f54da8cf7f5150812bf5903b6400739e445da03f36cb2.jpg new file mode 100644 index 00000000..b6fb2838 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-106_8ca532285e84d84b264f54da8cf7f5150812bf5903b6400739e445da03f36cb2.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-106_bb6e0ed9a7a03489648452c9efd3322f8a83442c12007537397170fd710468f2.jpg b/.raw/MidasFEAAnalysisManual/images/page-106_bb6e0ed9a7a03489648452c9efd3322f8a83442c12007537397170fd710468f2.jpg new file mode 100644 index 00000000..3ded5105 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-106_bb6e0ed9a7a03489648452c9efd3322f8a83442c12007537397170fd710468f2.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-106_ce93a84d6c1a698a5ca45bda76dba11f1fcfe1365d5b36c552101c9652ce586e.jpg b/.raw/MidasFEAAnalysisManual/images/page-106_ce93a84d6c1a698a5ca45bda76dba11f1fcfe1365d5b36c552101c9652ce586e.jpg new file mode 100644 index 00000000..bf98e298 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-106_ce93a84d6c1a698a5ca45bda76dba11f1fcfe1365d5b36c552101c9652ce586e.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-106_f9860bc49bcba63b569c9bc0ca542906bac456d635579fe29c2527e103661e96.jpg b/.raw/MidasFEAAnalysisManual/images/page-106_f9860bc49bcba63b569c9bc0ca542906bac456d635579fe29c2527e103661e96.jpg new file mode 100644 index 00000000..3345ec34 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-106_f9860bc49bcba63b569c9bc0ca542906bac456d635579fe29c2527e103661e96.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-107_09c0770a4e6a6f4c9dc0e8497564cff3b05437a872dbe043d8d42c9cc79dda38.jpg b/.raw/MidasFEAAnalysisManual/images/page-107_09c0770a4e6a6f4c9dc0e8497564cff3b05437a872dbe043d8d42c9cc79dda38.jpg new file mode 100644 index 00000000..6ee71031 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-107_09c0770a4e6a6f4c9dc0e8497564cff3b05437a872dbe043d8d42c9cc79dda38.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-107_2250621223930a220753d4c6e28deed32b2f67f0d3d007a3aebc8259180a5545.jpg b/.raw/MidasFEAAnalysisManual/images/page-107_2250621223930a220753d4c6e28deed32b2f67f0d3d007a3aebc8259180a5545.jpg new file mode 100644 index 00000000..4a1311c7 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-107_2250621223930a220753d4c6e28deed32b2f67f0d3d007a3aebc8259180a5545.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-107_2888f6275dacdb8ffc787f5ef7a9425f8cd9a32f6d06ee9af95abdaec94fe403.jpg b/.raw/MidasFEAAnalysisManual/images/page-107_2888f6275dacdb8ffc787f5ef7a9425f8cd9a32f6d06ee9af95abdaec94fe403.jpg new file mode 100644 index 00000000..0419c1dc Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-107_2888f6275dacdb8ffc787f5ef7a9425f8cd9a32f6d06ee9af95abdaec94fe403.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-107_4974033af209ea1287e112289dc4d64491b7f83cd0c4b2bf5b64338a18a4c53b.jpg b/.raw/MidasFEAAnalysisManual/images/page-107_4974033af209ea1287e112289dc4d64491b7f83cd0c4b2bf5b64338a18a4c53b.jpg new file mode 100644 index 00000000..f99e4757 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-107_4974033af209ea1287e112289dc4d64491b7f83cd0c4b2bf5b64338a18a4c53b.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-107_55821648d648c018bb8d02f063abda69658547ad16ed52c8f2b9d0d46d9b22a0.jpg b/.raw/MidasFEAAnalysisManual/images/page-107_55821648d648c018bb8d02f063abda69658547ad16ed52c8f2b9d0d46d9b22a0.jpg new file mode 100644 index 00000000..7daedb54 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-107_55821648d648c018bb8d02f063abda69658547ad16ed52c8f2b9d0d46d9b22a0.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-107_7c5a9495fb46852dc0a51c7c96dab1f7c9453cf5d9a06e9e44ff11b73c0e23fd.jpg b/.raw/MidasFEAAnalysisManual/images/page-107_7c5a9495fb46852dc0a51c7c96dab1f7c9453cf5d9a06e9e44ff11b73c0e23fd.jpg new file mode 100644 index 00000000..366a1869 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-107_7c5a9495fb46852dc0a51c7c96dab1f7c9453cf5d9a06e9e44ff11b73c0e23fd.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-107_86e528c6f1bad22f73d640dfcaa3fcf2a539e520e1ec0c917155bea9ec5a8c08.jpg b/.raw/MidasFEAAnalysisManual/images/page-107_86e528c6f1bad22f73d640dfcaa3fcf2a539e520e1ec0c917155bea9ec5a8c08.jpg new file mode 100644 index 00000000..26dfccdf Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-107_86e528c6f1bad22f73d640dfcaa3fcf2a539e520e1ec0c917155bea9ec5a8c08.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-107_9fe0fcda5c7b99967345703598a36a46895ce0fe736f9440507cbf166593239c.jpg b/.raw/MidasFEAAnalysisManual/images/page-107_9fe0fcda5c7b99967345703598a36a46895ce0fe736f9440507cbf166593239c.jpg new file mode 100644 index 00000000..88c5690a Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-107_9fe0fcda5c7b99967345703598a36a46895ce0fe736f9440507cbf166593239c.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-107_d5958c83449a71ac479a9c949f74682cc0f24a2f0f3f6a20a89713a0d804dcc1.jpg b/.raw/MidasFEAAnalysisManual/images/page-107_d5958c83449a71ac479a9c949f74682cc0f24a2f0f3f6a20a89713a0d804dcc1.jpg new file mode 100644 index 00000000..ef59a052 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-107_d5958c83449a71ac479a9c949f74682cc0f24a2f0f3f6a20a89713a0d804dcc1.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-107_e1656c33430966882ec12d34f7687a381306b0a438aa1a46711d5f83e2f331b1.jpg b/.raw/MidasFEAAnalysisManual/images/page-107_e1656c33430966882ec12d34f7687a381306b0a438aa1a46711d5f83e2f331b1.jpg new file mode 100644 index 00000000..1c70b625 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-107_e1656c33430966882ec12d34f7687a381306b0a438aa1a46711d5f83e2f331b1.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-107_f8c39fcd80e79e99545a3734063fc4c5d0a44e8e0c5c3779b09468163281c44f.jpg b/.raw/MidasFEAAnalysisManual/images/page-107_f8c39fcd80e79e99545a3734063fc4c5d0a44e8e0c5c3779b09468163281c44f.jpg new file mode 100644 index 00000000..55636764 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-107_f8c39fcd80e79e99545a3734063fc4c5d0a44e8e0c5c3779b09468163281c44f.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-107_faebb577588995dc2846c308784903e65fdd1ba3547bc370ab96962b6bafac79.jpg b/.raw/MidasFEAAnalysisManual/images/page-107_faebb577588995dc2846c308784903e65fdd1ba3547bc370ab96962b6bafac79.jpg new file mode 100644 index 00000000..d601e532 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-107_faebb577588995dc2846c308784903e65fdd1ba3547bc370ab96962b6bafac79.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-108_171dd031e19c29c97d82b36dce5f0df139e55473269c8bb2f267f3717c328436.jpg b/.raw/MidasFEAAnalysisManual/images/page-108_171dd031e19c29c97d82b36dce5f0df139e55473269c8bb2f267f3717c328436.jpg new file mode 100644 index 00000000..98e6ca1e Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-108_171dd031e19c29c97d82b36dce5f0df139e55473269c8bb2f267f3717c328436.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-108_1df331388846cc350a5445a80aabdcf64bf9ff8ec719d3a8994b65a5495e8804.jpg b/.raw/MidasFEAAnalysisManual/images/page-108_1df331388846cc350a5445a80aabdcf64bf9ff8ec719d3a8994b65a5495e8804.jpg new file mode 100644 index 00000000..a6ae565e Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-108_1df331388846cc350a5445a80aabdcf64bf9ff8ec719d3a8994b65a5495e8804.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-108_2d801aa97b18ce9508e64a0ee0c34d64771f428eddc4c94573b18b441f4296e6.jpg b/.raw/MidasFEAAnalysisManual/images/page-108_2d801aa97b18ce9508e64a0ee0c34d64771f428eddc4c94573b18b441f4296e6.jpg new file mode 100644 index 00000000..1d3447f9 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-108_2d801aa97b18ce9508e64a0ee0c34d64771f428eddc4c94573b18b441f4296e6.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-108_2df96149ec672efd79336184b1a6a494fec25b754af4de73f21b5c8d17112db8.jpg b/.raw/MidasFEAAnalysisManual/images/page-108_2df96149ec672efd79336184b1a6a494fec25b754af4de73f21b5c8d17112db8.jpg new file mode 100644 index 00000000..ff66a01a Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-108_2df96149ec672efd79336184b1a6a494fec25b754af4de73f21b5c8d17112db8.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-108_313ee286853db153932cfb0e86bd46b4022e97ab056c805b639e4a66ce4592d3.jpg b/.raw/MidasFEAAnalysisManual/images/page-108_313ee286853db153932cfb0e86bd46b4022e97ab056c805b639e4a66ce4592d3.jpg new file mode 100644 index 00000000..d9a18206 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-108_313ee286853db153932cfb0e86bd46b4022e97ab056c805b639e4a66ce4592d3.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-108_45fa126aee98518f2b6cae8c1178e236d8566f82ba5ddf0aee727ddcc6afeae9.jpg b/.raw/MidasFEAAnalysisManual/images/page-108_45fa126aee98518f2b6cae8c1178e236d8566f82ba5ddf0aee727ddcc6afeae9.jpg new file mode 100644 index 00000000..9fb4102e Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-108_45fa126aee98518f2b6cae8c1178e236d8566f82ba5ddf0aee727ddcc6afeae9.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-108_5ea368e029db94fb64835a1afc23fbbd9d571e633c549480687118e4d41eb21d.jpg b/.raw/MidasFEAAnalysisManual/images/page-108_5ea368e029db94fb64835a1afc23fbbd9d571e633c549480687118e4d41eb21d.jpg new file mode 100644 index 00000000..6c95c484 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-108_5ea368e029db94fb64835a1afc23fbbd9d571e633c549480687118e4d41eb21d.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-108_79199f0d8a13baaf0c6e2d69b60a497ac72085a83af3ba8096372cc283c261e5.jpg b/.raw/MidasFEAAnalysisManual/images/page-108_79199f0d8a13baaf0c6e2d69b60a497ac72085a83af3ba8096372cc283c261e5.jpg new file mode 100644 index 00000000..2f9e6c8a Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-108_79199f0d8a13baaf0c6e2d69b60a497ac72085a83af3ba8096372cc283c261e5.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-108_95587d9ae5d963c2df0d4b275484c5b62e495994acaa84c30589a88d82045867.jpg b/.raw/MidasFEAAnalysisManual/images/page-108_95587d9ae5d963c2df0d4b275484c5b62e495994acaa84c30589a88d82045867.jpg new file mode 100644 index 00000000..f8716a44 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-108_95587d9ae5d963c2df0d4b275484c5b62e495994acaa84c30589a88d82045867.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-108_aaec69a49cfa344a93aee1e8f7ef5a663917a48e080045ae9833ac78cd95ad14.jpg b/.raw/MidasFEAAnalysisManual/images/page-108_aaec69a49cfa344a93aee1e8f7ef5a663917a48e080045ae9833ac78cd95ad14.jpg new file mode 100644 index 00000000..d153c773 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-108_aaec69a49cfa344a93aee1e8f7ef5a663917a48e080045ae9833ac78cd95ad14.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-108_b0cf448cceaab86a5930b36cfd277c8049ee904a797d207fe3fbd73010893943.jpg b/.raw/MidasFEAAnalysisManual/images/page-108_b0cf448cceaab86a5930b36cfd277c8049ee904a797d207fe3fbd73010893943.jpg new file mode 100644 index 00000000..34dea79f Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-108_b0cf448cceaab86a5930b36cfd277c8049ee904a797d207fe3fbd73010893943.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-108_b9a2a6d77afee852e4811e1b9515304ffcaed0aeaeb681130e6f9d252b759364.jpg b/.raw/MidasFEAAnalysisManual/images/page-108_b9a2a6d77afee852e4811e1b9515304ffcaed0aeaeb681130e6f9d252b759364.jpg new file mode 100644 index 00000000..e4018d05 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-108_b9a2a6d77afee852e4811e1b9515304ffcaed0aeaeb681130e6f9d252b759364.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-108_c6e17fe1a18a644327f283b65b255ebe540908029c6be6816ff7f48d0ab8d6e0.jpg b/.raw/MidasFEAAnalysisManual/images/page-108_c6e17fe1a18a644327f283b65b255ebe540908029c6be6816ff7f48d0ab8d6e0.jpg new file mode 100644 index 00000000..f422cf24 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-108_c6e17fe1a18a644327f283b65b255ebe540908029c6be6816ff7f48d0ab8d6e0.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-109_35266c60ae42ba666f7d878b8b14f035053b5dfa0e56ae885bb6e81fb4e08d4c.jpg b/.raw/MidasFEAAnalysisManual/images/page-109_35266c60ae42ba666f7d878b8b14f035053b5dfa0e56ae885bb6e81fb4e08d4c.jpg new file mode 100644 index 00000000..8fe9e51b Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-109_35266c60ae42ba666f7d878b8b14f035053b5dfa0e56ae885bb6e81fb4e08d4c.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-109_5350942edee33f83a9c50a79ee89c37e8a750d31b59bc86d486b5c00d60f5f27.jpg b/.raw/MidasFEAAnalysisManual/images/page-109_5350942edee33f83a9c50a79ee89c37e8a750d31b59bc86d486b5c00d60f5f27.jpg new file mode 100644 index 00000000..af9a21af Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-109_5350942edee33f83a9c50a79ee89c37e8a750d31b59bc86d486b5c00d60f5f27.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-109_66ab41e0f6c10fa2a757b85f2a56a7d0c567e64c2cb26191140fdecbd963e755.jpg b/.raw/MidasFEAAnalysisManual/images/page-109_66ab41e0f6c10fa2a757b85f2a56a7d0c567e64c2cb26191140fdecbd963e755.jpg new file mode 100644 index 00000000..cee37b5d Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-109_66ab41e0f6c10fa2a757b85f2a56a7d0c567e64c2cb26191140fdecbd963e755.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-109_6d83879f0d49deb6f373429b2b5e6027ee4958d2e4ed2db86431f388a87e5a70.jpg b/.raw/MidasFEAAnalysisManual/images/page-109_6d83879f0d49deb6f373429b2b5e6027ee4958d2e4ed2db86431f388a87e5a70.jpg new file mode 100644 index 00000000..2873e2b0 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-109_6d83879f0d49deb6f373429b2b5e6027ee4958d2e4ed2db86431f388a87e5a70.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-109_a85a1bc0dccd60fff8f8616804a027b52e1f6622e2adae9bc3febf11158497cc.jpg b/.raw/MidasFEAAnalysisManual/images/page-109_a85a1bc0dccd60fff8f8616804a027b52e1f6622e2adae9bc3febf11158497cc.jpg new file mode 100644 index 00000000..2efc0361 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-109_a85a1bc0dccd60fff8f8616804a027b52e1f6622e2adae9bc3febf11158497cc.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-109_b4c116c5c05d045c40cb365bba1f47f33e09b6ebc4803e5a4c80b624fa064e59.jpg b/.raw/MidasFEAAnalysisManual/images/page-109_b4c116c5c05d045c40cb365bba1f47f33e09b6ebc4803e5a4c80b624fa064e59.jpg new file mode 100644 index 00000000..800fc3ac Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-109_b4c116c5c05d045c40cb365bba1f47f33e09b6ebc4803e5a4c80b624fa064e59.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-109_cccd1e2d52c4662bd3ddf2079657e7ac4d68d94ce8552f9d3406522563f20f30.jpg b/.raw/MidasFEAAnalysisManual/images/page-109_cccd1e2d52c4662bd3ddf2079657e7ac4d68d94ce8552f9d3406522563f20f30.jpg new file mode 100644 index 00000000..1b0324e0 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-109_cccd1e2d52c4662bd3ddf2079657e7ac4d68d94ce8552f9d3406522563f20f30.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-109_d5cac3c6a00454a8947007657921a503c1d9e3e10e70fbe4d18574d702a05d3a.jpg b/.raw/MidasFEAAnalysisManual/images/page-109_d5cac3c6a00454a8947007657921a503c1d9e3e10e70fbe4d18574d702a05d3a.jpg new file mode 100644 index 00000000..5b782506 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-109_d5cac3c6a00454a8947007657921a503c1d9e3e10e70fbe4d18574d702a05d3a.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-110_118c4824748e812677ce8c17a8b118c7a9bd321981bc0a7a750fe24707cd9f31.jpg b/.raw/MidasFEAAnalysisManual/images/page-110_118c4824748e812677ce8c17a8b118c7a9bd321981bc0a7a750fe24707cd9f31.jpg new file mode 100644 index 00000000..15fd05b6 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-110_118c4824748e812677ce8c17a8b118c7a9bd321981bc0a7a750fe24707cd9f31.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-110_2473b7e938dbf5d9ca143a902e50f3e51f9b4ea01fd23bdecd44a4957113689f.jpg b/.raw/MidasFEAAnalysisManual/images/page-110_2473b7e938dbf5d9ca143a902e50f3e51f9b4ea01fd23bdecd44a4957113689f.jpg new file mode 100644 index 00000000..14584c10 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-110_2473b7e938dbf5d9ca143a902e50f3e51f9b4ea01fd23bdecd44a4957113689f.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-110_6f49ed310722e74f1a81e5fb551145a6229f0d4175729a61c5f5e0ceed7c4424.jpg b/.raw/MidasFEAAnalysisManual/images/page-110_6f49ed310722e74f1a81e5fb551145a6229f0d4175729a61c5f5e0ceed7c4424.jpg new file mode 100644 index 00000000..731c99bb Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-110_6f49ed310722e74f1a81e5fb551145a6229f0d4175729a61c5f5e0ceed7c4424.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-110_99bba1b4dfde263720435ff77657593a6c65dfb82bca3607f0d159928d4e14e5.jpg b/.raw/MidasFEAAnalysisManual/images/page-110_99bba1b4dfde263720435ff77657593a6c65dfb82bca3607f0d159928d4e14e5.jpg new file mode 100644 index 00000000..cc65ab0d Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-110_99bba1b4dfde263720435ff77657593a6c65dfb82bca3607f0d159928d4e14e5.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-111_4929811c27bf887d96d8075286357419dc76ae1c3744de7dce9d50396101a4c9.jpg b/.raw/MidasFEAAnalysisManual/images/page-111_4929811c27bf887d96d8075286357419dc76ae1c3744de7dce9d50396101a4c9.jpg new file mode 100644 index 00000000..03e6c489 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-111_4929811c27bf887d96d8075286357419dc76ae1c3744de7dce9d50396101a4c9.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-111_55f36e1d8e63069eae0f5b52216f02efbc63d23d585e9180a4f22d2be896b1f2.jpg b/.raw/MidasFEAAnalysisManual/images/page-111_55f36e1d8e63069eae0f5b52216f02efbc63d23d585e9180a4f22d2be896b1f2.jpg new file mode 100644 index 00000000..c7342a71 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-111_55f36e1d8e63069eae0f5b52216f02efbc63d23d585e9180a4f22d2be896b1f2.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-111_748d8462f2fa684c58092c1a1b7dfa0ae5252043563245685fd98bb3015c17a9.jpg b/.raw/MidasFEAAnalysisManual/images/page-111_748d8462f2fa684c58092c1a1b7dfa0ae5252043563245685fd98bb3015c17a9.jpg new file mode 100644 index 00000000..cab99441 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-111_748d8462f2fa684c58092c1a1b7dfa0ae5252043563245685fd98bb3015c17a9.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-111_f03427f403ed2240166a308d1d162d841c684483d45781204ea60ad3e4e1235a.jpg b/.raw/MidasFEAAnalysisManual/images/page-111_f03427f403ed2240166a308d1d162d841c684483d45781204ea60ad3e4e1235a.jpg new file mode 100644 index 00000000..a9f23631 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-111_f03427f403ed2240166a308d1d162d841c684483d45781204ea60ad3e4e1235a.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-111_f255019e6be750e2fac236258d2bbaa398e8191c9cbc718cc4894dee40bd8b35.jpg b/.raw/MidasFEAAnalysisManual/images/page-111_f255019e6be750e2fac236258d2bbaa398e8191c9cbc718cc4894dee40bd8b35.jpg new file mode 100644 index 00000000..0b7f1312 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-111_f255019e6be750e2fac236258d2bbaa398e8191c9cbc718cc4894dee40bd8b35.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-111_f870f95d72fe7fee7e6a05db3175eb4cf6253fc5ffa1895bed6d96bf4ee11a91.jpg b/.raw/MidasFEAAnalysisManual/images/page-111_f870f95d72fe7fee7e6a05db3175eb4cf6253fc5ffa1895bed6d96bf4ee11a91.jpg new file mode 100644 index 00000000..d1ce0b45 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-111_f870f95d72fe7fee7e6a05db3175eb4cf6253fc5ffa1895bed6d96bf4ee11a91.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-112_234bfb132cbb20c9ac9357617d4a41636b83868f96df0aba7a611b680140134a.jpg b/.raw/MidasFEAAnalysisManual/images/page-112_234bfb132cbb20c9ac9357617d4a41636b83868f96df0aba7a611b680140134a.jpg new file mode 100644 index 00000000..2c4291f3 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-112_234bfb132cbb20c9ac9357617d4a41636b83868f96df0aba7a611b680140134a.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-112_3620732d334a6fd40b225da786f4bafcc4b4eeb346c78ffa8a311f2d893b6f1f.jpg b/.raw/MidasFEAAnalysisManual/images/page-112_3620732d334a6fd40b225da786f4bafcc4b4eeb346c78ffa8a311f2d893b6f1f.jpg new file mode 100644 index 00000000..188c07de Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-112_3620732d334a6fd40b225da786f4bafcc4b4eeb346c78ffa8a311f2d893b6f1f.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-112_afe23573123e666d6d880ef568f5a13bd8de094df9b5187e7672b0a19f95f66b.jpg b/.raw/MidasFEAAnalysisManual/images/page-112_afe23573123e666d6d880ef568f5a13bd8de094df9b5187e7672b0a19f95f66b.jpg new file mode 100644 index 00000000..4e05c27b Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-112_afe23573123e666d6d880ef568f5a13bd8de094df9b5187e7672b0a19f95f66b.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-112_bd8ae198b60ce4c51860e272e78b81080caa3770984ceb8f98e91b9749a76c08.jpg b/.raw/MidasFEAAnalysisManual/images/page-112_bd8ae198b60ce4c51860e272e78b81080caa3770984ceb8f98e91b9749a76c08.jpg new file mode 100644 index 00000000..a2d45a23 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-112_bd8ae198b60ce4c51860e272e78b81080caa3770984ceb8f98e91b9749a76c08.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-112_cbe7602d4653450582c0d1b9dd0bea17f2578dad55aa69261b0d4a3aed8f5e1f.jpg b/.raw/MidasFEAAnalysisManual/images/page-112_cbe7602d4653450582c0d1b9dd0bea17f2578dad55aa69261b0d4a3aed8f5e1f.jpg new file mode 100644 index 00000000..bd61fa86 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-112_cbe7602d4653450582c0d1b9dd0bea17f2578dad55aa69261b0d4a3aed8f5e1f.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-112_e3383bb23f4e70942c5dfdf97d3d78758e021143ea7cda1d43727aa8346e3868.jpg b/.raw/MidasFEAAnalysisManual/images/page-112_e3383bb23f4e70942c5dfdf97d3d78758e021143ea7cda1d43727aa8346e3868.jpg new file mode 100644 index 00000000..a0bce9fc Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-112_e3383bb23f4e70942c5dfdf97d3d78758e021143ea7cda1d43727aa8346e3868.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-113_faf9485da0e41ed9bddcfe72d42f2d7f5b47a8f4721709e908638851cd01f4d3.jpg b/.raw/MidasFEAAnalysisManual/images/page-113_faf9485da0e41ed9bddcfe72d42f2d7f5b47a8f4721709e908638851cd01f4d3.jpg new file mode 100644 index 00000000..3f6ade6e Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-113_faf9485da0e41ed9bddcfe72d42f2d7f5b47a8f4721709e908638851cd01f4d3.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-115_3b9e8e9d286e997ab5594e659d9e828aa0a33ac0033b6501532859a3058ca669.jpg b/.raw/MidasFEAAnalysisManual/images/page-115_3b9e8e9d286e997ab5594e659d9e828aa0a33ac0033b6501532859a3058ca669.jpg new file mode 100644 index 00000000..a390aacf Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-115_3b9e8e9d286e997ab5594e659d9e828aa0a33ac0033b6501532859a3058ca669.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-116_e3708cb4950bd2d947740f8cd6ece82d4b12544d95ab8254d2bf7ea6419366aa.jpg b/.raw/MidasFEAAnalysisManual/images/page-116_e3708cb4950bd2d947740f8cd6ece82d4b12544d95ab8254d2bf7ea6419366aa.jpg new file mode 100644 index 00000000..26c420e0 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-116_e3708cb4950bd2d947740f8cd6ece82d4b12544d95ab8254d2bf7ea6419366aa.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-117_a5f53cdd96697fe7286946e553eac893eddff5434c6e40cecbe885a9e0be55e1.jpg b/.raw/MidasFEAAnalysisManual/images/page-117_a5f53cdd96697fe7286946e553eac893eddff5434c6e40cecbe885a9e0be55e1.jpg new file mode 100644 index 00000000..bb40c6d9 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-117_a5f53cdd96697fe7286946e553eac893eddff5434c6e40cecbe885a9e0be55e1.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-119_0cafb295ba3cedab0dcf091cd408f183151fd9e4a136fa97bdb60bd5c3dc889a.jpg b/.raw/MidasFEAAnalysisManual/images/page-119_0cafb295ba3cedab0dcf091cd408f183151fd9e4a136fa97bdb60bd5c3dc889a.jpg new file mode 100644 index 00000000..173e7a73 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-119_0cafb295ba3cedab0dcf091cd408f183151fd9e4a136fa97bdb60bd5c3dc889a.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-119_7451781d3af4b49079b2a870e388e79e4a8c30471af4bbc4cac1f062652b1c5e.jpg b/.raw/MidasFEAAnalysisManual/images/page-119_7451781d3af4b49079b2a870e388e79e4a8c30471af4bbc4cac1f062652b1c5e.jpg new file mode 100644 index 00000000..85ba9037 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-119_7451781d3af4b49079b2a870e388e79e4a8c30471af4bbc4cac1f062652b1c5e.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-119_8835e5b0e687feb6e1cc1e61752ac19cabe779176c4f9d5cd237ba3da237f77d.jpg b/.raw/MidasFEAAnalysisManual/images/page-119_8835e5b0e687feb6e1cc1e61752ac19cabe779176c4f9d5cd237ba3da237f77d.jpg new file mode 100644 index 00000000..0fb1348f Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-119_8835e5b0e687feb6e1cc1e61752ac19cabe779176c4f9d5cd237ba3da237f77d.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-119_9a501d12fc99fc6a187ebf8d2fb97fe7e500c04980cf489243f8e7a703048be4.jpg b/.raw/MidasFEAAnalysisManual/images/page-119_9a501d12fc99fc6a187ebf8d2fb97fe7e500c04980cf489243f8e7a703048be4.jpg new file mode 100644 index 00000000..243a1270 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-119_9a501d12fc99fc6a187ebf8d2fb97fe7e500c04980cf489243f8e7a703048be4.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-119_ae57ce595b5de34c6f44ca5146bec5da7767d286b1f88b797b7c2caaf57d858b.jpg b/.raw/MidasFEAAnalysisManual/images/page-119_ae57ce595b5de34c6f44ca5146bec5da7767d286b1f88b797b7c2caaf57d858b.jpg new file mode 100644 index 00000000..c7f268fa Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-119_ae57ce595b5de34c6f44ca5146bec5da7767d286b1f88b797b7c2caaf57d858b.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-119_ca23f29d34c16d76794367c41a41d02cb9a8b233bece144bc2cf941f305f2725.jpg b/.raw/MidasFEAAnalysisManual/images/page-119_ca23f29d34c16d76794367c41a41d02cb9a8b233bece144bc2cf941f305f2725.jpg new file mode 100644 index 00000000..a0cf1f1d Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-119_ca23f29d34c16d76794367c41a41d02cb9a8b233bece144bc2cf941f305f2725.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-119_d35bec5819d67c92e8dc646b000e453ca0c711d72e2c7c3e9a02568b234bb6c7.jpg b/.raw/MidasFEAAnalysisManual/images/page-119_d35bec5819d67c92e8dc646b000e453ca0c711d72e2c7c3e9a02568b234bb6c7.jpg new file mode 100644 index 00000000..0caa5945 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-119_d35bec5819d67c92e8dc646b000e453ca0c711d72e2c7c3e9a02568b234bb6c7.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-119_e2fc0f3a9a44d260573011e59d97ebca4b0c17ae3d86b12cb0cca1647cb0e68c.jpg b/.raw/MidasFEAAnalysisManual/images/page-119_e2fc0f3a9a44d260573011e59d97ebca4b0c17ae3d86b12cb0cca1647cb0e68c.jpg new file mode 100644 index 00000000..42a68778 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-119_e2fc0f3a9a44d260573011e59d97ebca4b0c17ae3d86b12cb0cca1647cb0e68c.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-119_f2eae4bac4ff26a5290457a951dc01f988edd17e8777328ea3dfc6762a5492a4.jpg b/.raw/MidasFEAAnalysisManual/images/page-119_f2eae4bac4ff26a5290457a951dc01f988edd17e8777328ea3dfc6762a5492a4.jpg new file mode 100644 index 00000000..871fb035 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-119_f2eae4bac4ff26a5290457a951dc01f988edd17e8777328ea3dfc6762a5492a4.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-120_d6afb926aeb94be394324dd675cdbd82ca75fd07c2c7ebcefe61024918810710.jpg b/.raw/MidasFEAAnalysisManual/images/page-120_d6afb926aeb94be394324dd675cdbd82ca75fd07c2c7ebcefe61024918810710.jpg new file mode 100644 index 00000000..a91c5103 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-120_d6afb926aeb94be394324dd675cdbd82ca75fd07c2c7ebcefe61024918810710.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-120_d8b1dd1ae3ece5e9b419f71ca44d2235bbf896ff2f17c4196ddff65466ac91a5.jpg b/.raw/MidasFEAAnalysisManual/images/page-120_d8b1dd1ae3ece5e9b419f71ca44d2235bbf896ff2f17c4196ddff65466ac91a5.jpg new file mode 100644 index 00000000..4e711c77 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-120_d8b1dd1ae3ece5e9b419f71ca44d2235bbf896ff2f17c4196ddff65466ac91a5.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-120_ec20e0487fdb00f73ea78f318b298325dd82c0f888f1bf5e8ee2e9dc1f10ca45.jpg b/.raw/MidasFEAAnalysisManual/images/page-120_ec20e0487fdb00f73ea78f318b298325dd82c0f888f1bf5e8ee2e9dc1f10ca45.jpg new file mode 100644 index 00000000..cd7d3c6b Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-120_ec20e0487fdb00f73ea78f318b298325dd82c0f888f1bf5e8ee2e9dc1f10ca45.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-120_fdf605ce37e64c97a18dda58897c76cec0c6f33509d0a250e356187e303f1c45.jpg b/.raw/MidasFEAAnalysisManual/images/page-120_fdf605ce37e64c97a18dda58897c76cec0c6f33509d0a250e356187e303f1c45.jpg new file mode 100644 index 00000000..3f8ec7fc Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-120_fdf605ce37e64c97a18dda58897c76cec0c6f33509d0a250e356187e303f1c45.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-122_009ec7b516cd554fcb8146596056e2bb38cc58e0972aaa4dbb7b03b75c5e1342.jpg b/.raw/MidasFEAAnalysisManual/images/page-122_009ec7b516cd554fcb8146596056e2bb38cc58e0972aaa4dbb7b03b75c5e1342.jpg new file mode 100644 index 00000000..4b0efa0b Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-122_009ec7b516cd554fcb8146596056e2bb38cc58e0972aaa4dbb7b03b75c5e1342.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-122_e48a342a9a15da177444c3442daf8b9b9be764f44cd6d2484d4a6674957beb71.jpg b/.raw/MidasFEAAnalysisManual/images/page-122_e48a342a9a15da177444c3442daf8b9b9be764f44cd6d2484d4a6674957beb71.jpg new file mode 100644 index 00000000..f55d9ef7 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-122_e48a342a9a15da177444c3442daf8b9b9be764f44cd6d2484d4a6674957beb71.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-124_359c7d3bbc0a8d0427af0397297d639ae2fb685ca5a056f25674310152542efa.jpg b/.raw/MidasFEAAnalysisManual/images/page-124_359c7d3bbc0a8d0427af0397297d639ae2fb685ca5a056f25674310152542efa.jpg new file mode 100644 index 00000000..fbf3cf25 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-124_359c7d3bbc0a8d0427af0397297d639ae2fb685ca5a056f25674310152542efa.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-124_f16f876e2fefbe4b62d78fc44055c832ee54d5a8a525319f956cc9766f941b60.jpg b/.raw/MidasFEAAnalysisManual/images/page-124_f16f876e2fefbe4b62d78fc44055c832ee54d5a8a525319f956cc9766f941b60.jpg new file mode 100644 index 00000000..22e558a3 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-124_f16f876e2fefbe4b62d78fc44055c832ee54d5a8a525319f956cc9766f941b60.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-126_c4caa76cf4107f88c92abcba06309f56f1403682b2bb241f1f9ab6feabef8708.jpg b/.raw/MidasFEAAnalysisManual/images/page-126_c4caa76cf4107f88c92abcba06309f56f1403682b2bb241f1f9ab6feabef8708.jpg new file mode 100644 index 00000000..7adec090 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-126_c4caa76cf4107f88c92abcba06309f56f1403682b2bb241f1f9ab6feabef8708.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-128_423d7d3aabbeb5e8c24b46d2922da963f6f284b90cda0ff97f4dc8015c8f02f3.jpg b/.raw/MidasFEAAnalysisManual/images/page-128_423d7d3aabbeb5e8c24b46d2922da963f6f284b90cda0ff97f4dc8015c8f02f3.jpg new file mode 100644 index 00000000..80e41351 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-128_423d7d3aabbeb5e8c24b46d2922da963f6f284b90cda0ff97f4dc8015c8f02f3.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-128_516397b8c26b05c5b74095a3967c987e6295120d0e5e3e447b1e0b70d93e563d.jpg b/.raw/MidasFEAAnalysisManual/images/page-128_516397b8c26b05c5b74095a3967c987e6295120d0e5e3e447b1e0b70d93e563d.jpg new file mode 100644 index 00000000..53a06365 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-128_516397b8c26b05c5b74095a3967c987e6295120d0e5e3e447b1e0b70d93e563d.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-129_2b809ab68a5727f36e282223be9e14682ea926b074a013d0fd4e961016db5be5.jpg b/.raw/MidasFEAAnalysisManual/images/page-129_2b809ab68a5727f36e282223be9e14682ea926b074a013d0fd4e961016db5be5.jpg new file mode 100644 index 00000000..7a4c89fd Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-129_2b809ab68a5727f36e282223be9e14682ea926b074a013d0fd4e961016db5be5.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-129_3fc6eeb3ec90e5c881baa2859e559a48c226621e45e1355fcda4a033596b3f5f.jpg b/.raw/MidasFEAAnalysisManual/images/page-129_3fc6eeb3ec90e5c881baa2859e559a48c226621e45e1355fcda4a033596b3f5f.jpg new file mode 100644 index 00000000..a3d17458 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-129_3fc6eeb3ec90e5c881baa2859e559a48c226621e45e1355fcda4a033596b3f5f.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-129_489186395938e3019b2ba63a9dfcfeb4122f4c97f25fa2ecf747e06b8c9b705e.jpg b/.raw/MidasFEAAnalysisManual/images/page-129_489186395938e3019b2ba63a9dfcfeb4122f4c97f25fa2ecf747e06b8c9b705e.jpg new file mode 100644 index 00000000..4bfbed2a Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-129_489186395938e3019b2ba63a9dfcfeb4122f4c97f25fa2ecf747e06b8c9b705e.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-129_6418e1bbe2e0c1d92b27a0a715e5b003e5be26a37cf2a4b51bcd08e26797476f.jpg b/.raw/MidasFEAAnalysisManual/images/page-129_6418e1bbe2e0c1d92b27a0a715e5b003e5be26a37cf2a4b51bcd08e26797476f.jpg new file mode 100644 index 00000000..816b4a47 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-129_6418e1bbe2e0c1d92b27a0a715e5b003e5be26a37cf2a4b51bcd08e26797476f.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-130_5a3e60f7da6b137d7de7bc0438d09a8b72d8b0a82650772b418492116ed8ce52.jpg b/.raw/MidasFEAAnalysisManual/images/page-130_5a3e60f7da6b137d7de7bc0438d09a8b72d8b0a82650772b418492116ed8ce52.jpg new file mode 100644 index 00000000..7603b884 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-130_5a3e60f7da6b137d7de7bc0438d09a8b72d8b0a82650772b418492116ed8ce52.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-130_84f703cf28f23e82e3186b7e0d0fa1cc2202c324b6e0b83228e5d0bc0b15b58a.jpg b/.raw/MidasFEAAnalysisManual/images/page-130_84f703cf28f23e82e3186b7e0d0fa1cc2202c324b6e0b83228e5d0bc0b15b58a.jpg new file mode 100644 index 00000000..83f07df4 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-130_84f703cf28f23e82e3186b7e0d0fa1cc2202c324b6e0b83228e5d0bc0b15b58a.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-130_889bb0a0dea589c451d1c4c0f4c8d28c11f657b2d9cf6a3263c32e79ee41258c.jpg b/.raw/MidasFEAAnalysisManual/images/page-130_889bb0a0dea589c451d1c4c0f4c8d28c11f657b2d9cf6a3263c32e79ee41258c.jpg new file mode 100644 index 00000000..a8a6a273 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-130_889bb0a0dea589c451d1c4c0f4c8d28c11f657b2d9cf6a3263c32e79ee41258c.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-132_13bf5fdd82e2be27dac645575710efeca3262948fba8ea74dc4abf8836939066.jpg b/.raw/MidasFEAAnalysisManual/images/page-132_13bf5fdd82e2be27dac645575710efeca3262948fba8ea74dc4abf8836939066.jpg new file mode 100644 index 00000000..368925fd Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-132_13bf5fdd82e2be27dac645575710efeca3262948fba8ea74dc4abf8836939066.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-132_4028c3e74bf2ff682874044d42e8bd69b2d62b339e677a80ddc57ebbc4beed39.jpg b/.raw/MidasFEAAnalysisManual/images/page-132_4028c3e74bf2ff682874044d42e8bd69b2d62b339e677a80ddc57ebbc4beed39.jpg new file mode 100644 index 00000000..bbb092d4 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-132_4028c3e74bf2ff682874044d42e8bd69b2d62b339e677a80ddc57ebbc4beed39.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-132_44c92647ca2725aeabcbf94385aead61ed6ef1d2cd5e53a2cbf4696c9a90b6fe.jpg b/.raw/MidasFEAAnalysisManual/images/page-132_44c92647ca2725aeabcbf94385aead61ed6ef1d2cd5e53a2cbf4696c9a90b6fe.jpg new file mode 100644 index 00000000..83dfbe3b Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-132_44c92647ca2725aeabcbf94385aead61ed6ef1d2cd5e53a2cbf4696c9a90b6fe.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-132_636873c82683506ebf53b46a36bea73868466aea473c31b94ae8aaeb69213547.jpg b/.raw/MidasFEAAnalysisManual/images/page-132_636873c82683506ebf53b46a36bea73868466aea473c31b94ae8aaeb69213547.jpg new file mode 100644 index 00000000..eab2fe69 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-132_636873c82683506ebf53b46a36bea73868466aea473c31b94ae8aaeb69213547.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-132_c6c9cb201a26c6f284ecd8b7a4413bc8f4541ddbfa79d341ddfe5ed6b2dba154.jpg b/.raw/MidasFEAAnalysisManual/images/page-132_c6c9cb201a26c6f284ecd8b7a4413bc8f4541ddbfa79d341ddfe5ed6b2dba154.jpg new file mode 100644 index 00000000..cabc314f Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-132_c6c9cb201a26c6f284ecd8b7a4413bc8f4541ddbfa79d341ddfe5ed6b2dba154.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-132_ef5443b815aa2419b3f67533058e6c5d728b0ba3142e12ca51e1c916f232cbeb.jpg b/.raw/MidasFEAAnalysisManual/images/page-132_ef5443b815aa2419b3f67533058e6c5d728b0ba3142e12ca51e1c916f232cbeb.jpg new file mode 100644 index 00000000..318fbbc5 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-132_ef5443b815aa2419b3f67533058e6c5d728b0ba3142e12ca51e1c916f232cbeb.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-133_47b5bf1ce12c6a6c8b95b8de984a8ea898b9460f2ce7042ab86f1a0d0da25f4a.jpg b/.raw/MidasFEAAnalysisManual/images/page-133_47b5bf1ce12c6a6c8b95b8de984a8ea898b9460f2ce7042ab86f1a0d0da25f4a.jpg new file mode 100644 index 00000000..473f4251 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-133_47b5bf1ce12c6a6c8b95b8de984a8ea898b9460f2ce7042ab86f1a0d0da25f4a.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-133_7966a72181f62959712c6a5890bfb0c786397a1f2f2a9bd91bec1e3ed964e50d.jpg b/.raw/MidasFEAAnalysisManual/images/page-133_7966a72181f62959712c6a5890bfb0c786397a1f2f2a9bd91bec1e3ed964e50d.jpg new file mode 100644 index 00000000..938d8053 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-133_7966a72181f62959712c6a5890bfb0c786397a1f2f2a9bd91bec1e3ed964e50d.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-133_80b90684997535f42b3dc8dbc5f40a73699abf223afacd2d6bfb67800f4f671f.jpg b/.raw/MidasFEAAnalysisManual/images/page-133_80b90684997535f42b3dc8dbc5f40a73699abf223afacd2d6bfb67800f4f671f.jpg new file mode 100644 index 00000000..35650158 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-133_80b90684997535f42b3dc8dbc5f40a73699abf223afacd2d6bfb67800f4f671f.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-133_bd328e3290d611ff1fc1d7a48fac0021c2e28f4f4365db643a29b8daae3b5f6b.jpg b/.raw/MidasFEAAnalysisManual/images/page-133_bd328e3290d611ff1fc1d7a48fac0021c2e28f4f4365db643a29b8daae3b5f6b.jpg new file mode 100644 index 00000000..a74201d7 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-133_bd328e3290d611ff1fc1d7a48fac0021c2e28f4f4365db643a29b8daae3b5f6b.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-133_e76e0ec063e30d22b63f09b1562e45d9dcfff2bc8295f905987de1b15bb79e2f.jpg b/.raw/MidasFEAAnalysisManual/images/page-133_e76e0ec063e30d22b63f09b1562e45d9dcfff2bc8295f905987de1b15bb79e2f.jpg new file mode 100644 index 00000000..ce576617 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-133_e76e0ec063e30d22b63f09b1562e45d9dcfff2bc8295f905987de1b15bb79e2f.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-134_21b2a1fa2ba27532b716cf0ae38213e94cccd9438fcbb070315586d0c7b5ed01.jpg b/.raw/MidasFEAAnalysisManual/images/page-134_21b2a1fa2ba27532b716cf0ae38213e94cccd9438fcbb070315586d0c7b5ed01.jpg new file mode 100644 index 00000000..24d570b0 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-134_21b2a1fa2ba27532b716cf0ae38213e94cccd9438fcbb070315586d0c7b5ed01.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-134_2539df63e8cf7a039f68f2658188b3434de259f4264e51cd3e5ee07e71143538.jpg b/.raw/MidasFEAAnalysisManual/images/page-134_2539df63e8cf7a039f68f2658188b3434de259f4264e51cd3e5ee07e71143538.jpg new file mode 100644 index 00000000..5cf262f5 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-134_2539df63e8cf7a039f68f2658188b3434de259f4264e51cd3e5ee07e71143538.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-134_513c026e72a5e9a412d31ddb73ca2b1dadae6a4525b134c7ee7125004ecb850b.jpg b/.raw/MidasFEAAnalysisManual/images/page-134_513c026e72a5e9a412d31ddb73ca2b1dadae6a4525b134c7ee7125004ecb850b.jpg new file mode 100644 index 00000000..f1e44b5b Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-134_513c026e72a5e9a412d31ddb73ca2b1dadae6a4525b134c7ee7125004ecb850b.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-134_5a75857f1cacefa4691dd7968ed0e681085d52a47d961ae995b52749b03a7a1c.jpg b/.raw/MidasFEAAnalysisManual/images/page-134_5a75857f1cacefa4691dd7968ed0e681085d52a47d961ae995b52749b03a7a1c.jpg new file mode 100644 index 00000000..eb1b7d09 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-134_5a75857f1cacefa4691dd7968ed0e681085d52a47d961ae995b52749b03a7a1c.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-134_a3745bc4241fbe32d3f1ba5642e39afc279cc0c6e032052414c76cfc487b5e32.jpg b/.raw/MidasFEAAnalysisManual/images/page-134_a3745bc4241fbe32d3f1ba5642e39afc279cc0c6e032052414c76cfc487b5e32.jpg new file mode 100644 index 00000000..6cfe9122 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-134_a3745bc4241fbe32d3f1ba5642e39afc279cc0c6e032052414c76cfc487b5e32.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-135_03c916e66c94303d1239b899fd1de029183b6fbdc0b4beea08b6dc6187096866.jpg b/.raw/MidasFEAAnalysisManual/images/page-135_03c916e66c94303d1239b899fd1de029183b6fbdc0b4beea08b6dc6187096866.jpg new file mode 100644 index 00000000..d6c9a146 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-135_03c916e66c94303d1239b899fd1de029183b6fbdc0b4beea08b6dc6187096866.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-135_283a73962ce0742bb8ca58f1fd9b1e60ae6943857085940f02aedda3dd9a3f64.jpg b/.raw/MidasFEAAnalysisManual/images/page-135_283a73962ce0742bb8ca58f1fd9b1e60ae6943857085940f02aedda3dd9a3f64.jpg new file mode 100644 index 00000000..26d4df8d Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-135_283a73962ce0742bb8ca58f1fd9b1e60ae6943857085940f02aedda3dd9a3f64.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-135_3d7bfd29bbe9c7068d8ae3d4070c7d46400193691b36c8e35d839a4b0ab090fc.jpg b/.raw/MidasFEAAnalysisManual/images/page-135_3d7bfd29bbe9c7068d8ae3d4070c7d46400193691b36c8e35d839a4b0ab090fc.jpg new file mode 100644 index 00000000..4f853435 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-135_3d7bfd29bbe9c7068d8ae3d4070c7d46400193691b36c8e35d839a4b0ab090fc.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-135_4248377e8fdd1b86c3d2eb169c29f59e258d2d6636fec5a9c0fda46e34e16bcd.jpg b/.raw/MidasFEAAnalysisManual/images/page-135_4248377e8fdd1b86c3d2eb169c29f59e258d2d6636fec5a9c0fda46e34e16bcd.jpg new file mode 100644 index 00000000..5f96b994 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-135_4248377e8fdd1b86c3d2eb169c29f59e258d2d6636fec5a9c0fda46e34e16bcd.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-135_4c3337242d48f06a9316f7ea835be5bb9754334998a4b5d5b9cf9b756c91a23a.jpg b/.raw/MidasFEAAnalysisManual/images/page-135_4c3337242d48f06a9316f7ea835be5bb9754334998a4b5d5b9cf9b756c91a23a.jpg new file mode 100644 index 00000000..a30db445 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-135_4c3337242d48f06a9316f7ea835be5bb9754334998a4b5d5b9cf9b756c91a23a.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-136_41605825a435b205667fa00d4dfcb11aa5936d4841715bc01d287ef0547b77d1.jpg b/.raw/MidasFEAAnalysisManual/images/page-136_41605825a435b205667fa00d4dfcb11aa5936d4841715bc01d287ef0547b77d1.jpg new file mode 100644 index 00000000..db3b06cd Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-136_41605825a435b205667fa00d4dfcb11aa5936d4841715bc01d287ef0547b77d1.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-136_48297bcdbaa0f901c205671473fdb63dc90c03e6a29a5815671e333d9f5e9ca5.jpg b/.raw/MidasFEAAnalysisManual/images/page-136_48297bcdbaa0f901c205671473fdb63dc90c03e6a29a5815671e333d9f5e9ca5.jpg new file mode 100644 index 00000000..7c596996 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-136_48297bcdbaa0f901c205671473fdb63dc90c03e6a29a5815671e333d9f5e9ca5.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-136_736273e136d2b3720aea48976b1c3d1cb5aabb09eaead387a96a7ef1e32f82b1.jpg b/.raw/MidasFEAAnalysisManual/images/page-136_736273e136d2b3720aea48976b1c3d1cb5aabb09eaead387a96a7ef1e32f82b1.jpg new file mode 100644 index 00000000..bf152895 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-136_736273e136d2b3720aea48976b1c3d1cb5aabb09eaead387a96a7ef1e32f82b1.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-136_761b4d122fc94d7d0c423e3a012bd23f2c455821ea4a114751ae0070abb77f47.jpg b/.raw/MidasFEAAnalysisManual/images/page-136_761b4d122fc94d7d0c423e3a012bd23f2c455821ea4a114751ae0070abb77f47.jpg new file mode 100644 index 00000000..d5015a41 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-136_761b4d122fc94d7d0c423e3a012bd23f2c455821ea4a114751ae0070abb77f47.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-136_b6cd23c4530ea91ecdfc084f3406fd16a4132baeedb3a5b41dd06e3d6ce490a1.jpg b/.raw/MidasFEAAnalysisManual/images/page-136_b6cd23c4530ea91ecdfc084f3406fd16a4132baeedb3a5b41dd06e3d6ce490a1.jpg new file mode 100644 index 00000000..abaf1da5 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-136_b6cd23c4530ea91ecdfc084f3406fd16a4132baeedb3a5b41dd06e3d6ce490a1.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-136_d614fe302ac5c33f7f152ca495a15e1d40207c0ba95fefc0bbc97a7ff6a70c4b.jpg b/.raw/MidasFEAAnalysisManual/images/page-136_d614fe302ac5c33f7f152ca495a15e1d40207c0ba95fefc0bbc97a7ff6a70c4b.jpg new file mode 100644 index 00000000..54d7fcda Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-136_d614fe302ac5c33f7f152ca495a15e1d40207c0ba95fefc0bbc97a7ff6a70c4b.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-137_6637d4ae9698b315e96bc70aca34b0954bf566228461c9fbd51b7481f226cae9.jpg b/.raw/MidasFEAAnalysisManual/images/page-137_6637d4ae9698b315e96bc70aca34b0954bf566228461c9fbd51b7481f226cae9.jpg new file mode 100644 index 00000000..f254e07b Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-137_6637d4ae9698b315e96bc70aca34b0954bf566228461c9fbd51b7481f226cae9.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-137_9a86d538055ffbd21ec22bb7f32c77f78c78732f05652eba74c3d122ed02d722.jpg b/.raw/MidasFEAAnalysisManual/images/page-137_9a86d538055ffbd21ec22bb7f32c77f78c78732f05652eba74c3d122ed02d722.jpg new file mode 100644 index 00000000..1d833a82 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-137_9a86d538055ffbd21ec22bb7f32c77f78c78732f05652eba74c3d122ed02d722.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-138_27a348d72c3b2008f40675eb0b4f402adbf21e6028b828909068e7926a88eb69.jpg b/.raw/MidasFEAAnalysisManual/images/page-138_27a348d72c3b2008f40675eb0b4f402adbf21e6028b828909068e7926a88eb69.jpg new file mode 100644 index 00000000..a0c0ac63 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-138_27a348d72c3b2008f40675eb0b4f402adbf21e6028b828909068e7926a88eb69.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-138_75d3910f0b3ca769f2cd40cde556c172e1a3938c31639db2d686fe06b5f328b5.jpg b/.raw/MidasFEAAnalysisManual/images/page-138_75d3910f0b3ca769f2cd40cde556c172e1a3938c31639db2d686fe06b5f328b5.jpg new file mode 100644 index 00000000..938fc6cf Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-138_75d3910f0b3ca769f2cd40cde556c172e1a3938c31639db2d686fe06b5f328b5.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-138_8d7822207cc88b5d28bf78e3bbf3efbfc0404f77d572031e9ab4c1e44af626ea.jpg b/.raw/MidasFEAAnalysisManual/images/page-138_8d7822207cc88b5d28bf78e3bbf3efbfc0404f77d572031e9ab4c1e44af626ea.jpg new file mode 100644 index 00000000..74031227 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-138_8d7822207cc88b5d28bf78e3bbf3efbfc0404f77d572031e9ab4c1e44af626ea.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-139_754023eb9df91d3ac56ab7856a020fad6b61f1257620ec83f208fb8eacc825c0.jpg b/.raw/MidasFEAAnalysisManual/images/page-139_754023eb9df91d3ac56ab7856a020fad6b61f1257620ec83f208fb8eacc825c0.jpg new file mode 100644 index 00000000..a6ee396d Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-139_754023eb9df91d3ac56ab7856a020fad6b61f1257620ec83f208fb8eacc825c0.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-140_028fa209d65a3e8f13d4f175812e3aa388d87291e216810b6b5cd2579052ad2d.jpg b/.raw/MidasFEAAnalysisManual/images/page-140_028fa209d65a3e8f13d4f175812e3aa388d87291e216810b6b5cd2579052ad2d.jpg new file mode 100644 index 00000000..18637dcd Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-140_028fa209d65a3e8f13d4f175812e3aa388d87291e216810b6b5cd2579052ad2d.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-140_c5cc155b05408e27e0b5449f05ac429718134b94caa8a85c5d5d34d11adb6433.jpg b/.raw/MidasFEAAnalysisManual/images/page-140_c5cc155b05408e27e0b5449f05ac429718134b94caa8a85c5d5d34d11adb6433.jpg new file mode 100644 index 00000000..1cbf22eb Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-140_c5cc155b05408e27e0b5449f05ac429718134b94caa8a85c5d5d34d11adb6433.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-141_103997b1d2a01a457d504f41379ea708027b143a04d6014dc850ba20916c8ea8.jpg b/.raw/MidasFEAAnalysisManual/images/page-141_103997b1d2a01a457d504f41379ea708027b143a04d6014dc850ba20916c8ea8.jpg new file mode 100644 index 00000000..95913b0e Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-141_103997b1d2a01a457d504f41379ea708027b143a04d6014dc850ba20916c8ea8.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-141_ec802385f8d979c53e034463f89de7681e4461930463ed397d93cdd27d98c5fb.jpg b/.raw/MidasFEAAnalysisManual/images/page-141_ec802385f8d979c53e034463f89de7681e4461930463ed397d93cdd27d98c5fb.jpg new file mode 100644 index 00000000..35acb305 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-141_ec802385f8d979c53e034463f89de7681e4461930463ed397d93cdd27d98c5fb.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-142_1142977eb345240aa570e71e4ebc86e674c07ebaf97d39ac1edbb740ad2cb4d8.jpg b/.raw/MidasFEAAnalysisManual/images/page-142_1142977eb345240aa570e71e4ebc86e674c07ebaf97d39ac1edbb740ad2cb4d8.jpg new file mode 100644 index 00000000..395cc9fb Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-142_1142977eb345240aa570e71e4ebc86e674c07ebaf97d39ac1edbb740ad2cb4d8.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-143_1720486db79ee364458873f6c891cda351c27baff4d4f8d6a508a46bf1ee1759.jpg b/.raw/MidasFEAAnalysisManual/images/page-143_1720486db79ee364458873f6c891cda351c27baff4d4f8d6a508a46bf1ee1759.jpg new file mode 100644 index 00000000..ed4fb4b4 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-143_1720486db79ee364458873f6c891cda351c27baff4d4f8d6a508a46bf1ee1759.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-143_6357e809d4e71f762b4a65532372032007066b06b0b45bc8bde8d2bf46a6599c.jpg b/.raw/MidasFEAAnalysisManual/images/page-143_6357e809d4e71f762b4a65532372032007066b06b0b45bc8bde8d2bf46a6599c.jpg new file mode 100644 index 00000000..1c386730 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-143_6357e809d4e71f762b4a65532372032007066b06b0b45bc8bde8d2bf46a6599c.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-143_8693ec9ab035b65b5dcfee7da4a0a4369f87cb1a6c4ab59c665f332ed145437c.jpg b/.raw/MidasFEAAnalysisManual/images/page-143_8693ec9ab035b65b5dcfee7da4a0a4369f87cb1a6c4ab59c665f332ed145437c.jpg new file mode 100644 index 00000000..28a8bebe Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-143_8693ec9ab035b65b5dcfee7da4a0a4369f87cb1a6c4ab59c665f332ed145437c.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-143_f7a87c48837fc4ce69f31df4c8902e0c68a9864197800c835f59401bd3aa3669.jpg b/.raw/MidasFEAAnalysisManual/images/page-143_f7a87c48837fc4ce69f31df4c8902e0c68a9864197800c835f59401bd3aa3669.jpg new file mode 100644 index 00000000..91854921 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-143_f7a87c48837fc4ce69f31df4c8902e0c68a9864197800c835f59401bd3aa3669.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-144_05c4cae1abe2480774d00850c7c66722d81e309eaf1826cfaa2f4c762a89c5b8.jpg b/.raw/MidasFEAAnalysisManual/images/page-144_05c4cae1abe2480774d00850c7c66722d81e309eaf1826cfaa2f4c762a89c5b8.jpg new file mode 100644 index 00000000..47f23810 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-144_05c4cae1abe2480774d00850c7c66722d81e309eaf1826cfaa2f4c762a89c5b8.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-144_2380a75fd636e0eb14f2396c6b86c891d38c977c5a5ce1c89ecadfc4f6b05c74.jpg b/.raw/MidasFEAAnalysisManual/images/page-144_2380a75fd636e0eb14f2396c6b86c891d38c977c5a5ce1c89ecadfc4f6b05c74.jpg new file mode 100644 index 00000000..8caa9eda Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-144_2380a75fd636e0eb14f2396c6b86c891d38c977c5a5ce1c89ecadfc4f6b05c74.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-144_db602e6dea32d635502fbb92d6df573b037f0886e16c81338318f0945ac357c0.jpg b/.raw/MidasFEAAnalysisManual/images/page-144_db602e6dea32d635502fbb92d6df573b037f0886e16c81338318f0945ac357c0.jpg new file mode 100644 index 00000000..68fa5bd5 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-144_db602e6dea32d635502fbb92d6df573b037f0886e16c81338318f0945ac357c0.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-145_1c281e67bbf6fa77dfa4eeea0474b0c1295664d3e284b00ebf0c303e73507c42.jpg b/.raw/MidasFEAAnalysisManual/images/page-145_1c281e67bbf6fa77dfa4eeea0474b0c1295664d3e284b00ebf0c303e73507c42.jpg new file mode 100644 index 00000000..38351547 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-145_1c281e67bbf6fa77dfa4eeea0474b0c1295664d3e284b00ebf0c303e73507c42.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-145_fbf83d1d5d2d2c537dcdc2fcb47128cb8754f42e3ad396870e40394d89174795.jpg b/.raw/MidasFEAAnalysisManual/images/page-145_fbf83d1d5d2d2c537dcdc2fcb47128cb8754f42e3ad396870e40394d89174795.jpg new file mode 100644 index 00000000..a133cc91 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-145_fbf83d1d5d2d2c537dcdc2fcb47128cb8754f42e3ad396870e40394d89174795.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-146_3219bd7fe9cd4b975104f5a17eee4b420427bbe59bfda8756bd682c33a9bffae.jpg b/.raw/MidasFEAAnalysisManual/images/page-146_3219bd7fe9cd4b975104f5a17eee4b420427bbe59bfda8756bd682c33a9bffae.jpg new file mode 100644 index 00000000..48fe108c Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-146_3219bd7fe9cd4b975104f5a17eee4b420427bbe59bfda8756bd682c33a9bffae.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-147_4661ae664f5aaebeebd3d679452bdbdb3640aab36cd491e465ddc9a0afb492bc.jpg b/.raw/MidasFEAAnalysisManual/images/page-147_4661ae664f5aaebeebd3d679452bdbdb3640aab36cd491e465ddc9a0afb492bc.jpg new file mode 100644 index 00000000..2fe711d1 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-147_4661ae664f5aaebeebd3d679452bdbdb3640aab36cd491e465ddc9a0afb492bc.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-148_548382ea7b012cbe4a3a5d199e3b1deb6790190081ee8abb3cce7642edbda42e.jpg b/.raw/MidasFEAAnalysisManual/images/page-148_548382ea7b012cbe4a3a5d199e3b1deb6790190081ee8abb3cce7642edbda42e.jpg new file mode 100644 index 00000000..a7743f7d Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-148_548382ea7b012cbe4a3a5d199e3b1deb6790190081ee8abb3cce7642edbda42e.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-148_d77da82402c57ff70b7e15a38b6a0b2ff8e4c803ca493859e8cef7e721e7878d.jpg b/.raw/MidasFEAAnalysisManual/images/page-148_d77da82402c57ff70b7e15a38b6a0b2ff8e4c803ca493859e8cef7e721e7878d.jpg new file mode 100644 index 00000000..4735d177 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-148_d77da82402c57ff70b7e15a38b6a0b2ff8e4c803ca493859e8cef7e721e7878d.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-149_db38bacd7fd6208c8f279498ba96a9213bfcc04981f85f44bcaab1b33ff7662c.jpg b/.raw/MidasFEAAnalysisManual/images/page-149_db38bacd7fd6208c8f279498ba96a9213bfcc04981f85f44bcaab1b33ff7662c.jpg new file mode 100644 index 00000000..840f1c8d Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-149_db38bacd7fd6208c8f279498ba96a9213bfcc04981f85f44bcaab1b33ff7662c.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-151_446d4e8bf361ff3f3aa4aeb64d5ea1cbefdfd2dfd2dbef4e175bf50638fa92c6.jpg b/.raw/MidasFEAAnalysisManual/images/page-151_446d4e8bf361ff3f3aa4aeb64d5ea1cbefdfd2dfd2dbef4e175bf50638fa92c6.jpg new file mode 100644 index 00000000..fb07b1ca Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-151_446d4e8bf361ff3f3aa4aeb64d5ea1cbefdfd2dfd2dbef4e175bf50638fa92c6.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-152_55e947a94cc5809dd79f8e2daabdd83aa3cde90b478d712b70566128b6ff83a6.jpg b/.raw/MidasFEAAnalysisManual/images/page-152_55e947a94cc5809dd79f8e2daabdd83aa3cde90b478d712b70566128b6ff83a6.jpg new file mode 100644 index 00000000..e9b8f1a3 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-152_55e947a94cc5809dd79f8e2daabdd83aa3cde90b478d712b70566128b6ff83a6.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-152_a5f301cb5cfedd20b88ebf4a25d806c9ce3a4e4ff1daa4b54ea08004197f3c39.jpg b/.raw/MidasFEAAnalysisManual/images/page-152_a5f301cb5cfedd20b88ebf4a25d806c9ce3a4e4ff1daa4b54ea08004197f3c39.jpg new file mode 100644 index 00000000..928997f9 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-152_a5f301cb5cfedd20b88ebf4a25d806c9ce3a4e4ff1daa4b54ea08004197f3c39.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-153_7e77db69dbaa64ad72453cbbc2927cfed2c59551d2ce642b5bf1c9564291021e.jpg b/.raw/MidasFEAAnalysisManual/images/page-153_7e77db69dbaa64ad72453cbbc2927cfed2c59551d2ce642b5bf1c9564291021e.jpg new file mode 100644 index 00000000..616e56c4 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-153_7e77db69dbaa64ad72453cbbc2927cfed2c59551d2ce642b5bf1c9564291021e.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-154_8761e001c4eed19e4f5b77e2d6e9e8d4449caccbae647db8193246f70f468ac0.jpg b/.raw/MidasFEAAnalysisManual/images/page-154_8761e001c4eed19e4f5b77e2d6e9e8d4449caccbae647db8193246f70f468ac0.jpg new file mode 100644 index 00000000..7f1fe5ce Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-154_8761e001c4eed19e4f5b77e2d6e9e8d4449caccbae647db8193246f70f468ac0.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-155_0b20193b4040214b672fc6f1272d574230ca32766451aff5202883f414bb2a07.jpg b/.raw/MidasFEAAnalysisManual/images/page-155_0b20193b4040214b672fc6f1272d574230ca32766451aff5202883f414bb2a07.jpg new file mode 100644 index 00000000..baec9719 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-155_0b20193b4040214b672fc6f1272d574230ca32766451aff5202883f414bb2a07.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-155_2119c324a5a96acd081b51a9506cdb9271e3ddaeffc51439bab36b3055a1aefa.jpg b/.raw/MidasFEAAnalysisManual/images/page-155_2119c324a5a96acd081b51a9506cdb9271e3ddaeffc51439bab36b3055a1aefa.jpg new file mode 100644 index 00000000..e5314ba0 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-155_2119c324a5a96acd081b51a9506cdb9271e3ddaeffc51439bab36b3055a1aefa.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-155_6a02193a05c4cdce35a21c66626227140e2785421594679456bcfc4c422d1558.jpg b/.raw/MidasFEAAnalysisManual/images/page-155_6a02193a05c4cdce35a21c66626227140e2785421594679456bcfc4c422d1558.jpg new file mode 100644 index 00000000..e6a2d3ef Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-155_6a02193a05c4cdce35a21c66626227140e2785421594679456bcfc4c422d1558.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-155_d94cc7d19821aa03d8c1269b02ecd327b861ef8e4f65b3861e4f94758923b952.jpg b/.raw/MidasFEAAnalysisManual/images/page-155_d94cc7d19821aa03d8c1269b02ecd327b861ef8e4f65b3861e4f94758923b952.jpg new file mode 100644 index 00000000..ba3703ba Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-155_d94cc7d19821aa03d8c1269b02ecd327b861ef8e4f65b3861e4f94758923b952.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-155_e4708195cfc04f4b4671db7bb47f4f74b224cece3d83ba1d6b9cf0068b136ae1.jpg b/.raw/MidasFEAAnalysisManual/images/page-155_e4708195cfc04f4b4671db7bb47f4f74b224cece3d83ba1d6b9cf0068b136ae1.jpg new file mode 100644 index 00000000..f2663e43 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-155_e4708195cfc04f4b4671db7bb47f4f74b224cece3d83ba1d6b9cf0068b136ae1.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-155_f04e246fe3e10dd644969a1f1040b247e88ed7bf28057ba2ec205dc713ba6f34.jpg b/.raw/MidasFEAAnalysisManual/images/page-155_f04e246fe3e10dd644969a1f1040b247e88ed7bf28057ba2ec205dc713ba6f34.jpg new file mode 100644 index 00000000..db130250 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-155_f04e246fe3e10dd644969a1f1040b247e88ed7bf28057ba2ec205dc713ba6f34.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-156_950e1acfe5a1a07dc94caac44e09def47bea22dd6521f08e75b0057667ea05bf.jpg b/.raw/MidasFEAAnalysisManual/images/page-156_950e1acfe5a1a07dc94caac44e09def47bea22dd6521f08e75b0057667ea05bf.jpg new file mode 100644 index 00000000..bd9a241f Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-156_950e1acfe5a1a07dc94caac44e09def47bea22dd6521f08e75b0057667ea05bf.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-157_5588852d25a832b64a9e856f0b5f842aec570658de79eaf8fbffbbe558931d77.jpg b/.raw/MidasFEAAnalysisManual/images/page-157_5588852d25a832b64a9e856f0b5f842aec570658de79eaf8fbffbbe558931d77.jpg new file mode 100644 index 00000000..6c57f254 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-157_5588852d25a832b64a9e856f0b5f842aec570658de79eaf8fbffbbe558931d77.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-158_1ba1c647dbd66e7bb719cb5c51db6e045670693a1b4e85bd4b0e2156abeb17e5.jpg b/.raw/MidasFEAAnalysisManual/images/page-158_1ba1c647dbd66e7bb719cb5c51db6e045670693a1b4e85bd4b0e2156abeb17e5.jpg new file mode 100644 index 00000000..4954cca6 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-158_1ba1c647dbd66e7bb719cb5c51db6e045670693a1b4e85bd4b0e2156abeb17e5.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-158_604f70dd7e366526d6ded0cd17634aca20611a9b54465e1733dcc9adc6216162.jpg b/.raw/MidasFEAAnalysisManual/images/page-158_604f70dd7e366526d6ded0cd17634aca20611a9b54465e1733dcc9adc6216162.jpg new file mode 100644 index 00000000..567df37d Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-158_604f70dd7e366526d6ded0cd17634aca20611a9b54465e1733dcc9adc6216162.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-158_7477ebb3e8a09064c7d4d84ea0251e42271fa04e45180eded9b50bab448bf05a.jpg b/.raw/MidasFEAAnalysisManual/images/page-158_7477ebb3e8a09064c7d4d84ea0251e42271fa04e45180eded9b50bab448bf05a.jpg new file mode 100644 index 00000000..91a29a7f Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-158_7477ebb3e8a09064c7d4d84ea0251e42271fa04e45180eded9b50bab448bf05a.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-159_67c315d38f23cd1dc613dd7f4c2504409e7b685caaffad4d2f6c8efbe8d41b1e.jpg b/.raw/MidasFEAAnalysisManual/images/page-159_67c315d38f23cd1dc613dd7f4c2504409e7b685caaffad4d2f6c8efbe8d41b1e.jpg new file mode 100644 index 00000000..0c515088 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-159_67c315d38f23cd1dc613dd7f4c2504409e7b685caaffad4d2f6c8efbe8d41b1e.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-160_4e54fd4dcdcee1b9241a7439202d5e789ecf4f78a1a8ec2a5113f8342984458b.jpg b/.raw/MidasFEAAnalysisManual/images/page-160_4e54fd4dcdcee1b9241a7439202d5e789ecf4f78a1a8ec2a5113f8342984458b.jpg new file mode 100644 index 00000000..0518652e Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-160_4e54fd4dcdcee1b9241a7439202d5e789ecf4f78a1a8ec2a5113f8342984458b.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-160_86896bfb5c248914da04b68d529214287640df2071f30789016ae4c6a9f9e57e.jpg b/.raw/MidasFEAAnalysisManual/images/page-160_86896bfb5c248914da04b68d529214287640df2071f30789016ae4c6a9f9e57e.jpg new file mode 100644 index 00000000..98bdeb19 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-160_86896bfb5c248914da04b68d529214287640df2071f30789016ae4c6a9f9e57e.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-163_ada546f086cc3fec1fe6817b9f27fac912433da89d3d0984120e6bb5256fd222.jpg b/.raw/MidasFEAAnalysisManual/images/page-163_ada546f086cc3fec1fe6817b9f27fac912433da89d3d0984120e6bb5256fd222.jpg new file mode 100644 index 00000000..3297d777 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-163_ada546f086cc3fec1fe6817b9f27fac912433da89d3d0984120e6bb5256fd222.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-164_039e02f647f031e3fb74f8a89deb32a9843524cc6c8e471e377f2792766ab114.jpg b/.raw/MidasFEAAnalysisManual/images/page-164_039e02f647f031e3fb74f8a89deb32a9843524cc6c8e471e377f2792766ab114.jpg new file mode 100644 index 00000000..19995526 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-164_039e02f647f031e3fb74f8a89deb32a9843524cc6c8e471e377f2792766ab114.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-164_05cadd62c68ae478aeee65369db023db73496275433fa2dbac1afcdba8273b74.jpg b/.raw/MidasFEAAnalysisManual/images/page-164_05cadd62c68ae478aeee65369db023db73496275433fa2dbac1afcdba8273b74.jpg new file mode 100644 index 00000000..153cac2b Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-164_05cadd62c68ae478aeee65369db023db73496275433fa2dbac1afcdba8273b74.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-164_2293327c8f127e76b7a353fcbb4265d42a67d2e01cb1cea3f5aebffc4d20cb35.jpg b/.raw/MidasFEAAnalysisManual/images/page-164_2293327c8f127e76b7a353fcbb4265d42a67d2e01cb1cea3f5aebffc4d20cb35.jpg new file mode 100644 index 00000000..1cf4d4fa Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-164_2293327c8f127e76b7a353fcbb4265d42a67d2e01cb1cea3f5aebffc4d20cb35.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-164_363f1e3d33395fed4c2cfcf475a9ee8db731619097e9510ca92cf452c87c86c8.jpg b/.raw/MidasFEAAnalysisManual/images/page-164_363f1e3d33395fed4c2cfcf475a9ee8db731619097e9510ca92cf452c87c86c8.jpg new file mode 100644 index 00000000..f6ed159e Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-164_363f1e3d33395fed4c2cfcf475a9ee8db731619097e9510ca92cf452c87c86c8.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-164_829d4948ee01c5794be0e3cb404247d737986b54e65b1fef91c573b624da63e6.jpg b/.raw/MidasFEAAnalysisManual/images/page-164_829d4948ee01c5794be0e3cb404247d737986b54e65b1fef91c573b624da63e6.jpg new file mode 100644 index 00000000..5c8adf5c Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-164_829d4948ee01c5794be0e3cb404247d737986b54e65b1fef91c573b624da63e6.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-164_a49d1bfdf94d42e586782550206276663c75b3d48034c24651f7779127473ad0.jpg b/.raw/MidasFEAAnalysisManual/images/page-164_a49d1bfdf94d42e586782550206276663c75b3d48034c24651f7779127473ad0.jpg new file mode 100644 index 00000000..74f28e15 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-164_a49d1bfdf94d42e586782550206276663c75b3d48034c24651f7779127473ad0.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-164_b5f76b4b3335dfb2c12d5da576ae990987d394adfadd7c1f0cc74c518078d97d.jpg b/.raw/MidasFEAAnalysisManual/images/page-164_b5f76b4b3335dfb2c12d5da576ae990987d394adfadd7c1f0cc74c518078d97d.jpg new file mode 100644 index 00000000..c4c55b0d Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-164_b5f76b4b3335dfb2c12d5da576ae990987d394adfadd7c1f0cc74c518078d97d.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-165_09a8bfa79fbdf150ff9be495982623df86426f656aaedc9f893a3fc6203d85a3.jpg b/.raw/MidasFEAAnalysisManual/images/page-165_09a8bfa79fbdf150ff9be495982623df86426f656aaedc9f893a3fc6203d85a3.jpg new file mode 100644 index 00000000..34f605b2 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-165_09a8bfa79fbdf150ff9be495982623df86426f656aaedc9f893a3fc6203d85a3.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-165_671b23667f8db4896ee66133b26fb3f1d879a8e33efa1e386806360afab735e5.jpg b/.raw/MidasFEAAnalysisManual/images/page-165_671b23667f8db4896ee66133b26fb3f1d879a8e33efa1e386806360afab735e5.jpg new file mode 100644 index 00000000..2234dd8a Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-165_671b23667f8db4896ee66133b26fb3f1d879a8e33efa1e386806360afab735e5.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-165_8867545a13b616447baef441b1cc400426f9bd6be1bb074aa3c7d98870745626.jpg b/.raw/MidasFEAAnalysisManual/images/page-165_8867545a13b616447baef441b1cc400426f9bd6be1bb074aa3c7d98870745626.jpg new file mode 100644 index 00000000..767109c1 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-165_8867545a13b616447baef441b1cc400426f9bd6be1bb074aa3c7d98870745626.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-165_8f02381be6eefc30d113a2d51490bfb75703e1bc4cc3165c31e1f422331e4302.jpg b/.raw/MidasFEAAnalysisManual/images/page-165_8f02381be6eefc30d113a2d51490bfb75703e1bc4cc3165c31e1f422331e4302.jpg new file mode 100644 index 00000000..602e3070 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-165_8f02381be6eefc30d113a2d51490bfb75703e1bc4cc3165c31e1f422331e4302.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-165_9395a14600022b810b40ea31c392a8d5507a48bd636b85403796d32b867fa04f.jpg b/.raw/MidasFEAAnalysisManual/images/page-165_9395a14600022b810b40ea31c392a8d5507a48bd636b85403796d32b867fa04f.jpg new file mode 100644 index 00000000..a987cb55 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-165_9395a14600022b810b40ea31c392a8d5507a48bd636b85403796d32b867fa04f.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-166_57540bc30089657a5dcbe1aa16677c5971f5dd57fcd2e42c3e086218a637f7a5.jpg b/.raw/MidasFEAAnalysisManual/images/page-166_57540bc30089657a5dcbe1aa16677c5971f5dd57fcd2e42c3e086218a637f7a5.jpg new file mode 100644 index 00000000..a1442310 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-166_57540bc30089657a5dcbe1aa16677c5971f5dd57fcd2e42c3e086218a637f7a5.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-166_6e34f33208ee7c4a7c715d27169919c12c4031634961fc5c8361227d93d2bfa7.jpg b/.raw/MidasFEAAnalysisManual/images/page-166_6e34f33208ee7c4a7c715d27169919c12c4031634961fc5c8361227d93d2bfa7.jpg new file mode 100644 index 00000000..80e045c4 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-166_6e34f33208ee7c4a7c715d27169919c12c4031634961fc5c8361227d93d2bfa7.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-166_c6135755bc84fc5662afce53774a9f79f9c4f754c616f592ad2080fbb5ea2cc8.jpg b/.raw/MidasFEAAnalysisManual/images/page-166_c6135755bc84fc5662afce53774a9f79f9c4f754c616f592ad2080fbb5ea2cc8.jpg new file mode 100644 index 00000000..357478b6 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-166_c6135755bc84fc5662afce53774a9f79f9c4f754c616f592ad2080fbb5ea2cc8.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-167_5ad9bae6243c3e8e7d31e17faaf2f4cc8980675a1258bcd2a0d69386d36f88b2.jpg b/.raw/MidasFEAAnalysisManual/images/page-167_5ad9bae6243c3e8e7d31e17faaf2f4cc8980675a1258bcd2a0d69386d36f88b2.jpg new file mode 100644 index 00000000..d3084727 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-167_5ad9bae6243c3e8e7d31e17faaf2f4cc8980675a1258bcd2a0d69386d36f88b2.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-169_49e4f9ab8d371435a438e048cb5beec7019bb1c0fbfd0e135798846c4a7628fc.jpg b/.raw/MidasFEAAnalysisManual/images/page-169_49e4f9ab8d371435a438e048cb5beec7019bb1c0fbfd0e135798846c4a7628fc.jpg new file mode 100644 index 00000000..026f7f44 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-169_49e4f9ab8d371435a438e048cb5beec7019bb1c0fbfd0e135798846c4a7628fc.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-169_69afd5716fc0d80db17b86065fd50b03b0090ac33e75178ead3af4842c576c89.jpg b/.raw/MidasFEAAnalysisManual/images/page-169_69afd5716fc0d80db17b86065fd50b03b0090ac33e75178ead3af4842c576c89.jpg new file mode 100644 index 00000000..9254add7 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-169_69afd5716fc0d80db17b86065fd50b03b0090ac33e75178ead3af4842c576c89.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-169_884d281a9b307c25d6d58dc0eb8e614823de79aa06e10d5f5eefd26f7a705bc9.jpg b/.raw/MidasFEAAnalysisManual/images/page-169_884d281a9b307c25d6d58dc0eb8e614823de79aa06e10d5f5eefd26f7a705bc9.jpg new file mode 100644 index 00000000..03f0c03e Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-169_884d281a9b307c25d6d58dc0eb8e614823de79aa06e10d5f5eefd26f7a705bc9.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-169_9b45f150eb470716aa22f64cc44a851b0794afae7b5cc3e0cf3982e7927ad4e2.jpg b/.raw/MidasFEAAnalysisManual/images/page-169_9b45f150eb470716aa22f64cc44a851b0794afae7b5cc3e0cf3982e7927ad4e2.jpg new file mode 100644 index 00000000..61f8e2ad Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-169_9b45f150eb470716aa22f64cc44a851b0794afae7b5cc3e0cf3982e7927ad4e2.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-170_0ce08a47a7b0876971b0bf294b691813bfd8f5005a6999d3e6b3498cac909837.jpg b/.raw/MidasFEAAnalysisManual/images/page-170_0ce08a47a7b0876971b0bf294b691813bfd8f5005a6999d3e6b3498cac909837.jpg new file mode 100644 index 00000000..4a27e9cd Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-170_0ce08a47a7b0876971b0bf294b691813bfd8f5005a6999d3e6b3498cac909837.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-170_3801be51d649202dee8331e5e87a9d0b2fcb34316199091c48a388d2e930b5b6.jpg b/.raw/MidasFEAAnalysisManual/images/page-170_3801be51d649202dee8331e5e87a9d0b2fcb34316199091c48a388d2e930b5b6.jpg new file mode 100644 index 00000000..6998a5db Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-170_3801be51d649202dee8331e5e87a9d0b2fcb34316199091c48a388d2e930b5b6.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-170_4d92c566cca08d69aaacae35d89839836b386b86c7289af132a6a4ed07464efb.jpg b/.raw/MidasFEAAnalysisManual/images/page-170_4d92c566cca08d69aaacae35d89839836b386b86c7289af132a6a4ed07464efb.jpg new file mode 100644 index 00000000..c0404fdd Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-170_4d92c566cca08d69aaacae35d89839836b386b86c7289af132a6a4ed07464efb.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-171_061e57de22d3bd929a7210ff05e5e8764f8e4a8d1c361c5cf08a4316da701b25.jpg b/.raw/MidasFEAAnalysisManual/images/page-171_061e57de22d3bd929a7210ff05e5e8764f8e4a8d1c361c5cf08a4316da701b25.jpg new file mode 100644 index 00000000..2ba99710 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-171_061e57de22d3bd929a7210ff05e5e8764f8e4a8d1c361c5cf08a4316da701b25.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-171_0a51cf1303aee04cb9ebdcd4d921ce430fe4494e2a8ceda636d736d55bb103be.jpg b/.raw/MidasFEAAnalysisManual/images/page-171_0a51cf1303aee04cb9ebdcd4d921ce430fe4494e2a8ceda636d736d55bb103be.jpg new file mode 100644 index 00000000..e10cc1f5 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-171_0a51cf1303aee04cb9ebdcd4d921ce430fe4494e2a8ceda636d736d55bb103be.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-171_0cbf0e5bed4dcbac981e76773326b23a4834eeff9c6936e9fefff510855a3560.jpg b/.raw/MidasFEAAnalysisManual/images/page-171_0cbf0e5bed4dcbac981e76773326b23a4834eeff9c6936e9fefff510855a3560.jpg new file mode 100644 index 00000000..2027becf Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-171_0cbf0e5bed4dcbac981e76773326b23a4834eeff9c6936e9fefff510855a3560.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-171_2200ec44ee8186a6a649e7e7d0e189e3213a247ccc86895ba6e81344d7c3ea8b.jpg b/.raw/MidasFEAAnalysisManual/images/page-171_2200ec44ee8186a6a649e7e7d0e189e3213a247ccc86895ba6e81344d7c3ea8b.jpg new file mode 100644 index 00000000..360f4bc6 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-171_2200ec44ee8186a6a649e7e7d0e189e3213a247ccc86895ba6e81344d7c3ea8b.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-171_3fb7bf715eda20a9e5e2179d26b7b2ecef2ce64980c88882aa88bbf7be76125f.jpg b/.raw/MidasFEAAnalysisManual/images/page-171_3fb7bf715eda20a9e5e2179d26b7b2ecef2ce64980c88882aa88bbf7be76125f.jpg new file mode 100644 index 00000000..bbed027d Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-171_3fb7bf715eda20a9e5e2179d26b7b2ecef2ce64980c88882aa88bbf7be76125f.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-171_539c42af3dbbbefbac2cbac0ca2991c9861ddbceb8fd7ad078830ed984d1d22c.jpg b/.raw/MidasFEAAnalysisManual/images/page-171_539c42af3dbbbefbac2cbac0ca2991c9861ddbceb8fd7ad078830ed984d1d22c.jpg new file mode 100644 index 00000000..dd8f6b69 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-171_539c42af3dbbbefbac2cbac0ca2991c9861ddbceb8fd7ad078830ed984d1d22c.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-171_8357d370f27257a860264a42269bfdd615d83ccf7cdae9368b210e87e4e268b3.jpg b/.raw/MidasFEAAnalysisManual/images/page-171_8357d370f27257a860264a42269bfdd615d83ccf7cdae9368b210e87e4e268b3.jpg new file mode 100644 index 00000000..028b74e6 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-171_8357d370f27257a860264a42269bfdd615d83ccf7cdae9368b210e87e4e268b3.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-172_3adf12226003f198240092f672dacfbabe60ec94f967d34b4bc1c19ca6a856b0.jpg b/.raw/MidasFEAAnalysisManual/images/page-172_3adf12226003f198240092f672dacfbabe60ec94f967d34b4bc1c19ca6a856b0.jpg new file mode 100644 index 00000000..7e4d123a Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-172_3adf12226003f198240092f672dacfbabe60ec94f967d34b4bc1c19ca6a856b0.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-172_c03c9b1995cc4ace9c2d963f006df2e4daf6ff82e578366f14b8a0e0046d544e.jpg b/.raw/MidasFEAAnalysisManual/images/page-172_c03c9b1995cc4ace9c2d963f006df2e4daf6ff82e578366f14b8a0e0046d544e.jpg new file mode 100644 index 00000000..9b5565f2 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-172_c03c9b1995cc4ace9c2d963f006df2e4daf6ff82e578366f14b8a0e0046d544e.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-172_cc850b914f7c08f490fa37d54dad4220fa5bdff9099be837cae6287ed63924ad.jpg b/.raw/MidasFEAAnalysisManual/images/page-172_cc850b914f7c08f490fa37d54dad4220fa5bdff9099be837cae6287ed63924ad.jpg new file mode 100644 index 00000000..92b03903 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-172_cc850b914f7c08f490fa37d54dad4220fa5bdff9099be837cae6287ed63924ad.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-172_e218bfb1e0d10d8fe9615ab0866c62b237bbb17edb59d2f16af67763fa96ddbd.jpg b/.raw/MidasFEAAnalysisManual/images/page-172_e218bfb1e0d10d8fe9615ab0866c62b237bbb17edb59d2f16af67763fa96ddbd.jpg new file mode 100644 index 00000000..7827b6fe Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-172_e218bfb1e0d10d8fe9615ab0866c62b237bbb17edb59d2f16af67763fa96ddbd.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-172_ebfad05b962146d9d9954bd19b7460c1703b3b3d55679fe7d6c2e9c33d13be51.jpg b/.raw/MidasFEAAnalysisManual/images/page-172_ebfad05b962146d9d9954bd19b7460c1703b3b3d55679fe7d6c2e9c33d13be51.jpg new file mode 100644 index 00000000..581de5bb Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-172_ebfad05b962146d9d9954bd19b7460c1703b3b3d55679fe7d6c2e9c33d13be51.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-173_02b43cce76f1f8cca60aacf1f756a31d06af415caf967d37189733b98835eb14.jpg b/.raw/MidasFEAAnalysisManual/images/page-173_02b43cce76f1f8cca60aacf1f756a31d06af415caf967d37189733b98835eb14.jpg new file mode 100644 index 00000000..45947fbd Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-173_02b43cce76f1f8cca60aacf1f756a31d06af415caf967d37189733b98835eb14.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-173_91bd4ff49aad435d57e4d0a8cb62e79b3c5e0de04fa9a87e893ad771501a815f.jpg b/.raw/MidasFEAAnalysisManual/images/page-173_91bd4ff49aad435d57e4d0a8cb62e79b3c5e0de04fa9a87e893ad771501a815f.jpg new file mode 100644 index 00000000..8d0c894b Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-173_91bd4ff49aad435d57e4d0a8cb62e79b3c5e0de04fa9a87e893ad771501a815f.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-173_b9b8841efcbdd187471a09efe6373ef7698d5a1e39e04bfef20602c3242974c3.jpg b/.raw/MidasFEAAnalysisManual/images/page-173_b9b8841efcbdd187471a09efe6373ef7698d5a1e39e04bfef20602c3242974c3.jpg new file mode 100644 index 00000000..65b210aa Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-173_b9b8841efcbdd187471a09efe6373ef7698d5a1e39e04bfef20602c3242974c3.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-174_204091cd8289c6f85dd207452f7b7e567a67841694b5e32faeff1b90e38abdf8.jpg b/.raw/MidasFEAAnalysisManual/images/page-174_204091cd8289c6f85dd207452f7b7e567a67841694b5e32faeff1b90e38abdf8.jpg new file mode 100644 index 00000000..f331922c Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-174_204091cd8289c6f85dd207452f7b7e567a67841694b5e32faeff1b90e38abdf8.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-174_ac62f23427ad3d9d504bd893d51154b6fdbd96597ec326a8be31c535f1ed298d.jpg b/.raw/MidasFEAAnalysisManual/images/page-174_ac62f23427ad3d9d504bd893d51154b6fdbd96597ec326a8be31c535f1ed298d.jpg new file mode 100644 index 00000000..96ab775a Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-174_ac62f23427ad3d9d504bd893d51154b6fdbd96597ec326a8be31c535f1ed298d.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-174_cbc4fdc4510b3ee5e1635bd17fdd582e9f8c04978041054c20ca1ef33b65ce2f.jpg b/.raw/MidasFEAAnalysisManual/images/page-174_cbc4fdc4510b3ee5e1635bd17fdd582e9f8c04978041054c20ca1ef33b65ce2f.jpg new file mode 100644 index 00000000..2bf571f7 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-174_cbc4fdc4510b3ee5e1635bd17fdd582e9f8c04978041054c20ca1ef33b65ce2f.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-174_fd5d34cd7021a66ed9bda91499cb3d572cd2270ac4aafc295e2c3a159b586ef6.jpg b/.raw/MidasFEAAnalysisManual/images/page-174_fd5d34cd7021a66ed9bda91499cb3d572cd2270ac4aafc295e2c3a159b586ef6.jpg new file mode 100644 index 00000000..02451641 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-174_fd5d34cd7021a66ed9bda91499cb3d572cd2270ac4aafc295e2c3a159b586ef6.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-175_3a273e54c840daae64eca7d1e60d469c5d8ff3e4587b77a53b42a0e489e09374.jpg b/.raw/MidasFEAAnalysisManual/images/page-175_3a273e54c840daae64eca7d1e60d469c5d8ff3e4587b77a53b42a0e489e09374.jpg new file mode 100644 index 00000000..8c748355 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-175_3a273e54c840daae64eca7d1e60d469c5d8ff3e4587b77a53b42a0e489e09374.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-175_502c590d334359e1a9be6dd1e374f3ec83c16ad4346d7bef6df5aecd187f7de0.jpg b/.raw/MidasFEAAnalysisManual/images/page-175_502c590d334359e1a9be6dd1e374f3ec83c16ad4346d7bef6df5aecd187f7de0.jpg new file mode 100644 index 00000000..4475a5eb Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-175_502c590d334359e1a9be6dd1e374f3ec83c16ad4346d7bef6df5aecd187f7de0.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-175_5d76beeb222405c401d5f02a60c2279f39ae79ae0f1b7547fd103a0cb9354785.jpg b/.raw/MidasFEAAnalysisManual/images/page-175_5d76beeb222405c401d5f02a60c2279f39ae79ae0f1b7547fd103a0cb9354785.jpg new file mode 100644 index 00000000..4ea6cdd0 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-175_5d76beeb222405c401d5f02a60c2279f39ae79ae0f1b7547fd103a0cb9354785.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-175_9b3461c453320ac73cbd333afa9d84dadeb3ba8deaa3168c7d103a5a88c826b4.jpg b/.raw/MidasFEAAnalysisManual/images/page-175_9b3461c453320ac73cbd333afa9d84dadeb3ba8deaa3168c7d103a5a88c826b4.jpg new file mode 100644 index 00000000..0e3f521b Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-175_9b3461c453320ac73cbd333afa9d84dadeb3ba8deaa3168c7d103a5a88c826b4.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-175_aee27feb0ed751f46f522fa2cc6f0edaa5d03eb7109b4792a170351a8a0d55c3.jpg b/.raw/MidasFEAAnalysisManual/images/page-175_aee27feb0ed751f46f522fa2cc6f0edaa5d03eb7109b4792a170351a8a0d55c3.jpg new file mode 100644 index 00000000..f2ac8ca5 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-175_aee27feb0ed751f46f522fa2cc6f0edaa5d03eb7109b4792a170351a8a0d55c3.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-176_087f4a234d2964e6291bbfcc26a0eb2beceacebf8e2564115ad8c16d99305f2e.jpg b/.raw/MidasFEAAnalysisManual/images/page-176_087f4a234d2964e6291bbfcc26a0eb2beceacebf8e2564115ad8c16d99305f2e.jpg new file mode 100644 index 00000000..3e55925a Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-176_087f4a234d2964e6291bbfcc26a0eb2beceacebf8e2564115ad8c16d99305f2e.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-176_1385e58ec6c205750af70d22e16a34c40d799ad21851923570557656471b789c.jpg b/.raw/MidasFEAAnalysisManual/images/page-176_1385e58ec6c205750af70d22e16a34c40d799ad21851923570557656471b789c.jpg new file mode 100644 index 00000000..b7fff3ec Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-176_1385e58ec6c205750af70d22e16a34c40d799ad21851923570557656471b789c.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-176_5197af2754527c0bcd60e3960d19813d6aff72396c9737f87c44a536957b6bcf.jpg b/.raw/MidasFEAAnalysisManual/images/page-176_5197af2754527c0bcd60e3960d19813d6aff72396c9737f87c44a536957b6bcf.jpg new file mode 100644 index 00000000..3f19ad50 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-176_5197af2754527c0bcd60e3960d19813d6aff72396c9737f87c44a536957b6bcf.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-176_667acf9f855da332b7700469b325930a841d21296082b6c803cd28895d4649ec.jpg b/.raw/MidasFEAAnalysisManual/images/page-176_667acf9f855da332b7700469b325930a841d21296082b6c803cd28895d4649ec.jpg new file mode 100644 index 00000000..c8e38dc7 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-176_667acf9f855da332b7700469b325930a841d21296082b6c803cd28895d4649ec.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-176_9a6e4791922cb5e7d3e427d149ce604ac0dc0e7c01e26f83b3fa4cfaa992507a.jpg b/.raw/MidasFEAAnalysisManual/images/page-176_9a6e4791922cb5e7d3e427d149ce604ac0dc0e7c01e26f83b3fa4cfaa992507a.jpg new file mode 100644 index 00000000..a247ed12 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-176_9a6e4791922cb5e7d3e427d149ce604ac0dc0e7c01e26f83b3fa4cfaa992507a.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-176_aa14f845186617e330a050398aff0a595a19bb52ce0f7e400ab2d6c6f349841b.jpg b/.raw/MidasFEAAnalysisManual/images/page-176_aa14f845186617e330a050398aff0a595a19bb52ce0f7e400ab2d6c6f349841b.jpg new file mode 100644 index 00000000..244070e0 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-176_aa14f845186617e330a050398aff0a595a19bb52ce0f7e400ab2d6c6f349841b.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-176_bd47f41a5ec7b02960b6372d210aaa3856edda3906e9bbddd46424ebe2bcb54d.jpg b/.raw/MidasFEAAnalysisManual/images/page-176_bd47f41a5ec7b02960b6372d210aaa3856edda3906e9bbddd46424ebe2bcb54d.jpg new file mode 100644 index 00000000..d01bc53a Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-176_bd47f41a5ec7b02960b6372d210aaa3856edda3906e9bbddd46424ebe2bcb54d.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-176_d079f77ac9de44d1ce2efc93b872e269432a37b1615112bba8216e6f0f1cb26b.jpg b/.raw/MidasFEAAnalysisManual/images/page-176_d079f77ac9de44d1ce2efc93b872e269432a37b1615112bba8216e6f0f1cb26b.jpg new file mode 100644 index 00000000..f1fa7fa7 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-176_d079f77ac9de44d1ce2efc93b872e269432a37b1615112bba8216e6f0f1cb26b.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-176_e5576a1b588a1922c0a528d05781da1315969249f352653c25ae6a0d728f566f.jpg b/.raw/MidasFEAAnalysisManual/images/page-176_e5576a1b588a1922c0a528d05781da1315969249f352653c25ae6a0d728f566f.jpg new file mode 100644 index 00000000..306b4061 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-176_e5576a1b588a1922c0a528d05781da1315969249f352653c25ae6a0d728f566f.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-177_0827cb1fe1ec74c8550719859d779008ea8e73205969a4f85422ff6eb7ea3f16.jpg b/.raw/MidasFEAAnalysisManual/images/page-177_0827cb1fe1ec74c8550719859d779008ea8e73205969a4f85422ff6eb7ea3f16.jpg new file mode 100644 index 00000000..2c0ae19d Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-177_0827cb1fe1ec74c8550719859d779008ea8e73205969a4f85422ff6eb7ea3f16.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-177_0d0d54aa92dd3c06f379bff3d1bd7ce255ab583a666e89a0bacd267248c65e8c.jpg b/.raw/MidasFEAAnalysisManual/images/page-177_0d0d54aa92dd3c06f379bff3d1bd7ce255ab583a666e89a0bacd267248c65e8c.jpg new file mode 100644 index 00000000..6721e96e Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-177_0d0d54aa92dd3c06f379bff3d1bd7ce255ab583a666e89a0bacd267248c65e8c.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-177_70a588710ce31362f7f2021eeff58e2aafc3f087eba9a8a9382f42d214d1af25.jpg b/.raw/MidasFEAAnalysisManual/images/page-177_70a588710ce31362f7f2021eeff58e2aafc3f087eba9a8a9382f42d214d1af25.jpg new file mode 100644 index 00000000..60a90a03 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-177_70a588710ce31362f7f2021eeff58e2aafc3f087eba9a8a9382f42d214d1af25.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-177_783ee44c6bfe6a1a7409c94971a349a0be840eafb0c87b85f4329d05fe0ac458.jpg b/.raw/MidasFEAAnalysisManual/images/page-177_783ee44c6bfe6a1a7409c94971a349a0be840eafb0c87b85f4329d05fe0ac458.jpg new file mode 100644 index 00000000..0f723fd5 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-177_783ee44c6bfe6a1a7409c94971a349a0be840eafb0c87b85f4329d05fe0ac458.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-177_8cd9640922876b18416515300454629fe467332b45f03d165637d29d48e21734.jpg b/.raw/MidasFEAAnalysisManual/images/page-177_8cd9640922876b18416515300454629fe467332b45f03d165637d29d48e21734.jpg new file mode 100644 index 00000000..3e89f19a Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-177_8cd9640922876b18416515300454629fe467332b45f03d165637d29d48e21734.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-177_993fc038408391af3e6b74f5cd4d64dc50812306473a0aa06b308b9fae8d375c.jpg b/.raw/MidasFEAAnalysisManual/images/page-177_993fc038408391af3e6b74f5cd4d64dc50812306473a0aa06b308b9fae8d375c.jpg new file mode 100644 index 00000000..38f44b1f Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-177_993fc038408391af3e6b74f5cd4d64dc50812306473a0aa06b308b9fae8d375c.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-177_bf48cfab5797d7c3174f3253668665d921eb4c72a70388cf55c7f5074a515714.jpg b/.raw/MidasFEAAnalysisManual/images/page-177_bf48cfab5797d7c3174f3253668665d921eb4c72a70388cf55c7f5074a515714.jpg new file mode 100644 index 00000000..e03121be Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-177_bf48cfab5797d7c3174f3253668665d921eb4c72a70388cf55c7f5074a515714.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-178_1b0fe1af970ebb7b860969f0dd8ecc347056d938c33d13084b23e5b096af33cb.jpg b/.raw/MidasFEAAnalysisManual/images/page-178_1b0fe1af970ebb7b860969f0dd8ecc347056d938c33d13084b23e5b096af33cb.jpg new file mode 100644 index 00000000..19cccc12 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-178_1b0fe1af970ebb7b860969f0dd8ecc347056d938c33d13084b23e5b096af33cb.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-178_1eb6032b53fabc7d9253efa44cbc165160cd464b6ba31a39a3c7c75e67910622.jpg b/.raw/MidasFEAAnalysisManual/images/page-178_1eb6032b53fabc7d9253efa44cbc165160cd464b6ba31a39a3c7c75e67910622.jpg new file mode 100644 index 00000000..ec69cba3 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-178_1eb6032b53fabc7d9253efa44cbc165160cd464b6ba31a39a3c7c75e67910622.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-178_27e71e5e88bfbcf5b7002ded3659b3b326faba3872302793d8a9bc20035040cd.jpg b/.raw/MidasFEAAnalysisManual/images/page-178_27e71e5e88bfbcf5b7002ded3659b3b326faba3872302793d8a9bc20035040cd.jpg new file mode 100644 index 00000000..dc0fc2e5 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-178_27e71e5e88bfbcf5b7002ded3659b3b326faba3872302793d8a9bc20035040cd.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-178_2e989567ea03aac74c3e5e5b93187a60656522eaa373bad123d5c10d0c040b82.jpg b/.raw/MidasFEAAnalysisManual/images/page-178_2e989567ea03aac74c3e5e5b93187a60656522eaa373bad123d5c10d0c040b82.jpg new file mode 100644 index 00000000..40aa1d5d Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-178_2e989567ea03aac74c3e5e5b93187a60656522eaa373bad123d5c10d0c040b82.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-178_40d5e82c32eb5605356f9b537aec76fdf600f58b3007078affced42080a47f72.jpg b/.raw/MidasFEAAnalysisManual/images/page-178_40d5e82c32eb5605356f9b537aec76fdf600f58b3007078affced42080a47f72.jpg new file mode 100644 index 00000000..7bafd9dc Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-178_40d5e82c32eb5605356f9b537aec76fdf600f58b3007078affced42080a47f72.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-178_44aa1ecc999d9170f69a29b6e78d294374a914cdcccbea8767dcf56c562cb639.jpg b/.raw/MidasFEAAnalysisManual/images/page-178_44aa1ecc999d9170f69a29b6e78d294374a914cdcccbea8767dcf56c562cb639.jpg new file mode 100644 index 00000000..ef037627 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-178_44aa1ecc999d9170f69a29b6e78d294374a914cdcccbea8767dcf56c562cb639.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-178_49557821783474ba61285d45cf65cebc94ef064e2097713a3801fdc95b135d83.jpg b/.raw/MidasFEAAnalysisManual/images/page-178_49557821783474ba61285d45cf65cebc94ef064e2097713a3801fdc95b135d83.jpg new file mode 100644 index 00000000..94345887 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-178_49557821783474ba61285d45cf65cebc94ef064e2097713a3801fdc95b135d83.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-178_57ff5d5ea2dc15730144a8d077b7e3e3ddf9c31550650b65bf03ffe324970bec.jpg b/.raw/MidasFEAAnalysisManual/images/page-178_57ff5d5ea2dc15730144a8d077b7e3e3ddf9c31550650b65bf03ffe324970bec.jpg new file mode 100644 index 00000000..ba892d25 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-178_57ff5d5ea2dc15730144a8d077b7e3e3ddf9c31550650b65bf03ffe324970bec.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-178_5fb65b14ea4972704785083b1abb4aac19151843a674468822558f0916d3532d.jpg b/.raw/MidasFEAAnalysisManual/images/page-178_5fb65b14ea4972704785083b1abb4aac19151843a674468822558f0916d3532d.jpg new file mode 100644 index 00000000..67785e97 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-178_5fb65b14ea4972704785083b1abb4aac19151843a674468822558f0916d3532d.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-178_6b6f7b850efd9a67cc95301f41d9dabb81f91cd21dfa4a8eb1a91bcac9ac3775.jpg b/.raw/MidasFEAAnalysisManual/images/page-178_6b6f7b850efd9a67cc95301f41d9dabb81f91cd21dfa4a8eb1a91bcac9ac3775.jpg new file mode 100644 index 00000000..7871f5ef Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-178_6b6f7b850efd9a67cc95301f41d9dabb81f91cd21dfa4a8eb1a91bcac9ac3775.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-178_6e6df8fca340a98f0493c5b3cecc44c4fd4ed27fdcee5ad083ad590504f9c95a.jpg b/.raw/MidasFEAAnalysisManual/images/page-178_6e6df8fca340a98f0493c5b3cecc44c4fd4ed27fdcee5ad083ad590504f9c95a.jpg new file mode 100644 index 00000000..8db2c274 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-178_6e6df8fca340a98f0493c5b3cecc44c4fd4ed27fdcee5ad083ad590504f9c95a.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-178_7324fe6345048cd791a18d04b3e1fa22c6bc56993b19361d6012b58cbd806538.jpg b/.raw/MidasFEAAnalysisManual/images/page-178_7324fe6345048cd791a18d04b3e1fa22c6bc56993b19361d6012b58cbd806538.jpg new file mode 100644 index 00000000..2dac37da Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-178_7324fe6345048cd791a18d04b3e1fa22c6bc56993b19361d6012b58cbd806538.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-178_90fdc0a985c83d1169faeed1d47751b037e1ac8a8f653d6d079fea96dfa39ecb.jpg b/.raw/MidasFEAAnalysisManual/images/page-178_90fdc0a985c83d1169faeed1d47751b037e1ac8a8f653d6d079fea96dfa39ecb.jpg new file mode 100644 index 00000000..b61f6577 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-178_90fdc0a985c83d1169faeed1d47751b037e1ac8a8f653d6d079fea96dfa39ecb.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-178_98ef506d20eec03069e5da3a5f93807c466100748d2be3dd95dfc315702c9b4d.jpg b/.raw/MidasFEAAnalysisManual/images/page-178_98ef506d20eec03069e5da3a5f93807c466100748d2be3dd95dfc315702c9b4d.jpg new file mode 100644 index 00000000..46a97bcf Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-178_98ef506d20eec03069e5da3a5f93807c466100748d2be3dd95dfc315702c9b4d.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-178_9c1a8610425f172a3ba5fad811e79e246f014c7df40a5ec4d34597f1ff42e3c3.jpg b/.raw/MidasFEAAnalysisManual/images/page-178_9c1a8610425f172a3ba5fad811e79e246f014c7df40a5ec4d34597f1ff42e3c3.jpg new file mode 100644 index 00000000..bc41a6f0 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-178_9c1a8610425f172a3ba5fad811e79e246f014c7df40a5ec4d34597f1ff42e3c3.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-178_9e655b500d429cded9a2aac9578ee731092dd864414ad50bbfe8ec4ec13efa77.jpg b/.raw/MidasFEAAnalysisManual/images/page-178_9e655b500d429cded9a2aac9578ee731092dd864414ad50bbfe8ec4ec13efa77.jpg new file mode 100644 index 00000000..f77ad020 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-178_9e655b500d429cded9a2aac9578ee731092dd864414ad50bbfe8ec4ec13efa77.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-178_a4deeb7900fb510804dc7bd3ae05112fc43acf31d468782cacd2924c096c503e.jpg b/.raw/MidasFEAAnalysisManual/images/page-178_a4deeb7900fb510804dc7bd3ae05112fc43acf31d468782cacd2924c096c503e.jpg new file mode 100644 index 00000000..92f7d5f7 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-178_a4deeb7900fb510804dc7bd3ae05112fc43acf31d468782cacd2924c096c503e.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-178_c140e900bb6e819bff42c3993e071a2c45c5630730fa6e7fab0215641379d6a8.jpg b/.raw/MidasFEAAnalysisManual/images/page-178_c140e900bb6e819bff42c3993e071a2c45c5630730fa6e7fab0215641379d6a8.jpg new file mode 100644 index 00000000..79ac024d Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-178_c140e900bb6e819bff42c3993e071a2c45c5630730fa6e7fab0215641379d6a8.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-178_e3f8d2026081cf13be1f03a34ed336f327f279293c0dd0f4b467a04e04e32a65.jpg b/.raw/MidasFEAAnalysisManual/images/page-178_e3f8d2026081cf13be1f03a34ed336f327f279293c0dd0f4b467a04e04e32a65.jpg new file mode 100644 index 00000000..79346606 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-178_e3f8d2026081cf13be1f03a34ed336f327f279293c0dd0f4b467a04e04e32a65.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-178_e7132692accccfb70b2170ac4ea5d60adaeb2c7a63cabbabdbff150d5976071a.jpg b/.raw/MidasFEAAnalysisManual/images/page-178_e7132692accccfb70b2170ac4ea5d60adaeb2c7a63cabbabdbff150d5976071a.jpg new file mode 100644 index 00000000..b5a02ff9 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-178_e7132692accccfb70b2170ac4ea5d60adaeb2c7a63cabbabdbff150d5976071a.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-178_f1402ff46babf3b5278c31842f50ac8d390372af13511a823743466c3e927d0d.jpg b/.raw/MidasFEAAnalysisManual/images/page-178_f1402ff46babf3b5278c31842f50ac8d390372af13511a823743466c3e927d0d.jpg new file mode 100644 index 00000000..c8504fe2 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-178_f1402ff46babf3b5278c31842f50ac8d390372af13511a823743466c3e927d0d.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-178_f7812bf756d0298c3dbcba460fed3ea04d668d1844fbbd3d6a6b9ed478958246.jpg b/.raw/MidasFEAAnalysisManual/images/page-178_f7812bf756d0298c3dbcba460fed3ea04d668d1844fbbd3d6a6b9ed478958246.jpg new file mode 100644 index 00000000..c0c5a1d6 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-178_f7812bf756d0298c3dbcba460fed3ea04d668d1844fbbd3d6a6b9ed478958246.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-179_134a0b2418eb617cfc49b608c20c71a0cb057b1bdb0f5a818133bfaf34c47280.jpg b/.raw/MidasFEAAnalysisManual/images/page-179_134a0b2418eb617cfc49b608c20c71a0cb057b1bdb0f5a818133bfaf34c47280.jpg new file mode 100644 index 00000000..e1287783 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-179_134a0b2418eb617cfc49b608c20c71a0cb057b1bdb0f5a818133bfaf34c47280.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-179_349ce2904068f258e82fa51ee29b2d791ea3fd8da0d72ba67bdc476a5e6ffb71.jpg b/.raw/MidasFEAAnalysisManual/images/page-179_349ce2904068f258e82fa51ee29b2d791ea3fd8da0d72ba67bdc476a5e6ffb71.jpg new file mode 100644 index 00000000..48985ef7 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-179_349ce2904068f258e82fa51ee29b2d791ea3fd8da0d72ba67bdc476a5e6ffb71.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-179_5a16f0bede3e0a16b9ac2f9ce61458087705438bb6bc9a25ca775fbd41017498.jpg b/.raw/MidasFEAAnalysisManual/images/page-179_5a16f0bede3e0a16b9ac2f9ce61458087705438bb6bc9a25ca775fbd41017498.jpg new file mode 100644 index 00000000..ab9bf685 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-179_5a16f0bede3e0a16b9ac2f9ce61458087705438bb6bc9a25ca775fbd41017498.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-179_87a9aa7650a9d11d9a032225103f0c6dc2c8f27a5811cdf5414b02538f1ce25d.jpg b/.raw/MidasFEAAnalysisManual/images/page-179_87a9aa7650a9d11d9a032225103f0c6dc2c8f27a5811cdf5414b02538f1ce25d.jpg new file mode 100644 index 00000000..f2d5b51d Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-179_87a9aa7650a9d11d9a032225103f0c6dc2c8f27a5811cdf5414b02538f1ce25d.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-179_967435929b673e1602a54b1ac039dec42db016970195383415e8107993e23b25.jpg b/.raw/MidasFEAAnalysisManual/images/page-179_967435929b673e1602a54b1ac039dec42db016970195383415e8107993e23b25.jpg new file mode 100644 index 00000000..7e1b570a Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-179_967435929b673e1602a54b1ac039dec42db016970195383415e8107993e23b25.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-179_9735ebfc9df82e9523cbf99ef707f3801b99e384dbaea7cd08673fd81e943863.jpg b/.raw/MidasFEAAnalysisManual/images/page-179_9735ebfc9df82e9523cbf99ef707f3801b99e384dbaea7cd08673fd81e943863.jpg new file mode 100644 index 00000000..bedacd8a Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-179_9735ebfc9df82e9523cbf99ef707f3801b99e384dbaea7cd08673fd81e943863.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-179_a6b6aa7dc03b86bce593d3620040d6184a15e4d28e9259edfa11a40a7f109f61.jpg b/.raw/MidasFEAAnalysisManual/images/page-179_a6b6aa7dc03b86bce593d3620040d6184a15e4d28e9259edfa11a40a7f109f61.jpg new file mode 100644 index 00000000..d1cc36ec Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-179_a6b6aa7dc03b86bce593d3620040d6184a15e4d28e9259edfa11a40a7f109f61.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-179_a9bbcabb9e79269a82a108d1166a665f36cb45dfa021004dc6ed85db8f9fd5fd.jpg b/.raw/MidasFEAAnalysisManual/images/page-179_a9bbcabb9e79269a82a108d1166a665f36cb45dfa021004dc6ed85db8f9fd5fd.jpg new file mode 100644 index 00000000..ff7a0be3 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-179_a9bbcabb9e79269a82a108d1166a665f36cb45dfa021004dc6ed85db8f9fd5fd.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-179_abd9f09c37074dba8f1ec35859852c9a206c17d90687519f1bd247cca1929786.jpg b/.raw/MidasFEAAnalysisManual/images/page-179_abd9f09c37074dba8f1ec35859852c9a206c17d90687519f1bd247cca1929786.jpg new file mode 100644 index 00000000..8b406cb4 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-179_abd9f09c37074dba8f1ec35859852c9a206c17d90687519f1bd247cca1929786.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-179_c01b1be10492cfb640e4f30788ca31706c772735e51b135577bd9e696800fec6.jpg b/.raw/MidasFEAAnalysisManual/images/page-179_c01b1be10492cfb640e4f30788ca31706c772735e51b135577bd9e696800fec6.jpg new file mode 100644 index 00000000..3fd8b8b7 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-179_c01b1be10492cfb640e4f30788ca31706c772735e51b135577bd9e696800fec6.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-179_cfd2462ef024171eacf271183ef7b46e6a9d082a3c1f989cb84d16c7c8cfe1d9.jpg b/.raw/MidasFEAAnalysisManual/images/page-179_cfd2462ef024171eacf271183ef7b46e6a9d082a3c1f989cb84d16c7c8cfe1d9.jpg new file mode 100644 index 00000000..0fa36d41 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-179_cfd2462ef024171eacf271183ef7b46e6a9d082a3c1f989cb84d16c7c8cfe1d9.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-179_e816c3362eddfe58ffc5c96c8764340fe747912d62c2bfe9896cd98aa3d892a8.jpg b/.raw/MidasFEAAnalysisManual/images/page-179_e816c3362eddfe58ffc5c96c8764340fe747912d62c2bfe9896cd98aa3d892a8.jpg new file mode 100644 index 00000000..6d4de854 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-179_e816c3362eddfe58ffc5c96c8764340fe747912d62c2bfe9896cd98aa3d892a8.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-180_2007b16583d9d634391a9985878c698b7b0a4e6ff25a0b953deb6e5e9d4fd639.jpg b/.raw/MidasFEAAnalysisManual/images/page-180_2007b16583d9d634391a9985878c698b7b0a4e6ff25a0b953deb6e5e9d4fd639.jpg new file mode 100644 index 00000000..8c7af58a Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-180_2007b16583d9d634391a9985878c698b7b0a4e6ff25a0b953deb6e5e9d4fd639.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-180_31a28b7f5c8c331c9f1b0bbc0fe41d995f79e20f295011ab0a439b922bbb3fe5.jpg b/.raw/MidasFEAAnalysisManual/images/page-180_31a28b7f5c8c331c9f1b0bbc0fe41d995f79e20f295011ab0a439b922bbb3fe5.jpg new file mode 100644 index 00000000..83d1d666 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-180_31a28b7f5c8c331c9f1b0bbc0fe41d995f79e20f295011ab0a439b922bbb3fe5.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-180_3d793696c86d4b30751dc5dc3f44d6ae34edac2f34ffb67b358814c68eb5e6e4.jpg b/.raw/MidasFEAAnalysisManual/images/page-180_3d793696c86d4b30751dc5dc3f44d6ae34edac2f34ffb67b358814c68eb5e6e4.jpg new file mode 100644 index 00000000..7373b371 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-180_3d793696c86d4b30751dc5dc3f44d6ae34edac2f34ffb67b358814c68eb5e6e4.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-180_45d0c5773f88d5ebd6c6cf1de131def11925c5b640d01d3a29af410cdd1eda9d.jpg b/.raw/MidasFEAAnalysisManual/images/page-180_45d0c5773f88d5ebd6c6cf1de131def11925c5b640d01d3a29af410cdd1eda9d.jpg new file mode 100644 index 00000000..d64210d3 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-180_45d0c5773f88d5ebd6c6cf1de131def11925c5b640d01d3a29af410cdd1eda9d.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-180_6a161f4b22e5dc6f42d21d245888ea96e644dc43f21503296230dd45770f368a.jpg b/.raw/MidasFEAAnalysisManual/images/page-180_6a161f4b22e5dc6f42d21d245888ea96e644dc43f21503296230dd45770f368a.jpg new file mode 100644 index 00000000..fea1a5b3 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-180_6a161f4b22e5dc6f42d21d245888ea96e644dc43f21503296230dd45770f368a.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-180_82d4518b0d8d24c21b450e5207703dfca339005d5f8edbc75b84301af3f34374.jpg b/.raw/MidasFEAAnalysisManual/images/page-180_82d4518b0d8d24c21b450e5207703dfca339005d5f8edbc75b84301af3f34374.jpg new file mode 100644 index 00000000..7b75191f Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-180_82d4518b0d8d24c21b450e5207703dfca339005d5f8edbc75b84301af3f34374.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-180_a3dd566ad9a66e87dafeda0e37bd41d568444163c2e6b99aab61ceccc28ebdb4.jpg b/.raw/MidasFEAAnalysisManual/images/page-180_a3dd566ad9a66e87dafeda0e37bd41d568444163c2e6b99aab61ceccc28ebdb4.jpg new file mode 100644 index 00000000..bc770b2d Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-180_a3dd566ad9a66e87dafeda0e37bd41d568444163c2e6b99aab61ceccc28ebdb4.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-180_befd8fbf9427cb5c0bffaf140c60b3ddcd7fcfdeeae036cb2766777e7feff524.jpg b/.raw/MidasFEAAnalysisManual/images/page-180_befd8fbf9427cb5c0bffaf140c60b3ddcd7fcfdeeae036cb2766777e7feff524.jpg new file mode 100644 index 00000000..ffb955c3 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-180_befd8fbf9427cb5c0bffaf140c60b3ddcd7fcfdeeae036cb2766777e7feff524.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-180_cca95df006de21dd721886f592749396f4c290d2c840a360172d533f95378983.jpg b/.raw/MidasFEAAnalysisManual/images/page-180_cca95df006de21dd721886f592749396f4c290d2c840a360172d533f95378983.jpg new file mode 100644 index 00000000..3efedec1 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-180_cca95df006de21dd721886f592749396f4c290d2c840a360172d533f95378983.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-181_d82a03eac96d441c799cb1c06e1115ae454a85753c56f623b39fa3ae6edea19e.jpg b/.raw/MidasFEAAnalysisManual/images/page-181_d82a03eac96d441c799cb1c06e1115ae454a85753c56f623b39fa3ae6edea19e.jpg new file mode 100644 index 00000000..4eb7894d Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-181_d82a03eac96d441c799cb1c06e1115ae454a85753c56f623b39fa3ae6edea19e.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-181_d905aeab012cbae455c8727a3c34c1ce4e217b16d1dd050a3e253f4c48efffb0.jpg b/.raw/MidasFEAAnalysisManual/images/page-181_d905aeab012cbae455c8727a3c34c1ce4e217b16d1dd050a3e253f4c48efffb0.jpg new file mode 100644 index 00000000..46de1001 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-181_d905aeab012cbae455c8727a3c34c1ce4e217b16d1dd050a3e253f4c48efffb0.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-181_dd5d8ac74577ce654cdcf206e8c74bccf7c09e8256c4209f1ddc097606316291.jpg b/.raw/MidasFEAAnalysisManual/images/page-181_dd5d8ac74577ce654cdcf206e8c74bccf7c09e8256c4209f1ddc097606316291.jpg new file mode 100644 index 00000000..45935a2e Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-181_dd5d8ac74577ce654cdcf206e8c74bccf7c09e8256c4209f1ddc097606316291.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-181_ff0a031801c1852d074c4e91917ed3317ce088dadef324352e57cbbebe88677c.jpg b/.raw/MidasFEAAnalysisManual/images/page-181_ff0a031801c1852d074c4e91917ed3317ce088dadef324352e57cbbebe88677c.jpg new file mode 100644 index 00000000..2b6975e3 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-181_ff0a031801c1852d074c4e91917ed3317ce088dadef324352e57cbbebe88677c.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-182_171f67d7aed705432d7918d06c66f96f5748ed9e33163b4c89456944bd3dc107.jpg b/.raw/MidasFEAAnalysisManual/images/page-182_171f67d7aed705432d7918d06c66f96f5748ed9e33163b4c89456944bd3dc107.jpg new file mode 100644 index 00000000..c43ea336 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-182_171f67d7aed705432d7918d06c66f96f5748ed9e33163b4c89456944bd3dc107.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-182_c5d52150f38b733120a19c979f97b6766be55e547551d79c05914ae329116501.jpg b/.raw/MidasFEAAnalysisManual/images/page-182_c5d52150f38b733120a19c979f97b6766be55e547551d79c05914ae329116501.jpg new file mode 100644 index 00000000..cc93fca8 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-182_c5d52150f38b733120a19c979f97b6766be55e547551d79c05914ae329116501.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-183_1758dff7cea3d32f7c22ebe6480caf843fd5d8ad575d9c8fc1b62a681879518c.jpg b/.raw/MidasFEAAnalysisManual/images/page-183_1758dff7cea3d32f7c22ebe6480caf843fd5d8ad575d9c8fc1b62a681879518c.jpg new file mode 100644 index 00000000..3d0ac104 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-183_1758dff7cea3d32f7c22ebe6480caf843fd5d8ad575d9c8fc1b62a681879518c.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-184_38d73e83f194f210ef9e210b2b249abb88c565d5cfad2f3a4e30839d1f0792b4.jpg b/.raw/MidasFEAAnalysisManual/images/page-184_38d73e83f194f210ef9e210b2b249abb88c565d5cfad2f3a4e30839d1f0792b4.jpg new file mode 100644 index 00000000..dc3b9e41 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-184_38d73e83f194f210ef9e210b2b249abb88c565d5cfad2f3a4e30839d1f0792b4.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-184_9a506e7a4c98f55b4555e87925d3fa548a2dfb2ae54008d684806eb066828443.jpg b/.raw/MidasFEAAnalysisManual/images/page-184_9a506e7a4c98f55b4555e87925d3fa548a2dfb2ae54008d684806eb066828443.jpg new file mode 100644 index 00000000..4165c394 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-184_9a506e7a4c98f55b4555e87925d3fa548a2dfb2ae54008d684806eb066828443.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-184_ac0e368502cf3250aae4ba38c5809013e90ea58d100e22484ca7d128cf9ff311.jpg b/.raw/MidasFEAAnalysisManual/images/page-184_ac0e368502cf3250aae4ba38c5809013e90ea58d100e22484ca7d128cf9ff311.jpg new file mode 100644 index 00000000..281f95e3 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-184_ac0e368502cf3250aae4ba38c5809013e90ea58d100e22484ca7d128cf9ff311.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-184_cf3f679151060908eb8cdd626b9371334737d71c00e0b3b16cb6e099b0a901f2.jpg b/.raw/MidasFEAAnalysisManual/images/page-184_cf3f679151060908eb8cdd626b9371334737d71c00e0b3b16cb6e099b0a901f2.jpg new file mode 100644 index 00000000..7215fe62 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-184_cf3f679151060908eb8cdd626b9371334737d71c00e0b3b16cb6e099b0a901f2.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-185_0c9654efd284c3c47c749b21690564c5e85b9254583d4e7bd042aeef66df39a5.jpg b/.raw/MidasFEAAnalysisManual/images/page-185_0c9654efd284c3c47c749b21690564c5e85b9254583d4e7bd042aeef66df39a5.jpg new file mode 100644 index 00000000..b74ebb04 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-185_0c9654efd284c3c47c749b21690564c5e85b9254583d4e7bd042aeef66df39a5.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-185_2f764e4935c6e443f031e51223bb41d7a6ab1ec07fab8c88deadb3dad54658d9.jpg b/.raw/MidasFEAAnalysisManual/images/page-185_2f764e4935c6e443f031e51223bb41d7a6ab1ec07fab8c88deadb3dad54658d9.jpg new file mode 100644 index 00000000..2bf1cb40 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-185_2f764e4935c6e443f031e51223bb41d7a6ab1ec07fab8c88deadb3dad54658d9.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-185_3a879fd1253fa72c75dd21a9470944e8b6163724bb1aa9555c7e7a6b3cd00f17.jpg b/.raw/MidasFEAAnalysisManual/images/page-185_3a879fd1253fa72c75dd21a9470944e8b6163724bb1aa9555c7e7a6b3cd00f17.jpg new file mode 100644 index 00000000..61f3becf Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-185_3a879fd1253fa72c75dd21a9470944e8b6163724bb1aa9555c7e7a6b3cd00f17.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-185_76bfe14baabcc7fb56cb104e1bb8164748fe6220b9e5c7f10ee48df710f62472.jpg b/.raw/MidasFEAAnalysisManual/images/page-185_76bfe14baabcc7fb56cb104e1bb8164748fe6220b9e5c7f10ee48df710f62472.jpg new file mode 100644 index 00000000..6eb6f339 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-185_76bfe14baabcc7fb56cb104e1bb8164748fe6220b9e5c7f10ee48df710f62472.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-185_dcbc715567a275787e77c7bf15bda07d975fb1cd9e6ffd88250fd62b723b931d.jpg b/.raw/MidasFEAAnalysisManual/images/page-185_dcbc715567a275787e77c7bf15bda07d975fb1cd9e6ffd88250fd62b723b931d.jpg new file mode 100644 index 00000000..bf2626ee Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-185_dcbc715567a275787e77c7bf15bda07d975fb1cd9e6ffd88250fd62b723b931d.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-185_fe7f331912b0b0cdba95a22e072ec0aeb0f04f9155b9c15d0ed1e264288f46a2.jpg b/.raw/MidasFEAAnalysisManual/images/page-185_fe7f331912b0b0cdba95a22e072ec0aeb0f04f9155b9c15d0ed1e264288f46a2.jpg new file mode 100644 index 00000000..a956846c Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-185_fe7f331912b0b0cdba95a22e072ec0aeb0f04f9155b9c15d0ed1e264288f46a2.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-186_01c3f6ff0e0189d57c38058843d7fb9b23fcfaf8e808ecb0bb286911878f593a.jpg b/.raw/MidasFEAAnalysisManual/images/page-186_01c3f6ff0e0189d57c38058843d7fb9b23fcfaf8e808ecb0bb286911878f593a.jpg new file mode 100644 index 00000000..d73984e5 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-186_01c3f6ff0e0189d57c38058843d7fb9b23fcfaf8e808ecb0bb286911878f593a.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-186_1e920bbab6a0c2012e8445b0251de19186cf87006312dc61acb98f145c4026ab.jpg b/.raw/MidasFEAAnalysisManual/images/page-186_1e920bbab6a0c2012e8445b0251de19186cf87006312dc61acb98f145c4026ab.jpg new file mode 100644 index 00000000..1bf38b85 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-186_1e920bbab6a0c2012e8445b0251de19186cf87006312dc61acb98f145c4026ab.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-186_308d82c1d528a902db29f1731a39e0f68a5c48a077f9179cfa0d327d06313f4b.jpg b/.raw/MidasFEAAnalysisManual/images/page-186_308d82c1d528a902db29f1731a39e0f68a5c48a077f9179cfa0d327d06313f4b.jpg new file mode 100644 index 00000000..6e29f211 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-186_308d82c1d528a902db29f1731a39e0f68a5c48a077f9179cfa0d327d06313f4b.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-186_5c6f6ddfc9e49a65f07d5fcb9c6a243d838872b6235aa3f434c64696499b1388.jpg b/.raw/MidasFEAAnalysisManual/images/page-186_5c6f6ddfc9e49a65f07d5fcb9c6a243d838872b6235aa3f434c64696499b1388.jpg new file mode 100644 index 00000000..d7ab3254 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-186_5c6f6ddfc9e49a65f07d5fcb9c6a243d838872b6235aa3f434c64696499b1388.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-186_89f6bb742e5df0634e6d089a495232072f2a4b915935d38263ad625347b3ecbc.jpg b/.raw/MidasFEAAnalysisManual/images/page-186_89f6bb742e5df0634e6d089a495232072f2a4b915935d38263ad625347b3ecbc.jpg new file mode 100644 index 00000000..36856fc2 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-186_89f6bb742e5df0634e6d089a495232072f2a4b915935d38263ad625347b3ecbc.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-186_ed8fc50e55ecd072a84d6fba76dbb825e58efad044a68e60707f26e46d44dd9e.jpg b/.raw/MidasFEAAnalysisManual/images/page-186_ed8fc50e55ecd072a84d6fba76dbb825e58efad044a68e60707f26e46d44dd9e.jpg new file mode 100644 index 00000000..1dac706b Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-186_ed8fc50e55ecd072a84d6fba76dbb825e58efad044a68e60707f26e46d44dd9e.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-186_f4468a505eb6e2aa732de30334b2c542d00cd55b27402e3e19de3c5d27422590.jpg b/.raw/MidasFEAAnalysisManual/images/page-186_f4468a505eb6e2aa732de30334b2c542d00cd55b27402e3e19de3c5d27422590.jpg new file mode 100644 index 00000000..374cffd5 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-186_f4468a505eb6e2aa732de30334b2c542d00cd55b27402e3e19de3c5d27422590.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-186_f7ba9f5f09127ebb88aae3196d55616e3bb1340bf369825b34f8843bb4bc8cc7.jpg b/.raw/MidasFEAAnalysisManual/images/page-186_f7ba9f5f09127ebb88aae3196d55616e3bb1340bf369825b34f8843bb4bc8cc7.jpg new file mode 100644 index 00000000..492b6b52 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-186_f7ba9f5f09127ebb88aae3196d55616e3bb1340bf369825b34f8843bb4bc8cc7.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-186_f9472342bd557f15f2cbf5a7e44c4c7f08647206d48b4d1128d49d3ddf1c2a51.jpg b/.raw/MidasFEAAnalysisManual/images/page-186_f9472342bd557f15f2cbf5a7e44c4c7f08647206d48b4d1128d49d3ddf1c2a51.jpg new file mode 100644 index 00000000..4f29b2f6 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-186_f9472342bd557f15f2cbf5a7e44c4c7f08647206d48b4d1128d49d3ddf1c2a51.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-187_263338c11526dce0d6ecca4c8e274dd5a5c6e652cce3779cf67a47b7bef84fae.jpg b/.raw/MidasFEAAnalysisManual/images/page-187_263338c11526dce0d6ecca4c8e274dd5a5c6e652cce3779cf67a47b7bef84fae.jpg new file mode 100644 index 00000000..1d74f8c4 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-187_263338c11526dce0d6ecca4c8e274dd5a5c6e652cce3779cf67a47b7bef84fae.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-187_3e6ec253a5f07a39b33f585af3b51006f3076b6bbf9336a8e33ee38b69164e39.jpg b/.raw/MidasFEAAnalysisManual/images/page-187_3e6ec253a5f07a39b33f585af3b51006f3076b6bbf9336a8e33ee38b69164e39.jpg new file mode 100644 index 00000000..d4b3ffbd Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-187_3e6ec253a5f07a39b33f585af3b51006f3076b6bbf9336a8e33ee38b69164e39.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-187_5e2421ae42ca11effe1115af2362556a9dd4fc7e912e300b1e3e71d9eb191dae.jpg b/.raw/MidasFEAAnalysisManual/images/page-187_5e2421ae42ca11effe1115af2362556a9dd4fc7e912e300b1e3e71d9eb191dae.jpg new file mode 100644 index 00000000..9ff03921 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-187_5e2421ae42ca11effe1115af2362556a9dd4fc7e912e300b1e3e71d9eb191dae.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-187_7932055ece9fdded5328fd881ec4ea55efa2790265b3f37299391f66b6a4ed3f.jpg b/.raw/MidasFEAAnalysisManual/images/page-187_7932055ece9fdded5328fd881ec4ea55efa2790265b3f37299391f66b6a4ed3f.jpg new file mode 100644 index 00000000..9a2ff949 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-187_7932055ece9fdded5328fd881ec4ea55efa2790265b3f37299391f66b6a4ed3f.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-187_a06d9de975cdb0dadefc9f98392f9d6281507ffd60c9f1c8563a9c80507eec85.jpg b/.raw/MidasFEAAnalysisManual/images/page-187_a06d9de975cdb0dadefc9f98392f9d6281507ffd60c9f1c8563a9c80507eec85.jpg new file mode 100644 index 00000000..893039ce Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-187_a06d9de975cdb0dadefc9f98392f9d6281507ffd60c9f1c8563a9c80507eec85.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-187_bf7f2797dfd2d5a9bdffac8a4714aff31f651f75120183082a74908a464e6469.jpg b/.raw/MidasFEAAnalysisManual/images/page-187_bf7f2797dfd2d5a9bdffac8a4714aff31f651f75120183082a74908a464e6469.jpg new file mode 100644 index 00000000..05949526 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-187_bf7f2797dfd2d5a9bdffac8a4714aff31f651f75120183082a74908a464e6469.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-187_c9469424e146ab51c8dabd139700ca33995f6be2eac9dcf43971d60e191e949c.jpg b/.raw/MidasFEAAnalysisManual/images/page-187_c9469424e146ab51c8dabd139700ca33995f6be2eac9dcf43971d60e191e949c.jpg new file mode 100644 index 00000000..5a1b0f86 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-187_c9469424e146ab51c8dabd139700ca33995f6be2eac9dcf43971d60e191e949c.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-187_d481c4286eef15833c6570e670cea1425f4b988d53199a4aa608d00464bae176.jpg b/.raw/MidasFEAAnalysisManual/images/page-187_d481c4286eef15833c6570e670cea1425f4b988d53199a4aa608d00464bae176.jpg new file mode 100644 index 00000000..124b92e8 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-187_d481c4286eef15833c6570e670cea1425f4b988d53199a4aa608d00464bae176.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-187_fc51d399d4574dd4e8db8bf66aafefe28b9dffacac25e2c9857ac1e3ee3d5b3c.jpg b/.raw/MidasFEAAnalysisManual/images/page-187_fc51d399d4574dd4e8db8bf66aafefe28b9dffacac25e2c9857ac1e3ee3d5b3c.jpg new file mode 100644 index 00000000..591f93b5 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-187_fc51d399d4574dd4e8db8bf66aafefe28b9dffacac25e2c9857ac1e3ee3d5b3c.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-188_11def76c2c5ca33f09f700898e00fbece8147d566bfddf69b361d0939f63acc7.jpg b/.raw/MidasFEAAnalysisManual/images/page-188_11def76c2c5ca33f09f700898e00fbece8147d566bfddf69b361d0939f63acc7.jpg new file mode 100644 index 00000000..729a6f57 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-188_11def76c2c5ca33f09f700898e00fbece8147d566bfddf69b361d0939f63acc7.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-188_13181ebf3d26bc66e20ba450d7bc28886a7e10b275e542ad51c17d2044285222.jpg b/.raw/MidasFEAAnalysisManual/images/page-188_13181ebf3d26bc66e20ba450d7bc28886a7e10b275e542ad51c17d2044285222.jpg new file mode 100644 index 00000000..8ad6b096 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-188_13181ebf3d26bc66e20ba450d7bc28886a7e10b275e542ad51c17d2044285222.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-188_2d9570760489c9f300b02506721a9f4a1346786677d8125fa764a304efd56f63.jpg b/.raw/MidasFEAAnalysisManual/images/page-188_2d9570760489c9f300b02506721a9f4a1346786677d8125fa764a304efd56f63.jpg new file mode 100644 index 00000000..bc907c30 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-188_2d9570760489c9f300b02506721a9f4a1346786677d8125fa764a304efd56f63.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-188_4754ddde75d51bce6d9a2c55c474dae53a78fae6bbc8cfff142a0a868d01bc0e.jpg b/.raw/MidasFEAAnalysisManual/images/page-188_4754ddde75d51bce6d9a2c55c474dae53a78fae6bbc8cfff142a0a868d01bc0e.jpg new file mode 100644 index 00000000..d4a486d1 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-188_4754ddde75d51bce6d9a2c55c474dae53a78fae6bbc8cfff142a0a868d01bc0e.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-188_4a543eaaacb4bd9b70be7e4bf8c15448d2d2fe632de26dfde67514d34e7c6fee.jpg b/.raw/MidasFEAAnalysisManual/images/page-188_4a543eaaacb4bd9b70be7e4bf8c15448d2d2fe632de26dfde67514d34e7c6fee.jpg new file mode 100644 index 00000000..c5803de5 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-188_4a543eaaacb4bd9b70be7e4bf8c15448d2d2fe632de26dfde67514d34e7c6fee.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-188_4d3d7e8a8a65c785e80671fc1d783c11cd874feb243b1e69ec1729c01772309c.jpg b/.raw/MidasFEAAnalysisManual/images/page-188_4d3d7e8a8a65c785e80671fc1d783c11cd874feb243b1e69ec1729c01772309c.jpg new file mode 100644 index 00000000..56b2386e Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-188_4d3d7e8a8a65c785e80671fc1d783c11cd874feb243b1e69ec1729c01772309c.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-188_9b78726c2859d5acb4b283ff5725f14eb2e342271ad322c47f7cc25051fdd9a5.jpg b/.raw/MidasFEAAnalysisManual/images/page-188_9b78726c2859d5acb4b283ff5725f14eb2e342271ad322c47f7cc25051fdd9a5.jpg new file mode 100644 index 00000000..e2421718 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-188_9b78726c2859d5acb4b283ff5725f14eb2e342271ad322c47f7cc25051fdd9a5.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-189_0d3e537aac4bacbaeaecda626944da9c9cb3e997cb4ef8b30e5184fa1b1a8e9e.jpg b/.raw/MidasFEAAnalysisManual/images/page-189_0d3e537aac4bacbaeaecda626944da9c9cb3e997cb4ef8b30e5184fa1b1a8e9e.jpg new file mode 100644 index 00000000..017c595f Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-189_0d3e537aac4bacbaeaecda626944da9c9cb3e997cb4ef8b30e5184fa1b1a8e9e.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-189_242d1f90463ae6171d53e34cf6082b970d3603254e1fbad7be61699c78e1f002.jpg b/.raw/MidasFEAAnalysisManual/images/page-189_242d1f90463ae6171d53e34cf6082b970d3603254e1fbad7be61699c78e1f002.jpg new file mode 100644 index 00000000..e44b09d8 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-189_242d1f90463ae6171d53e34cf6082b970d3603254e1fbad7be61699c78e1f002.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-189_67044d1650ae1a2e020fc5fd85e7ee1ac012299cad5abf4af98584a863885070.jpg b/.raw/MidasFEAAnalysisManual/images/page-189_67044d1650ae1a2e020fc5fd85e7ee1ac012299cad5abf4af98584a863885070.jpg new file mode 100644 index 00000000..33ef4ea8 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-189_67044d1650ae1a2e020fc5fd85e7ee1ac012299cad5abf4af98584a863885070.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-189_69a556c0784504be7f9ba6fed00c837c654455add930664f022730973207640f.jpg b/.raw/MidasFEAAnalysisManual/images/page-189_69a556c0784504be7f9ba6fed00c837c654455add930664f022730973207640f.jpg new file mode 100644 index 00000000..4aa01f2e Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-189_69a556c0784504be7f9ba6fed00c837c654455add930664f022730973207640f.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-189_81a0f621e279b697731e770f0ed727080ac16470915d409fb412cca19284d1e2.jpg b/.raw/MidasFEAAnalysisManual/images/page-189_81a0f621e279b697731e770f0ed727080ac16470915d409fb412cca19284d1e2.jpg new file mode 100644 index 00000000..6289fc0e Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-189_81a0f621e279b697731e770f0ed727080ac16470915d409fb412cca19284d1e2.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-189_a7e68ed4590988a8f6bac6c051a608312660e8626bd3c764f27451a3af6e600e.jpg b/.raw/MidasFEAAnalysisManual/images/page-189_a7e68ed4590988a8f6bac6c051a608312660e8626bd3c764f27451a3af6e600e.jpg new file mode 100644 index 00000000..d8789e9b Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-189_a7e68ed4590988a8f6bac6c051a608312660e8626bd3c764f27451a3af6e600e.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-189_d3da62769800e3d839d37b69a79f30c35bca4786d94fa493facce541753e5f9c.jpg b/.raw/MidasFEAAnalysisManual/images/page-189_d3da62769800e3d839d37b69a79f30c35bca4786d94fa493facce541753e5f9c.jpg new file mode 100644 index 00000000..8df8f3c2 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-189_d3da62769800e3d839d37b69a79f30c35bca4786d94fa493facce541753e5f9c.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-189_f5343c008cca6c1d2724ae2c63f71b6aeec3432053b8a1c2a2f29a6b090b8c6d.jpg b/.raw/MidasFEAAnalysisManual/images/page-189_f5343c008cca6c1d2724ae2c63f71b6aeec3432053b8a1c2a2f29a6b090b8c6d.jpg new file mode 100644 index 00000000..7676854b Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-189_f5343c008cca6c1d2724ae2c63f71b6aeec3432053b8a1c2a2f29a6b090b8c6d.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-190_73736f7c3de775bd92b6ce89cf02fce39fef9719b75a6b8bf4c79b15973f0693.jpg b/.raw/MidasFEAAnalysisManual/images/page-190_73736f7c3de775bd92b6ce89cf02fce39fef9719b75a6b8bf4c79b15973f0693.jpg new file mode 100644 index 00000000..c7bb19cb Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-190_73736f7c3de775bd92b6ce89cf02fce39fef9719b75a6b8bf4c79b15973f0693.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-191_36686dc3e7a04b50c30686c998b2d62e3544a6255ab27895deafc6581a9e9bf9.jpg b/.raw/MidasFEAAnalysisManual/images/page-191_36686dc3e7a04b50c30686c998b2d62e3544a6255ab27895deafc6581a9e9bf9.jpg new file mode 100644 index 00000000..e1e6683d Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-191_36686dc3e7a04b50c30686c998b2d62e3544a6255ab27895deafc6581a9e9bf9.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-191_7a59bb8675547a99acb89b1bc997488823074d7f0de5e5b5693dc4712c1e85fb.jpg b/.raw/MidasFEAAnalysisManual/images/page-191_7a59bb8675547a99acb89b1bc997488823074d7f0de5e5b5693dc4712c1e85fb.jpg new file mode 100644 index 00000000..f0f8c835 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-191_7a59bb8675547a99acb89b1bc997488823074d7f0de5e5b5693dc4712c1e85fb.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-191_97ed257c7f99161f6cee666f96ed244bc4a5a9b825a7b91db9177da766545939.jpg b/.raw/MidasFEAAnalysisManual/images/page-191_97ed257c7f99161f6cee666f96ed244bc4a5a9b825a7b91db9177da766545939.jpg new file mode 100644 index 00000000..5e77fc9b Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-191_97ed257c7f99161f6cee666f96ed244bc4a5a9b825a7b91db9177da766545939.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-191_bfc02b227498565f19f7b7bfa182473aaaa24672d7cf3718d634893be0c36f66.jpg b/.raw/MidasFEAAnalysisManual/images/page-191_bfc02b227498565f19f7b7bfa182473aaaa24672d7cf3718d634893be0c36f66.jpg new file mode 100644 index 00000000..9e2e70a3 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-191_bfc02b227498565f19f7b7bfa182473aaaa24672d7cf3718d634893be0c36f66.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-191_c20969dff78f53dd7360c87a13e8d8c7f7794d987fea30eb15dca26ed9f74c29.jpg b/.raw/MidasFEAAnalysisManual/images/page-191_c20969dff78f53dd7360c87a13e8d8c7f7794d987fea30eb15dca26ed9f74c29.jpg new file mode 100644 index 00000000..d769eea7 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-191_c20969dff78f53dd7360c87a13e8d8c7f7794d987fea30eb15dca26ed9f74c29.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-191_e998d74e14b327ef281e10acb4536b21cd23134fd1ffcf3d10d568035476ad7f.jpg b/.raw/MidasFEAAnalysisManual/images/page-191_e998d74e14b327ef281e10acb4536b21cd23134fd1ffcf3d10d568035476ad7f.jpg new file mode 100644 index 00000000..97388385 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-191_e998d74e14b327ef281e10acb4536b21cd23134fd1ffcf3d10d568035476ad7f.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-192_592acb3ec2f2eb72363b992bae50bba39a0bfe338e71f99b20e31f2a54c6345b.jpg b/.raw/MidasFEAAnalysisManual/images/page-192_592acb3ec2f2eb72363b992bae50bba39a0bfe338e71f99b20e31f2a54c6345b.jpg new file mode 100644 index 00000000..5ae6fc6d Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-192_592acb3ec2f2eb72363b992bae50bba39a0bfe338e71f99b20e31f2a54c6345b.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-192_5d27ad4ef914a33fcc870765357721b53541cd8bf9342a3bed431485cfd47257.jpg b/.raw/MidasFEAAnalysisManual/images/page-192_5d27ad4ef914a33fcc870765357721b53541cd8bf9342a3bed431485cfd47257.jpg new file mode 100644 index 00000000..3177f473 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-192_5d27ad4ef914a33fcc870765357721b53541cd8bf9342a3bed431485cfd47257.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-192_840c8a1474ee8aa15dd6a4b66389bd6ad7633eeaf01558fcdf2e2ac80e68ed24.jpg b/.raw/MidasFEAAnalysisManual/images/page-192_840c8a1474ee8aa15dd6a4b66389bd6ad7633eeaf01558fcdf2e2ac80e68ed24.jpg new file mode 100644 index 00000000..26aff546 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-192_840c8a1474ee8aa15dd6a4b66389bd6ad7633eeaf01558fcdf2e2ac80e68ed24.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-192_93a2084f9bacf08c8964aee56836eedfa18dea01157e458deb2c37125c20c26d.jpg b/.raw/MidasFEAAnalysisManual/images/page-192_93a2084f9bacf08c8964aee56836eedfa18dea01157e458deb2c37125c20c26d.jpg new file mode 100644 index 00000000..b7797951 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-192_93a2084f9bacf08c8964aee56836eedfa18dea01157e458deb2c37125c20c26d.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-192_d4780d72c406e5fc1c1a5d1d59b46d3aead21c6deedf4cd3473313e708be3183.jpg b/.raw/MidasFEAAnalysisManual/images/page-192_d4780d72c406e5fc1c1a5d1d59b46d3aead21c6deedf4cd3473313e708be3183.jpg new file mode 100644 index 00000000..4eb833dc Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-192_d4780d72c406e5fc1c1a5d1d59b46d3aead21c6deedf4cd3473313e708be3183.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-192_deace92912c8547bddc584280f945d3ffabd478ed875157459a49539bb3e663e.jpg b/.raw/MidasFEAAnalysisManual/images/page-192_deace92912c8547bddc584280f945d3ffabd478ed875157459a49539bb3e663e.jpg new file mode 100644 index 00000000..4dff1586 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-192_deace92912c8547bddc584280f945d3ffabd478ed875157459a49539bb3e663e.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-193_0082c85b368d8b921f3ceef01be168c9000bac8613dda260935cce92951945fe.jpg b/.raw/MidasFEAAnalysisManual/images/page-193_0082c85b368d8b921f3ceef01be168c9000bac8613dda260935cce92951945fe.jpg new file mode 100644 index 00000000..f19829bb Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-193_0082c85b368d8b921f3ceef01be168c9000bac8613dda260935cce92951945fe.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-193_936a61260539e5f9d13a108611cb927279804fe1b278b7e0144e61f78af7ea3b.jpg b/.raw/MidasFEAAnalysisManual/images/page-193_936a61260539e5f9d13a108611cb927279804fe1b278b7e0144e61f78af7ea3b.jpg new file mode 100644 index 00000000..777b4809 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-193_936a61260539e5f9d13a108611cb927279804fe1b278b7e0144e61f78af7ea3b.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-193_acab957f61102fd11f450d11f32af16671d29aaf8a5753d79bb1f7a3945ea488.jpg b/.raw/MidasFEAAnalysisManual/images/page-193_acab957f61102fd11f450d11f32af16671d29aaf8a5753d79bb1f7a3945ea488.jpg new file mode 100644 index 00000000..e8d22f23 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-193_acab957f61102fd11f450d11f32af16671d29aaf8a5753d79bb1f7a3945ea488.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-193_bc79f7a0eb7afbd3ecdf570a247623cddf8ee896667e243130a8e9cbabf5a91b.jpg b/.raw/MidasFEAAnalysisManual/images/page-193_bc79f7a0eb7afbd3ecdf570a247623cddf8ee896667e243130a8e9cbabf5a91b.jpg new file mode 100644 index 00000000..5ff020ea Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-193_bc79f7a0eb7afbd3ecdf570a247623cddf8ee896667e243130a8e9cbabf5a91b.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-194_19cc34f16f3b52c9a919fdb97afd0c46766960670d985fa934033fbeff36a698.jpg b/.raw/MidasFEAAnalysisManual/images/page-194_19cc34f16f3b52c9a919fdb97afd0c46766960670d985fa934033fbeff36a698.jpg new file mode 100644 index 00000000..291b56e2 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-194_19cc34f16f3b52c9a919fdb97afd0c46766960670d985fa934033fbeff36a698.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-194_25767a5432d464a301448579b0195f8cd57b42e7675ef93e9c46808da2571fcf.jpg b/.raw/MidasFEAAnalysisManual/images/page-194_25767a5432d464a301448579b0195f8cd57b42e7675ef93e9c46808da2571fcf.jpg new file mode 100644 index 00000000..3e84fa61 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-194_25767a5432d464a301448579b0195f8cd57b42e7675ef93e9c46808da2571fcf.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-194_271728aa0012ee14c15b966b30e3388dc518672cfd0eff7d765e7998eb8986ef.jpg b/.raw/MidasFEAAnalysisManual/images/page-194_271728aa0012ee14c15b966b30e3388dc518672cfd0eff7d765e7998eb8986ef.jpg new file mode 100644 index 00000000..8d178022 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-194_271728aa0012ee14c15b966b30e3388dc518672cfd0eff7d765e7998eb8986ef.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-195_61c0890bfb53f88c9f15b1e0e0024dfcaab767d0da4d6bea743403fab7c27724.jpg b/.raw/MidasFEAAnalysisManual/images/page-195_61c0890bfb53f88c9f15b1e0e0024dfcaab767d0da4d6bea743403fab7c27724.jpg new file mode 100644 index 00000000..6b5ab093 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-195_61c0890bfb53f88c9f15b1e0e0024dfcaab767d0da4d6bea743403fab7c27724.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-195_8e846da2a4461b76d5bcc748f745463ec9a14d067140bdfaed324c6ab6157497.jpg b/.raw/MidasFEAAnalysisManual/images/page-195_8e846da2a4461b76d5bcc748f745463ec9a14d067140bdfaed324c6ab6157497.jpg new file mode 100644 index 00000000..faac7afb Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-195_8e846da2a4461b76d5bcc748f745463ec9a14d067140bdfaed324c6ab6157497.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-195_ba770b31172e9e103e847bbaac8ad2443877813805207d44bd5ff198a5ff0752.jpg b/.raw/MidasFEAAnalysisManual/images/page-195_ba770b31172e9e103e847bbaac8ad2443877813805207d44bd5ff198a5ff0752.jpg new file mode 100644 index 00000000..4d1b4368 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-195_ba770b31172e9e103e847bbaac8ad2443877813805207d44bd5ff198a5ff0752.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-196_20531541ec5a8e59dc704b1c837aef6bb6b47d6f401ae1f0dca318be1174c4e3.jpg b/.raw/MidasFEAAnalysisManual/images/page-196_20531541ec5a8e59dc704b1c837aef6bb6b47d6f401ae1f0dca318be1174c4e3.jpg new file mode 100644 index 00000000..8d1da3fb Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-196_20531541ec5a8e59dc704b1c837aef6bb6b47d6f401ae1f0dca318be1174c4e3.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-196_b3868be5f720c3eb5e27e54ae23558a33db8af49f2bab3da13147ebadcc312f9.jpg b/.raw/MidasFEAAnalysisManual/images/page-196_b3868be5f720c3eb5e27e54ae23558a33db8af49f2bab3da13147ebadcc312f9.jpg new file mode 100644 index 00000000..f7bd4018 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-196_b3868be5f720c3eb5e27e54ae23558a33db8af49f2bab3da13147ebadcc312f9.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-196_cd3c9bf9b682484164aa4c298392fe4332548e3a44fad81d7521343ae538e8ee.jpg b/.raw/MidasFEAAnalysisManual/images/page-196_cd3c9bf9b682484164aa4c298392fe4332548e3a44fad81d7521343ae538e8ee.jpg new file mode 100644 index 00000000..368a4a8d Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-196_cd3c9bf9b682484164aa4c298392fe4332548e3a44fad81d7521343ae538e8ee.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-197_2315d5656e9e7ecab86bc5306750619148275be2109073192766377a2c9b81c7.jpg b/.raw/MidasFEAAnalysisManual/images/page-197_2315d5656e9e7ecab86bc5306750619148275be2109073192766377a2c9b81c7.jpg new file mode 100644 index 00000000..0b4cd042 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-197_2315d5656e9e7ecab86bc5306750619148275be2109073192766377a2c9b81c7.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-197_3342f90fa4325e9509b92d5409c479d3743fe57fdfb6dc4e0f177f0aac4b2306.jpg b/.raw/MidasFEAAnalysisManual/images/page-197_3342f90fa4325e9509b92d5409c479d3743fe57fdfb6dc4e0f177f0aac4b2306.jpg new file mode 100644 index 00000000..eaca720a Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-197_3342f90fa4325e9509b92d5409c479d3743fe57fdfb6dc4e0f177f0aac4b2306.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-197_5298da6ca9530ae0327b3623e85e9dd0109293759aca2a1523dfd5bf52617406.jpg b/.raw/MidasFEAAnalysisManual/images/page-197_5298da6ca9530ae0327b3623e85e9dd0109293759aca2a1523dfd5bf52617406.jpg new file mode 100644 index 00000000..e8309cf6 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-197_5298da6ca9530ae0327b3623e85e9dd0109293759aca2a1523dfd5bf52617406.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-197_eb22ba2b3776eecdfca1fd0a23630d86bd77d6a957182c5b9a3fb1616204fd0b.jpg b/.raw/MidasFEAAnalysisManual/images/page-197_eb22ba2b3776eecdfca1fd0a23630d86bd77d6a957182c5b9a3fb1616204fd0b.jpg new file mode 100644 index 00000000..1244a5f3 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-197_eb22ba2b3776eecdfca1fd0a23630d86bd77d6a957182c5b9a3fb1616204fd0b.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-198_748c6940e3eccdbfe61f5daac3b8d29fbca507902271957b594d42e66996ccf4.jpg b/.raw/MidasFEAAnalysisManual/images/page-198_748c6940e3eccdbfe61f5daac3b8d29fbca507902271957b594d42e66996ccf4.jpg new file mode 100644 index 00000000..ae82bdb0 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-198_748c6940e3eccdbfe61f5daac3b8d29fbca507902271957b594d42e66996ccf4.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-199_6ae0a855ac23a0351c6ce7f9e41e8efb983a86a668b11630fa0a7b66ba30407d.jpg b/.raw/MidasFEAAnalysisManual/images/page-199_6ae0a855ac23a0351c6ce7f9e41e8efb983a86a668b11630fa0a7b66ba30407d.jpg new file mode 100644 index 00000000..741e3fca Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-199_6ae0a855ac23a0351c6ce7f9e41e8efb983a86a668b11630fa0a7b66ba30407d.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-200_0e63c3e8dd58315c06dc4f7afd41faa1227783a113287df6696c4e46efec6d09.jpg b/.raw/MidasFEAAnalysisManual/images/page-200_0e63c3e8dd58315c06dc4f7afd41faa1227783a113287df6696c4e46efec6d09.jpg new file mode 100644 index 00000000..ae1fd2d6 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-200_0e63c3e8dd58315c06dc4f7afd41faa1227783a113287df6696c4e46efec6d09.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-200_885f50dc28f41e061eeb03925975981329e4a383cf6e6fa95f369a5cba99312b.jpg b/.raw/MidasFEAAnalysisManual/images/page-200_885f50dc28f41e061eeb03925975981329e4a383cf6e6fa95f369a5cba99312b.jpg new file mode 100644 index 00000000..d10f512f Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-200_885f50dc28f41e061eeb03925975981329e4a383cf6e6fa95f369a5cba99312b.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-200_92d991a44d450b26a588ecc527c2f7f962257c4f483449aa4511ee7e077ae7e3.jpg b/.raw/MidasFEAAnalysisManual/images/page-200_92d991a44d450b26a588ecc527c2f7f962257c4f483449aa4511ee7e077ae7e3.jpg new file mode 100644 index 00000000..f9ea8e8f Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-200_92d991a44d450b26a588ecc527c2f7f962257c4f483449aa4511ee7e077ae7e3.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-200_971e27c278ccab53b34c7ffaa5a56a494579e84f46ce3e3dce45fecacb48707c.jpg b/.raw/MidasFEAAnalysisManual/images/page-200_971e27c278ccab53b34c7ffaa5a56a494579e84f46ce3e3dce45fecacb48707c.jpg new file mode 100644 index 00000000..38cf1608 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-200_971e27c278ccab53b34c7ffaa5a56a494579e84f46ce3e3dce45fecacb48707c.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-200_bac900ab2241e794cdeee53d3654bc605cf10145b6f4472021b4da430464b255.jpg b/.raw/MidasFEAAnalysisManual/images/page-200_bac900ab2241e794cdeee53d3654bc605cf10145b6f4472021b4da430464b255.jpg new file mode 100644 index 00000000..aaa75342 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-200_bac900ab2241e794cdeee53d3654bc605cf10145b6f4472021b4da430464b255.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-201_09a531a590b841f691207685fa9f1e359498aa0c3e364a414b2127d00f59ef48.jpg b/.raw/MidasFEAAnalysisManual/images/page-201_09a531a590b841f691207685fa9f1e359498aa0c3e364a414b2127d00f59ef48.jpg new file mode 100644 index 00000000..058c3cfa Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-201_09a531a590b841f691207685fa9f1e359498aa0c3e364a414b2127d00f59ef48.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-201_2c929362bf02b2c358d8178b441ab533e8266595d17b66e3ea9b187116629d5f.jpg b/.raw/MidasFEAAnalysisManual/images/page-201_2c929362bf02b2c358d8178b441ab533e8266595d17b66e3ea9b187116629d5f.jpg new file mode 100644 index 00000000..283c1db7 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-201_2c929362bf02b2c358d8178b441ab533e8266595d17b66e3ea9b187116629d5f.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-201_2fc84bf2e2bafe4490042aa8dcbed18b884c00ebdc0a04562737740cdb5dac53.jpg b/.raw/MidasFEAAnalysisManual/images/page-201_2fc84bf2e2bafe4490042aa8dcbed18b884c00ebdc0a04562737740cdb5dac53.jpg new file mode 100644 index 00000000..fd86f2e5 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-201_2fc84bf2e2bafe4490042aa8dcbed18b884c00ebdc0a04562737740cdb5dac53.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-201_7839793972dd7e7c59f2eddd23606348934ce79ff72795a44ceb0de8bea619f6.jpg b/.raw/MidasFEAAnalysisManual/images/page-201_7839793972dd7e7c59f2eddd23606348934ce79ff72795a44ceb0de8bea619f6.jpg new file mode 100644 index 00000000..d572b21a Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-201_7839793972dd7e7c59f2eddd23606348934ce79ff72795a44ceb0de8bea619f6.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-201_8c21fc36e58171b49f05c86289c0ea9fe283077fe9f8956ad34b11178197538a.jpg b/.raw/MidasFEAAnalysisManual/images/page-201_8c21fc36e58171b49f05c86289c0ea9fe283077fe9f8956ad34b11178197538a.jpg new file mode 100644 index 00000000..03a628b2 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-201_8c21fc36e58171b49f05c86289c0ea9fe283077fe9f8956ad34b11178197538a.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-201_ad8cceec2e518133905e0da9026bea770a70f4dd347660cfc544573dcb8774c9.jpg b/.raw/MidasFEAAnalysisManual/images/page-201_ad8cceec2e518133905e0da9026bea770a70f4dd347660cfc544573dcb8774c9.jpg new file mode 100644 index 00000000..6664f0a9 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-201_ad8cceec2e518133905e0da9026bea770a70f4dd347660cfc544573dcb8774c9.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-201_c0f83e8414c38a59aaae157dd8ad61b1f80e28e6f0395c91150c062099a4d284.jpg b/.raw/MidasFEAAnalysisManual/images/page-201_c0f83e8414c38a59aaae157dd8ad61b1f80e28e6f0395c91150c062099a4d284.jpg new file mode 100644 index 00000000..162f1c49 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-201_c0f83e8414c38a59aaae157dd8ad61b1f80e28e6f0395c91150c062099a4d284.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-202_4890ee01307da5693c8e52d7cc3d48e75d9c2656afdec9eaa418f7ca2ac97220.jpg b/.raw/MidasFEAAnalysisManual/images/page-202_4890ee01307da5693c8e52d7cc3d48e75d9c2656afdec9eaa418f7ca2ac97220.jpg new file mode 100644 index 00000000..cfd0fb73 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-202_4890ee01307da5693c8e52d7cc3d48e75d9c2656afdec9eaa418f7ca2ac97220.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-202_6497f02d5a524b46eb4a7edf5dde4a105812ba5d41fd7b9823e304c1d54170ad.jpg b/.raw/MidasFEAAnalysisManual/images/page-202_6497f02d5a524b46eb4a7edf5dde4a105812ba5d41fd7b9823e304c1d54170ad.jpg new file mode 100644 index 00000000..da193433 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-202_6497f02d5a524b46eb4a7edf5dde4a105812ba5d41fd7b9823e304c1d54170ad.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-202_a3008eb4a688b33a7ae6e623a96b48c5c74f062d3d21816088a955125b7bf4cc.jpg b/.raw/MidasFEAAnalysisManual/images/page-202_a3008eb4a688b33a7ae6e623a96b48c5c74f062d3d21816088a955125b7bf4cc.jpg new file mode 100644 index 00000000..a352a2c2 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-202_a3008eb4a688b33a7ae6e623a96b48c5c74f062d3d21816088a955125b7bf4cc.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-202_af5019deb0b3b2d7c35f79508ac0932ab3eec7fed947094df2cc1436b55972c1.jpg b/.raw/MidasFEAAnalysisManual/images/page-202_af5019deb0b3b2d7c35f79508ac0932ab3eec7fed947094df2cc1436b55972c1.jpg new file mode 100644 index 00000000..1a88013f Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-202_af5019deb0b3b2d7c35f79508ac0932ab3eec7fed947094df2cc1436b55972c1.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-202_dbc9d8711915a1e77439818426c1f686fdc401449f0beaa479a8d1e2652257fa.jpg b/.raw/MidasFEAAnalysisManual/images/page-202_dbc9d8711915a1e77439818426c1f686fdc401449f0beaa479a8d1e2652257fa.jpg new file mode 100644 index 00000000..79c039ea Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-202_dbc9d8711915a1e77439818426c1f686fdc401449f0beaa479a8d1e2652257fa.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-203_411382e9724cf92f01570a1c13c798dfe6f461c4acf0b80b9a1c02689b527d75.jpg b/.raw/MidasFEAAnalysisManual/images/page-203_411382e9724cf92f01570a1c13c798dfe6f461c4acf0b80b9a1c02689b527d75.jpg new file mode 100644 index 00000000..dcce30c3 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-203_411382e9724cf92f01570a1c13c798dfe6f461c4acf0b80b9a1c02689b527d75.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-203_58cd27f041956b38b276bc5c6695232a2a83c27277b807338655de9cd28473d6.jpg b/.raw/MidasFEAAnalysisManual/images/page-203_58cd27f041956b38b276bc5c6695232a2a83c27277b807338655de9cd28473d6.jpg new file mode 100644 index 00000000..ccfe4683 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-203_58cd27f041956b38b276bc5c6695232a2a83c27277b807338655de9cd28473d6.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-203_685510ff0db80370feee46df37f92b84fcb10d5ce02a5295921a53aada9e5348.jpg b/.raw/MidasFEAAnalysisManual/images/page-203_685510ff0db80370feee46df37f92b84fcb10d5ce02a5295921a53aada9e5348.jpg new file mode 100644 index 00000000..429fd274 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-203_685510ff0db80370feee46df37f92b84fcb10d5ce02a5295921a53aada9e5348.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-203_d4f925d0e46fa84b1a39a5e252b384b5fc4b8cb7c5613ebc879bb9c89bb63832.jpg b/.raw/MidasFEAAnalysisManual/images/page-203_d4f925d0e46fa84b1a39a5e252b384b5fc4b8cb7c5613ebc879bb9c89bb63832.jpg new file mode 100644 index 00000000..1e08a9c3 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-203_d4f925d0e46fa84b1a39a5e252b384b5fc4b8cb7c5613ebc879bb9c89bb63832.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-203_f66648eaf2da00a997e6100fec14a94a52babd85efb958d39c81d35f727924c2.jpg b/.raw/MidasFEAAnalysisManual/images/page-203_f66648eaf2da00a997e6100fec14a94a52babd85efb958d39c81d35f727924c2.jpg new file mode 100644 index 00000000..7029cc3f Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-203_f66648eaf2da00a997e6100fec14a94a52babd85efb958d39c81d35f727924c2.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-204_1708cca23c45cdec5a7c0c640f174572ffc1b5dae9ca1ed8c3f1da694c7467e0.jpg b/.raw/MidasFEAAnalysisManual/images/page-204_1708cca23c45cdec5a7c0c640f174572ffc1b5dae9ca1ed8c3f1da694c7467e0.jpg new file mode 100644 index 00000000..b79c855a Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-204_1708cca23c45cdec5a7c0c640f174572ffc1b5dae9ca1ed8c3f1da694c7467e0.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-204_2b00cfd60d8806fc567293ae13dca3e4b1fa1acd1c8efc74da43beeede04a810.jpg b/.raw/MidasFEAAnalysisManual/images/page-204_2b00cfd60d8806fc567293ae13dca3e4b1fa1acd1c8efc74da43beeede04a810.jpg new file mode 100644 index 00000000..8b33d190 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-204_2b00cfd60d8806fc567293ae13dca3e4b1fa1acd1c8efc74da43beeede04a810.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-204_3bf804f8c5bca57d63dc81282ebd80a8d84e0d611bff0f62f5d63ca1e93eb270.jpg b/.raw/MidasFEAAnalysisManual/images/page-204_3bf804f8c5bca57d63dc81282ebd80a8d84e0d611bff0f62f5d63ca1e93eb270.jpg new file mode 100644 index 00000000..8513a43a Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-204_3bf804f8c5bca57d63dc81282ebd80a8d84e0d611bff0f62f5d63ca1e93eb270.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-204_512d560987d9cb5a26f1892399266440ea119f6d6f6d2ea9f01c3d3d3dc62ad5.jpg b/.raw/MidasFEAAnalysisManual/images/page-204_512d560987d9cb5a26f1892399266440ea119f6d6f6d2ea9f01c3d3d3dc62ad5.jpg new file mode 100644 index 00000000..c44ce82d Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-204_512d560987d9cb5a26f1892399266440ea119f6d6f6d2ea9f01c3d3d3dc62ad5.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-204_68cb302cff51eef18f2cc41b67db1d9fe4863e7ef7d56e82852b5fc5583354be.jpg b/.raw/MidasFEAAnalysisManual/images/page-204_68cb302cff51eef18f2cc41b67db1d9fe4863e7ef7d56e82852b5fc5583354be.jpg new file mode 100644 index 00000000..6eb6c801 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-204_68cb302cff51eef18f2cc41b67db1d9fe4863e7ef7d56e82852b5fc5583354be.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-204_7c35c022c4d9da8bf076d94ff0c72676d0034dc4ccfc0269155b6e96f98bd64d.jpg b/.raw/MidasFEAAnalysisManual/images/page-204_7c35c022c4d9da8bf076d94ff0c72676d0034dc4ccfc0269155b6e96f98bd64d.jpg new file mode 100644 index 00000000..abdbcd94 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-204_7c35c022c4d9da8bf076d94ff0c72676d0034dc4ccfc0269155b6e96f98bd64d.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-205_587fca32d3da6f5895ad44ef29b94ffcf131ce1354e7dc57952bb17fbeb11018.jpg b/.raw/MidasFEAAnalysisManual/images/page-205_587fca32d3da6f5895ad44ef29b94ffcf131ce1354e7dc57952bb17fbeb11018.jpg new file mode 100644 index 00000000..6eaa2f7b Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-205_587fca32d3da6f5895ad44ef29b94ffcf131ce1354e7dc57952bb17fbeb11018.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-205_9936932f0cc29d896d0d39beb2e5e1da8327f50e745bc80baf1d33e078fed6ea.jpg b/.raw/MidasFEAAnalysisManual/images/page-205_9936932f0cc29d896d0d39beb2e5e1da8327f50e745bc80baf1d33e078fed6ea.jpg new file mode 100644 index 00000000..49432f55 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-205_9936932f0cc29d896d0d39beb2e5e1da8327f50e745bc80baf1d33e078fed6ea.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-205_db2b189d09b5e83ece7977a83669141dfe38f797f16db1e3edac16b237c50128.jpg b/.raw/MidasFEAAnalysisManual/images/page-205_db2b189d09b5e83ece7977a83669141dfe38f797f16db1e3edac16b237c50128.jpg new file mode 100644 index 00000000..ed9e832a Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-205_db2b189d09b5e83ece7977a83669141dfe38f797f16db1e3edac16b237c50128.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-205_f1ff5cf28b0731c70f53e0aa2be16227687297b1b44049a9069604c496b84384.jpg b/.raw/MidasFEAAnalysisManual/images/page-205_f1ff5cf28b0731c70f53e0aa2be16227687297b1b44049a9069604c496b84384.jpg new file mode 100644 index 00000000..d2f00935 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-205_f1ff5cf28b0731c70f53e0aa2be16227687297b1b44049a9069604c496b84384.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-205_f265b681b770c822a49eb8c8fc165cba324db297f52f783e7867ff1cb3cb76e4.jpg b/.raw/MidasFEAAnalysisManual/images/page-205_f265b681b770c822a49eb8c8fc165cba324db297f52f783e7867ff1cb3cb76e4.jpg new file mode 100644 index 00000000..36ce040c Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-205_f265b681b770c822a49eb8c8fc165cba324db297f52f783e7867ff1cb3cb76e4.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-206_0e36928321c838c323b9bfdaeb9fd1da27f9b93a497afd0dc2d7e3f3659dda75.jpg b/.raw/MidasFEAAnalysisManual/images/page-206_0e36928321c838c323b9bfdaeb9fd1da27f9b93a497afd0dc2d7e3f3659dda75.jpg new file mode 100644 index 00000000..ca5c324a Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-206_0e36928321c838c323b9bfdaeb9fd1da27f9b93a497afd0dc2d7e3f3659dda75.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-209_3b59918fe3e98d804458548e5b8cceed902b6dc7d0da4da6387e4d12f1b0147e.jpg b/.raw/MidasFEAAnalysisManual/images/page-209_3b59918fe3e98d804458548e5b8cceed902b6dc7d0da4da6387e4d12f1b0147e.jpg new file mode 100644 index 00000000..5d8c50f1 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-209_3b59918fe3e98d804458548e5b8cceed902b6dc7d0da4da6387e4d12f1b0147e.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-209_f2cfdfa5a748c191aa62faf0374f91e72ef64eb0d217f9d7b04eb8196bef31a1.jpg b/.raw/MidasFEAAnalysisManual/images/page-209_f2cfdfa5a748c191aa62faf0374f91e72ef64eb0d217f9d7b04eb8196bef31a1.jpg new file mode 100644 index 00000000..f211e992 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-209_f2cfdfa5a748c191aa62faf0374f91e72ef64eb0d217f9d7b04eb8196bef31a1.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-210_355525081b1e44f92f8f69aa44a9196e50f61ecc1cbf09c5d09004ca6a024b17.jpg b/.raw/MidasFEAAnalysisManual/images/page-210_355525081b1e44f92f8f69aa44a9196e50f61ecc1cbf09c5d09004ca6a024b17.jpg new file mode 100644 index 00000000..bdfd556b Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-210_355525081b1e44f92f8f69aa44a9196e50f61ecc1cbf09c5d09004ca6a024b17.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-210_4be8951e663c0728de94ba349766e5899da87daf53df247b304d0305c95ec35d.jpg b/.raw/MidasFEAAnalysisManual/images/page-210_4be8951e663c0728de94ba349766e5899da87daf53df247b304d0305c95ec35d.jpg new file mode 100644 index 00000000..6ee1373f Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-210_4be8951e663c0728de94ba349766e5899da87daf53df247b304d0305c95ec35d.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-210_fb484cae33f8b0437284b03349727b5bffb7a5896fb10866917514e650940506.jpg b/.raw/MidasFEAAnalysisManual/images/page-210_fb484cae33f8b0437284b03349727b5bffb7a5896fb10866917514e650940506.jpg new file mode 100644 index 00000000..1854f47d Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-210_fb484cae33f8b0437284b03349727b5bffb7a5896fb10866917514e650940506.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-211_b787ff7708f6eb83041546df1456f00b826f2429e8daec6adb5c9f5a12f38d77.jpg b/.raw/MidasFEAAnalysisManual/images/page-211_b787ff7708f6eb83041546df1456f00b826f2429e8daec6adb5c9f5a12f38d77.jpg new file mode 100644 index 00000000..e00d80ee Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-211_b787ff7708f6eb83041546df1456f00b826f2429e8daec6adb5c9f5a12f38d77.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-212_0125a0ba81492f903fcbe6a9cb18e9523adcc895e47b49249b8960ea15fa8c71.jpg b/.raw/MidasFEAAnalysisManual/images/page-212_0125a0ba81492f903fcbe6a9cb18e9523adcc895e47b49249b8960ea15fa8c71.jpg new file mode 100644 index 00000000..d6c8456d Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-212_0125a0ba81492f903fcbe6a9cb18e9523adcc895e47b49249b8960ea15fa8c71.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-212_5456ae91a327c33d3cd2c7f759ea853829d02a526da5a578441dc3608a93b9ea.jpg b/.raw/MidasFEAAnalysisManual/images/page-212_5456ae91a327c33d3cd2c7f759ea853829d02a526da5a578441dc3608a93b9ea.jpg new file mode 100644 index 00000000..8d66c41e Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-212_5456ae91a327c33d3cd2c7f759ea853829d02a526da5a578441dc3608a93b9ea.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-212_58e3405951a8d721d1c8e89131d19c11ec88f6cc246bc0e7f3b7794d83698ed8.jpg b/.raw/MidasFEAAnalysisManual/images/page-212_58e3405951a8d721d1c8e89131d19c11ec88f6cc246bc0e7f3b7794d83698ed8.jpg new file mode 100644 index 00000000..8d222abd Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-212_58e3405951a8d721d1c8e89131d19c11ec88f6cc246bc0e7f3b7794d83698ed8.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-212_5c8ad57c836a63aa432f392f66bb7dd6fd4bd4d40119ce59cc3163b6254885e5.jpg b/.raw/MidasFEAAnalysisManual/images/page-212_5c8ad57c836a63aa432f392f66bb7dd6fd4bd4d40119ce59cc3163b6254885e5.jpg new file mode 100644 index 00000000..4a530fb7 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-212_5c8ad57c836a63aa432f392f66bb7dd6fd4bd4d40119ce59cc3163b6254885e5.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-212_7e42f9b8a7006006dd4efaab66e1d6f472fd1c2f2d6fffbc49e5ccd615fd4ac4.jpg b/.raw/MidasFEAAnalysisManual/images/page-212_7e42f9b8a7006006dd4efaab66e1d6f472fd1c2f2d6fffbc49e5ccd615fd4ac4.jpg new file mode 100644 index 00000000..4aebcad4 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-212_7e42f9b8a7006006dd4efaab66e1d6f472fd1c2f2d6fffbc49e5ccd615fd4ac4.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-212_e5cb7642db2332bd3b1c577251ea728fcbb434e322f696125ef9dee3cea258ef.jpg b/.raw/MidasFEAAnalysisManual/images/page-212_e5cb7642db2332bd3b1c577251ea728fcbb434e322f696125ef9dee3cea258ef.jpg new file mode 100644 index 00000000..6281ea45 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-212_e5cb7642db2332bd3b1c577251ea728fcbb434e322f696125ef9dee3cea258ef.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-213_cc4a89f7912cbf20fef4decb7618c22fbdc22e3a865fb018dd9f5698a2587f9d.jpg b/.raw/MidasFEAAnalysisManual/images/page-213_cc4a89f7912cbf20fef4decb7618c22fbdc22e3a865fb018dd9f5698a2587f9d.jpg new file mode 100644 index 00000000..7c73cdac Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-213_cc4a89f7912cbf20fef4decb7618c22fbdc22e3a865fb018dd9f5698a2587f9d.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-213_eeed7bceb1f98c2a2077db90f501d4a68efbbfd9889d5213195c1b7f86426e86.jpg b/.raw/MidasFEAAnalysisManual/images/page-213_eeed7bceb1f98c2a2077db90f501d4a68efbbfd9889d5213195c1b7f86426e86.jpg new file mode 100644 index 00000000..fea4beac Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-213_eeed7bceb1f98c2a2077db90f501d4a68efbbfd9889d5213195c1b7f86426e86.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-213_f8051616e8b8cd9f914a3f1d4f51eec6485d7c08b795980bc70654e1e0a569f0.jpg b/.raw/MidasFEAAnalysisManual/images/page-213_f8051616e8b8cd9f914a3f1d4f51eec6485d7c08b795980bc70654e1e0a569f0.jpg new file mode 100644 index 00000000..87637bf4 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-213_f8051616e8b8cd9f914a3f1d4f51eec6485d7c08b795980bc70654e1e0a569f0.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-214_151d8dc33d48849cbcd85c98c1c9f1cc344d8508873b7ed1a786ea6bf8c1e9d3.jpg b/.raw/MidasFEAAnalysisManual/images/page-214_151d8dc33d48849cbcd85c98c1c9f1cc344d8508873b7ed1a786ea6bf8c1e9d3.jpg new file mode 100644 index 00000000..0d489b09 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-214_151d8dc33d48849cbcd85c98c1c9f1cc344d8508873b7ed1a786ea6bf8c1e9d3.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-214_5b6981bc1ba994450a07f82a3c0e7a931629dcccc4977c5a0a8136b65fa8c369.jpg b/.raw/MidasFEAAnalysisManual/images/page-214_5b6981bc1ba994450a07f82a3c0e7a931629dcccc4977c5a0a8136b65fa8c369.jpg new file mode 100644 index 00000000..c4723464 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-214_5b6981bc1ba994450a07f82a3c0e7a931629dcccc4977c5a0a8136b65fa8c369.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-214_71f97c15cb7b28a86075f61788b7fedd8abedf7130b137f3bbc8c18b8b1bf944.jpg b/.raw/MidasFEAAnalysisManual/images/page-214_71f97c15cb7b28a86075f61788b7fedd8abedf7130b137f3bbc8c18b8b1bf944.jpg new file mode 100644 index 00000000..4689b44b Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-214_71f97c15cb7b28a86075f61788b7fedd8abedf7130b137f3bbc8c18b8b1bf944.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-214_7c31960ec5869d0677717c92c4609c2bc0baf5ff438845bcf2530ee76d3b5061.jpg b/.raw/MidasFEAAnalysisManual/images/page-214_7c31960ec5869d0677717c92c4609c2bc0baf5ff438845bcf2530ee76d3b5061.jpg new file mode 100644 index 00000000..94d50cf8 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-214_7c31960ec5869d0677717c92c4609c2bc0baf5ff438845bcf2530ee76d3b5061.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-216_78ba4ad7561fbc8e1872908f666e35075fc3017c93866147d2571df0bb40b32f.jpg b/.raw/MidasFEAAnalysisManual/images/page-216_78ba4ad7561fbc8e1872908f666e35075fc3017c93866147d2571df0bb40b32f.jpg new file mode 100644 index 00000000..99baf587 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-216_78ba4ad7561fbc8e1872908f666e35075fc3017c93866147d2571df0bb40b32f.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-216_de77fcf03935c85230036e29654bc03b063f33dea75678c4fb4f4747b63778ad.jpg b/.raw/MidasFEAAnalysisManual/images/page-216_de77fcf03935c85230036e29654bc03b063f33dea75678c4fb4f4747b63778ad.jpg new file mode 100644 index 00000000..0b29145d Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-216_de77fcf03935c85230036e29654bc03b063f33dea75678c4fb4f4747b63778ad.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-217_512ebb5da44911534ea743048bf27b9e8a7a46c14064292e92cbd789e4048fd9.jpg b/.raw/MidasFEAAnalysisManual/images/page-217_512ebb5da44911534ea743048bf27b9e8a7a46c14064292e92cbd789e4048fd9.jpg new file mode 100644 index 00000000..daa8fd11 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-217_512ebb5da44911534ea743048bf27b9e8a7a46c14064292e92cbd789e4048fd9.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-217_88241e062ca84c426436c1fedf958ffb2cacad48788820b775e203be57a742af.jpg b/.raw/MidasFEAAnalysisManual/images/page-217_88241e062ca84c426436c1fedf958ffb2cacad48788820b775e203be57a742af.jpg new file mode 100644 index 00000000..0137078f Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-217_88241e062ca84c426436c1fedf958ffb2cacad48788820b775e203be57a742af.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-217_9213b3cf0a537d0565086e3e0d6122dadce50250bdfb9a1b4f3432599eac8100.jpg b/.raw/MidasFEAAnalysisManual/images/page-217_9213b3cf0a537d0565086e3e0d6122dadce50250bdfb9a1b4f3432599eac8100.jpg new file mode 100644 index 00000000..5835bebc Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-217_9213b3cf0a537d0565086e3e0d6122dadce50250bdfb9a1b4f3432599eac8100.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-217_9ba95523cad3979d9d589c6a5ec946788b76397a1431ab806592cece4c6ca708.jpg b/.raw/MidasFEAAnalysisManual/images/page-217_9ba95523cad3979d9d589c6a5ec946788b76397a1431ab806592cece4c6ca708.jpg new file mode 100644 index 00000000..a79b13e9 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-217_9ba95523cad3979d9d589c6a5ec946788b76397a1431ab806592cece4c6ca708.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-217_9f066efe0ec5dd843b403efd8b5913dce9b91fe4105435355b41218d6068bef7.jpg b/.raw/MidasFEAAnalysisManual/images/page-217_9f066efe0ec5dd843b403efd8b5913dce9b91fe4105435355b41218d6068bef7.jpg new file mode 100644 index 00000000..07156100 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-217_9f066efe0ec5dd843b403efd8b5913dce9b91fe4105435355b41218d6068bef7.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-217_c9fa5ce0420a4e508eedfb08c99e9a7d67005078b2901e7dbb2fb85b35077866.jpg b/.raw/MidasFEAAnalysisManual/images/page-217_c9fa5ce0420a4e508eedfb08c99e9a7d67005078b2901e7dbb2fb85b35077866.jpg new file mode 100644 index 00000000..60c42868 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-217_c9fa5ce0420a4e508eedfb08c99e9a7d67005078b2901e7dbb2fb85b35077866.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-218_1601407231c989c239fa07a2157b9c5bc3a0293722d624087ff2c3ed0368f61c.jpg b/.raw/MidasFEAAnalysisManual/images/page-218_1601407231c989c239fa07a2157b9c5bc3a0293722d624087ff2c3ed0368f61c.jpg new file mode 100644 index 00000000..873e63eb Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-218_1601407231c989c239fa07a2157b9c5bc3a0293722d624087ff2c3ed0368f61c.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-218_42c3765179bd545b5c3f398e8aed76e30763b7edfda2d587dd93d8669114cb01.jpg b/.raw/MidasFEAAnalysisManual/images/page-218_42c3765179bd545b5c3f398e8aed76e30763b7edfda2d587dd93d8669114cb01.jpg new file mode 100644 index 00000000..51c5bd0b Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-218_42c3765179bd545b5c3f398e8aed76e30763b7edfda2d587dd93d8669114cb01.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-218_efd394b61f8ba467c963f9b4c68023e6c2ae2a65ffa996d6e05bd5c37d332b82.jpg b/.raw/MidasFEAAnalysisManual/images/page-218_efd394b61f8ba467c963f9b4c68023e6c2ae2a65ffa996d6e05bd5c37d332b82.jpg new file mode 100644 index 00000000..d26a37a1 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-218_efd394b61f8ba467c963f9b4c68023e6c2ae2a65ffa996d6e05bd5c37d332b82.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-218_f27cb6743ac62478b1487a7a8179f1595d4c1ac6c7702e0b689758543a700785.jpg b/.raw/MidasFEAAnalysisManual/images/page-218_f27cb6743ac62478b1487a7a8179f1595d4c1ac6c7702e0b689758543a700785.jpg new file mode 100644 index 00000000..47e4e231 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-218_f27cb6743ac62478b1487a7a8179f1595d4c1ac6c7702e0b689758543a700785.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-219_327f28d7e37a1d5d55bb28cda07ab66915c51c2b54a46206762916993c0a1bd1.jpg b/.raw/MidasFEAAnalysisManual/images/page-219_327f28d7e37a1d5d55bb28cda07ab66915c51c2b54a46206762916993c0a1bd1.jpg new file mode 100644 index 00000000..4344b1fc Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-219_327f28d7e37a1d5d55bb28cda07ab66915c51c2b54a46206762916993c0a1bd1.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-219_36c854d2eb1d5a63040938b59d5ed0769d6f9166752aea0ce31f07107ddba5e6.jpg b/.raw/MidasFEAAnalysisManual/images/page-219_36c854d2eb1d5a63040938b59d5ed0769d6f9166752aea0ce31f07107ddba5e6.jpg new file mode 100644 index 00000000..ff9d5bd1 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-219_36c854d2eb1d5a63040938b59d5ed0769d6f9166752aea0ce31f07107ddba5e6.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-219_611581a0ef71f7f5f3ee3fdb0f6626a00591f040b4553d9cee5c4bd9a8002108.jpg b/.raw/MidasFEAAnalysisManual/images/page-219_611581a0ef71f7f5f3ee3fdb0f6626a00591f040b4553d9cee5c4bd9a8002108.jpg new file mode 100644 index 00000000..0ba2f749 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-219_611581a0ef71f7f5f3ee3fdb0f6626a00591f040b4553d9cee5c4bd9a8002108.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-219_67a94d78df03e96bb4017a369202ac49c300268f50a1a9c2a6e9d307f03a73a8.jpg b/.raw/MidasFEAAnalysisManual/images/page-219_67a94d78df03e96bb4017a369202ac49c300268f50a1a9c2a6e9d307f03a73a8.jpg new file mode 100644 index 00000000..58b709a6 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-219_67a94d78df03e96bb4017a369202ac49c300268f50a1a9c2a6e9d307f03a73a8.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-219_e98a5406bc5c3feb48322e6ceca6abddaf1120019de4deb12f83c6888ba2e535.jpg b/.raw/MidasFEAAnalysisManual/images/page-219_e98a5406bc5c3feb48322e6ceca6abddaf1120019de4deb12f83c6888ba2e535.jpg new file mode 100644 index 00000000..8fc5f70a Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-219_e98a5406bc5c3feb48322e6ceca6abddaf1120019de4deb12f83c6888ba2e535.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-220_2146637017358630ce7ab40ef57e452556b2db85ec05496dfed847d660e861b0.jpg b/.raw/MidasFEAAnalysisManual/images/page-220_2146637017358630ce7ab40ef57e452556b2db85ec05496dfed847d660e861b0.jpg new file mode 100644 index 00000000..89536b23 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-220_2146637017358630ce7ab40ef57e452556b2db85ec05496dfed847d660e861b0.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-220_426567630ba29363d4acd1eec93331766b99de8a8cdf62ba388eb5ffe889662f.jpg b/.raw/MidasFEAAnalysisManual/images/page-220_426567630ba29363d4acd1eec93331766b99de8a8cdf62ba388eb5ffe889662f.jpg new file mode 100644 index 00000000..d41d703b Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-220_426567630ba29363d4acd1eec93331766b99de8a8cdf62ba388eb5ffe889662f.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-220_518f8139606a49a3d1738922a7c0d4acbde678ea584ea489a74a158bc80eaab8.jpg b/.raw/MidasFEAAnalysisManual/images/page-220_518f8139606a49a3d1738922a7c0d4acbde678ea584ea489a74a158bc80eaab8.jpg new file mode 100644 index 00000000..189702d8 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-220_518f8139606a49a3d1738922a7c0d4acbde678ea584ea489a74a158bc80eaab8.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-220_cb4582278717a82160dd69a6954b855c39bcea9a2b3a6d9df70c381a7bcdd973.jpg b/.raw/MidasFEAAnalysisManual/images/page-220_cb4582278717a82160dd69a6954b855c39bcea9a2b3a6d9df70c381a7bcdd973.jpg new file mode 100644 index 00000000..47a23ba2 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-220_cb4582278717a82160dd69a6954b855c39bcea9a2b3a6d9df70c381a7bcdd973.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-221_01b89cb5b884fa8778835e90be90a38aaa5918f3937d0eeb9403c012027ba6e7.jpg b/.raw/MidasFEAAnalysisManual/images/page-221_01b89cb5b884fa8778835e90be90a38aaa5918f3937d0eeb9403c012027ba6e7.jpg new file mode 100644 index 00000000..c052dc4a Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-221_01b89cb5b884fa8778835e90be90a38aaa5918f3937d0eeb9403c012027ba6e7.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-221_0f0481c7f82b467bdbbf097c46bb3f779d3c3d851d42ee09ef9a846285b75439.jpg b/.raw/MidasFEAAnalysisManual/images/page-221_0f0481c7f82b467bdbbf097c46bb3f779d3c3d851d42ee09ef9a846285b75439.jpg new file mode 100644 index 00000000..43f30c07 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-221_0f0481c7f82b467bdbbf097c46bb3f779d3c3d851d42ee09ef9a846285b75439.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-221_75206ebc9d1c43e83cf8e7a8b2ee689c5fd9484dc1304af4a63cf9fc1fb99d39.jpg b/.raw/MidasFEAAnalysisManual/images/page-221_75206ebc9d1c43e83cf8e7a8b2ee689c5fd9484dc1304af4a63cf9fc1fb99d39.jpg new file mode 100644 index 00000000..26f6af48 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-221_75206ebc9d1c43e83cf8e7a8b2ee689c5fd9484dc1304af4a63cf9fc1fb99d39.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-221_dacbd7c0aae323d6eca1bbe9cc6e0d895bbb40dae79bc95b392d318356467f6d.jpg b/.raw/MidasFEAAnalysisManual/images/page-221_dacbd7c0aae323d6eca1bbe9cc6e0d895bbb40dae79bc95b392d318356467f6d.jpg new file mode 100644 index 00000000..29662224 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-221_dacbd7c0aae323d6eca1bbe9cc6e0d895bbb40dae79bc95b392d318356467f6d.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-222_02a4a3320f8048c2c4f278c60a9d3055944a9b4c3f93144e83888620d512ab99.jpg b/.raw/MidasFEAAnalysisManual/images/page-222_02a4a3320f8048c2c4f278c60a9d3055944a9b4c3f93144e83888620d512ab99.jpg new file mode 100644 index 00000000..f9edc391 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-222_02a4a3320f8048c2c4f278c60a9d3055944a9b4c3f93144e83888620d512ab99.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-222_0df8766bbc4d5b7ed40058b0ca6c30ba337e699a5cd28522a9efdd82464b4ea1.jpg b/.raw/MidasFEAAnalysisManual/images/page-222_0df8766bbc4d5b7ed40058b0ca6c30ba337e699a5cd28522a9efdd82464b4ea1.jpg new file mode 100644 index 00000000..76a5dc10 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-222_0df8766bbc4d5b7ed40058b0ca6c30ba337e699a5cd28522a9efdd82464b4ea1.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-222_1d94d23e2ebe16e566bef6dadd1c3c834882fab65774e85427a1633e10492914.jpg b/.raw/MidasFEAAnalysisManual/images/page-222_1d94d23e2ebe16e566bef6dadd1c3c834882fab65774e85427a1633e10492914.jpg new file mode 100644 index 00000000..8bd97b1c Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-222_1d94d23e2ebe16e566bef6dadd1c3c834882fab65774e85427a1633e10492914.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-222_2169aa7a7ec338869b54324d45806996423482c8ea4b7b34ca238e923c2c1490.jpg b/.raw/MidasFEAAnalysisManual/images/page-222_2169aa7a7ec338869b54324d45806996423482c8ea4b7b34ca238e923c2c1490.jpg new file mode 100644 index 00000000..6915b2ba Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-222_2169aa7a7ec338869b54324d45806996423482c8ea4b7b34ca238e923c2c1490.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-222_8e51b251fb3a6667610422e9212d008668d8d209c99856de99e8aea6d978309a.jpg b/.raw/MidasFEAAnalysisManual/images/page-222_8e51b251fb3a6667610422e9212d008668d8d209c99856de99e8aea6d978309a.jpg new file mode 100644 index 00000000..32a17fe1 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-222_8e51b251fb3a6667610422e9212d008668d8d209c99856de99e8aea6d978309a.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-222_a364802e20a31e27be09466ae11eca58888a591cf606eae4917dd6702e716a26.jpg b/.raw/MidasFEAAnalysisManual/images/page-222_a364802e20a31e27be09466ae11eca58888a591cf606eae4917dd6702e716a26.jpg new file mode 100644 index 00000000..7f95a541 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-222_a364802e20a31e27be09466ae11eca58888a591cf606eae4917dd6702e716a26.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-222_bcf4c1a68064df9f0f1f25f94c685dac9083d1427c9c3f99a87ff1dd921f2d70.jpg b/.raw/MidasFEAAnalysisManual/images/page-222_bcf4c1a68064df9f0f1f25f94c685dac9083d1427c9c3f99a87ff1dd921f2d70.jpg new file mode 100644 index 00000000..e1003121 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-222_bcf4c1a68064df9f0f1f25f94c685dac9083d1427c9c3f99a87ff1dd921f2d70.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-222_be1939e0ca40f6b1fe6ca3e28175786f611b2449610b31334999495c023a5464.jpg b/.raw/MidasFEAAnalysisManual/images/page-222_be1939e0ca40f6b1fe6ca3e28175786f611b2449610b31334999495c023a5464.jpg new file mode 100644 index 00000000..d1645a90 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-222_be1939e0ca40f6b1fe6ca3e28175786f611b2449610b31334999495c023a5464.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-222_ed2eca4f116d58d727bf49f463faee12485d0011a1a2e0b2b2530bdec2c58674.jpg b/.raw/MidasFEAAnalysisManual/images/page-222_ed2eca4f116d58d727bf49f463faee12485d0011a1a2e0b2b2530bdec2c58674.jpg new file mode 100644 index 00000000..9f914adb Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-222_ed2eca4f116d58d727bf49f463faee12485d0011a1a2e0b2b2530bdec2c58674.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-223_02b2de0d0e8cc43112f5bb3ce11aeb66f59df14dc3688d989a7b644b019dff23.jpg b/.raw/MidasFEAAnalysisManual/images/page-223_02b2de0d0e8cc43112f5bb3ce11aeb66f59df14dc3688d989a7b644b019dff23.jpg new file mode 100644 index 00000000..a291b435 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-223_02b2de0d0e8cc43112f5bb3ce11aeb66f59df14dc3688d989a7b644b019dff23.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-223_3c6c89b9bdf40abfeb2845b56724ddfc991e44ad401d36c0eb081faf54fdf9a7.jpg b/.raw/MidasFEAAnalysisManual/images/page-223_3c6c89b9bdf40abfeb2845b56724ddfc991e44ad401d36c0eb081faf54fdf9a7.jpg new file mode 100644 index 00000000..97e99a36 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-223_3c6c89b9bdf40abfeb2845b56724ddfc991e44ad401d36c0eb081faf54fdf9a7.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-223_9a43841375ef8906953f6edb1abaeddf589ae37759dbd0b462311606a00a9a3f.jpg b/.raw/MidasFEAAnalysisManual/images/page-223_9a43841375ef8906953f6edb1abaeddf589ae37759dbd0b462311606a00a9a3f.jpg new file mode 100644 index 00000000..61802168 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-223_9a43841375ef8906953f6edb1abaeddf589ae37759dbd0b462311606a00a9a3f.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-223_a03f7a5b060893262b45a97faeb1939ecb8cd5e4442371cab75af86749401df3.jpg b/.raw/MidasFEAAnalysisManual/images/page-223_a03f7a5b060893262b45a97faeb1939ecb8cd5e4442371cab75af86749401df3.jpg new file mode 100644 index 00000000..ac41d28c Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-223_a03f7a5b060893262b45a97faeb1939ecb8cd5e4442371cab75af86749401df3.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-224_258256c9b18ef99e8c1febd73a097e78519e966faee5f03a659a1cfbb49e0d2b.jpg b/.raw/MidasFEAAnalysisManual/images/page-224_258256c9b18ef99e8c1febd73a097e78519e966faee5f03a659a1cfbb49e0d2b.jpg new file mode 100644 index 00000000..e93aeb15 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-224_258256c9b18ef99e8c1febd73a097e78519e966faee5f03a659a1cfbb49e0d2b.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-224_71ae198a544f90b27097a345d55085fab5268022ef4aded26e74c2df41c6e262.jpg b/.raw/MidasFEAAnalysisManual/images/page-224_71ae198a544f90b27097a345d55085fab5268022ef4aded26e74c2df41c6e262.jpg new file mode 100644 index 00000000..8a167d48 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-224_71ae198a544f90b27097a345d55085fab5268022ef4aded26e74c2df41c6e262.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-224_a10256e16a0bc04d4a2b2289ca30bfa34e7b624b79e57872341e77435bb26c7b.jpg b/.raw/MidasFEAAnalysisManual/images/page-224_a10256e16a0bc04d4a2b2289ca30bfa34e7b624b79e57872341e77435bb26c7b.jpg new file mode 100644 index 00000000..cb853b15 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-224_a10256e16a0bc04d4a2b2289ca30bfa34e7b624b79e57872341e77435bb26c7b.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-224_a6fb215ab96b2f2b0b0b57213c890cd461c75c37145c91123b2c06a35ec0c7d0.jpg b/.raw/MidasFEAAnalysisManual/images/page-224_a6fb215ab96b2f2b0b0b57213c890cd461c75c37145c91123b2c06a35ec0c7d0.jpg new file mode 100644 index 00000000..70f3ddf4 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-224_a6fb215ab96b2f2b0b0b57213c890cd461c75c37145c91123b2c06a35ec0c7d0.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-225_19a360923ab99dec74cd25bac852c1a5ab577732f77fdd68e4059ebf155ab35b.jpg b/.raw/MidasFEAAnalysisManual/images/page-225_19a360923ab99dec74cd25bac852c1a5ab577732f77fdd68e4059ebf155ab35b.jpg new file mode 100644 index 00000000..6f96d0ef Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-225_19a360923ab99dec74cd25bac852c1a5ab577732f77fdd68e4059ebf155ab35b.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-225_62f731d139e586a4a4bb0098e4b931d27444f989dea986bc26bc781595d19f31.jpg b/.raw/MidasFEAAnalysisManual/images/page-225_62f731d139e586a4a4bb0098e4b931d27444f989dea986bc26bc781595d19f31.jpg new file mode 100644 index 00000000..67a32c3f Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-225_62f731d139e586a4a4bb0098e4b931d27444f989dea986bc26bc781595d19f31.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-225_6e62c3483165a6594f389fa15cac7c27cd71386b488ae4dac6373b332bf038fe.jpg b/.raw/MidasFEAAnalysisManual/images/page-225_6e62c3483165a6594f389fa15cac7c27cd71386b488ae4dac6373b332bf038fe.jpg new file mode 100644 index 00000000..4b1ba6b5 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-225_6e62c3483165a6594f389fa15cac7c27cd71386b488ae4dac6373b332bf038fe.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-225_b3ca56ce93189f7f7e67eff00cf3c42823a12078f7c52d2aebab1dd7f3039fda.jpg b/.raw/MidasFEAAnalysisManual/images/page-225_b3ca56ce93189f7f7e67eff00cf3c42823a12078f7c52d2aebab1dd7f3039fda.jpg new file mode 100644 index 00000000..146926ca Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-225_b3ca56ce93189f7f7e67eff00cf3c42823a12078f7c52d2aebab1dd7f3039fda.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-225_e514769f50ff105420af6362c210cc2267a3f689b36c37e4e564e4b9891bc2d8.jpg b/.raw/MidasFEAAnalysisManual/images/page-225_e514769f50ff105420af6362c210cc2267a3f689b36c37e4e564e4b9891bc2d8.jpg new file mode 100644 index 00000000..e0b62535 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-225_e514769f50ff105420af6362c210cc2267a3f689b36c37e4e564e4b9891bc2d8.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-226_1b4cb07644aebdd96fd8722d19bdeb5a093b9365cb205f0e3eac507b2658d930.jpg b/.raw/MidasFEAAnalysisManual/images/page-226_1b4cb07644aebdd96fd8722d19bdeb5a093b9365cb205f0e3eac507b2658d930.jpg new file mode 100644 index 00000000..63e3f57c Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-226_1b4cb07644aebdd96fd8722d19bdeb5a093b9365cb205f0e3eac507b2658d930.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-226_408509cf5fdbf1e817131ab2d16c2cb2ef8eca4ee34e09ba93bd1777fe9ef3b9.jpg b/.raw/MidasFEAAnalysisManual/images/page-226_408509cf5fdbf1e817131ab2d16c2cb2ef8eca4ee34e09ba93bd1777fe9ef3b9.jpg new file mode 100644 index 00000000..b8f4ec16 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-226_408509cf5fdbf1e817131ab2d16c2cb2ef8eca4ee34e09ba93bd1777fe9ef3b9.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-226_ccc2fe2c05a4af7170e23cea1dffe58037e67233b0d51c59e5990502b13a3733.jpg b/.raw/MidasFEAAnalysisManual/images/page-226_ccc2fe2c05a4af7170e23cea1dffe58037e67233b0d51c59e5990502b13a3733.jpg new file mode 100644 index 00000000..ad1d6183 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-226_ccc2fe2c05a4af7170e23cea1dffe58037e67233b0d51c59e5990502b13a3733.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-227_2cf6754116eee88f0ab269aa06e74f66e3fa53af11824f4014ea1f23aad68805.jpg b/.raw/MidasFEAAnalysisManual/images/page-227_2cf6754116eee88f0ab269aa06e74f66e3fa53af11824f4014ea1f23aad68805.jpg new file mode 100644 index 00000000..d377efb1 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-227_2cf6754116eee88f0ab269aa06e74f66e3fa53af11824f4014ea1f23aad68805.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-227_4262cf92f6f5b7ec1ca85ef83fc5fda56ca099541ad47cc2120dcc24d656c3cf.jpg b/.raw/MidasFEAAnalysisManual/images/page-227_4262cf92f6f5b7ec1ca85ef83fc5fda56ca099541ad47cc2120dcc24d656c3cf.jpg new file mode 100644 index 00000000..75bff0ce Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-227_4262cf92f6f5b7ec1ca85ef83fc5fda56ca099541ad47cc2120dcc24d656c3cf.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-227_7857b905c1dd384bf955cd4f8a01c481409bfd2bba1bc4ccb04954ea4bc3a23f.jpg b/.raw/MidasFEAAnalysisManual/images/page-227_7857b905c1dd384bf955cd4f8a01c481409bfd2bba1bc4ccb04954ea4bc3a23f.jpg new file mode 100644 index 00000000..523683f8 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-227_7857b905c1dd384bf955cd4f8a01c481409bfd2bba1bc4ccb04954ea4bc3a23f.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-228_54358e2291b015869307d2e4bbd216a4d5a5b82e2148a13d63afac78e589c0ae.jpg b/.raw/MidasFEAAnalysisManual/images/page-228_54358e2291b015869307d2e4bbd216a4d5a5b82e2148a13d63afac78e589c0ae.jpg new file mode 100644 index 00000000..85931e7a Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-228_54358e2291b015869307d2e4bbd216a4d5a5b82e2148a13d63afac78e589c0ae.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-228_7868e17301a8388ed0e5976a1ccf5c53a46bcaac481bdf80d89cdafa410a3c39.jpg b/.raw/MidasFEAAnalysisManual/images/page-228_7868e17301a8388ed0e5976a1ccf5c53a46bcaac481bdf80d89cdafa410a3c39.jpg new file mode 100644 index 00000000..410314c6 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-228_7868e17301a8388ed0e5976a1ccf5c53a46bcaac481bdf80d89cdafa410a3c39.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-228_b825271a3e6ee8cf62d6b0c765b22d11199dd4012ce7451fbca729e492c5a710.jpg b/.raw/MidasFEAAnalysisManual/images/page-228_b825271a3e6ee8cf62d6b0c765b22d11199dd4012ce7451fbca729e492c5a710.jpg new file mode 100644 index 00000000..f3aa8b9a Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-228_b825271a3e6ee8cf62d6b0c765b22d11199dd4012ce7451fbca729e492c5a710.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-228_d8620ffdb7116ec0f57fd0306decbce02121d3a3b305c9c374a5d3d573e969cb.jpg b/.raw/MidasFEAAnalysisManual/images/page-228_d8620ffdb7116ec0f57fd0306decbce02121d3a3b305c9c374a5d3d573e969cb.jpg new file mode 100644 index 00000000..abb5f113 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-228_d8620ffdb7116ec0f57fd0306decbce02121d3a3b305c9c374a5d3d573e969cb.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-228_ff7e36ea7c8db982e37ea36d4782b5c3f17ed4cfcb745a59617b1ca7c158d86a.jpg b/.raw/MidasFEAAnalysisManual/images/page-228_ff7e36ea7c8db982e37ea36d4782b5c3f17ed4cfcb745a59617b1ca7c158d86a.jpg new file mode 100644 index 00000000..ffd73617 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-228_ff7e36ea7c8db982e37ea36d4782b5c3f17ed4cfcb745a59617b1ca7c158d86a.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-229_5619967664071d63873a68ff3daf1ac6b175ce9727d1161fc5ca46314137a015.jpg b/.raw/MidasFEAAnalysisManual/images/page-229_5619967664071d63873a68ff3daf1ac6b175ce9727d1161fc5ca46314137a015.jpg new file mode 100644 index 00000000..384c351d Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-229_5619967664071d63873a68ff3daf1ac6b175ce9727d1161fc5ca46314137a015.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-229_aba92d0e35a0192380c89ead7d3052c331cc10739a4899ac28ef31fc4262fbc3.jpg b/.raw/MidasFEAAnalysisManual/images/page-229_aba92d0e35a0192380c89ead7d3052c331cc10739a4899ac28ef31fc4262fbc3.jpg new file mode 100644 index 00000000..87be172c Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-229_aba92d0e35a0192380c89ead7d3052c331cc10739a4899ac28ef31fc4262fbc3.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-229_dc03dd053e268710c567f4ac28ec4975461b798793065d0f6df9eb9f46379b47.jpg b/.raw/MidasFEAAnalysisManual/images/page-229_dc03dd053e268710c567f4ac28ec4975461b798793065d0f6df9eb9f46379b47.jpg new file mode 100644 index 00000000..06bbd03a Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-229_dc03dd053e268710c567f4ac28ec4975461b798793065d0f6df9eb9f46379b47.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-230_84e61f7b1961d0de0d5d9d16995175b2981cc29bf777aaf3c3084f8d53c7c60b.jpg b/.raw/MidasFEAAnalysisManual/images/page-230_84e61f7b1961d0de0d5d9d16995175b2981cc29bf777aaf3c3084f8d53c7c60b.jpg new file mode 100644 index 00000000..94ee3b88 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-230_84e61f7b1961d0de0d5d9d16995175b2981cc29bf777aaf3c3084f8d53c7c60b.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-230_927050cd0d64ec21f3c99e4dbd48e0de7dfa94a6f13251a2ad918c9d1283caf1.jpg b/.raw/MidasFEAAnalysisManual/images/page-230_927050cd0d64ec21f3c99e4dbd48e0de7dfa94a6f13251a2ad918c9d1283caf1.jpg new file mode 100644 index 00000000..b7585dbd Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-230_927050cd0d64ec21f3c99e4dbd48e0de7dfa94a6f13251a2ad918c9d1283caf1.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-230_d90f1d45b6a95e01281ef6e284326d81198af3292ad6fc2dba0b1c575518d772.jpg b/.raw/MidasFEAAnalysisManual/images/page-230_d90f1d45b6a95e01281ef6e284326d81198af3292ad6fc2dba0b1c575518d772.jpg new file mode 100644 index 00000000..5e671a07 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-230_d90f1d45b6a95e01281ef6e284326d81198af3292ad6fc2dba0b1c575518d772.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-231_3f8c3819c6a337fc0f2567ca92cd2e2e1f115eccf8a36bbba52633a29e98cd5a.jpg b/.raw/MidasFEAAnalysisManual/images/page-231_3f8c3819c6a337fc0f2567ca92cd2e2e1f115eccf8a36bbba52633a29e98cd5a.jpg new file mode 100644 index 00000000..0cc24b4e Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-231_3f8c3819c6a337fc0f2567ca92cd2e2e1f115eccf8a36bbba52633a29e98cd5a.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-231_4561a29f6141742971e641b4ff89449af39e0ddfb382c32113537d8cd58e9463.jpg b/.raw/MidasFEAAnalysisManual/images/page-231_4561a29f6141742971e641b4ff89449af39e0ddfb382c32113537d8cd58e9463.jpg new file mode 100644 index 00000000..886f2927 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-231_4561a29f6141742971e641b4ff89449af39e0ddfb382c32113537d8cd58e9463.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-232_06ed7ede4ec3b2c072aadd6f165c7afd893b9be50d4713672b5cce85112d9a21.jpg b/.raw/MidasFEAAnalysisManual/images/page-232_06ed7ede4ec3b2c072aadd6f165c7afd893b9be50d4713672b5cce85112d9a21.jpg new file mode 100644 index 00000000..becbb842 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-232_06ed7ede4ec3b2c072aadd6f165c7afd893b9be50d4713672b5cce85112d9a21.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-232_11558c63f70214e0ca2e153a75a255d30ac2452437ad7b768a46edf6a9cc1bf2.jpg b/.raw/MidasFEAAnalysisManual/images/page-232_11558c63f70214e0ca2e153a75a255d30ac2452437ad7b768a46edf6a9cc1bf2.jpg new file mode 100644 index 00000000..9aa288a6 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-232_11558c63f70214e0ca2e153a75a255d30ac2452437ad7b768a46edf6a9cc1bf2.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-232_f97d1c68a2e65fdb00c4349c4e9e7139a914937095e90ed05ef4d3f794275f6c.jpg b/.raw/MidasFEAAnalysisManual/images/page-232_f97d1c68a2e65fdb00c4349c4e9e7139a914937095e90ed05ef4d3f794275f6c.jpg new file mode 100644 index 00000000..01c8724f Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-232_f97d1c68a2e65fdb00c4349c4e9e7139a914937095e90ed05ef4d3f794275f6c.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-233_1dea67f1d23a9dfd6dd962622a82514f5dc807eaf2cf5eccce71824bae1ebf2a.jpg b/.raw/MidasFEAAnalysisManual/images/page-233_1dea67f1d23a9dfd6dd962622a82514f5dc807eaf2cf5eccce71824bae1ebf2a.jpg new file mode 100644 index 00000000..9185ff6d Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-233_1dea67f1d23a9dfd6dd962622a82514f5dc807eaf2cf5eccce71824bae1ebf2a.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-233_61062f757048f70d1a2738f8aa35a94e64467278287a000c2efaa88f54b1aaa0.jpg b/.raw/MidasFEAAnalysisManual/images/page-233_61062f757048f70d1a2738f8aa35a94e64467278287a000c2efaa88f54b1aaa0.jpg new file mode 100644 index 00000000..41d5949e Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-233_61062f757048f70d1a2738f8aa35a94e64467278287a000c2efaa88f54b1aaa0.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-233_7f4b9bfa1df96560c73ac0097b5041bca029ac151d004abc459d25956f4c6521.jpg b/.raw/MidasFEAAnalysisManual/images/page-233_7f4b9bfa1df96560c73ac0097b5041bca029ac151d004abc459d25956f4c6521.jpg new file mode 100644 index 00000000..ebe4ec37 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-233_7f4b9bfa1df96560c73ac0097b5041bca029ac151d004abc459d25956f4c6521.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-233_b5dcdaf0ff11e216f6292699918315521ef3162cdf78f85eb4be034047a0280a.jpg b/.raw/MidasFEAAnalysisManual/images/page-233_b5dcdaf0ff11e216f6292699918315521ef3162cdf78f85eb4be034047a0280a.jpg new file mode 100644 index 00000000..3ac3717f Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-233_b5dcdaf0ff11e216f6292699918315521ef3162cdf78f85eb4be034047a0280a.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-234_01378073ba7bd0435e8f1621b7f16c4bb5c7150e8dd22476730493fe800cfa68.jpg b/.raw/MidasFEAAnalysisManual/images/page-234_01378073ba7bd0435e8f1621b7f16c4bb5c7150e8dd22476730493fe800cfa68.jpg new file mode 100644 index 00000000..16e55f88 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-234_01378073ba7bd0435e8f1621b7f16c4bb5c7150e8dd22476730493fe800cfa68.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-234_419bd1733a138e9095e608bb6832ae108841f9a13101394b1f38be4135a9ae7b.jpg b/.raw/MidasFEAAnalysisManual/images/page-234_419bd1733a138e9095e608bb6832ae108841f9a13101394b1f38be4135a9ae7b.jpg new file mode 100644 index 00000000..28773b93 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-234_419bd1733a138e9095e608bb6832ae108841f9a13101394b1f38be4135a9ae7b.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-234_52021023e8330f09425a838a5233fc6345f17a9cb91f7d9a13fc93e20c7f00c9.jpg b/.raw/MidasFEAAnalysisManual/images/page-234_52021023e8330f09425a838a5233fc6345f17a9cb91f7d9a13fc93e20c7f00c9.jpg new file mode 100644 index 00000000..ca7892fb Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-234_52021023e8330f09425a838a5233fc6345f17a9cb91f7d9a13fc93e20c7f00c9.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-234_5d6be3f0582226a075ac30504f08ac03cabdb588d4152db3d6c3f206d75934c0.jpg b/.raw/MidasFEAAnalysisManual/images/page-234_5d6be3f0582226a075ac30504f08ac03cabdb588d4152db3d6c3f206d75934c0.jpg new file mode 100644 index 00000000..a020242e Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-234_5d6be3f0582226a075ac30504f08ac03cabdb588d4152db3d6c3f206d75934c0.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-234_90125676ef59c9eceb5d9aa197870196f01925b776d5e04c6bc0d5536bf4ae09.jpg b/.raw/MidasFEAAnalysisManual/images/page-234_90125676ef59c9eceb5d9aa197870196f01925b776d5e04c6bc0d5536bf4ae09.jpg new file mode 100644 index 00000000..04e84719 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-234_90125676ef59c9eceb5d9aa197870196f01925b776d5e04c6bc0d5536bf4ae09.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-234_c2073ee48f0d9a016d770b3bff7d1e8ededb0767bb52753af6a9b4a84667d075.jpg b/.raw/MidasFEAAnalysisManual/images/page-234_c2073ee48f0d9a016d770b3bff7d1e8ededb0767bb52753af6a9b4a84667d075.jpg new file mode 100644 index 00000000..149b349e Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-234_c2073ee48f0d9a016d770b3bff7d1e8ededb0767bb52753af6a9b4a84667d075.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-235_30a8bbebba286ca925fcc3cd4e426ed941b24ab7b95b5c6ac7d0c71b6276aff4.jpg b/.raw/MidasFEAAnalysisManual/images/page-235_30a8bbebba286ca925fcc3cd4e426ed941b24ab7b95b5c6ac7d0c71b6276aff4.jpg new file mode 100644 index 00000000..86bd7074 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-235_30a8bbebba286ca925fcc3cd4e426ed941b24ab7b95b5c6ac7d0c71b6276aff4.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-235_836a1b79cbb615af3a81a419a2222c8cc9424ed2e36684cd1117646d3200ff8e.jpg b/.raw/MidasFEAAnalysisManual/images/page-235_836a1b79cbb615af3a81a419a2222c8cc9424ed2e36684cd1117646d3200ff8e.jpg new file mode 100644 index 00000000..fa4ac6da Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-235_836a1b79cbb615af3a81a419a2222c8cc9424ed2e36684cd1117646d3200ff8e.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-235_bc60e4d490f1825691f8cde36d3842cbb243f78a73d7836acb9efefb9d943ad6.jpg b/.raw/MidasFEAAnalysisManual/images/page-235_bc60e4d490f1825691f8cde36d3842cbb243f78a73d7836acb9efefb9d943ad6.jpg new file mode 100644 index 00000000..2d239c1c Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-235_bc60e4d490f1825691f8cde36d3842cbb243f78a73d7836acb9efefb9d943ad6.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-235_be0b235443dfdbff0a9c03bda85548cf802b15c11285f56e0493dd02ab1715c1.jpg b/.raw/MidasFEAAnalysisManual/images/page-235_be0b235443dfdbff0a9c03bda85548cf802b15c11285f56e0493dd02ab1715c1.jpg new file mode 100644 index 00000000..2e93e4ad Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-235_be0b235443dfdbff0a9c03bda85548cf802b15c11285f56e0493dd02ab1715c1.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-235_cb0549ec3e8515332da1489d7b56d2ff522a1aadc73ae4ae14e49ba33a618152.jpg b/.raw/MidasFEAAnalysisManual/images/page-235_cb0549ec3e8515332da1489d7b56d2ff522a1aadc73ae4ae14e49ba33a618152.jpg new file mode 100644 index 00000000..768f40a6 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-235_cb0549ec3e8515332da1489d7b56d2ff522a1aadc73ae4ae14e49ba33a618152.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-235_d229aa351b5c7e3ce3e5b06ac3abd8e5305701845ab32459c85d79cf71175f15.jpg b/.raw/MidasFEAAnalysisManual/images/page-235_d229aa351b5c7e3ce3e5b06ac3abd8e5305701845ab32459c85d79cf71175f15.jpg new file mode 100644 index 00000000..e8923f1c Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-235_d229aa351b5c7e3ce3e5b06ac3abd8e5305701845ab32459c85d79cf71175f15.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-236_6f4f348090a62dd59e55c4e60d3bf2c324144de40421589dd8a65e171190395a.jpg b/.raw/MidasFEAAnalysisManual/images/page-236_6f4f348090a62dd59e55c4e60d3bf2c324144de40421589dd8a65e171190395a.jpg new file mode 100644 index 00000000..84ab5618 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-236_6f4f348090a62dd59e55c4e60d3bf2c324144de40421589dd8a65e171190395a.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-236_a0f04694831b4229a2285f02260e86e4f09975920a98eda6f40ac560a11f69f9.jpg b/.raw/MidasFEAAnalysisManual/images/page-236_a0f04694831b4229a2285f02260e86e4f09975920a98eda6f40ac560a11f69f9.jpg new file mode 100644 index 00000000..5bab782d Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-236_a0f04694831b4229a2285f02260e86e4f09975920a98eda6f40ac560a11f69f9.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-236_a37aca018fec57ff0862098462f2f2d6d3d58fcb0c3c31867401be53b62d873a.jpg b/.raw/MidasFEAAnalysisManual/images/page-236_a37aca018fec57ff0862098462f2f2d6d3d58fcb0c3c31867401be53b62d873a.jpg new file mode 100644 index 00000000..25407582 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-236_a37aca018fec57ff0862098462f2f2d6d3d58fcb0c3c31867401be53b62d873a.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-237_920045d8367afe0f91f395244a2a4bc066654c18884a9eddf7b30163e7bc700c.jpg b/.raw/MidasFEAAnalysisManual/images/page-237_920045d8367afe0f91f395244a2a4bc066654c18884a9eddf7b30163e7bc700c.jpg new file mode 100644 index 00000000..73390917 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-237_920045d8367afe0f91f395244a2a4bc066654c18884a9eddf7b30163e7bc700c.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-237_b83a220d76d1be073cc4aa9522bedc8a6d297bec63e5529cbc8b01704f157108.jpg b/.raw/MidasFEAAnalysisManual/images/page-237_b83a220d76d1be073cc4aa9522bedc8a6d297bec63e5529cbc8b01704f157108.jpg new file mode 100644 index 00000000..72b1ec0a Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-237_b83a220d76d1be073cc4aa9522bedc8a6d297bec63e5529cbc8b01704f157108.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-238_0a30f44e5f4fe60cba3572f8994f0e60507d421d3f04a8680a5d8249041519f9.jpg b/.raw/MidasFEAAnalysisManual/images/page-238_0a30f44e5f4fe60cba3572f8994f0e60507d421d3f04a8680a5d8249041519f9.jpg new file mode 100644 index 00000000..a0da144e Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-238_0a30f44e5f4fe60cba3572f8994f0e60507d421d3f04a8680a5d8249041519f9.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-238_6d6f1688385de02534810fdcefe7913fdfbb3f98b2c03d7c720fe8972d0b808a.jpg b/.raw/MidasFEAAnalysisManual/images/page-238_6d6f1688385de02534810fdcefe7913fdfbb3f98b2c03d7c720fe8972d0b808a.jpg new file mode 100644 index 00000000..ed37baa9 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-238_6d6f1688385de02534810fdcefe7913fdfbb3f98b2c03d7c720fe8972d0b808a.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-239_48c4042e195ce5dc605b87a91fdee471f6b254dfcac885476e4da79f6e64f624.jpg b/.raw/MidasFEAAnalysisManual/images/page-239_48c4042e195ce5dc605b87a91fdee471f6b254dfcac885476e4da79f6e64f624.jpg new file mode 100644 index 00000000..2d59c6d2 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-239_48c4042e195ce5dc605b87a91fdee471f6b254dfcac885476e4da79f6e64f624.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-239_4c1f1d32be3dbcf53d36b89ef6537cbf6f34121db29de75103119629ba1fa2ae.jpg b/.raw/MidasFEAAnalysisManual/images/page-239_4c1f1d32be3dbcf53d36b89ef6537cbf6f34121db29de75103119629ba1fa2ae.jpg new file mode 100644 index 00000000..6b19b703 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-239_4c1f1d32be3dbcf53d36b89ef6537cbf6f34121db29de75103119629ba1fa2ae.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-239_d2669f4fd259d3df6758145eb22bb70b392ffa0c85bad8f73fefb9470ba64cf6.jpg b/.raw/MidasFEAAnalysisManual/images/page-239_d2669f4fd259d3df6758145eb22bb70b392ffa0c85bad8f73fefb9470ba64cf6.jpg new file mode 100644 index 00000000..5686b001 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-239_d2669f4fd259d3df6758145eb22bb70b392ffa0c85bad8f73fefb9470ba64cf6.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-240_4f6b35ad793745520ad20bcef6c8c6fa31cacb740fa9f58add265a0bacac0fb5.jpg b/.raw/MidasFEAAnalysisManual/images/page-240_4f6b35ad793745520ad20bcef6c8c6fa31cacb740fa9f58add265a0bacac0fb5.jpg new file mode 100644 index 00000000..09812496 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-240_4f6b35ad793745520ad20bcef6c8c6fa31cacb740fa9f58add265a0bacac0fb5.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-240_723267b105b68084a34fa2ecabf726ca70381414a0346e0c13d6f7fc8d49380b.jpg b/.raw/MidasFEAAnalysisManual/images/page-240_723267b105b68084a34fa2ecabf726ca70381414a0346e0c13d6f7fc8d49380b.jpg new file mode 100644 index 00000000..2d8d0800 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-240_723267b105b68084a34fa2ecabf726ca70381414a0346e0c13d6f7fc8d49380b.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-240_c2b67daa6f564b63c7966427a1a498c52d42114e32a92033e12ff8d791ac12b3.jpg b/.raw/MidasFEAAnalysisManual/images/page-240_c2b67daa6f564b63c7966427a1a498c52d42114e32a92033e12ff8d791ac12b3.jpg new file mode 100644 index 00000000..4d4ac11d Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-240_c2b67daa6f564b63c7966427a1a498c52d42114e32a92033e12ff8d791ac12b3.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-240_df490055f13752070bce9630a70e6772cbbde9fff33cb56baf9ec609fa98cc3d.jpg b/.raw/MidasFEAAnalysisManual/images/page-240_df490055f13752070bce9630a70e6772cbbde9fff33cb56baf9ec609fa98cc3d.jpg new file mode 100644 index 00000000..65ba120d Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-240_df490055f13752070bce9630a70e6772cbbde9fff33cb56baf9ec609fa98cc3d.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-241_d3cdff7be71c6228b17f28e2ae437ce2b9dc4b949874e1aa2385883ec2f99950.jpg b/.raw/MidasFEAAnalysisManual/images/page-241_d3cdff7be71c6228b17f28e2ae437ce2b9dc4b949874e1aa2385883ec2f99950.jpg new file mode 100644 index 00000000..3137035e Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-241_d3cdff7be71c6228b17f28e2ae437ce2b9dc4b949874e1aa2385883ec2f99950.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-243_5f4e26dde7cccfd9cc82dea7956e96b44ae07bbe954a8613b3defee68834f5c7.jpg b/.raw/MidasFEAAnalysisManual/images/page-243_5f4e26dde7cccfd9cc82dea7956e96b44ae07bbe954a8613b3defee68834f5c7.jpg new file mode 100644 index 00000000..f551eb72 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-243_5f4e26dde7cccfd9cc82dea7956e96b44ae07bbe954a8613b3defee68834f5c7.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-243_c9fa4a91aa1e50e465ff8be16b57dfa2b109afd85ede4870ebad3029f9d101c1.jpg b/.raw/MidasFEAAnalysisManual/images/page-243_c9fa4a91aa1e50e465ff8be16b57dfa2b109afd85ede4870ebad3029f9d101c1.jpg new file mode 100644 index 00000000..5c5ae777 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-243_c9fa4a91aa1e50e465ff8be16b57dfa2b109afd85ede4870ebad3029f9d101c1.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-246_097c1afb51ea399b36abbbbe26a8082e24584ff0467466f67bffaae335b09c4f.jpg b/.raw/MidasFEAAnalysisManual/images/page-246_097c1afb51ea399b36abbbbe26a8082e24584ff0467466f67bffaae335b09c4f.jpg new file mode 100644 index 00000000..a73c5623 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-246_097c1afb51ea399b36abbbbe26a8082e24584ff0467466f67bffaae335b09c4f.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-246_4658a246d46bc11b1060b3ff922e3ffa80651689946409fcdbb82f36d55c268f.jpg b/.raw/MidasFEAAnalysisManual/images/page-246_4658a246d46bc11b1060b3ff922e3ffa80651689946409fcdbb82f36d55c268f.jpg new file mode 100644 index 00000000..fc248ec8 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-246_4658a246d46bc11b1060b3ff922e3ffa80651689946409fcdbb82f36d55c268f.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-246_6023f4061df6090bcb6d26389c99f51f9767078814149fa64e60246d7222a4dc.jpg b/.raw/MidasFEAAnalysisManual/images/page-246_6023f4061df6090bcb6d26389c99f51f9767078814149fa64e60246d7222a4dc.jpg new file mode 100644 index 00000000..3261c04d Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-246_6023f4061df6090bcb6d26389c99f51f9767078814149fa64e60246d7222a4dc.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-247_05bda7a6473aae339054a1d2a1b3c935297d95f8719a97553977ae73f93496b6.jpg b/.raw/MidasFEAAnalysisManual/images/page-247_05bda7a6473aae339054a1d2a1b3c935297d95f8719a97553977ae73f93496b6.jpg new file mode 100644 index 00000000..a4d40ee0 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-247_05bda7a6473aae339054a1d2a1b3c935297d95f8719a97553977ae73f93496b6.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-247_715faec2f2ca46cacefa93c4026cce82484e0c4d8876bcfb6bf6d32056bb51b8.jpg b/.raw/MidasFEAAnalysisManual/images/page-247_715faec2f2ca46cacefa93c4026cce82484e0c4d8876bcfb6bf6d32056bb51b8.jpg new file mode 100644 index 00000000..451402a1 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-247_715faec2f2ca46cacefa93c4026cce82484e0c4d8876bcfb6bf6d32056bb51b8.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-247_99bdbc80fe9d59302745dc887297e77c24997340048259b7b0354336916f228d.jpg b/.raw/MidasFEAAnalysisManual/images/page-247_99bdbc80fe9d59302745dc887297e77c24997340048259b7b0354336916f228d.jpg new file mode 100644 index 00000000..593af5d0 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-247_99bdbc80fe9d59302745dc887297e77c24997340048259b7b0354336916f228d.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-247_b32961aebcd775c6245ede6f2762a46bb43136c487ac407471c3390991f19002.jpg b/.raw/MidasFEAAnalysisManual/images/page-247_b32961aebcd775c6245ede6f2762a46bb43136c487ac407471c3390991f19002.jpg new file mode 100644 index 00000000..406de25e Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-247_b32961aebcd775c6245ede6f2762a46bb43136c487ac407471c3390991f19002.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-248_2720e762334f2a75d06fd54380661d32e2fd1c1b59c32d0f54d9276ebee8c46c.jpg b/.raw/MidasFEAAnalysisManual/images/page-248_2720e762334f2a75d06fd54380661d32e2fd1c1b59c32d0f54d9276ebee8c46c.jpg new file mode 100644 index 00000000..d07baf13 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-248_2720e762334f2a75d06fd54380661d32e2fd1c1b59c32d0f54d9276ebee8c46c.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-249_3a0729ee455cc0c396fd417a706a709ef5f6e3e54e3ad3fd78ad2a682f394bcb.jpg b/.raw/MidasFEAAnalysisManual/images/page-249_3a0729ee455cc0c396fd417a706a709ef5f6e3e54e3ad3fd78ad2a682f394bcb.jpg new file mode 100644 index 00000000..3eee0b5d Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-249_3a0729ee455cc0c396fd417a706a709ef5f6e3e54e3ad3fd78ad2a682f394bcb.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-249_61e4483d59752176fc427f492de8230b3b5597e60d713b46d9e333c92263b817.jpg b/.raw/MidasFEAAnalysisManual/images/page-249_61e4483d59752176fc427f492de8230b3b5597e60d713b46d9e333c92263b817.jpg new file mode 100644 index 00000000..11b9e62a Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-249_61e4483d59752176fc427f492de8230b3b5597e60d713b46d9e333c92263b817.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-249_7802c6d88828b850dc3b1fe582428e2a5502ec142e0f75ba2ed26cefbb944bf9.jpg b/.raw/MidasFEAAnalysisManual/images/page-249_7802c6d88828b850dc3b1fe582428e2a5502ec142e0f75ba2ed26cefbb944bf9.jpg new file mode 100644 index 00000000..e1a630e5 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-249_7802c6d88828b850dc3b1fe582428e2a5502ec142e0f75ba2ed26cefbb944bf9.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-249_caf28a937cceeef4062f0f9813ffa93ab86bdfb5f17d7227c5f382e6c91827a1.jpg b/.raw/MidasFEAAnalysisManual/images/page-249_caf28a937cceeef4062f0f9813ffa93ab86bdfb5f17d7227c5f382e6c91827a1.jpg new file mode 100644 index 00000000..9e52a61e Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-249_caf28a937cceeef4062f0f9813ffa93ab86bdfb5f17d7227c5f382e6c91827a1.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-249_f932d0a5b8112a2576d76fe1067d00cdbe8a5eabf3340b2a8b048247f9ca1882.jpg b/.raw/MidasFEAAnalysisManual/images/page-249_f932d0a5b8112a2576d76fe1067d00cdbe8a5eabf3340b2a8b048247f9ca1882.jpg new file mode 100644 index 00000000..18315b49 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-249_f932d0a5b8112a2576d76fe1067d00cdbe8a5eabf3340b2a8b048247f9ca1882.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-250_42ac99a25253090b385b3ea3dac7df6f747c113629d56aa5e184a569f41f56c2.jpg b/.raw/MidasFEAAnalysisManual/images/page-250_42ac99a25253090b385b3ea3dac7df6f747c113629d56aa5e184a569f41f56c2.jpg new file mode 100644 index 00000000..458dc37d Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-250_42ac99a25253090b385b3ea3dac7df6f747c113629d56aa5e184a569f41f56c2.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-250_7372d7a7a8edc2cf09f5b3f08a262088e564d8620355bb2d3355a82242b65aec.jpg b/.raw/MidasFEAAnalysisManual/images/page-250_7372d7a7a8edc2cf09f5b3f08a262088e564d8620355bb2d3355a82242b65aec.jpg new file mode 100644 index 00000000..ffa7b276 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-250_7372d7a7a8edc2cf09f5b3f08a262088e564d8620355bb2d3355a82242b65aec.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-250_7c473ecb26ebbf41992d78f0ac08936830d396f5742d03ca04c2cc173e2be707.jpg b/.raw/MidasFEAAnalysisManual/images/page-250_7c473ecb26ebbf41992d78f0ac08936830d396f5742d03ca04c2cc173e2be707.jpg new file mode 100644 index 00000000..32835490 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-250_7c473ecb26ebbf41992d78f0ac08936830d396f5742d03ca04c2cc173e2be707.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-251_209bcf0b7ef070ab439407bfa7fb014894c0f974d0eff7854849ff70097e43ed.jpg b/.raw/MidasFEAAnalysisManual/images/page-251_209bcf0b7ef070ab439407bfa7fb014894c0f974d0eff7854849ff70097e43ed.jpg new file mode 100644 index 00000000..fdf61f8f Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-251_209bcf0b7ef070ab439407bfa7fb014894c0f974d0eff7854849ff70097e43ed.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-251_5be7aa8f6d2b98add377d7aa11b5b0905b9d9ccf8807eb2cd3b6b199af52996a.jpg b/.raw/MidasFEAAnalysisManual/images/page-251_5be7aa8f6d2b98add377d7aa11b5b0905b9d9ccf8807eb2cd3b6b199af52996a.jpg new file mode 100644 index 00000000..ac9608af Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-251_5be7aa8f6d2b98add377d7aa11b5b0905b9d9ccf8807eb2cd3b6b199af52996a.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-251_866ed659086954183109d875cb9e9712f0d4d1a58488aa6614ecda6475221bbc.jpg b/.raw/MidasFEAAnalysisManual/images/page-251_866ed659086954183109d875cb9e9712f0d4d1a58488aa6614ecda6475221bbc.jpg new file mode 100644 index 00000000..55be798a Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-251_866ed659086954183109d875cb9e9712f0d4d1a58488aa6614ecda6475221bbc.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-251_88eb0c5ea8bde10a9d1a45ac9fe82cc899ca84e814a70473ddb51027ded7347e.jpg b/.raw/MidasFEAAnalysisManual/images/page-251_88eb0c5ea8bde10a9d1a45ac9fe82cc899ca84e814a70473ddb51027ded7347e.jpg new file mode 100644 index 00000000..ec9b482b Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-251_88eb0c5ea8bde10a9d1a45ac9fe82cc899ca84e814a70473ddb51027ded7347e.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-251_d40846b2f0889c6c6c165dfc8d656f78ae5545d08ceb14ea3639a93c87e04cd7.jpg b/.raw/MidasFEAAnalysisManual/images/page-251_d40846b2f0889c6c6c165dfc8d656f78ae5545d08ceb14ea3639a93c87e04cd7.jpg new file mode 100644 index 00000000..e43a7b95 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-251_d40846b2f0889c6c6c165dfc8d656f78ae5545d08ceb14ea3639a93c87e04cd7.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-252_0682b37d6998ff69147348cd54df632a28b0b804b73c6977eb46d95622db8ded.jpg b/.raw/MidasFEAAnalysisManual/images/page-252_0682b37d6998ff69147348cd54df632a28b0b804b73c6977eb46d95622db8ded.jpg new file mode 100644 index 00000000..2aa58e21 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-252_0682b37d6998ff69147348cd54df632a28b0b804b73c6977eb46d95622db8ded.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-252_3ca1c1a2214e7c78031c06d4f849d6c9834f240f4a34a786da141a0b75aba46d.jpg b/.raw/MidasFEAAnalysisManual/images/page-252_3ca1c1a2214e7c78031c06d4f849d6c9834f240f4a34a786da141a0b75aba46d.jpg new file mode 100644 index 00000000..5e4bea79 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-252_3ca1c1a2214e7c78031c06d4f849d6c9834f240f4a34a786da141a0b75aba46d.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-253_ae0e34b1a9ccecfb9bd6d18fff5c1790f96434aebe3fe0a5890573f426482fa6.jpg b/.raw/MidasFEAAnalysisManual/images/page-253_ae0e34b1a9ccecfb9bd6d18fff5c1790f96434aebe3fe0a5890573f426482fa6.jpg new file mode 100644 index 00000000..9ffd67c3 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-253_ae0e34b1a9ccecfb9bd6d18fff5c1790f96434aebe3fe0a5890573f426482fa6.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-253_f50f0354450bb5055054b65872c2c24ab4611ca3955d24360b4f8080993937ec.jpg b/.raw/MidasFEAAnalysisManual/images/page-253_f50f0354450bb5055054b65872c2c24ab4611ca3955d24360b4f8080993937ec.jpg new file mode 100644 index 00000000..0fac95a8 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-253_f50f0354450bb5055054b65872c2c24ab4611ca3955d24360b4f8080993937ec.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-254_05c406f956c8b893f7bf0f077aeb37b6aaeaed67da0e91338524dda2c93056e6.jpg b/.raw/MidasFEAAnalysisManual/images/page-254_05c406f956c8b893f7bf0f077aeb37b6aaeaed67da0e91338524dda2c93056e6.jpg new file mode 100644 index 00000000..29252b68 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-254_05c406f956c8b893f7bf0f077aeb37b6aaeaed67da0e91338524dda2c93056e6.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-254_1bcbf656489c1bc9606fad8a4a5824621c634d589eaf1cf173a612ef2e77cd0b.jpg b/.raw/MidasFEAAnalysisManual/images/page-254_1bcbf656489c1bc9606fad8a4a5824621c634d589eaf1cf173a612ef2e77cd0b.jpg new file mode 100644 index 00000000..cf5ad7fa Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-254_1bcbf656489c1bc9606fad8a4a5824621c634d589eaf1cf173a612ef2e77cd0b.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-254_50af1430e6b2a6b2cb5bdbbe9c4d537f8f39405160af073c067794b55324adb4.jpg b/.raw/MidasFEAAnalysisManual/images/page-254_50af1430e6b2a6b2cb5bdbbe9c4d537f8f39405160af073c067794b55324adb4.jpg new file mode 100644 index 00000000..a438090c Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-254_50af1430e6b2a6b2cb5bdbbe9c4d537f8f39405160af073c067794b55324adb4.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-254_884f694f81afd352607cf30b70c52644a368ff7aa7e0b0179ba1756fba55ba59.jpg b/.raw/MidasFEAAnalysisManual/images/page-254_884f694f81afd352607cf30b70c52644a368ff7aa7e0b0179ba1756fba55ba59.jpg new file mode 100644 index 00000000..0b8b4658 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-254_884f694f81afd352607cf30b70c52644a368ff7aa7e0b0179ba1756fba55ba59.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-255_04b74ddf61fb204d1cb794455a60028a6c0f733d4cfe55306b645e8a6bcbd4e8.jpg b/.raw/MidasFEAAnalysisManual/images/page-255_04b74ddf61fb204d1cb794455a60028a6c0f733d4cfe55306b645e8a6bcbd4e8.jpg new file mode 100644 index 00000000..9cc7cca1 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-255_04b74ddf61fb204d1cb794455a60028a6c0f733d4cfe55306b645e8a6bcbd4e8.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-255_338d3184f2288dc4653731b15f4431947d8479b5fdda5286ff7a7fa8bcb8e90e.jpg b/.raw/MidasFEAAnalysisManual/images/page-255_338d3184f2288dc4653731b15f4431947d8479b5fdda5286ff7a7fa8bcb8e90e.jpg new file mode 100644 index 00000000..a4698ed3 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-255_338d3184f2288dc4653731b15f4431947d8479b5fdda5286ff7a7fa8bcb8e90e.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-255_64b04376ef5456878115075bda6e074087c320f80616f2fd384403fc468bf09d.jpg b/.raw/MidasFEAAnalysisManual/images/page-255_64b04376ef5456878115075bda6e074087c320f80616f2fd384403fc468bf09d.jpg new file mode 100644 index 00000000..37f82d1b Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-255_64b04376ef5456878115075bda6e074087c320f80616f2fd384403fc468bf09d.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-255_876f257671cb16dc13bc0409dbb0ece25f1012e65e6b1eef88e8bec70c3fb4e1.jpg b/.raw/MidasFEAAnalysisManual/images/page-255_876f257671cb16dc13bc0409dbb0ece25f1012e65e6b1eef88e8bec70c3fb4e1.jpg new file mode 100644 index 00000000..c4ff32e3 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-255_876f257671cb16dc13bc0409dbb0ece25f1012e65e6b1eef88e8bec70c3fb4e1.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-255_8cac64025e9937be818517c71bff48c1ae64f12742a16a63b05a2636a5bdc9a0.jpg b/.raw/MidasFEAAnalysisManual/images/page-255_8cac64025e9937be818517c71bff48c1ae64f12742a16a63b05a2636a5bdc9a0.jpg new file mode 100644 index 00000000..19bc2484 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-255_8cac64025e9937be818517c71bff48c1ae64f12742a16a63b05a2636a5bdc9a0.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-256_1e20a04abcdb2fefd32fc1ed64b778115f25edf4ceedf61e70f11bfd7ceb7a0e.jpg b/.raw/MidasFEAAnalysisManual/images/page-256_1e20a04abcdb2fefd32fc1ed64b778115f25edf4ceedf61e70f11bfd7ceb7a0e.jpg new file mode 100644 index 00000000..36403161 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-256_1e20a04abcdb2fefd32fc1ed64b778115f25edf4ceedf61e70f11bfd7ceb7a0e.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-256_1ed7b8bc9cce8183d217393580a59a79d0a53ed2875e00ec277548c1a0deac3b.jpg b/.raw/MidasFEAAnalysisManual/images/page-256_1ed7b8bc9cce8183d217393580a59a79d0a53ed2875e00ec277548c1a0deac3b.jpg new file mode 100644 index 00000000..7b91b54c Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-256_1ed7b8bc9cce8183d217393580a59a79d0a53ed2875e00ec277548c1a0deac3b.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-256_28bd88ebe85c7f643d2d2a837ac9ddfa7e85940f16bb6ebf9b664e56d097f657.jpg b/.raw/MidasFEAAnalysisManual/images/page-256_28bd88ebe85c7f643d2d2a837ac9ddfa7e85940f16bb6ebf9b664e56d097f657.jpg new file mode 100644 index 00000000..43b69144 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-256_28bd88ebe85c7f643d2d2a837ac9ddfa7e85940f16bb6ebf9b664e56d097f657.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-256_4473312f1e81098f03b2c2d2343980f842c0dac77399c33fe616827e59887fad.jpg b/.raw/MidasFEAAnalysisManual/images/page-256_4473312f1e81098f03b2c2d2343980f842c0dac77399c33fe616827e59887fad.jpg new file mode 100644 index 00000000..a9b79ac7 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-256_4473312f1e81098f03b2c2d2343980f842c0dac77399c33fe616827e59887fad.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-256_5577adea6f4f3cb0135e9a68b06026f804fb5de9089cd32b65615d8fdc45e365.jpg b/.raw/MidasFEAAnalysisManual/images/page-256_5577adea6f4f3cb0135e9a68b06026f804fb5de9089cd32b65615d8fdc45e365.jpg new file mode 100644 index 00000000..ad2d0585 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-256_5577adea6f4f3cb0135e9a68b06026f804fb5de9089cd32b65615d8fdc45e365.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-256_a10e246fd15dd74f50891e3dcb4480ff0d64a643f5f88392473d8e98d660c827.jpg b/.raw/MidasFEAAnalysisManual/images/page-256_a10e246fd15dd74f50891e3dcb4480ff0d64a643f5f88392473d8e98d660c827.jpg new file mode 100644 index 00000000..06b2ee15 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-256_a10e246fd15dd74f50891e3dcb4480ff0d64a643f5f88392473d8e98d660c827.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-257_feabaee68de0f86abaf1d75dc78ee84b8ae9d726c762d53c159fbadf5b518756.jpg b/.raw/MidasFEAAnalysisManual/images/page-257_feabaee68de0f86abaf1d75dc78ee84b8ae9d726c762d53c159fbadf5b518756.jpg new file mode 100644 index 00000000..df20d809 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-257_feabaee68de0f86abaf1d75dc78ee84b8ae9d726c762d53c159fbadf5b518756.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-258_0edb5a01026f7de0e6d1433f89cbc52e18ee2773c2cb77a5fecee13842e208cc.jpg b/.raw/MidasFEAAnalysisManual/images/page-258_0edb5a01026f7de0e6d1433f89cbc52e18ee2773c2cb77a5fecee13842e208cc.jpg new file mode 100644 index 00000000..5cede268 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-258_0edb5a01026f7de0e6d1433f89cbc52e18ee2773c2cb77a5fecee13842e208cc.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-259_18d4e4e657039c9aab9a903a93585471079fb3d2135f02ba2c2cd9de1c37e107.jpg b/.raw/MidasFEAAnalysisManual/images/page-259_18d4e4e657039c9aab9a903a93585471079fb3d2135f02ba2c2cd9de1c37e107.jpg new file mode 100644 index 00000000..5163723b Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-259_18d4e4e657039c9aab9a903a93585471079fb3d2135f02ba2c2cd9de1c37e107.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-259_1a48be4030fdd27feaedbfe24af421ffb2352453e167c259d8e26beccb11d21d.jpg b/.raw/MidasFEAAnalysisManual/images/page-259_1a48be4030fdd27feaedbfe24af421ffb2352453e167c259d8e26beccb11d21d.jpg new file mode 100644 index 00000000..837b3771 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-259_1a48be4030fdd27feaedbfe24af421ffb2352453e167c259d8e26beccb11d21d.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-259_7f72ffc5bb2b7b401d91b10347c862617b5c9f17ebcb7e22c596164312e4f03e.jpg b/.raw/MidasFEAAnalysisManual/images/page-259_7f72ffc5bb2b7b401d91b10347c862617b5c9f17ebcb7e22c596164312e4f03e.jpg new file mode 100644 index 00000000..1d5a4906 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-259_7f72ffc5bb2b7b401d91b10347c862617b5c9f17ebcb7e22c596164312e4f03e.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-259_8f87b20c483cc00e26da09b3e82e2d12b8f7c830587ff028e9e6829054417e61.jpg b/.raw/MidasFEAAnalysisManual/images/page-259_8f87b20c483cc00e26da09b3e82e2d12b8f7c830587ff028e9e6829054417e61.jpg new file mode 100644 index 00000000..5c437007 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-259_8f87b20c483cc00e26da09b3e82e2d12b8f7c830587ff028e9e6829054417e61.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-259_b03a8ac99d961b684c3a83a21e511d28d84d041fb33b1db42b9c164d81d76d10.jpg b/.raw/MidasFEAAnalysisManual/images/page-259_b03a8ac99d961b684c3a83a21e511d28d84d041fb33b1db42b9c164d81d76d10.jpg new file mode 100644 index 00000000..e6a4b4e8 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-259_b03a8ac99d961b684c3a83a21e511d28d84d041fb33b1db42b9c164d81d76d10.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-259_cbaeba8345436fc03c4abb247a5f0f0e68a4cba73e4f378b648f18344cbe3b47.jpg b/.raw/MidasFEAAnalysisManual/images/page-259_cbaeba8345436fc03c4abb247a5f0f0e68a4cba73e4f378b648f18344cbe3b47.jpg new file mode 100644 index 00000000..7843b1f1 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-259_cbaeba8345436fc03c4abb247a5f0f0e68a4cba73e4f378b648f18344cbe3b47.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-259_d0d92bb4c3881194afbde6b0118c910325f2eeda29ecd0686ccbe1733c252c93.jpg b/.raw/MidasFEAAnalysisManual/images/page-259_d0d92bb4c3881194afbde6b0118c910325f2eeda29ecd0686ccbe1733c252c93.jpg new file mode 100644 index 00000000..9b14002b Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-259_d0d92bb4c3881194afbde6b0118c910325f2eeda29ecd0686ccbe1733c252c93.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-260_0e4c6c3665d560f8ec0a8f003a27b626e33771b0139c92aeb04a9c16c5788367.jpg b/.raw/MidasFEAAnalysisManual/images/page-260_0e4c6c3665d560f8ec0a8f003a27b626e33771b0139c92aeb04a9c16c5788367.jpg new file mode 100644 index 00000000..0b4ee400 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-260_0e4c6c3665d560f8ec0a8f003a27b626e33771b0139c92aeb04a9c16c5788367.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-260_341f44e07fdf46f8c4f9c6914eae7c008bf821a5d1c177ce8425893263937970.jpg b/.raw/MidasFEAAnalysisManual/images/page-260_341f44e07fdf46f8c4f9c6914eae7c008bf821a5d1c177ce8425893263937970.jpg new file mode 100644 index 00000000..a4997a17 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-260_341f44e07fdf46f8c4f9c6914eae7c008bf821a5d1c177ce8425893263937970.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-260_67d80a133bc3f18e1f5ab3f8043c9b5c371792f52605b1e6e607f1c0e9d73e3d.jpg b/.raw/MidasFEAAnalysisManual/images/page-260_67d80a133bc3f18e1f5ab3f8043c9b5c371792f52605b1e6e607f1c0e9d73e3d.jpg new file mode 100644 index 00000000..53df0082 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-260_67d80a133bc3f18e1f5ab3f8043c9b5c371792f52605b1e6e607f1c0e9d73e3d.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-260_cc7be5cc48bf0d2d4ef3e16869409b2ab88553052e73f190ac316d60485646cf.jpg b/.raw/MidasFEAAnalysisManual/images/page-260_cc7be5cc48bf0d2d4ef3e16869409b2ab88553052e73f190ac316d60485646cf.jpg new file mode 100644 index 00000000..a95968d8 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-260_cc7be5cc48bf0d2d4ef3e16869409b2ab88553052e73f190ac316d60485646cf.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-262_7779778fde418b061ad7b4b65f174d573d03117dcf755ac165aaeb965ad04d2d.jpg b/.raw/MidasFEAAnalysisManual/images/page-262_7779778fde418b061ad7b4b65f174d573d03117dcf755ac165aaeb965ad04d2d.jpg new file mode 100644 index 00000000..3be9c806 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-262_7779778fde418b061ad7b4b65f174d573d03117dcf755ac165aaeb965ad04d2d.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-262_8314a3e01f47a96e740a5b06a2182c182d4d4d88a76a3268a9deaae17ae5336a.jpg b/.raw/MidasFEAAnalysisManual/images/page-262_8314a3e01f47a96e740a5b06a2182c182d4d4d88a76a3268a9deaae17ae5336a.jpg new file mode 100644 index 00000000..c7040945 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-262_8314a3e01f47a96e740a5b06a2182c182d4d4d88a76a3268a9deaae17ae5336a.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-262_8e3d7d2c3e19733ba752c620efd5f706f6313fd2052c81c4f443aa53ea853096.jpg b/.raw/MidasFEAAnalysisManual/images/page-262_8e3d7d2c3e19733ba752c620efd5f706f6313fd2052c81c4f443aa53ea853096.jpg new file mode 100644 index 00000000..6485c2ad Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-262_8e3d7d2c3e19733ba752c620efd5f706f6313fd2052c81c4f443aa53ea853096.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-262_c07e0521137fed393af5dc5a26af0a67f4518c06759e2cf4998b5810d9dd3c12.jpg b/.raw/MidasFEAAnalysisManual/images/page-262_c07e0521137fed393af5dc5a26af0a67f4518c06759e2cf4998b5810d9dd3c12.jpg new file mode 100644 index 00000000..a69f8f75 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-262_c07e0521137fed393af5dc5a26af0a67f4518c06759e2cf4998b5810d9dd3c12.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-262_db4d8720652ef2950834a0e0f6af25e42fce51ad1286eb0b5e87d54bafb9d077.jpg b/.raw/MidasFEAAnalysisManual/images/page-262_db4d8720652ef2950834a0e0f6af25e42fce51ad1286eb0b5e87d54bafb9d077.jpg new file mode 100644 index 00000000..9687e691 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-262_db4d8720652ef2950834a0e0f6af25e42fce51ad1286eb0b5e87d54bafb9d077.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-263_692f9ab0156de3928ce7e4cb235ed0a9d0ee2d2f909aaaf8482c1e254efd5230.jpg b/.raw/MidasFEAAnalysisManual/images/page-263_692f9ab0156de3928ce7e4cb235ed0a9d0ee2d2f909aaaf8482c1e254efd5230.jpg new file mode 100644 index 00000000..1594fe28 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-263_692f9ab0156de3928ce7e4cb235ed0a9d0ee2d2f909aaaf8482c1e254efd5230.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-264_7fd9629ab3facb90257aab244eaf80d7982fb817c85286fda239dce010846a74.jpg b/.raw/MidasFEAAnalysisManual/images/page-264_7fd9629ab3facb90257aab244eaf80d7982fb817c85286fda239dce010846a74.jpg new file mode 100644 index 00000000..9abc8604 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-264_7fd9629ab3facb90257aab244eaf80d7982fb817c85286fda239dce010846a74.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-264_89a3d7cd7fc671707d01383081ec5ce456adaa5a8678278193982bbaea448c25.jpg b/.raw/MidasFEAAnalysisManual/images/page-264_89a3d7cd7fc671707d01383081ec5ce456adaa5a8678278193982bbaea448c25.jpg new file mode 100644 index 00000000..4797e518 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-264_89a3d7cd7fc671707d01383081ec5ce456adaa5a8678278193982bbaea448c25.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-264_c15826a658dd30a3a77afd23318d7cf6a2af2b95bc6f55a78d7110b226c1ad07.jpg b/.raw/MidasFEAAnalysisManual/images/page-264_c15826a658dd30a3a77afd23318d7cf6a2af2b95bc6f55a78d7110b226c1ad07.jpg new file mode 100644 index 00000000..63d017d1 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-264_c15826a658dd30a3a77afd23318d7cf6a2af2b95bc6f55a78d7110b226c1ad07.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-264_d7a6a28969f7735922f9d53dbfd2c78b027a398c9913d31bba2e4df49c0b710b.jpg b/.raw/MidasFEAAnalysisManual/images/page-264_d7a6a28969f7735922f9d53dbfd2c78b027a398c9913d31bba2e4df49c0b710b.jpg new file mode 100644 index 00000000..d3eb6e92 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-264_d7a6a28969f7735922f9d53dbfd2c78b027a398c9913d31bba2e4df49c0b710b.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-264_efee09dbfbc58d936f0136bf8fe3572d648fda6507ec771f07394b9923c5d395.jpg b/.raw/MidasFEAAnalysisManual/images/page-264_efee09dbfbc58d936f0136bf8fe3572d648fda6507ec771f07394b9923c5d395.jpg new file mode 100644 index 00000000..ae1b8ceb Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-264_efee09dbfbc58d936f0136bf8fe3572d648fda6507ec771f07394b9923c5d395.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-265_40364b631db21ab799311f97a64ad515ffebcdd4d18c23da06ad70b3f9f4cabd.jpg b/.raw/MidasFEAAnalysisManual/images/page-265_40364b631db21ab799311f97a64ad515ffebcdd4d18c23da06ad70b3f9f4cabd.jpg new file mode 100644 index 00000000..9f0fab1c Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-265_40364b631db21ab799311f97a64ad515ffebcdd4d18c23da06ad70b3f9f4cabd.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-265_504dac06a46cc90f105bf50a8643ce753b173480bbd4b579608114acc6bf24cb.jpg b/.raw/MidasFEAAnalysisManual/images/page-265_504dac06a46cc90f105bf50a8643ce753b173480bbd4b579608114acc6bf24cb.jpg new file mode 100644 index 00000000..04c96198 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-265_504dac06a46cc90f105bf50a8643ce753b173480bbd4b579608114acc6bf24cb.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-265_a8568e22a92c1e1b7262b473c835df61318a66c5bcd377921e8463f8ff34cc5f.jpg b/.raw/MidasFEAAnalysisManual/images/page-265_a8568e22a92c1e1b7262b473c835df61318a66c5bcd377921e8463f8ff34cc5f.jpg new file mode 100644 index 00000000..ba79cb5e Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-265_a8568e22a92c1e1b7262b473c835df61318a66c5bcd377921e8463f8ff34cc5f.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-265_ee74e57c0bf28023f962eb527e915a8417ae7c511b45b04b30372ef78d3ac930.jpg b/.raw/MidasFEAAnalysisManual/images/page-265_ee74e57c0bf28023f962eb527e915a8417ae7c511b45b04b30372ef78d3ac930.jpg new file mode 100644 index 00000000..493b3524 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-265_ee74e57c0bf28023f962eb527e915a8417ae7c511b45b04b30372ef78d3ac930.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-265_f9821600704adfab4299cb7e0a8da2bc82c728113551e263ff08a8602a8daae9.jpg b/.raw/MidasFEAAnalysisManual/images/page-265_f9821600704adfab4299cb7e0a8da2bc82c728113551e263ff08a8602a8daae9.jpg new file mode 100644 index 00000000..fd17f54a Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-265_f9821600704adfab4299cb7e0a8da2bc82c728113551e263ff08a8602a8daae9.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-266_a0cada7042a54eff768b0ca85da24e50e907a8d0a8ca0c46f502f691b66274d7.jpg b/.raw/MidasFEAAnalysisManual/images/page-266_a0cada7042a54eff768b0ca85da24e50e907a8d0a8ca0c46f502f691b66274d7.jpg new file mode 100644 index 00000000..13c765a7 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-266_a0cada7042a54eff768b0ca85da24e50e907a8d0a8ca0c46f502f691b66274d7.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-266_eb9495f9fba6041afa67ecbce9a4278dc725a8fa6ec34174cf99509f5ebab236.jpg b/.raw/MidasFEAAnalysisManual/images/page-266_eb9495f9fba6041afa67ecbce9a4278dc725a8fa6ec34174cf99509f5ebab236.jpg new file mode 100644 index 00000000..bbe15820 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-266_eb9495f9fba6041afa67ecbce9a4278dc725a8fa6ec34174cf99509f5ebab236.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-267_1533ee71c9dd2e949dbe004e8abd28bf5b52c74ac1b9fac1c0b571d1cd35bc0c.jpg b/.raw/MidasFEAAnalysisManual/images/page-267_1533ee71c9dd2e949dbe004e8abd28bf5b52c74ac1b9fac1c0b571d1cd35bc0c.jpg new file mode 100644 index 00000000..a215586a Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-267_1533ee71c9dd2e949dbe004e8abd28bf5b52c74ac1b9fac1c0b571d1cd35bc0c.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-267_51e8cd94122d2a362474dbf21a60cbd3e8387e129cb66fa6581f133324350dc8.jpg b/.raw/MidasFEAAnalysisManual/images/page-267_51e8cd94122d2a362474dbf21a60cbd3e8387e129cb66fa6581f133324350dc8.jpg new file mode 100644 index 00000000..bdaac6ec Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-267_51e8cd94122d2a362474dbf21a60cbd3e8387e129cb66fa6581f133324350dc8.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-267_76d32a2bb7a5c4504dfa1566c5c5dabfd4b5c6602bd1ec237f4a7c7e78939729.jpg b/.raw/MidasFEAAnalysisManual/images/page-267_76d32a2bb7a5c4504dfa1566c5c5dabfd4b5c6602bd1ec237f4a7c7e78939729.jpg new file mode 100644 index 00000000..addf2f07 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-267_76d32a2bb7a5c4504dfa1566c5c5dabfd4b5c6602bd1ec237f4a7c7e78939729.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-267_77470b6bfa42532ed48404bef83d11f2dc785ba381e57d969d6a4481d928c79a.jpg b/.raw/MidasFEAAnalysisManual/images/page-267_77470b6bfa42532ed48404bef83d11f2dc785ba381e57d969d6a4481d928c79a.jpg new file mode 100644 index 00000000..67454057 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-267_77470b6bfa42532ed48404bef83d11f2dc785ba381e57d969d6a4481d928c79a.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-267_7a4beb794317e29ef59ce0bf1f8dc549a5a39e3093419ee06852a56c75020c2b.jpg b/.raw/MidasFEAAnalysisManual/images/page-267_7a4beb794317e29ef59ce0bf1f8dc549a5a39e3093419ee06852a56c75020c2b.jpg new file mode 100644 index 00000000..64c2823d Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-267_7a4beb794317e29ef59ce0bf1f8dc549a5a39e3093419ee06852a56c75020c2b.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-267_bf495229118c44465802e8b91875065b192875e2c691843eafb9b29881023be8.jpg b/.raw/MidasFEAAnalysisManual/images/page-267_bf495229118c44465802e8b91875065b192875e2c691843eafb9b29881023be8.jpg new file mode 100644 index 00000000..17c453a3 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-267_bf495229118c44465802e8b91875065b192875e2c691843eafb9b29881023be8.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-267_dad178ee85648c9be341462b497f970c98dc2b4ec2c6dea96c0ca3268a8f24e6.jpg b/.raw/MidasFEAAnalysisManual/images/page-267_dad178ee85648c9be341462b497f970c98dc2b4ec2c6dea96c0ca3268a8f24e6.jpg new file mode 100644 index 00000000..32f0f38e Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-267_dad178ee85648c9be341462b497f970c98dc2b4ec2c6dea96c0ca3268a8f24e6.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-268_dc1be40451e3cbcf14cc0ac02b47ebd42ca7cbda7f470ff2aa3f702c6de57979.jpg b/.raw/MidasFEAAnalysisManual/images/page-268_dc1be40451e3cbcf14cc0ac02b47ebd42ca7cbda7f470ff2aa3f702c6de57979.jpg new file mode 100644 index 00000000..92b89c32 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-268_dc1be40451e3cbcf14cc0ac02b47ebd42ca7cbda7f470ff2aa3f702c6de57979.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-268_eee22d3212b54f5cc5a66c5dd396f6a9b1f27f217985af28bfdb44d17ec17ef3.jpg b/.raw/MidasFEAAnalysisManual/images/page-268_eee22d3212b54f5cc5a66c5dd396f6a9b1f27f217985af28bfdb44d17ec17ef3.jpg new file mode 100644 index 00000000..79b1abf3 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-268_eee22d3212b54f5cc5a66c5dd396f6a9b1f27f217985af28bfdb44d17ec17ef3.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-269_9dea558f8132671fa411745709240df49b9df4a14de6e7fa79e7559241fe1808.jpg b/.raw/MidasFEAAnalysisManual/images/page-269_9dea558f8132671fa411745709240df49b9df4a14de6e7fa79e7559241fe1808.jpg new file mode 100644 index 00000000..50eaf171 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-269_9dea558f8132671fa411745709240df49b9df4a14de6e7fa79e7559241fe1808.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-269_af834288b88d4d0a91674c1b507ea9fece8d4676e37f92b45bff687d025e4fe1.jpg b/.raw/MidasFEAAnalysisManual/images/page-269_af834288b88d4d0a91674c1b507ea9fece8d4676e37f92b45bff687d025e4fe1.jpg new file mode 100644 index 00000000..b9385669 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-269_af834288b88d4d0a91674c1b507ea9fece8d4676e37f92b45bff687d025e4fe1.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-269_e2dc274aec3763a94490802b8c1714d59e4ffd5ed2bd7db3ae941b2b5bd2f6c2.jpg b/.raw/MidasFEAAnalysisManual/images/page-269_e2dc274aec3763a94490802b8c1714d59e4ffd5ed2bd7db3ae941b2b5bd2f6c2.jpg new file mode 100644 index 00000000..31fa3573 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-269_e2dc274aec3763a94490802b8c1714d59e4ffd5ed2bd7db3ae941b2b5bd2f6c2.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-270_286afb968f2108bedee96cef75b805e02df49880d1b06895e9206672b7c49b05.jpg b/.raw/MidasFEAAnalysisManual/images/page-270_286afb968f2108bedee96cef75b805e02df49880d1b06895e9206672b7c49b05.jpg new file mode 100644 index 00000000..3ffd65d1 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-270_286afb968f2108bedee96cef75b805e02df49880d1b06895e9206672b7c49b05.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-270_67efe778ab461848268f4beaaab56b481781dccd180b10f0d0eeb8365499f00a.jpg b/.raw/MidasFEAAnalysisManual/images/page-270_67efe778ab461848268f4beaaab56b481781dccd180b10f0d0eeb8365499f00a.jpg new file mode 100644 index 00000000..69d3063f Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-270_67efe778ab461848268f4beaaab56b481781dccd180b10f0d0eeb8365499f00a.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-270_ca01377a0cc6b1cbad46e9b6a4b906fa8feb6d0429a76d6dc99439138f4b2c66.jpg b/.raw/MidasFEAAnalysisManual/images/page-270_ca01377a0cc6b1cbad46e9b6a4b906fa8feb6d0429a76d6dc99439138f4b2c66.jpg new file mode 100644 index 00000000..2d1b82b3 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-270_ca01377a0cc6b1cbad46e9b6a4b906fa8feb6d0429a76d6dc99439138f4b2c66.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-270_f39232dadd7104dcbad099289caaddbde9f8d4aea7fce885e3378e8582de2376.jpg b/.raw/MidasFEAAnalysisManual/images/page-270_f39232dadd7104dcbad099289caaddbde9f8d4aea7fce885e3378e8582de2376.jpg new file mode 100644 index 00000000..5201bc10 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-270_f39232dadd7104dcbad099289caaddbde9f8d4aea7fce885e3378e8582de2376.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-271_889fb4d7afddb1ab2d64ac1195b105ce110f5b473adeaf573fe4a206be024e11.jpg b/.raw/MidasFEAAnalysisManual/images/page-271_889fb4d7afddb1ab2d64ac1195b105ce110f5b473adeaf573fe4a206be024e11.jpg new file mode 100644 index 00000000..11cc20e9 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-271_889fb4d7afddb1ab2d64ac1195b105ce110f5b473adeaf573fe4a206be024e11.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-272_02acda8d23783ec3a755dccb3738e8c118359e9a56b2c647a0733a3f61875367.jpg b/.raw/MidasFEAAnalysisManual/images/page-272_02acda8d23783ec3a755dccb3738e8c118359e9a56b2c647a0733a3f61875367.jpg new file mode 100644 index 00000000..c1cd7fc0 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-272_02acda8d23783ec3a755dccb3738e8c118359e9a56b2c647a0733a3f61875367.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-272_32c6e14534de849399975abf1aa016e8562c95ddc9b69e0f697995b9b1fdd0ad.jpg b/.raw/MidasFEAAnalysisManual/images/page-272_32c6e14534de849399975abf1aa016e8562c95ddc9b69e0f697995b9b1fdd0ad.jpg new file mode 100644 index 00000000..83f5a098 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-272_32c6e14534de849399975abf1aa016e8562c95ddc9b69e0f697995b9b1fdd0ad.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-272_355ad9973ce6b51fc2047ee54d15fc7461e87c7b1eb22df3bc3366b8dcc35e63.jpg b/.raw/MidasFEAAnalysisManual/images/page-272_355ad9973ce6b51fc2047ee54d15fc7461e87c7b1eb22df3bc3366b8dcc35e63.jpg new file mode 100644 index 00000000..585b8bae Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-272_355ad9973ce6b51fc2047ee54d15fc7461e87c7b1eb22df3bc3366b8dcc35e63.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-272_68c293fcfb98b99d051228278bee72b10c50a6255f0695bcd7531435404720bc.jpg b/.raw/MidasFEAAnalysisManual/images/page-272_68c293fcfb98b99d051228278bee72b10c50a6255f0695bcd7531435404720bc.jpg new file mode 100644 index 00000000..8868ceff Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-272_68c293fcfb98b99d051228278bee72b10c50a6255f0695bcd7531435404720bc.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-272_a0e69c5fc9134f0d78821b5e652a7ea977492ba78fee810b86e81f78131b8798.jpg b/.raw/MidasFEAAnalysisManual/images/page-272_a0e69c5fc9134f0d78821b5e652a7ea977492ba78fee810b86e81f78131b8798.jpg new file mode 100644 index 00000000..7353eb26 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-272_a0e69c5fc9134f0d78821b5e652a7ea977492ba78fee810b86e81f78131b8798.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-273_24c4b3bb430e76f6339881e4a7b18d818a4c1f087071e5cf9f1ecd7b4a19a6bf.jpg b/.raw/MidasFEAAnalysisManual/images/page-273_24c4b3bb430e76f6339881e4a7b18d818a4c1f087071e5cf9f1ecd7b4a19a6bf.jpg new file mode 100644 index 00000000..2020b3ec Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-273_24c4b3bb430e76f6339881e4a7b18d818a4c1f087071e5cf9f1ecd7b4a19a6bf.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-273_a54a73cae5f83ccac5722a9c3ad33f879106622ddf756337fac882a915ca836b.jpg b/.raw/MidasFEAAnalysisManual/images/page-273_a54a73cae5f83ccac5722a9c3ad33f879106622ddf756337fac882a915ca836b.jpg new file mode 100644 index 00000000..0b20c1c3 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-273_a54a73cae5f83ccac5722a9c3ad33f879106622ddf756337fac882a915ca836b.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-273_d3b728365f1dd4c7f90f56bed43340cea320903cbf7c9b41e5df0cf1b3f4725d.jpg b/.raw/MidasFEAAnalysisManual/images/page-273_d3b728365f1dd4c7f90f56bed43340cea320903cbf7c9b41e5df0cf1b3f4725d.jpg new file mode 100644 index 00000000..07364092 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-273_d3b728365f1dd4c7f90f56bed43340cea320903cbf7c9b41e5df0cf1b3f4725d.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-273_ed558afd9c590ef8fd35c46c8a78ce4df51a1309b70bf451a7487a5a3b77109f.jpg b/.raw/MidasFEAAnalysisManual/images/page-273_ed558afd9c590ef8fd35c46c8a78ce4df51a1309b70bf451a7487a5a3b77109f.jpg new file mode 100644 index 00000000..16b0fe41 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-273_ed558afd9c590ef8fd35c46c8a78ce4df51a1309b70bf451a7487a5a3b77109f.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-274_4e029bc65e43fcec33c3889078418d3998c8cf1f7bbd65827ee9425498d4f8a4.jpg b/.raw/MidasFEAAnalysisManual/images/page-274_4e029bc65e43fcec33c3889078418d3998c8cf1f7bbd65827ee9425498d4f8a4.jpg new file mode 100644 index 00000000..46d70a64 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-274_4e029bc65e43fcec33c3889078418d3998c8cf1f7bbd65827ee9425498d4f8a4.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-274_7d952ec0fe056e367593511e7ded047e083f87025df0009115de63d2a455e9b9.jpg b/.raw/MidasFEAAnalysisManual/images/page-274_7d952ec0fe056e367593511e7ded047e083f87025df0009115de63d2a455e9b9.jpg new file mode 100644 index 00000000..140c5a2e Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-274_7d952ec0fe056e367593511e7ded047e083f87025df0009115de63d2a455e9b9.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-274_ef78f02386f061b40ad39128c94767d464211c649a3dba0ba905110374732acf.jpg b/.raw/MidasFEAAnalysisManual/images/page-274_ef78f02386f061b40ad39128c94767d464211c649a3dba0ba905110374732acf.jpg new file mode 100644 index 00000000..a5264f51 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-274_ef78f02386f061b40ad39128c94767d464211c649a3dba0ba905110374732acf.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-274_ef9a1542b41fe15ebf8895816abebe2d5fdcc735d00015503aa41d933e37f5ff.jpg b/.raw/MidasFEAAnalysisManual/images/page-274_ef9a1542b41fe15ebf8895816abebe2d5fdcc735d00015503aa41d933e37f5ff.jpg new file mode 100644 index 00000000..f52c1a15 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-274_ef9a1542b41fe15ebf8895816abebe2d5fdcc735d00015503aa41d933e37f5ff.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-275_505891818cdb62831bd90ef4212280ef3c36bb9b429cae8ad0297622bc20a86d.jpg b/.raw/MidasFEAAnalysisManual/images/page-275_505891818cdb62831bd90ef4212280ef3c36bb9b429cae8ad0297622bc20a86d.jpg new file mode 100644 index 00000000..d9432acf Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-275_505891818cdb62831bd90ef4212280ef3c36bb9b429cae8ad0297622bc20a86d.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-275_be6895cc8745e145aa683b3da3dffbbb0b748153a98607e51a76d41a10cd8746.jpg b/.raw/MidasFEAAnalysisManual/images/page-275_be6895cc8745e145aa683b3da3dffbbb0b748153a98607e51a76d41a10cd8746.jpg new file mode 100644 index 00000000..8cf2ff59 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-275_be6895cc8745e145aa683b3da3dffbbb0b748153a98607e51a76d41a10cd8746.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-275_ccc49f72c274a087ccc2bf9285e151c09b24552c67bf02edc60423e4be7453e7.jpg b/.raw/MidasFEAAnalysisManual/images/page-275_ccc49f72c274a087ccc2bf9285e151c09b24552c67bf02edc60423e4be7453e7.jpg new file mode 100644 index 00000000..2d552c90 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-275_ccc49f72c274a087ccc2bf9285e151c09b24552c67bf02edc60423e4be7453e7.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-276_2756fe9bce73d1341ef8bfc5be49400f95e0e734112cd5272b14e54b657fab3c.jpg b/.raw/MidasFEAAnalysisManual/images/page-276_2756fe9bce73d1341ef8bfc5be49400f95e0e734112cd5272b14e54b657fab3c.jpg new file mode 100644 index 00000000..84daea6c Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-276_2756fe9bce73d1341ef8bfc5be49400f95e0e734112cd5272b14e54b657fab3c.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-276_9b3d835d699b54cc003b1821a34abaaf3ddfb7a69730b57c0cfbceba0bf33a59.jpg b/.raw/MidasFEAAnalysisManual/images/page-276_9b3d835d699b54cc003b1821a34abaaf3ddfb7a69730b57c0cfbceba0bf33a59.jpg new file mode 100644 index 00000000..5a397b8e Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-276_9b3d835d699b54cc003b1821a34abaaf3ddfb7a69730b57c0cfbceba0bf33a59.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-277_626f26a2ce35144b7b34c9d185197dc2505dbc0bcd514fd28928698c3fd63c57.jpg b/.raw/MidasFEAAnalysisManual/images/page-277_626f26a2ce35144b7b34c9d185197dc2505dbc0bcd514fd28928698c3fd63c57.jpg new file mode 100644 index 00000000..32a524bd Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-277_626f26a2ce35144b7b34c9d185197dc2505dbc0bcd514fd28928698c3fd63c57.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-277_ee3d32bd347aabdd2460a06c60053aa9cfe5d07f6869fc3cc587187c0da94187.jpg b/.raw/MidasFEAAnalysisManual/images/page-277_ee3d32bd347aabdd2460a06c60053aa9cfe5d07f6869fc3cc587187c0da94187.jpg new file mode 100644 index 00000000..e79f7b6c Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-277_ee3d32bd347aabdd2460a06c60053aa9cfe5d07f6869fc3cc587187c0da94187.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-278_8261417b3da1096de190a05df6745b7a1a7e3021dc62a187ac24ef486bd691c8.jpg b/.raw/MidasFEAAnalysisManual/images/page-278_8261417b3da1096de190a05df6745b7a1a7e3021dc62a187ac24ef486bd691c8.jpg new file mode 100644 index 00000000..918a2e56 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-278_8261417b3da1096de190a05df6745b7a1a7e3021dc62a187ac24ef486bd691c8.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-278_ab91b5cc66ce7c3bfe2768a81ce94c1c566e5887b1cd583aff6a8c80a68beaaf.jpg b/.raw/MidasFEAAnalysisManual/images/page-278_ab91b5cc66ce7c3bfe2768a81ce94c1c566e5887b1cd583aff6a8c80a68beaaf.jpg new file mode 100644 index 00000000..f3551591 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-278_ab91b5cc66ce7c3bfe2768a81ce94c1c566e5887b1cd583aff6a8c80a68beaaf.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-278_b821515c9b87617d881eccfd6ff383ca929319c49348e7c7f01782262f72d146.jpg b/.raw/MidasFEAAnalysisManual/images/page-278_b821515c9b87617d881eccfd6ff383ca929319c49348e7c7f01782262f72d146.jpg new file mode 100644 index 00000000..05ccecc2 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-278_b821515c9b87617d881eccfd6ff383ca929319c49348e7c7f01782262f72d146.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-278_d4bf9937e51913a2f0b30e6c5e946c678e97c5760d7792eec37c7d00624f9453.jpg b/.raw/MidasFEAAnalysisManual/images/page-278_d4bf9937e51913a2f0b30e6c5e946c678e97c5760d7792eec37c7d00624f9453.jpg new file mode 100644 index 00000000..0b3d40f9 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-278_d4bf9937e51913a2f0b30e6c5e946c678e97c5760d7792eec37c7d00624f9453.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-279_0c3850886535dd80ca2b790ebbe3405069b8c1164a4dc29e4a3fed26993b0b64.jpg b/.raw/MidasFEAAnalysisManual/images/page-279_0c3850886535dd80ca2b790ebbe3405069b8c1164a4dc29e4a3fed26993b0b64.jpg new file mode 100644 index 00000000..2239f016 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-279_0c3850886535dd80ca2b790ebbe3405069b8c1164a4dc29e4a3fed26993b0b64.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-279_157702f223eff38cccf28d3283f43702abc57d8b32205709c7768362a1e44b6d.jpg b/.raw/MidasFEAAnalysisManual/images/page-279_157702f223eff38cccf28d3283f43702abc57d8b32205709c7768362a1e44b6d.jpg new file mode 100644 index 00000000..d0aefd60 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-279_157702f223eff38cccf28d3283f43702abc57d8b32205709c7768362a1e44b6d.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-279_1c8166de13ec7c18bc0ce0e650c1a320ebdfd75c90e0ac3c40e76972ecb67fd3.jpg b/.raw/MidasFEAAnalysisManual/images/page-279_1c8166de13ec7c18bc0ce0e650c1a320ebdfd75c90e0ac3c40e76972ecb67fd3.jpg new file mode 100644 index 00000000..10933c76 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-279_1c8166de13ec7c18bc0ce0e650c1a320ebdfd75c90e0ac3c40e76972ecb67fd3.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-279_7a888b1c6aace70b6fa4ada6512eea092960383a329ff179322351ce19f98b06.jpg b/.raw/MidasFEAAnalysisManual/images/page-279_7a888b1c6aace70b6fa4ada6512eea092960383a329ff179322351ce19f98b06.jpg new file mode 100644 index 00000000..c9fd1961 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-279_7a888b1c6aace70b6fa4ada6512eea092960383a329ff179322351ce19f98b06.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-279_816a0d0f911a2f77c80e253de77df2f2eddc7356416cf0b49245fbb61b8f3106.jpg b/.raw/MidasFEAAnalysisManual/images/page-279_816a0d0f911a2f77c80e253de77df2f2eddc7356416cf0b49245fbb61b8f3106.jpg new file mode 100644 index 00000000..42172d8c Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-279_816a0d0f911a2f77c80e253de77df2f2eddc7356416cf0b49245fbb61b8f3106.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-279_efc5d780d26cb7cc7dd355156c7c42b94baf3b25ec2b3d4c15ea8c73c9c7f12e.jpg b/.raw/MidasFEAAnalysisManual/images/page-279_efc5d780d26cb7cc7dd355156c7c42b94baf3b25ec2b3d4c15ea8c73c9c7f12e.jpg new file mode 100644 index 00000000..ba17e63d Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-279_efc5d780d26cb7cc7dd355156c7c42b94baf3b25ec2b3d4c15ea8c73c9c7f12e.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-280_5a2fb7366356eff1ff677a6cf87870a01e32e195b89ac3aff0751c0b04a35130.jpg b/.raw/MidasFEAAnalysisManual/images/page-280_5a2fb7366356eff1ff677a6cf87870a01e32e195b89ac3aff0751c0b04a35130.jpg new file mode 100644 index 00000000..b9691e54 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-280_5a2fb7366356eff1ff677a6cf87870a01e32e195b89ac3aff0751c0b04a35130.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-280_71caab4973f2d1beab398fd40784fe3592527658902868eb9b700c80b38b6d97.jpg b/.raw/MidasFEAAnalysisManual/images/page-280_71caab4973f2d1beab398fd40784fe3592527658902868eb9b700c80b38b6d97.jpg new file mode 100644 index 00000000..c0e8406b Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-280_71caab4973f2d1beab398fd40784fe3592527658902868eb9b700c80b38b6d97.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-280_bce1de66503c776de00b8668956edcf7f2ac5a121d9e85c00974b477599b34d6.jpg b/.raw/MidasFEAAnalysisManual/images/page-280_bce1de66503c776de00b8668956edcf7f2ac5a121d9e85c00974b477599b34d6.jpg new file mode 100644 index 00000000..ddf59f11 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-280_bce1de66503c776de00b8668956edcf7f2ac5a121d9e85c00974b477599b34d6.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-280_c606abf21f3b4378d0ea508e273b12f281f2522d8c56c23e84fd0b919c5a810c.jpg b/.raw/MidasFEAAnalysisManual/images/page-280_c606abf21f3b4378d0ea508e273b12f281f2522d8c56c23e84fd0b919c5a810c.jpg new file mode 100644 index 00000000..925962a2 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-280_c606abf21f3b4378d0ea508e273b12f281f2522d8c56c23e84fd0b919c5a810c.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-280_db4d0a1bb73da6c63bc0efc431b4d2c25316a728027291b0553a4fede77538c7.jpg b/.raw/MidasFEAAnalysisManual/images/page-280_db4d0a1bb73da6c63bc0efc431b4d2c25316a728027291b0553a4fede77538c7.jpg new file mode 100644 index 00000000..b13ce7b4 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-280_db4d0a1bb73da6c63bc0efc431b4d2c25316a728027291b0553a4fede77538c7.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-281_791782409efb1be341195390c0721f308af885f27de90661a4169f337ec509b1.jpg b/.raw/MidasFEAAnalysisManual/images/page-281_791782409efb1be341195390c0721f308af885f27de90661a4169f337ec509b1.jpg new file mode 100644 index 00000000..bdd61f86 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-281_791782409efb1be341195390c0721f308af885f27de90661a4169f337ec509b1.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-281_b8dbb94f1df1c50f81768cada44e374cd595b455021ef85b844553c8e3a0be38.jpg b/.raw/MidasFEAAnalysisManual/images/page-281_b8dbb94f1df1c50f81768cada44e374cd595b455021ef85b844553c8e3a0be38.jpg new file mode 100644 index 00000000..2e0a5883 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-281_b8dbb94f1df1c50f81768cada44e374cd595b455021ef85b844553c8e3a0be38.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-281_e7cd655986d55f767aaf1f9e039a78797d26b6dd00232d3469232608eff735a2.jpg b/.raw/MidasFEAAnalysisManual/images/page-281_e7cd655986d55f767aaf1f9e039a78797d26b6dd00232d3469232608eff735a2.jpg new file mode 100644 index 00000000..a815de2a Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-281_e7cd655986d55f767aaf1f9e039a78797d26b6dd00232d3469232608eff735a2.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-282_5e6ae6a6533577b001b08965182be97018b8e9231570b17571bc34d79c04d0b8.jpg b/.raw/MidasFEAAnalysisManual/images/page-282_5e6ae6a6533577b001b08965182be97018b8e9231570b17571bc34d79c04d0b8.jpg new file mode 100644 index 00000000..20be06f2 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-282_5e6ae6a6533577b001b08965182be97018b8e9231570b17571bc34d79c04d0b8.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-282_e79a7f4d94fd913c4680f329a450c903a0c62c66c644f52dcd2d609cefbc3240.jpg b/.raw/MidasFEAAnalysisManual/images/page-282_e79a7f4d94fd913c4680f329a450c903a0c62c66c644f52dcd2d609cefbc3240.jpg new file mode 100644 index 00000000..d29f092e Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-282_e79a7f4d94fd913c4680f329a450c903a0c62c66c644f52dcd2d609cefbc3240.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-283_05a95ac24aa2636388aeb94641735f0c5443de0394dbd1020c9dbdf73272dad2.jpg b/.raw/MidasFEAAnalysisManual/images/page-283_05a95ac24aa2636388aeb94641735f0c5443de0394dbd1020c9dbdf73272dad2.jpg new file mode 100644 index 00000000..8840daa9 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-283_05a95ac24aa2636388aeb94641735f0c5443de0394dbd1020c9dbdf73272dad2.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-283_b67cdd40fa76cea278dd1ccf09c47a2ced11fc6c87dd50c906b69c413503ec14.jpg b/.raw/MidasFEAAnalysisManual/images/page-283_b67cdd40fa76cea278dd1ccf09c47a2ced11fc6c87dd50c906b69c413503ec14.jpg new file mode 100644 index 00000000..65374e8a Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-283_b67cdd40fa76cea278dd1ccf09c47a2ced11fc6c87dd50c906b69c413503ec14.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-283_bd4bff7d272e578c17de355d07e4328a78c21ad0ec979001f54ae9d177e08a81.jpg b/.raw/MidasFEAAnalysisManual/images/page-283_bd4bff7d272e578c17de355d07e4328a78c21ad0ec979001f54ae9d177e08a81.jpg new file mode 100644 index 00000000..41deda83 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-283_bd4bff7d272e578c17de355d07e4328a78c21ad0ec979001f54ae9d177e08a81.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-283_f48208fa05a11e8c291305583be01d89a817a86eb099ed0a1a60b600ddb1b620.jpg b/.raw/MidasFEAAnalysisManual/images/page-283_f48208fa05a11e8c291305583be01d89a817a86eb099ed0a1a60b600ddb1b620.jpg new file mode 100644 index 00000000..1bc42df0 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-283_f48208fa05a11e8c291305583be01d89a817a86eb099ed0a1a60b600ddb1b620.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-285_1c036d6163084ecef491939f6aa8af26813d1d8a009c1032597a357d74982b27.jpg b/.raw/MidasFEAAnalysisManual/images/page-285_1c036d6163084ecef491939f6aa8af26813d1d8a009c1032597a357d74982b27.jpg new file mode 100644 index 00000000..50261814 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-285_1c036d6163084ecef491939f6aa8af26813d1d8a009c1032597a357d74982b27.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-285_4565cf4ee6f0f2b151e50738a2d8465ee1878db90d6296ab8f58c2bb8a7a6168.jpg b/.raw/MidasFEAAnalysisManual/images/page-285_4565cf4ee6f0f2b151e50738a2d8465ee1878db90d6296ab8f58c2bb8a7a6168.jpg new file mode 100644 index 00000000..241ecb70 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-285_4565cf4ee6f0f2b151e50738a2d8465ee1878db90d6296ab8f58c2bb8a7a6168.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-285_77929d5597913f1374e51275ad16e19211c77fdacf27ce9933cf1d4c008b116c.jpg b/.raw/MidasFEAAnalysisManual/images/page-285_77929d5597913f1374e51275ad16e19211c77fdacf27ce9933cf1d4c008b116c.jpg new file mode 100644 index 00000000..9cc8794a Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-285_77929d5597913f1374e51275ad16e19211c77fdacf27ce9933cf1d4c008b116c.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-285_9e95abc279419eed1236ecc47f12496a61d456ad467ef572979b7148b81edfe9.jpg b/.raw/MidasFEAAnalysisManual/images/page-285_9e95abc279419eed1236ecc47f12496a61d456ad467ef572979b7148b81edfe9.jpg new file mode 100644 index 00000000..c1c71e0c Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-285_9e95abc279419eed1236ecc47f12496a61d456ad467ef572979b7148b81edfe9.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-285_ecd893615e57bdf4b3d5bff34fd334958f50f5a5f7cdf8f2a712519304e02a8c.jpg b/.raw/MidasFEAAnalysisManual/images/page-285_ecd893615e57bdf4b3d5bff34fd334958f50f5a5f7cdf8f2a712519304e02a8c.jpg new file mode 100644 index 00000000..61170cd3 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-285_ecd893615e57bdf4b3d5bff34fd334958f50f5a5f7cdf8f2a712519304e02a8c.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-286_215bb600717280e53954f2d61223b89e4c5f18b215d4dbbc18e8d7d193dae097.jpg b/.raw/MidasFEAAnalysisManual/images/page-286_215bb600717280e53954f2d61223b89e4c5f18b215d4dbbc18e8d7d193dae097.jpg new file mode 100644 index 00000000..89539ba6 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-286_215bb600717280e53954f2d61223b89e4c5f18b215d4dbbc18e8d7d193dae097.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-286_377409e5f01fccb93de86ec0bee2c26494d964ae04d60fe0da950e6e36290d66.jpg b/.raw/MidasFEAAnalysisManual/images/page-286_377409e5f01fccb93de86ec0bee2c26494d964ae04d60fe0da950e6e36290d66.jpg new file mode 100644 index 00000000..1dc0d0bc Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-286_377409e5f01fccb93de86ec0bee2c26494d964ae04d60fe0da950e6e36290d66.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-286_4fcabffea2845cdf293ac87c2729af3641c881adb6622cbe812afc7853f4b9ab.jpg b/.raw/MidasFEAAnalysisManual/images/page-286_4fcabffea2845cdf293ac87c2729af3641c881adb6622cbe812afc7853f4b9ab.jpg new file mode 100644 index 00000000..4ff5a672 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-286_4fcabffea2845cdf293ac87c2729af3641c881adb6622cbe812afc7853f4b9ab.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-286_509ceb889e1777ee08b852c9fca5c9c5472f4f20fc5d98948ff15eebc51c1caa.jpg b/.raw/MidasFEAAnalysisManual/images/page-286_509ceb889e1777ee08b852c9fca5c9c5472f4f20fc5d98948ff15eebc51c1caa.jpg new file mode 100644 index 00000000..81c95202 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-286_509ceb889e1777ee08b852c9fca5c9c5472f4f20fc5d98948ff15eebc51c1caa.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-286_808362aee35577c72c2bebc52b7634d8821f5af1556927f999c5f2b8a19af8c2.jpg b/.raw/MidasFEAAnalysisManual/images/page-286_808362aee35577c72c2bebc52b7634d8821f5af1556927f999c5f2b8a19af8c2.jpg new file mode 100644 index 00000000..c41f7009 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-286_808362aee35577c72c2bebc52b7634d8821f5af1556927f999c5f2b8a19af8c2.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-286_a1cea66d2ef2f8171b14aa92f039665c75b102977cb9fefd60473a0e5866f67f.jpg b/.raw/MidasFEAAnalysisManual/images/page-286_a1cea66d2ef2f8171b14aa92f039665c75b102977cb9fefd60473a0e5866f67f.jpg new file mode 100644 index 00000000..aad77e1d Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-286_a1cea66d2ef2f8171b14aa92f039665c75b102977cb9fefd60473a0e5866f67f.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-286_d37df42e4d4a5f92b4e1721a2df1abe8f61b0078d8c9b11516f4f33d5558e395.jpg b/.raw/MidasFEAAnalysisManual/images/page-286_d37df42e4d4a5f92b4e1721a2df1abe8f61b0078d8c9b11516f4f33d5558e395.jpg new file mode 100644 index 00000000..6c855430 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-286_d37df42e4d4a5f92b4e1721a2df1abe8f61b0078d8c9b11516f4f33d5558e395.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-287_13a2fc862f62029b1b3ef05d0bc32584c4b785be7e0d444fbe3487629fa1e574.jpg b/.raw/MidasFEAAnalysisManual/images/page-287_13a2fc862f62029b1b3ef05d0bc32584c4b785be7e0d444fbe3487629fa1e574.jpg new file mode 100644 index 00000000..d5a4d772 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-287_13a2fc862f62029b1b3ef05d0bc32584c4b785be7e0d444fbe3487629fa1e574.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-287_31d09220a583f1a4f86b9372f049aeedeb7cbc9e090b0d1defe0cbe9da093274.jpg b/.raw/MidasFEAAnalysisManual/images/page-287_31d09220a583f1a4f86b9372f049aeedeb7cbc9e090b0d1defe0cbe9da093274.jpg new file mode 100644 index 00000000..8973edba Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-287_31d09220a583f1a4f86b9372f049aeedeb7cbc9e090b0d1defe0cbe9da093274.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-287_495a30ad6a1d4c04238b8bdadd8121f4d6a0c775febd982f593b710f90944327.jpg b/.raw/MidasFEAAnalysisManual/images/page-287_495a30ad6a1d4c04238b8bdadd8121f4d6a0c775febd982f593b710f90944327.jpg new file mode 100644 index 00000000..4f41a37f Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-287_495a30ad6a1d4c04238b8bdadd8121f4d6a0c775febd982f593b710f90944327.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-287_646b07650649c3f46e439bd3faa3c7fe2fe593a7bb511ee23221b50c279077ff.jpg b/.raw/MidasFEAAnalysisManual/images/page-287_646b07650649c3f46e439bd3faa3c7fe2fe593a7bb511ee23221b50c279077ff.jpg new file mode 100644 index 00000000..48f59dbe Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-287_646b07650649c3f46e439bd3faa3c7fe2fe593a7bb511ee23221b50c279077ff.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-288_6b156c2e12a45b810c19b46dce58b6a6d02438ee781dd930158d05f0469f4e14.jpg b/.raw/MidasFEAAnalysisManual/images/page-288_6b156c2e12a45b810c19b46dce58b6a6d02438ee781dd930158d05f0469f4e14.jpg new file mode 100644 index 00000000..95633c4d Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-288_6b156c2e12a45b810c19b46dce58b6a6d02438ee781dd930158d05f0469f4e14.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-288_c2f58ffc735ba00c73ca3c9aef263ff4dfb9424bbf4cead358b775eb4454ec93.jpg b/.raw/MidasFEAAnalysisManual/images/page-288_c2f58ffc735ba00c73ca3c9aef263ff4dfb9424bbf4cead358b775eb4454ec93.jpg new file mode 100644 index 00000000..6e15e944 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-288_c2f58ffc735ba00c73ca3c9aef263ff4dfb9424bbf4cead358b775eb4454ec93.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-289_1603b05262b2646242f1ca8da6d9e269ed242f58c64f6c37802842c6d7ced68b.jpg b/.raw/MidasFEAAnalysisManual/images/page-289_1603b05262b2646242f1ca8da6d9e269ed242f58c64f6c37802842c6d7ced68b.jpg new file mode 100644 index 00000000..a9a5b4c1 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-289_1603b05262b2646242f1ca8da6d9e269ed242f58c64f6c37802842c6d7ced68b.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-289_74bdb783eed90a6d9edc4a6852dd1b757e85e4d504a386a29f4405f918b9a690.jpg b/.raw/MidasFEAAnalysisManual/images/page-289_74bdb783eed90a6d9edc4a6852dd1b757e85e4d504a386a29f4405f918b9a690.jpg new file mode 100644 index 00000000..3dada461 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-289_74bdb783eed90a6d9edc4a6852dd1b757e85e4d504a386a29f4405f918b9a690.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-289_876b5528c4fb9d9db6e25ccf8077a0acaa0453ec0d9d859ab2959082d48ff47e.jpg b/.raw/MidasFEAAnalysisManual/images/page-289_876b5528c4fb9d9db6e25ccf8077a0acaa0453ec0d9d859ab2959082d48ff47e.jpg new file mode 100644 index 00000000..b68ed249 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-289_876b5528c4fb9d9db6e25ccf8077a0acaa0453ec0d9d859ab2959082d48ff47e.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-289_8cecb2458d9aac4d6430614dcdb1dc179f9c04c9735ec67fa555f096e6e6be10.jpg b/.raw/MidasFEAAnalysisManual/images/page-289_8cecb2458d9aac4d6430614dcdb1dc179f9c04c9735ec67fa555f096e6e6be10.jpg new file mode 100644 index 00000000..a9730021 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-289_8cecb2458d9aac4d6430614dcdb1dc179f9c04c9735ec67fa555f096e6e6be10.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-289_cf258a5e5f890a7cb41657be3fa95364b8e13259b7a561651601306cd6941367.jpg b/.raw/MidasFEAAnalysisManual/images/page-289_cf258a5e5f890a7cb41657be3fa95364b8e13259b7a561651601306cd6941367.jpg new file mode 100644 index 00000000..6b50b61d Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-289_cf258a5e5f890a7cb41657be3fa95364b8e13259b7a561651601306cd6941367.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-290_755f8fb708c280826e8a6888c79af02d586fdd772a0d10770aed2a8c77227338.jpg b/.raw/MidasFEAAnalysisManual/images/page-290_755f8fb708c280826e8a6888c79af02d586fdd772a0d10770aed2a8c77227338.jpg new file mode 100644 index 00000000..c8ed5b51 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-290_755f8fb708c280826e8a6888c79af02d586fdd772a0d10770aed2a8c77227338.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-290_fbe4417e778d1a3e9714485797a679dda7405e5b534b022de30263222498fa6d.jpg b/.raw/MidasFEAAnalysisManual/images/page-290_fbe4417e778d1a3e9714485797a679dda7405e5b534b022de30263222498fa6d.jpg new file mode 100644 index 00000000..1f636b88 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-290_fbe4417e778d1a3e9714485797a679dda7405e5b534b022de30263222498fa6d.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-291_1c777a9b3caf95c2df3528c348d1b01abb9e0e994e87038558448a469909ed4f.jpg b/.raw/MidasFEAAnalysisManual/images/page-291_1c777a9b3caf95c2df3528c348d1b01abb9e0e994e87038558448a469909ed4f.jpg new file mode 100644 index 00000000..0df1fe25 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-291_1c777a9b3caf95c2df3528c348d1b01abb9e0e994e87038558448a469909ed4f.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-291_320294bd313d9c8a0d497a1b68a582dfb2162f5d6bb74acf6d89adb59b74bdba.jpg b/.raw/MidasFEAAnalysisManual/images/page-291_320294bd313d9c8a0d497a1b68a582dfb2162f5d6bb74acf6d89adb59b74bdba.jpg new file mode 100644 index 00000000..e6be8b36 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-291_320294bd313d9c8a0d497a1b68a582dfb2162f5d6bb74acf6d89adb59b74bdba.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-291_47218894c54cddd4dc462cce4e588cf530e176913f942cf03ce25beb9c0d503c.jpg b/.raw/MidasFEAAnalysisManual/images/page-291_47218894c54cddd4dc462cce4e588cf530e176913f942cf03ce25beb9c0d503c.jpg new file mode 100644 index 00000000..4a535e15 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-291_47218894c54cddd4dc462cce4e588cf530e176913f942cf03ce25beb9c0d503c.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-291_f222fd0bc72fddf2c249760eb1e9b6516e881a9c78c9ee56b81fb3853a4a50d8.jpg b/.raw/MidasFEAAnalysisManual/images/page-291_f222fd0bc72fddf2c249760eb1e9b6516e881a9c78c9ee56b81fb3853a4a50d8.jpg new file mode 100644 index 00000000..a6c27bc1 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-291_f222fd0bc72fddf2c249760eb1e9b6516e881a9c78c9ee56b81fb3853a4a50d8.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-292_ff5b8803536c50262e0bfcd3299e394d4e8f83af5f1c94454838329cc12a40ae.jpg b/.raw/MidasFEAAnalysisManual/images/page-292_ff5b8803536c50262e0bfcd3299e394d4e8f83af5f1c94454838329cc12a40ae.jpg new file mode 100644 index 00000000..f9a5cf54 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-292_ff5b8803536c50262e0bfcd3299e394d4e8f83af5f1c94454838329cc12a40ae.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-293_0fca90ba71db0b76f818278cd92b2d5481a774a4016d233675e0863955813f9b.jpg b/.raw/MidasFEAAnalysisManual/images/page-293_0fca90ba71db0b76f818278cd92b2d5481a774a4016d233675e0863955813f9b.jpg new file mode 100644 index 00000000..3361c0a1 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-293_0fca90ba71db0b76f818278cd92b2d5481a774a4016d233675e0863955813f9b.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-293_5f62ff4e98387d67917475163dd0918603a3d3ff2a3ed536f8dc039bd4c3c179.jpg b/.raw/MidasFEAAnalysisManual/images/page-293_5f62ff4e98387d67917475163dd0918603a3d3ff2a3ed536f8dc039bd4c3c179.jpg new file mode 100644 index 00000000..e60396b2 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-293_5f62ff4e98387d67917475163dd0918603a3d3ff2a3ed536f8dc039bd4c3c179.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-294_de4c2bd0442f158e84911781bca3a2a66140b1737567bca1bd5973853ec45cb2.jpg b/.raw/MidasFEAAnalysisManual/images/page-294_de4c2bd0442f158e84911781bca3a2a66140b1737567bca1bd5973853ec45cb2.jpg new file mode 100644 index 00000000..ab0348fd Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-294_de4c2bd0442f158e84911781bca3a2a66140b1737567bca1bd5973853ec45cb2.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-295_2ae8e8aac5e33f58335a5e2c4ccb8f00fe3f37e18afab1356b3a92c3d9c6beeb.jpg b/.raw/MidasFEAAnalysisManual/images/page-295_2ae8e8aac5e33f58335a5e2c4ccb8f00fe3f37e18afab1356b3a92c3d9c6beeb.jpg new file mode 100644 index 00000000..e6a6cc2c Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-295_2ae8e8aac5e33f58335a5e2c4ccb8f00fe3f37e18afab1356b3a92c3d9c6beeb.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-295_e6d542d7d0494a0fdfcf2336204aa9f03bf95a60743f2a8c7bbfa7672412f5c4.jpg b/.raw/MidasFEAAnalysisManual/images/page-295_e6d542d7d0494a0fdfcf2336204aa9f03bf95a60743f2a8c7bbfa7672412f5c4.jpg new file mode 100644 index 00000000..700de30e Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-295_e6d542d7d0494a0fdfcf2336204aa9f03bf95a60743f2a8c7bbfa7672412f5c4.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-296_1005889e813807eb87c98ec4b902ab7595fa400ceb0f2a7b6a6430799e3061d8.jpg b/.raw/MidasFEAAnalysisManual/images/page-296_1005889e813807eb87c98ec4b902ab7595fa400ceb0f2a7b6a6430799e3061d8.jpg new file mode 100644 index 00000000..570cf646 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-296_1005889e813807eb87c98ec4b902ab7595fa400ceb0f2a7b6a6430799e3061d8.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-296_15a4a32b513363f39335d84adf6422c9f61c04c64621476138ec68c66c8ff467.jpg b/.raw/MidasFEAAnalysisManual/images/page-296_15a4a32b513363f39335d84adf6422c9f61c04c64621476138ec68c66c8ff467.jpg new file mode 100644 index 00000000..fbc4e82e Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-296_15a4a32b513363f39335d84adf6422c9f61c04c64621476138ec68c66c8ff467.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-296_65ab9f72537e67b2ef9c419b9bb9d8cce8a495f73e20689211fa3bb70f518c64.jpg b/.raw/MidasFEAAnalysisManual/images/page-296_65ab9f72537e67b2ef9c419b9bb9d8cce8a495f73e20689211fa3bb70f518c64.jpg new file mode 100644 index 00000000..7b8180a7 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-296_65ab9f72537e67b2ef9c419b9bb9d8cce8a495f73e20689211fa3bb70f518c64.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-296_bbe4cd88bb379ef3c14498b8acff0c394fd314f94e1edee1cf4d2ed6dfc18590.jpg b/.raw/MidasFEAAnalysisManual/images/page-296_bbe4cd88bb379ef3c14498b8acff0c394fd314f94e1edee1cf4d2ed6dfc18590.jpg new file mode 100644 index 00000000..bfe99dd2 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-296_bbe4cd88bb379ef3c14498b8acff0c394fd314f94e1edee1cf4d2ed6dfc18590.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-296_c0c43cd959921e8a1028e77d1a50103829839fe81bd80c4879e7995367debeb9.jpg b/.raw/MidasFEAAnalysisManual/images/page-296_c0c43cd959921e8a1028e77d1a50103829839fe81bd80c4879e7995367debeb9.jpg new file mode 100644 index 00000000..36ef168c Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-296_c0c43cd959921e8a1028e77d1a50103829839fe81bd80c4879e7995367debeb9.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-296_c7e9080df4a3bf011199b7fc0010f7099dd0a2b22a4141fe5cac4b8b15df143e.jpg b/.raw/MidasFEAAnalysisManual/images/page-296_c7e9080df4a3bf011199b7fc0010f7099dd0a2b22a4141fe5cac4b8b15df143e.jpg new file mode 100644 index 00000000..392cf0b3 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-296_c7e9080df4a3bf011199b7fc0010f7099dd0a2b22a4141fe5cac4b8b15df143e.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-296_c817e376cd21c71a6da58240afd776a123acc0e2ac125bef66c3a291e9aa08b0.jpg b/.raw/MidasFEAAnalysisManual/images/page-296_c817e376cd21c71a6da58240afd776a123acc0e2ac125bef66c3a291e9aa08b0.jpg new file mode 100644 index 00000000..12a7da7d Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-296_c817e376cd21c71a6da58240afd776a123acc0e2ac125bef66c3a291e9aa08b0.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-296_df24ff1b390c895dc09cb9eab1dd870e755373e44de6d4b87064f7693e22b052.jpg b/.raw/MidasFEAAnalysisManual/images/page-296_df24ff1b390c895dc09cb9eab1dd870e755373e44de6d4b87064f7693e22b052.jpg new file mode 100644 index 00000000..21e2cf1e Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-296_df24ff1b390c895dc09cb9eab1dd870e755373e44de6d4b87064f7693e22b052.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-296_ea418ec98c8745875656aefb9848cb67917090404234f2542969bc3350dafa16.jpg b/.raw/MidasFEAAnalysisManual/images/page-296_ea418ec98c8745875656aefb9848cb67917090404234f2542969bc3350dafa16.jpg new file mode 100644 index 00000000..3b569119 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-296_ea418ec98c8745875656aefb9848cb67917090404234f2542969bc3350dafa16.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-296_fcd3d44f810ea57bcd77bc6c6f2e564771d28bf5abb391f847270b8d2a254c09.jpg b/.raw/MidasFEAAnalysisManual/images/page-296_fcd3d44f810ea57bcd77bc6c6f2e564771d28bf5abb391f847270b8d2a254c09.jpg new file mode 100644 index 00000000..e06089df Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-296_fcd3d44f810ea57bcd77bc6c6f2e564771d28bf5abb391f847270b8d2a254c09.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-296_ffff5f14b505296645c2e3b712fab44896bc3c375bd4d2320ef9473ae8ed680c.jpg b/.raw/MidasFEAAnalysisManual/images/page-296_ffff5f14b505296645c2e3b712fab44896bc3c375bd4d2320ef9473ae8ed680c.jpg new file mode 100644 index 00000000..3f811bd4 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-296_ffff5f14b505296645c2e3b712fab44896bc3c375bd4d2320ef9473ae8ed680c.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-297_269b1253e3747f8b805e347c67b28be6be3cda0f96b84ef6a3a066be8f59203c.jpg b/.raw/MidasFEAAnalysisManual/images/page-297_269b1253e3747f8b805e347c67b28be6be3cda0f96b84ef6a3a066be8f59203c.jpg new file mode 100644 index 00000000..52570ee4 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-297_269b1253e3747f8b805e347c67b28be6be3cda0f96b84ef6a3a066be8f59203c.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-297_3b88e182fb90aba11ede3f571d3ecc800c8cbc8fe4606c77e6801234f60be4ba.jpg b/.raw/MidasFEAAnalysisManual/images/page-297_3b88e182fb90aba11ede3f571d3ecc800c8cbc8fe4606c77e6801234f60be4ba.jpg new file mode 100644 index 00000000..7da1d6e2 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-297_3b88e182fb90aba11ede3f571d3ecc800c8cbc8fe4606c77e6801234f60be4ba.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-297_a08185e6b30f7bfc620cc56c9e8346f143df34c443dbe8034108f86ca838b86a.jpg b/.raw/MidasFEAAnalysisManual/images/page-297_a08185e6b30f7bfc620cc56c9e8346f143df34c443dbe8034108f86ca838b86a.jpg new file mode 100644 index 00000000..ed926588 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-297_a08185e6b30f7bfc620cc56c9e8346f143df34c443dbe8034108f86ca838b86a.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-298_08400bd364c94d00b93021aa4579501ad0d5927e6f766ee5803a25cd0a1d7c2d.jpg b/.raw/MidasFEAAnalysisManual/images/page-298_08400bd364c94d00b93021aa4579501ad0d5927e6f766ee5803a25cd0a1d7c2d.jpg new file mode 100644 index 00000000..33b47733 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-298_08400bd364c94d00b93021aa4579501ad0d5927e6f766ee5803a25cd0a1d7c2d.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-298_8651aeefc9614c433236f3d1f2adddcecfb075f69c7a73f3a2393cad00eb8f3b.jpg b/.raw/MidasFEAAnalysisManual/images/page-298_8651aeefc9614c433236f3d1f2adddcecfb075f69c7a73f3a2393cad00eb8f3b.jpg new file mode 100644 index 00000000..34e61870 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-298_8651aeefc9614c433236f3d1f2adddcecfb075f69c7a73f3a2393cad00eb8f3b.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-298_9db2700686c88cc7b0369c3a649bdc13950d3c9c0286d01ff121ec3e20a72c57.jpg b/.raw/MidasFEAAnalysisManual/images/page-298_9db2700686c88cc7b0369c3a649bdc13950d3c9c0286d01ff121ec3e20a72c57.jpg new file mode 100644 index 00000000..8ac50b7f Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-298_9db2700686c88cc7b0369c3a649bdc13950d3c9c0286d01ff121ec3e20a72c57.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-298_e26bd2d055f0e96130a2fdd6a2b8efb25395a69b9672a93ba4f1001538e4d7c8.jpg b/.raw/MidasFEAAnalysisManual/images/page-298_e26bd2d055f0e96130a2fdd6a2b8efb25395a69b9672a93ba4f1001538e4d7c8.jpg new file mode 100644 index 00000000..3affc4b0 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-298_e26bd2d055f0e96130a2fdd6a2b8efb25395a69b9672a93ba4f1001538e4d7c8.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-298_fd13c4413fea5b96459f8a564e28c9f2fe3c37dab1645a14a7598598eb545ff3.jpg b/.raw/MidasFEAAnalysisManual/images/page-298_fd13c4413fea5b96459f8a564e28c9f2fe3c37dab1645a14a7598598eb545ff3.jpg new file mode 100644 index 00000000..cbc0f73c Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-298_fd13c4413fea5b96459f8a564e28c9f2fe3c37dab1645a14a7598598eb545ff3.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-299_08cf2e68e4fadccb473737fd69d91704b0649a8970d590e588e64da259c8f452.jpg b/.raw/MidasFEAAnalysisManual/images/page-299_08cf2e68e4fadccb473737fd69d91704b0649a8970d590e588e64da259c8f452.jpg new file mode 100644 index 00000000..b6c25a8d Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-299_08cf2e68e4fadccb473737fd69d91704b0649a8970d590e588e64da259c8f452.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-299_6c720550af06086882c4a45338c3b4423138e84d0ec45c19b4ef59e7e838747e.jpg b/.raw/MidasFEAAnalysisManual/images/page-299_6c720550af06086882c4a45338c3b4423138e84d0ec45c19b4ef59e7e838747e.jpg new file mode 100644 index 00000000..83de6918 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-299_6c720550af06086882c4a45338c3b4423138e84d0ec45c19b4ef59e7e838747e.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-299_764437dae600e249092ba5ec0f195e625c827f6c8f9038cd5b8a51ffdffc760d.jpg b/.raw/MidasFEAAnalysisManual/images/page-299_764437dae600e249092ba5ec0f195e625c827f6c8f9038cd5b8a51ffdffc760d.jpg new file mode 100644 index 00000000..d3cf4d34 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-299_764437dae600e249092ba5ec0f195e625c827f6c8f9038cd5b8a51ffdffc760d.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-299_c112ed426e775670df9cd4bf47082b5e217166513d95ba81b9fca57377c273d0.jpg b/.raw/MidasFEAAnalysisManual/images/page-299_c112ed426e775670df9cd4bf47082b5e217166513d95ba81b9fca57377c273d0.jpg new file mode 100644 index 00000000..29563d05 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-299_c112ed426e775670df9cd4bf47082b5e217166513d95ba81b9fca57377c273d0.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-299_c31a07a8adee19c5b326e7fb7284a441082002db25239d0cfe4d1efdbad6e46a.jpg b/.raw/MidasFEAAnalysisManual/images/page-299_c31a07a8adee19c5b326e7fb7284a441082002db25239d0cfe4d1efdbad6e46a.jpg new file mode 100644 index 00000000..ba959adf Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-299_c31a07a8adee19c5b326e7fb7284a441082002db25239d0cfe4d1efdbad6e46a.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-300_2d59ec1b579c4df7f31ab317f959a8e065f80591c3fc6941ab6afa0bcadc248a.jpg b/.raw/MidasFEAAnalysisManual/images/page-300_2d59ec1b579c4df7f31ab317f959a8e065f80591c3fc6941ab6afa0bcadc248a.jpg new file mode 100644 index 00000000..bf584302 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-300_2d59ec1b579c4df7f31ab317f959a8e065f80591c3fc6941ab6afa0bcadc248a.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-300_47a55bc3852748f963eed2b1da67ee91410f960e1ffd0f03c15f628526e44f62.jpg b/.raw/MidasFEAAnalysisManual/images/page-300_47a55bc3852748f963eed2b1da67ee91410f960e1ffd0f03c15f628526e44f62.jpg new file mode 100644 index 00000000..9956c60c Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-300_47a55bc3852748f963eed2b1da67ee91410f960e1ffd0f03c15f628526e44f62.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-300_6e12e571429846b8b7cf12cb1064961c1ad1609df07df7db4e80ebefde223f9c.jpg b/.raw/MidasFEAAnalysisManual/images/page-300_6e12e571429846b8b7cf12cb1064961c1ad1609df07df7db4e80ebefde223f9c.jpg new file mode 100644 index 00000000..d6093378 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-300_6e12e571429846b8b7cf12cb1064961c1ad1609df07df7db4e80ebefde223f9c.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-301_41d5952e83a2e3eddb0fa3b6d5ed2c85f9779015b0b54174afecdeaf3e3c5fe2.jpg b/.raw/MidasFEAAnalysisManual/images/page-301_41d5952e83a2e3eddb0fa3b6d5ed2c85f9779015b0b54174afecdeaf3e3c5fe2.jpg new file mode 100644 index 00000000..ebb02668 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-301_41d5952e83a2e3eddb0fa3b6d5ed2c85f9779015b0b54174afecdeaf3e3c5fe2.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-302_9403787ed3a0a9e63ef1c17d12c534036fa06da8dc24071ac4ee9855db351b0a.jpg b/.raw/MidasFEAAnalysisManual/images/page-302_9403787ed3a0a9e63ef1c17d12c534036fa06da8dc24071ac4ee9855db351b0a.jpg new file mode 100644 index 00000000..9e806b2a Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-302_9403787ed3a0a9e63ef1c17d12c534036fa06da8dc24071ac4ee9855db351b0a.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-302_e025b2cb58f5d049b9b63a7ef9afa610167cdffa2304ad9a2a25c7499f202fdb.jpg b/.raw/MidasFEAAnalysisManual/images/page-302_e025b2cb58f5d049b9b63a7ef9afa610167cdffa2304ad9a2a25c7499f202fdb.jpg new file mode 100644 index 00000000..627345c4 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-302_e025b2cb58f5d049b9b63a7ef9afa610167cdffa2304ad9a2a25c7499f202fdb.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-303_16f8ba6bbd6e5edcd8196f0fc2f7a5325abf677ee37c59de8a508b56ce3c7a2b.jpg b/.raw/MidasFEAAnalysisManual/images/page-303_16f8ba6bbd6e5edcd8196f0fc2f7a5325abf677ee37c59de8a508b56ce3c7a2b.jpg new file mode 100644 index 00000000..57360439 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-303_16f8ba6bbd6e5edcd8196f0fc2f7a5325abf677ee37c59de8a508b56ce3c7a2b.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-303_350387c282d053bb637e2a34333ac93db3d47b668946910008e8343447d9abe3.jpg b/.raw/MidasFEAAnalysisManual/images/page-303_350387c282d053bb637e2a34333ac93db3d47b668946910008e8343447d9abe3.jpg new file mode 100644 index 00000000..2cc13195 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-303_350387c282d053bb637e2a34333ac93db3d47b668946910008e8343447d9abe3.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-303_9564013e6c157ced602f0dcf714d86f0e0f7711eb446caf93ff6f53eca4c5f8e.jpg b/.raw/MidasFEAAnalysisManual/images/page-303_9564013e6c157ced602f0dcf714d86f0e0f7711eb446caf93ff6f53eca4c5f8e.jpg new file mode 100644 index 00000000..e8d9946a Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-303_9564013e6c157ced602f0dcf714d86f0e0f7711eb446caf93ff6f53eca4c5f8e.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-304_7abf5edba6d944f7cb173762ac71264479cd3e3af258abdde6f60bca31628d3e.jpg b/.raw/MidasFEAAnalysisManual/images/page-304_7abf5edba6d944f7cb173762ac71264479cd3e3af258abdde6f60bca31628d3e.jpg new file mode 100644 index 00000000..f6674c0b Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-304_7abf5edba6d944f7cb173762ac71264479cd3e3af258abdde6f60bca31628d3e.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-304_e5c1ec7a9772794e69a838f0dcdb2adf5f844c666e435df8f570727a115a0fa0.jpg b/.raw/MidasFEAAnalysisManual/images/page-304_e5c1ec7a9772794e69a838f0dcdb2adf5f844c666e435df8f570727a115a0fa0.jpg new file mode 100644 index 00000000..663f6198 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-304_e5c1ec7a9772794e69a838f0dcdb2adf5f844c666e435df8f570727a115a0fa0.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-305_54ec9a47f18ed3532083fb7f8a95bd31bcbe81161e0a74c8b61c8a788b548a9a.jpg b/.raw/MidasFEAAnalysisManual/images/page-305_54ec9a47f18ed3532083fb7f8a95bd31bcbe81161e0a74c8b61c8a788b548a9a.jpg new file mode 100644 index 00000000..eaf244a7 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-305_54ec9a47f18ed3532083fb7f8a95bd31bcbe81161e0a74c8b61c8a788b548a9a.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-305_97defe82786e52fe4a7616d66a4fe1e569c2690e61547e0d3e6fd4605f2d42f6.jpg b/.raw/MidasFEAAnalysisManual/images/page-305_97defe82786e52fe4a7616d66a4fe1e569c2690e61547e0d3e6fd4605f2d42f6.jpg new file mode 100644 index 00000000..f19e71be Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-305_97defe82786e52fe4a7616d66a4fe1e569c2690e61547e0d3e6fd4605f2d42f6.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-305_98b4ad45dccc7f96829b863b2e3e8fb23b2be31c10487499044d6c418d760f03.jpg b/.raw/MidasFEAAnalysisManual/images/page-305_98b4ad45dccc7f96829b863b2e3e8fb23b2be31c10487499044d6c418d760f03.jpg new file mode 100644 index 00000000..4b0598f9 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-305_98b4ad45dccc7f96829b863b2e3e8fb23b2be31c10487499044d6c418d760f03.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-305_de517dff6b590d93845f55119675158843dcd7bacf07a1448b1d38b4e09c186e.jpg b/.raw/MidasFEAAnalysisManual/images/page-305_de517dff6b590d93845f55119675158843dcd7bacf07a1448b1d38b4e09c186e.jpg new file mode 100644 index 00000000..6f520474 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-305_de517dff6b590d93845f55119675158843dcd7bacf07a1448b1d38b4e09c186e.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-306_4b711471cb3666f6e29d974a1136ce91dbb99167dcc042a6082c7255d2fb0efd.jpg b/.raw/MidasFEAAnalysisManual/images/page-306_4b711471cb3666f6e29d974a1136ce91dbb99167dcc042a6082c7255d2fb0efd.jpg new file mode 100644 index 00000000..a81af19b Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-306_4b711471cb3666f6e29d974a1136ce91dbb99167dcc042a6082c7255d2fb0efd.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-306_69fa0c6e48e6c2011d0ab79386c75a843f352bdf164df552d1165216d5d24473.jpg b/.raw/MidasFEAAnalysisManual/images/page-306_69fa0c6e48e6c2011d0ab79386c75a843f352bdf164df552d1165216d5d24473.jpg new file mode 100644 index 00000000..f821a9cd Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-306_69fa0c6e48e6c2011d0ab79386c75a843f352bdf164df552d1165216d5d24473.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-306_f513d290a6621dd4d54341a0ce544ce83e8b829b5f08c4de2089d5484d6f1aaf.jpg b/.raw/MidasFEAAnalysisManual/images/page-306_f513d290a6621dd4d54341a0ce544ce83e8b829b5f08c4de2089d5484d6f1aaf.jpg new file mode 100644 index 00000000..c1736520 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-306_f513d290a6621dd4d54341a0ce544ce83e8b829b5f08c4de2089d5484d6f1aaf.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-307_0009bde5bcff91379fcf60e8dc23ab3984cc60e4789ea90e8f2d7b159f0a35e3.jpg b/.raw/MidasFEAAnalysisManual/images/page-307_0009bde5bcff91379fcf60e8dc23ab3984cc60e4789ea90e8f2d7b159f0a35e3.jpg new file mode 100644 index 00000000..81cae078 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-307_0009bde5bcff91379fcf60e8dc23ab3984cc60e4789ea90e8f2d7b159f0a35e3.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-307_2137da0f07f50ea97020a24f14e209d1bdf61ae6fbe2de9f12b980896cfed192.jpg b/.raw/MidasFEAAnalysisManual/images/page-307_2137da0f07f50ea97020a24f14e209d1bdf61ae6fbe2de9f12b980896cfed192.jpg new file mode 100644 index 00000000..fd69490a Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-307_2137da0f07f50ea97020a24f14e209d1bdf61ae6fbe2de9f12b980896cfed192.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-307_502f9b3a0b3240bee4c13df0768787a2bf91bcf77dfd3caf62416111b0051b27.jpg b/.raw/MidasFEAAnalysisManual/images/page-307_502f9b3a0b3240bee4c13df0768787a2bf91bcf77dfd3caf62416111b0051b27.jpg new file mode 100644 index 00000000..66960031 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-307_502f9b3a0b3240bee4c13df0768787a2bf91bcf77dfd3caf62416111b0051b27.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-307_dff17eaafaaac2faf5f0cf0585483d124d8fdeae0819e1fb0cb43e6fec8cffec.jpg b/.raw/MidasFEAAnalysisManual/images/page-307_dff17eaafaaac2faf5f0cf0585483d124d8fdeae0819e1fb0cb43e6fec8cffec.jpg new file mode 100644 index 00000000..476e593a Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-307_dff17eaafaaac2faf5f0cf0585483d124d8fdeae0819e1fb0cb43e6fec8cffec.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-308_01a39b9a0c4c2b5fdff772634d338c015b411978cb9be15abc973425396043ba.jpg b/.raw/MidasFEAAnalysisManual/images/page-308_01a39b9a0c4c2b5fdff772634d338c015b411978cb9be15abc973425396043ba.jpg new file mode 100644 index 00000000..bbea8316 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-308_01a39b9a0c4c2b5fdff772634d338c015b411978cb9be15abc973425396043ba.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-308_0bac13d3293c27a48869e2bd9bb4e2d9263c8c15fdbac495251a36104d786d8b.jpg b/.raw/MidasFEAAnalysisManual/images/page-308_0bac13d3293c27a48869e2bd9bb4e2d9263c8c15fdbac495251a36104d786d8b.jpg new file mode 100644 index 00000000..cb1f4843 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-308_0bac13d3293c27a48869e2bd9bb4e2d9263c8c15fdbac495251a36104d786d8b.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-308_34deb1f18882d2e07b84cf1e4949aa3ea3e784527894ba99d35496bfd11b77fe.jpg b/.raw/MidasFEAAnalysisManual/images/page-308_34deb1f18882d2e07b84cf1e4949aa3ea3e784527894ba99d35496bfd11b77fe.jpg new file mode 100644 index 00000000..9f1d9dba Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-308_34deb1f18882d2e07b84cf1e4949aa3ea3e784527894ba99d35496bfd11b77fe.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-308_3e76cf9835e7f40143ab124f75e908a0c708e5f27e734f698117de6ef88c42a7.jpg b/.raw/MidasFEAAnalysisManual/images/page-308_3e76cf9835e7f40143ab124f75e908a0c708e5f27e734f698117de6ef88c42a7.jpg new file mode 100644 index 00000000..f2bf50de Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-308_3e76cf9835e7f40143ab124f75e908a0c708e5f27e734f698117de6ef88c42a7.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-308_7a16ac63decbee7b700082390746b230321d39bc284e4e3b8ee7e766c6e7ce43.jpg b/.raw/MidasFEAAnalysisManual/images/page-308_7a16ac63decbee7b700082390746b230321d39bc284e4e3b8ee7e766c6e7ce43.jpg new file mode 100644 index 00000000..0f69d936 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-308_7a16ac63decbee7b700082390746b230321d39bc284e4e3b8ee7e766c6e7ce43.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-308_97b5e5c3ad297d0c121fe71e51c975eb7784d7cb984647fbff6982e2ba0a5c3d.jpg b/.raw/MidasFEAAnalysisManual/images/page-308_97b5e5c3ad297d0c121fe71e51c975eb7784d7cb984647fbff6982e2ba0a5c3d.jpg new file mode 100644 index 00000000..b59ff691 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-308_97b5e5c3ad297d0c121fe71e51c975eb7784d7cb984647fbff6982e2ba0a5c3d.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-308_a8253cdd1110d9dd0eadc433bee83ed015b83d506d6d821454fcb1c256d9e448.jpg b/.raw/MidasFEAAnalysisManual/images/page-308_a8253cdd1110d9dd0eadc433bee83ed015b83d506d6d821454fcb1c256d9e448.jpg new file mode 100644 index 00000000..56006a87 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-308_a8253cdd1110d9dd0eadc433bee83ed015b83d506d6d821454fcb1c256d9e448.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-308_bde56440c2305390f8cc53bd4419da374b08289f215886a94192956ec41d93c0.jpg b/.raw/MidasFEAAnalysisManual/images/page-308_bde56440c2305390f8cc53bd4419da374b08289f215886a94192956ec41d93c0.jpg new file mode 100644 index 00000000..438f13b2 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-308_bde56440c2305390f8cc53bd4419da374b08289f215886a94192956ec41d93c0.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-310_19d4344dc5337c855529bad20aba56a6b5b8104b8f80e437a95a607e377b4b32.jpg b/.raw/MidasFEAAnalysisManual/images/page-310_19d4344dc5337c855529bad20aba56a6b5b8104b8f80e437a95a607e377b4b32.jpg new file mode 100644 index 00000000..3d428425 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-310_19d4344dc5337c855529bad20aba56a6b5b8104b8f80e437a95a607e377b4b32.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-310_5e46c1e4128f813a1da17720b70e09ed120e0eb6ad8c612a68545f252863a9e8.jpg b/.raw/MidasFEAAnalysisManual/images/page-310_5e46c1e4128f813a1da17720b70e09ed120e0eb6ad8c612a68545f252863a9e8.jpg new file mode 100644 index 00000000..85dabc44 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-310_5e46c1e4128f813a1da17720b70e09ed120e0eb6ad8c612a68545f252863a9e8.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-311_1973f7988ac00e0605f667bb14f3fa9a59d07bdc985e0d5effcfb05039e47424.jpg b/.raw/MidasFEAAnalysisManual/images/page-311_1973f7988ac00e0605f667bb14f3fa9a59d07bdc985e0d5effcfb05039e47424.jpg new file mode 100644 index 00000000..643337af Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-311_1973f7988ac00e0605f667bb14f3fa9a59d07bdc985e0d5effcfb05039e47424.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-311_79ff7db728e0bbffe8547e6c2a9544b91b1e38b1c6025772e692106af2c38fb7.jpg b/.raw/MidasFEAAnalysisManual/images/page-311_79ff7db728e0bbffe8547e6c2a9544b91b1e38b1c6025772e692106af2c38fb7.jpg new file mode 100644 index 00000000..dc9651be Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-311_79ff7db728e0bbffe8547e6c2a9544b91b1e38b1c6025772e692106af2c38fb7.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-311_9821724f319b5d6e83f12faaf3170b719e432cff719e5f1382526344410ea223.jpg b/.raw/MidasFEAAnalysisManual/images/page-311_9821724f319b5d6e83f12faaf3170b719e432cff719e5f1382526344410ea223.jpg new file mode 100644 index 00000000..10602094 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-311_9821724f319b5d6e83f12faaf3170b719e432cff719e5f1382526344410ea223.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-311_f8cd791a35d67dff7606542b127adcef1bf386316c2f0df4c793fee21538d2d2.jpg b/.raw/MidasFEAAnalysisManual/images/page-311_f8cd791a35d67dff7606542b127adcef1bf386316c2f0df4c793fee21538d2d2.jpg new file mode 100644 index 00000000..2663fe15 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-311_f8cd791a35d67dff7606542b127adcef1bf386316c2f0df4c793fee21538d2d2.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-312_859e4e0290669193454b28aac9fe67c04f02571548bb3c770f0a9a2a08633710.jpg b/.raw/MidasFEAAnalysisManual/images/page-312_859e4e0290669193454b28aac9fe67c04f02571548bb3c770f0a9a2a08633710.jpg new file mode 100644 index 00000000..c94f9415 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-312_859e4e0290669193454b28aac9fe67c04f02571548bb3c770f0a9a2a08633710.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-312_8774034a2c482b72c388b248eb01a0a2c9bc120298701965c905a97713186ff2.jpg b/.raw/MidasFEAAnalysisManual/images/page-312_8774034a2c482b72c388b248eb01a0a2c9bc120298701965c905a97713186ff2.jpg new file mode 100644 index 00000000..6f8d403f Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-312_8774034a2c482b72c388b248eb01a0a2c9bc120298701965c905a97713186ff2.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-312_ed708a9f51d32a6cd038f5a675aa623ba4877d7a89c0fcc6ceba72e0038bdd98.jpg b/.raw/MidasFEAAnalysisManual/images/page-312_ed708a9f51d32a6cd038f5a675aa623ba4877d7a89c0fcc6ceba72e0038bdd98.jpg new file mode 100644 index 00000000..7330bffb Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-312_ed708a9f51d32a6cd038f5a675aa623ba4877d7a89c0fcc6ceba72e0038bdd98.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-313_19bcffe7b7ec061d61f2ead255938395ed2d7788799c7a6441d624944637aa16.jpg b/.raw/MidasFEAAnalysisManual/images/page-313_19bcffe7b7ec061d61f2ead255938395ed2d7788799c7a6441d624944637aa16.jpg new file mode 100644 index 00000000..a47a0bdf Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-313_19bcffe7b7ec061d61f2ead255938395ed2d7788799c7a6441d624944637aa16.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-313_7854acb5df905940f0c6e488236d8241e37e098e82a58d8df6dea4e18a5fcf34.jpg b/.raw/MidasFEAAnalysisManual/images/page-313_7854acb5df905940f0c6e488236d8241e37e098e82a58d8df6dea4e18a5fcf34.jpg new file mode 100644 index 00000000..84af633f Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-313_7854acb5df905940f0c6e488236d8241e37e098e82a58d8df6dea4e18a5fcf34.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-313_9379286658acb8007c39ef0393de70c4b213eff3f550db75a6833098bf8096d4.jpg b/.raw/MidasFEAAnalysisManual/images/page-313_9379286658acb8007c39ef0393de70c4b213eff3f550db75a6833098bf8096d4.jpg new file mode 100644 index 00000000..0b8ba995 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-313_9379286658acb8007c39ef0393de70c4b213eff3f550db75a6833098bf8096d4.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-313_93cc84679cad818176e50d91e7e7407350f99ddaa3fcb48ab3951b0733401655.jpg b/.raw/MidasFEAAnalysisManual/images/page-313_93cc84679cad818176e50d91e7e7407350f99ddaa3fcb48ab3951b0733401655.jpg new file mode 100644 index 00000000..4c8a1b8f Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-313_93cc84679cad818176e50d91e7e7407350f99ddaa3fcb48ab3951b0733401655.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-313_9df8b7f185bfb7510d6ce3766880f0f5b195021071186b0f22d558a8633e98b3.jpg b/.raw/MidasFEAAnalysisManual/images/page-313_9df8b7f185bfb7510d6ce3766880f0f5b195021071186b0f22d558a8633e98b3.jpg new file mode 100644 index 00000000..89f6ea2c Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-313_9df8b7f185bfb7510d6ce3766880f0f5b195021071186b0f22d558a8633e98b3.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-313_d0a8e136f32e3e7817aa7537ec1853db8deb979a4db86174251d2044d20ba448.jpg b/.raw/MidasFEAAnalysisManual/images/page-313_d0a8e136f32e3e7817aa7537ec1853db8deb979a4db86174251d2044d20ba448.jpg new file mode 100644 index 00000000..61388470 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-313_d0a8e136f32e3e7817aa7537ec1853db8deb979a4db86174251d2044d20ba448.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-314_0ed05c5042bbe9f285ff1e7ccf2b27d952ebc1ea70718906a3b9c9ddb568954f.jpg b/.raw/MidasFEAAnalysisManual/images/page-314_0ed05c5042bbe9f285ff1e7ccf2b27d952ebc1ea70718906a3b9c9ddb568954f.jpg new file mode 100644 index 00000000..016dec85 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-314_0ed05c5042bbe9f285ff1e7ccf2b27d952ebc1ea70718906a3b9c9ddb568954f.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-314_40cbe4dbd92a06f2b47b055b7b0a0ff615e13db829a1d69d32c58fdbb3b264be.jpg b/.raw/MidasFEAAnalysisManual/images/page-314_40cbe4dbd92a06f2b47b055b7b0a0ff615e13db829a1d69d32c58fdbb3b264be.jpg new file mode 100644 index 00000000..b128d1c7 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-314_40cbe4dbd92a06f2b47b055b7b0a0ff615e13db829a1d69d32c58fdbb3b264be.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-314_94f1e4a84c8b60f63e78bdb8d6d8df913a5d9f2e259ee851ba345a034f162b23.jpg b/.raw/MidasFEAAnalysisManual/images/page-314_94f1e4a84c8b60f63e78bdb8d6d8df913a5d9f2e259ee851ba345a034f162b23.jpg new file mode 100644 index 00000000..4eea558d Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-314_94f1e4a84c8b60f63e78bdb8d6d8df913a5d9f2e259ee851ba345a034f162b23.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-315_3a55757c4d48755ec54026f81ba5ae0ffdba32a332ef4d4cbaafb0cc32985135.jpg b/.raw/MidasFEAAnalysisManual/images/page-315_3a55757c4d48755ec54026f81ba5ae0ffdba32a332ef4d4cbaafb0cc32985135.jpg new file mode 100644 index 00000000..190f14f9 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-315_3a55757c4d48755ec54026f81ba5ae0ffdba32a332ef4d4cbaafb0cc32985135.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-315_46706599ba6e458f917af1c5d73deef707fc1588c951f68a78954c813707caf2.jpg b/.raw/MidasFEAAnalysisManual/images/page-315_46706599ba6e458f917af1c5d73deef707fc1588c951f68a78954c813707caf2.jpg new file mode 100644 index 00000000..9b2abc9d Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-315_46706599ba6e458f917af1c5d73deef707fc1588c951f68a78954c813707caf2.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-315_6915164c3ea50752285bb4deb4e6ea963291cd0962ee0d9b822a2c0c1cb98368.jpg b/.raw/MidasFEAAnalysisManual/images/page-315_6915164c3ea50752285bb4deb4e6ea963291cd0962ee0d9b822a2c0c1cb98368.jpg new file mode 100644 index 00000000..f0d7f911 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-315_6915164c3ea50752285bb4deb4e6ea963291cd0962ee0d9b822a2c0c1cb98368.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-315_788ecbcfe4f69b1d91b0aad7eb44bda32c2c05b22a55f6549b57b334a207e61c.jpg b/.raw/MidasFEAAnalysisManual/images/page-315_788ecbcfe4f69b1d91b0aad7eb44bda32c2c05b22a55f6549b57b334a207e61c.jpg new file mode 100644 index 00000000..19e9a500 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-315_788ecbcfe4f69b1d91b0aad7eb44bda32c2c05b22a55f6549b57b334a207e61c.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-315_951d22eb75b4d3352ccb0a7fcc294c34027db0aa6c268487709126a6806dec17.jpg b/.raw/MidasFEAAnalysisManual/images/page-315_951d22eb75b4d3352ccb0a7fcc294c34027db0aa6c268487709126a6806dec17.jpg new file mode 100644 index 00000000..36e73f1c Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-315_951d22eb75b4d3352ccb0a7fcc294c34027db0aa6c268487709126a6806dec17.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-315_d478252c3fa034e65b4179c082c31673d7dce074d794ffb7006496d55c540651.jpg b/.raw/MidasFEAAnalysisManual/images/page-315_d478252c3fa034e65b4179c082c31673d7dce074d794ffb7006496d55c540651.jpg new file mode 100644 index 00000000..34a9e027 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-315_d478252c3fa034e65b4179c082c31673d7dce074d794ffb7006496d55c540651.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-315_d4afbe112faaeaa53f6e44afe4998f568e0b9d917813d4bad35d0333bdef1498.jpg b/.raw/MidasFEAAnalysisManual/images/page-315_d4afbe112faaeaa53f6e44afe4998f568e0b9d917813d4bad35d0333bdef1498.jpg new file mode 100644 index 00000000..f7e47021 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-315_d4afbe112faaeaa53f6e44afe4998f568e0b9d917813d4bad35d0333bdef1498.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-315_e70ccbd9b6d4f7d3115ae9694c8213875b73f7b778f8020d00e294af797f5020.jpg b/.raw/MidasFEAAnalysisManual/images/page-315_e70ccbd9b6d4f7d3115ae9694c8213875b73f7b778f8020d00e294af797f5020.jpg new file mode 100644 index 00000000..359c64c5 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-315_e70ccbd9b6d4f7d3115ae9694c8213875b73f7b778f8020d00e294af797f5020.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-315_fb1ff1a972fb2279cb7fdca0496ce8a40bbf36984e683d4d29aee8b568fe1f9e.jpg b/.raw/MidasFEAAnalysisManual/images/page-315_fb1ff1a972fb2279cb7fdca0496ce8a40bbf36984e683d4d29aee8b568fe1f9e.jpg new file mode 100644 index 00000000..e58a8c5b Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-315_fb1ff1a972fb2279cb7fdca0496ce8a40bbf36984e683d4d29aee8b568fe1f9e.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-316_0dad016c35825779519389b9e9792269eded15f6dd292d8bda6b59056c660875.jpg b/.raw/MidasFEAAnalysisManual/images/page-316_0dad016c35825779519389b9e9792269eded15f6dd292d8bda6b59056c660875.jpg new file mode 100644 index 00000000..7d901818 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-316_0dad016c35825779519389b9e9792269eded15f6dd292d8bda6b59056c660875.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-316_606195d21abd4d245bc05b98c14ef2ab3ebd43b1b69350e2367498e023591dc6.jpg b/.raw/MidasFEAAnalysisManual/images/page-316_606195d21abd4d245bc05b98c14ef2ab3ebd43b1b69350e2367498e023591dc6.jpg new file mode 100644 index 00000000..5e1c817d Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-316_606195d21abd4d245bc05b98c14ef2ab3ebd43b1b69350e2367498e023591dc6.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-316_aabf0971053fb445ab9428d88cf655ab41425a26f52ba1881abf74f050a100d6.jpg b/.raw/MidasFEAAnalysisManual/images/page-316_aabf0971053fb445ab9428d88cf655ab41425a26f52ba1881abf74f050a100d6.jpg new file mode 100644 index 00000000..3729ad3b Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-316_aabf0971053fb445ab9428d88cf655ab41425a26f52ba1881abf74f050a100d6.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-316_c924b924bef026b59e53cdbd7c1bbbf6db84f13393bfdd0a10fd79e405456ef6.jpg b/.raw/MidasFEAAnalysisManual/images/page-316_c924b924bef026b59e53cdbd7c1bbbf6db84f13393bfdd0a10fd79e405456ef6.jpg new file mode 100644 index 00000000..cb0926db Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-316_c924b924bef026b59e53cdbd7c1bbbf6db84f13393bfdd0a10fd79e405456ef6.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-317_1094e0fc0d14912f0675f2a81e36ebc836363e003ba4af413b2dbe6ff9d466e9.jpg b/.raw/MidasFEAAnalysisManual/images/page-317_1094e0fc0d14912f0675f2a81e36ebc836363e003ba4af413b2dbe6ff9d466e9.jpg new file mode 100644 index 00000000..3a9eabd3 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-317_1094e0fc0d14912f0675f2a81e36ebc836363e003ba4af413b2dbe6ff9d466e9.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-317_46814005d1be5c1f03d721a5a3c78f8ba920434c0f5f432e2a68ee866ecbe2d4.jpg b/.raw/MidasFEAAnalysisManual/images/page-317_46814005d1be5c1f03d721a5a3c78f8ba920434c0f5f432e2a68ee866ecbe2d4.jpg new file mode 100644 index 00000000..6c7d7815 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-317_46814005d1be5c1f03d721a5a3c78f8ba920434c0f5f432e2a68ee866ecbe2d4.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-317_840afe4dafaf3f977ac689dadedb7f3cb7fa0954c72f0ddfa8f3d659c93d14d9.jpg b/.raw/MidasFEAAnalysisManual/images/page-317_840afe4dafaf3f977ac689dadedb7f3cb7fa0954c72f0ddfa8f3d659c93d14d9.jpg new file mode 100644 index 00000000..2b43c1a2 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-317_840afe4dafaf3f977ac689dadedb7f3cb7fa0954c72f0ddfa8f3d659c93d14d9.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-317_8fb0d4e0eddf15ba4e1511933c35351026539faaec04b7f3427c296b5bc650b2.jpg b/.raw/MidasFEAAnalysisManual/images/page-317_8fb0d4e0eddf15ba4e1511933c35351026539faaec04b7f3427c296b5bc650b2.jpg new file mode 100644 index 00000000..c6350ea1 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-317_8fb0d4e0eddf15ba4e1511933c35351026539faaec04b7f3427c296b5bc650b2.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-318_5eb01301ae07753fdc8abd7450f0464bb4e3fc3e78e69e9211c9d9da65d24b30.jpg b/.raw/MidasFEAAnalysisManual/images/page-318_5eb01301ae07753fdc8abd7450f0464bb4e3fc3e78e69e9211c9d9da65d24b30.jpg new file mode 100644 index 00000000..cb0bee14 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-318_5eb01301ae07753fdc8abd7450f0464bb4e3fc3e78e69e9211c9d9da65d24b30.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-318_6bd260bb8f6ee61cc018c13a9d4c1cc70e94cfe98283fc3a7a41fb40e59313ed.jpg b/.raw/MidasFEAAnalysisManual/images/page-318_6bd260bb8f6ee61cc018c13a9d4c1cc70e94cfe98283fc3a7a41fb40e59313ed.jpg new file mode 100644 index 00000000..0df1b21d Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-318_6bd260bb8f6ee61cc018c13a9d4c1cc70e94cfe98283fc3a7a41fb40e59313ed.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-318_7810607553d7b9b7d53b92f1740c56f1b4861a72e4ffc1509787e63630734499.jpg b/.raw/MidasFEAAnalysisManual/images/page-318_7810607553d7b9b7d53b92f1740c56f1b4861a72e4ffc1509787e63630734499.jpg new file mode 100644 index 00000000..5151d650 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-318_7810607553d7b9b7d53b92f1740c56f1b4861a72e4ffc1509787e63630734499.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-318_b6a3b907a787bb20d9b1e6ee120a92a61909d45de8835e8d04a6e3e67b355043.jpg b/.raw/MidasFEAAnalysisManual/images/page-318_b6a3b907a787bb20d9b1e6ee120a92a61909d45de8835e8d04a6e3e67b355043.jpg new file mode 100644 index 00000000..889f653c Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-318_b6a3b907a787bb20d9b1e6ee120a92a61909d45de8835e8d04a6e3e67b355043.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-318_d8f14bb63cc5c2a87b2889c5d73078dccedf1fbc4f8eaf388c6e7706b366bbef.jpg b/.raw/MidasFEAAnalysisManual/images/page-318_d8f14bb63cc5c2a87b2889c5d73078dccedf1fbc4f8eaf388c6e7706b366bbef.jpg new file mode 100644 index 00000000..27025c9a Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-318_d8f14bb63cc5c2a87b2889c5d73078dccedf1fbc4f8eaf388c6e7706b366bbef.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-318_f4316cb5f9f32162d71762acb9c89aa09e5f83b0de94beb3f731872bb56f2c20.jpg b/.raw/MidasFEAAnalysisManual/images/page-318_f4316cb5f9f32162d71762acb9c89aa09e5f83b0de94beb3f731872bb56f2c20.jpg new file mode 100644 index 00000000..c8c79bdf Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-318_f4316cb5f9f32162d71762acb9c89aa09e5f83b0de94beb3f731872bb56f2c20.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-321_10e88087016c348fe05a487d7a875259e7d23d1fc62c4487b58c8e5ead7b70c3.jpg b/.raw/MidasFEAAnalysisManual/images/page-321_10e88087016c348fe05a487d7a875259e7d23d1fc62c4487b58c8e5ead7b70c3.jpg new file mode 100644 index 00000000..02201cf1 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-321_10e88087016c348fe05a487d7a875259e7d23d1fc62c4487b58c8e5ead7b70c3.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-323_24bce34b260fa921ea3c8beb173e812ba02f21c77fab94a310dc667a2382b9a5.jpg b/.raw/MidasFEAAnalysisManual/images/page-323_24bce34b260fa921ea3c8beb173e812ba02f21c77fab94a310dc667a2382b9a5.jpg new file mode 100644 index 00000000..e7a81282 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-323_24bce34b260fa921ea3c8beb173e812ba02f21c77fab94a310dc667a2382b9a5.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-323_e1d4479804093cafc46522a94d102ac21f508425d911f92d3a84f4b16e29c9f9.jpg b/.raw/MidasFEAAnalysisManual/images/page-323_e1d4479804093cafc46522a94d102ac21f508425d911f92d3a84f4b16e29c9f9.jpg new file mode 100644 index 00000000..b61cf113 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-323_e1d4479804093cafc46522a94d102ac21f508425d911f92d3a84f4b16e29c9f9.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-324_3ad19cbeb0cc1f80a00375c30dadabcde6a5750a6e65a6e4e0c06aa8ad12305f.jpg b/.raw/MidasFEAAnalysisManual/images/page-324_3ad19cbeb0cc1f80a00375c30dadabcde6a5750a6e65a6e4e0c06aa8ad12305f.jpg new file mode 100644 index 00000000..028956e6 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-324_3ad19cbeb0cc1f80a00375c30dadabcde6a5750a6e65a6e4e0c06aa8ad12305f.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-325_20065987a8022657216048ec22698db0ce0b7fc0633b96e5125646dc58ba4a58.jpg b/.raw/MidasFEAAnalysisManual/images/page-325_20065987a8022657216048ec22698db0ce0b7fc0633b96e5125646dc58ba4a58.jpg new file mode 100644 index 00000000..5a79385b Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-325_20065987a8022657216048ec22698db0ce0b7fc0633b96e5125646dc58ba4a58.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-325_3399486968fd39943e2a59a3c933c02dde63edd70c3ab2cd38f97d3d42c5aff4.jpg b/.raw/MidasFEAAnalysisManual/images/page-325_3399486968fd39943e2a59a3c933c02dde63edd70c3ab2cd38f97d3d42c5aff4.jpg new file mode 100644 index 00000000..50a30ba6 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-325_3399486968fd39943e2a59a3c933c02dde63edd70c3ab2cd38f97d3d42c5aff4.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-325_4e000e66b2f2a73645af18a7a7b871363645035edd71266ae2cd8dc7ed4a682e.jpg b/.raw/MidasFEAAnalysisManual/images/page-325_4e000e66b2f2a73645af18a7a7b871363645035edd71266ae2cd8dc7ed4a682e.jpg new file mode 100644 index 00000000..45d275cf Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-325_4e000e66b2f2a73645af18a7a7b871363645035edd71266ae2cd8dc7ed4a682e.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-325_615a52c7258f02350e5613b1f6687c8df39614e25af274f52a9f4c1dcba27e7e.jpg b/.raw/MidasFEAAnalysisManual/images/page-325_615a52c7258f02350e5613b1f6687c8df39614e25af274f52a9f4c1dcba27e7e.jpg new file mode 100644 index 00000000..26341d59 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-325_615a52c7258f02350e5613b1f6687c8df39614e25af274f52a9f4c1dcba27e7e.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-325_63e7a432fd74fe6e6a74145ef227110549ae12bdaa463a7282e403f2f83fcf9f.jpg b/.raw/MidasFEAAnalysisManual/images/page-325_63e7a432fd74fe6e6a74145ef227110549ae12bdaa463a7282e403f2f83fcf9f.jpg new file mode 100644 index 00000000..1123607b Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-325_63e7a432fd74fe6e6a74145ef227110549ae12bdaa463a7282e403f2f83fcf9f.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-325_c5d624769abe920806e661f16b0029ca2c1df8a6a0caf95b41e14b32803657e8.jpg b/.raw/MidasFEAAnalysisManual/images/page-325_c5d624769abe920806e661f16b0029ca2c1df8a6a0caf95b41e14b32803657e8.jpg new file mode 100644 index 00000000..585bb77c Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-325_c5d624769abe920806e661f16b0029ca2c1df8a6a0caf95b41e14b32803657e8.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-325_d212eea82666c3bbe19550a4a92f5db0597c9ad659e11f45ac19f756d3732d11.jpg b/.raw/MidasFEAAnalysisManual/images/page-325_d212eea82666c3bbe19550a4a92f5db0597c9ad659e11f45ac19f756d3732d11.jpg new file mode 100644 index 00000000..c2316e70 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-325_d212eea82666c3bbe19550a4a92f5db0597c9ad659e11f45ac19f756d3732d11.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-326_07b5ca144581aaaad3b37a2e99b5254ae728314dee6cb6856c9e0b801e025b86.jpg b/.raw/MidasFEAAnalysisManual/images/page-326_07b5ca144581aaaad3b37a2e99b5254ae728314dee6cb6856c9e0b801e025b86.jpg new file mode 100644 index 00000000..1830b12b Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-326_07b5ca144581aaaad3b37a2e99b5254ae728314dee6cb6856c9e0b801e025b86.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-326_0ca2a6b7659a5255620cb8c843ae9af5eff3d161bb5f666168fb7a34acde67fd.jpg b/.raw/MidasFEAAnalysisManual/images/page-326_0ca2a6b7659a5255620cb8c843ae9af5eff3d161bb5f666168fb7a34acde67fd.jpg new file mode 100644 index 00000000..4644fb3f Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-326_0ca2a6b7659a5255620cb8c843ae9af5eff3d161bb5f666168fb7a34acde67fd.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-326_e0970ea76f76092a9481f85bdb1c952e4f211fe4500b9a53961392beb23c600b.jpg b/.raw/MidasFEAAnalysisManual/images/page-326_e0970ea76f76092a9481f85bdb1c952e4f211fe4500b9a53961392beb23c600b.jpg new file mode 100644 index 00000000..b13735bb Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-326_e0970ea76f76092a9481f85bdb1c952e4f211fe4500b9a53961392beb23c600b.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-327_5075a6f63f2ca23970e76d1e48c05df22e95b0d2e69d3194b0965fc21ba79ce5.jpg b/.raw/MidasFEAAnalysisManual/images/page-327_5075a6f63f2ca23970e76d1e48c05df22e95b0d2e69d3194b0965fc21ba79ce5.jpg new file mode 100644 index 00000000..4855bb95 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-327_5075a6f63f2ca23970e76d1e48c05df22e95b0d2e69d3194b0965fc21ba79ce5.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-329_866770365b45ca21d755fc3d05c45494ef039431856e41304a8e0e54fc95f5da.jpg b/.raw/MidasFEAAnalysisManual/images/page-329_866770365b45ca21d755fc3d05c45494ef039431856e41304a8e0e54fc95f5da.jpg new file mode 100644 index 00000000..2b430417 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-329_866770365b45ca21d755fc3d05c45494ef039431856e41304a8e0e54fc95f5da.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-330_355a625c023af1db0932e82f21a441147474d0bd00484e804148eb8a5f90bdfb.jpg b/.raw/MidasFEAAnalysisManual/images/page-330_355a625c023af1db0932e82f21a441147474d0bd00484e804148eb8a5f90bdfb.jpg new file mode 100644 index 00000000..733405d2 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-330_355a625c023af1db0932e82f21a441147474d0bd00484e804148eb8a5f90bdfb.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-330_f78a2765e1fd7e3f2e49acce44f32a795b83ea758f7c2c93addd8c29097fac23.jpg b/.raw/MidasFEAAnalysisManual/images/page-330_f78a2765e1fd7e3f2e49acce44f32a795b83ea758f7c2c93addd8c29097fac23.jpg new file mode 100644 index 00000000..dd508d42 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-330_f78a2765e1fd7e3f2e49acce44f32a795b83ea758f7c2c93addd8c29097fac23.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-331_d5b35f66a5014e6a5137e70a04198f6eb71fbef7ae690b7b3f1610713c916d12.jpg b/.raw/MidasFEAAnalysisManual/images/page-331_d5b35f66a5014e6a5137e70a04198f6eb71fbef7ae690b7b3f1610713c916d12.jpg new file mode 100644 index 00000000..d04e5092 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-331_d5b35f66a5014e6a5137e70a04198f6eb71fbef7ae690b7b3f1610713c916d12.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-331_ec05d4926089b233bb84eb0daa5d2cdad65003f63785d953bd0e95a60639dcb0.jpg b/.raw/MidasFEAAnalysisManual/images/page-331_ec05d4926089b233bb84eb0daa5d2cdad65003f63785d953bd0e95a60639dcb0.jpg new file mode 100644 index 00000000..7f5414d8 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-331_ec05d4926089b233bb84eb0daa5d2cdad65003f63785d953bd0e95a60639dcb0.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-333_af07284671837207515cae06ea9a727a351c33981652cb9e6dbf67dc86c41d54.jpg b/.raw/MidasFEAAnalysisManual/images/page-333_af07284671837207515cae06ea9a727a351c33981652cb9e6dbf67dc86c41d54.jpg new file mode 100644 index 00000000..90cd567d Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-333_af07284671837207515cae06ea9a727a351c33981652cb9e6dbf67dc86c41d54.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-333_dbc9ba66ec9b0c416dae026d9ba83b1642e2d870bcb0ff42bffad48281b3f41b.jpg b/.raw/MidasFEAAnalysisManual/images/page-333_dbc9ba66ec9b0c416dae026d9ba83b1642e2d870bcb0ff42bffad48281b3f41b.jpg new file mode 100644 index 00000000..25c85d4c Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-333_dbc9ba66ec9b0c416dae026d9ba83b1642e2d870bcb0ff42bffad48281b3f41b.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-334_d8048b33c3afeac3239ac6707a3987797d7092dbb306a0f3ca50d4ffdfa137ea.jpg b/.raw/MidasFEAAnalysisManual/images/page-334_d8048b33c3afeac3239ac6707a3987797d7092dbb306a0f3ca50d4ffdfa137ea.jpg new file mode 100644 index 00000000..7c8649e6 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-334_d8048b33c3afeac3239ac6707a3987797d7092dbb306a0f3ca50d4ffdfa137ea.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-334_f26d41136a699851cc2e279b36556decaee3e84a1595a090084a04cb189f8099.jpg b/.raw/MidasFEAAnalysisManual/images/page-334_f26d41136a699851cc2e279b36556decaee3e84a1595a090084a04cb189f8099.jpg new file mode 100644 index 00000000..1c7f6fea Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-334_f26d41136a699851cc2e279b36556decaee3e84a1595a090084a04cb189f8099.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-335_3b00f74d5b7d9d9f0062dbacb10f3fa5af9f4d8d0b4e1764053f33e392e20c24.jpg b/.raw/MidasFEAAnalysisManual/images/page-335_3b00f74d5b7d9d9f0062dbacb10f3fa5af9f4d8d0b4e1764053f33e392e20c24.jpg new file mode 100644 index 00000000..50becaa5 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-335_3b00f74d5b7d9d9f0062dbacb10f3fa5af9f4d8d0b4e1764053f33e392e20c24.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-335_42b24dca799115a6ecc69f0c5271c0d7a90fcaad7ab5887086dd976e50d7330f.jpg b/.raw/MidasFEAAnalysisManual/images/page-335_42b24dca799115a6ecc69f0c5271c0d7a90fcaad7ab5887086dd976e50d7330f.jpg new file mode 100644 index 00000000..e619e3d9 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-335_42b24dca799115a6ecc69f0c5271c0d7a90fcaad7ab5887086dd976e50d7330f.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-335_847739a4c48c283a2ee95cccbe827b5f0c4f6e274c51829b9678e0e60e7b0ca3.jpg b/.raw/MidasFEAAnalysisManual/images/page-335_847739a4c48c283a2ee95cccbe827b5f0c4f6e274c51829b9678e0e60e7b0ca3.jpg new file mode 100644 index 00000000..d55c4556 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-335_847739a4c48c283a2ee95cccbe827b5f0c4f6e274c51829b9678e0e60e7b0ca3.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-335_88148ae40166979e96d7ff033771359ab98cb6d741a10fc74c94d6714a0ff509.jpg b/.raw/MidasFEAAnalysisManual/images/page-335_88148ae40166979e96d7ff033771359ab98cb6d741a10fc74c94d6714a0ff509.jpg new file mode 100644 index 00000000..c8114eab Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-335_88148ae40166979e96d7ff033771359ab98cb6d741a10fc74c94d6714a0ff509.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-335_f46511dc3108930a72b2989b6ff0590ed5123527d7fb3524d7edad146e2b1ca7.jpg b/.raw/MidasFEAAnalysisManual/images/page-335_f46511dc3108930a72b2989b6ff0590ed5123527d7fb3524d7edad146e2b1ca7.jpg new file mode 100644 index 00000000..e61327ce Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-335_f46511dc3108930a72b2989b6ff0590ed5123527d7fb3524d7edad146e2b1ca7.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-336_45f064d22f5ebedd3e7707867a077afe015776a79f5144ad43c3d3d9aeb9addc.jpg b/.raw/MidasFEAAnalysisManual/images/page-336_45f064d22f5ebedd3e7707867a077afe015776a79f5144ad43c3d3d9aeb9addc.jpg new file mode 100644 index 00000000..ef3aa87c Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-336_45f064d22f5ebedd3e7707867a077afe015776a79f5144ad43c3d3d9aeb9addc.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-336_6f56290e19ae6881537e0396f9e7de39fb80d1883106a03cc4c89344a2da9695.jpg b/.raw/MidasFEAAnalysisManual/images/page-336_6f56290e19ae6881537e0396f9e7de39fb80d1883106a03cc4c89344a2da9695.jpg new file mode 100644 index 00000000..6e7e62e2 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-336_6f56290e19ae6881537e0396f9e7de39fb80d1883106a03cc4c89344a2da9695.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-337_0d8d5b7e8a66eb5932010e0eb836385d51ed7ba717d3fa1dcebaf21868e9b721.jpg b/.raw/MidasFEAAnalysisManual/images/page-337_0d8d5b7e8a66eb5932010e0eb836385d51ed7ba717d3fa1dcebaf21868e9b721.jpg new file mode 100644 index 00000000..d08bd8f8 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-337_0d8d5b7e8a66eb5932010e0eb836385d51ed7ba717d3fa1dcebaf21868e9b721.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-337_1e450d5f22a9df3376d3edbd712167ccff5d9940dd3aead3709ac21b03a196e8.jpg b/.raw/MidasFEAAnalysisManual/images/page-337_1e450d5f22a9df3376d3edbd712167ccff5d9940dd3aead3709ac21b03a196e8.jpg new file mode 100644 index 00000000..dc9aecc0 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-337_1e450d5f22a9df3376d3edbd712167ccff5d9940dd3aead3709ac21b03a196e8.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-337_5019a032ee9225c49a340450fe71e92287e75acb143a1bed244ffee349fc6fb0.jpg b/.raw/MidasFEAAnalysisManual/images/page-337_5019a032ee9225c49a340450fe71e92287e75acb143a1bed244ffee349fc6fb0.jpg new file mode 100644 index 00000000..7e806a90 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-337_5019a032ee9225c49a340450fe71e92287e75acb143a1bed244ffee349fc6fb0.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-337_50563e255f09d253fa1efb14a85597a2a897f67d5d83eb685ff5bbda0d6e67ab.jpg b/.raw/MidasFEAAnalysisManual/images/page-337_50563e255f09d253fa1efb14a85597a2a897f67d5d83eb685ff5bbda0d6e67ab.jpg new file mode 100644 index 00000000..c6865952 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-337_50563e255f09d253fa1efb14a85597a2a897f67d5d83eb685ff5bbda0d6e67ab.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-337_90267ea0ec7dd77b77e9e7febf2ebb75daedc946059f7a21fdb9b399debeeafb.jpg b/.raw/MidasFEAAnalysisManual/images/page-337_90267ea0ec7dd77b77e9e7febf2ebb75daedc946059f7a21fdb9b399debeeafb.jpg new file mode 100644 index 00000000..c942f3e7 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-337_90267ea0ec7dd77b77e9e7febf2ebb75daedc946059f7a21fdb9b399debeeafb.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-337_95ae705196bb57f733b06e972d3eb21a61e604d4aa4dc508d6c67b272d212af8.jpg b/.raw/MidasFEAAnalysisManual/images/page-337_95ae705196bb57f733b06e972d3eb21a61e604d4aa4dc508d6c67b272d212af8.jpg new file mode 100644 index 00000000..3accc888 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-337_95ae705196bb57f733b06e972d3eb21a61e604d4aa4dc508d6c67b272d212af8.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-337_c5ba36dd5d309961a3801c6c30f29301d164a38b7bd610ba8f9e86367baf6575.jpg b/.raw/MidasFEAAnalysisManual/images/page-337_c5ba36dd5d309961a3801c6c30f29301d164a38b7bd610ba8f9e86367baf6575.jpg new file mode 100644 index 00000000..881164ae Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-337_c5ba36dd5d309961a3801c6c30f29301d164a38b7bd610ba8f9e86367baf6575.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-338_1dd41c95dc1275ee736c6129cbefc3d3ced89db012e7ac138fbb37d896b6fdaa.jpg b/.raw/MidasFEAAnalysisManual/images/page-338_1dd41c95dc1275ee736c6129cbefc3d3ced89db012e7ac138fbb37d896b6fdaa.jpg new file mode 100644 index 00000000..398aa8d5 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-338_1dd41c95dc1275ee736c6129cbefc3d3ced89db012e7ac138fbb37d896b6fdaa.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-338_fd9b517a93fb015cfd11d998f2d786d16bc2a41e9212e597a315b0c3de6ad242.jpg b/.raw/MidasFEAAnalysisManual/images/page-338_fd9b517a93fb015cfd11d998f2d786d16bc2a41e9212e597a315b0c3de6ad242.jpg new file mode 100644 index 00000000..0eceb833 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-338_fd9b517a93fb015cfd11d998f2d786d16bc2a41e9212e597a315b0c3de6ad242.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-340_932a9046bd3fd1e062b95cf9663ed15e10d4dc7f1ee8cbd253945760f82884f3.jpg b/.raw/MidasFEAAnalysisManual/images/page-340_932a9046bd3fd1e062b95cf9663ed15e10d4dc7f1ee8cbd253945760f82884f3.jpg new file mode 100644 index 00000000..d9626214 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-340_932a9046bd3fd1e062b95cf9663ed15e10d4dc7f1ee8cbd253945760f82884f3.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-341_723abcd3a0a0da9b9613c2c3ffd76aa78f433b4340156e144f9b26d8a20a9535.jpg b/.raw/MidasFEAAnalysisManual/images/page-341_723abcd3a0a0da9b9613c2c3ffd76aa78f433b4340156e144f9b26d8a20a9535.jpg new file mode 100644 index 00000000..91a392d7 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-341_723abcd3a0a0da9b9613c2c3ffd76aa78f433b4340156e144f9b26d8a20a9535.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-341_fad5fae90f6c2ba6aeaca69cbbc4c6edb1a91fe1fdd5d167c0a0ec4d286c5a93.jpg b/.raw/MidasFEAAnalysisManual/images/page-341_fad5fae90f6c2ba6aeaca69cbbc4c6edb1a91fe1fdd5d167c0a0ec4d286c5a93.jpg new file mode 100644 index 00000000..5c33eebb Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-341_fad5fae90f6c2ba6aeaca69cbbc4c6edb1a91fe1fdd5d167c0a0ec4d286c5a93.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-342_16b990ca614aae3714820b104950ed7ebd1e748a78d652023a59b143bd6f4f03.jpg b/.raw/MidasFEAAnalysisManual/images/page-342_16b990ca614aae3714820b104950ed7ebd1e748a78d652023a59b143bd6f4f03.jpg new file mode 100644 index 00000000..dc0deabe Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-342_16b990ca614aae3714820b104950ed7ebd1e748a78d652023a59b143bd6f4f03.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-342_1cdba34a338f67f42604abdf04cc8d294d362dde366373e3c5b462941e88b7a0.jpg b/.raw/MidasFEAAnalysisManual/images/page-342_1cdba34a338f67f42604abdf04cc8d294d362dde366373e3c5b462941e88b7a0.jpg new file mode 100644 index 00000000..a2772285 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-342_1cdba34a338f67f42604abdf04cc8d294d362dde366373e3c5b462941e88b7a0.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-342_1fe50e055e6e4de5ffa39c89dcd007a665e63e9b0b0204ce337888a0261e0c80.jpg b/.raw/MidasFEAAnalysisManual/images/page-342_1fe50e055e6e4de5ffa39c89dcd007a665e63e9b0b0204ce337888a0261e0c80.jpg new file mode 100644 index 00000000..6145d6ca Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-342_1fe50e055e6e4de5ffa39c89dcd007a665e63e9b0b0204ce337888a0261e0c80.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-342_3f67936e8c1190100c42d2642ad79d13bc58b0117e452d1ba80ce0b63abb791c.jpg b/.raw/MidasFEAAnalysisManual/images/page-342_3f67936e8c1190100c42d2642ad79d13bc58b0117e452d1ba80ce0b63abb791c.jpg new file mode 100644 index 00000000..0dcca586 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-342_3f67936e8c1190100c42d2642ad79d13bc58b0117e452d1ba80ce0b63abb791c.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-342_5ced3218d39b302e775cbb7081ca82f52a698ab42c38590e3b8f98bc48c11bf2.jpg b/.raw/MidasFEAAnalysisManual/images/page-342_5ced3218d39b302e775cbb7081ca82f52a698ab42c38590e3b8f98bc48c11bf2.jpg new file mode 100644 index 00000000..433a85b6 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-342_5ced3218d39b302e775cbb7081ca82f52a698ab42c38590e3b8f98bc48c11bf2.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-342_8544612cba5c1a44e1b55ece837ebce178ccd25983083c8cc2430e3d2d76c681.jpg b/.raw/MidasFEAAnalysisManual/images/page-342_8544612cba5c1a44e1b55ece837ebce178ccd25983083c8cc2430e3d2d76c681.jpg new file mode 100644 index 00000000..29b55e11 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-342_8544612cba5c1a44e1b55ece837ebce178ccd25983083c8cc2430e3d2d76c681.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-344_0cd1f2eced78a23c6c46509ee561146c6bcb363bd0335be93a67ba24b1c3472f.jpg b/.raw/MidasFEAAnalysisManual/images/page-344_0cd1f2eced78a23c6c46509ee561146c6bcb363bd0335be93a67ba24b1c3472f.jpg new file mode 100644 index 00000000..4b832dc0 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-344_0cd1f2eced78a23c6c46509ee561146c6bcb363bd0335be93a67ba24b1c3472f.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-345_46a45693ff48d5e040b10a41bea7537428787770ebf9356d4c9575674a2fb9fe.jpg b/.raw/MidasFEAAnalysisManual/images/page-345_46a45693ff48d5e040b10a41bea7537428787770ebf9356d4c9575674a2fb9fe.jpg new file mode 100644 index 00000000..96955ec1 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-345_46a45693ff48d5e040b10a41bea7537428787770ebf9356d4c9575674a2fb9fe.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-346_9bebe6780ec4a8d4e8fbbff3281eccb2106135d3bfca6432680af1e7d9f604c6.jpg b/.raw/MidasFEAAnalysisManual/images/page-346_9bebe6780ec4a8d4e8fbbff3281eccb2106135d3bfca6432680af1e7d9f604c6.jpg new file mode 100644 index 00000000..7d88bd2b Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-346_9bebe6780ec4a8d4e8fbbff3281eccb2106135d3bfca6432680af1e7d9f604c6.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-347_1450762538d55c5f722981c48c7f8f8485f2de3a54a3e83d205f64cadb509783.jpg b/.raw/MidasFEAAnalysisManual/images/page-347_1450762538d55c5f722981c48c7f8f8485f2de3a54a3e83d205f64cadb509783.jpg new file mode 100644 index 00000000..0f41c488 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-347_1450762538d55c5f722981c48c7f8f8485f2de3a54a3e83d205f64cadb509783.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-347_75921e25308a0b650c7a275a018cf7df4809bbe0adaafb796915c37cc5a94fed.jpg b/.raw/MidasFEAAnalysisManual/images/page-347_75921e25308a0b650c7a275a018cf7df4809bbe0adaafb796915c37cc5a94fed.jpg new file mode 100644 index 00000000..b4faa3ae Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-347_75921e25308a0b650c7a275a018cf7df4809bbe0adaafb796915c37cc5a94fed.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-347_dd583585a3bbd2171c964f94b03d704c51928e0bc6b3de40a41f5731891638ce.jpg b/.raw/MidasFEAAnalysisManual/images/page-347_dd583585a3bbd2171c964f94b03d704c51928e0bc6b3de40a41f5731891638ce.jpg new file mode 100644 index 00000000..f37aae76 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-347_dd583585a3bbd2171c964f94b03d704c51928e0bc6b3de40a41f5731891638ce.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-348_2e61d37c9091f9cf8298396fdac24724aec53e31097e9f926a88f60632a320bc.jpg b/.raw/MidasFEAAnalysisManual/images/page-348_2e61d37c9091f9cf8298396fdac24724aec53e31097e9f926a88f60632a320bc.jpg new file mode 100644 index 00000000..d585ec71 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-348_2e61d37c9091f9cf8298396fdac24724aec53e31097e9f926a88f60632a320bc.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-348_3c103a1b42c959bbc871dcdb60db8e6713c517c45e8ff4513bc7e59a48297a44.jpg b/.raw/MidasFEAAnalysisManual/images/page-348_3c103a1b42c959bbc871dcdb60db8e6713c517c45e8ff4513bc7e59a48297a44.jpg new file mode 100644 index 00000000..8425626e Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-348_3c103a1b42c959bbc871dcdb60db8e6713c517c45e8ff4513bc7e59a48297a44.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-348_62b4cdab1a3f2fc93e742ee2aa4f61db16969544b22a5a03ebfe0a600c5e8e59.jpg b/.raw/MidasFEAAnalysisManual/images/page-348_62b4cdab1a3f2fc93e742ee2aa4f61db16969544b22a5a03ebfe0a600c5e8e59.jpg new file mode 100644 index 00000000..977e97bd Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-348_62b4cdab1a3f2fc93e742ee2aa4f61db16969544b22a5a03ebfe0a600c5e8e59.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-348_98b6476469b8ea23467be0aab4587ffcf2b869a56a10e5b772d7ca8f117baa59.jpg b/.raw/MidasFEAAnalysisManual/images/page-348_98b6476469b8ea23467be0aab4587ffcf2b869a56a10e5b772d7ca8f117baa59.jpg new file mode 100644 index 00000000..19b2032c Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-348_98b6476469b8ea23467be0aab4587ffcf2b869a56a10e5b772d7ca8f117baa59.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-348_bd052206ba36160ee24ea2a172f6a9686e5ec099207d6c08a46de928350c613e.jpg b/.raw/MidasFEAAnalysisManual/images/page-348_bd052206ba36160ee24ea2a172f6a9686e5ec099207d6c08a46de928350c613e.jpg new file mode 100644 index 00000000..507d87f3 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-348_bd052206ba36160ee24ea2a172f6a9686e5ec099207d6c08a46de928350c613e.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-348_ca4a5fe05aa37b10190c1376c3596efe9981717ada03eb988d5beb5cdb8b6b25.jpg b/.raw/MidasFEAAnalysisManual/images/page-348_ca4a5fe05aa37b10190c1376c3596efe9981717ada03eb988d5beb5cdb8b6b25.jpg new file mode 100644 index 00000000..b2585006 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-348_ca4a5fe05aa37b10190c1376c3596efe9981717ada03eb988d5beb5cdb8b6b25.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-349_1db592eccae0605274b65ff6480a54c9090a81629099eeb105bd09ad41789789.jpg b/.raw/MidasFEAAnalysisManual/images/page-349_1db592eccae0605274b65ff6480a54c9090a81629099eeb105bd09ad41789789.jpg new file mode 100644 index 00000000..e82fb11b Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-349_1db592eccae0605274b65ff6480a54c9090a81629099eeb105bd09ad41789789.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-349_2d02982bc735ff540ba5b37b41f6c2dc22e31846c5d557ca82b814ba38241533.jpg b/.raw/MidasFEAAnalysisManual/images/page-349_2d02982bc735ff540ba5b37b41f6c2dc22e31846c5d557ca82b814ba38241533.jpg new file mode 100644 index 00000000..a9363799 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-349_2d02982bc735ff540ba5b37b41f6c2dc22e31846c5d557ca82b814ba38241533.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-349_a9a076d153806e974a669423efcdccc8accd5df5832d373a517f757f8f625b40.jpg b/.raw/MidasFEAAnalysisManual/images/page-349_a9a076d153806e974a669423efcdccc8accd5df5832d373a517f757f8f625b40.jpg new file mode 100644 index 00000000..2ad391b4 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-349_a9a076d153806e974a669423efcdccc8accd5df5832d373a517f757f8f625b40.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-349_d2dbc4c9f0396b7380c9da6d0255f57a1fa6b5aa4656ff517537b7a09c71f2e5.jpg b/.raw/MidasFEAAnalysisManual/images/page-349_d2dbc4c9f0396b7380c9da6d0255f57a1fa6b5aa4656ff517537b7a09c71f2e5.jpg new file mode 100644 index 00000000..8e1a73ed Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-349_d2dbc4c9f0396b7380c9da6d0255f57a1fa6b5aa4656ff517537b7a09c71f2e5.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-349_d81a0bfbef05537096ebd0a56ef5af7030ea1b78f4d3366a93c693b73bea4836.jpg b/.raw/MidasFEAAnalysisManual/images/page-349_d81a0bfbef05537096ebd0a56ef5af7030ea1b78f4d3366a93c693b73bea4836.jpg new file mode 100644 index 00000000..bc365be5 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-349_d81a0bfbef05537096ebd0a56ef5af7030ea1b78f4d3366a93c693b73bea4836.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-349_e127ae8883161c6833c619d5efdaac2c7ef1014c69a816f8d1df36aecf8efa26.jpg b/.raw/MidasFEAAnalysisManual/images/page-349_e127ae8883161c6833c619d5efdaac2c7ef1014c69a816f8d1df36aecf8efa26.jpg new file mode 100644 index 00000000..cfb428d2 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-349_e127ae8883161c6833c619d5efdaac2c7ef1014c69a816f8d1df36aecf8efa26.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-350_38643122b3a01e26d88ba109c0bfe98fb27c0a538b79292d92927725824e57ef.jpg b/.raw/MidasFEAAnalysisManual/images/page-350_38643122b3a01e26d88ba109c0bfe98fb27c0a538b79292d92927725824e57ef.jpg new file mode 100644 index 00000000..e0549754 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-350_38643122b3a01e26d88ba109c0bfe98fb27c0a538b79292d92927725824e57ef.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-350_699044fa685cf1906636f3b8a7196b3cff74544191174833bf7c91b967831e7e.jpg b/.raw/MidasFEAAnalysisManual/images/page-350_699044fa685cf1906636f3b8a7196b3cff74544191174833bf7c91b967831e7e.jpg new file mode 100644 index 00000000..4f4fde90 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-350_699044fa685cf1906636f3b8a7196b3cff74544191174833bf7c91b967831e7e.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-350_72fc9c5f44d2f0ecc88f950ce2f1f9e56d8f717ec592d9c8de6af73b299d8342.jpg b/.raw/MidasFEAAnalysisManual/images/page-350_72fc9c5f44d2f0ecc88f950ce2f1f9e56d8f717ec592d9c8de6af73b299d8342.jpg new file mode 100644 index 00000000..7cc8e24b Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-350_72fc9c5f44d2f0ecc88f950ce2f1f9e56d8f717ec592d9c8de6af73b299d8342.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-350_a2e7778e11f2aee4707239fff35b227b15c0f474396dcb6295e5e96387788031.jpg b/.raw/MidasFEAAnalysisManual/images/page-350_a2e7778e11f2aee4707239fff35b227b15c0f474396dcb6295e5e96387788031.jpg new file mode 100644 index 00000000..7393ff33 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-350_a2e7778e11f2aee4707239fff35b227b15c0f474396dcb6295e5e96387788031.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-350_ab7f44856b35e558ba48aa876e54358adfa02b3f70aea1e42c51a31f833ed5df.jpg b/.raw/MidasFEAAnalysisManual/images/page-350_ab7f44856b35e558ba48aa876e54358adfa02b3f70aea1e42c51a31f833ed5df.jpg new file mode 100644 index 00000000..78e116a5 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-350_ab7f44856b35e558ba48aa876e54358adfa02b3f70aea1e42c51a31f833ed5df.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-350_bae5b3961dca967e31df0444b1fb88de737d26cecc01869333b966b4bb478244.jpg b/.raw/MidasFEAAnalysisManual/images/page-350_bae5b3961dca967e31df0444b1fb88de737d26cecc01869333b966b4bb478244.jpg new file mode 100644 index 00000000..960dd820 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-350_bae5b3961dca967e31df0444b1fb88de737d26cecc01869333b966b4bb478244.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-350_d24e27892e0dacae0e412c1da9f37fdd790fb9f48a8e1e273214b1e1048d2356.jpg b/.raw/MidasFEAAnalysisManual/images/page-350_d24e27892e0dacae0e412c1da9f37fdd790fb9f48a8e1e273214b1e1048d2356.jpg new file mode 100644 index 00000000..c3206d3d Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-350_d24e27892e0dacae0e412c1da9f37fdd790fb9f48a8e1e273214b1e1048d2356.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-351_28459b408026fc8e2fc23e723b8b5b036edc460aebd69103d720a90882b35a8d.jpg b/.raw/MidasFEAAnalysisManual/images/page-351_28459b408026fc8e2fc23e723b8b5b036edc460aebd69103d720a90882b35a8d.jpg new file mode 100644 index 00000000..704524f2 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-351_28459b408026fc8e2fc23e723b8b5b036edc460aebd69103d720a90882b35a8d.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-351_60dca97555c04ef3a33867ddeb6d6d77b2763bab485e50b6f58c6535a53c0cd0.jpg b/.raw/MidasFEAAnalysisManual/images/page-351_60dca97555c04ef3a33867ddeb6d6d77b2763bab485e50b6f58c6535a53c0cd0.jpg new file mode 100644 index 00000000..d2daa99f Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-351_60dca97555c04ef3a33867ddeb6d6d77b2763bab485e50b6f58c6535a53c0cd0.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-351_7658c2040cdf6ef1ed9009b30e6b471f37cca9c520d11a5b4cd4415c4fd1a4f6.jpg b/.raw/MidasFEAAnalysisManual/images/page-351_7658c2040cdf6ef1ed9009b30e6b471f37cca9c520d11a5b4cd4415c4fd1a4f6.jpg new file mode 100644 index 00000000..0e7cc074 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-351_7658c2040cdf6ef1ed9009b30e6b471f37cca9c520d11a5b4cd4415c4fd1a4f6.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-352_2a3023c9d8cfa9804a914a7ae381bd43ce0e41c1faf78e2db750a0a6284a138a.jpg b/.raw/MidasFEAAnalysisManual/images/page-352_2a3023c9d8cfa9804a914a7ae381bd43ce0e41c1faf78e2db750a0a6284a138a.jpg new file mode 100644 index 00000000..0d327c03 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-352_2a3023c9d8cfa9804a914a7ae381bd43ce0e41c1faf78e2db750a0a6284a138a.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-352_856b2ba4900a1c2cea7e81fb41975a0f298b0809ffafa2b1f8b0ecba9ea57f24.jpg b/.raw/MidasFEAAnalysisManual/images/page-352_856b2ba4900a1c2cea7e81fb41975a0f298b0809ffafa2b1f8b0ecba9ea57f24.jpg new file mode 100644 index 00000000..e8feda3d Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-352_856b2ba4900a1c2cea7e81fb41975a0f298b0809ffafa2b1f8b0ecba9ea57f24.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-355_a406731126e1eed79c71f8533202f1e4edba26b5e84783af5cb6af6acc191c80.jpg b/.raw/MidasFEAAnalysisManual/images/page-355_a406731126e1eed79c71f8533202f1e4edba26b5e84783af5cb6af6acc191c80.jpg new file mode 100644 index 00000000..98d4b679 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-355_a406731126e1eed79c71f8533202f1e4edba26b5e84783af5cb6af6acc191c80.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-356_5dee2ad7d127e18ee67fa96a8080619fc4826d2f666da1b23e6f1223a2ea252b.jpg b/.raw/MidasFEAAnalysisManual/images/page-356_5dee2ad7d127e18ee67fa96a8080619fc4826d2f666da1b23e6f1223a2ea252b.jpg new file mode 100644 index 00000000..15a47df1 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-356_5dee2ad7d127e18ee67fa96a8080619fc4826d2f666da1b23e6f1223a2ea252b.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-357_5d0c5f206885ae810609680b7dc77533db8b024f866a40a56e15b9aeb5e793c7.jpg b/.raw/MidasFEAAnalysisManual/images/page-357_5d0c5f206885ae810609680b7dc77533db8b024f866a40a56e15b9aeb5e793c7.jpg new file mode 100644 index 00000000..f210cf60 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-357_5d0c5f206885ae810609680b7dc77533db8b024f866a40a56e15b9aeb5e793c7.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-357_853a85297b46dc0b2f4b3c217e076a3023e268876c0fc2c0fcc2f811a76ac2da.jpg b/.raw/MidasFEAAnalysisManual/images/page-357_853a85297b46dc0b2f4b3c217e076a3023e268876c0fc2c0fcc2f811a76ac2da.jpg new file mode 100644 index 00000000..062210b8 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-357_853a85297b46dc0b2f4b3c217e076a3023e268876c0fc2c0fcc2f811a76ac2da.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-357_a33c2c3d9efb1fe4826368eb9573018bc5451f1136a6b8c32898115d2950c837.jpg b/.raw/MidasFEAAnalysisManual/images/page-357_a33c2c3d9efb1fe4826368eb9573018bc5451f1136a6b8c32898115d2950c837.jpg new file mode 100644 index 00000000..9b19a2c3 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-357_a33c2c3d9efb1fe4826368eb9573018bc5451f1136a6b8c32898115d2950c837.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-357_af93246cbcc56c9f5070b705a0ce01c5871f321653be6d7823d43043364073ee.jpg b/.raw/MidasFEAAnalysisManual/images/page-357_af93246cbcc56c9f5070b705a0ce01c5871f321653be6d7823d43043364073ee.jpg new file mode 100644 index 00000000..7338dd99 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-357_af93246cbcc56c9f5070b705a0ce01c5871f321653be6d7823d43043364073ee.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-357_bf32e0972bb1dd0a76ad273470f80c9e73be8e2131e6c7b7fbb0570d6840ca2e.jpg b/.raw/MidasFEAAnalysisManual/images/page-357_bf32e0972bb1dd0a76ad273470f80c9e73be8e2131e6c7b7fbb0570d6840ca2e.jpg new file mode 100644 index 00000000..e51d7021 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-357_bf32e0972bb1dd0a76ad273470f80c9e73be8e2131e6c7b7fbb0570d6840ca2e.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-358_0d053f49ef534d5258c9b73c7969f155a55090a77debe7ca9c3083c6fcb41f97.jpg b/.raw/MidasFEAAnalysisManual/images/page-358_0d053f49ef534d5258c9b73c7969f155a55090a77debe7ca9c3083c6fcb41f97.jpg new file mode 100644 index 00000000..c557c408 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-358_0d053f49ef534d5258c9b73c7969f155a55090a77debe7ca9c3083c6fcb41f97.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-358_5a1d06c767d19fd9e08edd69004bad0ddf9c604619d7cd71bc010c41b0235f7b.jpg b/.raw/MidasFEAAnalysisManual/images/page-358_5a1d06c767d19fd9e08edd69004bad0ddf9c604619d7cd71bc010c41b0235f7b.jpg new file mode 100644 index 00000000..ac4da0e7 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-358_5a1d06c767d19fd9e08edd69004bad0ddf9c604619d7cd71bc010c41b0235f7b.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-358_915921761e79c41e8326c25405c3b9d58c83b5b82674c3fb38ce1747b77b426e.jpg b/.raw/MidasFEAAnalysisManual/images/page-358_915921761e79c41e8326c25405c3b9d58c83b5b82674c3fb38ce1747b77b426e.jpg new file mode 100644 index 00000000..eb346ff6 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-358_915921761e79c41e8326c25405c3b9d58c83b5b82674c3fb38ce1747b77b426e.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-358_de969ca3eadd85c0bf6ac0e92c875edfbdc87b91d839e35372e7c03dc029f691.jpg b/.raw/MidasFEAAnalysisManual/images/page-358_de969ca3eadd85c0bf6ac0e92c875edfbdc87b91d839e35372e7c03dc029f691.jpg new file mode 100644 index 00000000..99f2d687 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-358_de969ca3eadd85c0bf6ac0e92c875edfbdc87b91d839e35372e7c03dc029f691.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-358_f5caafd0f25895ac3cd2e43443ab69be4a034c7e945e97df63dfe3602e1d9215.jpg b/.raw/MidasFEAAnalysisManual/images/page-358_f5caafd0f25895ac3cd2e43443ab69be4a034c7e945e97df63dfe3602e1d9215.jpg new file mode 100644 index 00000000..ed35c305 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-358_f5caafd0f25895ac3cd2e43443ab69be4a034c7e945e97df63dfe3602e1d9215.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-359_12729333db715c2e7b5aa5a90b302a306251bd195e0d1c5fb75cae5d01286f8f.jpg b/.raw/MidasFEAAnalysisManual/images/page-359_12729333db715c2e7b5aa5a90b302a306251bd195e0d1c5fb75cae5d01286f8f.jpg new file mode 100644 index 00000000..43b250a5 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-359_12729333db715c2e7b5aa5a90b302a306251bd195e0d1c5fb75cae5d01286f8f.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-359_6a338d0c26da11d7b403b0540596cea92b739955a16fc184db58245854293243.jpg b/.raw/MidasFEAAnalysisManual/images/page-359_6a338d0c26da11d7b403b0540596cea92b739955a16fc184db58245854293243.jpg new file mode 100644 index 00000000..2ced7a48 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-359_6a338d0c26da11d7b403b0540596cea92b739955a16fc184db58245854293243.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-359_7c207a64afa177815ae499a3346be26e8d730b48084c7c282ae183e72839a874.jpg b/.raw/MidasFEAAnalysisManual/images/page-359_7c207a64afa177815ae499a3346be26e8d730b48084c7c282ae183e72839a874.jpg new file mode 100644 index 00000000..17aba064 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-359_7c207a64afa177815ae499a3346be26e8d730b48084c7c282ae183e72839a874.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-359_82417a0a80ed3120c283a2a620d604cc18cf6a9cdb5d1daa0cb69c7ea4a5bb24.jpg b/.raw/MidasFEAAnalysisManual/images/page-359_82417a0a80ed3120c283a2a620d604cc18cf6a9cdb5d1daa0cb69c7ea4a5bb24.jpg new file mode 100644 index 00000000..a6e60fc1 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-359_82417a0a80ed3120c283a2a620d604cc18cf6a9cdb5d1daa0cb69c7ea4a5bb24.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-359_b779d9fd422b3207ae78574c9aefd65c65d78779cbc19848dd40e00fa6ff6701.jpg b/.raw/MidasFEAAnalysisManual/images/page-359_b779d9fd422b3207ae78574c9aefd65c65d78779cbc19848dd40e00fa6ff6701.jpg new file mode 100644 index 00000000..3aaf5a2f Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-359_b779d9fd422b3207ae78574c9aefd65c65d78779cbc19848dd40e00fa6ff6701.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-359_f71daee90ffc07d28095dd364e6580d9a04eaa50c844597d682daa428afd9e32.jpg b/.raw/MidasFEAAnalysisManual/images/page-359_f71daee90ffc07d28095dd364e6580d9a04eaa50c844597d682daa428afd9e32.jpg new file mode 100644 index 00000000..8411d9ef Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-359_f71daee90ffc07d28095dd364e6580d9a04eaa50c844597d682daa428afd9e32.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-360_071955228869fe89a2feb6119ef2793380dc793fdf1b7930128ea511c68b5beb.jpg b/.raw/MidasFEAAnalysisManual/images/page-360_071955228869fe89a2feb6119ef2793380dc793fdf1b7930128ea511c68b5beb.jpg new file mode 100644 index 00000000..df32906f Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-360_071955228869fe89a2feb6119ef2793380dc793fdf1b7930128ea511c68b5beb.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-360_1fd3b8516c55d085a1edcd9710784f01d33873c104fa080d5ac505d4e6d50808.jpg b/.raw/MidasFEAAnalysisManual/images/page-360_1fd3b8516c55d085a1edcd9710784f01d33873c104fa080d5ac505d4e6d50808.jpg new file mode 100644 index 00000000..e5f1c56b Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-360_1fd3b8516c55d085a1edcd9710784f01d33873c104fa080d5ac505d4e6d50808.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-360_8bee234d2c6cb3105420a5460b1dad20d97dc9ad8dab02aa83b52a6422b3b2b1.jpg b/.raw/MidasFEAAnalysisManual/images/page-360_8bee234d2c6cb3105420a5460b1dad20d97dc9ad8dab02aa83b52a6422b3b2b1.jpg new file mode 100644 index 00000000..4fbbd8ec Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-360_8bee234d2c6cb3105420a5460b1dad20d97dc9ad8dab02aa83b52a6422b3b2b1.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-360_b46ee0ea22a513f5d3ee46c0fa6be628a8eba761680c6161a75d0cadb2e35e98.jpg b/.raw/MidasFEAAnalysisManual/images/page-360_b46ee0ea22a513f5d3ee46c0fa6be628a8eba761680c6161a75d0cadb2e35e98.jpg new file mode 100644 index 00000000..1e877351 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-360_b46ee0ea22a513f5d3ee46c0fa6be628a8eba761680c6161a75d0cadb2e35e98.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-361_b7acf06336e37a9f8a5e6b682904b652b462ad4d00a1ce0f75a762be1851df97.jpg b/.raw/MidasFEAAnalysisManual/images/page-361_b7acf06336e37a9f8a5e6b682904b652b462ad4d00a1ce0f75a762be1851df97.jpg new file mode 100644 index 00000000..a0cf65b7 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-361_b7acf06336e37a9f8a5e6b682904b652b462ad4d00a1ce0f75a762be1851df97.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-362_9ec5f06597798a4d06d4adeeedbc5c8cb824b68c919adaab679d2f2db8ddc5f1.jpg b/.raw/MidasFEAAnalysisManual/images/page-362_9ec5f06597798a4d06d4adeeedbc5c8cb824b68c919adaab679d2f2db8ddc5f1.jpg new file mode 100644 index 00000000..714015f0 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-362_9ec5f06597798a4d06d4adeeedbc5c8cb824b68c919adaab679d2f2db8ddc5f1.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-362_cb2819f1dde23e5af8c268a80a84b46078a666916d6e150b5a32872a79280c00.jpg b/.raw/MidasFEAAnalysisManual/images/page-362_cb2819f1dde23e5af8c268a80a84b46078a666916d6e150b5a32872a79280c00.jpg new file mode 100644 index 00000000..ec388cfb Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-362_cb2819f1dde23e5af8c268a80a84b46078a666916d6e150b5a32872a79280c00.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-363_4b27400d7c6ddf448c5de77232825b0373c38e0e2ace921dbcff3c8ef3c8d9ed.jpg b/.raw/MidasFEAAnalysisManual/images/page-363_4b27400d7c6ddf448c5de77232825b0373c38e0e2ace921dbcff3c8ef3c8d9ed.jpg new file mode 100644 index 00000000..33272a1b Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-363_4b27400d7c6ddf448c5de77232825b0373c38e0e2ace921dbcff3c8ef3c8d9ed.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-363_8fd37442f77a7342210a9322bd384ce2ccaaa290d194c2cb1c092379d386f3e4.jpg b/.raw/MidasFEAAnalysisManual/images/page-363_8fd37442f77a7342210a9322bd384ce2ccaaa290d194c2cb1c092379d386f3e4.jpg new file mode 100644 index 00000000..be6ddecf Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-363_8fd37442f77a7342210a9322bd384ce2ccaaa290d194c2cb1c092379d386f3e4.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-363_99aa34398353704213410c18bb9b114ab2b4f11d0071745ee788e0ad57e7e58b.jpg b/.raw/MidasFEAAnalysisManual/images/page-363_99aa34398353704213410c18bb9b114ab2b4f11d0071745ee788e0ad57e7e58b.jpg new file mode 100644 index 00000000..04eeaa37 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-363_99aa34398353704213410c18bb9b114ab2b4f11d0071745ee788e0ad57e7e58b.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-363_dff4c0dd228a92ebbe70db0e78e7be7cdc6ac2a073f067d43235fdcdd18cf735.jpg b/.raw/MidasFEAAnalysisManual/images/page-363_dff4c0dd228a92ebbe70db0e78e7be7cdc6ac2a073f067d43235fdcdd18cf735.jpg new file mode 100644 index 00000000..0fd9a238 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-363_dff4c0dd228a92ebbe70db0e78e7be7cdc6ac2a073f067d43235fdcdd18cf735.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-364_90b1c0232241c1169e0de505a63d3917936317483e74d606fc722b78edda1f87.jpg b/.raw/MidasFEAAnalysisManual/images/page-364_90b1c0232241c1169e0de505a63d3917936317483e74d606fc722b78edda1f87.jpg new file mode 100644 index 00000000..c3e75647 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-364_90b1c0232241c1169e0de505a63d3917936317483e74d606fc722b78edda1f87.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-365_2151cb534eec83186770c6f7025ac8f727cf11cb295d4d8f0888221e23fa0354.jpg b/.raw/MidasFEAAnalysisManual/images/page-365_2151cb534eec83186770c6f7025ac8f727cf11cb295d4d8f0888221e23fa0354.jpg new file mode 100644 index 00000000..e0553100 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-365_2151cb534eec83186770c6f7025ac8f727cf11cb295d4d8f0888221e23fa0354.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-366_b00ef754b887d00ff6247caade26a7e2c9f4ee27399bf6aa4b55eb69353c7449.jpg b/.raw/MidasFEAAnalysisManual/images/page-366_b00ef754b887d00ff6247caade26a7e2c9f4ee27399bf6aa4b55eb69353c7449.jpg new file mode 100644 index 00000000..b00007e4 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-366_b00ef754b887d00ff6247caade26a7e2c9f4ee27399bf6aa4b55eb69353c7449.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-366_d2c28fe27574e31158106430a02f4a9d10b1bdcba1e694f166a4de4e8f5448bd.jpg b/.raw/MidasFEAAnalysisManual/images/page-366_d2c28fe27574e31158106430a02f4a9d10b1bdcba1e694f166a4de4e8f5448bd.jpg new file mode 100644 index 00000000..9ab3b014 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-366_d2c28fe27574e31158106430a02f4a9d10b1bdcba1e694f166a4de4e8f5448bd.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-366_f7cc228ecf2ac917ddf6037ba874c30bb1dc477738c0bdcd512cb6faba515316.jpg b/.raw/MidasFEAAnalysisManual/images/page-366_f7cc228ecf2ac917ddf6037ba874c30bb1dc477738c0bdcd512cb6faba515316.jpg new file mode 100644 index 00000000..61bda89d Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-366_f7cc228ecf2ac917ddf6037ba874c30bb1dc477738c0bdcd512cb6faba515316.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-367_359bd7fbcbd5123502def2d3878d335a03305169c1de3ee9f7bdc2c6bb005f5e.jpg b/.raw/MidasFEAAnalysisManual/images/page-367_359bd7fbcbd5123502def2d3878d335a03305169c1de3ee9f7bdc2c6bb005f5e.jpg new file mode 100644 index 00000000..9befd733 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-367_359bd7fbcbd5123502def2d3878d335a03305169c1de3ee9f7bdc2c6bb005f5e.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-367_b25f19ae997e1b3efcf04c7aff935ed993114b6cb3e67594b7880eb32427f6a2.jpg b/.raw/MidasFEAAnalysisManual/images/page-367_b25f19ae997e1b3efcf04c7aff935ed993114b6cb3e67594b7880eb32427f6a2.jpg new file mode 100644 index 00000000..6020fbb0 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-367_b25f19ae997e1b3efcf04c7aff935ed993114b6cb3e67594b7880eb32427f6a2.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-368_3106bf1879af5809ca737b77d2fd4c51401e4e1fb29087ff3937c7471884a319.jpg b/.raw/MidasFEAAnalysisManual/images/page-368_3106bf1879af5809ca737b77d2fd4c51401e4e1fb29087ff3937c7471884a319.jpg new file mode 100644 index 00000000..cf667f94 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-368_3106bf1879af5809ca737b77d2fd4c51401e4e1fb29087ff3937c7471884a319.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-369_3915d2946f7fd0ad0391de6510432c4f9be4241053e4170f59b802189c6d4111.jpg b/.raw/MidasFEAAnalysisManual/images/page-369_3915d2946f7fd0ad0391de6510432c4f9be4241053e4170f59b802189c6d4111.jpg new file mode 100644 index 00000000..812072aa Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-369_3915d2946f7fd0ad0391de6510432c4f9be4241053e4170f59b802189c6d4111.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-370_06767cde03674b7573e30e6e5e46afe1b0a3c2ff948bced7ac03628bb243f875.jpg b/.raw/MidasFEAAnalysisManual/images/page-370_06767cde03674b7573e30e6e5e46afe1b0a3c2ff948bced7ac03628bb243f875.jpg new file mode 100644 index 00000000..7286fa7c Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-370_06767cde03674b7573e30e6e5e46afe1b0a3c2ff948bced7ac03628bb243f875.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-370_0b4930e7fc5ec3435cd91d5cfe69f6c1a71351d057f207362544e5e9b97fb219.jpg b/.raw/MidasFEAAnalysisManual/images/page-370_0b4930e7fc5ec3435cd91d5cfe69f6c1a71351d057f207362544e5e9b97fb219.jpg new file mode 100644 index 00000000..4914313f Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-370_0b4930e7fc5ec3435cd91d5cfe69f6c1a71351d057f207362544e5e9b97fb219.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-370_147ab1ca6c6090273f0756da4e0dc7366546bae8eb6f0fbf25aa3366e33642f8.jpg b/.raw/MidasFEAAnalysisManual/images/page-370_147ab1ca6c6090273f0756da4e0dc7366546bae8eb6f0fbf25aa3366e33642f8.jpg new file mode 100644 index 00000000..fb18dbf1 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-370_147ab1ca6c6090273f0756da4e0dc7366546bae8eb6f0fbf25aa3366e33642f8.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-370_2c782fb85c88f2c9a316ed2279d43ba1eb8b822c2b1a7f08352785e1ffc6cd0b.jpg b/.raw/MidasFEAAnalysisManual/images/page-370_2c782fb85c88f2c9a316ed2279d43ba1eb8b822c2b1a7f08352785e1ffc6cd0b.jpg new file mode 100644 index 00000000..055a00a8 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-370_2c782fb85c88f2c9a316ed2279d43ba1eb8b822c2b1a7f08352785e1ffc6cd0b.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-370_2dfdb39e36a1c2976bec573a9bcaad75310cc1fadb26f77d5544645c37082a89.jpg b/.raw/MidasFEAAnalysisManual/images/page-370_2dfdb39e36a1c2976bec573a9bcaad75310cc1fadb26f77d5544645c37082a89.jpg new file mode 100644 index 00000000..9a099a8c Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-370_2dfdb39e36a1c2976bec573a9bcaad75310cc1fadb26f77d5544645c37082a89.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-370_47bea622c9e20fc75a5774e57fc8c81c52c0c15d6930c79f74399de1f8413e13.jpg b/.raw/MidasFEAAnalysisManual/images/page-370_47bea622c9e20fc75a5774e57fc8c81c52c0c15d6930c79f74399de1f8413e13.jpg new file mode 100644 index 00000000..3c5bbaa0 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-370_47bea622c9e20fc75a5774e57fc8c81c52c0c15d6930c79f74399de1f8413e13.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-370_5db87d02c269dfd177dbef4b745e96718a59a483f7e960b7d73644828fa94ca2.jpg b/.raw/MidasFEAAnalysisManual/images/page-370_5db87d02c269dfd177dbef4b745e96718a59a483f7e960b7d73644828fa94ca2.jpg new file mode 100644 index 00000000..0176a4f5 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-370_5db87d02c269dfd177dbef4b745e96718a59a483f7e960b7d73644828fa94ca2.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-371_287ffa54b545bcc8a0863e6f7166175bde485ae56fc7c74650e4613d093dd2fe.jpg b/.raw/MidasFEAAnalysisManual/images/page-371_287ffa54b545bcc8a0863e6f7166175bde485ae56fc7c74650e4613d093dd2fe.jpg new file mode 100644 index 00000000..e0470090 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-371_287ffa54b545bcc8a0863e6f7166175bde485ae56fc7c74650e4613d093dd2fe.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-371_4ff1947c79f186bbefaf28f67eccab0d8d8a820e9686f9c835f2ed92f5c5df3b.jpg b/.raw/MidasFEAAnalysisManual/images/page-371_4ff1947c79f186bbefaf28f67eccab0d8d8a820e9686f9c835f2ed92f5c5df3b.jpg new file mode 100644 index 00000000..36c98285 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-371_4ff1947c79f186bbefaf28f67eccab0d8d8a820e9686f9c835f2ed92f5c5df3b.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-371_6952436ac66c9501b60384fbd7e72f5fd87e90b7f0294b4265d53fac3037371e.jpg b/.raw/MidasFEAAnalysisManual/images/page-371_6952436ac66c9501b60384fbd7e72f5fd87e90b7f0294b4265d53fac3037371e.jpg new file mode 100644 index 00000000..119d3aa0 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-371_6952436ac66c9501b60384fbd7e72f5fd87e90b7f0294b4265d53fac3037371e.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-371_8e71d347f91b2917d9a912fe1721e65b48bb3e2bd380a76a7a8ded4993b4babf.jpg b/.raw/MidasFEAAnalysisManual/images/page-371_8e71d347f91b2917d9a912fe1721e65b48bb3e2bd380a76a7a8ded4993b4babf.jpg new file mode 100644 index 00000000..f845c410 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-371_8e71d347f91b2917d9a912fe1721e65b48bb3e2bd380a76a7a8ded4993b4babf.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-371_98876dfd8f598ca4d9ab3b2ee68de19630e3a8bb7dcd88b79a91511064fcda37.jpg b/.raw/MidasFEAAnalysisManual/images/page-371_98876dfd8f598ca4d9ab3b2ee68de19630e3a8bb7dcd88b79a91511064fcda37.jpg new file mode 100644 index 00000000..7e7ac372 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-371_98876dfd8f598ca4d9ab3b2ee68de19630e3a8bb7dcd88b79a91511064fcda37.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-371_9ccc8a7626512c2fa65408b8de7d8133a46d788fb99f537b4eca447c7e371737.jpg b/.raw/MidasFEAAnalysisManual/images/page-371_9ccc8a7626512c2fa65408b8de7d8133a46d788fb99f537b4eca447c7e371737.jpg new file mode 100644 index 00000000..2a61aef0 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-371_9ccc8a7626512c2fa65408b8de7d8133a46d788fb99f537b4eca447c7e371737.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-371_9d65215588e2c5ed581731aa2dfe5f883a7758b424c86eb3d71b006b14584722.jpg b/.raw/MidasFEAAnalysisManual/images/page-371_9d65215588e2c5ed581731aa2dfe5f883a7758b424c86eb3d71b006b14584722.jpg new file mode 100644 index 00000000..b362947d Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-371_9d65215588e2c5ed581731aa2dfe5f883a7758b424c86eb3d71b006b14584722.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-371_b60fb4de27e64ff968ba6e2e1745634b38e13d741373b882f7930442c56badb6.jpg b/.raw/MidasFEAAnalysisManual/images/page-371_b60fb4de27e64ff968ba6e2e1745634b38e13d741373b882f7930442c56badb6.jpg new file mode 100644 index 00000000..8fabe461 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-371_b60fb4de27e64ff968ba6e2e1745634b38e13d741373b882f7930442c56badb6.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-371_c7c5856caf5f730cb0c2422c809d5716191389d1a63f4a72387f43a620d2be07.jpg b/.raw/MidasFEAAnalysisManual/images/page-371_c7c5856caf5f730cb0c2422c809d5716191389d1a63f4a72387f43a620d2be07.jpg new file mode 100644 index 00000000..04c8853e Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-371_c7c5856caf5f730cb0c2422c809d5716191389d1a63f4a72387f43a620d2be07.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-371_f58682a8b5fcdc22fd710329c639b7a558df4a364e278928bb48015b26d327a9.jpg b/.raw/MidasFEAAnalysisManual/images/page-371_f58682a8b5fcdc22fd710329c639b7a558df4a364e278928bb48015b26d327a9.jpg new file mode 100644 index 00000000..ea0a95ce Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-371_f58682a8b5fcdc22fd710329c639b7a558df4a364e278928bb48015b26d327a9.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-372_06c91834ecfa2dfc7dc49d344766716bd291fba2803538b52eb62c000e6a4a3c.jpg b/.raw/MidasFEAAnalysisManual/images/page-372_06c91834ecfa2dfc7dc49d344766716bd291fba2803538b52eb62c000e6a4a3c.jpg new file mode 100644 index 00000000..6e0b2131 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-372_06c91834ecfa2dfc7dc49d344766716bd291fba2803538b52eb62c000e6a4a3c.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-372_0f3a3013a4f082f0fb476272d1579d4add0b56bf4b3f20262deee5cec1daefe9.jpg b/.raw/MidasFEAAnalysisManual/images/page-372_0f3a3013a4f082f0fb476272d1579d4add0b56bf4b3f20262deee5cec1daefe9.jpg new file mode 100644 index 00000000..1ba01c49 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-372_0f3a3013a4f082f0fb476272d1579d4add0b56bf4b3f20262deee5cec1daefe9.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-372_4b91c9f9fb289260ad4087e7f155c8e26e1a3cfe9fe618d713e55de519887ad1.jpg b/.raw/MidasFEAAnalysisManual/images/page-372_4b91c9f9fb289260ad4087e7f155c8e26e1a3cfe9fe618d713e55de519887ad1.jpg new file mode 100644 index 00000000..bba0c372 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-372_4b91c9f9fb289260ad4087e7f155c8e26e1a3cfe9fe618d713e55de519887ad1.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-372_8e76c2cc98058e9e56888c45ebf5e69dbdfc21720864f82713745f865ede6569.jpg b/.raw/MidasFEAAnalysisManual/images/page-372_8e76c2cc98058e9e56888c45ebf5e69dbdfc21720864f82713745f865ede6569.jpg new file mode 100644 index 00000000..3ea60812 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-372_8e76c2cc98058e9e56888c45ebf5e69dbdfc21720864f82713745f865ede6569.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-372_95cb4a5584a29a822a676ee5e5a33abb67a77e426fc61e2bfb3c4140c5177310.jpg b/.raw/MidasFEAAnalysisManual/images/page-372_95cb4a5584a29a822a676ee5e5a33abb67a77e426fc61e2bfb3c4140c5177310.jpg new file mode 100644 index 00000000..5c8031e1 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-372_95cb4a5584a29a822a676ee5e5a33abb67a77e426fc61e2bfb3c4140c5177310.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-372_a1cb29c324c1213de41b39adace909a11584bc53d89d22fa5f108c396d92b5a6.jpg b/.raw/MidasFEAAnalysisManual/images/page-372_a1cb29c324c1213de41b39adace909a11584bc53d89d22fa5f108c396d92b5a6.jpg new file mode 100644 index 00000000..cf450831 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-372_a1cb29c324c1213de41b39adace909a11584bc53d89d22fa5f108c396d92b5a6.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-372_ac32e2b9938c66ee4529e956dfbb9a75c7fc019328ce74ec102ae4a4265d4cf8.jpg b/.raw/MidasFEAAnalysisManual/images/page-372_ac32e2b9938c66ee4529e956dfbb9a75c7fc019328ce74ec102ae4a4265d4cf8.jpg new file mode 100644 index 00000000..0d6ce815 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-372_ac32e2b9938c66ee4529e956dfbb9a75c7fc019328ce74ec102ae4a4265d4cf8.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-373_5c705a140345ad0083017deb55a49bb067ab2923637237b3f0247b6f2760906b.jpg b/.raw/MidasFEAAnalysisManual/images/page-373_5c705a140345ad0083017deb55a49bb067ab2923637237b3f0247b6f2760906b.jpg new file mode 100644 index 00000000..a54d6b1f Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-373_5c705a140345ad0083017deb55a49bb067ab2923637237b3f0247b6f2760906b.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-373_5fab2f238c0acff1cac72908e2ff96ee2755d952239b2ff435c4f68dfd60c2bf.jpg b/.raw/MidasFEAAnalysisManual/images/page-373_5fab2f238c0acff1cac72908e2ff96ee2755d952239b2ff435c4f68dfd60c2bf.jpg new file mode 100644 index 00000000..d4f0c77c Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-373_5fab2f238c0acff1cac72908e2ff96ee2755d952239b2ff435c4f68dfd60c2bf.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-373_8f5a912cc5996e85728516e56b6f5e5f83165239fcf8753ad8ca57bc67808429.jpg b/.raw/MidasFEAAnalysisManual/images/page-373_8f5a912cc5996e85728516e56b6f5e5f83165239fcf8753ad8ca57bc67808429.jpg new file mode 100644 index 00000000..cedac6e9 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-373_8f5a912cc5996e85728516e56b6f5e5f83165239fcf8753ad8ca57bc67808429.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-373_b3cdf5631c82511ee5120f5b0ccf850ecefeed9e5ddc95f998ad934de9a47700.jpg b/.raw/MidasFEAAnalysisManual/images/page-373_b3cdf5631c82511ee5120f5b0ccf850ecefeed9e5ddc95f998ad934de9a47700.jpg new file mode 100644 index 00000000..411d8917 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-373_b3cdf5631c82511ee5120f5b0ccf850ecefeed9e5ddc95f998ad934de9a47700.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-374_07fdeac894e8308a023e1c47bbf5c6f7d24971b2643aa1a5bfb5c4f6d791b142.jpg b/.raw/MidasFEAAnalysisManual/images/page-374_07fdeac894e8308a023e1c47bbf5c6f7d24971b2643aa1a5bfb5c4f6d791b142.jpg new file mode 100644 index 00000000..8f78604c Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-374_07fdeac894e8308a023e1c47bbf5c6f7d24971b2643aa1a5bfb5c4f6d791b142.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-374_08df674b035c87d5c1b0484f917a6a579ea72893a2860da99437a76805d6c05d.jpg b/.raw/MidasFEAAnalysisManual/images/page-374_08df674b035c87d5c1b0484f917a6a579ea72893a2860da99437a76805d6c05d.jpg new file mode 100644 index 00000000..ad38d080 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-374_08df674b035c87d5c1b0484f917a6a579ea72893a2860da99437a76805d6c05d.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-374_10a4b3d7e76a4f2491739ffdcb1c6ce857ed8b587a11b68d17638add94b9bb3d.jpg b/.raw/MidasFEAAnalysisManual/images/page-374_10a4b3d7e76a4f2491739ffdcb1c6ce857ed8b587a11b68d17638add94b9bb3d.jpg new file mode 100644 index 00000000..a2839f68 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-374_10a4b3d7e76a4f2491739ffdcb1c6ce857ed8b587a11b68d17638add94b9bb3d.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-374_466c54fb6583047ef3f2072dc4718ab03ae162836144b615f42277c14ec0548a.jpg b/.raw/MidasFEAAnalysisManual/images/page-374_466c54fb6583047ef3f2072dc4718ab03ae162836144b615f42277c14ec0548a.jpg new file mode 100644 index 00000000..9bd39eaf Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-374_466c54fb6583047ef3f2072dc4718ab03ae162836144b615f42277c14ec0548a.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-374_4eb80a7290f6ecabbbc81a458d78c7da984cf29895a55587eee94935e6e57d7d.jpg b/.raw/MidasFEAAnalysisManual/images/page-374_4eb80a7290f6ecabbbc81a458d78c7da984cf29895a55587eee94935e6e57d7d.jpg new file mode 100644 index 00000000..a626ea2d Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-374_4eb80a7290f6ecabbbc81a458d78c7da984cf29895a55587eee94935e6e57d7d.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-374_58aa80c074efb31e44899be1886bacf6b7aa240c1b259b066b2e2446f56f42d9.jpg b/.raw/MidasFEAAnalysisManual/images/page-374_58aa80c074efb31e44899be1886bacf6b7aa240c1b259b066b2e2446f56f42d9.jpg new file mode 100644 index 00000000..5329ef24 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-374_58aa80c074efb31e44899be1886bacf6b7aa240c1b259b066b2e2446f56f42d9.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-374_7323d37d647474ba215ca86b25f49a0f9c2bd60a0a88d4e60b0073b9aae175e2.jpg b/.raw/MidasFEAAnalysisManual/images/page-374_7323d37d647474ba215ca86b25f49a0f9c2bd60a0a88d4e60b0073b9aae175e2.jpg new file mode 100644 index 00000000..463270c7 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-374_7323d37d647474ba215ca86b25f49a0f9c2bd60a0a88d4e60b0073b9aae175e2.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-374_942e12c019cfc9189e74acf18e78526135ffc848ca994ecbc5f53e46028b6ed6.jpg b/.raw/MidasFEAAnalysisManual/images/page-374_942e12c019cfc9189e74acf18e78526135ffc848ca994ecbc5f53e46028b6ed6.jpg new file mode 100644 index 00000000..c996ca74 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-374_942e12c019cfc9189e74acf18e78526135ffc848ca994ecbc5f53e46028b6ed6.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-374_b40628be5c819fc4c8e3da430f4bf94d83aa0dbb35b20bea80977212183de5ce.jpg b/.raw/MidasFEAAnalysisManual/images/page-374_b40628be5c819fc4c8e3da430f4bf94d83aa0dbb35b20bea80977212183de5ce.jpg new file mode 100644 index 00000000..7a75706f Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-374_b40628be5c819fc4c8e3da430f4bf94d83aa0dbb35b20bea80977212183de5ce.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-374_cd8480e233fc25ed2bc1b9ed08ecb8b8336572d956a36d2f75137de5928c22d9.jpg b/.raw/MidasFEAAnalysisManual/images/page-374_cd8480e233fc25ed2bc1b9ed08ecb8b8336572d956a36d2f75137de5928c22d9.jpg new file mode 100644 index 00000000..b22a8c95 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-374_cd8480e233fc25ed2bc1b9ed08ecb8b8336572d956a36d2f75137de5928c22d9.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-375_09e44bc083f53bb76a05e34c818bde65e8fdee69a3a360678649b41d5bd293af.jpg b/.raw/MidasFEAAnalysisManual/images/page-375_09e44bc083f53bb76a05e34c818bde65e8fdee69a3a360678649b41d5bd293af.jpg new file mode 100644 index 00000000..10ca437d Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-375_09e44bc083f53bb76a05e34c818bde65e8fdee69a3a360678649b41d5bd293af.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-375_1f9f9bb9c56fe21f42678aeddf65ad3467de97d401c4678207e0aac5e0a4b77c.jpg b/.raw/MidasFEAAnalysisManual/images/page-375_1f9f9bb9c56fe21f42678aeddf65ad3467de97d401c4678207e0aac5e0a4b77c.jpg new file mode 100644 index 00000000..42af7f80 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-375_1f9f9bb9c56fe21f42678aeddf65ad3467de97d401c4678207e0aac5e0a4b77c.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-375_20a4c5191ceab30c1629b75a1c22beaae3931d8d4da11d9bf52737c90938f193.jpg b/.raw/MidasFEAAnalysisManual/images/page-375_20a4c5191ceab30c1629b75a1c22beaae3931d8d4da11d9bf52737c90938f193.jpg new file mode 100644 index 00000000..2243cdac Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-375_20a4c5191ceab30c1629b75a1c22beaae3931d8d4da11d9bf52737c90938f193.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-375_7f0b771c5ec397b5d0ab55f41f8b5d3c66c0dd1ccc3ff84debb8b58f3a5416b6.jpg b/.raw/MidasFEAAnalysisManual/images/page-375_7f0b771c5ec397b5d0ab55f41f8b5d3c66c0dd1ccc3ff84debb8b58f3a5416b6.jpg new file mode 100644 index 00000000..c63f26a3 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-375_7f0b771c5ec397b5d0ab55f41f8b5d3c66c0dd1ccc3ff84debb8b58f3a5416b6.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-376_862e610a71647901230a897b54ee998ec3bc9008f0a7ea22d12e07b67f4696d3.jpg b/.raw/MidasFEAAnalysisManual/images/page-376_862e610a71647901230a897b54ee998ec3bc9008f0a7ea22d12e07b67f4696d3.jpg new file mode 100644 index 00000000..1f4d6f4c Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-376_862e610a71647901230a897b54ee998ec3bc9008f0a7ea22d12e07b67f4696d3.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-376_e005572ff6e98e04ca1bbc1b442ba618db783d69a96c3a39265ef53a4226b822.jpg b/.raw/MidasFEAAnalysisManual/images/page-376_e005572ff6e98e04ca1bbc1b442ba618db783d69a96c3a39265ef53a4226b822.jpg new file mode 100644 index 00000000..6e755359 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-376_e005572ff6e98e04ca1bbc1b442ba618db783d69a96c3a39265ef53a4226b822.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-379_1b416ebb2b02cea1647ec7e9051c791251754d86bceaa30f6e758488d8d3beb1.jpg b/.raw/MidasFEAAnalysisManual/images/page-379_1b416ebb2b02cea1647ec7e9051c791251754d86bceaa30f6e758488d8d3beb1.jpg new file mode 100644 index 00000000..442fbcfa Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-379_1b416ebb2b02cea1647ec7e9051c791251754d86bceaa30f6e758488d8d3beb1.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-379_4b187b4e03afe1e99a988d7a5b5bdf4f48cb33423885461fa0ac761966f7bd5d.jpg b/.raw/MidasFEAAnalysisManual/images/page-379_4b187b4e03afe1e99a988d7a5b5bdf4f48cb33423885461fa0ac761966f7bd5d.jpg new file mode 100644 index 00000000..137dcf56 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-379_4b187b4e03afe1e99a988d7a5b5bdf4f48cb33423885461fa0ac761966f7bd5d.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-380_19342e75cb54f8821c7415cdace1f2718afb72880405b59f64c5f8b89aea59dd.jpg b/.raw/MidasFEAAnalysisManual/images/page-380_19342e75cb54f8821c7415cdace1f2718afb72880405b59f64c5f8b89aea59dd.jpg new file mode 100644 index 00000000..87562456 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-380_19342e75cb54f8821c7415cdace1f2718afb72880405b59f64c5f8b89aea59dd.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-380_1f5c19735c3a0d871f134d726984b8423fd7ebede10e635e86b536aff7c3dae2.jpg b/.raw/MidasFEAAnalysisManual/images/page-380_1f5c19735c3a0d871f134d726984b8423fd7ebede10e635e86b536aff7c3dae2.jpg new file mode 100644 index 00000000..3d4b163f Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-380_1f5c19735c3a0d871f134d726984b8423fd7ebede10e635e86b536aff7c3dae2.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-380_5ae9f225afa7c864ba3bf901cc8ee392a5240a01a8027842549b9b1ae0ef9469.jpg b/.raw/MidasFEAAnalysisManual/images/page-380_5ae9f225afa7c864ba3bf901cc8ee392a5240a01a8027842549b9b1ae0ef9469.jpg new file mode 100644 index 00000000..59214b50 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-380_5ae9f225afa7c864ba3bf901cc8ee392a5240a01a8027842549b9b1ae0ef9469.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-380_6d4bdd3f338a4cde232abffc24d1ead9f55f4783e17a2b8993c1a005041f9bb3.jpg b/.raw/MidasFEAAnalysisManual/images/page-380_6d4bdd3f338a4cde232abffc24d1ead9f55f4783e17a2b8993c1a005041f9bb3.jpg new file mode 100644 index 00000000..0b4219c4 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-380_6d4bdd3f338a4cde232abffc24d1ead9f55f4783e17a2b8993c1a005041f9bb3.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-380_9b3a2cf2372227903dcbd2b85255c8a5cfa46d02144b9ffa3aad98386c25737f.jpg b/.raw/MidasFEAAnalysisManual/images/page-380_9b3a2cf2372227903dcbd2b85255c8a5cfa46d02144b9ffa3aad98386c25737f.jpg new file mode 100644 index 00000000..d6ff977d Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-380_9b3a2cf2372227903dcbd2b85255c8a5cfa46d02144b9ffa3aad98386c25737f.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-380_9b4ffd700f67a68b5b96ec0d44aa5f55e3ad8fc82b001968bbaf7944b756fe98.jpg b/.raw/MidasFEAAnalysisManual/images/page-380_9b4ffd700f67a68b5b96ec0d44aa5f55e3ad8fc82b001968bbaf7944b756fe98.jpg new file mode 100644 index 00000000..9ce8a874 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-380_9b4ffd700f67a68b5b96ec0d44aa5f55e3ad8fc82b001968bbaf7944b756fe98.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-380_b8703e5c55f2cb364f722db3729b06e5e1dccdade2db8bdb4333dcb800d7d061.jpg b/.raw/MidasFEAAnalysisManual/images/page-380_b8703e5c55f2cb364f722db3729b06e5e1dccdade2db8bdb4333dcb800d7d061.jpg new file mode 100644 index 00000000..804e9048 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-380_b8703e5c55f2cb364f722db3729b06e5e1dccdade2db8bdb4333dcb800d7d061.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-381_20b709704136ed42752eced7df3d49f850b2a2494b52ec4029c32b2a93e39184.jpg b/.raw/MidasFEAAnalysisManual/images/page-381_20b709704136ed42752eced7df3d49f850b2a2494b52ec4029c32b2a93e39184.jpg new file mode 100644 index 00000000..78df75f4 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-381_20b709704136ed42752eced7df3d49f850b2a2494b52ec4029c32b2a93e39184.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-381_2ffb84a590d98a0c42c7061357aad86266a8cf012c4abb1a84376e5d9b8fa0db.jpg b/.raw/MidasFEAAnalysisManual/images/page-381_2ffb84a590d98a0c42c7061357aad86266a8cf012c4abb1a84376e5d9b8fa0db.jpg new file mode 100644 index 00000000..237394f7 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-381_2ffb84a590d98a0c42c7061357aad86266a8cf012c4abb1a84376e5d9b8fa0db.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-381_603c4e34d2cc3da290ddcba5d1e8a3fd6c5556c05de801371b746dc99a17f17e.jpg b/.raw/MidasFEAAnalysisManual/images/page-381_603c4e34d2cc3da290ddcba5d1e8a3fd6c5556c05de801371b746dc99a17f17e.jpg new file mode 100644 index 00000000..c4381bda Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-381_603c4e34d2cc3da290ddcba5d1e8a3fd6c5556c05de801371b746dc99a17f17e.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-381_911de06402e05ab243273943e5ce763fb1da85f97a7d4be44ac2a41d59887a31.jpg b/.raw/MidasFEAAnalysisManual/images/page-381_911de06402e05ab243273943e5ce763fb1da85f97a7d4be44ac2a41d59887a31.jpg new file mode 100644 index 00000000..3d80c97a Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-381_911de06402e05ab243273943e5ce763fb1da85f97a7d4be44ac2a41d59887a31.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-382_285442a52cf33b82f4115d47cf4fb9ca53d9d7088f015d080017b726abc9bef3.jpg b/.raw/MidasFEAAnalysisManual/images/page-382_285442a52cf33b82f4115d47cf4fb9ca53d9d7088f015d080017b726abc9bef3.jpg new file mode 100644 index 00000000..b58c483f Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-382_285442a52cf33b82f4115d47cf4fb9ca53d9d7088f015d080017b726abc9bef3.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-382_79b61c8a827a0abfe5e67fcd2ba5e1c6ffe430172013994c2eea90fcd5a641d3.jpg b/.raw/MidasFEAAnalysisManual/images/page-382_79b61c8a827a0abfe5e67fcd2ba5e1c6ffe430172013994c2eea90fcd5a641d3.jpg new file mode 100644 index 00000000..797b4d79 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-382_79b61c8a827a0abfe5e67fcd2ba5e1c6ffe430172013994c2eea90fcd5a641d3.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-382_8b5a9d337593ff3007b5dc05fd7499d89afafcb09e08d10e27ea1429655ca944.jpg b/.raw/MidasFEAAnalysisManual/images/page-382_8b5a9d337593ff3007b5dc05fd7499d89afafcb09e08d10e27ea1429655ca944.jpg new file mode 100644 index 00000000..f23d2e95 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-382_8b5a9d337593ff3007b5dc05fd7499d89afafcb09e08d10e27ea1429655ca944.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-382_addb8cc3d372d1526d4dd7b40a51af170d67beba0f9d9f476a48d59c12b6a97a.jpg b/.raw/MidasFEAAnalysisManual/images/page-382_addb8cc3d372d1526d4dd7b40a51af170d67beba0f9d9f476a48d59c12b6a97a.jpg new file mode 100644 index 00000000..dd1a4707 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-382_addb8cc3d372d1526d4dd7b40a51af170d67beba0f9d9f476a48d59c12b6a97a.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-382_b7d5ccdb2836e21ddcb76703b292afc09cd60e3a5febd0e54cf1a2ce970f7c3a.jpg b/.raw/MidasFEAAnalysisManual/images/page-382_b7d5ccdb2836e21ddcb76703b292afc09cd60e3a5febd0e54cf1a2ce970f7c3a.jpg new file mode 100644 index 00000000..9f6bf50b Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-382_b7d5ccdb2836e21ddcb76703b292afc09cd60e3a5febd0e54cf1a2ce970f7c3a.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-382_d8d8904f71954e31e9e805b6352cff91ad91182be6715ea0fd17a54094a80749.jpg b/.raw/MidasFEAAnalysisManual/images/page-382_d8d8904f71954e31e9e805b6352cff91ad91182be6715ea0fd17a54094a80749.jpg new file mode 100644 index 00000000..f2b07cd6 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-382_d8d8904f71954e31e9e805b6352cff91ad91182be6715ea0fd17a54094a80749.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-382_ed743b134862b3e3d5ed2798da89407a329e87c58195634e01a78ce0e25660e6.jpg b/.raw/MidasFEAAnalysisManual/images/page-382_ed743b134862b3e3d5ed2798da89407a329e87c58195634e01a78ce0e25660e6.jpg new file mode 100644 index 00000000..0a57effb Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-382_ed743b134862b3e3d5ed2798da89407a329e87c58195634e01a78ce0e25660e6.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-383_192d09456923cdfd264abbbe63356e7cb568b75a78968241a37157ca3925ca18.jpg b/.raw/MidasFEAAnalysisManual/images/page-383_192d09456923cdfd264abbbe63356e7cb568b75a78968241a37157ca3925ca18.jpg new file mode 100644 index 00000000..15e4b809 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-383_192d09456923cdfd264abbbe63356e7cb568b75a78968241a37157ca3925ca18.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-383_3656bc0cf18dc4e8fb261a13fc1cd4c809d3003cd7734f3a62ff4fee26f8af6b.jpg b/.raw/MidasFEAAnalysisManual/images/page-383_3656bc0cf18dc4e8fb261a13fc1cd4c809d3003cd7734f3a62ff4fee26f8af6b.jpg new file mode 100644 index 00000000..349acbf8 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-383_3656bc0cf18dc4e8fb261a13fc1cd4c809d3003cd7734f3a62ff4fee26f8af6b.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-383_4df11d9fdcc79f58c9f1932d5caaddf8ca5eae35fde941d53d0d7a276346dff2.jpg b/.raw/MidasFEAAnalysisManual/images/page-383_4df11d9fdcc79f58c9f1932d5caaddf8ca5eae35fde941d53d0d7a276346dff2.jpg new file mode 100644 index 00000000..452004eb Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-383_4df11d9fdcc79f58c9f1932d5caaddf8ca5eae35fde941d53d0d7a276346dff2.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-383_5e87d0354327bdbd0d3cc9bff3a71ba40173ffcee3acc0e4527f3122bb33ea63.jpg b/.raw/MidasFEAAnalysisManual/images/page-383_5e87d0354327bdbd0d3cc9bff3a71ba40173ffcee3acc0e4527f3122bb33ea63.jpg new file mode 100644 index 00000000..6b3b18db Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-383_5e87d0354327bdbd0d3cc9bff3a71ba40173ffcee3acc0e4527f3122bb33ea63.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-383_a8e0c90265ae715788d4175ad9ba56cd0a8efba05dc8600c0681c218dcc9addd.jpg b/.raw/MidasFEAAnalysisManual/images/page-383_a8e0c90265ae715788d4175ad9ba56cd0a8efba05dc8600c0681c218dcc9addd.jpg new file mode 100644 index 00000000..e12ecf14 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-383_a8e0c90265ae715788d4175ad9ba56cd0a8efba05dc8600c0681c218dcc9addd.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-383_b05897df4014e83d646dd0db2f32643e951f281b46a1823240152bc536da4219.jpg b/.raw/MidasFEAAnalysisManual/images/page-383_b05897df4014e83d646dd0db2f32643e951f281b46a1823240152bc536da4219.jpg new file mode 100644 index 00000000..a7867103 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-383_b05897df4014e83d646dd0db2f32643e951f281b46a1823240152bc536da4219.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-383_b4d6cb6623e8488f69f9498ad313abf22245837c10a875d54f05ce2950bc4b81.jpg b/.raw/MidasFEAAnalysisManual/images/page-383_b4d6cb6623e8488f69f9498ad313abf22245837c10a875d54f05ce2950bc4b81.jpg new file mode 100644 index 00000000..db6ed0b5 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-383_b4d6cb6623e8488f69f9498ad313abf22245837c10a875d54f05ce2950bc4b81.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-383_c63e7690fd6e6a639abd79ec3796c33fa91f106a162a95d5724a20038c8aa87d.jpg b/.raw/MidasFEAAnalysisManual/images/page-383_c63e7690fd6e6a639abd79ec3796c33fa91f106a162a95d5724a20038c8aa87d.jpg new file mode 100644 index 00000000..7db0804c Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-383_c63e7690fd6e6a639abd79ec3796c33fa91f106a162a95d5724a20038c8aa87d.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-384_2facbfeb55e8143ef06299af5d8f5037f35dd190fdcd170e65dd54c63d7ee057.jpg b/.raw/MidasFEAAnalysisManual/images/page-384_2facbfeb55e8143ef06299af5d8f5037f35dd190fdcd170e65dd54c63d7ee057.jpg new file mode 100644 index 00000000..43e0f75f Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-384_2facbfeb55e8143ef06299af5d8f5037f35dd190fdcd170e65dd54c63d7ee057.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-384_801a747631aa0c2e5a6a77c5629b635725829132574991b406bc5b1ea5f07784.jpg b/.raw/MidasFEAAnalysisManual/images/page-384_801a747631aa0c2e5a6a77c5629b635725829132574991b406bc5b1ea5f07784.jpg new file mode 100644 index 00000000..5bda5b63 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-384_801a747631aa0c2e5a6a77c5629b635725829132574991b406bc5b1ea5f07784.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-385_5d072bb2a20043d4d437c39727f23d58cd837dc7613d6757a9e57873c8a6d94d.jpg b/.raw/MidasFEAAnalysisManual/images/page-385_5d072bb2a20043d4d437c39727f23d58cd837dc7613d6757a9e57873c8a6d94d.jpg new file mode 100644 index 00000000..ec1f45f8 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-385_5d072bb2a20043d4d437c39727f23d58cd837dc7613d6757a9e57873c8a6d94d.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-387_29333529022e0723b23a9f82a6b211e6d42708c2674193daef1c25f9b88ea524.jpg b/.raw/MidasFEAAnalysisManual/images/page-387_29333529022e0723b23a9f82a6b211e6d42708c2674193daef1c25f9b88ea524.jpg new file mode 100644 index 00000000..c48cd628 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-387_29333529022e0723b23a9f82a6b211e6d42708c2674193daef1c25f9b88ea524.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-387_6207a258d872e5548a3b2485aeac2518a634048482fc0021912832170b9afdfc.jpg b/.raw/MidasFEAAnalysisManual/images/page-387_6207a258d872e5548a3b2485aeac2518a634048482fc0021912832170b9afdfc.jpg new file mode 100644 index 00000000..877a6b2f Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-387_6207a258d872e5548a3b2485aeac2518a634048482fc0021912832170b9afdfc.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-387_e54776594350c88fb8dbdc4fdaa660acd3a55f06b87515b56081433bb6112fde.jpg b/.raw/MidasFEAAnalysisManual/images/page-387_e54776594350c88fb8dbdc4fdaa660acd3a55f06b87515b56081433bb6112fde.jpg new file mode 100644 index 00000000..c21f3164 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-387_e54776594350c88fb8dbdc4fdaa660acd3a55f06b87515b56081433bb6112fde.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-388_1b2071f98c4db6a51bfcf9414c8a35064484a2ea9d952099e4748da0e8a3c2e5.jpg b/.raw/MidasFEAAnalysisManual/images/page-388_1b2071f98c4db6a51bfcf9414c8a35064484a2ea9d952099e4748da0e8a3c2e5.jpg new file mode 100644 index 00000000..d02ffad9 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-388_1b2071f98c4db6a51bfcf9414c8a35064484a2ea9d952099e4748da0e8a3c2e5.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-388_49ecbf3c0d6f4251f0b86ad7bf36c550b20d66189b0c7e5997c02d4f24784cc3.jpg b/.raw/MidasFEAAnalysisManual/images/page-388_49ecbf3c0d6f4251f0b86ad7bf36c550b20d66189b0c7e5997c02d4f24784cc3.jpg new file mode 100644 index 00000000..60c649d9 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-388_49ecbf3c0d6f4251f0b86ad7bf36c550b20d66189b0c7e5997c02d4f24784cc3.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-388_4eeda66a5ff9d0f422cb6ce09dad9b5743eafb09daf4c83c30760e0853c10796.jpg b/.raw/MidasFEAAnalysisManual/images/page-388_4eeda66a5ff9d0f422cb6ce09dad9b5743eafb09daf4c83c30760e0853c10796.jpg new file mode 100644 index 00000000..357e7c57 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-388_4eeda66a5ff9d0f422cb6ce09dad9b5743eafb09daf4c83c30760e0853c10796.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-388_c25eeb9bf8471e5fc23df02367ccf6641a24109090a7429db0a259ed90769a6a.jpg b/.raw/MidasFEAAnalysisManual/images/page-388_c25eeb9bf8471e5fc23df02367ccf6641a24109090a7429db0a259ed90769a6a.jpg new file mode 100644 index 00000000..2fbac557 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-388_c25eeb9bf8471e5fc23df02367ccf6641a24109090a7429db0a259ed90769a6a.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-388_e373029cdca13cb81619d76fb20512ce6a15a3d22ec08007c0306879cb6c19b0.jpg b/.raw/MidasFEAAnalysisManual/images/page-388_e373029cdca13cb81619d76fb20512ce6a15a3d22ec08007c0306879cb6c19b0.jpg new file mode 100644 index 00000000..05337ece Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-388_e373029cdca13cb81619d76fb20512ce6a15a3d22ec08007c0306879cb6c19b0.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-389_9862038f871deddbb9815debedf92e98929b7d0f0b5df4c94c3d700842a9b8f0.jpg b/.raw/MidasFEAAnalysisManual/images/page-389_9862038f871deddbb9815debedf92e98929b7d0f0b5df4c94c3d700842a9b8f0.jpg new file mode 100644 index 00000000..0162ad00 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-389_9862038f871deddbb9815debedf92e98929b7d0f0b5df4c94c3d700842a9b8f0.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-390_2236a836c74c3190aad1bb9bec151ccc52e33ee34e009a81db1794ecd3bca267.jpg b/.raw/MidasFEAAnalysisManual/images/page-390_2236a836c74c3190aad1bb9bec151ccc52e33ee34e009a81db1794ecd3bca267.jpg new file mode 100644 index 00000000..ee17cfa9 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-390_2236a836c74c3190aad1bb9bec151ccc52e33ee34e009a81db1794ecd3bca267.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-390_47bd67dd8f15a10ab9406cb220395b9a4ae6da4037f389c6ef3b868bc7fd88f4.jpg b/.raw/MidasFEAAnalysisManual/images/page-390_47bd67dd8f15a10ab9406cb220395b9a4ae6da4037f389c6ef3b868bc7fd88f4.jpg new file mode 100644 index 00000000..d65f67b8 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-390_47bd67dd8f15a10ab9406cb220395b9a4ae6da4037f389c6ef3b868bc7fd88f4.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-390_6069cea554eb5b2eaeb98666806b938e3e02ecaa6a332e553757b5dbf46e09b7.jpg b/.raw/MidasFEAAnalysisManual/images/page-390_6069cea554eb5b2eaeb98666806b938e3e02ecaa6a332e553757b5dbf46e09b7.jpg new file mode 100644 index 00000000..52d3e95d Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-390_6069cea554eb5b2eaeb98666806b938e3e02ecaa6a332e553757b5dbf46e09b7.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-391_5cd0e71e74bb57f9184fb6313cff3a93844b77475272d3b39a40ae321a083449.jpg b/.raw/MidasFEAAnalysisManual/images/page-391_5cd0e71e74bb57f9184fb6313cff3a93844b77475272d3b39a40ae321a083449.jpg new file mode 100644 index 00000000..6eae0466 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-391_5cd0e71e74bb57f9184fb6313cff3a93844b77475272d3b39a40ae321a083449.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-391_6a818136657cdd6a654ba3af2bddb0bffb2a5ab64c12973cd0a4f0716ba1c6ec.jpg b/.raw/MidasFEAAnalysisManual/images/page-391_6a818136657cdd6a654ba3af2bddb0bffb2a5ab64c12973cd0a4f0716ba1c6ec.jpg new file mode 100644 index 00000000..f331e2d8 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-391_6a818136657cdd6a654ba3af2bddb0bffb2a5ab64c12973cd0a4f0716ba1c6ec.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-391_b82dec70a0d4926964ac8dad1951c5aca471f89ab66d262b43713b7476b2b7b3.jpg b/.raw/MidasFEAAnalysisManual/images/page-391_b82dec70a0d4926964ac8dad1951c5aca471f89ab66d262b43713b7476b2b7b3.jpg new file mode 100644 index 00000000..527497f9 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-391_b82dec70a0d4926964ac8dad1951c5aca471f89ab66d262b43713b7476b2b7b3.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-391_d66c7a773693fc946dcd896aa8970d57ecdd1b36488051d9fecf05b0af5f13af.jpg b/.raw/MidasFEAAnalysisManual/images/page-391_d66c7a773693fc946dcd896aa8970d57ecdd1b36488051d9fecf05b0af5f13af.jpg new file mode 100644 index 00000000..cf398cd8 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-391_d66c7a773693fc946dcd896aa8970d57ecdd1b36488051d9fecf05b0af5f13af.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-391_e5a0347b74a100c2c52b76afed1ad325f8b7ba60dc562ec162c2332f04bad09a.jpg b/.raw/MidasFEAAnalysisManual/images/page-391_e5a0347b74a100c2c52b76afed1ad325f8b7ba60dc562ec162c2332f04bad09a.jpg new file mode 100644 index 00000000..3c3ea5d5 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-391_e5a0347b74a100c2c52b76afed1ad325f8b7ba60dc562ec162c2332f04bad09a.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-392_01d74489a1f718c6067db9a23c567fa30def40a7646240b1a1f1eca2639ccf7d.jpg b/.raw/MidasFEAAnalysisManual/images/page-392_01d74489a1f718c6067db9a23c567fa30def40a7646240b1a1f1eca2639ccf7d.jpg new file mode 100644 index 00000000..65b49908 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-392_01d74489a1f718c6067db9a23c567fa30def40a7646240b1a1f1eca2639ccf7d.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-392_17d43cfb5abc734b9ed7208faba3ccf2edd9079802c48a20924e906324485f77.jpg b/.raw/MidasFEAAnalysisManual/images/page-392_17d43cfb5abc734b9ed7208faba3ccf2edd9079802c48a20924e906324485f77.jpg new file mode 100644 index 00000000..61352d25 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-392_17d43cfb5abc734b9ed7208faba3ccf2edd9079802c48a20924e906324485f77.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-392_599f207dba500c3d430defbcd6eb050ea86d670f44b671d3fc5543ffe55b2114.jpg b/.raw/MidasFEAAnalysisManual/images/page-392_599f207dba500c3d430defbcd6eb050ea86d670f44b671d3fc5543ffe55b2114.jpg new file mode 100644 index 00000000..a584f099 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-392_599f207dba500c3d430defbcd6eb050ea86d670f44b671d3fc5543ffe55b2114.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-392_8ef60938bedb640d3e8bc18b7344eeaef4c548ba2de4f909626bef184df2b93f.jpg b/.raw/MidasFEAAnalysisManual/images/page-392_8ef60938bedb640d3e8bc18b7344eeaef4c548ba2de4f909626bef184df2b93f.jpg new file mode 100644 index 00000000..79996e65 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-392_8ef60938bedb640d3e8bc18b7344eeaef4c548ba2de4f909626bef184df2b93f.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-392_acfd111dbe5ae08324260a18de5e5018196326aa22664022003a8f92bc6dbfa9.jpg b/.raw/MidasFEAAnalysisManual/images/page-392_acfd111dbe5ae08324260a18de5e5018196326aa22664022003a8f92bc6dbfa9.jpg new file mode 100644 index 00000000..985f2a7f Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-392_acfd111dbe5ae08324260a18de5e5018196326aa22664022003a8f92bc6dbfa9.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-392_fba35ee8b3a6af8f2d763329b1cf429261d9dcf056e8200ca316294540ab9a60.jpg b/.raw/MidasFEAAnalysisManual/images/page-392_fba35ee8b3a6af8f2d763329b1cf429261d9dcf056e8200ca316294540ab9a60.jpg new file mode 100644 index 00000000..5f3a1733 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-392_fba35ee8b3a6af8f2d763329b1cf429261d9dcf056e8200ca316294540ab9a60.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-392_fcc16bcdc593ab63cbb7a321190e71088eb9e165dd7576e8435a15d3ce4d7c4f.jpg b/.raw/MidasFEAAnalysisManual/images/page-392_fcc16bcdc593ab63cbb7a321190e71088eb9e165dd7576e8435a15d3ce4d7c4f.jpg new file mode 100644 index 00000000..ee9266ff Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-392_fcc16bcdc593ab63cbb7a321190e71088eb9e165dd7576e8435a15d3ce4d7c4f.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-393_54b19d631472852b5c6501d967ba7f836fea889ac3d4a0bb430a0614f93c6a99.jpg b/.raw/MidasFEAAnalysisManual/images/page-393_54b19d631472852b5c6501d967ba7f836fea889ac3d4a0bb430a0614f93c6a99.jpg new file mode 100644 index 00000000..8cf3e540 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-393_54b19d631472852b5c6501d967ba7f836fea889ac3d4a0bb430a0614f93c6a99.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-393_cc94f0c388deab678e58396013a3f6970bd64e27c69156c1cbb51ef4cf422978.jpg b/.raw/MidasFEAAnalysisManual/images/page-393_cc94f0c388deab678e58396013a3f6970bd64e27c69156c1cbb51ef4cf422978.jpg new file mode 100644 index 00000000..5a8394b1 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-393_cc94f0c388deab678e58396013a3f6970bd64e27c69156c1cbb51ef4cf422978.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-393_f90f7eaabad73022d949c8829f6e69ad479e48f39b03b2cf7eaf621233d59977.jpg b/.raw/MidasFEAAnalysisManual/images/page-393_f90f7eaabad73022d949c8829f6e69ad479e48f39b03b2cf7eaf621233d59977.jpg new file mode 100644 index 00000000..482d4e5c Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-393_f90f7eaabad73022d949c8829f6e69ad479e48f39b03b2cf7eaf621233d59977.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-394_2934b45d99d75b2351da08a8858f1d314ef12881da86844a0004fd2bd2df6fee.jpg b/.raw/MidasFEAAnalysisManual/images/page-394_2934b45d99d75b2351da08a8858f1d314ef12881da86844a0004fd2bd2df6fee.jpg new file mode 100644 index 00000000..e918c656 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-394_2934b45d99d75b2351da08a8858f1d314ef12881da86844a0004fd2bd2df6fee.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-394_4ec78bdbd9776fd68154098a8b73169d207b2f07e7be915e48ae85601642db38.jpg b/.raw/MidasFEAAnalysisManual/images/page-394_4ec78bdbd9776fd68154098a8b73169d207b2f07e7be915e48ae85601642db38.jpg new file mode 100644 index 00000000..eec5e8a7 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-394_4ec78bdbd9776fd68154098a8b73169d207b2f07e7be915e48ae85601642db38.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-394_50e5ff6e8a002ff732972a7d49b830df7a853e1a3965854a19cab0c87ce03aee.jpg b/.raw/MidasFEAAnalysisManual/images/page-394_50e5ff6e8a002ff732972a7d49b830df7a853e1a3965854a19cab0c87ce03aee.jpg new file mode 100644 index 00000000..6b17b348 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-394_50e5ff6e8a002ff732972a7d49b830df7a853e1a3965854a19cab0c87ce03aee.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-394_567142cee40e1c8742bde222985e64f00a2887cf96a0d72c179bd62f9aa8d271.jpg b/.raw/MidasFEAAnalysisManual/images/page-394_567142cee40e1c8742bde222985e64f00a2887cf96a0d72c179bd62f9aa8d271.jpg new file mode 100644 index 00000000..b61c0fcc Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-394_567142cee40e1c8742bde222985e64f00a2887cf96a0d72c179bd62f9aa8d271.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-394_67b50723a6911d4eb3f48484811aca8f487a1b9330a7c66c9240ef53c966b611.jpg b/.raw/MidasFEAAnalysisManual/images/page-394_67b50723a6911d4eb3f48484811aca8f487a1b9330a7c66c9240ef53c966b611.jpg new file mode 100644 index 00000000..be8641e6 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-394_67b50723a6911d4eb3f48484811aca8f487a1b9330a7c66c9240ef53c966b611.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-395_30ae0204c00d6effab634d800b7f99716a8ca37a9803a2b7eff9dd7802d6c799.jpg b/.raw/MidasFEAAnalysisManual/images/page-395_30ae0204c00d6effab634d800b7f99716a8ca37a9803a2b7eff9dd7802d6c799.jpg new file mode 100644 index 00000000..358e6483 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-395_30ae0204c00d6effab634d800b7f99716a8ca37a9803a2b7eff9dd7802d6c799.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-395_68e9247ca772a54bef09a59a96dd2a92f0ca335f2a1cfd4e53bb97a3d48274f9.jpg b/.raw/MidasFEAAnalysisManual/images/page-395_68e9247ca772a54bef09a59a96dd2a92f0ca335f2a1cfd4e53bb97a3d48274f9.jpg new file mode 100644 index 00000000..8b98c290 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-395_68e9247ca772a54bef09a59a96dd2a92f0ca335f2a1cfd4e53bb97a3d48274f9.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-395_769d7c3f89a5f0d4784aef116730fa7b9efae6d96147515c7c778b95f1269211.jpg b/.raw/MidasFEAAnalysisManual/images/page-395_769d7c3f89a5f0d4784aef116730fa7b9efae6d96147515c7c778b95f1269211.jpg new file mode 100644 index 00000000..acb1b141 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-395_769d7c3f89a5f0d4784aef116730fa7b9efae6d96147515c7c778b95f1269211.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-396_2312527a5d2b180aa9e368e6079a25d2ca25de5613bed40b4110e9f2d0ada035.jpg b/.raw/MidasFEAAnalysisManual/images/page-396_2312527a5d2b180aa9e368e6079a25d2ca25de5613bed40b4110e9f2d0ada035.jpg new file mode 100644 index 00000000..a8220d13 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-396_2312527a5d2b180aa9e368e6079a25d2ca25de5613bed40b4110e9f2d0ada035.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-396_7b2a37be00c0e4d185b83860373bb00ef53634bf7fc5af815946f1a3c172a6a2.jpg b/.raw/MidasFEAAnalysisManual/images/page-396_7b2a37be00c0e4d185b83860373bb00ef53634bf7fc5af815946f1a3c172a6a2.jpg new file mode 100644 index 00000000..9799f4d2 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-396_7b2a37be00c0e4d185b83860373bb00ef53634bf7fc5af815946f1a3c172a6a2.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-396_ea7ca277bc36419fdc0641fbbe18d7cbce972ef323e51c80c760466f8cd355ba.jpg b/.raw/MidasFEAAnalysisManual/images/page-396_ea7ca277bc36419fdc0641fbbe18d7cbce972ef323e51c80c760466f8cd355ba.jpg new file mode 100644 index 00000000..6f277368 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-396_ea7ca277bc36419fdc0641fbbe18d7cbce972ef323e51c80c760466f8cd355ba.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-396_edf58246515342f9cd947da44c9e71fc70fc27c40b71f0f7e420525302505ad6.jpg b/.raw/MidasFEAAnalysisManual/images/page-396_edf58246515342f9cd947da44c9e71fc70fc27c40b71f0f7e420525302505ad6.jpg new file mode 100644 index 00000000..2246fc3b Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-396_edf58246515342f9cd947da44c9e71fc70fc27c40b71f0f7e420525302505ad6.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-396_efaf2ff0c0f25f7d4d6a9a90feafe77e453d117c4fd156461200af0a70e59818.jpg b/.raw/MidasFEAAnalysisManual/images/page-396_efaf2ff0c0f25f7d4d6a9a90feafe77e453d117c4fd156461200af0a70e59818.jpg new file mode 100644 index 00000000..6e5a2879 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-396_efaf2ff0c0f25f7d4d6a9a90feafe77e453d117c4fd156461200af0a70e59818.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-397_c2d790227bcf34d69815cd4dd309184d2b805ce771a1fda25b23724b006efa14.jpg b/.raw/MidasFEAAnalysisManual/images/page-397_c2d790227bcf34d69815cd4dd309184d2b805ce771a1fda25b23724b006efa14.jpg new file mode 100644 index 00000000..7c8754ba Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-397_c2d790227bcf34d69815cd4dd309184d2b805ce771a1fda25b23724b006efa14.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-398_e3c44420913a4a8ce26f31dc2c3b5b1865e2a3c6fda11705669b2d930beb151f.jpg b/.raw/MidasFEAAnalysisManual/images/page-398_e3c44420913a4a8ce26f31dc2c3b5b1865e2a3c6fda11705669b2d930beb151f.jpg new file mode 100644 index 00000000..7fbf1efa Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-398_e3c44420913a4a8ce26f31dc2c3b5b1865e2a3c6fda11705669b2d930beb151f.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-404_b97e9c670dcc1f56ea2d7312ac210cd7c38b11a2f4159c00c63074e97df18afb.jpg b/.raw/MidasFEAAnalysisManual/images/page-404_b97e9c670dcc1f56ea2d7312ac210cd7c38b11a2f4159c00c63074e97df18afb.jpg new file mode 100644 index 00000000..2844e716 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-404_b97e9c670dcc1f56ea2d7312ac210cd7c38b11a2f4159c00c63074e97df18afb.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-407_7fb27dfefc4c1e58e84d74ab2798174e9c9bef4bf1a8c9964bb3f6e6487cead5.jpg b/.raw/MidasFEAAnalysisManual/images/page-407_7fb27dfefc4c1e58e84d74ab2798174e9c9bef4bf1a8c9964bb3f6e6487cead5.jpg new file mode 100644 index 00000000..41061871 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-407_7fb27dfefc4c1e58e84d74ab2798174e9c9bef4bf1a8c9964bb3f6e6487cead5.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-408_3b889c9eb459dff8a66b892febe1971fca666b2bda315ca52ccaaa6d70f1ad0f.jpg b/.raw/MidasFEAAnalysisManual/images/page-408_3b889c9eb459dff8a66b892febe1971fca666b2bda315ca52ccaaa6d70f1ad0f.jpg new file mode 100644 index 00000000..1e715d42 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-408_3b889c9eb459dff8a66b892febe1971fca666b2bda315ca52ccaaa6d70f1ad0f.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-408_82c9f0dfb02a73f7fa00a82f583fd1287867c8a1c0649952f33a22cb8e6b665e.jpg b/.raw/MidasFEAAnalysisManual/images/page-408_82c9f0dfb02a73f7fa00a82f583fd1287867c8a1c0649952f33a22cb8e6b665e.jpg new file mode 100644 index 00000000..0714d969 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-408_82c9f0dfb02a73f7fa00a82f583fd1287867c8a1c0649952f33a22cb8e6b665e.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-409_28dd6d2d94b41a04619dfbdd7272ee3a2767c8a35058f78fc076feaecc341db7.jpg b/.raw/MidasFEAAnalysisManual/images/page-409_28dd6d2d94b41a04619dfbdd7272ee3a2767c8a35058f78fc076feaecc341db7.jpg new file mode 100644 index 00000000..efc1e699 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-409_28dd6d2d94b41a04619dfbdd7272ee3a2767c8a35058f78fc076feaecc341db7.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-409_3ce11deefc6da89926edc6555dd66593ada3b326eca882a8d18378c88754e82a.jpg b/.raw/MidasFEAAnalysisManual/images/page-409_3ce11deefc6da89926edc6555dd66593ada3b326eca882a8d18378c88754e82a.jpg new file mode 100644 index 00000000..16f61421 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-409_3ce11deefc6da89926edc6555dd66593ada3b326eca882a8d18378c88754e82a.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-409_5a023412e127e347ee87e2e5f0ebadfa406e1b58cfc3de0afa526c5995c6a03c.jpg b/.raw/MidasFEAAnalysisManual/images/page-409_5a023412e127e347ee87e2e5f0ebadfa406e1b58cfc3de0afa526c5995c6a03c.jpg new file mode 100644 index 00000000..fe74bf4e Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-409_5a023412e127e347ee87e2e5f0ebadfa406e1b58cfc3de0afa526c5995c6a03c.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-409_7a65fc44488f4b4947a0a121cebc2968e8933d7e5e4f11882ca51746b69a31b1.jpg b/.raw/MidasFEAAnalysisManual/images/page-409_7a65fc44488f4b4947a0a121cebc2968e8933d7e5e4f11882ca51746b69a31b1.jpg new file mode 100644 index 00000000..abed191b Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-409_7a65fc44488f4b4947a0a121cebc2968e8933d7e5e4f11882ca51746b69a31b1.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-414_01eab59a932bee69cd0af3fe3f7f4b86efa862959fa256926ef9d666740c4c9e.jpg b/.raw/MidasFEAAnalysisManual/images/page-414_01eab59a932bee69cd0af3fe3f7f4b86efa862959fa256926ef9d666740c4c9e.jpg new file mode 100644 index 00000000..93501561 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-414_01eab59a932bee69cd0af3fe3f7f4b86efa862959fa256926ef9d666740c4c9e.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-414_3ec5b42ca3cbab686041c314651ebfd945449551fac209dbb4b65956bd97e101.jpg b/.raw/MidasFEAAnalysisManual/images/page-414_3ec5b42ca3cbab686041c314651ebfd945449551fac209dbb4b65956bd97e101.jpg new file mode 100644 index 00000000..f0ec6298 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-414_3ec5b42ca3cbab686041c314651ebfd945449551fac209dbb4b65956bd97e101.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-414_40a7849adf2c136161ee0addb178f92cbafb5cbd499c1dcfda1df4fd5ab722d6.jpg b/.raw/MidasFEAAnalysisManual/images/page-414_40a7849adf2c136161ee0addb178f92cbafb5cbd499c1dcfda1df4fd5ab722d6.jpg new file mode 100644 index 00000000..b8671899 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-414_40a7849adf2c136161ee0addb178f92cbafb5cbd499c1dcfda1df4fd5ab722d6.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-414_4ef4df854ba557f58c38edd0a7de0d1d92e43198bcdf792690c62fa080b8d1cd.jpg b/.raw/MidasFEAAnalysisManual/images/page-414_4ef4df854ba557f58c38edd0a7de0d1d92e43198bcdf792690c62fa080b8d1cd.jpg new file mode 100644 index 00000000..1adcc6fb Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-414_4ef4df854ba557f58c38edd0a7de0d1d92e43198bcdf792690c62fa080b8d1cd.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-414_50483ef1bf2bb0c255d9ede400df35e925cbaaf36062c820e3816e763f8a0878.jpg b/.raw/MidasFEAAnalysisManual/images/page-414_50483ef1bf2bb0c255d9ede400df35e925cbaaf36062c820e3816e763f8a0878.jpg new file mode 100644 index 00000000..eecc4b97 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-414_50483ef1bf2bb0c255d9ede400df35e925cbaaf36062c820e3816e763f8a0878.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-414_964d5a731bc8af03934781366a2fdc9bf501890b7b444bd03905ae4b8284d50b.jpg b/.raw/MidasFEAAnalysisManual/images/page-414_964d5a731bc8af03934781366a2fdc9bf501890b7b444bd03905ae4b8284d50b.jpg new file mode 100644 index 00000000..18fdd4d4 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-414_964d5a731bc8af03934781366a2fdc9bf501890b7b444bd03905ae4b8284d50b.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-414_a0619d7d2b3f79609b1239e8902b25dd94825314d3acc0d553e6fef47a28503c.jpg b/.raw/MidasFEAAnalysisManual/images/page-414_a0619d7d2b3f79609b1239e8902b25dd94825314d3acc0d553e6fef47a28503c.jpg new file mode 100644 index 00000000..faf38082 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-414_a0619d7d2b3f79609b1239e8902b25dd94825314d3acc0d553e6fef47a28503c.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-415_1805ba7c42d5e457573fb69c4e09de96285d72dfb86329152df8713d6958bc47.jpg b/.raw/MidasFEAAnalysisManual/images/page-415_1805ba7c42d5e457573fb69c4e09de96285d72dfb86329152df8713d6958bc47.jpg new file mode 100644 index 00000000..89af67c0 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-415_1805ba7c42d5e457573fb69c4e09de96285d72dfb86329152df8713d6958bc47.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-415_1c69aff3205e0b1b309ebcd0a80d8a2d6090638dfb5f4f873b30920014441a8b.jpg b/.raw/MidasFEAAnalysisManual/images/page-415_1c69aff3205e0b1b309ebcd0a80d8a2d6090638dfb5f4f873b30920014441a8b.jpg new file mode 100644 index 00000000..2220a615 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-415_1c69aff3205e0b1b309ebcd0a80d8a2d6090638dfb5f4f873b30920014441a8b.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-415_303e7143282782139fa84a7c7b7cf3e16b09af656c63404315330b8e80d1aeb5.jpg b/.raw/MidasFEAAnalysisManual/images/page-415_303e7143282782139fa84a7c7b7cf3e16b09af656c63404315330b8e80d1aeb5.jpg new file mode 100644 index 00000000..0c6d2875 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-415_303e7143282782139fa84a7c7b7cf3e16b09af656c63404315330b8e80d1aeb5.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-415_3e1b43f9191ee6d664088d0ca07206d62d60ced6997e1b7957e47c5f8a11fe19.jpg b/.raw/MidasFEAAnalysisManual/images/page-415_3e1b43f9191ee6d664088d0ca07206d62d60ced6997e1b7957e47c5f8a11fe19.jpg new file mode 100644 index 00000000..ee02fd50 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-415_3e1b43f9191ee6d664088d0ca07206d62d60ced6997e1b7957e47c5f8a11fe19.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-415_e95b5cfacfe8261dd62a9ce08aa892f705c2ae9f238a23530c37e85bc3d45c0b.jpg b/.raw/MidasFEAAnalysisManual/images/page-415_e95b5cfacfe8261dd62a9ce08aa892f705c2ae9f238a23530c37e85bc3d45c0b.jpg new file mode 100644 index 00000000..9b51b050 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-415_e95b5cfacfe8261dd62a9ce08aa892f705c2ae9f238a23530c37e85bc3d45c0b.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-416_0782ee55a1db723a089e8e7ff1228b877ce7e5728eb70c01ebc9ac952985b0ad.jpg b/.raw/MidasFEAAnalysisManual/images/page-416_0782ee55a1db723a089e8e7ff1228b877ce7e5728eb70c01ebc9ac952985b0ad.jpg new file mode 100644 index 00000000..8291bc14 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-416_0782ee55a1db723a089e8e7ff1228b877ce7e5728eb70c01ebc9ac952985b0ad.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-416_594326d36f1a13b0c92c1f76d05b15036242281934daab600b111302589910b9.jpg b/.raw/MidasFEAAnalysisManual/images/page-416_594326d36f1a13b0c92c1f76d05b15036242281934daab600b111302589910b9.jpg new file mode 100644 index 00000000..6c6eb358 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-416_594326d36f1a13b0c92c1f76d05b15036242281934daab600b111302589910b9.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-416_5f3bed2a3dd5fb56066cccb845075f5d1f243996810b0d34380187fefac57bb1.jpg b/.raw/MidasFEAAnalysisManual/images/page-416_5f3bed2a3dd5fb56066cccb845075f5d1f243996810b0d34380187fefac57bb1.jpg new file mode 100644 index 00000000..0d128722 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-416_5f3bed2a3dd5fb56066cccb845075f5d1f243996810b0d34380187fefac57bb1.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-416_65f6e522d6aa8db3d3a9f4292fea656ac968f3ad94f72125222f1444b956eafa.jpg b/.raw/MidasFEAAnalysisManual/images/page-416_65f6e522d6aa8db3d3a9f4292fea656ac968f3ad94f72125222f1444b956eafa.jpg new file mode 100644 index 00000000..15961428 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-416_65f6e522d6aa8db3d3a9f4292fea656ac968f3ad94f72125222f1444b956eafa.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-416_7b088660ee965336a35d248f1c3a5681435fb288a4c647a6b8af58b874b9b819.jpg b/.raw/MidasFEAAnalysisManual/images/page-416_7b088660ee965336a35d248f1c3a5681435fb288a4c647a6b8af58b874b9b819.jpg new file mode 100644 index 00000000..a2333a1f Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-416_7b088660ee965336a35d248f1c3a5681435fb288a4c647a6b8af58b874b9b819.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-416_82e582f905c222cbf4bd758b067cf0c0540776acb3f032bb63f82a1b879d9998.jpg b/.raw/MidasFEAAnalysisManual/images/page-416_82e582f905c222cbf4bd758b067cf0c0540776acb3f032bb63f82a1b879d9998.jpg new file mode 100644 index 00000000..a16d9621 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-416_82e582f905c222cbf4bd758b067cf0c0540776acb3f032bb63f82a1b879d9998.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-416_86de487d8584bc5ccb5844edfe5ab308e00fba6f523ce3311b98f84f808e45ed.jpg b/.raw/MidasFEAAnalysisManual/images/page-416_86de487d8584bc5ccb5844edfe5ab308e00fba6f523ce3311b98f84f808e45ed.jpg new file mode 100644 index 00000000..c6b3fd21 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-416_86de487d8584bc5ccb5844edfe5ab308e00fba6f523ce3311b98f84f808e45ed.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-416_942bf712968f5073988207dbce138919bb86bb001d1e5219597f2578cc31e473.jpg b/.raw/MidasFEAAnalysisManual/images/page-416_942bf712968f5073988207dbce138919bb86bb001d1e5219597f2578cc31e473.jpg new file mode 100644 index 00000000..d5d9d70c Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-416_942bf712968f5073988207dbce138919bb86bb001d1e5219597f2578cc31e473.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-416_9925818ae93263ddc2f4b8ee0aefb8f9a92b5c4eddc8e1f6a3b2699805687378.jpg b/.raw/MidasFEAAnalysisManual/images/page-416_9925818ae93263ddc2f4b8ee0aefb8f9a92b5c4eddc8e1f6a3b2699805687378.jpg new file mode 100644 index 00000000..a013dc84 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-416_9925818ae93263ddc2f4b8ee0aefb8f9a92b5c4eddc8e1f6a3b2699805687378.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-416_b5b4f7983e3fa87a7419efb873cd83c784aeef2231e356371e4c0f6b6f0371c0.jpg b/.raw/MidasFEAAnalysisManual/images/page-416_b5b4f7983e3fa87a7419efb873cd83c784aeef2231e356371e4c0f6b6f0371c0.jpg new file mode 100644 index 00000000..6b884a22 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-416_b5b4f7983e3fa87a7419efb873cd83c784aeef2231e356371e4c0f6b6f0371c0.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-416_c0084908b89e3af9ea1c3d4591701fbfefa832d886cc2a087e3c2294d0b1686f.jpg b/.raw/MidasFEAAnalysisManual/images/page-416_c0084908b89e3af9ea1c3d4591701fbfefa832d886cc2a087e3c2294d0b1686f.jpg new file mode 100644 index 00000000..5e6af7e0 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-416_c0084908b89e3af9ea1c3d4591701fbfefa832d886cc2a087e3c2294d0b1686f.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-416_d6776644c21c921e105c2f6de64306964413790c31b98dcb2677f821e6c77622.jpg b/.raw/MidasFEAAnalysisManual/images/page-416_d6776644c21c921e105c2f6de64306964413790c31b98dcb2677f821e6c77622.jpg new file mode 100644 index 00000000..97459522 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-416_d6776644c21c921e105c2f6de64306964413790c31b98dcb2677f821e6c77622.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-416_d9667ba2818d1ba656229bf86f47571a55d8404645740a4b586a49dc1f847e6e.jpg b/.raw/MidasFEAAnalysisManual/images/page-416_d9667ba2818d1ba656229bf86f47571a55d8404645740a4b586a49dc1f847e6e.jpg new file mode 100644 index 00000000..be33dda0 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-416_d9667ba2818d1ba656229bf86f47571a55d8404645740a4b586a49dc1f847e6e.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-416_e6d9032504991141f52f2fb172786f190757e6d245078636c98dd44d3e9e0f43.jpg b/.raw/MidasFEAAnalysisManual/images/page-416_e6d9032504991141f52f2fb172786f190757e6d245078636c98dd44d3e9e0f43.jpg new file mode 100644 index 00000000..ae2677e8 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-416_e6d9032504991141f52f2fb172786f190757e6d245078636c98dd44d3e9e0f43.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-418_b656882326d99b85b260452d2b24532b40179c2a28cbe40acd3297668ccce544.jpg b/.raw/MidasFEAAnalysisManual/images/page-418_b656882326d99b85b260452d2b24532b40179c2a28cbe40acd3297668ccce544.jpg new file mode 100644 index 00000000..f22a4772 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-418_b656882326d99b85b260452d2b24532b40179c2a28cbe40acd3297668ccce544.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-418_d0f61c50bcb990bdc05af3b7e606c55ba7e73fdd32a0151e5f91e9262c60a905.jpg b/.raw/MidasFEAAnalysisManual/images/page-418_d0f61c50bcb990bdc05af3b7e606c55ba7e73fdd32a0151e5f91e9262c60a905.jpg new file mode 100644 index 00000000..b06263c3 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-418_d0f61c50bcb990bdc05af3b7e606c55ba7e73fdd32a0151e5f91e9262c60a905.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-419_05b9bb462df0f92805a3ca7c516c101b1cb5f039fa816d3bd4fbc33db791de72.jpg b/.raw/MidasFEAAnalysisManual/images/page-419_05b9bb462df0f92805a3ca7c516c101b1cb5f039fa816d3bd4fbc33db791de72.jpg new file mode 100644 index 00000000..4813f92a Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-419_05b9bb462df0f92805a3ca7c516c101b1cb5f039fa816d3bd4fbc33db791de72.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-419_1e20b161d527e4f7429c1326e6e008a386e0fcb7ee39be00b8cb1ff02399d754.jpg b/.raw/MidasFEAAnalysisManual/images/page-419_1e20b161d527e4f7429c1326e6e008a386e0fcb7ee39be00b8cb1ff02399d754.jpg new file mode 100644 index 00000000..288bc94a Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-419_1e20b161d527e4f7429c1326e6e008a386e0fcb7ee39be00b8cb1ff02399d754.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-419_6e8ac3a41b77a6eb6ce8cb5a37907774394cedad31d99f2f068d34e7d575e760.jpg b/.raw/MidasFEAAnalysisManual/images/page-419_6e8ac3a41b77a6eb6ce8cb5a37907774394cedad31d99f2f068d34e7d575e760.jpg new file mode 100644 index 00000000..2a4cd311 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-419_6e8ac3a41b77a6eb6ce8cb5a37907774394cedad31d99f2f068d34e7d575e760.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-419_cff4ad22445fe390c519dad527661cdc1cd8cb6eb697b67560cdddf61a84b6e7.jpg b/.raw/MidasFEAAnalysisManual/images/page-419_cff4ad22445fe390c519dad527661cdc1cd8cb6eb697b67560cdddf61a84b6e7.jpg new file mode 100644 index 00000000..84dc58c9 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-419_cff4ad22445fe390c519dad527661cdc1cd8cb6eb697b67560cdddf61a84b6e7.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-420_127ba1006a7dc53bd9db3ccb13c4a1d4f7a8a1147aefb5f8b72c8c31d4e0aca0.jpg b/.raw/MidasFEAAnalysisManual/images/page-420_127ba1006a7dc53bd9db3ccb13c4a1d4f7a8a1147aefb5f8b72c8c31d4e0aca0.jpg new file mode 100644 index 00000000..05aa482e Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-420_127ba1006a7dc53bd9db3ccb13c4a1d4f7a8a1147aefb5f8b72c8c31d4e0aca0.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-420_246fe44218113dbf00c7990018fdcff05f6347f611b7c766c7cacc17d07e6dfd.jpg b/.raw/MidasFEAAnalysisManual/images/page-420_246fe44218113dbf00c7990018fdcff05f6347f611b7c766c7cacc17d07e6dfd.jpg new file mode 100644 index 00000000..0408482c Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-420_246fe44218113dbf00c7990018fdcff05f6347f611b7c766c7cacc17d07e6dfd.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-420_4ac11c1cdb644b38d476fce76b43a0d2dec31f18b7eae9374b93f5f15fab2640.jpg b/.raw/MidasFEAAnalysisManual/images/page-420_4ac11c1cdb644b38d476fce76b43a0d2dec31f18b7eae9374b93f5f15fab2640.jpg new file mode 100644 index 00000000..b6465016 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-420_4ac11c1cdb644b38d476fce76b43a0d2dec31f18b7eae9374b93f5f15fab2640.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-420_6bd93306d62d42a50004eb6b58879b0031875ab9d9f57001c026f8e9f638284c.jpg b/.raw/MidasFEAAnalysisManual/images/page-420_6bd93306d62d42a50004eb6b58879b0031875ab9d9f57001c026f8e9f638284c.jpg new file mode 100644 index 00000000..07ce7d61 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-420_6bd93306d62d42a50004eb6b58879b0031875ab9d9f57001c026f8e9f638284c.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-420_86346ff47613285242779891c93fbb7808b2afac1cbeeb861219279c1029ab77.jpg b/.raw/MidasFEAAnalysisManual/images/page-420_86346ff47613285242779891c93fbb7808b2afac1cbeeb861219279c1029ab77.jpg new file mode 100644 index 00000000..0a95cbae Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-420_86346ff47613285242779891c93fbb7808b2afac1cbeeb861219279c1029ab77.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-420_b25c77819d57c2be2f3568b553dd840d3c3dc774b00ad2743c97ebe6adf4f5b6.jpg b/.raw/MidasFEAAnalysisManual/images/page-420_b25c77819d57c2be2f3568b553dd840d3c3dc774b00ad2743c97ebe6adf4f5b6.jpg new file mode 100644 index 00000000..5315a0ee Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-420_b25c77819d57c2be2f3568b553dd840d3c3dc774b00ad2743c97ebe6adf4f5b6.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-420_b3a0fa20b561d34be759ffd805563e5584644baef5a23ed70fd77d895b4b3d50.jpg b/.raw/MidasFEAAnalysisManual/images/page-420_b3a0fa20b561d34be759ffd805563e5584644baef5a23ed70fd77d895b4b3d50.jpg new file mode 100644 index 00000000..996eb6c2 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-420_b3a0fa20b561d34be759ffd805563e5584644baef5a23ed70fd77d895b4b3d50.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-420_b642022134421bef9136935bcabff5f5496744836783aabb31780024ec242f97.jpg b/.raw/MidasFEAAnalysisManual/images/page-420_b642022134421bef9136935bcabff5f5496744836783aabb31780024ec242f97.jpg new file mode 100644 index 00000000..24fac4fb Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-420_b642022134421bef9136935bcabff5f5496744836783aabb31780024ec242f97.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-420_e7252cd7c8f9228f1506d8aba84ab4f27f90f9687473f0319c520015269d00b1.jpg b/.raw/MidasFEAAnalysisManual/images/page-420_e7252cd7c8f9228f1506d8aba84ab4f27f90f9687473f0319c520015269d00b1.jpg new file mode 100644 index 00000000..2cd8c80c Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-420_e7252cd7c8f9228f1506d8aba84ab4f27f90f9687473f0319c520015269d00b1.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-422_f14bf3ba692844d67c2225c8610b47141c19297fabfd9fdd39d376dcdf482fb9.jpg b/.raw/MidasFEAAnalysisManual/images/page-422_f14bf3ba692844d67c2225c8610b47141c19297fabfd9fdd39d376dcdf482fb9.jpg new file mode 100644 index 00000000..39404b8b Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-422_f14bf3ba692844d67c2225c8610b47141c19297fabfd9fdd39d376dcdf482fb9.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-423_2fb9d486a02f9039f574b43ea0571ccc80bc31459273a5f5f0cc421842d430a3.jpg b/.raw/MidasFEAAnalysisManual/images/page-423_2fb9d486a02f9039f574b43ea0571ccc80bc31459273a5f5f0cc421842d430a3.jpg new file mode 100644 index 00000000..f0192130 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-423_2fb9d486a02f9039f574b43ea0571ccc80bc31459273a5f5f0cc421842d430a3.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-423_7716f2d7d44607b074b3200b82e54315870cd5f170c6ca2f89cf69516cec0c31.jpg b/.raw/MidasFEAAnalysisManual/images/page-423_7716f2d7d44607b074b3200b82e54315870cd5f170c6ca2f89cf69516cec0c31.jpg new file mode 100644 index 00000000..f26413d0 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-423_7716f2d7d44607b074b3200b82e54315870cd5f170c6ca2f89cf69516cec0c31.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-424_33a61cdbe4a720b5e8858a91ce2e5e71a6ed83246b49df6de89cf78d9342c385.jpg b/.raw/MidasFEAAnalysisManual/images/page-424_33a61cdbe4a720b5e8858a91ce2e5e71a6ed83246b49df6de89cf78d9342c385.jpg new file mode 100644 index 00000000..7bad47bc Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-424_33a61cdbe4a720b5e8858a91ce2e5e71a6ed83246b49df6de89cf78d9342c385.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-424_4bc33848d2ea2935b3645f230774e909cc45033db0ed7bec4be7437ed55fa46e.jpg b/.raw/MidasFEAAnalysisManual/images/page-424_4bc33848d2ea2935b3645f230774e909cc45033db0ed7bec4be7437ed55fa46e.jpg new file mode 100644 index 00000000..7041b212 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-424_4bc33848d2ea2935b3645f230774e909cc45033db0ed7bec4be7437ed55fa46e.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-424_62d2952905403fb277f24d45322c9415f4e7a0402df595c3fa6c6cb3f744f82c.jpg b/.raw/MidasFEAAnalysisManual/images/page-424_62d2952905403fb277f24d45322c9415f4e7a0402df595c3fa6c6cb3f744f82c.jpg new file mode 100644 index 00000000..bb17cf19 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-424_62d2952905403fb277f24d45322c9415f4e7a0402df595c3fa6c6cb3f744f82c.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-424_7636f84aa1f515b8589e75996de9800b2909b3132658c0517db3478dde2de43b.jpg b/.raw/MidasFEAAnalysisManual/images/page-424_7636f84aa1f515b8589e75996de9800b2909b3132658c0517db3478dde2de43b.jpg new file mode 100644 index 00000000..f5b21623 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-424_7636f84aa1f515b8589e75996de9800b2909b3132658c0517db3478dde2de43b.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-424_a83fc3da85b682e23f27ad45639c935aeffdc73b2d0911f75e4f45171763b02b.jpg b/.raw/MidasFEAAnalysisManual/images/page-424_a83fc3da85b682e23f27ad45639c935aeffdc73b2d0911f75e4f45171763b02b.jpg new file mode 100644 index 00000000..796344fb Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-424_a83fc3da85b682e23f27ad45639c935aeffdc73b2d0911f75e4f45171763b02b.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-424_af9d94aadc6268144a896c539ba8a005c156fb41ee153a3f434f7fa56fa87b25.jpg b/.raw/MidasFEAAnalysisManual/images/page-424_af9d94aadc6268144a896c539ba8a005c156fb41ee153a3f434f7fa56fa87b25.jpg new file mode 100644 index 00000000..c5b22e84 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-424_af9d94aadc6268144a896c539ba8a005c156fb41ee153a3f434f7fa56fa87b25.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-424_b06cafaf07143fb3f694d1c3491855fa0cc3782670bb3224163afd660c737def.jpg b/.raw/MidasFEAAnalysisManual/images/page-424_b06cafaf07143fb3f694d1c3491855fa0cc3782670bb3224163afd660c737def.jpg new file mode 100644 index 00000000..24d28e29 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-424_b06cafaf07143fb3f694d1c3491855fa0cc3782670bb3224163afd660c737def.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-424_cbf8f17d4c4c88491646abb2463e8bdee670c22bb3ccacf0868970c7d983a622.jpg b/.raw/MidasFEAAnalysisManual/images/page-424_cbf8f17d4c4c88491646abb2463e8bdee670c22bb3ccacf0868970c7d983a622.jpg new file mode 100644 index 00000000..18d3819d Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-424_cbf8f17d4c4c88491646abb2463e8bdee670c22bb3ccacf0868970c7d983a622.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-424_cd412e27da9869ac4e10a40aeb44b3c38877028da548afbd47884544a27960a4.jpg b/.raw/MidasFEAAnalysisManual/images/page-424_cd412e27da9869ac4e10a40aeb44b3c38877028da548afbd47884544a27960a4.jpg new file mode 100644 index 00000000..12aad6e0 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-424_cd412e27da9869ac4e10a40aeb44b3c38877028da548afbd47884544a27960a4.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-424_e6e1070e8a7e41327e011bec4c4691bb5b720dc5936394a26d94b84fc0c118c6.jpg b/.raw/MidasFEAAnalysisManual/images/page-424_e6e1070e8a7e41327e011bec4c4691bb5b720dc5936394a26d94b84fc0c118c6.jpg new file mode 100644 index 00000000..536a7b6c Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-424_e6e1070e8a7e41327e011bec4c4691bb5b720dc5936394a26d94b84fc0c118c6.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-424_f16216bb2b0068a5aa2622667bab737f0528ca5bd62c6f65673d34be5cbb0314.jpg b/.raw/MidasFEAAnalysisManual/images/page-424_f16216bb2b0068a5aa2622667bab737f0528ca5bd62c6f65673d34be5cbb0314.jpg new file mode 100644 index 00000000..642153fd Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-424_f16216bb2b0068a5aa2622667bab737f0528ca5bd62c6f65673d34be5cbb0314.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-425_3d3fb8c384be3066af2f498fad67c56866174363f7778b6712a4a53840ad66c2.jpg b/.raw/MidasFEAAnalysisManual/images/page-425_3d3fb8c384be3066af2f498fad67c56866174363f7778b6712a4a53840ad66c2.jpg new file mode 100644 index 00000000..9ac9b416 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-425_3d3fb8c384be3066af2f498fad67c56866174363f7778b6712a4a53840ad66c2.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-425_7e3d9ba05a47df58927c27aba71e93f09df02a5005c2954407799cb346220d62.jpg b/.raw/MidasFEAAnalysisManual/images/page-425_7e3d9ba05a47df58927c27aba71e93f09df02a5005c2954407799cb346220d62.jpg new file mode 100644 index 00000000..3b2edcf0 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-425_7e3d9ba05a47df58927c27aba71e93f09df02a5005c2954407799cb346220d62.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-431_24a6f994be28cb6a939b1d3b7c8ee3b9c3a35f3a96bda8baf0a5888122804cbb.jpg b/.raw/MidasFEAAnalysisManual/images/page-431_24a6f994be28cb6a939b1d3b7c8ee3b9c3a35f3a96bda8baf0a5888122804cbb.jpg new file mode 100644 index 00000000..82634772 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-431_24a6f994be28cb6a939b1d3b7c8ee3b9c3a35f3a96bda8baf0a5888122804cbb.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-431_4ee5a52f25e0110c6af294468815331b092ec3c5d88f432345c3f170a5efc337.jpg b/.raw/MidasFEAAnalysisManual/images/page-431_4ee5a52f25e0110c6af294468815331b092ec3c5d88f432345c3f170a5efc337.jpg new file mode 100644 index 00000000..00074423 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-431_4ee5a52f25e0110c6af294468815331b092ec3c5d88f432345c3f170a5efc337.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-431_550fa5534db2c8883a8799cea70dc48591f9d732937285ec906271d1c2b8fdcc.jpg b/.raw/MidasFEAAnalysisManual/images/page-431_550fa5534db2c8883a8799cea70dc48591f9d732937285ec906271d1c2b8fdcc.jpg new file mode 100644 index 00000000..58aee1d6 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-431_550fa5534db2c8883a8799cea70dc48591f9d732937285ec906271d1c2b8fdcc.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-431_65a50cfb9c1b31392ae20d54c839e68badaef915f4c1d326d5992c24400e84e5.jpg b/.raw/MidasFEAAnalysisManual/images/page-431_65a50cfb9c1b31392ae20d54c839e68badaef915f4c1d326d5992c24400e84e5.jpg new file mode 100644 index 00000000..28a0e1b7 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-431_65a50cfb9c1b31392ae20d54c839e68badaef915f4c1d326d5992c24400e84e5.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-432_a2349847f7dea102f597a6b147c21994059d35956362dad7e75d08d233e56157.jpg b/.raw/MidasFEAAnalysisManual/images/page-432_a2349847f7dea102f597a6b147c21994059d35956362dad7e75d08d233e56157.jpg new file mode 100644 index 00000000..b77295fb Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-432_a2349847f7dea102f597a6b147c21994059d35956362dad7e75d08d233e56157.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-433_18b692e813e95cffc98d8aff26226dd2ee4ce0d6266c54481f40f326755485da.jpg b/.raw/MidasFEAAnalysisManual/images/page-433_18b692e813e95cffc98d8aff26226dd2ee4ce0d6266c54481f40f326755485da.jpg new file mode 100644 index 00000000..17dc54d1 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-433_18b692e813e95cffc98d8aff26226dd2ee4ce0d6266c54481f40f326755485da.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-433_3e7cfa7389c292b2340f2fda7b7e8d30e9edd10116fbe14be83ad9b1dbf10cec.jpg b/.raw/MidasFEAAnalysisManual/images/page-433_3e7cfa7389c292b2340f2fda7b7e8d30e9edd10116fbe14be83ad9b1dbf10cec.jpg new file mode 100644 index 00000000..8192a130 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-433_3e7cfa7389c292b2340f2fda7b7e8d30e9edd10116fbe14be83ad9b1dbf10cec.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-433_4c356f6676862edfc7fe04829be1d12b77e2d80a9f0492f88ac0bf3fab2ecefd.jpg b/.raw/MidasFEAAnalysisManual/images/page-433_4c356f6676862edfc7fe04829be1d12b77e2d80a9f0492f88ac0bf3fab2ecefd.jpg new file mode 100644 index 00000000..d1f84db9 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-433_4c356f6676862edfc7fe04829be1d12b77e2d80a9f0492f88ac0bf3fab2ecefd.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-433_74f0f7f669750bfaed43295d7f482cd1075174671c65c1719586c50c7d11ac91.jpg b/.raw/MidasFEAAnalysisManual/images/page-433_74f0f7f669750bfaed43295d7f482cd1075174671c65c1719586c50c7d11ac91.jpg new file mode 100644 index 00000000..c8202577 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-433_74f0f7f669750bfaed43295d7f482cd1075174671c65c1719586c50c7d11ac91.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-433_8f2e3ace89d1cddbb54f70d9380107347de1b0cecc12d05c7a6424ce7d26fa55.jpg b/.raw/MidasFEAAnalysisManual/images/page-433_8f2e3ace89d1cddbb54f70d9380107347de1b0cecc12d05c7a6424ce7d26fa55.jpg new file mode 100644 index 00000000..3ec63a14 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-433_8f2e3ace89d1cddbb54f70d9380107347de1b0cecc12d05c7a6424ce7d26fa55.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-433_caf3c9e967d751653a79af38459bff542d1322e56c21f90ec7018856d379581e.jpg b/.raw/MidasFEAAnalysisManual/images/page-433_caf3c9e967d751653a79af38459bff542d1322e56c21f90ec7018856d379581e.jpg new file mode 100644 index 00000000..9985bdf9 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-433_caf3c9e967d751653a79af38459bff542d1322e56c21f90ec7018856d379581e.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-433_daca368d57f9660f455648d9021d5712b25c9d25b20dfd01a69a96e0ed0831eb.jpg b/.raw/MidasFEAAnalysisManual/images/page-433_daca368d57f9660f455648d9021d5712b25c9d25b20dfd01a69a96e0ed0831eb.jpg new file mode 100644 index 00000000..749c14ab Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-433_daca368d57f9660f455648d9021d5712b25c9d25b20dfd01a69a96e0ed0831eb.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-433_dcf2be6301321e1c3c6927bacddc20887d453ea5798da1a8931ab0dbcfb35e95.jpg b/.raw/MidasFEAAnalysisManual/images/page-433_dcf2be6301321e1c3c6927bacddc20887d453ea5798da1a8931ab0dbcfb35e95.jpg new file mode 100644 index 00000000..ef204c11 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-433_dcf2be6301321e1c3c6927bacddc20887d453ea5798da1a8931ab0dbcfb35e95.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-434_427c649e149602fc501a9e9c3a3f0ab93ad0785b7e06d41b63b3099fc499e1b5.jpg b/.raw/MidasFEAAnalysisManual/images/page-434_427c649e149602fc501a9e9c3a3f0ab93ad0785b7e06d41b63b3099fc499e1b5.jpg new file mode 100644 index 00000000..4f0acad7 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-434_427c649e149602fc501a9e9c3a3f0ab93ad0785b7e06d41b63b3099fc499e1b5.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-434_7080c5683a8b2114f716e258c9593e267e47f1e431aab22effbb30fa4b4d460c.jpg b/.raw/MidasFEAAnalysisManual/images/page-434_7080c5683a8b2114f716e258c9593e267e47f1e431aab22effbb30fa4b4d460c.jpg new file mode 100644 index 00000000..db0b8e49 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-434_7080c5683a8b2114f716e258c9593e267e47f1e431aab22effbb30fa4b4d460c.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-434_7dd6580cecf4f053917b1572728fd9bda54cb425eda0eaeb134675ce91d058d5.jpg b/.raw/MidasFEAAnalysisManual/images/page-434_7dd6580cecf4f053917b1572728fd9bda54cb425eda0eaeb134675ce91d058d5.jpg new file mode 100644 index 00000000..7e21968c Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-434_7dd6580cecf4f053917b1572728fd9bda54cb425eda0eaeb134675ce91d058d5.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-434_dd53a5f683e50234af1e514eb9ea9f99439a830e4c12d7dc32d3151c217d5af5.jpg b/.raw/MidasFEAAnalysisManual/images/page-434_dd53a5f683e50234af1e514eb9ea9f99439a830e4c12d7dc32d3151c217d5af5.jpg new file mode 100644 index 00000000..1cb2178a Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-434_dd53a5f683e50234af1e514eb9ea9f99439a830e4c12d7dc32d3151c217d5af5.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-436_4c89a593bff73ceeae1ac1ab5fbc9a33fb7a067c8b66a52d2cf611cd47ec5ef8.jpg b/.raw/MidasFEAAnalysisManual/images/page-436_4c89a593bff73ceeae1ac1ab5fbc9a33fb7a067c8b66a52d2cf611cd47ec5ef8.jpg new file mode 100644 index 00000000..40e0ec8c Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-436_4c89a593bff73ceeae1ac1ab5fbc9a33fb7a067c8b66a52d2cf611cd47ec5ef8.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-438_0e5eac6942ab37062f5c10dbc407a2fff9d2c73d339190baae03fa1202c7e1db.jpg b/.raw/MidasFEAAnalysisManual/images/page-438_0e5eac6942ab37062f5c10dbc407a2fff9d2c73d339190baae03fa1202c7e1db.jpg new file mode 100644 index 00000000..50c038c8 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-438_0e5eac6942ab37062f5c10dbc407a2fff9d2c73d339190baae03fa1202c7e1db.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-438_1c17a57533716ba859587d496f4a86c48c96b8f89b9fd45125dc44ec694c749f.jpg b/.raw/MidasFEAAnalysisManual/images/page-438_1c17a57533716ba859587d496f4a86c48c96b8f89b9fd45125dc44ec694c749f.jpg new file mode 100644 index 00000000..759fb27e Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-438_1c17a57533716ba859587d496f4a86c48c96b8f89b9fd45125dc44ec694c749f.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-438_7e040b78a21c21aa0ec5a29ad9e7d0e790647f9a8fcd9caec869aaf580ea9d38.jpg b/.raw/MidasFEAAnalysisManual/images/page-438_7e040b78a21c21aa0ec5a29ad9e7d0e790647f9a8fcd9caec869aaf580ea9d38.jpg new file mode 100644 index 00000000..f6baa7cd Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-438_7e040b78a21c21aa0ec5a29ad9e7d0e790647f9a8fcd9caec869aaf580ea9d38.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-438_ef82774f3ed8b22e716c1466e5f8a92e6b78752b15973480289b2a5f24af632f.jpg b/.raw/MidasFEAAnalysisManual/images/page-438_ef82774f3ed8b22e716c1466e5f8a92e6b78752b15973480289b2a5f24af632f.jpg new file mode 100644 index 00000000..56205bd9 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-438_ef82774f3ed8b22e716c1466e5f8a92e6b78752b15973480289b2a5f24af632f.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-439_2d6cba53e621bd244772f5fe6cb446bfa488e7167c283eed9bb8144b5267b659.jpg b/.raw/MidasFEAAnalysisManual/images/page-439_2d6cba53e621bd244772f5fe6cb446bfa488e7167c283eed9bb8144b5267b659.jpg new file mode 100644 index 00000000..38de2577 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-439_2d6cba53e621bd244772f5fe6cb446bfa488e7167c283eed9bb8144b5267b659.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-439_8e6fb5645921f62b83f701114016a5c53b82c3ac9bda812962e56ee8ef64643e.jpg b/.raw/MidasFEAAnalysisManual/images/page-439_8e6fb5645921f62b83f701114016a5c53b82c3ac9bda812962e56ee8ef64643e.jpg new file mode 100644 index 00000000..04220b23 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-439_8e6fb5645921f62b83f701114016a5c53b82c3ac9bda812962e56ee8ef64643e.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-439_afc9718336edf1a83daafe926b8dc669d95f1f55e979b1f6bb61b4c4ae51c3af.jpg b/.raw/MidasFEAAnalysisManual/images/page-439_afc9718336edf1a83daafe926b8dc669d95f1f55e979b1f6bb61b4c4ae51c3af.jpg new file mode 100644 index 00000000..608f57e1 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-439_afc9718336edf1a83daafe926b8dc669d95f1f55e979b1f6bb61b4c4ae51c3af.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-439_de720500d99c4d22ff06b6dda6a494c3953f49cbae94f09ffc2a2ec6a3ab13fe.jpg b/.raw/MidasFEAAnalysisManual/images/page-439_de720500d99c4d22ff06b6dda6a494c3953f49cbae94f09ffc2a2ec6a3ab13fe.jpg new file mode 100644 index 00000000..691f249f Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-439_de720500d99c4d22ff06b6dda6a494c3953f49cbae94f09ffc2a2ec6a3ab13fe.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-440_8316e8a49473232905a0f9b6079e8d5ec1fbf5185e0195a4c506c865689e54df.jpg b/.raw/MidasFEAAnalysisManual/images/page-440_8316e8a49473232905a0f9b6079e8d5ec1fbf5185e0195a4c506c865689e54df.jpg new file mode 100644 index 00000000..ac095323 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-440_8316e8a49473232905a0f9b6079e8d5ec1fbf5185e0195a4c506c865689e54df.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-441_2c689ac303ab3c892b2d0d2455d21ca4f54181383d7de84f08da4926f99c7251.jpg b/.raw/MidasFEAAnalysisManual/images/page-441_2c689ac303ab3c892b2d0d2455d21ca4f54181383d7de84f08da4926f99c7251.jpg new file mode 100644 index 00000000..c2f9a1d2 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-441_2c689ac303ab3c892b2d0d2455d21ca4f54181383d7de84f08da4926f99c7251.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-441_9b3d77cf26d782d999b326735768d0ec257bda28aca8255629b468be5d773b89.jpg b/.raw/MidasFEAAnalysisManual/images/page-441_9b3d77cf26d782d999b326735768d0ec257bda28aca8255629b468be5d773b89.jpg new file mode 100644 index 00000000..3bb354dd Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-441_9b3d77cf26d782d999b326735768d0ec257bda28aca8255629b468be5d773b89.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-442_10af45d06fe6d23532ff8bac3eb9ccfc1d197b10c5b3d34991ece12724b0a153.jpg b/.raw/MidasFEAAnalysisManual/images/page-442_10af45d06fe6d23532ff8bac3eb9ccfc1d197b10c5b3d34991ece12724b0a153.jpg new file mode 100644 index 00000000..80490b5d Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-442_10af45d06fe6d23532ff8bac3eb9ccfc1d197b10c5b3d34991ece12724b0a153.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-442_1d223862ba51774442ee2a2a7280078abd5184aa017a5dfddbd5719a966344e4.jpg b/.raw/MidasFEAAnalysisManual/images/page-442_1d223862ba51774442ee2a2a7280078abd5184aa017a5dfddbd5719a966344e4.jpg new file mode 100644 index 00000000..3f6c9f7b Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-442_1d223862ba51774442ee2a2a7280078abd5184aa017a5dfddbd5719a966344e4.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-442_3931ffe96df2b7434b36e5db3cf68ed6b07b307ddfb134536b6dff1112419759.jpg b/.raw/MidasFEAAnalysisManual/images/page-442_3931ffe96df2b7434b36e5db3cf68ed6b07b307ddfb134536b6dff1112419759.jpg new file mode 100644 index 00000000..e9ea7e85 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-442_3931ffe96df2b7434b36e5db3cf68ed6b07b307ddfb134536b6dff1112419759.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-442_5e0a14fade78c0f5e5f3928fd4e81c28f7446560d4679c8056307aa263806454.jpg b/.raw/MidasFEAAnalysisManual/images/page-442_5e0a14fade78c0f5e5f3928fd4e81c28f7446560d4679c8056307aa263806454.jpg new file mode 100644 index 00000000..b357f9f6 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-442_5e0a14fade78c0f5e5f3928fd4e81c28f7446560d4679c8056307aa263806454.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-442_71b1fd69070cfd70a2e69ee06b039ed66e5c6b119c3b9fbee2f058bf2f845a58.jpg b/.raw/MidasFEAAnalysisManual/images/page-442_71b1fd69070cfd70a2e69ee06b039ed66e5c6b119c3b9fbee2f058bf2f845a58.jpg new file mode 100644 index 00000000..4139958d Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-442_71b1fd69070cfd70a2e69ee06b039ed66e5c6b119c3b9fbee2f058bf2f845a58.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-442_7a5ee48e96753da034411080281935ac2fa2c0ae316a3ff862bbeef882f5b079.jpg b/.raw/MidasFEAAnalysisManual/images/page-442_7a5ee48e96753da034411080281935ac2fa2c0ae316a3ff862bbeef882f5b079.jpg new file mode 100644 index 00000000..084c2ef4 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-442_7a5ee48e96753da034411080281935ac2fa2c0ae316a3ff862bbeef882f5b079.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-442_ae8e6d2a9cc51a2e89f15233a2a4ea5eb73b4d66b41ed73dbc726516e6abde91.jpg b/.raw/MidasFEAAnalysisManual/images/page-442_ae8e6d2a9cc51a2e89f15233a2a4ea5eb73b4d66b41ed73dbc726516e6abde91.jpg new file mode 100644 index 00000000..c9ba6bf3 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-442_ae8e6d2a9cc51a2e89f15233a2a4ea5eb73b4d66b41ed73dbc726516e6abde91.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-442_d35da2d3a420200f14b77675e9c521b6a9ca9504d134104ff87df5621835294d.jpg b/.raw/MidasFEAAnalysisManual/images/page-442_d35da2d3a420200f14b77675e9c521b6a9ca9504d134104ff87df5621835294d.jpg new file mode 100644 index 00000000..687faed1 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-442_d35da2d3a420200f14b77675e9c521b6a9ca9504d134104ff87df5621835294d.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-442_d5807caadc42201fcb3ae805bb2440cf3cea957cc81fcd4bbef0fa5168ae8829.jpg b/.raw/MidasFEAAnalysisManual/images/page-442_d5807caadc42201fcb3ae805bb2440cf3cea957cc81fcd4bbef0fa5168ae8829.jpg new file mode 100644 index 00000000..c70e7c87 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-442_d5807caadc42201fcb3ae805bb2440cf3cea957cc81fcd4bbef0fa5168ae8829.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-442_ec8c005207492a3f281460d43fb66915e645a9a2e61bba0aca991b7c1c7cf662.jpg b/.raw/MidasFEAAnalysisManual/images/page-442_ec8c005207492a3f281460d43fb66915e645a9a2e61bba0aca991b7c1c7cf662.jpg new file mode 100644 index 00000000..8959ff51 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-442_ec8c005207492a3f281460d43fb66915e645a9a2e61bba0aca991b7c1c7cf662.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-443_42a5e715c56489f121b1a5c738174006bbd27f39b181777c930a78c08fd5035a.jpg b/.raw/MidasFEAAnalysisManual/images/page-443_42a5e715c56489f121b1a5c738174006bbd27f39b181777c930a78c08fd5035a.jpg new file mode 100644 index 00000000..e01e9091 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-443_42a5e715c56489f121b1a5c738174006bbd27f39b181777c930a78c08fd5035a.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-443_43f53aa91448ecbaeeb8f03bb80bdb6aa2251482b15b9cad198a191f62854afe.jpg b/.raw/MidasFEAAnalysisManual/images/page-443_43f53aa91448ecbaeeb8f03bb80bdb6aa2251482b15b9cad198a191f62854afe.jpg new file mode 100644 index 00000000..6dfd19ab Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-443_43f53aa91448ecbaeeb8f03bb80bdb6aa2251482b15b9cad198a191f62854afe.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-443_9366d435a9dbbf2b105cc10ada7b15f9b55bf0065c185ad8abbe20e4b3f44ac5.jpg b/.raw/MidasFEAAnalysisManual/images/page-443_9366d435a9dbbf2b105cc10ada7b15f9b55bf0065c185ad8abbe20e4b3f44ac5.jpg new file mode 100644 index 00000000..c6b053e6 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-443_9366d435a9dbbf2b105cc10ada7b15f9b55bf0065c185ad8abbe20e4b3f44ac5.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-443_a70d2349d19bf851dd0a2450f74b574690ea4bf61b08b3169deb679acc0dc8e2.jpg b/.raw/MidasFEAAnalysisManual/images/page-443_a70d2349d19bf851dd0a2450f74b574690ea4bf61b08b3169deb679acc0dc8e2.jpg new file mode 100644 index 00000000..f2237447 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-443_a70d2349d19bf851dd0a2450f74b574690ea4bf61b08b3169deb679acc0dc8e2.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-443_d406d3bf076618ed04489cecace57fa108abece0caa1b60e8bf50965b6351ed4.jpg b/.raw/MidasFEAAnalysisManual/images/page-443_d406d3bf076618ed04489cecace57fa108abece0caa1b60e8bf50965b6351ed4.jpg new file mode 100644 index 00000000..3d5d7721 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-443_d406d3bf076618ed04489cecace57fa108abece0caa1b60e8bf50965b6351ed4.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-443_d566993635d10521a53367dd266bf6bd4507923ddd4b129a4ee4ea9118f0cc30.jpg b/.raw/MidasFEAAnalysisManual/images/page-443_d566993635d10521a53367dd266bf6bd4507923ddd4b129a4ee4ea9118f0cc30.jpg new file mode 100644 index 00000000..907ed51f Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-443_d566993635d10521a53367dd266bf6bd4507923ddd4b129a4ee4ea9118f0cc30.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-444_3f085455a70e187bfc9e37f4ccadc8f2cfbdbae1e95ba78413e02ad95db05d48.jpg b/.raw/MidasFEAAnalysisManual/images/page-444_3f085455a70e187bfc9e37f4ccadc8f2cfbdbae1e95ba78413e02ad95db05d48.jpg new file mode 100644 index 00000000..05617163 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-444_3f085455a70e187bfc9e37f4ccadc8f2cfbdbae1e95ba78413e02ad95db05d48.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-444_63fbad64628303f0538e4f9c98e9245b706ae2fafa83e2a6a0a76e6057f793f7.jpg b/.raw/MidasFEAAnalysisManual/images/page-444_63fbad64628303f0538e4f9c98e9245b706ae2fafa83e2a6a0a76e6057f793f7.jpg new file mode 100644 index 00000000..f893f707 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-444_63fbad64628303f0538e4f9c98e9245b706ae2fafa83e2a6a0a76e6057f793f7.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-444_bdd42dbb00b743f7fb95c72ebe27e5e52228ded6166eefc2324f06407f2ac879.jpg b/.raw/MidasFEAAnalysisManual/images/page-444_bdd42dbb00b743f7fb95c72ebe27e5e52228ded6166eefc2324f06407f2ac879.jpg new file mode 100644 index 00000000..3a6a806b Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-444_bdd42dbb00b743f7fb95c72ebe27e5e52228ded6166eefc2324f06407f2ac879.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-444_e6affadae66685c1767ce5a590bd610a4530675274153e6cc590966f7561a468.jpg b/.raw/MidasFEAAnalysisManual/images/page-444_e6affadae66685c1767ce5a590bd610a4530675274153e6cc590966f7561a468.jpg new file mode 100644 index 00000000..88d35091 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-444_e6affadae66685c1767ce5a590bd610a4530675274153e6cc590966f7561a468.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-444_e6ba1cc509462eaf888518743dc8aeb9e3f89f72e0addaa90c2c5c7aceaa7023.jpg b/.raw/MidasFEAAnalysisManual/images/page-444_e6ba1cc509462eaf888518743dc8aeb9e3f89f72e0addaa90c2c5c7aceaa7023.jpg new file mode 100644 index 00000000..142a705f Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-444_e6ba1cc509462eaf888518743dc8aeb9e3f89f72e0addaa90c2c5c7aceaa7023.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-444_f18692d554729473128d2817c0c3e6a457b9cd1fc6a83f488f1b23cbc903374e.jpg b/.raw/MidasFEAAnalysisManual/images/page-444_f18692d554729473128d2817c0c3e6a457b9cd1fc6a83f488f1b23cbc903374e.jpg new file mode 100644 index 00000000..23ba0190 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-444_f18692d554729473128d2817c0c3e6a457b9cd1fc6a83f488f1b23cbc903374e.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-445_95dcb4822bd453700d55cd6e65a4cace3a15b81cd8c1eeaf63a1e2dbd43dc653.jpg b/.raw/MidasFEAAnalysisManual/images/page-445_95dcb4822bd453700d55cd6e65a4cace3a15b81cd8c1eeaf63a1e2dbd43dc653.jpg new file mode 100644 index 00000000..1e7c2693 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-445_95dcb4822bd453700d55cd6e65a4cace3a15b81cd8c1eeaf63a1e2dbd43dc653.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-445_ab087996c1bb058a0eeae965918caf4da3704b7867eaa1dd510207486f1a7bad.jpg b/.raw/MidasFEAAnalysisManual/images/page-445_ab087996c1bb058a0eeae965918caf4da3704b7867eaa1dd510207486f1a7bad.jpg new file mode 100644 index 00000000..52e655e9 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-445_ab087996c1bb058a0eeae965918caf4da3704b7867eaa1dd510207486f1a7bad.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-445_b165d8c2a0e38e43298010c6a6a409634089407caadc115834da57199e9d12d7.jpg b/.raw/MidasFEAAnalysisManual/images/page-445_b165d8c2a0e38e43298010c6a6a409634089407caadc115834da57199e9d12d7.jpg new file mode 100644 index 00000000..d432029e Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-445_b165d8c2a0e38e43298010c6a6a409634089407caadc115834da57199e9d12d7.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-445_d48c25e89d8a202d01dba981a9e931cc65916c0ca43c2393f1d02537d6da9f0e.jpg b/.raw/MidasFEAAnalysisManual/images/page-445_d48c25e89d8a202d01dba981a9e931cc65916c0ca43c2393f1d02537d6da9f0e.jpg new file mode 100644 index 00000000..84f71286 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-445_d48c25e89d8a202d01dba981a9e931cc65916c0ca43c2393f1d02537d6da9f0e.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-446_a8e585547ce12cc01accf5d5111fc1620d0758081462e9d8b270e154ff054c2f.jpg b/.raw/MidasFEAAnalysisManual/images/page-446_a8e585547ce12cc01accf5d5111fc1620d0758081462e9d8b270e154ff054c2f.jpg new file mode 100644 index 00000000..567c8cb9 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-446_a8e585547ce12cc01accf5d5111fc1620d0758081462e9d8b270e154ff054c2f.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-446_d956230505ea8a95d6c84f555f20bd30f290897e4b822ba09c8cb7de4f2ea442.jpg b/.raw/MidasFEAAnalysisManual/images/page-446_d956230505ea8a95d6c84f555f20bd30f290897e4b822ba09c8cb7de4f2ea442.jpg new file mode 100644 index 00000000..4532bd46 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-446_d956230505ea8a95d6c84f555f20bd30f290897e4b822ba09c8cb7de4f2ea442.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-446_fdfcf24832141069ed316903371d1084e7c8902246c545013bd4042139508a58.jpg b/.raw/MidasFEAAnalysisManual/images/page-446_fdfcf24832141069ed316903371d1084e7c8902246c545013bd4042139508a58.jpg new file mode 100644 index 00000000..b483a5ee Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-446_fdfcf24832141069ed316903371d1084e7c8902246c545013bd4042139508a58.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-447_94842f2d0ad0defb6890ab6dc48f3bfc83cd973297cf164cd7257a623ec08cc2.jpg b/.raw/MidasFEAAnalysisManual/images/page-447_94842f2d0ad0defb6890ab6dc48f3bfc83cd973297cf164cd7257a623ec08cc2.jpg new file mode 100644 index 00000000..38292d92 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-447_94842f2d0ad0defb6890ab6dc48f3bfc83cd973297cf164cd7257a623ec08cc2.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-448_3e60a14182160d18c852fef92d6679a0b87df9e366d5f31ec6d4660c1b6bd14d.jpg b/.raw/MidasFEAAnalysisManual/images/page-448_3e60a14182160d18c852fef92d6679a0b87df9e366d5f31ec6d4660c1b6bd14d.jpg new file mode 100644 index 00000000..7e0055b7 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-448_3e60a14182160d18c852fef92d6679a0b87df9e366d5f31ec6d4660c1b6bd14d.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-448_5047f1a0c6e007bfb13dd84f67e7b465e8807c4c189083823f46f225daeed47d.jpg b/.raw/MidasFEAAnalysisManual/images/page-448_5047f1a0c6e007bfb13dd84f67e7b465e8807c4c189083823f46f225daeed47d.jpg new file mode 100644 index 00000000..b2449c0e Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-448_5047f1a0c6e007bfb13dd84f67e7b465e8807c4c189083823f46f225daeed47d.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-448_5dd6e067da752d61d75d0faa4a7df0269f78d951372667312bc9332b68dc7c24.jpg b/.raw/MidasFEAAnalysisManual/images/page-448_5dd6e067da752d61d75d0faa4a7df0269f78d951372667312bc9332b68dc7c24.jpg new file mode 100644 index 00000000..02db97be Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-448_5dd6e067da752d61d75d0faa4a7df0269f78d951372667312bc9332b68dc7c24.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-448_6eedcc25ed90a26aaf964040e77c8622bfecf8e2ae31f9c06da7cf846a7daca7.jpg b/.raw/MidasFEAAnalysisManual/images/page-448_6eedcc25ed90a26aaf964040e77c8622bfecf8e2ae31f9c06da7cf846a7daca7.jpg new file mode 100644 index 00000000..6d3aca3a Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-448_6eedcc25ed90a26aaf964040e77c8622bfecf8e2ae31f9c06da7cf846a7daca7.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-448_74b46028a29a9cb6a2e24163f7ba6d0bea93b4bb1ae0c3ed0fe0e0380997cecd.jpg b/.raw/MidasFEAAnalysisManual/images/page-448_74b46028a29a9cb6a2e24163f7ba6d0bea93b4bb1ae0c3ed0fe0e0380997cecd.jpg new file mode 100644 index 00000000..35446445 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-448_74b46028a29a9cb6a2e24163f7ba6d0bea93b4bb1ae0c3ed0fe0e0380997cecd.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-448_ab92b1427e793bf12d6354330c862e65351dde0034efafa253f5f481ff405c33.jpg b/.raw/MidasFEAAnalysisManual/images/page-448_ab92b1427e793bf12d6354330c862e65351dde0034efafa253f5f481ff405c33.jpg new file mode 100644 index 00000000..6d98324a Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-448_ab92b1427e793bf12d6354330c862e65351dde0034efafa253f5f481ff405c33.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-448_ae5f1243dfd8c52c87b8d7541ecc71d31bd9fb1222a7cc5f88699b760962c40a.jpg b/.raw/MidasFEAAnalysisManual/images/page-448_ae5f1243dfd8c52c87b8d7541ecc71d31bd9fb1222a7cc5f88699b760962c40a.jpg new file mode 100644 index 00000000..9ccc20ef Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-448_ae5f1243dfd8c52c87b8d7541ecc71d31bd9fb1222a7cc5f88699b760962c40a.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-448_c0134d0ecc080c3e2a59d2fce5f67c645e07c82066551320f1079add0cb0675b.jpg b/.raw/MidasFEAAnalysisManual/images/page-448_c0134d0ecc080c3e2a59d2fce5f67c645e07c82066551320f1079add0cb0675b.jpg new file mode 100644 index 00000000..89292525 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-448_c0134d0ecc080c3e2a59d2fce5f67c645e07c82066551320f1079add0cb0675b.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-448_ebe65409013259627705612de2e68ae078726b76495a10637847ffb2deb1fcac.jpg b/.raw/MidasFEAAnalysisManual/images/page-448_ebe65409013259627705612de2e68ae078726b76495a10637847ffb2deb1fcac.jpg new file mode 100644 index 00000000..290b0a4f Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-448_ebe65409013259627705612de2e68ae078726b76495a10637847ffb2deb1fcac.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-448_fe3651eb06373b2ceb871bad626736fc8ba9538c968aa4c24a3df77da96b7e82.jpg b/.raw/MidasFEAAnalysisManual/images/page-448_fe3651eb06373b2ceb871bad626736fc8ba9538c968aa4c24a3df77da96b7e82.jpg new file mode 100644 index 00000000..f401d16c Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-448_fe3651eb06373b2ceb871bad626736fc8ba9538c968aa4c24a3df77da96b7e82.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-449_0a3e21fa2a6b0aa41622584fe618872d58c8a063a1aa6eed1d58b336b5349db2.jpg b/.raw/MidasFEAAnalysisManual/images/page-449_0a3e21fa2a6b0aa41622584fe618872d58c8a063a1aa6eed1d58b336b5349db2.jpg new file mode 100644 index 00000000..dde7e067 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-449_0a3e21fa2a6b0aa41622584fe618872d58c8a063a1aa6eed1d58b336b5349db2.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-449_0e576b05a412c9152258fb9dba2002a56b99bb2ef22e27274b3470591378773b.jpg b/.raw/MidasFEAAnalysisManual/images/page-449_0e576b05a412c9152258fb9dba2002a56b99bb2ef22e27274b3470591378773b.jpg new file mode 100644 index 00000000..aca46797 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-449_0e576b05a412c9152258fb9dba2002a56b99bb2ef22e27274b3470591378773b.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-449_2194e62cac0c147c6a3e71b70fad3ebe19ac9d6c159bfd87478dc814125561e0.jpg b/.raw/MidasFEAAnalysisManual/images/page-449_2194e62cac0c147c6a3e71b70fad3ebe19ac9d6c159bfd87478dc814125561e0.jpg new file mode 100644 index 00000000..8048d821 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-449_2194e62cac0c147c6a3e71b70fad3ebe19ac9d6c159bfd87478dc814125561e0.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-449_3d07c89fd993606b2c8c8db5a1572a7575bb413e3e5a16ae39da9528cb7291c6.jpg b/.raw/MidasFEAAnalysisManual/images/page-449_3d07c89fd993606b2c8c8db5a1572a7575bb413e3e5a16ae39da9528cb7291c6.jpg new file mode 100644 index 00000000..4072d9aa Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-449_3d07c89fd993606b2c8c8db5a1572a7575bb413e3e5a16ae39da9528cb7291c6.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-449_717121a5a32413619c4f0b6aa3dba76a3c0cb8e3667157e3d837212711e298e2.jpg b/.raw/MidasFEAAnalysisManual/images/page-449_717121a5a32413619c4f0b6aa3dba76a3c0cb8e3667157e3d837212711e298e2.jpg new file mode 100644 index 00000000..a6941e27 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-449_717121a5a32413619c4f0b6aa3dba76a3c0cb8e3667157e3d837212711e298e2.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-450_4ced52959dc504972d80a4503fb222912612ba0a11b846ab8dc1136117a84915.jpg b/.raw/MidasFEAAnalysisManual/images/page-450_4ced52959dc504972d80a4503fb222912612ba0a11b846ab8dc1136117a84915.jpg new file mode 100644 index 00000000..56bb37c4 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-450_4ced52959dc504972d80a4503fb222912612ba0a11b846ab8dc1136117a84915.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-450_4f75ca46905585d61ae67a4fca10256aeb1c74fab74b92cd9280d62057d6268d.jpg b/.raw/MidasFEAAnalysisManual/images/page-450_4f75ca46905585d61ae67a4fca10256aeb1c74fab74b92cd9280d62057d6268d.jpg new file mode 100644 index 00000000..7e1833c5 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-450_4f75ca46905585d61ae67a4fca10256aeb1c74fab74b92cd9280d62057d6268d.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-450_689acc48835977a6b8f66f0c0cc04ae250c082c30746cb154d1900b044c0c199.jpg b/.raw/MidasFEAAnalysisManual/images/page-450_689acc48835977a6b8f66f0c0cc04ae250c082c30746cb154d1900b044c0c199.jpg new file mode 100644 index 00000000..f86ee84e Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-450_689acc48835977a6b8f66f0c0cc04ae250c082c30746cb154d1900b044c0c199.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-450_76dd4653b63d87e28b81433df31a672162dce0f5891b91a66d8ccb4edb620518.jpg b/.raw/MidasFEAAnalysisManual/images/page-450_76dd4653b63d87e28b81433df31a672162dce0f5891b91a66d8ccb4edb620518.jpg new file mode 100644 index 00000000..f45adcb0 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-450_76dd4653b63d87e28b81433df31a672162dce0f5891b91a66d8ccb4edb620518.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-450_9811c4815c5ae3033fa34aa9690a2fb15ba586ace32450c8ed98fc6ef3d7cffd.jpg b/.raw/MidasFEAAnalysisManual/images/page-450_9811c4815c5ae3033fa34aa9690a2fb15ba586ace32450c8ed98fc6ef3d7cffd.jpg new file mode 100644 index 00000000..e5c22b24 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-450_9811c4815c5ae3033fa34aa9690a2fb15ba586ace32450c8ed98fc6ef3d7cffd.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-450_9acb8264087dc58a3782583ba8dca2ab2744d20ad3c539b400a868c949f7710d.jpg b/.raw/MidasFEAAnalysisManual/images/page-450_9acb8264087dc58a3782583ba8dca2ab2744d20ad3c539b400a868c949f7710d.jpg new file mode 100644 index 00000000..f24f5fec Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-450_9acb8264087dc58a3782583ba8dca2ab2744d20ad3c539b400a868c949f7710d.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-450_a62187f9fac193b5e39e20fad940d77d78e7eda22b911481726329de6a441ba8.jpg b/.raw/MidasFEAAnalysisManual/images/page-450_a62187f9fac193b5e39e20fad940d77d78e7eda22b911481726329de6a441ba8.jpg new file mode 100644 index 00000000..463d2301 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-450_a62187f9fac193b5e39e20fad940d77d78e7eda22b911481726329de6a441ba8.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-450_be85b27f436ddb8acb26f70d2ed4adf9b5217b0f3d12642e9d24296aeedfc410.jpg b/.raw/MidasFEAAnalysisManual/images/page-450_be85b27f436ddb8acb26f70d2ed4adf9b5217b0f3d12642e9d24296aeedfc410.jpg new file mode 100644 index 00000000..4a3e1eef Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-450_be85b27f436ddb8acb26f70d2ed4adf9b5217b0f3d12642e9d24296aeedfc410.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-450_c2396bdc4058eed3ab14bdb1c0cd71e8ed86fa41a7c17259297917bedb58d915.jpg b/.raw/MidasFEAAnalysisManual/images/page-450_c2396bdc4058eed3ab14bdb1c0cd71e8ed86fa41a7c17259297917bedb58d915.jpg new file mode 100644 index 00000000..6a36fd12 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-450_c2396bdc4058eed3ab14bdb1c0cd71e8ed86fa41a7c17259297917bedb58d915.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-450_db5a127ae1a2c1e6fc086b85909e68b1ca45fb99cca4075f8dca2bc99863ca45.jpg b/.raw/MidasFEAAnalysisManual/images/page-450_db5a127ae1a2c1e6fc086b85909e68b1ca45fb99cca4075f8dca2bc99863ca45.jpg new file mode 100644 index 00000000..fb11e746 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-450_db5a127ae1a2c1e6fc086b85909e68b1ca45fb99cca4075f8dca2bc99863ca45.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-451_2b0f13da686874be962415c99a9d0f37483e082483dae5b4d5b7824784012b94.jpg b/.raw/MidasFEAAnalysisManual/images/page-451_2b0f13da686874be962415c99a9d0f37483e082483dae5b4d5b7824784012b94.jpg new file mode 100644 index 00000000..0e109982 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-451_2b0f13da686874be962415c99a9d0f37483e082483dae5b4d5b7824784012b94.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-451_2c192a1493ac8e59409373a0ee15c1d2c0a0524de6f433246e7db6678e33cfa3.jpg b/.raw/MidasFEAAnalysisManual/images/page-451_2c192a1493ac8e59409373a0ee15c1d2c0a0524de6f433246e7db6678e33cfa3.jpg new file mode 100644 index 00000000..7bb885f0 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-451_2c192a1493ac8e59409373a0ee15c1d2c0a0524de6f433246e7db6678e33cfa3.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-451_515a4f4c24d054b4e59c83f1d796e09e2f086816efd328e94fc0a3b35723d165.jpg b/.raw/MidasFEAAnalysisManual/images/page-451_515a4f4c24d054b4e59c83f1d796e09e2f086816efd328e94fc0a3b35723d165.jpg new file mode 100644 index 00000000..d91dbd61 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-451_515a4f4c24d054b4e59c83f1d796e09e2f086816efd328e94fc0a3b35723d165.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-451_53399a593ce1b4fd1dd47e2a1432c9d3a3f7e0d4f120e1b0701c4a838d6ef3de.jpg b/.raw/MidasFEAAnalysisManual/images/page-451_53399a593ce1b4fd1dd47e2a1432c9d3a3f7e0d4f120e1b0701c4a838d6ef3de.jpg new file mode 100644 index 00000000..44e947ae Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-451_53399a593ce1b4fd1dd47e2a1432c9d3a3f7e0d4f120e1b0701c4a838d6ef3de.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-451_572610f8afaa6e24935960d87049096414fc9e7fa861dafb7f5069b33d4168d8.jpg b/.raw/MidasFEAAnalysisManual/images/page-451_572610f8afaa6e24935960d87049096414fc9e7fa861dafb7f5069b33d4168d8.jpg new file mode 100644 index 00000000..5c65adb0 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-451_572610f8afaa6e24935960d87049096414fc9e7fa861dafb7f5069b33d4168d8.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-451_743eea29ba2ccb82f6b66a8fe1f5ca96b422db806fc23547846acec810f5147d.jpg b/.raw/MidasFEAAnalysisManual/images/page-451_743eea29ba2ccb82f6b66a8fe1f5ca96b422db806fc23547846acec810f5147d.jpg new file mode 100644 index 00000000..08c27dcf Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-451_743eea29ba2ccb82f6b66a8fe1f5ca96b422db806fc23547846acec810f5147d.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-451_9dac19381ae26459e6ccfb627f06237131496cc76105f2b9a50db6120f0c9161.jpg b/.raw/MidasFEAAnalysisManual/images/page-451_9dac19381ae26459e6ccfb627f06237131496cc76105f2b9a50db6120f0c9161.jpg new file mode 100644 index 00000000..f56cdba0 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-451_9dac19381ae26459e6ccfb627f06237131496cc76105f2b9a50db6120f0c9161.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-452_1a363e10a68fbde617b87f1ae291a1be06f39aee71648a4c90a3e856fede206d.jpg b/.raw/MidasFEAAnalysisManual/images/page-452_1a363e10a68fbde617b87f1ae291a1be06f39aee71648a4c90a3e856fede206d.jpg new file mode 100644 index 00000000..a470e64d Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-452_1a363e10a68fbde617b87f1ae291a1be06f39aee71648a4c90a3e856fede206d.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-452_1a3d1ad57a2b1073193b42148f5fe1f21e5fd84120114561747b42bb8cee8d91.jpg b/.raw/MidasFEAAnalysisManual/images/page-452_1a3d1ad57a2b1073193b42148f5fe1f21e5fd84120114561747b42bb8cee8d91.jpg new file mode 100644 index 00000000..ce6f6f06 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-452_1a3d1ad57a2b1073193b42148f5fe1f21e5fd84120114561747b42bb8cee8d91.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-452_3ff0714e2658dfaa5d2caa4a870920f5cf798cfdd5953bdbac4ef7b592a4d6b4.jpg b/.raw/MidasFEAAnalysisManual/images/page-452_3ff0714e2658dfaa5d2caa4a870920f5cf798cfdd5953bdbac4ef7b592a4d6b4.jpg new file mode 100644 index 00000000..659c32ac Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-452_3ff0714e2658dfaa5d2caa4a870920f5cf798cfdd5953bdbac4ef7b592a4d6b4.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-453_22e86c1442bc71de35ada3e46e7d948f4772b98fe6bed2ae4a9c09b08bcf614c.jpg b/.raw/MidasFEAAnalysisManual/images/page-453_22e86c1442bc71de35ada3e46e7d948f4772b98fe6bed2ae4a9c09b08bcf614c.jpg new file mode 100644 index 00000000..4c5b26f9 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-453_22e86c1442bc71de35ada3e46e7d948f4772b98fe6bed2ae4a9c09b08bcf614c.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-453_3d5aff29367391d94b132b9c1c57abe0f78f6a9cc32948299e558bde90f23e25.jpg b/.raw/MidasFEAAnalysisManual/images/page-453_3d5aff29367391d94b132b9c1c57abe0f78f6a9cc32948299e558bde90f23e25.jpg new file mode 100644 index 00000000..345db29b Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-453_3d5aff29367391d94b132b9c1c57abe0f78f6a9cc32948299e558bde90f23e25.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-453_3f65e721227dda62e58897922b001c5367b88185cf32c4de5693b458af5afeba.jpg b/.raw/MidasFEAAnalysisManual/images/page-453_3f65e721227dda62e58897922b001c5367b88185cf32c4de5693b458af5afeba.jpg new file mode 100644 index 00000000..8d230604 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-453_3f65e721227dda62e58897922b001c5367b88185cf32c4de5693b458af5afeba.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-453_50337fb7afd15bd0e3f3b489becf9081789351f24e29c82de64910dc9c52f4fa.jpg b/.raw/MidasFEAAnalysisManual/images/page-453_50337fb7afd15bd0e3f3b489becf9081789351f24e29c82de64910dc9c52f4fa.jpg new file mode 100644 index 00000000..bc7cdf59 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-453_50337fb7afd15bd0e3f3b489becf9081789351f24e29c82de64910dc9c52f4fa.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-453_647b3c1ce7f4a856b1f6468e15c448dfde950db208fb3e1bf042a6a2d3e9b0d7.jpg b/.raw/MidasFEAAnalysisManual/images/page-453_647b3c1ce7f4a856b1f6468e15c448dfde950db208fb3e1bf042a6a2d3e9b0d7.jpg new file mode 100644 index 00000000..742345e4 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-453_647b3c1ce7f4a856b1f6468e15c448dfde950db208fb3e1bf042a6a2d3e9b0d7.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-453_68931a82722949900d62eeb69c55a3f95e33880f556baa98cf63144126a0b94c.jpg b/.raw/MidasFEAAnalysisManual/images/page-453_68931a82722949900d62eeb69c55a3f95e33880f556baa98cf63144126a0b94c.jpg new file mode 100644 index 00000000..cfd3cdd3 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-453_68931a82722949900d62eeb69c55a3f95e33880f556baa98cf63144126a0b94c.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-453_7663a4d38efbe7ca218f543bae8b4d4582963c715d1d197b631394d268ea1d16.jpg b/.raw/MidasFEAAnalysisManual/images/page-453_7663a4d38efbe7ca218f543bae8b4d4582963c715d1d197b631394d268ea1d16.jpg new file mode 100644 index 00000000..635053c6 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-453_7663a4d38efbe7ca218f543bae8b4d4582963c715d1d197b631394d268ea1d16.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-453_89a95ebb9c91f62502c43674fc5381ba9c2f7b56ee237fd76202be605a19f33d.jpg b/.raw/MidasFEAAnalysisManual/images/page-453_89a95ebb9c91f62502c43674fc5381ba9c2f7b56ee237fd76202be605a19f33d.jpg new file mode 100644 index 00000000..8e66b4f8 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-453_89a95ebb9c91f62502c43674fc5381ba9c2f7b56ee237fd76202be605a19f33d.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-453_8cb2fa8b80c5327ed74ec759eff02bf7f817c2469bc506813361cdd4ee39fb6e.jpg b/.raw/MidasFEAAnalysisManual/images/page-453_8cb2fa8b80c5327ed74ec759eff02bf7f817c2469bc506813361cdd4ee39fb6e.jpg new file mode 100644 index 00000000..d2cc0255 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-453_8cb2fa8b80c5327ed74ec759eff02bf7f817c2469bc506813361cdd4ee39fb6e.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-453_9f6c0b5e676fb62ef354799b5e43c7263bbead425491eef99dcea02bee22af56.jpg b/.raw/MidasFEAAnalysisManual/images/page-453_9f6c0b5e676fb62ef354799b5e43c7263bbead425491eef99dcea02bee22af56.jpg new file mode 100644 index 00000000..f31bf68d Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-453_9f6c0b5e676fb62ef354799b5e43c7263bbead425491eef99dcea02bee22af56.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-454_16493100447a8518e63324952a01adfc125ff3df99309de2d8d222914da95371.jpg b/.raw/MidasFEAAnalysisManual/images/page-454_16493100447a8518e63324952a01adfc125ff3df99309de2d8d222914da95371.jpg new file mode 100644 index 00000000..8243b72c Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-454_16493100447a8518e63324952a01adfc125ff3df99309de2d8d222914da95371.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-454_23b76c53f56f56c56d9246e75c9f4bdef8411d6a31e1a545ecd920b74af59736.jpg b/.raw/MidasFEAAnalysisManual/images/page-454_23b76c53f56f56c56d9246e75c9f4bdef8411d6a31e1a545ecd920b74af59736.jpg new file mode 100644 index 00000000..3619b154 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-454_23b76c53f56f56c56d9246e75c9f4bdef8411d6a31e1a545ecd920b74af59736.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-454_2a5e5ed0b6ac6974f7f75bba99124c89296fed6a73090ac04d07fd75ce1e31db.jpg b/.raw/MidasFEAAnalysisManual/images/page-454_2a5e5ed0b6ac6974f7f75bba99124c89296fed6a73090ac04d07fd75ce1e31db.jpg new file mode 100644 index 00000000..52d72f51 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-454_2a5e5ed0b6ac6974f7f75bba99124c89296fed6a73090ac04d07fd75ce1e31db.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-454_4eb90b0f99d5ec220c8739660013007f5d931319e2a9f6864b968bcaf716ab07.jpg b/.raw/MidasFEAAnalysisManual/images/page-454_4eb90b0f99d5ec220c8739660013007f5d931319e2a9f6864b968bcaf716ab07.jpg new file mode 100644 index 00000000..10c0ebd4 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-454_4eb90b0f99d5ec220c8739660013007f5d931319e2a9f6864b968bcaf716ab07.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-454_638d2d6382c0adf820df623941fad6dd6071c2e13d592363674cbdbe3a0f7634.jpg b/.raw/MidasFEAAnalysisManual/images/page-454_638d2d6382c0adf820df623941fad6dd6071c2e13d592363674cbdbe3a0f7634.jpg new file mode 100644 index 00000000..aad372c5 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-454_638d2d6382c0adf820df623941fad6dd6071c2e13d592363674cbdbe3a0f7634.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-454_6c8652f12b38bf141e9d08f29e73dba0f1ad81bd2b46d2f7ace9a7c8c88c6013.jpg b/.raw/MidasFEAAnalysisManual/images/page-454_6c8652f12b38bf141e9d08f29e73dba0f1ad81bd2b46d2f7ace9a7c8c88c6013.jpg new file mode 100644 index 00000000..1e009dbc Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-454_6c8652f12b38bf141e9d08f29e73dba0f1ad81bd2b46d2f7ace9a7c8c88c6013.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-454_8246ee8da0ed1e04faa5128a2df5b4578e263a9aa8751714201587188886f56e.jpg b/.raw/MidasFEAAnalysisManual/images/page-454_8246ee8da0ed1e04faa5128a2df5b4578e263a9aa8751714201587188886f56e.jpg new file mode 100644 index 00000000..695a7350 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-454_8246ee8da0ed1e04faa5128a2df5b4578e263a9aa8751714201587188886f56e.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-454_90a7f8f78d4ad90e666003fd912d5541e7d778bf81048d822a33c80fb327e454.jpg b/.raw/MidasFEAAnalysisManual/images/page-454_90a7f8f78d4ad90e666003fd912d5541e7d778bf81048d822a33c80fb327e454.jpg new file mode 100644 index 00000000..b7840ff0 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-454_90a7f8f78d4ad90e666003fd912d5541e7d778bf81048d822a33c80fb327e454.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-454_aff7dadb729432ed2361fa86257e999397becb322b0ab7ce58753b374d52fe22.jpg b/.raw/MidasFEAAnalysisManual/images/page-454_aff7dadb729432ed2361fa86257e999397becb322b0ab7ce58753b374d52fe22.jpg new file mode 100644 index 00000000..3d00229a Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-454_aff7dadb729432ed2361fa86257e999397becb322b0ab7ce58753b374d52fe22.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-454_b849ea2dae47f811b6fff3e9b9720ba3ee3a0b5d90b7fbdc7ea6c3e21c371f6f.jpg b/.raw/MidasFEAAnalysisManual/images/page-454_b849ea2dae47f811b6fff3e9b9720ba3ee3a0b5d90b7fbdc7ea6c3e21c371f6f.jpg new file mode 100644 index 00000000..5d9b9b30 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-454_b849ea2dae47f811b6fff3e9b9720ba3ee3a0b5d90b7fbdc7ea6c3e21c371f6f.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-454_e7d2662410ab6ab5af7a6e0d4fbd8f46828221f50b10e580a3871a8328e7e29b.jpg b/.raw/MidasFEAAnalysisManual/images/page-454_e7d2662410ab6ab5af7a6e0d4fbd8f46828221f50b10e580a3871a8328e7e29b.jpg new file mode 100644 index 00000000..c4238e16 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-454_e7d2662410ab6ab5af7a6e0d4fbd8f46828221f50b10e580a3871a8328e7e29b.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-454_ea728de01163c12833a0266a17754e501f124e2f409600902a561d5e2c1a98a6.jpg b/.raw/MidasFEAAnalysisManual/images/page-454_ea728de01163c12833a0266a17754e501f124e2f409600902a561d5e2c1a98a6.jpg new file mode 100644 index 00000000..daa5d76b Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-454_ea728de01163c12833a0266a17754e501f124e2f409600902a561d5e2c1a98a6.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-454_ee4ff4426ea66b816e3f8ba4f7d0fa821693b644527a220ca49852e1b93cf3d6.jpg b/.raw/MidasFEAAnalysisManual/images/page-454_ee4ff4426ea66b816e3f8ba4f7d0fa821693b644527a220ca49852e1b93cf3d6.jpg new file mode 100644 index 00000000..1af361e9 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-454_ee4ff4426ea66b816e3f8ba4f7d0fa821693b644527a220ca49852e1b93cf3d6.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-455_05710973b64961b10a9c62c68db9eae16a1b362023c336748e54ff8f3a3d7cad.jpg b/.raw/MidasFEAAnalysisManual/images/page-455_05710973b64961b10a9c62c68db9eae16a1b362023c336748e54ff8f3a3d7cad.jpg new file mode 100644 index 00000000..8c20fded Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-455_05710973b64961b10a9c62c68db9eae16a1b362023c336748e54ff8f3a3d7cad.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-455_0e5f4538c28bae1ad61eadcd1edef0bd152db7b7052e7b0a1d10b3a2912e84fa.jpg b/.raw/MidasFEAAnalysisManual/images/page-455_0e5f4538c28bae1ad61eadcd1edef0bd152db7b7052e7b0a1d10b3a2912e84fa.jpg new file mode 100644 index 00000000..09fde517 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-455_0e5f4538c28bae1ad61eadcd1edef0bd152db7b7052e7b0a1d10b3a2912e84fa.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-455_19771a68e9cca6cf0d30f07eb9ecb2619d426f49554fd06530153c948e1dad2f.jpg b/.raw/MidasFEAAnalysisManual/images/page-455_19771a68e9cca6cf0d30f07eb9ecb2619d426f49554fd06530153c948e1dad2f.jpg new file mode 100644 index 00000000..2f21f35c Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-455_19771a68e9cca6cf0d30f07eb9ecb2619d426f49554fd06530153c948e1dad2f.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-455_2482ec87c58bdbc1a744909420a99c1e2341840286947948dbd455972c17bf00.jpg b/.raw/MidasFEAAnalysisManual/images/page-455_2482ec87c58bdbc1a744909420a99c1e2341840286947948dbd455972c17bf00.jpg new file mode 100644 index 00000000..ee127319 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-455_2482ec87c58bdbc1a744909420a99c1e2341840286947948dbd455972c17bf00.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-455_2e1f53a2d3e762b61be94a715d6549a35f67f38fc6969914d9107d86f9cc3b6a.jpg b/.raw/MidasFEAAnalysisManual/images/page-455_2e1f53a2d3e762b61be94a715d6549a35f67f38fc6969914d9107d86f9cc3b6a.jpg new file mode 100644 index 00000000..ccc37a3d Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-455_2e1f53a2d3e762b61be94a715d6549a35f67f38fc6969914d9107d86f9cc3b6a.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-455_366b5e375a3cbf3e534d3ffa08d0c142ba0d82f3359287450affd63c8b34997b.jpg b/.raw/MidasFEAAnalysisManual/images/page-455_366b5e375a3cbf3e534d3ffa08d0c142ba0d82f3359287450affd63c8b34997b.jpg new file mode 100644 index 00000000..85b9a555 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-455_366b5e375a3cbf3e534d3ffa08d0c142ba0d82f3359287450affd63c8b34997b.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-455_3afc67fc0f01aa1186bb86fbd3af2eef7be7b2a8d23588025a8affdbbab8e425.jpg b/.raw/MidasFEAAnalysisManual/images/page-455_3afc67fc0f01aa1186bb86fbd3af2eef7be7b2a8d23588025a8affdbbab8e425.jpg new file mode 100644 index 00000000..dd010304 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-455_3afc67fc0f01aa1186bb86fbd3af2eef7be7b2a8d23588025a8affdbbab8e425.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-455_3cf04838471be4f7b063b81237f43f6afc1fd561dc4b8576d29ec52b1124b932.jpg b/.raw/MidasFEAAnalysisManual/images/page-455_3cf04838471be4f7b063b81237f43f6afc1fd561dc4b8576d29ec52b1124b932.jpg new file mode 100644 index 00000000..547f7b0d Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-455_3cf04838471be4f7b063b81237f43f6afc1fd561dc4b8576d29ec52b1124b932.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-455_46a75647fcd95091590efdcd1fb39005cebd19bbcd9c8e17c927285b4ba25c82.jpg b/.raw/MidasFEAAnalysisManual/images/page-455_46a75647fcd95091590efdcd1fb39005cebd19bbcd9c8e17c927285b4ba25c82.jpg new file mode 100644 index 00000000..8c84a0e3 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-455_46a75647fcd95091590efdcd1fb39005cebd19bbcd9c8e17c927285b4ba25c82.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-455_6cbe856ea1d33c52c602dca93bc9056df345d1fe8f8e65dc1a8e0389d3ebbbcb.jpg b/.raw/MidasFEAAnalysisManual/images/page-455_6cbe856ea1d33c52c602dca93bc9056df345d1fe8f8e65dc1a8e0389d3ebbbcb.jpg new file mode 100644 index 00000000..9f258772 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-455_6cbe856ea1d33c52c602dca93bc9056df345d1fe8f8e65dc1a8e0389d3ebbbcb.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-455_7db46e6347b9965d5016297637d54ea406173234d78764129009b5390487e02c.jpg b/.raw/MidasFEAAnalysisManual/images/page-455_7db46e6347b9965d5016297637d54ea406173234d78764129009b5390487e02c.jpg new file mode 100644 index 00000000..1ee0d2e7 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-455_7db46e6347b9965d5016297637d54ea406173234d78764129009b5390487e02c.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-455_9cf3ef1b4600c933d472814c1bcb7cd04b5a5afb0034f506072ac33f99b402e8.jpg b/.raw/MidasFEAAnalysisManual/images/page-455_9cf3ef1b4600c933d472814c1bcb7cd04b5a5afb0034f506072ac33f99b402e8.jpg new file mode 100644 index 00000000..59b523fb Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-455_9cf3ef1b4600c933d472814c1bcb7cd04b5a5afb0034f506072ac33f99b402e8.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-455_a4246dfede8be730ee75da869d1cabdb5dfc861fa6d9e2112d2d660bedec68ea.jpg b/.raw/MidasFEAAnalysisManual/images/page-455_a4246dfede8be730ee75da869d1cabdb5dfc861fa6d9e2112d2d660bedec68ea.jpg new file mode 100644 index 00000000..5a3a0317 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-455_a4246dfede8be730ee75da869d1cabdb5dfc861fa6d9e2112d2d660bedec68ea.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-455_a5b673d5df854e2cdaaf694b3c721f465c808a1b2479ef14061b86361089ac00.jpg b/.raw/MidasFEAAnalysisManual/images/page-455_a5b673d5df854e2cdaaf694b3c721f465c808a1b2479ef14061b86361089ac00.jpg new file mode 100644 index 00000000..4c59925a Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-455_a5b673d5df854e2cdaaf694b3c721f465c808a1b2479ef14061b86361089ac00.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-455_a79ad66d4f1a32474be177b6feeff6f63a65a93a8cdb33f3cfc94eac977c00e2.jpg b/.raw/MidasFEAAnalysisManual/images/page-455_a79ad66d4f1a32474be177b6feeff6f63a65a93a8cdb33f3cfc94eac977c00e2.jpg new file mode 100644 index 00000000..14db9a51 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-455_a79ad66d4f1a32474be177b6feeff6f63a65a93a8cdb33f3cfc94eac977c00e2.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-455_bde214582ccf2071013d03d5c00457f27fc57969bc8a6858bbf95acd55cbb463.jpg b/.raw/MidasFEAAnalysisManual/images/page-455_bde214582ccf2071013d03d5c00457f27fc57969bc8a6858bbf95acd55cbb463.jpg new file mode 100644 index 00000000..2c554117 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-455_bde214582ccf2071013d03d5c00457f27fc57969bc8a6858bbf95acd55cbb463.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-455_c511bd1618f64e1d08333944a1b85c2dab97123ac6befd0ec9b541b2ac47c329.jpg b/.raw/MidasFEAAnalysisManual/images/page-455_c511bd1618f64e1d08333944a1b85c2dab97123ac6befd0ec9b541b2ac47c329.jpg new file mode 100644 index 00000000..049550b2 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-455_c511bd1618f64e1d08333944a1b85c2dab97123ac6befd0ec9b541b2ac47c329.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-455_d733ef836d554958ce97c725386d2f3ffac4c08c3c10438a3b12ee4a7c586e24.jpg b/.raw/MidasFEAAnalysisManual/images/page-455_d733ef836d554958ce97c725386d2f3ffac4c08c3c10438a3b12ee4a7c586e24.jpg new file mode 100644 index 00000000..50ca2122 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-455_d733ef836d554958ce97c725386d2f3ffac4c08c3c10438a3b12ee4a7c586e24.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-455_e283208aaeae2da977a8a56c9ad00c7ae3652aa2696850c904be01373839a54a.jpg b/.raw/MidasFEAAnalysisManual/images/page-455_e283208aaeae2da977a8a56c9ad00c7ae3652aa2696850c904be01373839a54a.jpg new file mode 100644 index 00000000..e7c43fb3 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-455_e283208aaeae2da977a8a56c9ad00c7ae3652aa2696850c904be01373839a54a.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-455_e5055e5f384b2391232742d76fa9a6b574558c5115c62188372385cfb06a59bf.jpg b/.raw/MidasFEAAnalysisManual/images/page-455_e5055e5f384b2391232742d76fa9a6b574558c5115c62188372385cfb06a59bf.jpg new file mode 100644 index 00000000..363d3360 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-455_e5055e5f384b2391232742d76fa9a6b574558c5115c62188372385cfb06a59bf.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-455_e6c4d7917a41973415f2c480d6c94e7a466619ade21770851159dd123476248f.jpg b/.raw/MidasFEAAnalysisManual/images/page-455_e6c4d7917a41973415f2c480d6c94e7a466619ade21770851159dd123476248f.jpg new file mode 100644 index 00000000..b62c85ff Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-455_e6c4d7917a41973415f2c480d6c94e7a466619ade21770851159dd123476248f.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-455_e6d24a39f7a825e02e137df3a60969a64b95bdc71847d5845518b6cedad5e692.jpg b/.raw/MidasFEAAnalysisManual/images/page-455_e6d24a39f7a825e02e137df3a60969a64b95bdc71847d5845518b6cedad5e692.jpg new file mode 100644 index 00000000..9b2e540d Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-455_e6d24a39f7a825e02e137df3a60969a64b95bdc71847d5845518b6cedad5e692.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-456_18d18ed66e81b498533a10cf2991f2ba5604910224a97ef4f13792fde5090061.jpg b/.raw/MidasFEAAnalysisManual/images/page-456_18d18ed66e81b498533a10cf2991f2ba5604910224a97ef4f13792fde5090061.jpg new file mode 100644 index 00000000..61e3b69b Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-456_18d18ed66e81b498533a10cf2991f2ba5604910224a97ef4f13792fde5090061.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-456_2de0c5d6bb6ad84d4c992cda9bbfe801567b31224d49a86a44042386066887cc.jpg b/.raw/MidasFEAAnalysisManual/images/page-456_2de0c5d6bb6ad84d4c992cda9bbfe801567b31224d49a86a44042386066887cc.jpg new file mode 100644 index 00000000..3cc6cbde Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-456_2de0c5d6bb6ad84d4c992cda9bbfe801567b31224d49a86a44042386066887cc.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-456_33e3d12e15c02cc0d0668921e68e364e8224e7968c3dd7b1b34eb8dd3e854a60.jpg b/.raw/MidasFEAAnalysisManual/images/page-456_33e3d12e15c02cc0d0668921e68e364e8224e7968c3dd7b1b34eb8dd3e854a60.jpg new file mode 100644 index 00000000..78c05be7 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-456_33e3d12e15c02cc0d0668921e68e364e8224e7968c3dd7b1b34eb8dd3e854a60.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-456_35eb73ed86320a428a6f08bcb3b9646357fba544b85eb34dce3390ec170f7578.jpg b/.raw/MidasFEAAnalysisManual/images/page-456_35eb73ed86320a428a6f08bcb3b9646357fba544b85eb34dce3390ec170f7578.jpg new file mode 100644 index 00000000..41b0def4 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-456_35eb73ed86320a428a6f08bcb3b9646357fba544b85eb34dce3390ec170f7578.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-456_4909eb2cf306ac6a7c2a40438bf741a8d8f7262bec9ad8f50f02974a03808af1.jpg b/.raw/MidasFEAAnalysisManual/images/page-456_4909eb2cf306ac6a7c2a40438bf741a8d8f7262bec9ad8f50f02974a03808af1.jpg new file mode 100644 index 00000000..35805a50 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-456_4909eb2cf306ac6a7c2a40438bf741a8d8f7262bec9ad8f50f02974a03808af1.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-456_5880eedf4bebb8c46e4c571b48012a146cf62467991f1a2b243646ba15209855.jpg b/.raw/MidasFEAAnalysisManual/images/page-456_5880eedf4bebb8c46e4c571b48012a146cf62467991f1a2b243646ba15209855.jpg new file mode 100644 index 00000000..c9c741bf Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-456_5880eedf4bebb8c46e4c571b48012a146cf62467991f1a2b243646ba15209855.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-456_67c5fde81c8278cfc4bb0ccfb691d0dcb964899eceb9b9ff47ec66546617a851.jpg b/.raw/MidasFEAAnalysisManual/images/page-456_67c5fde81c8278cfc4bb0ccfb691d0dcb964899eceb9b9ff47ec66546617a851.jpg new file mode 100644 index 00000000..2a1f5b71 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-456_67c5fde81c8278cfc4bb0ccfb691d0dcb964899eceb9b9ff47ec66546617a851.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-456_735ad42c36ccfe432ce6997a6d9e4525c5f7e082eb11559f4341bb5bcbb9ad79.jpg b/.raw/MidasFEAAnalysisManual/images/page-456_735ad42c36ccfe432ce6997a6d9e4525c5f7e082eb11559f4341bb5bcbb9ad79.jpg new file mode 100644 index 00000000..02fa1408 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-456_735ad42c36ccfe432ce6997a6d9e4525c5f7e082eb11559f4341bb5bcbb9ad79.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-456_7c633c196969fca083746e4f67c8f3f4ab5261d0db00ee216fe4527e09ccb6ad.jpg b/.raw/MidasFEAAnalysisManual/images/page-456_7c633c196969fca083746e4f67c8f3f4ab5261d0db00ee216fe4527e09ccb6ad.jpg new file mode 100644 index 00000000..167e0c22 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-456_7c633c196969fca083746e4f67c8f3f4ab5261d0db00ee216fe4527e09ccb6ad.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-456_920688aa7ac79f66ba31fbfd6917df91054d1da1612e73f87acee9b0bc82481a.jpg b/.raw/MidasFEAAnalysisManual/images/page-456_920688aa7ac79f66ba31fbfd6917df91054d1da1612e73f87acee9b0bc82481a.jpg new file mode 100644 index 00000000..b045410a Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-456_920688aa7ac79f66ba31fbfd6917df91054d1da1612e73f87acee9b0bc82481a.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-456_95338767cbb5da6e19808ccfbf25ba1c454a7f95be38e638de0991b9a0317734.jpg b/.raw/MidasFEAAnalysisManual/images/page-456_95338767cbb5da6e19808ccfbf25ba1c454a7f95be38e638de0991b9a0317734.jpg new file mode 100644 index 00000000..3bf1439e Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-456_95338767cbb5da6e19808ccfbf25ba1c454a7f95be38e638de0991b9a0317734.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-456_9807648cdb985ce5d96f6023309661518ff295388669e1367048709a5a25fa48.jpg b/.raw/MidasFEAAnalysisManual/images/page-456_9807648cdb985ce5d96f6023309661518ff295388669e1367048709a5a25fa48.jpg new file mode 100644 index 00000000..87573f6a Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-456_9807648cdb985ce5d96f6023309661518ff295388669e1367048709a5a25fa48.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-456_ac5614d1afc0de66b79916c5a54227deb059a44e21cc878bd68bbcf9b5e7a77c.jpg b/.raw/MidasFEAAnalysisManual/images/page-456_ac5614d1afc0de66b79916c5a54227deb059a44e21cc878bd68bbcf9b5e7a77c.jpg new file mode 100644 index 00000000..9aa47d00 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-456_ac5614d1afc0de66b79916c5a54227deb059a44e21cc878bd68bbcf9b5e7a77c.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-456_ba6bec20a6d8536da0f2b260a76fbb99b3e3774e89f2ef74a160c424fabe7c86.jpg b/.raw/MidasFEAAnalysisManual/images/page-456_ba6bec20a6d8536da0f2b260a76fbb99b3e3774e89f2ef74a160c424fabe7c86.jpg new file mode 100644 index 00000000..49470dcc Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-456_ba6bec20a6d8536da0f2b260a76fbb99b3e3774e89f2ef74a160c424fabe7c86.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-456_c8c03146d2b0e3dc67b230c7f277c591092cbba4bc279faec30b877b6b085304.jpg b/.raw/MidasFEAAnalysisManual/images/page-456_c8c03146d2b0e3dc67b230c7f277c591092cbba4bc279faec30b877b6b085304.jpg new file mode 100644 index 00000000..78bcfbf2 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-456_c8c03146d2b0e3dc67b230c7f277c591092cbba4bc279faec30b877b6b085304.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-456_e85247017259873a03f6df2654b36dc50e4b56fbbde3ab5773c7aeada9c50803.jpg b/.raw/MidasFEAAnalysisManual/images/page-456_e85247017259873a03f6df2654b36dc50e4b56fbbde3ab5773c7aeada9c50803.jpg new file mode 100644 index 00000000..4da4752f Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-456_e85247017259873a03f6df2654b36dc50e4b56fbbde3ab5773c7aeada9c50803.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-456_ec052d42e428e06d3818f9a39734fbba4b0de294c21269d9a87029ea5443c1aa.jpg b/.raw/MidasFEAAnalysisManual/images/page-456_ec052d42e428e06d3818f9a39734fbba4b0de294c21269d9a87029ea5443c1aa.jpg new file mode 100644 index 00000000..18fe9598 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-456_ec052d42e428e06d3818f9a39734fbba4b0de294c21269d9a87029ea5443c1aa.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-456_f09a210ef750f8d30fd6961e21d4f0115e66ea4296956c3b95e1677417e18e63.jpg b/.raw/MidasFEAAnalysisManual/images/page-456_f09a210ef750f8d30fd6961e21d4f0115e66ea4296956c3b95e1677417e18e63.jpg new file mode 100644 index 00000000..35dc6448 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-456_f09a210ef750f8d30fd6961e21d4f0115e66ea4296956c3b95e1677417e18e63.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-457_0a38c90070011532f1dd1fa46ad1ccd0075a36c8ea234e26f4fa67c836ea3810.jpg b/.raw/MidasFEAAnalysisManual/images/page-457_0a38c90070011532f1dd1fa46ad1ccd0075a36c8ea234e26f4fa67c836ea3810.jpg new file mode 100644 index 00000000..038bfe04 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-457_0a38c90070011532f1dd1fa46ad1ccd0075a36c8ea234e26f4fa67c836ea3810.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-457_941de31f9075cee35630615b25b1a0dbed2713294d15973e543f56a3d9703b0a.jpg b/.raw/MidasFEAAnalysisManual/images/page-457_941de31f9075cee35630615b25b1a0dbed2713294d15973e543f56a3d9703b0a.jpg new file mode 100644 index 00000000..1a9859b5 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-457_941de31f9075cee35630615b25b1a0dbed2713294d15973e543f56a3d9703b0a.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-458_a398be8fd3aa134876ceb1e5afcc34e7a2a9e753f404f94d573de7178dab79d7.jpg b/.raw/MidasFEAAnalysisManual/images/page-458_a398be8fd3aa134876ceb1e5afcc34e7a2a9e753f404f94d573de7178dab79d7.jpg new file mode 100644 index 00000000..e599c5e3 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-458_a398be8fd3aa134876ceb1e5afcc34e7a2a9e753f404f94d573de7178dab79d7.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-458_dd6c3902bae07ceef5bebf3849d2b15166c5da7b445507c47ea5685f92554ab3.jpg b/.raw/MidasFEAAnalysisManual/images/page-458_dd6c3902bae07ceef5bebf3849d2b15166c5da7b445507c47ea5685f92554ab3.jpg new file mode 100644 index 00000000..42d71028 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-458_dd6c3902bae07ceef5bebf3849d2b15166c5da7b445507c47ea5685f92554ab3.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-458_ecc1c9a5d7d8ddd565a2f77710b0c901a2195af91c1bf21fd792fe47cd0adcb1.jpg b/.raw/MidasFEAAnalysisManual/images/page-458_ecc1c9a5d7d8ddd565a2f77710b0c901a2195af91c1bf21fd792fe47cd0adcb1.jpg new file mode 100644 index 00000000..1476981d Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-458_ecc1c9a5d7d8ddd565a2f77710b0c901a2195af91c1bf21fd792fe47cd0adcb1.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-459_b47f91ff3e609178c17632a897c9b507e5e865b7468c992fd1708337002af819.jpg b/.raw/MidasFEAAnalysisManual/images/page-459_b47f91ff3e609178c17632a897c9b507e5e865b7468c992fd1708337002af819.jpg new file mode 100644 index 00000000..e27b1d1c Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-459_b47f91ff3e609178c17632a897c9b507e5e865b7468c992fd1708337002af819.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-464_b9148a2e2204b6039f18667c8670172d4abdc95a218cbb6befd4bd9c94deedd8.jpg b/.raw/MidasFEAAnalysisManual/images/page-464_b9148a2e2204b6039f18667c8670172d4abdc95a218cbb6befd4bd9c94deedd8.jpg new file mode 100644 index 00000000..5f3944b9 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-464_b9148a2e2204b6039f18667c8670172d4abdc95a218cbb6befd4bd9c94deedd8.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-465_1be3ddeb10086200a0181d36a8af653bbda584cde4389f7a968d6af8bdaf4525.jpg b/.raw/MidasFEAAnalysisManual/images/page-465_1be3ddeb10086200a0181d36a8af653bbda584cde4389f7a968d6af8bdaf4525.jpg new file mode 100644 index 00000000..613326a0 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-465_1be3ddeb10086200a0181d36a8af653bbda584cde4389f7a968d6af8bdaf4525.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-465_396bd5e401366a05b40017fa68240f5239410a1d4aafc784332f235d66550356.jpg b/.raw/MidasFEAAnalysisManual/images/page-465_396bd5e401366a05b40017fa68240f5239410a1d4aafc784332f235d66550356.jpg new file mode 100644 index 00000000..49329fa5 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-465_396bd5e401366a05b40017fa68240f5239410a1d4aafc784332f235d66550356.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-465_a71e63ce46cb1996bbd2845039109fdb55d646badf491fb1e9993f68e4ffa0fd.jpg b/.raw/MidasFEAAnalysisManual/images/page-465_a71e63ce46cb1996bbd2845039109fdb55d646badf491fb1e9993f68e4ffa0fd.jpg new file mode 100644 index 00000000..c7a11c73 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-465_a71e63ce46cb1996bbd2845039109fdb55d646badf491fb1e9993f68e4ffa0fd.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-466_36def956ba24d3a89af8b786e48705af85486282c876d05a67bb4293c51f5708.jpg b/.raw/MidasFEAAnalysisManual/images/page-466_36def956ba24d3a89af8b786e48705af85486282c876d05a67bb4293c51f5708.jpg new file mode 100644 index 00000000..8cc55be8 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-466_36def956ba24d3a89af8b786e48705af85486282c876d05a67bb4293c51f5708.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-466_67595b6e3a58f0f93faea67d6519b5bf40c0e5150b131c6cf1a0828bc238ba5c.jpg b/.raw/MidasFEAAnalysisManual/images/page-466_67595b6e3a58f0f93faea67d6519b5bf40c0e5150b131c6cf1a0828bc238ba5c.jpg new file mode 100644 index 00000000..7b1fe4ed Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-466_67595b6e3a58f0f93faea67d6519b5bf40c0e5150b131c6cf1a0828bc238ba5c.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-466_701d163e5244dce7bd9c947bcf70876068f94c942349368550bbd3b74e71dbc0.jpg b/.raw/MidasFEAAnalysisManual/images/page-466_701d163e5244dce7bd9c947bcf70876068f94c942349368550bbd3b74e71dbc0.jpg new file mode 100644 index 00000000..588f63f9 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-466_701d163e5244dce7bd9c947bcf70876068f94c942349368550bbd3b74e71dbc0.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-466_bb76d4b59146aac10949f43e334b0789a7f3a2c98c344ef8d4253baf19b7a5ed.jpg b/.raw/MidasFEAAnalysisManual/images/page-466_bb76d4b59146aac10949f43e334b0789a7f3a2c98c344ef8d4253baf19b7a5ed.jpg new file mode 100644 index 00000000..d16bc2b7 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-466_bb76d4b59146aac10949f43e334b0789a7f3a2c98c344ef8d4253baf19b7a5ed.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-466_cdcbb4210eacd3750541f46389e6d35606324ba444105b9e4e8bffbc3bf6c544.jpg b/.raw/MidasFEAAnalysisManual/images/page-466_cdcbb4210eacd3750541f46389e6d35606324ba444105b9e4e8bffbc3bf6c544.jpg new file mode 100644 index 00000000..eeb3000e Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-466_cdcbb4210eacd3750541f46389e6d35606324ba444105b9e4e8bffbc3bf6c544.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-467_54e930cb1838e38db6b4bbd8d31a5ecec3ba683626d2095b968e2ecdd1772c45.jpg b/.raw/MidasFEAAnalysisManual/images/page-467_54e930cb1838e38db6b4bbd8d31a5ecec3ba683626d2095b968e2ecdd1772c45.jpg new file mode 100644 index 00000000..bc6254c5 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-467_54e930cb1838e38db6b4bbd8d31a5ecec3ba683626d2095b968e2ecdd1772c45.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-467_5fd020f2f0bb3d2dae5b494f41f15127d88da28717513129b61ed8aa1bea11e1.jpg b/.raw/MidasFEAAnalysisManual/images/page-467_5fd020f2f0bb3d2dae5b494f41f15127d88da28717513129b61ed8aa1bea11e1.jpg new file mode 100644 index 00000000..991b69b0 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-467_5fd020f2f0bb3d2dae5b494f41f15127d88da28717513129b61ed8aa1bea11e1.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-467_6fdc6f66013a1bc42b57756a84ac5e0b79d5b577d56333d02a9970e5a9e44beb.jpg b/.raw/MidasFEAAnalysisManual/images/page-467_6fdc6f66013a1bc42b57756a84ac5e0b79d5b577d56333d02a9970e5a9e44beb.jpg new file mode 100644 index 00000000..a694e2ec Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-467_6fdc6f66013a1bc42b57756a84ac5e0b79d5b577d56333d02a9970e5a9e44beb.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-467_95e29bded4f15e4bcf239646ad5a5dfba30c78bfe0385c728a102f5d6042f989.jpg b/.raw/MidasFEAAnalysisManual/images/page-467_95e29bded4f15e4bcf239646ad5a5dfba30c78bfe0385c728a102f5d6042f989.jpg new file mode 100644 index 00000000..f79614ce Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-467_95e29bded4f15e4bcf239646ad5a5dfba30c78bfe0385c728a102f5d6042f989.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-467_eeb327c7d502bdb278825c4325b081d5fe694e006ac7c8235d9b22bf082be163.jpg b/.raw/MidasFEAAnalysisManual/images/page-467_eeb327c7d502bdb278825c4325b081d5fe694e006ac7c8235d9b22bf082be163.jpg new file mode 100644 index 00000000..cf44f154 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-467_eeb327c7d502bdb278825c4325b081d5fe694e006ac7c8235d9b22bf082be163.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-468_0d1c2faf7d437ca5ea23e78ebf4db14fe3d4b28c27951348a87fa0ff379da1bf.jpg b/.raw/MidasFEAAnalysisManual/images/page-468_0d1c2faf7d437ca5ea23e78ebf4db14fe3d4b28c27951348a87fa0ff379da1bf.jpg new file mode 100644 index 00000000..d359cc25 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-468_0d1c2faf7d437ca5ea23e78ebf4db14fe3d4b28c27951348a87fa0ff379da1bf.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-468_2a6945859ccdd1b5b5f75cc8172ae0edc7fc3b7ffb53a64a7ef6bbff7432fe02.jpg b/.raw/MidasFEAAnalysisManual/images/page-468_2a6945859ccdd1b5b5f75cc8172ae0edc7fc3b7ffb53a64a7ef6bbff7432fe02.jpg new file mode 100644 index 00000000..a25e8662 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-468_2a6945859ccdd1b5b5f75cc8172ae0edc7fc3b7ffb53a64a7ef6bbff7432fe02.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-468_837603eec56b391aa6f18fa9664c3d19873a7a5b7200fe4ed5dffa69ccac719f.jpg b/.raw/MidasFEAAnalysisManual/images/page-468_837603eec56b391aa6f18fa9664c3d19873a7a5b7200fe4ed5dffa69ccac719f.jpg new file mode 100644 index 00000000..9b2ea659 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-468_837603eec56b391aa6f18fa9664c3d19873a7a5b7200fe4ed5dffa69ccac719f.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-468_839b8b1f30e4427d056ede4b484159da9db995fda8ff5b3fc75edd8e0bcd4e05.jpg b/.raw/MidasFEAAnalysisManual/images/page-468_839b8b1f30e4427d056ede4b484159da9db995fda8ff5b3fc75edd8e0bcd4e05.jpg new file mode 100644 index 00000000..fb112a6b Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-468_839b8b1f30e4427d056ede4b484159da9db995fda8ff5b3fc75edd8e0bcd4e05.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-468_c3f79735fe16bca88b1a3833591f16fe3c86296d901b636bc3f208bb2f410741.jpg b/.raw/MidasFEAAnalysisManual/images/page-468_c3f79735fe16bca88b1a3833591f16fe3c86296d901b636bc3f208bb2f410741.jpg new file mode 100644 index 00000000..84f0cd36 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-468_c3f79735fe16bca88b1a3833591f16fe3c86296d901b636bc3f208bb2f410741.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-468_cc97781117251ed3fd94b0d7554ff4992bd43d521dfd49e29073c3f9c39f8af2.jpg b/.raw/MidasFEAAnalysisManual/images/page-468_cc97781117251ed3fd94b0d7554ff4992bd43d521dfd49e29073c3f9c39f8af2.jpg new file mode 100644 index 00000000..74f175b0 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-468_cc97781117251ed3fd94b0d7554ff4992bd43d521dfd49e29073c3f9c39f8af2.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-468_cedff9dd68282011d025e7a640d360f9443774e4d690f34c0c7422d2d4d65ded.jpg b/.raw/MidasFEAAnalysisManual/images/page-468_cedff9dd68282011d025e7a640d360f9443774e4d690f34c0c7422d2d4d65ded.jpg new file mode 100644 index 00000000..2d2ceba8 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-468_cedff9dd68282011d025e7a640d360f9443774e4d690f34c0c7422d2d4d65ded.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-468_d53b75378553debd47f2dffb0f04bf0f6167863367f5b06ecef420b29b11f8af.jpg b/.raw/MidasFEAAnalysisManual/images/page-468_d53b75378553debd47f2dffb0f04bf0f6167863367f5b06ecef420b29b11f8af.jpg new file mode 100644 index 00000000..5441adda Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-468_d53b75378553debd47f2dffb0f04bf0f6167863367f5b06ecef420b29b11f8af.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-468_dd07dc66a99eab2f6da86a473777b4959187afdbdf857016257bd0dbf77f98fb.jpg b/.raw/MidasFEAAnalysisManual/images/page-468_dd07dc66a99eab2f6da86a473777b4959187afdbdf857016257bd0dbf77f98fb.jpg new file mode 100644 index 00000000..96818e85 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-468_dd07dc66a99eab2f6da86a473777b4959187afdbdf857016257bd0dbf77f98fb.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-468_e12fdce0cc0df5246436c28583fb66f4a6f499deb6baa1866b216b5d96b8bb6d.jpg b/.raw/MidasFEAAnalysisManual/images/page-468_e12fdce0cc0df5246436c28583fb66f4a6f499deb6baa1866b216b5d96b8bb6d.jpg new file mode 100644 index 00000000..01a2084f Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-468_e12fdce0cc0df5246436c28583fb66f4a6f499deb6baa1866b216b5d96b8bb6d.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-468_e663b667e75e674682ad7147d5ec615fd1c024e287b71251c0739505b440bd9e.jpg b/.raw/MidasFEAAnalysisManual/images/page-468_e663b667e75e674682ad7147d5ec615fd1c024e287b71251c0739505b440bd9e.jpg new file mode 100644 index 00000000..fb1d0fea Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-468_e663b667e75e674682ad7147d5ec615fd1c024e287b71251c0739505b440bd9e.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-469_0d62c0b9cb50b3f5cacc4f365f10f358f78a53991f19891bc9245bb5800476bd.jpg b/.raw/MidasFEAAnalysisManual/images/page-469_0d62c0b9cb50b3f5cacc4f365f10f358f78a53991f19891bc9245bb5800476bd.jpg new file mode 100644 index 00000000..17083d0c Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-469_0d62c0b9cb50b3f5cacc4f365f10f358f78a53991f19891bc9245bb5800476bd.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-469_30cabe88c45bfcf73cacbe5eb0a459e898e0679a33554a53dff638114bcf9f51.jpg b/.raw/MidasFEAAnalysisManual/images/page-469_30cabe88c45bfcf73cacbe5eb0a459e898e0679a33554a53dff638114bcf9f51.jpg new file mode 100644 index 00000000..f59d6255 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-469_30cabe88c45bfcf73cacbe5eb0a459e898e0679a33554a53dff638114bcf9f51.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-469_3deeae222ee223c6ca8ec8921aa30f72ed0753b5f04eee60471caed8699092f2.jpg b/.raw/MidasFEAAnalysisManual/images/page-469_3deeae222ee223c6ca8ec8921aa30f72ed0753b5f04eee60471caed8699092f2.jpg new file mode 100644 index 00000000..2d826183 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-469_3deeae222ee223c6ca8ec8921aa30f72ed0753b5f04eee60471caed8699092f2.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-469_70047139de2a5f10956b201d1dd36ac712e62d695dad96891e96166dfd634a1f.jpg b/.raw/MidasFEAAnalysisManual/images/page-469_70047139de2a5f10956b201d1dd36ac712e62d695dad96891e96166dfd634a1f.jpg new file mode 100644 index 00000000..a1000b83 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-469_70047139de2a5f10956b201d1dd36ac712e62d695dad96891e96166dfd634a1f.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-469_8fdb03ccde4bd20ee1f3dda0adec7c5390374fb81db1f8469d4bbe56d64cfe8c.jpg b/.raw/MidasFEAAnalysisManual/images/page-469_8fdb03ccde4bd20ee1f3dda0adec7c5390374fb81db1f8469d4bbe56d64cfe8c.jpg new file mode 100644 index 00000000..2a2a1064 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-469_8fdb03ccde4bd20ee1f3dda0adec7c5390374fb81db1f8469d4bbe56d64cfe8c.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-469_c7041729d217593caacc12806032bcee1d820d108076dd1d2bd12255c26b0966.jpg b/.raw/MidasFEAAnalysisManual/images/page-469_c7041729d217593caacc12806032bcee1d820d108076dd1d2bd12255c26b0966.jpg new file mode 100644 index 00000000..2122d4b4 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-469_c7041729d217593caacc12806032bcee1d820d108076dd1d2bd12255c26b0966.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-469_eb2e1f43d03e290d9d2078a8f9df96921752b7ed8ea03ea81d11cdf2ccac9694.jpg b/.raw/MidasFEAAnalysisManual/images/page-469_eb2e1f43d03e290d9d2078a8f9df96921752b7ed8ea03ea81d11cdf2ccac9694.jpg new file mode 100644 index 00000000..d9dbff1e Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-469_eb2e1f43d03e290d9d2078a8f9df96921752b7ed8ea03ea81d11cdf2ccac9694.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-469_f80ba3178f46dbbc5a5794ae64e68bec3c1233cac04ba94d894b3c9848a2d769.jpg b/.raw/MidasFEAAnalysisManual/images/page-469_f80ba3178f46dbbc5a5794ae64e68bec3c1233cac04ba94d894b3c9848a2d769.jpg new file mode 100644 index 00000000..1ffbed49 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-469_f80ba3178f46dbbc5a5794ae64e68bec3c1233cac04ba94d894b3c9848a2d769.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-470_3a2c307d50d31180a5162c3935efa271bf0c248620100986d408d22c51d75f1b.jpg b/.raw/MidasFEAAnalysisManual/images/page-470_3a2c307d50d31180a5162c3935efa271bf0c248620100986d408d22c51d75f1b.jpg new file mode 100644 index 00000000..98d2fb7f Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-470_3a2c307d50d31180a5162c3935efa271bf0c248620100986d408d22c51d75f1b.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-470_610fbb523993965fc30a89e72a24f2db28430a1ec9c8ee5a400d5d68a0d21a75.jpg b/.raw/MidasFEAAnalysisManual/images/page-470_610fbb523993965fc30a89e72a24f2db28430a1ec9c8ee5a400d5d68a0d21a75.jpg new file mode 100644 index 00000000..5f78713f Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-470_610fbb523993965fc30a89e72a24f2db28430a1ec9c8ee5a400d5d68a0d21a75.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-470_69f94dd41f2a1fe0548effd008ec500c2dd812ad5a23c36754b82cddaf6e87f2.jpg b/.raw/MidasFEAAnalysisManual/images/page-470_69f94dd41f2a1fe0548effd008ec500c2dd812ad5a23c36754b82cddaf6e87f2.jpg new file mode 100644 index 00000000..c1a0ef70 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-470_69f94dd41f2a1fe0548effd008ec500c2dd812ad5a23c36754b82cddaf6e87f2.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-470_798c61dd731d80095465d9ac632213da3f88207d387c0dbdd01a22349962fe02.jpg b/.raw/MidasFEAAnalysisManual/images/page-470_798c61dd731d80095465d9ac632213da3f88207d387c0dbdd01a22349962fe02.jpg new file mode 100644 index 00000000..cc7fa122 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-470_798c61dd731d80095465d9ac632213da3f88207d387c0dbdd01a22349962fe02.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-470_90ce83921f792912ff4e1168acfdd5c435128f4f8d1f208d770f327c53acce63.jpg b/.raw/MidasFEAAnalysisManual/images/page-470_90ce83921f792912ff4e1168acfdd5c435128f4f8d1f208d770f327c53acce63.jpg new file mode 100644 index 00000000..ac528f15 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-470_90ce83921f792912ff4e1168acfdd5c435128f4f8d1f208d770f327c53acce63.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-470_bcef88c3ef920228d51e71bd42bf41169051ab05e35aa48f3289423e40063d5e.jpg b/.raw/MidasFEAAnalysisManual/images/page-470_bcef88c3ef920228d51e71bd42bf41169051ab05e35aa48f3289423e40063d5e.jpg new file mode 100644 index 00000000..7e013e00 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-470_bcef88c3ef920228d51e71bd42bf41169051ab05e35aa48f3289423e40063d5e.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-470_da751381ff96bd00a655dc1052e5a43fa2af505580487ca73dd84b326bbcd7f7.jpg b/.raw/MidasFEAAnalysisManual/images/page-470_da751381ff96bd00a655dc1052e5a43fa2af505580487ca73dd84b326bbcd7f7.jpg new file mode 100644 index 00000000..c5824592 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-470_da751381ff96bd00a655dc1052e5a43fa2af505580487ca73dd84b326bbcd7f7.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-470_e2e8f66098edd763b662a01856050477fb626bf5f209094c2a6ccc80490bb227.jpg b/.raw/MidasFEAAnalysisManual/images/page-470_e2e8f66098edd763b662a01856050477fb626bf5f209094c2a6ccc80490bb227.jpg new file mode 100644 index 00000000..ff61a115 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-470_e2e8f66098edd763b662a01856050477fb626bf5f209094c2a6ccc80490bb227.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-470_e3c35fbaab1891a7812dc75dc950da2d6a686400586d2653b51790650941a915.jpg b/.raw/MidasFEAAnalysisManual/images/page-470_e3c35fbaab1891a7812dc75dc950da2d6a686400586d2653b51790650941a915.jpg new file mode 100644 index 00000000..42a44735 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-470_e3c35fbaab1891a7812dc75dc950da2d6a686400586d2653b51790650941a915.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-470_e54ba14f7923cd7de4ff60bea7b658feae151fef59260f02afb823054d2a6f43.jpg b/.raw/MidasFEAAnalysisManual/images/page-470_e54ba14f7923cd7de4ff60bea7b658feae151fef59260f02afb823054d2a6f43.jpg new file mode 100644 index 00000000..4c74a43c Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-470_e54ba14f7923cd7de4ff60bea7b658feae151fef59260f02afb823054d2a6f43.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-471_b118621c57bba521a61da84d063ef6b345c062e6fcb9712adc2fba87ccfdbc17.jpg b/.raw/MidasFEAAnalysisManual/images/page-471_b118621c57bba521a61da84d063ef6b345c062e6fcb9712adc2fba87ccfdbc17.jpg new file mode 100644 index 00000000..af6eca9d Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-471_b118621c57bba521a61da84d063ef6b345c062e6fcb9712adc2fba87ccfdbc17.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-478_07c853448b29e9e292334c2a1d47038c43af8833bc89571934124760828c96f7.jpg b/.raw/MidasFEAAnalysisManual/images/page-478_07c853448b29e9e292334c2a1d47038c43af8833bc89571934124760828c96f7.jpg new file mode 100644 index 00000000..80ae4ca2 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-478_07c853448b29e9e292334c2a1d47038c43af8833bc89571934124760828c96f7.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-478_23c58da6f7468b03677f8c95d4825c6a77266887653839461760ff5a2b8acc54.jpg b/.raw/MidasFEAAnalysisManual/images/page-478_23c58da6f7468b03677f8c95d4825c6a77266887653839461760ff5a2b8acc54.jpg new file mode 100644 index 00000000..d28e2f03 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-478_23c58da6f7468b03677f8c95d4825c6a77266887653839461760ff5a2b8acc54.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-478_517720b74a38f4a0f04f87ab739af6a94523cad111e0e4f0a3f6c448a5f07e5c.jpg b/.raw/MidasFEAAnalysisManual/images/page-478_517720b74a38f4a0f04f87ab739af6a94523cad111e0e4f0a3f6c448a5f07e5c.jpg new file mode 100644 index 00000000..4267ba3d Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-478_517720b74a38f4a0f04f87ab739af6a94523cad111e0e4f0a3f6c448a5f07e5c.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-479_11fa8b2bcfccae68bc148d259cf9463eae39971b5c09e23fb83ad9fbab370d65.jpg b/.raw/MidasFEAAnalysisManual/images/page-479_11fa8b2bcfccae68bc148d259cf9463eae39971b5c09e23fb83ad9fbab370d65.jpg new file mode 100644 index 00000000..4d79ef71 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-479_11fa8b2bcfccae68bc148d259cf9463eae39971b5c09e23fb83ad9fbab370d65.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-480_89f4983800afd66b9619229eea0baa0ae5d6a06b7b464229e28c61606e2fec47.jpg b/.raw/MidasFEAAnalysisManual/images/page-480_89f4983800afd66b9619229eea0baa0ae5d6a06b7b464229e28c61606e2fec47.jpg new file mode 100644 index 00000000..6e72d824 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-480_89f4983800afd66b9619229eea0baa0ae5d6a06b7b464229e28c61606e2fec47.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-480_d5d6aaa0db5a0061764837b26142560665fbc05904e49af3877143d2425e3389.jpg b/.raw/MidasFEAAnalysisManual/images/page-480_d5d6aaa0db5a0061764837b26142560665fbc05904e49af3877143d2425e3389.jpg new file mode 100644 index 00000000..3722712a Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-480_d5d6aaa0db5a0061764837b26142560665fbc05904e49af3877143d2425e3389.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-480_dacfa66ba0d69087187704ea8592e7bb0cc2e61c41b873b0324b2b0b6200df7b.jpg b/.raw/MidasFEAAnalysisManual/images/page-480_dacfa66ba0d69087187704ea8592e7bb0cc2e61c41b873b0324b2b0b6200df7b.jpg new file mode 100644 index 00000000..2945e6ee Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-480_dacfa66ba0d69087187704ea8592e7bb0cc2e61c41b873b0324b2b0b6200df7b.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-481_0ceca902f67e2adff41f8340ea80f8d24bfc67ec442b5d13ddfe97a0533368b7.jpg b/.raw/MidasFEAAnalysisManual/images/page-481_0ceca902f67e2adff41f8340ea80f8d24bfc67ec442b5d13ddfe97a0533368b7.jpg new file mode 100644 index 00000000..a9bb9fac Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-481_0ceca902f67e2adff41f8340ea80f8d24bfc67ec442b5d13ddfe97a0533368b7.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-481_8951a1af6abf1a439203f84a2d0b27c8766e2ab9429a803685223511d5b3de25.jpg b/.raw/MidasFEAAnalysisManual/images/page-481_8951a1af6abf1a439203f84a2d0b27c8766e2ab9429a803685223511d5b3de25.jpg new file mode 100644 index 00000000..ef574414 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-481_8951a1af6abf1a439203f84a2d0b27c8766e2ab9429a803685223511d5b3de25.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-481_b6b1a2ce72913e17b36b1f527ea46d63aff8cabd3daddbe49c932821dff8240e.jpg b/.raw/MidasFEAAnalysisManual/images/page-481_b6b1a2ce72913e17b36b1f527ea46d63aff8cabd3daddbe49c932821dff8240e.jpg new file mode 100644 index 00000000..ce1b4532 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-481_b6b1a2ce72913e17b36b1f527ea46d63aff8cabd3daddbe49c932821dff8240e.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-481_c978fa87322a84b67537420080de862479a51e5e916d39ac1536928cbc85b31b.jpg b/.raw/MidasFEAAnalysisManual/images/page-481_c978fa87322a84b67537420080de862479a51e5e916d39ac1536928cbc85b31b.jpg new file mode 100644 index 00000000..eb70526f Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-481_c978fa87322a84b67537420080de862479a51e5e916d39ac1536928cbc85b31b.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-482_c63206d3b9a057ee1a345ed8d0bf9498c139fe87eb82b4278407cb516b4182f3.jpg b/.raw/MidasFEAAnalysisManual/images/page-482_c63206d3b9a057ee1a345ed8d0bf9498c139fe87eb82b4278407cb516b4182f3.jpg new file mode 100644 index 00000000..65a90a8b Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-482_c63206d3b9a057ee1a345ed8d0bf9498c139fe87eb82b4278407cb516b4182f3.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-483_40e67f99a0679d277f4ba46e3b35817693354de579e43acb315f7c39654d1fb9.jpg b/.raw/MidasFEAAnalysisManual/images/page-483_40e67f99a0679d277f4ba46e3b35817693354de579e43acb315f7c39654d1fb9.jpg new file mode 100644 index 00000000..290a1c7c Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-483_40e67f99a0679d277f4ba46e3b35817693354de579e43acb315f7c39654d1fb9.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-483_8fb0b30fe0d443feb1c1d24ca21d5001f830ab7417ea829b298ef1c2f0b87ec8.jpg b/.raw/MidasFEAAnalysisManual/images/page-483_8fb0b30fe0d443feb1c1d24ca21d5001f830ab7417ea829b298ef1c2f0b87ec8.jpg new file mode 100644 index 00000000..52a0aafa Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-483_8fb0b30fe0d443feb1c1d24ca21d5001f830ab7417ea829b298ef1c2f0b87ec8.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-484_5936ba8c2a9e6e57ebb504a856e3cba65cf07597fb45ba674a3ead94395623ae.jpg b/.raw/MidasFEAAnalysisManual/images/page-484_5936ba8c2a9e6e57ebb504a856e3cba65cf07597fb45ba674a3ead94395623ae.jpg new file mode 100644 index 00000000..892e281d Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-484_5936ba8c2a9e6e57ebb504a856e3cba65cf07597fb45ba674a3ead94395623ae.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-484_6e68053c6ffd4f24231350f2ad477e224993ab4c40b12c3cf051eb4f7ca94a26.jpg b/.raw/MidasFEAAnalysisManual/images/page-484_6e68053c6ffd4f24231350f2ad477e224993ab4c40b12c3cf051eb4f7ca94a26.jpg new file mode 100644 index 00000000..18429e7d Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-484_6e68053c6ffd4f24231350f2ad477e224993ab4c40b12c3cf051eb4f7ca94a26.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-484_b40ce4708d97a5e287e6dbdac75aefc1d37e893994eb2d6a62f8a59e0572db38.jpg b/.raw/MidasFEAAnalysisManual/images/page-484_b40ce4708d97a5e287e6dbdac75aefc1d37e893994eb2d6a62f8a59e0572db38.jpg new file mode 100644 index 00000000..28ef7fb6 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-484_b40ce4708d97a5e287e6dbdac75aefc1d37e893994eb2d6a62f8a59e0572db38.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-484_c136ea92ead8403e771628403b616d1fb930af466a1878e892f386513e2b59ab.jpg b/.raw/MidasFEAAnalysisManual/images/page-484_c136ea92ead8403e771628403b616d1fb930af466a1878e892f386513e2b59ab.jpg new file mode 100644 index 00000000..bd189eb3 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-484_c136ea92ead8403e771628403b616d1fb930af466a1878e892f386513e2b59ab.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-485_2e411ce57e07053c2e9f71ce5c49be222b4c00caaa6e45a8a9bc607c81ed7af5.jpg b/.raw/MidasFEAAnalysisManual/images/page-485_2e411ce57e07053c2e9f71ce5c49be222b4c00caaa6e45a8a9bc607c81ed7af5.jpg new file mode 100644 index 00000000..2d080b53 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-485_2e411ce57e07053c2e9f71ce5c49be222b4c00caaa6e45a8a9bc607c81ed7af5.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-485_33501c985ab2207dfb228cb0b2eb07bb1c7ec03d6e122b0e8ee464c565a9bee3.jpg b/.raw/MidasFEAAnalysisManual/images/page-485_33501c985ab2207dfb228cb0b2eb07bb1c7ec03d6e122b0e8ee464c565a9bee3.jpg new file mode 100644 index 00000000..3ddf0dfa Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-485_33501c985ab2207dfb228cb0b2eb07bb1c7ec03d6e122b0e8ee464c565a9bee3.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-486_ea42121132e01298efd824c5b79a061a8b5a6d4f790f14ea7bd28f6c77f47317.jpg b/.raw/MidasFEAAnalysisManual/images/page-486_ea42121132e01298efd824c5b79a061a8b5a6d4f790f14ea7bd28f6c77f47317.jpg new file mode 100644 index 00000000..028a2bef Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-486_ea42121132e01298efd824c5b79a061a8b5a6d4f790f14ea7bd28f6c77f47317.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-487_15293f9b70aea67f375a95ad34e39a65d5e093be54d319886cc3c294ec893b7e.jpg b/.raw/MidasFEAAnalysisManual/images/page-487_15293f9b70aea67f375a95ad34e39a65d5e093be54d319886cc3c294ec893b7e.jpg new file mode 100644 index 00000000..9650a7db Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-487_15293f9b70aea67f375a95ad34e39a65d5e093be54d319886cc3c294ec893b7e.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-487_82af530fbafe6da38ad5ba2fe6329fa7930fef5590ad61b497765289d2637620.jpg b/.raw/MidasFEAAnalysisManual/images/page-487_82af530fbafe6da38ad5ba2fe6329fa7930fef5590ad61b497765289d2637620.jpg new file mode 100644 index 00000000..e54da56c Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-487_82af530fbafe6da38ad5ba2fe6329fa7930fef5590ad61b497765289d2637620.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-487_e4593ed470ab40e426a0a74bbba2954c37b3a6f32281c96c8b2dcc8f3a6d3ebb.jpg b/.raw/MidasFEAAnalysisManual/images/page-487_e4593ed470ab40e426a0a74bbba2954c37b3a6f32281c96c8b2dcc8f3a6d3ebb.jpg new file mode 100644 index 00000000..2b09adec Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-487_e4593ed470ab40e426a0a74bbba2954c37b3a6f32281c96c8b2dcc8f3a6d3ebb.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-490_911c7e7865312e1c719c52240c1ee049f179718bbcf977fa463bee0ff3fe6188.jpg b/.raw/MidasFEAAnalysisManual/images/page-490_911c7e7865312e1c719c52240c1ee049f179718bbcf977fa463bee0ff3fe6188.jpg new file mode 100644 index 00000000..d1b56dc1 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-490_911c7e7865312e1c719c52240c1ee049f179718bbcf977fa463bee0ff3fe6188.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-490_a993f19f4a7e807766bdcef9d2f1b03bf435d448c8181385d194b607d929f897.jpg b/.raw/MidasFEAAnalysisManual/images/page-490_a993f19f4a7e807766bdcef9d2f1b03bf435d448c8181385d194b607d929f897.jpg new file mode 100644 index 00000000..b30d5c1d Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-490_a993f19f4a7e807766bdcef9d2f1b03bf435d448c8181385d194b607d929f897.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-490_ec5453dd57d3ebf20d52486c249da2db9eaae658f4a3fbc4d3023d0635569d60.jpg b/.raw/MidasFEAAnalysisManual/images/page-490_ec5453dd57d3ebf20d52486c249da2db9eaae658f4a3fbc4d3023d0635569d60.jpg new file mode 100644 index 00000000..e2a3b3a0 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-490_ec5453dd57d3ebf20d52486c249da2db9eaae658f4a3fbc4d3023d0635569d60.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-490_ef3d36f2a05425285bcfd0074b02f72ac886674773d0e65bec6d95f6e1db2fe5.jpg b/.raw/MidasFEAAnalysisManual/images/page-490_ef3d36f2a05425285bcfd0074b02f72ac886674773d0e65bec6d95f6e1db2fe5.jpg new file mode 100644 index 00000000..d411fd9f Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-490_ef3d36f2a05425285bcfd0074b02f72ac886674773d0e65bec6d95f6e1db2fe5.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-491_0ac010d6174ce9a7a24fe02fa30f1dd43e46f5eae18c9f2ee01040e982d4b225.jpg b/.raw/MidasFEAAnalysisManual/images/page-491_0ac010d6174ce9a7a24fe02fa30f1dd43e46f5eae18c9f2ee01040e982d4b225.jpg new file mode 100644 index 00000000..e08b88ba Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-491_0ac010d6174ce9a7a24fe02fa30f1dd43e46f5eae18c9f2ee01040e982d4b225.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-491_8f89c7a43a7857628faea375e1c2ac1d10c8e340b13d01a89a1e0475be57f421.jpg b/.raw/MidasFEAAnalysisManual/images/page-491_8f89c7a43a7857628faea375e1c2ac1d10c8e340b13d01a89a1e0475be57f421.jpg new file mode 100644 index 00000000..4dcd45a9 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-491_8f89c7a43a7857628faea375e1c2ac1d10c8e340b13d01a89a1e0475be57f421.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-492_19c4a72462899c5ee206e91f9962d9d5051b9442409056fe5d76100f3f92ad48.jpg b/.raw/MidasFEAAnalysisManual/images/page-492_19c4a72462899c5ee206e91f9962d9d5051b9442409056fe5d76100f3f92ad48.jpg new file mode 100644 index 00000000..9b92b600 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-492_19c4a72462899c5ee206e91f9962d9d5051b9442409056fe5d76100f3f92ad48.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-492_f275544b42f656d7703e1747f4c8c473e66d3239467b76a5980e4a8bf85817f4.jpg b/.raw/MidasFEAAnalysisManual/images/page-492_f275544b42f656d7703e1747f4c8c473e66d3239467b76a5980e4a8bf85817f4.jpg new file mode 100644 index 00000000..4846805f Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-492_f275544b42f656d7703e1747f4c8c473e66d3239467b76a5980e4a8bf85817f4.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-494_b0782609a23a12e9961e773ea3ac8dcaaee7d08e65b1f4913344b18bcc4c8e19.jpg b/.raw/MidasFEAAnalysisManual/images/page-494_b0782609a23a12e9961e773ea3ac8dcaaee7d08e65b1f4913344b18bcc4c8e19.jpg new file mode 100644 index 00000000..bd5dd884 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-494_b0782609a23a12e9961e773ea3ac8dcaaee7d08e65b1f4913344b18bcc4c8e19.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-497_063a4c2a7954ffac61e814e6a2f369c37927ab17917530f46557bece553a36a7.jpg b/.raw/MidasFEAAnalysisManual/images/page-497_063a4c2a7954ffac61e814e6a2f369c37927ab17917530f46557bece553a36a7.jpg new file mode 100644 index 00000000..35569f20 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-497_063a4c2a7954ffac61e814e6a2f369c37927ab17917530f46557bece553a36a7.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-498_0b8ab18e8ce6cdd87df5644652eab7dca874c5a5f27bba3caf2c7bf145c469c2.jpg b/.raw/MidasFEAAnalysisManual/images/page-498_0b8ab18e8ce6cdd87df5644652eab7dca874c5a5f27bba3caf2c7bf145c469c2.jpg new file mode 100644 index 00000000..7e1c61ea Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-498_0b8ab18e8ce6cdd87df5644652eab7dca874c5a5f27bba3caf2c7bf145c469c2.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-498_3b2d4ebbd86a7f630053b2a8883d5d7522d406a95963c18a0e5d3027fd5883d7.jpg b/.raw/MidasFEAAnalysisManual/images/page-498_3b2d4ebbd86a7f630053b2a8883d5d7522d406a95963c18a0e5d3027fd5883d7.jpg new file mode 100644 index 00000000..edbace0a Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-498_3b2d4ebbd86a7f630053b2a8883d5d7522d406a95963c18a0e5d3027fd5883d7.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-498_5944362cd978a99f0dec78c082c8a26e03fe3b07d4bae646797c4eaa16eab359.jpg b/.raw/MidasFEAAnalysisManual/images/page-498_5944362cd978a99f0dec78c082c8a26e03fe3b07d4bae646797c4eaa16eab359.jpg new file mode 100644 index 00000000..98742a62 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-498_5944362cd978a99f0dec78c082c8a26e03fe3b07d4bae646797c4eaa16eab359.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-498_90cc30fff88952fa2444966a53774bd2d86db108ee787d5de072c22c9be99542.jpg b/.raw/MidasFEAAnalysisManual/images/page-498_90cc30fff88952fa2444966a53774bd2d86db108ee787d5de072c22c9be99542.jpg new file mode 100644 index 00000000..783f29d1 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-498_90cc30fff88952fa2444966a53774bd2d86db108ee787d5de072c22c9be99542.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-498_b3c0523616d541d2e9a1fa3fa9382693212a84fbb44eb52a45e0c334644c1539.jpg b/.raw/MidasFEAAnalysisManual/images/page-498_b3c0523616d541d2e9a1fa3fa9382693212a84fbb44eb52a45e0c334644c1539.jpg new file mode 100644 index 00000000..36660453 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-498_b3c0523616d541d2e9a1fa3fa9382693212a84fbb44eb52a45e0c334644c1539.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-499_2e09a4656bc51fa2c544391810cbcc6584a014367d0c6701bd78ece916f3f7e0.jpg b/.raw/MidasFEAAnalysisManual/images/page-499_2e09a4656bc51fa2c544391810cbcc6584a014367d0c6701bd78ece916f3f7e0.jpg new file mode 100644 index 00000000..e3bf5c3c Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-499_2e09a4656bc51fa2c544391810cbcc6584a014367d0c6701bd78ece916f3f7e0.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-499_77cd43410bb29a6fc8d0918af97b42e799f6b54ee512235a18b03e3bc0ac3327.jpg b/.raw/MidasFEAAnalysisManual/images/page-499_77cd43410bb29a6fc8d0918af97b42e799f6b54ee512235a18b03e3bc0ac3327.jpg new file mode 100644 index 00000000..89c84c03 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-499_77cd43410bb29a6fc8d0918af97b42e799f6b54ee512235a18b03e3bc0ac3327.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-500_97cc9c22350b18f6f8a8ec15897d029e2c1d19410f6b6152a12c3f5639372351.jpg b/.raw/MidasFEAAnalysisManual/images/page-500_97cc9c22350b18f6f8a8ec15897d029e2c1d19410f6b6152a12c3f5639372351.jpg new file mode 100644 index 00000000..99ee3f7e Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-500_97cc9c22350b18f6f8a8ec15897d029e2c1d19410f6b6152a12c3f5639372351.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-501_54ce3240425c323f86b89465d09b6453ad6d1382bb0da680530d6a681e9a85df.jpg b/.raw/MidasFEAAnalysisManual/images/page-501_54ce3240425c323f86b89465d09b6453ad6d1382bb0da680530d6a681e9a85df.jpg new file mode 100644 index 00000000..25c07406 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-501_54ce3240425c323f86b89465d09b6453ad6d1382bb0da680530d6a681e9a85df.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-501_80e4b1648529a5cf1a978b75b966ad04f58c9d7d351bf0d72c309200bfe88ef1.jpg b/.raw/MidasFEAAnalysisManual/images/page-501_80e4b1648529a5cf1a978b75b966ad04f58c9d7d351bf0d72c309200bfe88ef1.jpg new file mode 100644 index 00000000..d9588db6 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-501_80e4b1648529a5cf1a978b75b966ad04f58c9d7d351bf0d72c309200bfe88ef1.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-501_c76605e6cf3aada98c53d5a1f36bf3911e1c3ae4b15d5103e395e077dcd0bff4.jpg b/.raw/MidasFEAAnalysisManual/images/page-501_c76605e6cf3aada98c53d5a1f36bf3911e1c3ae4b15d5103e395e077dcd0bff4.jpg new file mode 100644 index 00000000..6a65719c Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-501_c76605e6cf3aada98c53d5a1f36bf3911e1c3ae4b15d5103e395e077dcd0bff4.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-501_e855280d1856fdd167990f0b9f0f11ec03705144cf377be78beb7b61231f0d8e.jpg b/.raw/MidasFEAAnalysisManual/images/page-501_e855280d1856fdd167990f0b9f0f11ec03705144cf377be78beb7b61231f0d8e.jpg new file mode 100644 index 00000000..7674ba02 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-501_e855280d1856fdd167990f0b9f0f11ec03705144cf377be78beb7b61231f0d8e.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-502_359ea82647119527852c94c6ee38066f1a22f7cf7fb73febc3ffbc495035e158.jpg b/.raw/MidasFEAAnalysisManual/images/page-502_359ea82647119527852c94c6ee38066f1a22f7cf7fb73febc3ffbc495035e158.jpg new file mode 100644 index 00000000..e87632fd Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-502_359ea82647119527852c94c6ee38066f1a22f7cf7fb73febc3ffbc495035e158.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-503_17a783bf745c8e3f68a914033bbea35fafc55175fe880ff51711fb25a45d8d29.jpg b/.raw/MidasFEAAnalysisManual/images/page-503_17a783bf745c8e3f68a914033bbea35fafc55175fe880ff51711fb25a45d8d29.jpg new file mode 100644 index 00000000..8591904c Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-503_17a783bf745c8e3f68a914033bbea35fafc55175fe880ff51711fb25a45d8d29.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-503_9dd9fad63c4562f1cee8fb7d02449fc2d848757a84b6987aaaf9bad5fe112ce3.jpg b/.raw/MidasFEAAnalysisManual/images/page-503_9dd9fad63c4562f1cee8fb7d02449fc2d848757a84b6987aaaf9bad5fe112ce3.jpg new file mode 100644 index 00000000..dc182968 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-503_9dd9fad63c4562f1cee8fb7d02449fc2d848757a84b6987aaaf9bad5fe112ce3.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-503_b6562173bc49c3fbafb4d8a552fd7931693162df62378bedbb28d4e27a12e6a8.jpg b/.raw/MidasFEAAnalysisManual/images/page-503_b6562173bc49c3fbafb4d8a552fd7931693162df62378bedbb28d4e27a12e6a8.jpg new file mode 100644 index 00000000..f811fbf7 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-503_b6562173bc49c3fbafb4d8a552fd7931693162df62378bedbb28d4e27a12e6a8.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-503_dde619c12bac62b7f01161035ae4ba95f9e745edca067b120da168b90e0c564c.jpg b/.raw/MidasFEAAnalysisManual/images/page-503_dde619c12bac62b7f01161035ae4ba95f9e745edca067b120da168b90e0c564c.jpg new file mode 100644 index 00000000..5b24c168 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-503_dde619c12bac62b7f01161035ae4ba95f9e745edca067b120da168b90e0c564c.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-504_3010c9ec8783aa8585c070c7230a78e2762017d63742ac30474d4a91aa14e7db.jpg b/.raw/MidasFEAAnalysisManual/images/page-504_3010c9ec8783aa8585c070c7230a78e2762017d63742ac30474d4a91aa14e7db.jpg new file mode 100644 index 00000000..a29d028a Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-504_3010c9ec8783aa8585c070c7230a78e2762017d63742ac30474d4a91aa14e7db.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-504_d9f6ed99f77960330afe0a5a54c16137252a70e8fcd0b96689ef56fdedb393aa.jpg b/.raw/MidasFEAAnalysisManual/images/page-504_d9f6ed99f77960330afe0a5a54c16137252a70e8fcd0b96689ef56fdedb393aa.jpg new file mode 100644 index 00000000..6e909f24 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-504_d9f6ed99f77960330afe0a5a54c16137252a70e8fcd0b96689ef56fdedb393aa.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-504_e5a9fab14c3ca374565d454f4467a2bf9348248d422cbd1d29cb215b6a3fd3b1.jpg b/.raw/MidasFEAAnalysisManual/images/page-504_e5a9fab14c3ca374565d454f4467a2bf9348248d422cbd1d29cb215b6a3fd3b1.jpg new file mode 100644 index 00000000..4a327d5c Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-504_e5a9fab14c3ca374565d454f4467a2bf9348248d422cbd1d29cb215b6a3fd3b1.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-504_f63ae20f5515411fa101c3de87ef0dea87dba7ece33919e2c615102e5192e527.jpg b/.raw/MidasFEAAnalysisManual/images/page-504_f63ae20f5515411fa101c3de87ef0dea87dba7ece33919e2c615102e5192e527.jpg new file mode 100644 index 00000000..f2d4ea6d Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-504_f63ae20f5515411fa101c3de87ef0dea87dba7ece33919e2c615102e5192e527.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-505_5a8f4c1079a5dbfbf8c0e5179876b360f6534599ae561819012bf5d8be49342b.jpg b/.raw/MidasFEAAnalysisManual/images/page-505_5a8f4c1079a5dbfbf8c0e5179876b360f6534599ae561819012bf5d8be49342b.jpg new file mode 100644 index 00000000..ebb41596 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-505_5a8f4c1079a5dbfbf8c0e5179876b360f6534599ae561819012bf5d8be49342b.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-505_bf1f28e8c72980b952d4e70cf862c8efe568e60e10fd32d1d55f58b2c3323e3c.jpg b/.raw/MidasFEAAnalysisManual/images/page-505_bf1f28e8c72980b952d4e70cf862c8efe568e60e10fd32d1d55f58b2c3323e3c.jpg new file mode 100644 index 00000000..9aea5e60 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-505_bf1f28e8c72980b952d4e70cf862c8efe568e60e10fd32d1d55f58b2c3323e3c.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-505_c3f840b3ea76e4ab17b9d4537253a1cd22eafc92a8c8a5d7eb3720a3ae933466.jpg b/.raw/MidasFEAAnalysisManual/images/page-505_c3f840b3ea76e4ab17b9d4537253a1cd22eafc92a8c8a5d7eb3720a3ae933466.jpg new file mode 100644 index 00000000..6bbe128d Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-505_c3f840b3ea76e4ab17b9d4537253a1cd22eafc92a8c8a5d7eb3720a3ae933466.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-506_00784f675a01952ea0ac5f6332e62d4414fbaf7c862fc91171551434a685c879.jpg b/.raw/MidasFEAAnalysisManual/images/page-506_00784f675a01952ea0ac5f6332e62d4414fbaf7c862fc91171551434a685c879.jpg new file mode 100644 index 00000000..229a6d40 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-506_00784f675a01952ea0ac5f6332e62d4414fbaf7c862fc91171551434a685c879.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-506_3dff672a6045678bbf6955edc8ac48f2718501ee4bacbc366dcf95061037a2c6.jpg b/.raw/MidasFEAAnalysisManual/images/page-506_3dff672a6045678bbf6955edc8ac48f2718501ee4bacbc366dcf95061037a2c6.jpg new file mode 100644 index 00000000..990415fc Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-506_3dff672a6045678bbf6955edc8ac48f2718501ee4bacbc366dcf95061037a2c6.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-506_66be8dc1734cab45c463371b8fa215cf7e4c53ac88fbd4b2c733c32c39cc0102.jpg b/.raw/MidasFEAAnalysisManual/images/page-506_66be8dc1734cab45c463371b8fa215cf7e4c53ac88fbd4b2c733c32c39cc0102.jpg new file mode 100644 index 00000000..5257b100 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-506_66be8dc1734cab45c463371b8fa215cf7e4c53ac88fbd4b2c733c32c39cc0102.jpg differ diff --git a/.raw/MidasFEAAnalysisManual/images/page-509_1573edfeaa6142cbc6e4323ebfa7e659cb876dacdac64b1db399bf0b9ddd4aa4.jpg b/.raw/MidasFEAAnalysisManual/images/page-509_1573edfeaa6142cbc6e4323ebfa7e659cb876dacdac64b1db399bf0b9ddd4aa4.jpg new file mode 100644 index 00000000..43f35251 Binary files /dev/null and b/.raw/MidasFEAAnalysisManual/images/page-509_1573edfeaa6142cbc6e4323ebfa7e659cb876dacdac64b1db399bf0b9ddd4aa4.jpg differ